%%% -*-BibTeX-*- %%% ==================================================================== %%% BibTeX-file{ %%% author = "Nelson H. F. Beebe", %%% version = "5.333", %%% date = "06 March 2026", %%% time = "07:26:32 MDT", %%% filename = "elefunt.bib", %%% address = "University of Utah %%% Department of Mathematics, 110 LCB %%% 155 S 1400 E RM 233 %%% Salt Lake City, UT 84112-0090 %%% USA", %%% telephone = "+1 801 581 5254", %%% URL = "http://waww.math.utah.edu/~beebe", %%% checksum = "45392 94689 430275 4324436", %%% email = "beebe at math.utah.edu, beebe at acm.org, %%% beebe at computer.org (Internet)", %%% codetable = "ISO/ASCII", %%% keywords = "BibTeX; bibliography; convergence %%% acceleration; elementary functions; sequence %%% acceleration; special functions", %%% license = "public domain", %%% supported = "yes", %%% docstring = "This is a bibliography of publications about %%% the computation of the elementary functions %%% (square root, exponential, logarithm, %%% trigonometric, inverse trigonometric, %%% hyperbolic, inverse hyperbolic, ...), and %%% some selected special functions (Bessel, %%% cumulative normal distribution, elliptic %%% integral, exponential integral, Gamma, %%% inverse normal distribution, log-Gamma, psi, %%% ...), in computer programming languages. %%% Additional references to articles, books, and %%% conference proceedings provide more %%% mathematical background. %%% %%% At version 5.274 (21 October 2023), about a %%% dozen references have been added on %%% polynomial evaluation: see the remarks, and %%% references, in entry Todd:1955:MWN, and the %%% keyword phrase ``number of multiplications to %%% evaluate a polynomial'' in other entries. %%% %%% At version 5.333, the year coverage looked %%% like this: %%% %%% 1624 ( 1) 1759 ( 0) 1894 ( 0) %%% 1627 ( 0) 1762 ( 0) 1897 ( 1) %%% 1644 ( 0) 1779 ( 0) 1914 ( 1) %%% 1649 ( 0) 1784 ( 0) 1919 ( 1) %%% 1651 ( 0) 1786 ( 0) 1921 ( 2) %%% 1654 ( 0) 1789 ( 0) 1924 ( 1) %%% 1655 ( 0) 1790 ( 0) 1925 ( 2) %%% 1656 ( 0) 1791 ( 0) 1926 ( 1) %%% 1660 ( 0) 1795 ( 0) 1930 ( 1) %%% 1661 ( 0) 1796 ( 0) 1931 ( 1) %%% 1664 ( 0) 1799 ( 0) 1934 ( 2) %%% 1665 ( 0) 1800 ( 0) 1935 ( 1) %%% 1667 ( 0) 1802 ( 0) 1937 ( 3) %%% 1668 ( 0) 1803 ( 0) 1938 ( 1) %%% 1670 ( 0) 1805 ( 0) 1940 ( 1) %%% 1671 ( 0) 1806 ( 0) 1941 ( 2) %%% 1672 ( 0) 1807 ( 0) 1942 ( 2) %%% 1673 ( 0) 1808 ( 0) 1943 ( 3) %%% 1674 ( 0) 1809 ( 0) 1944 ( 3) %%% 1675 ( 0) 1810 ( 0) 1945 ( 4) %%% 1676 ( 0) 1811 ( 0) 1946 ( 1) %%% 1678 ( 0) 1813 ( 0) 1948 ( 4) %%% 1679 ( 0) 1814 ( 0) 1949 ( 6) %%% 1680 ( 0) 1815 ( 0) 1950 ( 4) %%% 1681 ( 0) 1816 ( 0) 1951 ( 7) %%% 1682 ( 0) 1817 ( 0) 1952 ( 4) %%% 1683 ( 0) 1818 ( 0) 1953 ( 7) %%% 1684 ( 0) 1819 ( 1) 1954 ( 11) %%% 1685 ( 0) 1820 ( 0) 1955 ( 14) %%% 1686 ( 0) 1821 ( 0) 1956 ( 11) %%% 1687 ( 0) 1822 ( 0) 1957 ( 13) %%% 1688 ( 0) 1823 ( 0) 1958 ( 12) %%% 1689 ( 0) 1824 ( 0) 1959 ( 21) %%% 1690 ( 0) 1825 ( 0) 1960 ( 25) %%% 1691 ( 0) 1826 ( 0) 1961 ( 29) %%% 1692 ( 0) 1827 ( 0) 1962 ( 25) %%% 1693 ( 0) 1828 ( 0) 1963 ( 38) %%% 1694 ( 0) 1829 ( 0) 1964 ( 48) %%% 1695 ( 0) 1830 ( 0) 1965 ( 37) %%% 1696 ( 0) 1831 ( 0) 1966 ( 27) %%% 1697 ( 0) 1832 ( 0) 1967 ( 37) %%% 1698 ( 0) 1833 ( 0) 1968 ( 37) %%% 1699 ( 0) 1834 ( 0) 1969 ( 38) %%% 1700 ( 0) 1835 ( 0) 1970 ( 44) %%% 1701 ( 0) 1836 ( 0) 1971 ( 35) %%% 1702 ( 0) 1837 ( 0) 1972 ( 34) %%% 1703 ( 0) 1838 ( 0) 1973 ( 35) %%% 1704 ( 0) 1839 ( 0) 1974 ( 27) %%% 1705 ( 0) 1840 ( 0) 1975 ( 39) %%% 1706 ( 0) 1841 ( 0) 1976 ( 42) %%% 1707 ( 0) 1842 ( 0) 1977 ( 62) %%% 1708 ( 0) 1843 ( 1) 1978 ( 39) %%% 1709 ( 0) 1844 ( 0) 1979 ( 41) %%% 1710 ( 0) 1845 ( 0) 1980 ( 36) %%% 1711 ( 0) 1846 ( 0) 1981 ( 53) %%% 1712 ( 0) 1847 ( 0) 1982 ( 50) %%% 1713 ( 0) 1848 ( 0) 1983 ( 48) %%% 1714 ( 0) 1849 ( 0) 1984 ( 54) %%% 1715 ( 0) 1850 ( 0) 1985 ( 39) %%% 1716 ( 0) 1851 ( 0) 1986 ( 42) %%% 1717 ( 0) 1852 ( 0) 1987 ( 41) %%% 1718 ( 0) 1853 ( 0) 1988 ( 59) %%% 1719 ( 0) 1854 ( 0) 1989 ( 98) %%% 1720 ( 0) 1855 ( 0) 1990 ( 53) %%% 1721 ( 0) 1856 ( 0) 1991 ( 58) %%% 1722 ( 0) 1857 ( 0) 1992 ( 47) %%% 1723 ( 0) 1858 ( 0) 1993 ( 53) %%% 1724 ( 0) 1859 ( 0) 1994 ( 61) %%% 1725 ( 0) 1860 ( 0) 1995 ( 52) %%% 1726 ( 0) 1861 ( 0) 1996 ( 48) %%% 1727 ( 0) 1862 ( 0) 1997 ( 42) %%% 1728 ( 0) 1863 ( 0) 1998 ( 41) %%% 1729 ( 0) 1864 ( 0) 1999 ( 59) %%% 1730 ( 0) 1865 ( 0) 2000 ( 48) %%% 1731 ( 0) 1866 ( 0) 2001 ( 45) %%% 1732 ( 0) 1867 ( 0) 2002 ( 34) %%% 1733 ( 0) 1868 ( 0) 2003 ( 39) %%% 1734 ( 0) 1869 ( 0) 2004 ( 48) %%% 1735 ( 0) 1870 ( 0) 2005 ( 36) %%% 1736 ( 0) 1871 ( 0) 2006 ( 30) %%% 1737 ( 0) 1872 ( 0) 2007 ( 40) %%% 1738 ( 0) 1873 ( 0) 2008 ( 38) %%% 1739 ( 0) 1874 ( 0) 2009 ( 37) %%% 1740 ( 0) 1875 ( 0) 2010 ( 39) %%% 1741 ( 0) 1876 ( 0) 2011 ( 51) %%% 1742 ( 0) 1877 ( 0) 2012 ( 38) %%% 1743 ( 0) 1878 ( 0) 2013 ( 39) %%% 1744 ( 0) 1879 ( 0) 2014 ( 42) %%% 1745 ( 0) 1880 ( 0) 2015 ( 43) %%% 1746 ( 0) 1881 ( 0) 2016 ( 47) %%% 1747 ( 0) 1882 ( 0) 2017 ( 24) %%% 1748 ( 0) 1883 ( 0) 2018 ( 37) %%% 1749 ( 0) 1884 ( 0) 2019 ( 30) %%% 1750 ( 0) 1885 ( 0) 2020 ( 28) %%% 1751 ( 0) 1886 ( 0) 2021 ( 18) %%% 1752 ( 0) 1887 ( 0) 2022 ( 17) %%% 1753 ( 0) 1888 ( 0) 2023 ( 15) %%% 1754 ( 0) 1889 ( 0) 2024 ( 20) %%% 1755 ( 0) 1890 ( 0) 2025 ( 17) %%% 1756 ( 0) 1891 ( 0) 2026 ( 1) %%% 1757 ( 0) 1892 ( 1) %%% 20xx ( 1) %%% %%% Article: 1946 %%% Book: 247 %%% InBook: 5 %%% InCollection: 60 %%% InProceedings: 297 %%% Manual: 2 %%% MastersThesis: 5 %%% Misc: 28 %%% Periodical: 1 %%% PhdThesis: 22 %%% Proceedings: 103 %%% TechReport: 106 %%% Unpublished: 5 %%% %%% Total entries: 2827 %%% %%% At version 4.00, this bibliography was %%% significantly extended by merging in the 262 %%% entries from the bibliography given in entry %%% Fullerton:1980:BEM. That document was not %%% available electronically on the Internet, and %%% the original bib/refer bibliography data from %%% which the report was derived may have been %%% lost with the death of its author, so a %%% printed copy of the report was scanned and %%% converted to searchable text. Its entries %%% were then compared with the previous contents %%% of this file, and the extensive collections %%% in the TeX User Group and BibNet Project %%% archives. All of the publications that %%% Fullerton covered are marked with a citedby %%% keyword, and his notes are included in remark %%% keyword values prefixed by his name. %%% %%% At version 5.00 of 1-Dec-2011, numerous %%% references on hypergeometric functions and on %%% convergence acceleration of sequences were %%% added. Particularly with special functions, %%% many standard series expansions converge too %%% slowly to be useful, but acceleration %%% techniques can sometimes provide a dramatic %%% improvement, making such sums numerically %%% practical. The new entries in that area all %%% contain the phrase `convergence acceleration' %%% in their keywords values. See entry %%% Willis:2012:AGH for a good recent survey of %%% both areas. %%% %%% This bibliography has been collected from %%% bibliographies in the author's personal %%% files, from the OCLC WorldCat and %%% Contents1st databases, from the American %%% Mathematical Society MathSciNet database, %%% from the ACM Computing Archive CD-ROM, %%% and from the computer science bibliography %%% collection on ftp.ira.uka.de in %%% /pub/bibliography to which many people of %%% have contributed. %%% %%% Numerous errors in the sources noted above %%% have been corrected. Spelling has been %%% verified with the UNIX spell and GNU ispell %%% programs using the exception dictionary %%% stored in the companion file with extension %%% .sok. %%% %%% BibTeX citation tags are uniformly chosen as %%% name:year:abbrev, where name is the family %%% name of the first author or editor, year is a %%% 4-digit number, and abbrev is a 3-letter %%% condensation of important title %%% words. Citation tags were automatically %%% generated by software developed for the %%% BibNet Project. %%% %%% This bibliography is sorted by year, and %%% within each year, by author and title key, %%% using ``bibsort -byyear''. Cross-referenced %%% proceedings entries appear at the end, %%% because of a restriction in the current %%% BibTeX. %%% %%% The checksum field above contains a CRC-16 %%% checksum as the first value, followed by the %%% equivalent of the standard UNIX wc (word %%% count) utility output of lines, words, and %%% characters. This is produced by Robert %%% Solovay's checksum utility.", %%% } %%% ==================================================================== %%% A delimited macro \toenglish ... \endtoenglish is NECESSARY here. %%% The more conventional undelimited form \toenglish{...} has braces %%% that prevent BibTeX's downcasing operation, and the alternate form %%% {\toenglish{...}} is considered a `special character' by BibTeX, %%% and all of {...} gets downcased. We avoid the name \english to %%% prevent conflicts with language options in packages like Babel. %%% %%% To suppress output of English translations of non-English titles, %%% use %%% "\def \toenglish #1\endtoenglish{\unskip}" %%% instead. %%% %%% Alternative Cyrillic definitions are supplied in the event that %%% cyracc.def is not loaded. @Preamble{"\input bibnames.sty" # "\def \toenglish #1\endtoenglish{[{\em English:} #1\unskip]} " # "\ifx \undefined \arccoth \def \arccoth \operatorname{arccoth} \fi" # "\ifx \undefined \booktitle \def \booktitle #1{{{\em #1}}} \fi" # "\ifx \undefined \cyr \let \cyr = \relax \fi" # "\ifx \undefined \cdprime \def \cdprime {''} \fi" # "\ifx \undefined \k \let \k = \c \fi" # "\ifx \undefined \erfc \def \erfc #1{{\rm #1}} \fi" # "\ifx \undefined \mathrm \def \mathrm #1{{\rm #1}} \fi" # "\ifx \undefined \operatorname \def \operatorname #1{{\rm #1}} \fi" # "\ifx \undefined \pkg \def \pkg #1{{{\tt #1}}} \fi" # "\ifx \undefined \TM \def \TM {${}^{\sc TM}$} \fi" # "\hyphenation{ Rich-ard }" } %%% ==================================================================== %%% Acknowledgement abbreviations: @String{ack-nhfb = "Nelson H. F. Beebe, University of Utah, Department of Mathematics, 110 LCB, 155 S 1400 E RM 233, Salt Lake City, UT 84112-0090, USA, Tel: +1 801 581 5254, e-mail: \path|beebe@math.utah.edu|, \path|beebe@acm.org|, \path|beebe@computer.org| (Internet), URL: \path|https://www.math.utah.edu/~beebe/|"} @String{ack-mv = "Matti Vuorinen, Department of Mathematics and Statistics, University of Turku, Vesilinnantie 5, 20014 Turku, Finland, e-mail: \path|vuorinen@utu.fi|, URL: \path|http://users.utu.fi/vuorinen/|"} @String{ack-nj = "Norbert Juffa, 2445 Mission College Blvd., Santa Clara, CA 95054, USA, e-mail: \path|norbert@@iit.com|"} @String{ack-rfb = "Ronald F. Boisvert, Applied and Computational Mathematics Division, National Institute of Standards and Technology, Gaithersburg, MD 20899, USA, Tel: +1 301 975 3812, e-mail: \path=boisvert@cam.nist.gov="} %%% ==================================================================== %%% Institution abbreviations: @String{inst-ANL = "Argonne National Laboratory"} @String{inst-ANL:adr = "9700 South Cass Avenue, Argonne, IL 60439-4801, USA"} @String{inst-ATT-BELL = "AT\&T Bell Laboratories"} @String{inst-ATT-BELL:adr = "Murray Hill, NJ, USA"} @String{inst-BERKELEY-CS = "Department of Computer Science, University of California"} @String{inst-BERKELEY-CS:adr = "Berkeley, CA, USA"} @String{inst-CECM = "Centre for Experimental and Constructive Mathematics (CECM) at Simon Fraser University (SFU)"} @String{inst-CECM:adr = "Burnaby, BC V5A 1S6, Canada"} @String{inst-CPAM-UCB = "Center for Pure and Applied Mathematics, University of California, Berkeley"} @String{inst-CPAM-UCB:adr = "Berkeley, CA, USA"} @String{inst-CSC = "Center for Scientific Computing, Department of Mathematics, University of Utah"} @String{inst-CSC:adr = "Salt Lake City, UT 84112, USA"} @String{inst-INST-ADV-STUDY = "Institute for Advanced Study"} @String{inst-INST-ADV-STUDY:adr = "Princeton, NJ, USA"} @String{inst-IPTC = "{Institut f{\"u}r Physikalische und Theoretische Chemie}"} @String{inst-IPTC:adr = "{Universit{\"a}t Regensburg, D-93040 Regensburg}"} @String{inst-LASL = "Los Alamos Scientific Laboratory"} @String{inst-LASL:adr = "Los Alamos, NM, USA"} @String{inst-LORIA-INRIA-LORRAINE = "LORIA/INRIA Lorraine"} @String{inst-LORIA-INRIA-LORRAINE:adr = "B{\^a}timent A, Technop{\^o}le de Nancy-Brabois, 615 rue du jardin botanique, F-54602 Villers-l{\`e}s-Nancy Cedex, France"} @String{inst-MATH-NPS = "Department of Mathematics, Naval Postgraduate School"} @String{inst-MATH-NPS:adr = "Monterey CA 93943, USA"} @String{inst-PRINCETON = "Princeton University"} @String{inst-PRINCETON:adr = "Princeton, NJ, USA"} @String{inst-STAN-CS = "Stanford University, Department of Computer Science"} @String{inst-STAN-CS:adr = "Stanford, CA, USA"} %%% ==================================================================== %%% Journal abbreviations: @String{j-ACM-COMM-COMP-ALGEBRA = "ACM Communications in Computer Algebra"} @String{j-ACTA-INFO = "Acta Informatica"} @String{j-ACTA-MATH = "Acta Mathematica"} @String{j-ACTA-NUMERICA = "Acta Numerica"} @String{j-ADV-APPL-MATH = "Advances in Applied Mathematics"} @String{j-ADV-COMPUT-MATH = "Advances in Computational Mathematics"} @String{j-ADV-QUANTUM-CHEM = "Advances in Quantum Chemistry"} @String{j-ALGORITHMS-BASEL = "Algorithms ({Basel})"} @String{j-AM-J-MATH = "American Journal of Mathematics"} @String{j-AMER-MATH-MONTHLY = "American Mathematical Monthly"} @String{j-AMER-STAT = "The American Statistician"} @String{j-ANAL-APPL = "Analysis and Applications (Singapore)"} @String{j-ANN-APPL-STAT = "Annals of Applied Statistics"} @String{j-ANN-INST-STAT-MATH-TOKYO = "Annals of the Institute of Statistical Mathematics"} @String{j-ANN-MATH-ARTIF-INTELL = "Annals of Mathematics and Artificial Intelligence"} @String{j-ANN-STAT = "Annals of Statistics"} @String{j-ANZIAM-J = "The ANZIAM Journal"} @String{j-APPL-COMPUT-HARMON-ANAL = "Applied and Computational Harmonic Analysis. Time-Frequency and Time-Scale Analysis, Wavelets, Numerical Algorithms, and Applications"} @String{j-APPL-MATH-COMP = "Applied Mathematics and Computation"} @String{j-APPL-MATH-LETT = "Applied Mathematics Letters"} @String{j-APPL-MATH-SCI-RUSE = "Applied Mathematical Sciences (Ruse)"} @String{j-APPL-NUM-MATH = "Applied Numerical Mathematics: Transactions of IMACS"} @String{j-APPL-OPTICS = "Applied Optics"} @String{j-APPL-STAT = "Applied Statistics"} @String{j-ARCH-HIST-EXACT-SCI = "Archive for History of Exact Sciences"} @String{j-ARCH-RAT-MECH-ANAL = "Archive for Rational Mechanics and Analysis"} @String{j-ASTROPHYS-SPACE-SCI = "Astrophysics and Space Science"} @String{j-ATMOS-SCI-LETT = "Atmospheric Science Letters"} @String{j-AUST-J-STAT = "Australian Journal of Statistics"} @String{j-AUSTRALIAN-J-PHYS = "Australian Journal of Physics"} @String{j-AUTOMATICA = "Automatica: the journal of IFAC, the International Federation of Automatic Control"} @String{j-BELL-SYST-TECH-J = "The Bell System Technical Journal"} @String{j-BIOMETRIKA = "Biometrika"} @String{j-BIT = "BIT (Nordisk tidskrift for informationsbehandling)"} @String{j-BIT-NUM-MATH = "BIT Numerical Mathematics"} @String{j-BRITISH-J-HIST-MATH = "British Journal for the History of Mathematics"} @String{j-BRITISH-J-PHILOS-SCI = "British Journal for the Philosophy of Science"} @String{j-BULL-AMS = "Bulletin of the American Mathematical Society"} @String{j-C-R-ACAD-BULGARE-SCI = "Comptes rendus de l'Acad{\'e}mie bulgare des sciences"} @String{j-CACM = "Communications of the ACM"} @String{j-CALCOLO = "Calcolo"} @String{j-CAN-J-MATH = "Canadian Journal of Mathematics = Journal canadien de math{\'e}matiques"} @String{j-CAN-MATH-BULL = "Bulletin canadien de math\-{\'e}\-mat\-iques = Canadian Mathematical Bulletin"} @String{j-CCCUJ = "C/C++ Users Journal"} @String{j-CELEST-MECH-DYN-ASTR = "Celestial Mechanics and Dynamical Astronomy"} @String{j-CENTAURUS = "Centaurus: An International Journal of the History of Science and its Cultural Aspects"} @String{j-COLLEGE-MATH-J = "College Mathematics Journal"} @String{j-COMM-PURE-APPL-MATH = "Communications on Pure and Applied Mathematics (New York)"} @String{j-COMMUN-STAT-SIMUL-COMPUT = "Communications in Statistics: Simulation and Computation"} @String{j-COMMUN-STAT-THEORY-METH = "Communications in Statistics: Theory and Methods"} @String{j-COMP-ARCH-NEWS = "ACM SIGARCH Computer Architecture News"} @String{j-COMP-J = "The Computer Journal"} @String{j-COMP-PHYS-COMM = "Computer Physics Communications"} @String{j-COMPUT-APPL-MATH = "Journal of Computational and Applied Mathematics"} @String{j-COMPUT-MATH-APPL = "Computers and Mathematics with Applications"} @String{j-COMPUT-MATH-MATH-PHYS = "Computational Mathematics and Mathematical Physics"} @String{j-COMPUT-PHYS = "Computers in Physics"} @String{j-COMPUT-PHYS-REP = "Computer Physics Reports"} @String{j-COMPUT-SCI-ENG = "Computing in Science and Engineering"} @String{j-COMPUT-STAT-DATA-ANAL = "Computational Statistics \& Data Analysis"} @String{j-COMPUTER = "Computer"} @String{j-COMPUTING = "Computing: Archiv fur informatik und numerik"} @String{j-COMPUTING-SUPPLEMENTUM = "Computing. Supplementum"} @String{j-CONST-APPROX = "Constructive Approximation"} @String{j-CSSP = "Circuits, systems, and signal processing: {CSSP}"} @String{j-CYBER = "Cybernetics"} @String{j-DDJ = "Dr. Dobb's Journal of Software Tools"} @String{j-DESIGNS-CODES-CRYPTOGR = "Designs, Codes, and Cryptography"} @String{j-DOKL-AKAD-NAUK = "Doklady Akademii nauk SSSR"} @String{j-EDN = "EDN"} @String{j-ELECT-LETTERS = "Electronics Letters"} @String{j-ELECT-NOTES-THEOR-COMP-SCI = "Electronic Notes in Theoretical Computer Science"} @String{j-ELECTRON-COMMUN-JPN = "Electronics and communications in Japan"} @String{j-ELECTRON-TRANS-NUMER-ANAL = "Electronic Transactions on Numerical Analysis (ETNA)"} @String{j-ELECTRONIC-DESIGN = "Electronic Design"} @String{j-ELECTRONICS = "Electronics"} @String{j-ELECTRONIK = "Elektronik"} @String{j-ELEKTRONIKER = "Elektroniker (Switzerland)"} @String{j-ELEK-RECHENANLAGEN = "Elektronische Rechenanlagen"} @String{j-EMBED-SYS-PROG = "Embedded Systems Programming"} @String{j-ENTROPY = "Entropy"} @String{j-EXP-MATH = "Experimental mathematics"} @String{j-FIB-QUART = "Fibonacci Quarterly"} @String{j-FORM-METHODS-SYST-DES = "Formal Methods in System Design"} @String{j-HEWLETT-PACKARD-J = "Hew\-lett-Pack\-ard Journal: technical information from the laboratories of Hew\-lett-Pack\-ard Company"} @String{j-HIST-MATH = "Historia Mathematica"} @String{j-HIST-SCI-2 = "Historia Scientiarum. Second Series. International Journal of the History of Science Society of Japan"} @String{j-IBM-JRD = "IBM Journal of Research and Development"} @String{j-IBM-SYS-J = "IBM Systems Journal"} @String{j-IBM-TDB = "IBM Technical Disclosure Bulletin"} @String{j-IEE-PROC-COMPUT-DIGIT-TECH = "IEE Proceedings. Computers and Digital Techniques"} @String{j-IEEE-ACCESS = "IEEE Access"} @String{j-IEEE-CGA = "IEEE Computer Graphics and Applications"} @String{j-IEEE-COMMUN-LET = "IEEE Communications Letters"} @String{j-IEEE-J-SOLID-STATE-CIRCUITS = "IEEE Journal of Solid-State Circuits"} @String{j-IEEE-MICRO = "IEEE Micro"} @String{j-IEEE-SIGNAL-PROCESS-LETT = "IEEE Signal Processing Letters"} @String{j-IEEE-SIGNAL-PROCESS-MAG = "IEEE Signal Processing Magazine"} @String{j-IEEE-SPECTRUM = "IEEE Spectrum"} @String{j-IEEE-TRANS-CIRCUITS-SYST-1 = "IEEE Transactions on Circuits and Systems I: Regular Papers"} @String{j-IEEE-TRANS-CIRCUITS-SYST-2 = "IEEE Transactions on Circuits and Systems. 2, Analog and Digital Signal Processing"} @String{j-IEEE-TRANS-CIRCUITS-SYST-II-EXPRESS-BRIEFS = "IEEE Transactions on Circuits and Systems II: Express Briefs"} @String{j-IEEE-TRANS-COMM = "IEEE Transactions on Communications"} @String{j-IEEE-TRANS-COMPUT = "IEEE Transactions on Computers"} @String{j-IEEE-TRANS-ELEC-COMPUT = "IEEE Transactions on Electronic Computers"} @String{j-IEEE-TRANS-EMERG-TOP-COMPUT = "IEEE Transactions on Emerging Topics in Computing"} @String{j-IEEE-TRANS-INF-THEORY = "IEEE Transactions on Information Theory"} @String{j-IEEE-TRANS-MICROWAVE-THEORY-TECH = "IEEE Transactions on Microwave Theory and Techniques"} @String{j-IEEE-TRANS-PAR-DIST-SYS = "IEEE Transactions on Parallel and Distributed Systems"} @String{j-IEEE-TRANS-SIG-PROC = "IEEE Transactions on Signal Processing"} @String{j-IEEE-TRANS-VLSI-SYST = "IEEE Transactions on Very Large Scale Integration (VLSI) Systems"} @String{j-IEEE-TRANS-VEH-TECHNOL = "IEEE Transactions on Vehicular Technology"} @String{j-IEEE-TRANS-WIREL-COMMUN = "IEEE Transactions on Wireless Communications"} @String{j-IJQC = "International Journal of Quantum Chemistry"} @String{j-IJSAHPC = "The International Journal of Supercomputer Applications and High Performance Computing"} @String{j-IMA-J-NUMER-ANAL = "IMA Journal of Numerical Analysis"} @String{j-INFO-PROC-LETT = "Information Processing Letters"} @String{j-INT-J-COMPUT-INF-SCI = "International Journal of Computer and Information Sciences"} @String{j-INT-J-COMPUT-MATH = "International Journal of Computer Mathematics"} @String{j-INT-J-HIGH-SPEED-COMPUTING = "International Journal of High Speed Computing"} @String{j-INT-J-MATH-EDU-SCI-TECH = "International journal of mathematical education in science and technology"} @String{j-INT-J-MATH-MATH-SCI = "International Journal of Mathematics and Mathematical Sciences"} @String{j-INT-J-NUMER-METHODS-ENG = "International Journal for Numerical Methods in Engineering"} @String{j-INT-J-SOFTW-TOOLS-TECHNOL-TRANSFER = "International Journal on Software Tools for Technology Transfer (STTT)"} @String{j-INTEGRATION-VLSI-J = "Integration, the VLSI journal"} @String{j-INTEL-TECH-J = "Intel Technology Journal"} @String{j-INTERNET-J-CHEM = "Internet Journal of Chemistry"} @String{j-INTERVAL-COMP = "Interval Computations = Interval'nye vychisleniia"} @String{j-IRE-TRANS-ELEC-COMPUT = "IRE Transactions on Electronic Computers"} @String{j-ISIS = "Isis"} @String{j-J-ACM = "Journal of the Association for Computing Machinery"} @String{j-J-ACOUST-SOC-AM = "Journal of the Acoustical Society of America"} @String{j-J-ALG = "Journal of Algorithms"} @String{j-J-AM-STAT-ASSOC = "Journal of the American Statistical Association"} @String{j-J-APPL-STAT = "Journal of Applied Statistics"} @String{j-J-APPROX-THEORY = "Journal of Approximation Theory"} @String{j-J-AUSTRAL-MATH-SOC-SER-B = "Journal of the Australian Mathematical Society: Series B, Applied Mathematics"} @String{j-J-AUTOM-REASON = "Journal of Automated Reasoning"} @String{j-J-CHINESE-INST-ENG = "Journal of the Chinese Institute of Engineers = Chung-kuo kung ch'eng hsueh kan"} @String{j-J-COMB-THEORY-A = "Journal of Combinatorial Theory (Series A)"} @String{j-J-COMPUT-APPL-MATH = "Journal of Computational and Applied Mathematics"} @String{j-J-COMPUT-CHEM = 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Brown Company Publishers"} @String{pub-WCB:adr = "Dubuque, IA, USA"} @String{pub-WI = "Wiley-In{\-}ter{\-}sci{\-}ence"} @String{pub-WI:adr = "New York, NY, USA"} @String{pub-WILEY = "Wiley"} @String{pub-WILEY:adr = "New York, NY, USA"} @String{pub-WORLD-SCI = "World Scientific Publishing Co. Pte. Ltd."} @String{pub-WORLD-SCI:adr = "P. O. Box 128, Farrer Road, Singapore 9128"} %%% ==================================================================== %%% Series abbreviations: @String{ser-APPL-MATH-SER-NBS = "Applied Mathematics Series / National Bureau of Standards"} @String{ser-LECT-NOTES-MATH = "Lecture Notes in Mathematics"} @String{ser-LNAI = "Lecture Notes in Artificial Intelligence"} @String{ser-LNCS = "Lecture Notes in Computer Science"} %%% ==================================================================== %%% Bibliography entries, sorted by year and citation label: @Book{Briggs:1624:ALL, author = "Henry Briggs", title = "Arithmetica Logarithmica. ({Latin}) [{Logarithmic} arithmetic]", publisher = "Excudebat Gulielmus Iones", address = "London, UK", year = "1624", bibdate = "Mon Nov 10 08:25:38 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "This may be the earliest use of a CORDIC-like algorithm for computing tables of logarithms, although it seems to have been unknown to those who are later credited with the CORDIC invention \cite{Volder:1956:BCA,Bemer:1958:SMC,Volder:1959:CCT,Volder:1959:CTC,Walther:1971:UAE}. See \cite{Roegel:2011:RTB} for an analyis and description of Briggs' computational methods, and a list of 120 publications about Briggs' tables", URL = "http://www.17centurymaths.com; https://archive.org/details/bub_gb_L88WAAAAQAAJ; https://en.wikipedia.org/wiki/Henry_Briggs_(mathematician); https://old.maa.org/press/periodicals/convergence/mathematical-treasure-iarithmetica-logarithmicai-of-henry-briggs; https://www.google.com/books/edition/Arithmetica_logarithmica/L88WAAAAQAAJ?hl=en", acknowledgement = ack-nhfb, author-dates = "1 February 1561--26 January 1630", language = "Latin", remark-1 = "Cited briefly in \cite[page 75]{Andrews:1978:EFM}. Other sources suggest years of 1616--1617. Denis Roegel \cite{Roegel:2011:RTB} says that Briggs published in 1617 a 16-page booklet with a 15-page table of 14-digit base-10 logarithms of the integers from 1 to 1000. This 1624 publication continues that work, supplying 14-digit logarithms of the integers 1 to 20000, and 90001 to 100000.", remark-2 = "Full Latin title is \booktitle{Arithmetica logarithmica, sive, Logarithmorum chiliades triginta: pro numeris naturali serie crescentibus ab vnitate ad 20,000, et a 90,000 ad 100,000: quorum ope multa perficiuntur arithmetica problemata et geometrica: hos numeros primus invenit clarissimus vir Iohannes Neperus, Baro Merchistonij: eos autem ex eiusdem sententia mutavit / eorumque ortum et vsum illustravit Henricus Briggius \ldots{}}. A machine translation is \booktitle{Logarithmic arithmetic, or, Thirty thousand logarithms: for natural numbers in series increasing from unity to 20,000, and from 90,000 to 100,000: by means of which many arithmetical and geometric problems are accomplished: these numbers were first invented by the most illustrious man John Neperus, Baron Merchiston: but they were changed according to his own opinion / and their origin and use were illustrated by Henry Briggs \ldots{}}.", } @Article{Horner:1819:XNM, author = "William George Horner", title = "{XXI}. {A} new method of solving numerical equations of all orders, by continuous approximation", journal = j-PHILOS-TRANS-R-SOC-LOND, volume = "109", pages = "308--335", year = "1819", CODEN = "PTRSAV", DOI = "https://doi.org/10.1098/rstl.1819.0023", ISSN = "0370-2316 (print), 2053-9207 (electronic)", bibdate = "Sat Oct 21 12:27:25 2023", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "https://royalsocietypublishing.org/doi/pdf/10.1098/rstl.1819.0023", acknowledgement = ack-nhfb, fjournal = "Philosophical Transactions of the Royal Society of London", journal-URL = "http://rsta.royalsocietypublishing.org/", keywords = "Horner's nested form; number of multiplications to evaluate a polynomial", read = "1 July 1819", remark-1 = "Communicated by Davies Gilbert, Esq. F.R.S.", remark-2 = "On page 310 of this paper, Horner gives the steps of the nested form as a chain of fused multiply-add operations, and credits this idea to Joseph-Louis Lagrange (1736--1813) in his 1813 book \booktitle{Th{\'e}orie des fonctions analytiques}.", remark-3 = "Knuth \cite[p. 486]{Knuth:1998:SA} gives this paper as the reference for Horner's nested form, but also reports that Isaac Newton used it in unpublished notes 150 years earlier, and that it was employed by the Chinese in the 13th century CE.", } @Article{Lovelace:1843:SAE, author = "Ada Augusta Lovelace", title = "Sketch of the {Analytical Engine}", journal = "Scientific Memoirs", volume = "3", number = "??", pages = "666--731", month = "????", year = "1843", bibdate = "Sun Aug 18 09:31:28 2013", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/l/lovelace-ada-augusta.bib; https://www.math.utah.edu/pub/tex/bib/adabooks.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "Reprinted in \cite{Lovelace:1989:SAE}.", acknowledgement = ack-nhfb, keywords = "Bernoulli numbers", remark = "This paper contains what some view as possibly the world's first computer program, a recipe for computing Bernoulli numbers on Charles Babbage's Analytical Engine, which was not successfully constructed until more than a century after their deaths, in 1852 and 1871, respectively. It is not, however, the world's first computational algorithm: that credit is given to Euclid's procedure for fast computation of the greatest common denominator, about 300 BCE, but possibly known a few hundred years earlier.", } @Book{Greenhill:1892:AEF, author = "Alfred George Greenhill", title = "The Applications of Elliptic Functions", publisher = pub-MACMILLAN, address = pub-MACMILLAN:adr, pages = "xi + 357", year = "1892", bibdate = "Wed Mar 15 08:21:33 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, author-dates = "1847--1927", remark = "Reprinted in \cite{Greenhill:1959:AEF}.", } @Article{Dawson:1897:NV, author = "H. G. Dawson", title = "On the Numerical Value of $ \int_0^h e^{x^2} \, d x $", journal = j-PROC-LONDON-MATH-SOC-1, volume = "s1-29", number = "1", pages = "519--522", month = nov, year = "1897", CODEN = "PLMTAL", DOI = "https://doi.org/10.1112/plms/s1-29.1.519", ISSN = "0024-6115 (print), 1460-244X (electronic)", ISSN-L = "0024-6115", bibdate = "Sat Jun 12:08:16 2024", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", fjournal = "Proceedings of the London Mathematical Society. First Series", journal-URL = "http://plms.oxfordjournals.org/content/by/year", remark = "This paper is the origin of Dawson's integral and the function dawson(x).", } @Book{Kennelly:1914:TCH, author = "Arthur E. (Arthur Edwin) Kennelly", title = "Tables of Complex Hyperbolic and Circular Functions", publisher = pub-HARVARD, address = pub-HARVARD:adr, pages = "iii + 212", year = "1914", LCCN = "QA342 .K45", bibdate = "Sat Apr 1 14:49:41 MDT 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; z3950.loc.gov:7090/Voyager", acknowledgement = ack-nhfb, author-dates = "1861--1939", subject = "Exponential functions", } @Book{Pairman:1919:TDT, author = "Eleanor Pairman", title = "Tables of the Digamma and Trigamma Functions", volume = "I", publisher = pub-CAMBRIDGE, address = pub-CAMBRIDGE:adr, pages = "9 + 11", year = "1919", LCCN = "QA47 .T7 no.1", bibdate = "Sat Mar 25 16:17:54 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcw.bib; z3950.loc.gov:7090/Voyager", series = "Tracts for computers", acknowledgement = ack-nhfb, remark = "Edited by Karl Pearson. According to \cite{Davis:1935:EPF}, the author coined the phrase `polygamma function' in this booklet. English dictionaries usually do not include the word `polygamma'.", subject = "Gamma functions", } @Book{Kennelly:1921:TCH, author = "Arthur Edwin Kennelly", title = "Tables of Complex Hyperbolic and Circular Functions", publisher = pub-HARVARD, address = pub-HARVARD:adr, pages = "iii + 240", year = "1921", LCCN = "QA342 .K45 1921", bibdate = "Sat Apr 1 14:49:41 MDT 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; z3950.loc.gov:7090/Voyager", acknowledgement = ack-nhfb, author-dates = "1861--", subject = "Functions, Exponential", } @Article{King:1921:SNF, author = "Louis Vessot King", title = "On Some New Formulae for the Numerical Calculation of the Mutual Induction of Coaxial Circles", journal = j-PROC-R-SOC-LOND-SER-A-MATH-PHYS, volume = "100", number = "702", pages = "60--66", day = "4", month = oct, year = "1921", DOI = "https://doi.org/10.1098/rspa.1921.0070", ISSN = "0950-1207 (print), 2053-9150 (electronic)", bibdate = "Wed Feb 03 09:07:10 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcw.bib", note = "This is the first known publication of the AGM method, discovered by the author in 1913, for computing Jacobian elliptic functions. See also \cite{King:1924:DNC,King:2007:DNC}.", URL = "http://www.jstor.org/stable/93861", acknowledgement = ack-nhfb, fjournal = "Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character", journal-URL = "http://rspa.royalsocietypublishing.org/", } @Book{King:1924:DNC, author = "Louis Vessot King", title = "On the Direct Numerical Calculation of Elliptic Functions and Integrals", publisher = pub-CAMBRIDGE, address = pub-CAMBRIDGE:adr, pages = "viii + 42", year = "1924", LCCN = "QA343", bibdate = "Wed Feb 03 08:53:04 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcw.bib", acknowledgement = ack-nhfb, remark = "The AGM method for Jacobian elliptic functions was discovered by this book's author at McGill University in 1913, first published in \cite{King:1921:SNF}, and then in this monograph (reprinted in \cite{King:2007:DNC}).", } @Book{Johnson:1925:TCT, author = "K. S. (Kenneth Simonds) Johnson", title = "Transmission Circuits for Telephonic Communication: Methods of Analysis and Design", publisher = "D. Van Nostrand Company", address = "New York, NY, USA", pages = "334", year = "1925", bibdate = "Tue Apr 29 09:51:33 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "Second Printing --- Corrected (1925); Third printing, with corrections and additions (1927).", URL = "TK6301 .J6 1927", acknowledgement = ack-nhfb, remark = "According to \cite{Dempsey:2024:PBI}, this book, based on a series of lectures at Bell Telephone Laboratories, was a key source of tabulations of branch cuts for various inverse functions.", } @Article{Ritt:1925:EFT, author = "J. F. Ritt", title = "Elementary functions and their inverses", journal = j-TRANS-AM-MATH-SOC, volume = "27", number = "1", pages = "68--90", year = "1925", CODEN = "TAMTAM", DOI = "https://doi.org/10.1090/S0002-9947-1925-1501299-9", ISSN = "0002-9947 (print), 1088-6850 (electronic)", ISSN-L = "0002-9947", MRclass = "30A05 (33B10)", MRnumber = "MR1501299", bibdate = "Wed Apr 13 06:46:35 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.ams.org/journals/tran/1925-027-01/S0002-9947-1925-1501299-9/", acknowledgement = ack-nhfb, fjournal = "Transactions of the American Mathematical Society", journal-URL = "http://www.ams.org/journals/tran/", } @Article{Dederick:1926:QDDc, author = "L. S. Dederick", title = "Questions and Discussions: Discussions: a Modified Method for Cube Roots and Fifth Roots", journal = j-AMER-MATH-MONTHLY, volume = "33", number = "9", pages = "469--472", month = nov, year = "1926", CODEN = "AMMYAE", ISSN = "0002-9890 (print), 1930-0972 (electronic)", ISSN-L = "0002-9890", bibdate = "Mon Jun 28 12:38:12 MDT 1999", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", URL = "http://www.jstor.org/stable/2299613", acknowledgement = ack-nhfb, fjournal = "American Mathematical Monthly", journal-URL = "https://www.jstor.org/journals/00029890.htm", } @Article{Mahler:1930:NUG, author = "K. Mahler", title = "{{\"U}ber die Nullstellen der unvollstaendigen Gammafunktionen}. ({German}) [{On} the zeros of the incomplete gamma-functions]", journal = j-REND-CIRC-MAT, volume = "54", number = "??", pages = "1--41", month = "????", year = "1930", CODEN = "RCMMAR", ISSN = "0009-725X (print), 1973-4409 (electronic)", ISSN-L = "0009-725X", bibdate = "Sat Feb 18 14:57:12 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "https://carma.newcastle.edu.au/mahler/collected.html; https://carma.newcastle.edu.au/mahler/docs/002.pdf", acknowledgement = ack-nhfb, fjournal = "Rendiconti del Circolo matematico di Palermo", language = "German", remark = "Based on doctoral dissertation, Frankfurt, Germany (1927).", } @Book{Hobson:1931:TSE, author = "Ernest William Hobson", title = "The Theory of Spherical and Ellipsoidal Harmonics", publisher = pub-CAMBRIDGE, address = pub-CAMBRIDGE:adr, pages = "xi + 500", year = "1931", LCCN = "QA406 .H7", bibdate = "Sat Apr 1 14:40:56 MDT 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; z3950.loc.gov:7090/Voyager", acknowledgement = ack-nhfb, author-dates = "1856--1933", subject = "Spherical harmonics; Lam{\'e}'s functions", } @Article{Kalbfell:1934:QDN, author = "D. C. Kalbfell", title = "Questions, Discussions and Notes: On a Method for Calculating Square Roots", journal = j-AMER-MATH-MONTHLY, volume = "41", number = "8", pages = "504--506", month = oct, year = "1934", CODEN = "AMMYAE", ISSN = "0002-9890 (print), 1930-0972 (electronic)", ISSN-L = "0002-9890", bibdate = "Mon Jun 28 12:37:31 MDT 1999", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; JSTOR database", URL = "http://www.jstor.org/stable/2300417", acknowledgement = ack-nhfb, fjournal = "American Mathematical Monthly", journal-URL = "https://www.jstor.org/journals/00029890.htm", } @Book{McLachlan:1934:BFE, author = "N. W. (Norman William) McLachlan", title = "{Bessel} Functions for Engineers", publisher = pub-CLARENDON, address = pub-CLARENDON:adr, pages = "xi + 1 + 192", year = "1934", LCCN = "QA408 .M3", bibdate = "Sat Apr 1 14:44:36 MDT 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; z3950.loc.gov:7090/Voyager", series = "The Oxford engineering science series", acknowledgement = ack-nhfb, author-dates = "1888--", subject = "Bessel functions", } @Article{Davis:1935:EPF, author = "H. T. Davis", title = "An extension to polygamma functions of a theorem of {Gauss}", journal = j-BULL-AMS, volume = "41", number = "4", pages = "243--248", month = apr, year = "1935", CODEN = "BAMOAD", DOI = "https://doi.org/10.1090/s0002-9904-1935-06055-0", ISSN = "0002-9904 (print), 1936-881X (electronic)", ISSN-L = "0002-9904", bibdate = "Sat Mar 25 16:16:08 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcw.bib", acknowledgement = ack-nhfb, fjournal = "Bulletin of the American Mathematical Society", journal-URL = "http://www.ams.org/journals/bull/all_issues.html", remark = "A footnote on the first page says ``The name polygamma is suggested by the paper, \booktitle{Tables of the Digamma and Trigamma Functions}, by Eleanor Pairman, Tracts for Computers, No. 1, 1919'' \cite{Pairman:1919:TDT}.", } @Article{Airey:1937:CFA, author = "J. R. Airey", title = "The ``converging factor'' in asymptotic series and the calculation of {Bessel}, {Laguerre} and other functions", journal = j-PHILOS-MAG, volume = "24", number = "162", pages = "521--552", month = "????", year = "1937", CODEN = "PHMAA4", DOI = "https://doi.org/10.1080/14786443708565133", ISSN = "0031-8086", ISSN-L = "0031-8086", bibdate = "Thu Dec 01 14:26:37 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Philosophical Magazine", journal-URL = "http://www.tandfonline.com/loi/tphm19", } @Article{Escott:1937:QDN, author = "E. B. Escott", title = "Questions, Discussions, and Notes: Rapid Method for Extracting a Square Root", journal = j-AMER-MATH-MONTHLY, volume = "44", number = "10", pages = "644--646", month = dec, year = "1937", CODEN = "AMMYAE", ISSN = "0002-9890 (print), 1930-0972 (electronic)", ISSN-L = "0002-9890", bibdate = "Mon Jun 28 12:38:44 MDT 1999", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; JSTOR database", URL = "http://www.jstor.org/stable/2301484", acknowledgement = ack-nhfb, fjournal = "American Mathematical Monthly", journal-URL = "https://www.jstor.org/journals/00029890.htm", } @Article{Ostrowski:1937:KAN, author = "A. M. Ostrowski", title = "{{\"U}ber die Konvergenz und die Abr{\"u}ndungsfestigkeit des Newtonschen Verfahren}. ({German}) [{On} the convergence and rounding strength of {Newton}'s method]", journal = "Rec. Math.", volume = "2", number = "??", pages = "1073--1098", month = "????", year = "1937", bibdate = "Mon Oct 23 14:58:35 2023", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, language = "German", } @Article{Ostrowski:1938:NMA, author = "A. M. Ostrowski", title = "On {Newton}'s method of approximation", journal = "British Association for the Advancement of Science", volume = "??", number = "??", pages = "392--??", month = "????", year = "1938", bibdate = "Mon Oct 23 14:57:22 2023", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, } @Book{Emde:1940:TEF, author = "Fritz Emde", title = "{Tafeln Elementarer Funktionen} ({German}) [Tables of Elementary Functions]", publisher = "B. T. Teubner", address = "Leipzig, Germany and Berlin, Germany", pages = "xii + 181", year = "1940", LCCN = "QA47 .E5", bibdate = "Fri Jun 11 12:34:09 1999", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://lccn.loc.gov/45006177", acknowledgement = ack-nhfb, author-dates = "1873--1951", language = "German", } @Article{Stoner:1941:FEF, author = "Paul Matthew Stoner", title = "Fitting the Exponential Function and the {Gompertz} Function by the Method of Least Squares", journal = j-J-AM-STAT-ASSOC, volume = "36", number = "216", pages = "515--518", month = dec, year = "1941", CODEN = "JSTNAL", ISSN = "0162-1459 (print), 1537-274X (electronic)", ISSN-L = "0162-1459", bibdate = "Wed Jan 25 08:05:24 MST 2012", bibsource = "http://www.jstor.org/journals/01621459.html; http://www.jstor.org/stable/i314095; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jamstatassoc1940.bib", URL = "http://www.jstor.org/stable/2278959", acknowledgement = ack-nhfb, fjournal = "Journal of the American Statistical Association", journal-URL = "http://www.tandfonline.com/loi/uasa20", } @Book{Stratton:1941:ECS, author = "Julius Adams Stratton and Philip M. (Philip McCord) Morse and Lan Jen Chu and Reina Albagli Hutner", title = "Elliptic cylinder and spheroidal wave functions, including tables of separation constants and coefficients", publisher = pub-WILEY, address = pub-WILEY:adr, pages = "xii + 127", year = "1941", LCCN = "QC174.2 .S78", bibdate = "Sat Apr 1 14:32:29 MDT 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; z3950.loc.gov:7090/Voyager", acknowledgement = ack-nhfb, author-dates = "1901--1994", remark = "A publication of the Technology Press, Massachusetts Institute of Technology.", subject = "Wave mechanics; Mathematics; Tables", } @PhdThesis{Dopper:1942:AOV, author = "Herman Pieter Dopper", title = "Asymptotische Ontwikkelingen van de Onvolledige Gammafuncties. ({Dutch}) [{Asymptotic} developments of the Incomplete Gamma Functions]", type = "{Ph.D.} thesis", school = "Rijksuniversiteit Groningen", address = "Groningen, The Netherlands", pages = "????", year = "1942", bibdate = "Sat Feb 18 14:32:59 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, language = "Dutch", remark = "Cited in \cite{Paris:2016:UAE}.", } @Article{Lancaster:1942:MME, author = "Otis E. Lancaster", title = "Machine Method for the Extraction of Cube Root", journal = j-J-AM-STAT-ASSOC, volume = "37", number = "217", pages = "112--115", month = mar, year = "1942", CODEN = "JSTNAL", ISSN = "0162-1459 (print), 1537-274X (electronic)", ISSN-L = "0162-1459", bibdate = "Wed Jan 25 08:05:24 MST 2012", bibsource = "http://www.jstor.org/journals/01621459.html; http://www.jstor.org/stable/i314096; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jamstatassoc1940.bib", URL = "http://www.jstor.org/stable/2279437", acknowledgement = ack-nhfb, fjournal = "Journal of the American Statistical Association", journal-URL = "http://www.tandfonline.com/loi/uasa20", } @Article{Archibald:1943:TTF, author = "Raymond Clare Archibald", title = "Tables of Trigonometric Functions in Non-Sexagesimal Arguments", journal = j-MATH-TABLES-AIDS-COMPUT, volume = "1", number = "2", pages = "33--44", month = apr, year = "1943", CODEN = "MTTCAS", DOI = "https://doi.org/10.1090/S0025-5718-43-99136-7", ISSN = "0891-6837 (print), 2326-4853 (electronic)", ISSN-L = "0891-6837", bibdate = "Tue Oct 13 08:44:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, journal-URL = "http://www.ams.org/mcom/", remark = "Original journal has only `R. C. A.' as author: possibly R. C. Archibald. AMS metadata now shows the full name.", } @Book{Tolke:1943:PFE, author = "Friedrich T{\"o}lke", title = "{Praktische Funktionenlehre. 1. Elementare und elementare transzendente Funktionen, Unterstufe}. ({German}) [{Practical} functional theory. 1. {Elementary} and elementary transcendental functions, lower stage]", publisher = pub-SV, address = pub-SV:adr, pages = "viii + 261", year = "1943", bibdate = "Mon Feb 13 19:15:35 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, language = "German", } @Article{Turing:1943:MCZ, author = "A. M. Turing", title = "A method for the calculation of the zeta-function", journal = j-PROC-LONDON-MATH-SOC-2, volume = "48", pages = "180--197", year = "1943", ISSN = "0024-6115 (print), 1460-244X (electronic)", ISSN-L = "0024-6115", MRclass = "10.0X", MRnumber = "MR0009612 (5,173a)", MRreviewer = "C. L. Siegel", bibdate = "Sat Nov 19 13:23:32 2005", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/t/turing-alan-mathison.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://turing.ecs.soton.ac.uk/browse.php/B/17", ZMnumber = "0061.08304", acknowledgement = ack-nhfb, fjournal = "Proceedings of the London Mathematical Society. Second Series", received = "7 March 1939", remark = "According to \cite[page 260]{Newman:1955:AMT}, publication was delayed four years by war-time difficulties.", } @Article{Bateman:1944:GTB, author = "Harry Bateman and Raymond Clare Archibald", title = "A Guide to Tables of {Bessel} Functions", journal = j-MATH-TABLES-OTHER-AIDS-COMPUT, volume = "1", number = "7", pages = "205--308", month = jul, year = "1944", CODEN = "MTTCAS", DOI = "https://doi.org/10.1090/S0025-5718-1944-0011175-4", ISSN = "0891-6837 (print), 2326-4853 (electronic)", ISSN-L = "0891-6837", MRnumber = "6,132b", MRreviewer = "G. Szeg{\H{o}}", bibdate = "Tue Oct 13 08:44:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "Mathematical Tables and Other Aids to Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Lehmer:1944:NCB, author = "Derrick Henry Lehmer", title = "Note on the Computation of the {Bessel} Function {$ I_n(X) $}", journal = j-MATH-TABLES-OTHER-AIDS-COMPUT, volume = "1", number = "5", pages = "133--135", month = apr, year = "1944", CODEN = "MTTCAS", DOI = "https://doi.org/10.1090/S0025-5718-44-99053-8", ISSN = "0891-6837 (print), 2326-4853 (electronic)", ISSN-L = "0891-6837", bibdate = "Tue Oct 13 08:44:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "Mathematical Tables and Other Aids to Computation", journal-URL = "http://www.ams.org/mcom/", remark = "Original journal has only `D. H. L.' as author: probably D. H. Lehmer. AMS metadata now shows the full name.", } @Book{Lewis:1944:SCH, author = "Charles J. Lewis", title = "A survey of the confluent hypergeometric function", publisher = "????", address = "Washington, DC, USA", pages = "155", year = "1944", LCCN = "QA351 .L53", bibdate = "Sat Oct 30 21:06:31 MDT 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; z3950.loc.gov:7090/Voyager", acknowledgement = ack-nhfb, subject = "Hypergeometric functions", } @Article{Abramowitz:1945:ZCB, author = "Milton Abramowitz", title = "Zeros of certain {Bessel} functions of fractional order", journal = j-MATH-TABLES-OTHER-AIDS-COMPUT, volume = "1", number = "9", pages = "353--354", month = jan, year = "1945", CODEN = "MTTCAS", DOI = "https://doi.org/10.1090/S0025-5718-1945-0011176-7", ISSN = "0891-6837 (print), 2326-4853 (electronic)", ISSN-L = "0891-6837", MRclass = "65.0X", MRnumber = "6,132c", bibdate = "Tue Oct 13 08:44:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "Mathematical Tables and Other Aids to Computation", journal-URL = "http://www.ams.org/mcom/", } @Book{Briggs:1945:TAL, author = "Lyman J. Briggs and Arnold N. Lowan", title = "Tables of Associated {Legendre} Functions", publisher = pub-U-COLUMBIA, address = pub-U-COLUMBIA:adr, pages = "xlvi + 303 + 3", year = "1945", LCCN = "QA406 .U5 1945", bibdate = "Sat Apr 1 14:47:02 MDT 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; z3950.loc.gov:7090/Voyager", acknowledgement = ack-nhfb, remark = "Prepared by the Mathematical Tables Project and conducted under the sponsorship of the National bureau of Standards. Present volume begun under the auspices of the Work Projects Administration for the City of New York and completed with the support of the Office of Scientific Research and Development. Lyman J. Briggs, Director, National Bureau of Standards and official sponsor. Arnold N. Lowan, Project Director, Mathematical Tables Project. Reproduced by a photo offset process.", subject = "Legendre's functions", } @Book{Emde:1945:TEF, author = "Fritz Emde", title = "{Tafeln Elementarer Funktionen} ({German}) [Tables of Elementary Functions]", publisher = "Edwards Bros.", address = "Ann Arbor, MI, USA", pages = "xii + 181", year = "1945", LCCN = "????", bibdate = "Fri Jun 11 12:34:09 1999", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "Original edition published in 1940.", acknowledgement = ack-nhfb, language = "German", } @Article{Anonymous:1946:MZC, author = "Anonymous", title = "More Zeros of Certain {Bessel} Functions of Fractional Order", journal = j-MATH-TABLES-OTHER-AIDS-COMPUT, volume = "2", number = "15", pages = "118--119", month = jul, year = "1946", CODEN = "MTTCAS", DOI = "https://doi.org/10.1090/S0025-5718-1946-0016689-0", ISSN = "0891-6837 (print), 2326-4853 (electronic)", ISSN-L = "0891-6837", bibdate = "Tue Oct 13 08:44:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "Mathematical Tables and Other Aids to Computation", journal-URL = "http://www.ams.org/mcom/", } @Book{Anonymous:1948:TBF, author = "Anonymous", title = "Tables of the {Bessel} Functions {$ Y_0 (x) $}, {$ Y_1 (x) $}, {$ K_0 (x) $}, {$ K_1 (x) $}, $ 0 \leq x \leq 1 $", volume = "1", publisher = pub-US-GPO, address = pub-US-GPO:adr, pages = "ix + 60", year = "1948", LCCN = "QA3 .U5 no. 1", bibdate = "Sat Nov 04 16:47:30 2023", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", series = ser-APPL-MATH-SER-NBS, acknowledgement = ack-nhfb, } @Book{Emde:1948:TEF, author = "Fritz Emde", title = "{Tafeln Elementarer Funktionen} ({German}) [Tables of Elementary Functions]", publisher = pub-TEUBNER, address = pub-TEUBNER:adr, edition = "Second", pages = "xii + 181", year = "1948", LCCN = "QA55 .E5 1948", MRclass = "65.0X", MRnumber = "11,263k", bibdate = "Sat Jan 11 17:44:01 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, language = "German", } @TechReport{Tukey:1948:NSR, author = "John W. Tukey", title = "A note on the square-root iteration", type = "SRG Memorandum report", number = "10", institution = inst-PRINCETON, address = inst-PRINCETON:adr, pages = "18", year = "1948", bibdate = "Tue May 15 08:00:09 2012", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/t/tukey-john-w.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, } @Article{Turing:1945:MCZ, author = "A. M. Turing", title = "A Method for the Calculation of the Zeta-Function", journal = j-PROC-LONDON-MATH-SOC-2, volume = "s2-48", number = "1", pages = "180--197", year = "1945", CODEN = "PLMTAL", DOI = "https://doi.org/10.1112/plms/s2-48.1.180", ISSN = "0024-6115 (print), 1460-244X (electronic)", ISSN-L = "0024-6115", bibdate = "Wed Dec 3 13:33:49 2025", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/t/turing-alan-mathison.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Proceedings of the London Mathematical Society. Second Series", journal-URL = "http://plms.oxfordjournals.org/content/by/year", read = "16 March 1935", received = "7 March 1939", remark = "NB: Notice the 6.5 year delay in publication, due to wartime conditions. The running page headers carry the date as 16 March 1939.", } @Article{Wise:1948:IBF, author = "M. E. Wise", title = "The Incomplete Beta Function and the Incomplete Gamma Function: An Acknowledgment", journal = j-J-R-STAT-SOC-SER-B-METHODOL, volume = "10", number = "2", pages = "264--264", month = "????", year = "1948", CODEN = "JSTBAJ", DOI = "https://doi.org/10.2307/2983781", ISSN = "0035-9246", ISSN-L = "0035-9246", bibdate = "Fri Jan 23 11:53:21 MST 2015", bibsource = "http://www.jstor.org/stable/i349688; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jrss-b.bib", URL = "http://www.jstor.org/stable/2983781", acknowledgement = ack-nhfb, fjournal = "Journal of the Royal Statistical Society. Series B (Methodological)", journal-URL = "http://www.jstor.org/journals/00359246.html", } @Article{Hartree:1949:NIP, author = "D. R. Hartree", title = "Notes on iterative processes", journal = j-PROC-CAMBRIDGE-PHIL-SOC, volume = "45", number = "2", pages = "230--236", month = apr, year = "1949", CODEN = "PCPSA4", DOI = "https://doi.org/10.1017/s0305004100024762", ISSN = "0008-1981", ISSN-L = "0008-1981", MRclass = "65.0X", MRnumber = "29268", MRreviewer = "E. Bodewig", bibdate = "Thu Aug 3 09:15:52 2023", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/h/hartree-douglas-r.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "Correction: in equation (29) at the bottom of page 233, replace the denominator term $ 2 y_0 $ by $ 2 y_1 $, matching the denominator in equation (32) on page 234.", URL = "https://ui.adsabs.harvard.edu/abs/1949PCPS...45..230H", ZMnumber = "0033.19003", acknowledgement = ack-nhfb, author-dates = "Douglas Rayner Hartree (27 March 1897--12 February 1958)", fjournal = "Proceedings of the Cambridge Philosophical Society", journal-URL = "http://journals.cambridge.org/action/displayJournal?jid=PSP", keywords = "$1/p$-th root; iterative process; reciprocal square root; square root", ZBmath = "3051117", } @Article{Lowan:1949:CLN, author = "Arnold N. Lowan", title = "The {Computation Laboratory of the National Bureau of Standards}", journal = j-SCRIPTA-MATH, volume = "15", number = "??", pages = "33--63", month = "????", year = "1949", ISSN = "0036-9713", bibdate = "Thu Oct 26 11:15:25 2023", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/scripta-math.bib", ZMnumber = "0034.07002", acknowledgement = ack-nhfb, ajournal = "Scripta Math.", fjournal = "Scripta Mathematica: A Quarterly Journal Devoted to the Philosophy, History, and Expository Treatment of Mathematics", ZBmath = "3052129", } @Book{Magnus:1949:FTS, author = "Wilhelm Magnus and Fritz Oberhettinger", title = "Formulas and theorems for the special functions of mathematical physics", publisher = "Chelsea Pub. Co.", address = "New York, NY, USA", pages = "172", year = "1949", LCCN = "QA41 M19e", bibdate = "Sat Oct 30 18:44:51 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "Translated from the German by John Wermer.", acknowledgement = ack-nhfb, } @Article{Mitchell:1949:TFA, author = "K. Mitchell", title = "Tables of the function $ \int_0^z - \log |1 - y| / y \, d y $ with an account of some properties of this and related functions", journal = j-PHILOS-MAG, volume = "40", number = "302", pages = "351--368", year = "1949", CODEN = "PHMAA4", DOI = "https://doi.org/10.1080/14786444908561256", ISSN = "0031-8086", bibdate = "Sat Jun 17 17:47:16 2023", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "https://www.tandfonline.com/doi/abs/10.1080/14786444908561256", acknowledgement = ack-nhfb, fjournal = "Philosophical Magazine", journal-URL = "http://www.tandfonline.com/loi/tphm19", received = "9 April 1948", } @InProceedings{Polya:1949:RCP, author = "G. P{\'o}lya", editor = "J. Neyman", booktitle = "Proceedings of the First Berkeley Symposium on Mathematical Statistics and Probability", title = "Remarks on computing the probability integral in one and two dimensions", publisher = pub-U-CALIFORNIA-PRESS, address = pub-U-CALIFORNIA-PRESS:adr, pages = "63--78", year = "1949", bibdate = "Sat Dec 16 17:23:49 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, } @Book{USNBSCL:1949:TCH, author = "{United States National Bureau of Standards Computation Laboratory }", title = "Tables of the confluent hypergeometric function {$ F(n / 2, 1 / 2, x) $ and related functions}", volume = "3", publisher = pub-US-GPO, address = pub-US-GPO:adr, pages = "xxii + 73", year = "1949", LCCN = "QA3 .U5 no. 3", bibdate = "Sat Oct 30 21:06:31 MDT 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; z3950.loc.gov:7090/Voyager", series = "Applied mathematics series", URL = "http://openlibrary.org/works/OL1218358W/Tables_of_the_confluent_hypergeometric_function_F%28n_2_1_2_x%29_and_related_functions", acknowledgement = ack-nhfb, subject = "Hypergeometric functions; Mathematics; Tables", } @Article{Norlund:1950:HF, author = "Niels Erik N{\o}rlund", title = "Hypergeometric functions", journal = "Mat. Tidsskr. B.", volume = "1950", number = "??", pages = "18--21", year = "1950", MRclass = "33.0X", MRnumber = "MR0045259 (13,554e)", MRreviewer = "S. C. van Veen", bibdate = "Thu Dec 01 12:41:44 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", ZMnumber = "0067.29402", acknowledgement = ack-nhfb, language = "Danish", } @TechReport{Stanley:1950:TRG, author = "John Pearson Stanley and Maurice V. Wilkes", title = "Table of the reciprocal of the gamma function for complex argument", type = "Report", institution = "Computation Centre, University of Toronto", address = "Toronto, ON, Canada", pages = "????", year = "1950", MRclass = "65.0X", MRnumber = "48144", MRreviewer = "S. C. van Veen", bibdate = "Sat Sep 07 16:52:05 2024", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/w/wilkes-maurice-v.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", ZMnumber = "0040.06903", abstract = "Toronto, ON, Canada", acknowledgement = ack-nhfb, author-dates = "Sir Maurice Vincent Wilkes (26 June 1913--29 November 2010)", RSBM-number = "22", ZBmath = "3060432", } @Book{Tolke:1950:PFE, author = "Friedrich T{\"o}lke", title = "{Praktische Funktionenlehre. 1. Elementare und elementare transzendente Funktionen}. ({German}) [{Practical} functional theory. 1. {Elementary} and elementary transcendental functions]", publisher = pub-SV, address = pub-SV:adr, pages = "xi + 440", year = "1950", bibdate = "Mon Feb 13 19:12:35 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, language = "German", } @Article{Tricomi:1950:AEU, author = "F. G. Tricomi", title = "{Asymptotische Eigenschaften der unvollst{\"a}ndigen Gammafunktion}. ({German}) [{Asymptotic} properties of the incomplete gamma function]", journal = j-MATH-Z, volume = "53", number = "2", pages = "136--148", month = apr, year = "1950", CODEN = "MAZEAX", DOI = "https://doi.org/10.1007/BF01162409", ISSN = "0025-5874 (print), 1432-1823 (electronic)", ISSN-L = "0025-5874", bibdate = "Sat Feb 18 14:47:24 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://link.springer.com/article/10.1007/BF01162409", acknowledgement = ack-nhfb, fjournal = "Mathematische Zeitschrift", journal-URL = "http://link.springer.com/journal/209", language = "German", } @Article{Cadwell:1951:BNI, author = "J. H. Cadwell", title = "The Bivariate Normal Integral", journal = j-BIOMETRIKA, volume = "38", number = "3/4", pages = "475--479", month = dec, year = "1951", CODEN = "BIOKAX", DOI = "https://doi.org/10.2307/2332596", ISSN = "0006-3444 (print), 1464-3510 (electronic)", ISSN-L = "0006-3444", MRclass = "60.0X", MRnumber = "0045960 (13,662h)", MRreviewer = "G. E. Noether", bibdate = "Sat Jun 21 14:32:38 MDT 2014", bibsource = "http://www.jstor.org/journals/00063444.html; http://www.jstor.org/stable/i315418; https://www.math.utah.edu/pub/tex/bib/biometrika1950.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.jstor.org/stable/2332596", acknowledgement = ack-nhfb, fjournal = "Biometrika", journal-URL = "http://biomet.oxfordjournals.org/content/by/year; http://www.jstor.org/journals/00063444.html", } @Article{Fogel:1951:FTE, author = "{\`E}. K. Fogel'", title = "A finite theory of elementary functions. {I}. {Logarithmic} and exponential functions. ({Russian})", journal = "Latvijas PSR Zin{\=a}t{\c{n}}u Akad. V{\=e}stis", volume = "5", number = "46", pages = "801--813", year = "1951", MRclass = "33.0X", MRnumber = "15,218b", MRreviewer = "H. N. Shapiro", bibdate = "Sat Apr 25 13:05:19 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, language = "Russian", } @InProceedings{Gellman:1951:CCH, author = "Harvey Gellman", booktitle = "Proceedings of a Computation Seminar [{IBM Department of Education, Endicot, NY, from December 5 to 9, 1949}]", title = "The Calculation of Complex Hypergeometric Functions with the {IBM Type 602-A} Calculating Punch", publisher = "IBM", address = "New York, NY, USA", pages = "161--168", year = "1951", bibdate = "Mon Jun 18 06:09:24 2018", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, remark = "A Computation Seminar, sponsored by the International Business Machines Corporation, was held in the IBM Department of Education, Endicot, NY, from December 5 to 9, 1949. Attending the Seminar were one hundred and seven research engineers and scientists who are experienced both in applying mathematical methods to the solution of physical problems and in the associated punched card methods of computation.", } @Article{Rutishauser:1951:BAK, author = "Heinz Rutishauser", title = "{Bemerkungen zur Arbeit von K. Emden ``Eine L{\"o}sung f{\"u}r $ \int e^{b(x + a \cos x)} \, d x $}''. ({German}) [{Remarks} on the work by {K. Emden, ``A solution for $ \int e^{b (x + a \ cos x)} \, d x $''}]", journal = j-Z-ANGE-MATH-PHYS, volume = "2", number = "4", pages = "292--293", month = jul, year = "1951", CODEN = "ZAMPDB", DOI = "https://doi.org/10.1007/bf02579691", ISSN = "0044-2275 (print), 1420-9039 (electronic)", ISSN-L = "0044-2275", MRclass = "26.1X", MRnumber = "44598", MRreviewer = "F. J. Murray", bibdate = "Mon Aug 24 21:56:15 2020", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/r/rutishauser-heinz.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, author-dates = "Heinz Rutishauser (30 January 1918--10 November 1970)", fjournal = "Zeitschrift f{\"u}r Angewandte Mathematik und Physik. ZAMP. Journal of Applied Mathematics and Physics. Journal de Math\'{e}matiques et de Physique Appliqu\'{e}es", journal-URL = "http://link.springer.com/journal/33", language = "German", } @Article{Salzer:1951:FCE, author = "H. E. Salzer", title = "Formulas for Calculating the Error Function of a Complex Variable", journal = j-MATH-TABLES-OTHER-AIDS-COMPUT, volume = "5", number = "34", pages = "67--70", month = apr, year = "1951", CODEN = "MTTCAS", DOI = "https://doi.org/10.1090/S0025-5718-1951-0048150-3", ISSN = "0891-6837 (print), 2326-4853 (electronic)", ISSN-L = "0891-6837", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "Mathematical Tables and Other Aids to Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Salzer:1951:RTT, author = "H. E. Salzer", title = "Radix Table for Trigonometric Functions and their Inverses to High Accuracy", journal = j-MATH-TABLES-OTHER-AIDS-COMPUT, volume = "5", number = "33", pages = "9--11", month = jan, year = "1951", CODEN = "MTTCAS", DOI = "https://doi.org/10.1090/S0025-5718-51-99447-1", ISSN = "0891-6837 (print), 2326-4853 (electronic)", ISSN-L = "0891-6837", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "Mathematical Tables and Other Aids to Computation", journal-URL = "http://www.ams.org/mcom/", } @Book{Wilkes:1951:PPE, author = "Maurice V. Wilkes and David J. Wheeler and Stanley Gill", title = "The Preparation of Programs for an Electronic Digital Computer", publisher = pub-AW, address = pub-AW:adr, pages = "167", year = "1951", LCCN = "QA76.5 .W55 1951", bibdate = "Mon Feb 10 09:42:47 2020", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/h/hartree-douglas-r.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", note = "See also second edition \cite{Wilkes:1957:PPE}, and reprint \cite{Wilkes:1982:PPE}.", acknowledgement = ack-nhfb, tableofcontents = "Part I \\ Chapter 1. The Design of Programs for Electronic Computing Machines / 1 \\ 1-1 Introduction / 1 \\ 1-2 Types of automatic computing machines / 2 \\ 1-3 Description of the EDSAC / 3 \\ 1-4 The EDSAC order code / 5 \\ 1-5 Notes on the order code / 6 \\ 1-6 The use of conditional orders / 7 \\ 1-7 Modification of orders by the program / 8 \\ 1-8 Multiaddress codes / 11 \\ 1-9 Binary--decimal conversion / 12 \\ 1-10 Checking facilities / 14 \\ Chapter 2. Input of Orders / 15 \\ 2-1 Initial orders / 15 \\ 2-2 Pseudo-orders / 17 \\ 2-3 Examples / 17 \\ 2-4 Control combinations / 17 \\ 2-5 Starting the program / 18 \\ 2-6 Use of code letters / 19 \\ 2-7 Constants / 20 \\ 2-8 Notation / 20 \\ Chapter 3. Subroutines and Parameters / 22 \\ 3-1 Open subroutines / 22 \\ 3-2 closed subroutines / 22 \\ 3-3 preset parameters / 23 \\ 3-4 program parameters / 23 \\ Chapter 4. Library Subroutines and their Use in Constructing Programs / 25 \\ 4-1 Library catalog / 25 \\ 4-2 Input and output subroutines / 25 \\ 4-3 Division subroutines / 27 \\ 4-4 Trigonometrical and other functions / 27 \\ 4-5 Quadrature / 27 \\ 4-6 Assembly subroutines / 27 \\ 4-7 Integration of differential equations / 32 \\ 4-8 Processes, Interpretive subroutines / 34 \\ Chapter 5. Pitfalls / 38 \\ 5-1 Proofreading of programs, points to be checked / 38 \\ 5-2 Location of mistakes in a program / 39 \\ 5-3 Counting operations / 41 \\ Chapter 6. Use of the EDSAC \& Its Associated Equipment / 42 \\ 6-1 Tape Punching \& editing facilities / 42 \\ 6-2 Storage of library subroutines / 43 \\ 6-3 EDSAC organization / 43 \\ 6-4 EDSAC controls / 43 \\ Chapter 7. Examples / 45 \\ 7-1 Example 1. Calculation of $\exp(-\sin x)$ / 45 \\ 7-2 Example 2. Calculation of $\pi$ by evaluation of definite integral / 48 \\ 7-3 Alternative method for Example 2 / 52 \\ 7-4 Example 2, with extra print orders for checking / 53 \\ 7-5 Application of checking subroutine C11 to Example 2 / 54 \\ 7-6 Example of integration of an ordinary differential equation / 46 \\ 7-7 Evaluation of a definite integral / 61 \\ 7-8 Program to facilitate the solution of algebraic equation / 66 \\ Part II. Specifications of Library Subroutines / 72 \\ A. Subroutines to carry out floating point arithmetic / 73 \\ B. Subroutines to carry out arithmetical operations on complex numbers / 78 \\ C. Checking subroutines / 79 \\ D. Division subroutines / 82 \\ E. Exponential subroutines / 83 \\ F. General routines relating to functions / 84 \\ G. Subroutines for integration of ordinary differential equations / 86 \\ J. Subroutines for calculating special functions [Legendre polynomials] / 88 \\ K. Subroutines for the summation of power series / 88 \\ L. Subroutines for evaluating logarithms / 91 \\ M. Miscellaneous subroutines / 91 \\ P. Print subroutines / 92 \\ Q. Quadrature subroutines / 95 \\ R. Input subroutines / 96 \\ S. Subroutines for evaluation of fractional powers / 98 \\ T. Subroutines for calculating trigonometrical functions / 99 \\ U. Subroutines for counting operations / 101 \\ V1. Multiplication of vector by symmetric matrix / 102 \\ V2. Addition and subtraction of $n$ dimensional vectors / 103 \\ Part III. Programs of Selected Library Subroutines / 104 \\ Appendix A. Keyboard perforator code, etc. / 158 \\ Appendix B. The initial orders / 159 \\ Appendix C. Control combinations / 161 \\ Appendix D. Interpretive subroutines: example of packing of orders / 162 \\ Appendix E. Methods of counting in a simple cycle / 164 \\ Index", } @Book{Anonymous:1952:BFP, author = "Anonymous", title = "{Bessel} Functions, {Part II}", volume = "10", publisher = pub-CAMBRIDGE, address = pub-CAMBRIDGE:adr, pages = "????", year = "1952", bibdate = "Tue Nov 14 14:59:31 2023", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/f/fox-leslie.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", series = "British Association Mathematical Tables", acknowledgement = ack-nhfb, author-dates = "Leslie Fox (30 September 1918--1 August 1992)", remark = "TO DO: was Leslie Fox a co-author?? Who were the authors?? Page count??", } @InBook{Goncarov:1952:EFC, author = "V. L. Gon{\v{c}}arov", booktitle = "Encyclopaedia of elementary mathematics. {Book III}. {Functions} and limits (the foundations of analysis)", title = "Elementary functions of a complex variable", publisher = "Gosudarstv. Izdat. Tehn-Teoret. Lit.", address = "Moscow-Leningrad, USSR", pages = "491--552", year = "1952", MRclass = "30.0X", MRnumber = "14,1073c", MRreviewer = "R. P. Boas, Jr.", bibdate = "Sat Jan 11 17:44:01 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, } @InCollection{Goncarov:1952:EFR, author = "V. L. Gon{\v{c}}arov", booktitle = "Encyclopaedia of elementary mathematics. {Book III}. {Functions} and limits (the foundations of analysis)", title = "Elementary functions of a real variable. Limits of sequences and functions. {The} general concept of a function", publisher = "Gosudarstv. Izdat. Tehn-Teoret. Lit.", address = "Moscow-Leningrad, USSR", pages = "9--296", year = "1952", MRclass = "27.2X", MRnumber = "14,1070d", MRreviewer = "R. P. Boas, Jr.", bibdate = "Sat Jan 11 17:44:01 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, } @Article{Kuipers:1952:PSE, author = "L. Kuipers", title = "Properties of some elementary functions", journal = "Nederl. Akad. Wetensch. Proc. Ser. A. = Indagationes Math.", volume = "55", number = "14", pages = "388--393", year = "1952", MRclass = "27.0X", MRnumber = "14,360e", MRreviewer = "E. Frank", bibdate = "Sat Jan 11 17:44:01 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, } @Book{Bowman:1953:IEF, author = "Frank Bowman", title = "Introduction to Elliptic Functions with Applications", publisher = "English Universities Press", address = "London, UK", pages = "115", year = "1953", LCCN = "QA343 .B76 1953", bibdate = "Wed Mar 15 06:50:49 MDT 2017", bibsource = "fsz3950.oclc.org:210/WorldCat; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, subject = "Elliptic functions; Elliptische functies", } @InProceedings{Lovelace:1953:AET, author = "Ada Augusta Lovelace", title = "{Appendix 1}: {Extracts} From {{\booktitle{Taylor's Scientific Memoirs}}, Vol. III}", crossref = "Bowden:1953:FTT", pages = "341--408", year = "1953", bibdate = "Fri Jun 08 08:33:30 2018", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/l/lovelace-ada-augusta.bib; https://www.math.utah.edu/pub/tex/bib/adabooks.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "Reprint of \cite{Lovelace:1843:SAE}. Pages 400--408 describe the computation of the Bernoulli numbers.", acknowledgement = ack-nhfb, } @Article{Salzer:1953:RTO, author = "Herbert E. Salzer", title = "Radix Table for Obtaining Hyperbolic and Inverse Hyperbolic Functions to Many Places", journal = j-J-MATH-PHYS-MIT, volume = "32", number = "1--4", pages = "197--202", month = apr, year = "1953", CODEN = "JMPHA9", DOI = "https://doi.org/10.1002/sapm1953321197", ISSN = "0097-1421", ISSN-L = "0097-1421", bibdate = "Sat Aug 19 13:35:59 MDT 2023", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jmathphysmit.bib", URL = "https://onlinelibrary.wiley.com/doi/epdf/10.1002/sapm1953321197", acknowledgement = ack-nhfb, ajournal = "J. Math. Phys. (MIT)", fjournal = "Journal of Mathematics and Physics (MIT)", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1111/(ISSN)1467-9590", onlinedate = "April 1953", } @Article{Scherberg:1953:ACP, author = "Max G. Scherberg and John F. Riordan", title = "Analogue Calculation of Polynomial and Trigonometric Expansions (in {Other Aids to Computation})", journal = j-MATH-TABLES-OTHER-AIDS-COMPUT, volume = "7", number = "41", pages = "61--65", month = jan, year = "1953", CODEN = "MTTCAS", DOI = "https://doi.org/10.1090/S0025-5718-53-99373-9", ISSN = "0891-6837 (print), 2326-4853 (electronic)", ISSN-L = "0891-6837", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "Mathematical Tables and Other Aids to Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Stiefel:1953:ITF, author = "E. Stiefel", title = "{Zur Interpolation von tabellierten Funktionen durch Exponentialsummen und zur Berechnung von Eigenwerten aus den Schwarzschen Konstanten}. ({German}) [{On} interpolation of tabulated functions by exponential sums and on the calculation of eigenvalues from the {Schwarz}'s constants]", journal = j-Z-ANGE-MATH-MECH, volume = "33", pages = "260--262", year = "1953", CODEN = "ZAMMAX", DOI = "https://doi.org/10.1002/zamm.19530330806", ISSN = "0044-2267 (print), 1521-4001 (electronic)", ISSN-L = "0044-2267", MRclass = "65.0X", MRnumber = "59061", MRreviewer = "D. C. Gilles", bibdate = "Wed Sep 2 16:23:13 2020", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/s/stiefel-eduard.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, author-dates = "Eduard Stiefel (21 April 1909--25 November 1978)", fjournal = "Zeitschrift f{\"{u}}r Angewandte Mathematik und Mechanik. Ingenieurwissenschaftliche Forschungsarbeiten", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)1521-4001", language = "German", } @Article{Turing:1953:SCR, author = "A. M. Turing", title = "Some calculations of the {Riemann} zeta-function", journal = j-PROC-LONDON-MATH-SOC-3, volume = "3", number = "3", pages = "99--117", year = "1953", CODEN = "PLMTAL", ISSN = "0024-6115 (print), 1460-244X (electronic)", ISSN-L = "0024-6115", MRclass = "65.0X", MRnumber = "MR0055785 (14,1126e)", MRreviewer = "D. H. Lehmer", bibdate = "Sat Nov 19 13:23:32 2005", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/t/turing-alan-mathison.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "See corrections and improvements \cite{Lehman:1970:DZR}, and \cite{Trudgian:2011:ITM}. The latter comments: ``Turing's Method has become the standard technique used in modern verification of the Riemann hypothesis.'' See also \cite{Lehmer:1956:RRZ}.", URL = "http://turing.ecs.soton.ac.uk/browse.php/B/21", ZMnumber = "0050.08101", acknowledgement = ack-nhfb, fjournal = "Proceedings of the London Mathematical Society. Third Series", } @Article{Aaboe:1954:AKI, author = "Asger Aaboe", title = "{Al-K{\=a}sh{\v{\i}}}'s iteration method for the determination of $ \sin 1^\circ $", journal = j-SCRIPTA-MATH, volume = "20", number = "??", pages = "24--29", month = "????", year = "1954", ISSN = "0036-9713", ISSN-L = "0036-9713", MRclass = "01.0X", MRnumber = "62046", bibdate = "Thu Oct 26 11:15:25 2023", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/scripta-math.bib", ZMnumber = "0055.00104", acknowledgement = ack-nhfb, ajournal = "Scripta Math.", fjournal = "Scripta Mathematica: A Quarterly Journal Devoted to the Philosophy, History, and Expository Treatment of Mathematics", ZBmath = "3086736", } @Article{Atta:1954:CGH, author = "Susie E. Atta and Ward C. Sangren", title = "Calculation of Generalized Hypergeometric Series", journal = j-J-ACM, volume = "1", number = "4", pages = "170--172", month = oct, year = "1954", CODEN = "JACOAH", DOI = "https://doi.org/10.1145/320783.320785", ISSN = "0004-5411 (print), 1557-735X (electronic)", ISSN-L = "0004-5411", bibdate = "Tue Nov 08 21:50:00 1994", bibsource = "http://portal.acm.org/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jacm.bib", acknowledgement = ack-nhfb, ajournal = "J. Assoc. Comput. Mach.", fjournal = "Journal of the Association for Computing Machinery", journal-URL = "https://dl.acm.org/loi/jacm", } @Article{Cahill:1954:PCH, author = "W. F. Cahill", title = "Programs for Computing the Hypergeometric Series (in Automatic Computing Machinery; Discussions)", journal = j-MATH-TABLES-OTHER-AIDS-COMPUT, volume = "8", number = "45", pages = "36--37", month = jan, year = "1954", CODEN = "MTTCAS", DOI = "https://doi.org/10.1090/S0025-5718-54-99344-8", ISSN = "0891-6837 (print), 2326-4853 (electronic)", ISSN-L = "0891-6837", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp1950.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "Mathematical Tables and Other Aids to Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Clenshaw:1954:PAE, author = "C. W. Clenshaw", title = "Polynomial approximations to elementary functions", journal = j-MATH-TABLES-OTHER-AIDS-COMPUT, volume = "8", number = "47", pages = "143--147", month = jul, year = "1954", CODEN = "MTTCAS", DOI = "https://doi.org/10.1090/S0025-5718-1954-0063487-2", ISSN = "0891-6837 (print), 2326-4853 (electronic)", ISSN-L = "0891-6837", MRclass = "41.1X", MRnumber = "16,128f", bibdate = "Sat Jan 11 17:44:01 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Mathematical Tables and Other Aids to Computation", journal-URL = "http://www.ams.org/mcom/", } @Book{Fox:1954:STB, author = "L. Fox", title = "A Short Table for {Bessel} Functions of Integer Orders and Large Arguments", volume = "3", publisher = pub-CAMBRIDGE, address = pub-CAMBRIDGE:adr, pages = "28", year = "1954", MRclass = "65.0X", MRnumber = "65245", MRreviewer = "R. C. Archibald", bibdate = "Mon Nov 13 14:02:18 2023", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/f/fox-leslie.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", series = "Royal Society Shorter Mathematical Tables", acknowledgement = ack-nhfb, author-dates = "Leslie Fox (30 September 1918--1 August 1992)", } @TechReport{Franklin:1954:CARa, author = "J. Franklin and B. Friedman", title = "A convergent asymptotic representation for integrals", institution = "Division of Electromagnetic Research, Institute of Mathematical Sciences, New York University", address = "New York, NY, USA", pages = "i + 17", year = "1954", MRclass = "40.0X", MRnumber = "0068019", MRreviewer = "J. G. van der Corput", bibdate = "Tue Feb 06 15:03:36 2018", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", series = "Res. Rep. No. BR-9", acknowledgement = ack-nhfb, remark = "See applications in \cite{Temme:2015:AMI,Navas-Palencia:2018:HPC}.", } @Article{LaFara:1954:MCI, author = "Robert L. LaFara", title = "A Method for Calculating Inverse Trigonometric Functions", journal = j-MATH-TABLES-OTHER-AIDS-COMPUT, volume = "8", number = "47", pages = "132--139", month = jul, year = "1954", CODEN = "MTTCAS", DOI = "https://doi.org/10.1090/S0025-5718-1954-0063150-8", ISSN = "0891-6837 (print), 2326-4853 (electronic)", ISSN-L = "0891-6837", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "Mathematical Tables and Other Aids to Computation", journal-URL = "http://www.ams.org/mcom/", } @InCollection{Ostrowski:1954:TPA, author = "A. M. Ostrowski", title = "On two problems in abstract algebra connected with {Horner}'s rule", crossref = "Birkhoff:1954:SMM", pages = "40--48", year = "1954", bibdate = "Fri Oct 20 10:13:10 2023", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, keywords = "number of multiplications to evaluate a polynomial", remark = "TO DO: Find copy of this book section.", } @Article{Shenton:1954:INI, author = "L. R. Shenton", title = "Inequalities for the Normal Integral Including a New Continued Fraction", journal = j-BIOMETRIKA, volume = "41", number = "1/2", pages = "177--189", month = jun, year = "1954", CODEN = "BIOKAX", DOI = "https://doi.org/10.2307/2333015", ISSN = "0006-3444 (print), 1464-3510 (electronic)", ISSN-L = "0006-3444", MRclass = "62.0X", MRnumber = "0061785 (15,884e)", MRreviewer = "E. Lukacs", bibdate = "Sat Jun 21 14:32:43 MDT 2014", bibsource = "http://www.jstor.org/journals/00063444.html; http://www.jstor.org/stable/i315422; https://www.math.utah.edu/pub/tex/bib/biometrika1950.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.jstor.org/stable/2333015", acknowledgement = ack-nhfb, fjournal = "Biometrika", journal-URL = "http://biomet.oxfordjournals.org/content/by/year; http://www.jstor.org/journals/00063444.html", } @InProceedings{Todd:1954:MWN, author = "John Todd", editor = "????", booktitle = "Transactions of {2nd Symposium on Applied Mathematics, 29 April 1954, University of Chicago}", title = "Motivation for working on numerical analysis", publisher = pub-AMS, address = pub-AMS:adr, pages = "????", year = "1954", bibdate = "Fri Oct 20 13:28:45 2023", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, remark = "Sponsored by the American Mathematical Society and the Office of Ordnance Research", } @Article{Booth:1955:NAP, author = "A. D. Booth", title = "A note on approximating polynomials for trigonometric functions", journal = j-MATH-TABLES-OTHER-AIDS-COMPUT, volume = "9", number = "49", pages = "21--23", month = jan, year = "1955", CODEN = "MTTCAS", DOI = "https://doi.org/10.1090/S0025-5718-1955-0069579-7", ISSN = "0891-6837 (print), 2326-4853 (electronic)", ISSN-L = "0891-6837", MRclass = "65.0X", MRnumber = "69579", MRreviewer = "L. Fox", bibdate = "Tue Nov 14 17:19:58 2023", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/f/fox-leslie.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp1950.bib", acknowledgement = ack-nhfb, fjournal = "Mathematical Tables and other Aids to Computation", journal-URL = "http://www.ams.org/mcom/", reviewer-dates = "Leslie Fox (30 September 1918--1 August 1992)", } @TechReport{Carlson:1955:RAF, author = "Bengt Carlson and Max Goldstein", title = "Rational Approximations of Functions", type = "Report", number = "LA-1943", institution = inst-LASL, address = inst-LASL:adr, pages = "iv + 46", month = aug, year = "1955", DOI = "https://doi.org/10.2172/4374577", bibdate = "Sat Dec 27 09:41:36 2014", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.osti.gov/scitech/biblio/4374577-0deJO9/; http://www.osti.gov/scitech/servlets/purl/4374577", acknowledgement = ack-nhfb, keywords = "$-\ln(x)/(1 - x)$; $\art(x) / x$; $\cos(x)$; $\exp(-x)$; $\sin(x) / x$; $\tan(x) / x$; $x \cot(x)$; $x**(1/2)$; $x**(1/3)$; $x**(1/4)$; $x**(1/5)$; $x**(1/6)$; $x**(1/7)$; continued fractions; rational approximations", remark = "Cited in \cite[page 71]{Abramowitz:1964:HMF}.", } @Article{Clenshaw:1955:NSC, author = "C. W. Clenshaw", title = "A Note on the Summation of {Chebyshev} Series", journal = j-MATH-TABLES-OTHER-AIDS-COMPUT, volume = "9", number = "51", pages = "118--120", month = jul, year = "1955", CODEN = "MTTCAS", DOI = "https://doi.org/10.1090/S0025-5718-1955-0071856-0", ISSN = "0891-6837 (print), 2326-4853 (electronic)", ISSN-L = "0891-6837", MRclass = "65.0X", MRnumber = "0071856", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/c/clenshaw-charles-w.bib; https://www.math.utah.edu/pub/bibnet/subjects/acc-stab-num-alg-2ed.bib; https://www.math.utah.edu/pub/bibnet/subjects/acc-stab-num-alg.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp1950.bib; JSTOR database", acknowledgement = ack-nhfb, author-dates = "Charles William Clenshaw (15 March 1926--23 September 2004)", fjournal = "Mathematical Tables and Other Aids to Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "Chebyshev series; Clenshaw algorithm; Clenshaw summation; Horner polynomial evaluation", remark = "Hidden inside \cite{Brenner:1955:TNS}, but important in its own right for commentary on the recursive algorithm for summation of Chebyshev series, and a brief analysis of its accuracy.", } @Article{Froberg:1955:NTC, author = "Carl-Erik Fr{\"o}berg", title = "Numerical Treatment of {Coulomb} Wave Functions", journal = j-REV-MOD-PHYS, volume = "27", number = "4", pages = "399--411", month = oct, year = "1955", CODEN = "RMPHAT", DOI = "https://doi.org/10.1103/RevModPhys.27.399", ISSN = "0034-6861 (print), 1538-4527 (electronic), 1539-0756", ISSN-L = "0034-6861", bibdate = "Tue May 22 16:36:44 MDT 2012", bibsource = "http://rmp.aps.org/toc/RMP/v27/i4; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/revmodphys1950.bib", URL = "http://link.aps.org/doi/10.1103/RevModPhys.27.399; http://rmp.aps.org/abstract/RMP/v27/i4/p399_1", acknowledgement = ack-nhfb, fjournal = "Reviews of Modern Physics", journal-URL = "http://rmp.aps.org/browse", } @Book{Hastings:1955:ADC, author = "Cecil {Hastings, Jr.}", title = "Approximations for Digital Computers", publisher = pub-PRINCETON, address = pub-PRINCETON:adr, pages = "viii + 201", year = "1955", ISBN = "0-691-07914-5", ISBN-13 = "978-0-691-07914-1", LCCN = "QA76 .H37", bibdate = "Mon Oct 01 15:59:48 2007", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/master.bib; https://www.math.utah.edu/pub/tex/bib/prng.bib; z3950.loc.gov:7090/Voyager", note = "Assisted by Jeanne T. Hayward and James P. Wong, Jr.", series = "The Rand series", acknowledgement = ack-nhfb, remark-1 = "Reprinted 1957, 1959, 1962, 1966, and 1970. I have fourth printing (1962).", remark-2 = "Hastings gives a polynomial approximation for computing random numbers from a normal distribution.", subject = "Electronic digital computers; Numerical analysis", tableofcontents = "Preface / v \\ Part I \\ 1: Concerning Best Fit / 3 \\ 2: Linear Weights / 19 \\ 3: An Iterative Procedure / 27 \\ 4: Solution of Equations / 35 \\ 5: Chebyshev Polynomials / 47 \\ 6: Concerning Weights / 65 \\ 7: Function With a Peak / 75 \\ 8: Rates of Convergence / 83 \\ 9: Choice of Form / 95 \\ 10: A Scoring-Camera Problem / 115 \\ Part II \\ 1: $\log_{10} x$ / 125 \\ 5: $\phi(x) = (1 - e^{-x}) / x$ / 129 \\ 8: $\arctan x$ / 132 \\ 14: $\sin (\pi/2) x$ / 138 \\ 17: $10^x$ / 141 \\ 21: $W(x) = e^{-x} / (1 + e^{-x})^2$ / 145 \\ 24: $P_k(x) = 1.72 + 42 x^2$ or $0.136 / x^2$ / 148 \\ 27: $E'(x) = (1 / \sqrt{2 \pi}) e^{-(1/2)x^2}$ / 151 \\ 30: ``Total Klein-Nishina Cross Section'' Function / 154 \\ 31: $\Gamma(1 + x)$ / 155 \\ 35: $\arcsin x$ / 159 \\ 40: $\log_2 x$ / 164 \\ 43: $\Phi(x) = (2 / \sqrt{\pi}) \int_0^x e^{-t^2} \, dt$ / 167 \\ 46: $K(k) = \int_0^{\pi/2} (1 / \sqrt{1 - k^2 \sin^2 \phi}) \, d\phi$ / 170 \\ 49: $E(k) = \int_0^{\pi/2} (\sqrt{1 - k^2 \sin^2 \phi}) \, d\phi$ / 173 \\ 52: $\ln(1 + x)$ / 176 \\ 57: $e^{-x}$ / 181 \\ 61: $\Phi(x) = (2 / \sqrt{\pi}) \int_0^x e^{-t^2} \, dt$ / 185 \\ 64: $-{\rm Ei}(-x) = \int_x^\infty (e^{-t} / t) \, dt$ / 188 \\ 67: $q = (1 / \sqrt{2 \pi}) \int_{x(q)}^\infty e^{-(1/2)t^2} \, dt$ / 191 \\ 69: $W(z) = \int_0^\infty (e^(-u z) / (K_1^2(u) + \pi^2 I_1^2(u))) (1/u) \, du$ / 193 \\ 71: $P(x) = \int_x^\infty (\sin(t - x) / t) \, dt$ / 195 \\ 74: $Q(x) = \int_x^\infty (\cos(t - x) / t) \, dt$ / 195 \\ References for Part II / 201", } @Book{Hobson:1955:TSE, author = "Ernest William Hobson", title = "The Theory of Spherical and Ellipsoidal Harmonics", publisher = "Chelsea Pub. Co.", address = "New York, NY, USA", pages = "500", year = "1955", LCCN = "QA406 .H7 1955", bibdate = "Sat Apr 1 14:40:56 MDT 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; z3950.loc.gov:7090/Voyager", acknowledgement = ack-nhfb, author-dates = "1856--1933", subject = "Spherical harmonics; Lam{\'e}'s functions", } @Book{McLachlan:1955:BFE, author = "N. W. (Norman William) McLachlan", title = "{Bessel} Functions for Engineers", publisher = pub-CLARENDON, address = pub-CLARENDON:adr, edition = "Second", pages = "xii + 239", year = "1955", LCCN = "QA408 .M3 1955", bibdate = "Sat Apr 1 14:44:36 MDT 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; z3950.loc.gov:7090/Voyager", series = "The Oxford engineering science series", acknowledgement = ack-nhfb, author-dates = "1888--", subject = "Bessel functions", } @Article{Motzkin:1955:EP, author = "T. S. Motzkin", title = "Evaluation of polynomials", journal = j-BULL-AMS, volume = "61", number = "2", pages = "163--163", month = mar, year = "1955", CODEN = "BAMOAD", ISSN = "0002-9904 (print), 1936-881X (electronic)", ISSN-L = "0002-9904", bibdate = "Fri Oct 20 09:06:44 2023", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Bulletin of the American Mathematical Society", issue = "635", journal-URL = "http://www.ams.org/journals/bull/all_issues.html", keywords = "number of multiplications to evaluate a polynomial", received = "12 November 1954", remark = "One-paragraph abstract only, with reference-less mention of Ostrowski.", } @Article{Motzkin:1955:ERF, author = "T. S. Motzkin", title = "Evaluation of rational functions", journal = j-BULL-AMS, volume = "61", number = "2", pages = "163--163", month = mar, year = "1955", CODEN = "BAMOAD", ISSN = "0002-9904 (print), 1936-881X (electronic)", ISSN-L = "0002-9904", bibdate = "Fri Oct 20 09:06:44 2023", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Bulletin of the American Mathematical Society", issue = "635", journal-URL = "http://www.ams.org/journals/bull/all_issues.html", keywords = "number of multiplications to evaluate a polynomial", received = "12 November 1954", remark = "One-paragraph abstract only, with reference to Ostrowski.", } @Article{Norlund:1955:HF, author = "Niels Erik N{\o}rlund", title = "Hypergeometric functions", journal = j-ACTA-MATH, volume = "94", number = "??", pages = "289--349", month = "????", year = "1955", CODEN = "ACMAA8", DOI = "https://doi.org/10.1007/BF02392494", ISSN = "0001-5962 (print), 1871-2509 (electronic)", ISSN-L = "0001-5962", MRclass = "33.0X", MRnumber = "MR0074585 (17,610d)", MRreviewer = "A. Erd{\'e}lyi", bibdate = "Thu Dec 01 10:09:47 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Acta Mathematica", journal-URL = "http://link.springer.com/journal/11511", } @Article{Preston:1955:ACS, author = "F. S. Preston", title = "An Analog Computer for the Solution of Tangents", journal = j-IRE-TRANS-ELEC-COMPUT, volume = "EC-4", number = "3", pages = "101--106", month = "????", year = "1955", CODEN = "IRELAO", DOI = "https://doi.org/10.1109/IRETELC.1955.507908", ISSN = "0367-9950", ISSN-L = "0367-9950", bibdate = "Thu Jun 30 15:10:37 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", URL = "https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5407908", acknowledgement = ack-nhfb, fjournal = "IRE Transactions on Electronic Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=5407885", } @Article{Robinson:1955:EAC, author = "A. S. Robinson", title = "An Electronic Analog Computing Technique for the Solution of Trigonometric Problems", journal = j-IRE-TRANS-ELEC-COMPUT, volume = "EC-4", number = "3", pages = "95--101", month = "????", year = "1955", CODEN = "IRELAO", DOI = "https://doi.org/10.1109/IRETELC.1955.5407907", ISSN = "0367-9950", ISSN-L = "0367-9950", bibdate = "Thu Jun 30 15:10:37 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", URL = "https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5407907", acknowledgement = ack-nhfb, fjournal = "IRE Transactions on Electronic Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=5407885", } @Article{Salzer:1955:CZE, author = "Herbert E. Salzer", title = "Complex zeros of the error function", journal = j-J-FRANKLIN-INST, volume = "260", number = "3", pages = "209--211", month = sep, year = "1955", CODEN = "JFINAB", DOI = "https://doi.org/10.1016/0016-0032(55)90732-8", ISSN = "0016-0032 (print), 1879-2693 (electronic)", ISSN-L = "0016-0032", MRclass = "65.1X", MRnumber = "71880", MRreviewer = "L. Fox", bibdate = "Tue Nov 14 17:19:58 2023", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/f/fox-leslie.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Journal of the Franklin Institute", journal-URL = "http://www.sciencedirect.com/science/journal/00160032", reviewer-dates = "Leslie Fox (30 September 1918--1 August 1992)", } @Article{Todd:1955:MWN, author = "John Todd", title = "Motivation for working in numerical analysis", journal = j-COMM-PURE-APPL-MATH, volume = "8", number = "1", pages = "97--116", month = feb, year = "1955", CODEN = "CPAMAT, CPMAMV", DOI = "https://doi.org/10.1002/cpa.3160080107", ISSN = "0010-3640 (print), 1097-0312 (electronic)", ISSN-L = "0010-3640", MRclass = "65.0X", MRnumber = "70251", MRreviewer = "G. E. Forsythe", bibdate = "Fri Oct 20 08:38:37 2023", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/t/todd-john.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", ZMnumber = "0064.37402", acknowledgement = ack-nhfb, ajournal = "Comm. Pure Appl. Math.", author-dates = "John Todd (16 May 1911--21 June 2007)", fjournal = "Communications on Pure and Applied Mathematics (New York)", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)1097-0312", keywords = "characteristic roots (eigenvalues) of finite matrices; game theory; integral equations; modified differences; Monte Carlo; number of multiplications to evaluate a polynomial; polynomial evaluation; quadrature; recent activity in numerical analysis; sequence convergence acceleration; theory of machines (automata)", remark-1 = "This may be the earliest paper to note that a polynomial of degree $n$ can be evaluated with fewer than $n$ multiplications, but requiring more than $n$ additions. The normal Horner form requires $n$ multiply and $n$ add operations: $ h = a_n $, $ h = x h + a_k $ (for $ k = n - 1 $ to $0$), and on modern hardware can be conveniently evaluated in $n$ consecutive fused multiply-add operations. However, the evaluation of the altered forms often has terms of differing signs, and may be subject to catastrophic leading digit loss, when the original polynomial, if it had coefficients of the same sign, might have been computed stably for $ x > 0$. In some cases, the new coefficients are complex, even when those of the original polynomial are real numbers. See also related publications \cite{Ostrowski:1954:TPA, Todd:1954:MWN, Motzkin:1955:EP, Motzkin:1955:ERF, Belaga:1958:SPI, Pan:1959:CSC, Pan:1959:SCP, Floyd:1961:ACE, Dorn:1962:GHR, Knuth:1962:EPC, Eisman:1963:PER, Eve:1964:EP, Rice:1965:CPR, Winograd:1970:NMN, Rabin:1972:FEP, Miller:1975:CCN, Knuth:1998:EP, Kusterer:1979:SEP, Ceberio:2002:HRI, Cameron:2024:AHM}. Rice reports extreme numerical instability of the Belaga and Motzkin forms, and moderate instability of the Pan forms, while the Chebyshev form is never unstable. Todd cites Motzkin's work as ``to appear'', and those two one-paragraph abstracts were received 12 November 1954 and published in March 1955, but Todd's paper has no received date, so we cannot determine their relative priority. Entry \cite{Ostrowski:1954:TPA} may be prior art, but a copy of that work has not yet been located. The quotation in entry \cite{Eve:1964:E} summarizes the bounds on the number of add and multiply operations.", remark-2 = "Knuth's treatment (Knuth:1962:EPC) concentrates on operation counts, because the polynomial variable need not be a real scalar floating-point number: it could be complex, multiple precision, matrix, series, ..., where multiplication is relatively expensive. Knuth remarks on page 485 that ``numerical analysis of the accuracy achieved \ldots{} is beyond the scope of this book: the reader should be careful to investigate the accuracy of any calculations undertaken with floating-point arithmetic.'' On page 486, Knuth notes that the nested form is often attributed to Horner:1819:XNM, but that Isaac Newton used it in unpublished notes 150 years earlier, and it was employed by the Chinese in the 13th century CE.", remark-3 = "The year of this paper is erroneously cited in reference lists of several sources as 1951, rather than the correct 1955.", ZBmath = "3109832", } @Book{Achieser:1956:TA, author = "N. I. Achieser", title = "Theory of Approximation", publisher = "Frederick Ungar Publishing Company", address = "New York, NY, USA", pages = "x + 307", year = "1956", LCCN = "QA221 .A533 1956", bibdate = "Fri Oct 20 08:06:59 MDT 2023", bibsource = "fsz3950.oclc.org:210/WorldCat; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, remark = "Translation of Russian original, Lek{\"e}t{\`\i}sii po teorii approksima{\"e}t{\`\i}sii. Reprinted in \cite{Achieser:1992:TA}.", subject = "Mathematical analysis", tableofcontents = "Approximation Problems in Linear Normalized Spaces \\ Formulation of the Principal Problem in the Theory of Approximation / 1 \\ The Concept of Metric Space / 1 \\ The Concept of Linear Normalized Space / 2 \\ Examples of Linear Normalized Spaces / 3 \\ The Inequalities of Holder and Minkowski / 4 \\ Additional Examples of Linear Normalized Spaces / 7 \\ Hilbert Space / 8 \\ The Fundamental Theorem of Approximation Theory in Linear Normalized Spaces / 10 \\ Strictly Normalized Spaces / 11 \\ An Example of Approximation in the Space $L^p$ / 12 \\ Geometric Interpretation / 13 \\ Separable and Complete Spaces / 14 \\ Approximation Theorems in Hilbert Space / 15 \\ An Example of Approximation in Hilbert Space / 19 \\ More About the Approximation Problem in Hilbert Space / 21 \\ Orthonormalized Vector Systems in Hilbert Space / 22 \\ Orthogonalization of Vector Systems / 23 \\ Infinite Orthonormalized Systems / 25 \\ An Example of a Non-Separable System / 29 \\ Weierstrass' First Theorem / 29 \\ Weierstrass' Second Theorem / 32 \\ The Separability of the Space C / 33 \\ The Separability of the Space $L^p$ / 34 \\ Generalization of Weierstrass' Theorem to the Space $L^p$ / 37 \\ The Completeness of the Space $L^p$ / 38 \\ Examples of Complete Orthonormalized Systems in L[superscript 2] / 40 \\ Muntz's Theorem / 43 \\ The Concept of the Linear Functional / 46 \\ F. Riesz's Theorem / 47 \\ A Criterion for the Closure of a Set of Vectors in Linear Normalized Spaces / 49 \\ P. L. Tchebysheff's Domain of Ideas \\ Statement of the Problem / 51 \\ A Generalization of the Theorem of de la Vallee-Poussin / 52 \\ The Existence Theorem / 53 \\ Tchebysheff's Theorem / 55 \\ A Special Case of Tchebysheff's Theorem / 57 \\ The Tchebysheff Polynomials of Least Deviation from Zero / 57 \\ A Further Example of P. Tchebysheff's Theorem / 58 \\ An Example for the Application of the General Theorem of de la Vallee-Poussin / 60 \\ An Example for the Application of P. L. Tchebysheff's General Theorem / 62 \\ The Passage to Periodic Functions / 64 \\ An Example of Approximating with the Aid of Periodic Functions / 66 \\ The Weierstrass Function / 66 \\ Haar's Problem / 67 \\ Proof of the Necessity of Haar's Condition / 68 \\ Proof of the Sufficiency of Haar's Condition / 69 \\ An Example Related to Haar's Problem / 72 \\ P. L. Tchebysheff's Systems of Functions / 73 \\ Generalization of P. L. Tchebysheff's Theorem / 74 \\ On a Question Pertaining to the Approximation of a Continuous Function in the Space $L$ / 76 \\ A. A. Markoff's Theorem / 82 \\ Special Cases of the Theorem of A. A. Markoff / 85 \\ Elements of Harmonic Analysis \\ The Simplest Properties of Fourier Series / 89 \\ Fourier Series for Functions of Bounded Variation / 93 \\ The Parseval Equation for Fourier Series / 97 \\ Examples of Fourier Series / 98 \\ Trigonometric Integrals / 101 \\ The Riemann--Lebesgue Theorem / 103 \\ Plancherel's Theory / 104 \\ Watson's Theorem / 106 \\ Plancherel's Theorem / 108 \\ Fejer's Theorem / 110 \\ Integral-Operators of the Fejer Type / 113 \\ The Theorem of Young and Hardy / 116 \\ Examples of Kernels of the Fejer Type / 118 \\ The Fourier Transformation of Integrable Functions / 120 \\ The Faltung of two Functions / 122 \\ V. A. Stekloff's Functions / 123 \\ Multimonotonic Functions / 125 \\ Conjugate Functions / 126 \\ Certain Extremal Properties of Integral Transcendental Functions of the Exponential Type \\ Integral Functions of the Exponential Type / 130 \\ The Borel Transformation / 132 \\ The Theorem of Wiener and Paley / 134 \\ Integral Functions of the Exponential Type which are Bounded along the Real Axis / 137 \\ S. N. Bernstein's Inequality / 140 \\ B. M. Levitan's Polynomials / 146 \\ The Theorem of Fejer and Riesz. A Generalization of This Theorem / 152 \\ A Criterion for the Representation of Continuous Functions as Fourier--Stieltjes Integrals / 154 \\ Questions Regarding the Best Harmonic Approximation of Functions Preliminary Remarks / 160 \\ The Modulus of Continuity / 161 \\ The Generalization to the Space $L^p$ ($p \geq 1$) / 162 \\ An Example of Harmonic Approximation / 165 \\ Some Estimates for Fourier Coefficients / 169 \\ More about V. A. Stekloff's Functions / 173 \\ Two Lemmas / 175 \\ The Direct Problem of Harmonic Approximation / 176 \\ A Criterion due to B. Sz.-Nagy / 183 \\ The Best Approximation of Differentiable Functions / 187 \\ Direct Observations Concerning Periodic Functions / 195 \\ Jackson's Second Theorem / 199 \\ The Generalized Fejer Method / 201 \\ Berstein's Theorem / 206 \\ Priwaloff's Theorem / 210 \\ Generalizations of Bernstein's Theorems to the Space $L^p$ ($p \geq 1$) / 211 \\ The Best Harmonic Approximation of Analytic Functions / 214 \\ A Different Formulation of the Result of the Preceding Section / 218 \\ The Converse of Bernstein's Theorem / 221 \\ Wiener's Theorem on Approximation \\ Wiener's Problem / 224 \\ The Necessity of Wiener's Condition / 224 \\ Some Definitions and Notation / 225 \\ Several Lemmas / 227 \\ The Wiener--Levy Theorem / 230 \\ Proof of the Sufficiency of Wiener's Condition / 233 \\ Wiener's General Tauber Theorem / 234 \\ Weakly Decreasing Functions / 235 \\ Remarks on the Terminology / 237 \\ Ikehara's Theorem / 238 \\ Carleman's Tauber Theorem / 241 \\ Various Addenda and Problems \\ Elementary Extremal Problems and Certain Closure Criteria / 243 \\ Szego's Theorem and Some of Its Applications / 256 \\ Further Examples of Closed Sequences of Functions / 267 \\ The Caratheodory--Fejer Problem and Similar Problems / 270 \\ Solotareff's Problems and Related Problems / 280 \\ The Best Harmonic Approximation of the Simplest Analytic Functions / 289 \\ Notes / 296 \\ Index / 306", } @TechReport{Fox:1956:NLS, author = "Phyllis Fox", title = "The {NYU} Library of Subroutines", type = "Report", number = "NYO-6483", institution = "AEC Computing Facility, Institute of Mathematical Sciences, New York University", address = "New York, NY, USA", pages = "11", day = "1", month = jan, year = "1956", bibdate = "Wed Dec 17 10:07:24 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "https://dn790009.ca.archive.org/0/items/nyulibraryofsubr00foxp/nyulibraryofsubr00foxp.pdf", abstract = "A set of subroutines has been developed at the AEC Computing Facility at New York University for evaluating simple functions and for performing certain standard procedures. Detailed descriptions of the routines are available, and a library tape containing the routines is kept at the Univac. The routines are written to be compatible with the NYU Compiler System (NYO-6478). The number of routines is constantly being increased and the routines themselves may be modified from time Lo Time on the basis of experience in applying them to various problems. The list presented here includes those routines currently in use, together with a brief description of each. A revised index of current routines will be issued at intervals, but before using any of the routines the programmer should check the library subroutine book near the computer for the latest corrections and modifications.", acknowledgement = ack-nhfb, pdfpages = "26", remark = "This library was developed before the first high-level language, Fortran, and the report contains only a few lines of description for each component, without any mention of computational algorithms. Its contents nevertheless identify the computational needs of early digital computer users. Its author went on to co-develop the Bell Labs PORT Library \cite{Fox:1977:PPM,Fox:1978:AFP,Fox:1978:PMS,Fox:1979:RFP}.", tableofcontents = "Introduction / 3 \\ Index / 4 \\ Arithmetic Routines / 4 \\ Block Output Routines / 5 \\ Debugging Routines / 5 \\ Differential Equation Routines / 7 \\ Edit Routines / 7 \\ Exponentials and Roots / 8 \\ Fill Routines / 9 \\ Ignore Squeeze Routines / 9 \\ Integration Routines / 9 \\ Logarithm Routines / 10 \\ Polynomial Routines / 11 \\ Sorting Routines / 11 \\ Trigonometric Routines / 11", } @InProceedings{Haynes:1956:EIE, author = "John G. Haynes", editor = "????", booktitle = "{ACM'56: Proceedings of the 1956 11th ACM national meeting}", title = "Evaluation of incomplete elliptic integrals by {Gaussian} integration", publisher = pub-ACM, address = pub-ACM:adr, pages = "56--59", year = "1956", DOI = "https://doi.org/10.1145/800258.808948", bibdate = "Fri Dec 21 08:53:15 2018", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "https://dl.acm.org/ft_gateway.cfm?id=808948", acknowledgement = ack-nhfb, } @Book{Jeffreys:1956:MMP, author = "Harold Jeffreys and Bertha {Swirles Jeffreys}", title = "Methods of Mathematical Physics", publisher = pub-CAMBRIDGE, address = pub-CAMBRIDGE:adr, edition = "Third", pages = "714", year = "1956", LCCN = "QA401 .J4 1956", bibdate = "Thu Aug 17 10:48:45 2023", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/h/hartree-douglas-r.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "https://en.wikipedia.org/wiki/Bertha_Swirles; https://en.wikipedia.org/wiki/Harold_Jeffreys", acknowledgement = ack-nhfb, author-dates = "Sir Harold Jeffreys (22 April 1891--18 March 1989); Lady Bertha Swirles Jeffreys (22 May 1903--18 December 1999)", remark = "References to Douglas Hartree in", remark-1 = "First edition 1946, second edition 1950, third edition 1956, first paperback edition 1972, reprinted 1978, 1980, 1988, 1992, 1999, 2001. Third edition preface is dated April 1953. Second edition preface is dated 15 November 1948. First edition preface is dated 1946. Reprinted in \cite{Jeffreys:1999:MMP}.", subject-dates = "Douglas Rayner Hartree (27 March 1897--12 February 1958)", tableofcontents = "Preface \\ Authors' Notes \\ 1: The Real Variable \\ 2: Scalars and Vectors \\ 3: Tensors \\ 4: Matrices \\ 5: Multiple Integrals \\ 6: Potential Theory \\ 7: Operational Methods \\ 8: Physical Applications of the Operational Method \\ 9: Numerical Methods \\ 10: Calculus of Variations \\ 11: Functions of a Complex Variable \\ 12: Contour Integration and Bromwich's Integral \\ 13: Conformal Representation \\ 14: Fourier's Theorem \\ 15: The Factorial and Related Functions \\ 16: Solution of Linear Differential Equation \\ 17: Asymptotic Expansions \\ 18: The Equations of Potential, Waves, and Heat Conduction \\ 19: Waves in One Dimension and Waves With Spherical Symmetry \\ 20: Conduction of Heat in One and Three Dimensions \\ 21: Bessel Functions \\ 22: Applications of Bessel Functions \\ 23: The Confluent Hypergeometric Function \\ 24: Legendre Functions and Associated Functions \\ 25: Elliptic Functions \\ Notes \\ Appendix on Notation \\ Index", } @Article{Lehmer:1956:RRZ, author = "D. H. Lehmer", title = "On the roots of the {Riemann} zeta-function", journal = j-ACTA-MATH, volume = "95", number = "1", pages = "291--298", month = dec, year = "1956", CODEN = "ACMAA8", DOI = "https://doi.org/10.1007/BF02401102", ISSN = "0001-5962 (print), 1871-2509 (electronic)", ISSN-L = "0001-5962", MRclass = "10.1X", MRnumber = "0086082 (19,121a)", MRreviewer = "L. Schoenfeld", bibdate = "Mon Sep 28 16:18:23 2015", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/t/turing-alan-mathison.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Acta Mathematica", journal-URL = "http://link.springer.com/journal/11511", remark-1 = "See \cite[references 38--39, page 54]{Bullynck:2015:CPT} for Turing's role in this work, published two years after Turing's death in 1954. Turing's incomplete work appears in \cite{Turing:1953:SCR}.", remark-2 = "From page 293: ``Plans to extend the work of Titchmarsh [on the zeros of the Riemann zeta function] by use of a differential analyzer were made in 1939 by the late A. M. Turing. These were interrupted by the war and later rendered obsolete by the advent of the electronic digital computers.''", remark-3 = "From page 293: ``In 1947 the writer programmed an extension of the work of Titchmarsh [on the zeros of the Riemann zeta function] for the ENIAC, the only electronic computer then in operation. However, before the program could be run, the ENIAC was drastically modified thus rendering it useless for the problem.''", remark-4 = "From page 293: ``In June 1950, Turing used the Manchester University Mark 1 electronic digital computer to examine the zeta-function for $24,937.96 < t < 25,735.93$ (that is for $63 < \sqrt{\tau} < 6.4$) and found in this region of the critical strip that there are about 1070 simple zeros all with $a = 1/2$. In another short run the validity of the Riemann Hypothesis was verified between Titchmarsh's upper limit of $t = 1468$ and $t = 1540$. Only some twenty hours of machine time was used. Unfortunately no further time was made available and these incomplete results were published in 1953.''", } @InProceedings{Luke:1956:RFAa, author = "Yudell L. Luke", editor = "????", booktitle = "{ACM'56: Proceedings of the 1956 11th ACM national meeting}", title = "On rational function approximations to the exponential function with application to the practical solution of linear differential difference equations with constant coefficients", publisher = pub-ACM, address = pub-ACM:adr, pages = "13--16", year = "1956", DOI = "https://doi.org/10.1145/800258.808937", bibdate = "Fri Dec 21 08:53:15 2018", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "See errata and addenda \cite{Luke:1956:RFAb}.", URL = "https://dl.acm.org/ft_gateway.cfm?id=808937", acknowledgement = ack-nhfb, } @InProceedings{Luke:1956:RFAb, author = "Yudell L. Luke", editor = "????", booktitle = "{ACM'56: Proceedings of the 1956 11th ACM national meeting}", title = "On rational function approximations to the exponential function with application to the practical solution of linear differential difference equations with constant coefficients: Errata and addenda", publisher = pub-ACM, address = pub-ACM:adr, pages = "177--178", year = "1956", DOI = "https://doi.org/10.1145/800258.808979", bibdate = "Fri Dec 21 08:53:15 2018", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "See \cite{Luke:1956:RFAa}.", URL = "https://dl.acm.org/ft_gateway.cfm?id=808979", acknowledgement = ack-nhfb, } @Article{Marx:1956:ABL, author = "Helmut Marx", title = "{Additionsverfahren zur Berechnung des Logarithmus und der Exponentialfunktion (I)}. ({German}) [{Addition} method for calculating the logarithm and the exponential function ({I})]", journal = "{Mitteilungen des Mathematischen Seminars der Universit{\"a}t Gie{\ss}en}", number = "54", pages = "i + 26", day = "??", month = "????", year = "1956", MRclass = "68.0X", MRnumber = "82745", bibdate = "Mon Nov 10 11:53:20 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, language = "German", remark-1 = "From the MathSciNet review by D. H. Lehmer: ``$exp(x + i y)$ can be computed to nearly 10 decimal places with at most 131 additions.'' Until the full paper can be located, it is uncertain whether Marx had any contact with Volder, whose 1956 report was then still a corporate secret.", remark-2 = "Cited in \cite{Ercegovac:1973:REC}. According to https://onlinebooks.library.upenn.edu/webbin/serial?id=mittmathsemgies, only numbers 1--28 have been found and are available online.", remark-3 = "Were there later papers, II, III, \ldots{}?", xxjournal = "Mitteilungen aus dem Mathematische Seminar Giessen", } @Article{Morrison:1956:MCC, author = "D. R. Morrison", title = "A Method for Computing Certain Inverse Functions", journal = j-MATH-TABLES-OTHER-AIDS-COMPUT, volume = "10", number = "56", pages = "202--208", month = oct, year = "1956", CODEN = "MTTCAS", ISSN = "0891-6837 (print), 2326-4853 (electronic)", ISSN-L = "0891-6837", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp1950.bib; JSTOR database", acknowledgement = ack-nhfb, ajournal = "Math. Tables Other Aids Comput.", fjournal = "Mathematical Tables and Other Aids to Computation", journal-URL = "http://www.ams.org/mcom/", } @Book{Sneddon:1956:SFM, author = "Ian Naismith Sneddon", title = "Special Functions of Mathematical Physics and Chemistry", volume = "19", publisher = "Oliver and Boyd", address = "Edinburgh, UK", edition = "Third", pages = "viii + 164", year = "1956", LCCN = "QA1 U588 v. 19", bibdate = "Sat Oct 30 18:41:48 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; z3950.loc.gov:7090/Voyager", series = "University mathematical texts", acknowledgement = ack-nhfb, remark = "See second edition \cite{Sneddon:1961:SFM} and third edition \cite{Sneddon:1980:SFM}.", subject = "Functions, Special", } @Book{Stratton:1956:SWF, author = "Julius Adams Stratton", title = "Spheroidal Wave Functions, Including Tables of Separation Constants and Coefficients", publisher = pub-WILEY, address = pub-WILEY:adr, pages = "xiii + 613", year = "1956", LCCN = "QA405 .S8", bibdate = "Sat Apr 1 14:32:29 MDT 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; z3950.loc.gov:7090/Voyager", acknowledgement = ack-nhfb, author-dates = "1901--1994", subject = "Wave mechanics; Spheroidal functions", } @Book{Edmonds:1957:AMQ, author = "A. R. (Alan Robert) Edmonds", title = "Angular Momentum in Quantum Mechanics", publisher = pub-PRINCETON, address = pub-PRINCETON:adr, pages = "viii + 146", year = "1957", LCCN = "????", bibdate = "Mon Aug 4 15:01:00 MDT 2025", bibsource = "fsz3950.oclc.org:210/WorldCat; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", series = "Investigations in physics", acknowledgement = ack-nhfb, } @Book{Flammer:1957:SWF, author = "Carson Flammer", title = "Spheroidal Wave Functions", publisher = pub-STANFORD, address = pub-STANFORD:adr, pages = "ix + 220", year = "1957", LCCN = "QA405 .F55", bibdate = "Sat Apr 1 14:32:29 MDT 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; z3950.loc.gov:7090/Voyager", series = "A Stanford Research Institute monograph", acknowledgement = ack-nhfb, remark = "The basis for this monograph was work done at Stanford Research Institute for the United States Air Force Cambridge Research Center under Contract AF 19 (604)-1296.", subject = "Spheroidal functions", } @Article{Franklin:1957:CARb, author = "Joel Franklin and Bernard Friedman", title = "A convergent asymptotic representation for integrals", journal = j-PROC-CAMBRIDGE-PHIL-SOC, volume = "53", pages = "612--619", year = "1957", CODEN = "PCPSA4", ISSN = "0008-1981", MRclass = "42.1X", MRnumber = "0090691", MRreviewer = "P. Henrici", bibdate = "Tue Feb 06 15:03:36 2018", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "https://www.cambridge.org/core/services/aop-cambridge-core/content/view/2F167020D3A1F98182882E99549E7751/S0305004100032667a.pdf/a-convergent-asymptotic-representation-for-integrals.pdf", abstract = "This paper represents a new method for obtaining an asymptotic representation for integrals of the form $ \int_0^\infty e^{-p x} x^{c - 1} f(x) \, d x $ when $p$ is large. It is shown that if $ f(x)$ satisfies certain conditions this representation is also convergent. Numerical calculations seem to show that the first term of the representation gives a close approximation to the value of the integral for a wide range of values of $p$.", acknowledgement = ack-nhfb, fjournal = "Proceedings of the Cambridge Philosophical Society. Mathematical and physical sciences", journal-URL = "http://journals.cambridge.org/action/displayJournal?jid=PSP", remark = "See applications in \cite{Temme:2015:AMI,Navas-Palencia:2018:HPC}.", } @Article{Hitchcock:1957:PAB, author = "A. J. M. Hitchcock", title = "Polynomial Approximations to {Bessel} Functions of Order Zero and One and to Related Functions", journal = j-MATH-TABLES-OTHER-AIDS-COMPUT, volume = "11", number = "58", pages = "86--88", month = apr, year = "1957", CODEN = "MTTCAS", ISSN = "0891-6837 (print), 2326-4853 (electronic)", ISSN-L = "0891-6837", bibdate = "Mon Feb 27 08:05:17 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp1950.bib; JSTOR database", URL = "http://www.jstor.org/stable/2002156", acknowledgement = ack-nhfb, fjournal = "Mathematical Tables and Other Aids to Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Kogbetliantz:1957:CUE, author = "E. G. (Ervand George) Kogbetliantz", title = "Computation of {$ e^N $} for $ - \infty < {N} < + \infty $ Using an Electronic Computer", journal = j-IBM-JRD, volume = "1", number = "2", pages = "110--115", month = apr, year = "1957", CODEN = "IBMJAE", DOI = "https://doi.org/10.1147/rd.12.0110", ISSN = "0018-8646 (print), 2151-8556 (electronic)", ISSN-L = "0018-8646", MRclass = "68.0X", MRnumber = "19,775d", bibdate = "Tue Sep 06 20:55:54 1994", bibsource = "http://www.research.ibm.com/journal/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/ibmjrd.bib", URL = "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5392731", acknowledgement = ack-nhfb, ajournal = "IBM J. Res. Develop.", author-dates = "22 February 1888--5 November 1974", fjournal = "IBM Journal of Research and Development", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=5288520", reviewer = "W. F. Freiberger", } @Article{Linskii:1957:CEF, author = "V. S. Linski{\u{\i}}", title = "Calculation of elementary functions on automatic digital machines. ({Russian})", journal = "Vy{\v{c}}isl. Mat.", volume = "2", pages = "90--119", year = "1957", MRclass = "68.00", MRnumber = "21 \#982", MRreviewer = "J. W. Carr, III", bibdate = "Sat Jan 11 17:44:01 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, language = "Russian", } @Article{Luke:1957:C, author = "Yudell L. Luke", title = "On the Computation of $ \log {Z} $ and $ \operatorname {arc} \tan {Z} $", journal = j-MATH-TABLES-OTHER-AIDS-COMPUT, volume = "11", number = "57", pages = "16--18", month = jan, year = "1957", CODEN = "MTTCAS", DOI = "https://doi.org/10.1090/S0025-5718-1957-0084855-1", ISSN = "0891-6837 (print), 2326-4853 (electronic)", ISSN-L = "0891-6837", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "Mathematical Tables and Other Aids to Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Miller:1957:NGS, author = "J. C. P. Miller", title = "Note on the General Solution of the Confluent Hypergeometric Equation", journal = j-MATH-TABLES-OTHER-AIDS-COMPUT, volume = "11", number = "58", pages = "97--99", month = apr, year = "1957", CODEN = "MTTCAS", ISSN = "0891-6837 (print), 2326-4853 (electronic)", ISSN-L = "0891-6837", bibdate = "Mon Feb 27 08:05:17 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp1950.bib; JSTOR database", URL = "http://www.jstor.org/stable/2002156", acknowledgement = ack-nhfb, fjournal = "Mathematical Tables and Other Aids to Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Southard:1957:ATW, author = "Thomas H. Southard", title = "Approximation and Table of the {Weierstrass} $ \wp $ Function in the Equianharmonic Case for Real Argument", journal = j-MATH-TABLES-OTHER-AIDS-COMPUT, volume = "11", number = "58", pages = "99--100", month = apr, year = "1957", CODEN = "MTTCAS", ISSN = "0891-6837 (print), 2326-4853 (electronic)", ISSN-L = "0891-6837", bibdate = "Mon Feb 27 08:05:17 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp1950.bib; JSTOR database", URL = "http://www.jstor.org/stable/2002156", acknowledgement = ack-nhfb, fjournal = "Mathematical Tables and Other Aids to Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Stegun:1957:GBF, author = "Irene A. Stegun and Milton Abramowitz", title = "Generation of {Bessel} Functions on High Speed Computers", journal = j-MATH-TABLES-OTHER-AIDS-COMPUT, volume = "11", number = "60", pages = "255--257", month = oct, year = "1957", CODEN = "MTTCAS", DOI = "https://doi.org/10.1090/S0025-5718-1957-0093939-3", ISSN = "0891-6837 (print), 2326-4853 (electronic)", ISSN-L = "0891-6837", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "Mathematical Tables and Other Aids to Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{T:1957:CHS, author = "C. B. T.", title = "Comment: {T. H. Southard, \booktitle{Approximation and table of the Weierstrass $ \wp $ function in the equianharmonic case for real argument}. [MTAC, this issue, p. 99--100]}", journal = j-MATH-TABLES-OTHER-AIDS-COMPUT, volume = "11", number = "58", pages = "110--110", month = apr, year = "1957", CODEN = "MTTCAS", ISSN = "0891-6837 (print), 2326-4853 (electronic)", ISSN-L = "0891-6837", bibdate = "Mon Feb 27 08:05:17 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp1950.bib; JSTOR database", URL = "http://www.jstor.org/stable/2002157", acknowledgement = ack-nhfb, fjournal = "Mathematical Tables and Other Aids to Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Wilkes:1957:NMC, author = "M. V. Wilkes and D. J. Wheeler", title = "Note on ``{A} Method for Computing Certain Inverse Functions'' (in {Technical Notes and Short Papers})", journal = j-MATH-TABLES-OTHER-AIDS-COMPUT, volume = "11", number = "59", pages = "204--204", month = jul, year = "1957", CODEN = "MTTCAS", DOI = "https://doi.org/10.2307/2002084", ISSN = "0891-6837 (print), 2326-4853 (electronic)", ISSN-L = "0891-6837", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/w/wilkes-maurice-v.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp1950.bib; JSTOR database", URL = "https://www.jstor.org/stable/2002084", ZMnumber = "0081.12704", acknowledgement = ack-nhfb, ajournal = "Math. Tables Other Aids Comput.", author-dates = "Sir Maurice Vincent Wilkes (26 June 1913--29 November 2010)", fjournal = "Mathematical Tables and Other Aids to Computation", journal-URL = "http://www.ams.org/mcom/", RSBM-number = "45", ZBmath = "3133093", } @Book{Wilkes:1957:PPE, author = "Maurice V. Wilkes and David J. Wheeler and Stanley Gill", title = "The Preparation of Programs for an Electronic Digital Computer", publisher = pub-AW, address = pub-AW:adr, edition = "Second", pages = "xiv + 238", year = "1957", LCCN = "QA76.5 .W52 1957", bibdate = "Mon Feb 10 09:42:47 2020", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", note = "See also first edition \cite{Wilkes:1951:PPE}.", URL = "https://b-ok.org/book/3668116/b363ff", acknowledgement = ack-nhfb, remark-1 = "According to \cite{Anderson:2019:SAM}, this book discusses the computation of integer population counts on the Electronic Delay Storage Automatic Calculator (EDSAC) computer using a recursive divide-and-conquer algorithm. See also somewhat negative 1958 review by Fernando J. Corbat{\'o} \cite{https://doi.org/10.1063/1.3062687}. Floating-point arithmetic is discussed on pages 60, 90--91, and 135--137.", remark-2 = "From page 5: ``Each storage location in the EDSAC holds 17 binary digits. In words representing numbers, the binary point is regarded as being to the right of the extreme left-hand digit; this digit (the most significant digit) is used as a sign indicator and is referred to as the sign digit. \ldots{} the capacity of the accumulator is 70 digits; there is, therefore, plenty of room to hold the full 33-digit product of two 17-digit numbers. \ldots{} A negative number $-x$ (where $O < x \leq 1$) is represented by a $1$ in the sign-digit position, followed by the digits of $(1 - x)$; for example, $1.1100\ldots{}$ represents $-(1 - 3/4) = -1/4$. \ldots{}. Another way of explaining the representation of negative numbers is to regard the sign digit as an ordinary numerical digit, and to say that $-x$ is stored as the number $(2 - x)$. Note in particular that $1.0000\ldots{}$ represents $-1$.'' [Page 59 calls this a {\em True complements} representation, distinguished from one's complement.]", remark-3 = "From page 35: ``The EDSAC has a facility which enables an even-numbered storage location and the following odd-numbered storage to be used as a single storage location holding 35 binary digits.'' [This suggests the word size in 18, not 17 as page 5 suggests. The Wikipedia article on the EDSAC reports: ``The EDSAC's main memory consisted of 1024 locations, though only 512 locations were initially installed. Each contained 18 bits, but the topmost bit was always unavailable due to timing problems, so only 17 bits were used.'']", remark-4 = "From page 36: ``The multiplier register of the arithmetical unit is of sufficient capacity to hold a long number, and the accumulator is of sufficient capacity to hold the complete (69) binary digit [including the sign bit] product of two long numbers.''", remark-5 = "From page 36: ``In some calculations, long numbers may not provide sufficient precision. In such cases, the programmer may make use of what is known as double-length or double-precision working, in which two long storage locations are used to hold the digits of a single number.'' [this would be a quad-word number holding 69 bits, including the sign bit.].", remark-6 = "From page 60: ``\ldots{} two double-length numbers, each stored in two locations, can be added and the result put in two locations in the store, by means of six orders''.", remark-7 = "From page 90: ``Each number is expressed in the form $a \cdot 10^p$, where $-10 \leq a \leq 10$ and $63 \leq p < 63$ and is represented in the store by $a \cdot 2^{-11} + p \cdot 2^{-6}$.''", remark-8 = "From page 91: ``Numbers are expressed in the form $a \cdot 10^p$, where $a$ and $p$ are packed into a single storage location. The number of digits defining $p$ may be varied from 4 to 15 by means of a preset parameter, so that a suitable value for the permissible range of variation of numbers may be selected for a given calculation.''", remark-9 = "From page 91: ``Although the use of floating-point operation can simplify the programmer's task by relieving him of undue preoccupation with scaling, it must not be thought that it solves all his difficulties. In particular, the loss of significant digits resulting from the subtraction of a number from a nearly equal number can have serious consequences unless proper precautions are taken.''", tableofcontents = "CHAPTER 1. THE ELEMENTS OF PROGRAM DESIGN / 1 \\ 1-1 Introduction / 1 \\ 1-2 Types of automatic computing machine / 1 \\ 1-3 The EDSAC / 3 \\ 1-4 Store / 5 \\ 1-5 Arithmetical unit / 5 \\ 1-6 Form of numbers in the machine / 5 \\ 1-7 Form of orders in the machine / 6 \\ 1-8 Storage of orders / 6 \\ 1-9 Written form of orders / 7 \\ 1-10 Some simple examples / 7 \\ Exercises A / 9 \\ 1-11 Jump orders / 9 \\ Exercises B / 11 \\ 1-12 Repeated groups of orders / 11 \\ 1-13 The use of the B-register / 15 \\ Exercises C / 18 \\ 1-14 Equivalence between orders and numbers; pseudo-orders / 18 \\ 1-15 Use of the arithmetical unit for constructing or modifying orders / 20 \\ 1-16 The mix order / 23 \\ Exercises D / 24 \\ CHAPTER 2. SUBROUTINES / 25 \\ 2-1 Introduction / 25 \\ 2-2 Relative numbering of addresses / 25 \\ 2-3 Internal and external forms of orders / 26 \\ 2-4 Reading of orders from the input tape / 28 \\ 2-5 Open and closed subroutines / 29 \\ 2-6 Entering and leaving a closed subroutine / 29 \\ 2-7 Closed B subroutines / 30 \\ 2-8 Closed A subroutines / 31 \\ 2-9 Use of library subroutines / 32 \\ Exercises E / 33 \\ 2-10 Long numbers / 35 \\ 2-11 Some further orders in the order code / 36 \\ 2-12 Scale factors / 38 \\ 2-13 Control combinations / 39 \\ Exercises F / 40 \\ 2-14 Relative addresses in control combinations / 41 \\ 2-15 Extension of the use of relative addresses / 41 \\ 2-16 Setting of the constants to be added by terminal code letters / 43 \\ 2-17 Complete table of terminal code letters / 44 \\ 2-18 Parameters / 45 \\ 2-19 Preset parameters / 46 \\ 2-20 Program parameters / 46 \\ 2-21 Standard procedure for setting preset parameters / 46 \\ 2-22 Interpretive subroutines / 47 \\ Exercises G / 49 \\ CHAPTER 3. PROGRAMMING FOR OTHER MACHINES / 51 \\ 3-1 Introduction / 51 \\ 3-2 Single-address codes / 52 \\ 3-3 Multi-address codes / 53 \\ 3-4 Multiplication and division / 56 \\ 3-5 Source-destination codes / 57 \\ 3-6 Representation of negative numbers / 59 \\ 3-7 Miscellaneous facilities / 60 \\ 3-8 Minimum-access coding / 61 \\ 3-9 The evaluation of an order code / 63 \\ 3-10 Use of an auxiliary store / 64 \\ CHAPTER 4. INPUT AND OUTPUT / 66 \\ 4-1 Introduction / 66 \\ 4-2 Input of numbers / 66 \\ 4-3 Output of numbers / 67 \\ 4-4 Input of orders / 69 \\ 4-5 Recognition of the code letter S / 72 \\ 4-6 Economy of input and output time / 72 \\ 4-7 Some features of input systems used with other machines / 73 \\ 4-8 Punched tape / 73 \\ 4-9 Punched cards / 75 \\ CHAPTER 5. THE LIBRARY OF SUBROUTINES / 80 \\ 5-1 Introduction / 80 \\ 5-2 Library catalog / 80 \\ 5-3 Input subroutines / 81 \\ 5-4 Output subroutines / 81 \\ 5-5 Division subroutines / 82 \\ 5-6 Trigonometric and other functions / 82 \\ 5-7 The economization of a power series by the use of Chebyshev polynomials / 83 \\ 5-8 Quadrature / 86 \\ 5-9 Integration of ordinary differential equations / 87 \\ 5-10 Library subroutines Gl2 and G13: Runge--Kutta processes / 88 \\ < 5-11 The independent variable / 88 \\ 5-12 Definition of the Runge--Kutta--Gill process / 89 \\ 5-13 Taylor-series method / 90 \\ 5-14 Interpretive subroutines / 90 \\ 5-15 Floating-point subroutines / 90 \\ CHAPTER 6. DIAGNOSIS OF ERRORS IN PROGRAM / 92 \\ 6-1 Introduction / 92 \\ 6-2 Proofreading of programs / 93 \\ 6-3 Punching / 93 \\ 6-4 Locating mistakes in a program- / 94 \\ 6-5 Subroutines for checking programs / 96 \\ 6-6 The development of a program / 97 \\ CHAPTER 7. EXAMPLES OF COMPLETE PROGRAMS FOR THE EDSAC / 99 \\ EXAMPLE 1 Calculation of $e^{-\sin x}$ / 99 \\ EXAMPLE 2 The evaluation of a definite integral / 102 \\ EXAMPLE 3 Integration of an ordinary differential equation / 108 \\ EXAMPLE 4 Evaluation of a Fourier transform / 113 \\ EXAMPLE 5 Evaluation of a definite integral / 118 \\ CHAPTER 8. AUTOMATIC PROGRAMMING / 126 \\ 8-1 Introduction / 126 \\ 8-2 Conversion versus interpretation / 127 \\ 8-3 Assembly of a program / 127 \\ 8-4 Floating addresses / 129 \\ 8-5 Formula recognition / 136 \\ Part Two: SPECIFICATIONS OF EDSAC LIBRARY SUBROUTINES / 139 \\ CATEGORY A. Subroutines to carry out floating-point arithmetic / 140 \\ CATEGORY B. Subroutines to perform arithmetical operations on complex numbers / 142 \\ CATEGORY C. Error-diagnosis subroutines / 144 \\ CATEGORY D. Division subroutines / 146 \\ CATEGORY E. Exponential subroutines / 148 \\ CATEGORY F. General subroutines relating to functions / 148 \\ CATEGORY G. Subroutines for the integration of differential equations / 150 \\ CATEGORY L. Subroutines for evaluating logarithms / 153 \\ CATEGORY M. Miscellaneous subroutines / 154 \\ CATEGORY N. Operations on double-length numbers / 156 \\ CATEGORY P. Print subroutines / 158 \\ CATEGORY Q. Quadrature subroutines / 162 \\ CATEGORY R. Input subroutines / 164 \\ CATEGORY s. Subroutines for evaluating fractional powers / 168 \\ CATEGORY T. Subroutines for calculating trigonometric functions / 169 \\ CATEGORY Z. Post-mortem routines / 170 \\ PART THREE: PROGRAMS OF SELECTED EDSAC LIBRARY SUBROUTINES / 173 \\ APPENDIX 1. Input and output codes of the EDSAC / 212 \\ APPENDIX 2. Order code and controls of the EDSAC / 214 \\ APPENDIX 3. The initial input routine of the EDSAC / 218 \\ APPENDIX 4. Control combinations / 221 \\ APPENDIX 5. Specimen solutions to programming exercises / 223 \\ BIBLIOGRAPHY / 233 \\ INDEX / 237", } @Article{Beattie:1958:TFZ, author = "Curtis L. Beattie", title = "Table of First 700 Zeros of {Bessel} Functions --- {$ J_l(x) $} and {$ J^{\prime }_l(x) $}", journal = j-BELL-SYST-TECH-J, volume = "37", number = "3", pages = "689--697", month = may, year = "1958", CODEN = "BSTJAN", ISSN = "0005-8580", MRclass = "65.00", MRnumber = "0093928 (20 \#448)", MRreviewer = "J. C. P. Miller", bibdate = "Tue Nov 9 11:15:54 MST 2010", bibsource = "http://bstj.bell-labs.com/oldfiles/year.1958/BSTJ.1958.3703.html; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://bstj.bell-labs.com/BSTJ/images/Vol37/bstj37-3-689.pdf", acknowledgement = ack-nhfb, fjournal = "The Bell System Technical Journal", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)1538-7305/issues/", } @Article{Belaga:1958:SPI, author = "{\`E}. G. Belaga", title = "Some problems involved in the calculation of polynomials", journal = j-DOKL-AKAD-NAUK, volume = "123", pages = "775--777", year = "1958", CODEN = "DANKAS", ISSN = "0002-3264", MRclass = "65.00", MRnumber = "105192", MRreviewer = "John Todd", bibdate = "Fri Oct 20 10:34:44 2023", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Doklady Akademii Nauk SSSR", journal-URL = "http://istina.msu.ru/journals/366838/", keywords = "number of multiplications to evaluate a polynomial", } @Book{Bowman:1958:IBF, author = "Frank Bowman", title = "Introduction to {Bessel} Functions", publisher = pub-DOVER, address = pub-DOVER:adr, pages = "x + 135", year = "1958", LCCN = "QA408 .B68i 1958", bibdate = "Sat Jan 15 17:24:26 MST 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; melvyl.cdlib.org:210/CDL90", acknowledgement = ack-nhfb, remark = "An unabridged and unaltered republication of the first edition.", subject = "Bessel functions", } @Article{Dingle:1958:AEC, author = "R. B. Dingle", title = "Asymptotic Expansions and Converging Factors. {III}. Gamma, Psi and Polygamma Functions, and {Fermi--Dirac} and {Bose--Einstein} Integrals", journal = j-PROC-R-SOC-LOND-SER-A-MATH-PHYS-SCI, volume = "244", number = "1239", pages = "484--490", day = "22", month = apr, year = "1958", CODEN = "PRLAAZ", ISSN = "0080-4630", bibdate = "Mon Jun 18 07:22:24 MDT 2012", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/d/dirac-p-a-m.bib; https://www.math.utah.edu/pub/bibnet/authors/f/fermi-enrico.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", URL = "http://www.jstor.org/stable/100264", acknowledgement = ack-nhfb, fjournal = "Proceedings of the Royal Society of London. Series A, Mathematical and physical sciences", journal-URL = "http://rspa.royalsocietypublishing.org/content/current", } @Article{Gill:1958:BFH, author = "S. Gill", title = "A Binary Form of {Horner}'s Method", journal = j-COMP-J, volume = "1", number = "2", pages = "84--86", month = jul, year = "1958", CODEN = "CMPJA6", DOI = "https://doi.org/10.1093/comjnl/1.2.84", ISSN = "0010-4620 (print), 1460-2067 (electronic)", ISSN-L = "0010-4620", bibdate = "Tue Dec 4 14:47:23 MST 2012", bibsource = "http://comjnl.oxfordjournals.org/content/1/2.toc; http://www.math.utah.edu/pub/tex/bib/compj1950.bib; http://www3.oup.co.uk/computer_journal/hdb/Volume_01/Issue_02/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "http://comjnl.oxfordjournals.org/content/1/2/84.full.pdf+html; http://www3.oup.co.uk/computer_journal/hdb/Volume_01/Issue_02/010084.sgm.abs.html; http://www3.oup.co.uk/computer_journal/hdb/Volume_01/Issue_02/tiff/84.tif; http://www3.oup.co.uk/computer_journal/hdb/Volume_01/Issue_02/tiff/85.tif; http://www3.oup.co.uk/computer_journal/hdb/Volume_01/Issue_02/tiff/86.tif", acknowledgement = ack-nhfb, fjournal = "The Computer Journal", journal-URL = "http://comjnl.oxfordjournals.org/", } @Article{Goldstein:1958:BFL, author = "M. Goldstein and R. M. Thaler", title = "{Bessel} Functions for Large Arguments", journal = j-MATH-TABLES-OTHER-AIDS-COMPUT, volume = "12", number = "61", pages = "18--26", month = jan, year = "1958", CODEN = "MTTCAS", DOI = "https://doi.org/10.1090/S0025-5718-1958-0102906-3", ISSN = "0891-6837 (print), 2326-4853 (electronic)", ISSN-L = "0891-6837", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "Mathematical Tables and Other Aids to Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Kogbetliantz:1958:CANb, author = "E. G. (Ervand George) Kogbetliantz", title = "Computation of Arcsin {N} for $ 0 < {N} < 1 $ Using an Electronic Computer", journal = j-IBM-JRD, volume = "2", number = "3", pages = "218--222", month = jul, year = "1958", CODEN = "IBMJAE", DOI = "https://doi.org/10.1147/rd.23.0218", ISSN = "0018-8646 (print), 2151-8556 (electronic)", ISSN-L = "0018-8646", MRclass = "65.3X", MRnumber = "19,1197c", bibdate = "Wed Aug 31 13:41:37 1994", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "IBM Journal of Research and Development", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=5288520", reviewer = "C. B. Haselgrove", } @Article{Kogbetliantz:1958:CAUa, author = "E. G. (Ervand George) Kogbetliantz", title = "Computation of Arctan {N} for $ - \infty < {N} < + \infty $ Using an Electronic Computer", journal = j-IBM-JRD, volume = "2", number = "1", pages = "43--53", month = jan, year = "1958", CODEN = "IBMJAE", DOI = "https://doi.org/10.1147/rd.21.0043", ISSN = "0018-8646 (print), 2151-8556 (electronic)", ISSN-L = "0018-8646", MRclass = "65.3X", MRnumber = "19,982e", bibdate = "Wed Aug 31 13:40:00 1994", bibsource = "http://www.research.ibm.com/journal/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/ibmjrd.bib", URL = "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5392657", acknowledgement = ack-nhfb, ajournal = "IBM J. Res. Develop.", fjournal = "IBM Journal of Research and Development", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=5288520", reviewer = "C. B. Haselgrove", } @Book{Lewin:1958:DAF, author = "Leonard Lewin", title = "Dilogarithms and Associated Functions", publisher = "Macdonald", address = "London, UK", pages = "353", year = "1958", LCCN = "QA351 .L5", bibdate = "Fri Jun 16 13:51:36 MDT 2023", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; z3950.loc.gov:7090/Voyager", acknowledgement = ack-nhfb, author-dates = "22-Jul-1919--13-Aug-2007", author-url = "https://en.wikipedia.org/wiki/Leonard_Lewin_(telecommunications_engineer)", remark = "Foreword by J. C. P. Miller", subject = "Dilogarithms", } @TechReport{Miller:1958:LNN, author = "J. C. P. Miller", title = "Lecture Notes on Numerical Analysis", type = "Report", institution = "Cambridge University", address = "Cambridge, England", year = "1958", bibdate = "Fri Sep 20 14:46:35 2024", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/k/kahan-william-m.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "https://www.youtube.com/watch?v=81YrRSeReKo", acknowledgement = ack-nhfb, remark-1 = "Cited in \cite[Reference 21]{Agarwal:1986:NSV} in elefunt.bib and fparith.bib.", remark-2 = "Page 135 of the Agarwal paper says ``Kahan has also informed us that Miller [21] wrote about the extra-accurate table idea in 1958.'' William M. Kahan spent 1958--1960 in Cambridge, as a pro-forma student of J. C. P. Miller --- Kahan already had a Ph.D. from the University of Toronto, but it was not recognized by Cambridge University! See the video interview in the URL.", } @Article{Sugai:1958:ERR, author = "Iwao Sugai", title = "Extraction of Roots by Repeated Subtractions for Digital Computers", journal = j-CACM, volume = "1", number = "12", pages = "6--8", month = dec, year = "1958", CODEN = "CACMA2", DOI = "https://doi.org/10.1145/377924.377928", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Wed Jul 14 15:48:22 MDT 2004", bibsource = "http://dblp.uni-trier.de/db/journals/cacm/cacm1.html#Sugai58; https://www.math.utah.edu/pub/tex/bib/cacm1950.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", oldlabel = "Sugai58", XMLdata = "ftp://ftp.informatik.uni-trier.de/pub/users/Ley/bib/records.tar.gz#journals/cacm/Sugai58", } @Article{Wadey:1958:TSR, author = "W. G. Wadey", title = "Two Square-Root Approximations", journal = j-CACM, volume = "1", number = "11", pages = "13--14", month = nov, year = "1958", CODEN = "CACMA2", DOI = "https://doi.org/10.1145/368932.368936", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Wed Jul 14 15:48:22 MDT 2004", bibsource = "http://dblp.uni-trier.de/db/journals/cacm/cacm1.html#Wadey58; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", oldlabel = "Wadey58", XMLdata = "ftp://ftp.informatik.uni-trier.de/pub/users/Ley/bib/records.tar.gz#journals/cacm/Wadey58", } @Article{Corbato:1959:GSB, author = "Fernando J. Corbat{\'o} and Jack L. Uretsky", title = "Generation of Spherical {Bessel} Functions in Digital Computers", journal = j-J-ACM, volume = "6", number = "3", pages = "366--375", month = jul, year = "1959", CODEN = "JACOAH", ISSN = "0004-5411 (print), 1557-735X (electronic)", ISSN-L = "0004-5411", bibdate = "Mon Dec 05 20:07:17 1994", bibsource = "http://portal.acm.org/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jacm.bib", acknowledgement = ack-nhfb, ajournal = "J. Assoc. Comput. Mach.", fjournal = "Journal of the Association for Computing Machinery", journal-URL = "https://dl.acm.org/loi/jacm", } @Article{Davis:1959:LEI, author = "Philip J. Davis", title = "{Leonhard Euler}'s Integral: a Historical Profile of the Gamma Function: In Memoriam: {Milton Abramowitz}", journal = j-AMER-MATH-MONTHLY, volume = "66", number = "10", pages = "849--869", month = dec, year = "1959", CODEN = "AMMYAE", DOI = "https://doi.org/10.2307/2309786", ISSN = "0002-9890 (print), 1930-0972 (electronic)", ISSN-L = "0002-9890", MRclass = "00 (33.00)", MRnumber = "MR0106810 (21 #5540)", bibdate = "Mon Jun 28 12:39:33 MDT 1999", bibsource = "https://www.math.utah.edu/pub/tex/bib/amermathmonthly1950.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.jstor.org/stable/2309786", acknowledgement = ack-nhfb, author-dates = "Philip J. Davis (2 January 1923--14 March 2018)", fjournal = "American Mathematical Monthly", journal-URL = "https://www.jstor.org/journals/00029890.htm", } @Article{DiDonato:1959:NFC, author = "A. R. DiDonato and A. V. Hershey", title = "New Formulas for Computing Incomplete Elliptic Integrals of the First and Second Kind", journal = j-J-ACM, volume = "6", number = "4", pages = "515--526", month = oct, year = "1959", CODEN = "JACOAH", DOI = "https://doi.org/10.1145/320998.321005", ISSN = "0004-5411 (print), 1557-735X (electronic)", ISSN-L = "0004-5411", MRclass = "65.00", MRnumber = "0107353", MRreviewer = "F. Stallmann", bibdate = "Mon Dec 05 20:10:59 1994", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, ajournal = "J. Assoc. Comput. Mach.", fjournal = "Journal of the ACM", journal-URL = "https://dl.acm.org/loi/jacm", } @Book{Emde:1959:TEF, author = "Fritz Emde", title = "{Tafeln Elementarer Funktionen} ({German}) [Tables of Elementary Functions]", publisher = "B. T. Teubner", address = "Leipzig, Germany and Berlin, Germany", edition = "Third", pages = "xii + 181", year = "1959", LCCN = "QA47 .E5", bibdate = "Fri Jun 11 12:34:09 1999", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://lccn.loc.gov/61020314", acknowledgement = ack-nhfb, author-dates = "1873--1951", language = "German", } @Article{Gautschi:1959:EIL, author = "W. Gautschi", title = "Exponential integral $ \int_1^\infty e^{-x t} t^{-n} \, d t $ for large values of $n$", journal = j-J-RES-NATL-BUR-STAND-1934, volume = "62", number = "3", pages = "123--125", month = mar, year = "1959", DOI = "https://doi.org/10.6028/jres.062.022", ISSN = "0091-0635", bibdate = "Sat Feb 18 14:39:27 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Journal of Research of the National Bureau of Standards (1934)", journal-URL = "https://www.nist.gov/nist-research-library/journal-research-nist/past-papers", } @Article{Goldstein:1959:RTC, author = "M. Goldstein and R. M. Thaler", title = "Recurrence Techniques for the Calculation of {Bessel} Functions", journal = j-MATH-TABLES-OTHER-AIDS-COMPUT, volume = "13", number = "66", pages = "102--108", month = apr, year = "1959", CODEN = "MTTCAS", DOI = "https://doi.org/10.1090/S0025-5718-1959-0105794-5", ISSN = "0891-6837 (print), 2326-4853 (electronic)", ISSN-L = "0891-6837", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "Mathematical Tables and Other Aids to Computation", journal-URL = "http://www.ams.org/mcom/", } @Book{Greenhill:1959:AEF, author = "Alfred George Greenhill", title = "The Applications of Elliptic Functions", publisher = pub-DOVER, address = pub-DOVER:adr, pages = "xi + 357", year = "1959", bibdate = "Wed Mar 15 08:21:33 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "See \cite{Greenhill:1892:AEF}.", acknowledgement = ack-nhfb, author-dates = "1847--1927", remark = "Reprint of \cite{Greenhill:1892:AEF}.", } @Article{Kogbetliantz:1959:CSC, author = "E. G. (Ervand George) Kogbetliantz", title = "Computation of $ \sin {N} $, $ \cos {N} $, and $ {M} $ th Root of $ {N} $ Using an Electronic Computer", journal = j-IBM-JRD, volume = "3", number = "2", pages = "147--152", month = apr, year = "1959", CODEN = "IBMJAE", DOI = "https://doi.org/10.1147/rd.32.0147", ISSN = "0018-8646 (print), 2151-8556 (electronic)", ISSN-L = "0018-8646", MRclass = "65.00", MRnumber = "21 \#964", bibdate = "Thu Sep 1 10:15:56 1994", bibsource = "garbo.uwasa.fi:/pc/doc-soft/fpbiblio.txt; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "IBM Journal of Research and Development", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=5288520", reviewer = "B. A. Chartres", } @Article{Kreyszig:1959:RUE, author = "Erwin Kreyszig and John Todd", title = "The radius of univalence of the error function", journal = j-NUM-MATH, volume = "1", pages = "78--89", month = dec, year = "1959", CODEN = "NUMMA7", ISSN = "0029-599X (print), 0945-3245 (electronic)", ISSN-L = "0029-599X", bibdate = "Sun Oct 17 20:47:18 MDT 1999", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/nummath.bib", acknowledgement = ack-nhfb, fjournal = "Numerische Mathematik", journal-URL = "http://link.springer.com/journal/211", } @Article{Longman:1959:STT, author = "I. M. Longman", title = "A Short Table of $ \int^\infty_x {J}_0 (t)t^{-n} d t $ and $ \int^\infty_x {J}_1 (t) t^{-n} d t $ (in {Technical Notes and Short Papers})", journal = j-MATH-TABLES-OTHER-AIDS-COMPUT, volume = "13", number = "68", pages = "306--311", month = oct, year = "1959", CODEN = "MTTCAS", DOI = "https://doi.org/10.1090/S0025-5718-1959-0108892-5", ISSN = "0891-6837 (print), 2326-4853 (electronic)", ISSN-L = "0891-6837", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Mathematical Tables and Other Aids to Computation", journal-URL = "http://www.ams.org/mcom/", remark = "Fullerton: 6-digit values for $ n = 1, 2, \ldots {}, 22 $ and for $ x = 1, 2, \ldots {}, 10 $.", } @Article{Luke:1959:ECH, author = "Yudell L. Luke", title = "Expansion of the Confluent Hypergeometric Function in Series of {Bessel} Functions", journal = j-MATH-TABLES-OTHER-AIDS-COMPUT, volume = "13", number = "68", pages = "261--271", month = oct, year = "1959", CODEN = "MTTCAS", DOI = "https://doi.org/10.1090/S0025-5718-1959-0107027-2", ISSN = "0891-6837 (print), 2326-4853 (electronic)", ISSN-L = "0891-6837", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp1950.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "Mathematical Tables and Other Aids to Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Pan:1959:CSC, author = "V. Ya. Pan", title = "Certain schemes for the calculation of values of polynomials with real coefficients", journal = "Problemy Kibernetiki", volume = "5", number = "??", pages = "17--29", month = "????", year = "1959", bibdate = "Fri Oct 20 10:40:33 2023", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "Also available as English translation JPRS No. 11045.", acknowledgement = ack-nhfb, ajournal = "Probl. Kibernetiki", keywords = "number of multiplications to evaluate a polynomial", language = "Russian", remark = "Check: MathSciNet does not cover this volume, or record this paper, and the year may be wrong (based on available volume/year list entries). Cited in \cite[ref. 5, p. 178]{Fike:1967:MEP}.", } @Article{Pan:1959:SCP, author = "V. Ya. Pan", title = "Schemes for the computation of polynomials with real coefficients", journal = j-DOKL-AKAD-NAUK, volume = "127", number = "2", pages = "266--269", month = "????", year = "1959", CODEN = "DANKAS", ISSN = "0002-3264", bibdate = "Fri Oct 20 10:38:12 2023", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Doklady Akademii nauk SSSR", journal-URL = "http://istina.msu.ru/journals/366838/", keywords = "number of multiplications to evaluate a polynomial", language = "Russian", } @Article{Sarafyan:1959:NMC, author = "Diran Sarafyan", title = "A New Method of Computation of Square Roots Without Using Division", journal = j-CACM, volume = "2", number = "11", pages = "23--24", month = nov, year = "1959", CODEN = "CACMA2", DOI = "https://doi.org/10.1145/368481.368511", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Wed Jul 14 15:48:24 MDT 2004", bibsource = "http://dblp.uni-trier.de/db/journals/cacm/cacm2.html#Sarafyan59; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "See comments \cite{Traub:1960:CNM}.", acknowledgement = ack-nhfb, fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", oldlabel = "Sarafyan59", XMLdata = "ftp://ftp.informatik.uni-trier.de/pub/users/Ley/bib/records.tar.gz#journals/cacm/Sarafyan59", } @Article{Sherry:1959:CGF, author = "M. E. Sherry and S. Fulda", title = "Calculation of Gamma Functions to High Accuracy (in {Technical Notes and Short Papers})", journal = j-MATH-TABLES-OTHER-AIDS-COMPUT, volume = "13", number = "68", pages = "314--315", month = oct, year = "1959", CODEN = "MTTCAS", DOI = "https://doi.org/10.1090/S0025-5718-1959-0108891-3", ISSN = "0891-6837 (print), 2326-4853 (electronic)", ISSN-L = "0891-6837", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "Mathematical Tables and Other Aids to Computation", journal-URL = "http://www.ams.org/mcom/", } @InProceedings{Stiefel:1959:NMT, author = "Eduard L. Stiefel", title = "Numerical methods of {Tchebycheff} approximation", crossref = "Langer:1959:NAP", pages = "217--232", year = "1959", MRclass = "65.00 (41.00)", MRnumber = "0107961", MRreviewer = "M. R. Hestenes", bibdate = "Wed Sep 2 16:23:13 2020", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/s/stiefel-eduard.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, author-dates = "Eduard Stiefel (21 April 1909--25 November 1978)", } @Article{Strachey:1959:TSR, author = "C. Strachey", title = "On taking the square root of a complex number", journal = j-COMP-J, volume = "2", number = "2", pages = "89--89", month = jul, year = "1959", CODEN = "CMPJA6", DOI = "https://doi.org/10.1093/comjnl/2.2.89", ISSN = "0010-4620 (print), 1460-2067 (electronic)", ISSN-L = "0010-4620", bibdate = "Fri Sep 29 08:55:11 MDT 2000", bibsource = "http://www3.oup.co.uk/computer_journal/hdb/Volume_02/Issue_02/; https://www.math.utah.edu/pub/tex/bib/compj1950.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "http://www3.oup.co.uk/computer_journal/hdb/Volume_02/Issue_02/020089.sgm.abs.html; http://www3.oup.co.uk/computer_journal/hdb/Volume_02/Issue_02/tiff/89.tif", acknowledgement = ack-nhfb, fjournal = "The Computer Journal", journal-URL = "http://comjnl.oxfordjournals.org/", } @Article{Volder:1959:CCT, author = "Jack Volder", title = "The {CORDIC} Computing Technique", journal = "Proceedings of the Western Joint Computer Conference", pages = "257--261", year = "1959", DOI = "https://doi.org/10.1145/1457838.1457886", bibdate = "Mon May 19 13:30:58 1997", bibsource = "ftp://ftp.ira.uka.de/pub/bibliography/Graphics/siggraph/Pre.1975.bib.gz; https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, keywords = "elementary functions", } @Article{Volder:1959:CTC, author = "Jack E. Volder", title = "The {CORDIC} Trigonometric Computing Technique", journal = j-IRE-TRANS-ELEC-COMPUT, volume = "EC-8", number = "3", pages = "330--334", month = sep, year = "1959", CODEN = "IRELAO", DOI = "https://doi.org/10.1109/TEC.1959.5222693", ISSN = "0367-9950", ISSN-L = "0367-9950", bibdate = "Thu Jul 14 15:56:45 MDT 2011", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", URL = "https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5222693", acknowledgement = ack-nj # "\slash " # ack-nhfb, fjournal = "IRE Transactions on Electronic Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=5407885", } @Article{Wensley:1959:CNA, author = "J. H. Wensley", title = "A Class of Non-Analytical Iterative Processes", journal = j-COMP-J, volume = "1", number = "4", pages = "163--167", month = jan, year = "1959", CODEN = "CMPJA6", DOI = "https://doi.org/10.1093/comjnl/1.4.163", ISSN = "0010-4620 (print), 1460-2067 (electronic)", ISSN-L = "0010-4620", bibdate = "Tue Dec 4 14:47:23 MST 2012", bibsource = "Compiler/semantics.bib; http://comjnl.oxfordjournals.org/content/1/4.toc; http://www.math.utah.edu/pub/tex/bib/compj1950.bib; http://www3.oup.co.uk/computer_journal/hdb/Volume_01/Issue_04/; https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "http://comjnl.oxfordjournals.org/content/1/4/163.full.pdf+html; http://www3.oup.co.uk/computer_journal/hdb/Volume_01/Issue_04/010163.sgm.abs.html; http://www3.oup.co.uk/computer_journal/hdb/Volume_01/Issue_04/tiff/163.tif; http://www3.oup.co.uk/computer_journal/hdb/Volume_01/Issue_04/tiff/164.tif; http://www3.oup.co.uk/computer_journal/hdb/Volume_01/Issue_04/tiff/165.tif; http://www3.oup.co.uk/computer_journal/hdb/Volume_01/Issue_04/tiff/166.tif; http://www3.oup.co.uk/computer_journal/hdb/Volume_01/Issue_04/tiff/167.tif", acknowledgement = ack-nhfb, fjournal = "The Computer Journal", journal-URL = "http://comjnl.oxfordjournals.org/", remark = "The author describes digit-at-a-time algorithms for divide, square root, fourth root, inverse cosine, inverse tangent, inverse Jacobi elliptic function, and logarithm.", } @InProceedings{Akushsky:1960:MSO, author = "I. Y. Akushsky and L. B. Emelianow-Yaroslavsky and E. A. Klyamko and V. S. Linsky and G. D. Monakhov", editor = "????", booktitle = "Information Processing, Proceedings of the {International Conference on Information Processing, UNESCO, Paris, 15--20 June 1959}", title = "Methods of speeding-up the operation of digital computers", publisher = "UNESCO / R. Oldenbourg / Butterworths", address = "Paris, France / Munich, West Germany / London, UK", pages = "382--389 (or 382--388??)", year = "1960", bibdate = "Mon Nov 10 12:12:52 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "https://www.vldb.org/dblp/db/conf/ifip/ifip1959.html", acknowledgement = ack-nhfb, remark = "Cited in \cite{Ercegovac:1972:RES} as work on shift-add algorithms for function computation.", } @Book{Anonymous:1960:BFP, author = "Anonymous", title = "{Bessel} Functions. {Part III}: {Zeros} and Associated Values", volume = "7", publisher = pub-CAMBRIDGE, address = pub-CAMBRIDGE:adr, pages = "lx + 79", year = "1960", MRclass = "65.00", MRnumber = "119441", MRreviewer = "L. Fox", bibdate = "Tue Nov 14 17:19:58 2023", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/f/fox-leslie.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "Prepared under the direction of the Bessel Functions Panel of the Mathematical Tables Committee", series = "Royal Society Mathematical Tables", acknowledgement = ack-nhfb, reviewer-dates = "Leslie Fox (30 September 1918--1 August 1992)", } @Article{Anonymous:1960:EFG, author = "Anonymous", title = "Errata to {Fisher} and {Gupta}", journal = j-TECHNOMETRICS, volume = "2", number = "4", pages = "523--524", month = nov, year = "1960", CODEN = "TCMTA2", DOI = "https://doi.org/10.2307/1266462", ISSN = "0040-1706 (print), 1537-2723 (electronic)", ISSN-L = "0040-1706", bibdate = "Sat Jun 21 13:17:29 MDT 2014", bibsource = "http://www.jstor.org/journals/00401706.html; http://www.jstor.org/stable/i254224; https://www.math.utah.edu/pub/bibnet/authors/f/fisher-ronald-aylmer.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/technometrics1960.bib", note = "See \cite{Fisher:1960:PPD,Gupta:1960:OSG}.", URL = "http://www.jstor.org/stable/1266462", acknowledgement = ack-nhfb, fjournal = "Technometrics", journal-URL = "http://www.jstor.org/journals/00401706.html", subject-dates = "Sir Ronald Aylmer Fisher (17 February 1890--29 July 1962)", } @Article{Beam:1960:ACE, author = "A. Beam", title = "{Algorithm 14}: {Complex} exponential integral", journal = j-CACM, volume = "3", number = "7", pages = "406--406", month = jul, year = "1960", CODEN = "CACMA2", DOI = "https://doi.org/10.1145/367349.367351", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Fri Nov 25 18:19:27 MST 2005", bibsource = "http://www.acm.org/pubs/contents/journals/cacm/; https://www.math.utah.edu/pub/tex/bib/cacm1960.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "$\Ei(z)$; special functions", remark = "Fullerton: 30-line Algol procedure incorrectly labelled algorithm 13.", } @Book{Edmonds:1960:AMQ, author = "A. R. (Alan Robert) Edmonds", title = "Angular Momentum in Quantum Mechanics", volume = "4", publisher = pub-PRINCETON, address = pub-PRINCETON:adr, edition = "Second", pages = "viii + 146", year = "1960", ISBN = "0-691-02589-4 (hardcover), 0-691-07912-9 (paperback), 1-4008-8418-7 (online)", ISBN-13 = "978-0-691-02589-6 (hardcover), 978-0-691-07912-7 (paperback), 978-1-4008-8418-6 (online)", ISSN = "0075-0220", LCCN = "QC174; QC793.3.A5 E36 1960", bibdate = "Mon Aug 4 15:01:00 MDT 2025", bibsource = "fsz3950.oclc.org:210/WorldCat; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", series = "Investigations in physics", acknowledgement = ack-nhfb, shorttableofcontents = "Group theoretical preliminaries / 3 \\ The quantization of angular momentum / 10 \\ The coupling of angular momentum vectors / 31 \\ The representations of finite rotations / 53 \\ Spherical tensors and tensor operators / 68 \\ The construction of invariants from the vector-coupling coefficients / 90 \\ The evaluation of matrix elements in actual problems / 109 \\ Appendix 2: Approximate expressions for vector-coupling coefficients and 6-j symbols / 122 \\ Tables 1--5 / 124 \\ Cited References and Bibliography / 133 \\ Index / 141", tableofcontents = "GROUP THEORETICAL PRELIMINARIES / 3 \\ Introduction \\ Elementary theory of groups \\ The Euler angles \\ Representation theory \\ THE QUANTIZATION OF ANGULAR MOMENTUM / 10 \\ Definition of angular momentum in quantum mechanics \\ Angular momentum of a system of particles \\ Representation of the angular momentum operators \\ The physical significance of the quantization of angular momentum \\ The eigenvectors of the angular momentum operators [symbols] \\ The spin eigenvectors \\ Angular momentum eigenfunctions in the case of large $l$ \\ Time reversal and the angular momentum operators \\ THE COUPLING OF ANGULAR MOMENTUM VECTORS / 31\\ The addition of angular momenta \\ Commutation relations between components of $J_1$, $J_2$, and $J$ \\ Selection rules for the matrix elements of $J_1$ and $J_2$ \\ The choice of the phases of the states [symbols] \\ The vector coupling coefficients \\ Computation of the vector coupling coefficients \\ The Wigner 3-j symbol \\ Tabulation of formulas and numerical values for vector-coupling coefficients \\ Time reversal and the eigenvectors resulting from vector coupling \\ THE REPRESENTATIONS OF FINITE ROTATIONS / 53 \\ The transformations of the angular momentum eigenvectors under finite rotations \\ The symmetries of the [symbols] \\ Products of the [symbols] \\ Recursion relations for the [symbols] \\ Computation of the [symbols] \\ Integrals involving the [symbols] \\ The [symbols] as angular momentum eigenfunctions \\ The symmetric top \\ SPHERICAL TENSORS AND TENSOR OPERATORS / 68 \\ Spherical tensors \\ The tensor operators in quantum mechanics \\ Factorization of the matrix elements of tensor operators (Wigner--Eckart theorem) \\ The reduced matrix elements of a tensor operator \\ Hermitian adjoint of tensor operators \\ Electric quadrupole moment of proton or electron \\ The gradient formula \\ Expansion of a plane wave in spherical waves \\ Vector spherical harmonics \\ Spin spherical harmonics \\ Emission and absorption of particles \\ THE CONSTRUCTION OF INVARIANTS FROM THE VECTOR COUPLING COEFFICIENTS / 90 \\ The recoupling of three angular momenta \\ The properties of the 6-j symbol \\ The 9-j symbol \\ THE EVALUATION OF MATRIX ELEMENTS IN ACTUAL PROBLEMS / 109 \\ Matrix elements of the tensor product of two tensor operators \\ Selected examples for atomic, molecular and nuclear physics \\ APPENDICES \\ Appendix 1: Theorems used in chapter 3 / 121 \\ Appendix 2: Approximate expressions for vector-coupling coefficients and 6-j symbols / 122 \\ Tables 1--5 / 124 \\ Cited References and Bibliography / 133 \\ Index / 141", } @Article{Fisher:1960:PPD, author = "Ronald A. Fisher and E. A. Cornish", title = "The Percentile Points of Distributions Having Known Cumulants", journal = j-TECHNOMETRICS, volume = "2", number = "2", pages = "209--225", month = may, year = "1960", CODEN = "TCMTA2", DOI = "https://doi.org/10.2307/1266546", ISSN = "0040-1706 (print), 1537-2723 (electronic)", ISSN-L = "0040-1706", bibdate = "Sat Jun 21 13:17:27 MDT 2014", bibsource = "http://www.jstor.org/journals/00401706.html; http://www.jstor.org/stable/i254222; https://www.math.utah.edu/pub/bibnet/authors/f/fisher-ronald-aylmer.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/technometrics1960.bib", note = "See errata \cite{Anonymous:1960:EFG}.", URL = "http://www.jstor.org/stable/1266546", acknowledgement = ack-nhfb, author-dates = "Sir Ronald Aylmer Fisher (17 February 1890--29 July 1962)", fjournal = "Technometrics", journal-URL = "http://www.jstor.org/journals/00401706.html", } @Book{Fox:1960:TWP, author = "L. Fox", title = "Tables of {Weber} Parabolic Cylinder Functions and Other Functions for Large Arguments", volume = "4", publisher = pub-HMSO, address = pub-HMSO:adr, pages = "iii + 40", year = "1960", MRclass = "65.00 (33.00)", MRnumber = "120778", MRreviewer = "L. J. Slater", bibdate = "Mon Nov 13 14:02:18 2023", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/f/fox-leslie.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "Department of Scientific and Industrial Research", series = "National Physical Laboratory Mathematical Tables", acknowledgement = ack-nhfb, author-dates = "Leslie Fox (30 September 1918--1 August 1992)", } @Article{Gupta:1960:OSG, author = "Shanti S. Gupta", title = "Order Statistics from the Gamma Distribution", journal = j-TECHNOMETRICS, volume = "2", number = "2", pages = "243--262", month = may, year = "1960", CODEN = "TCMTA2", DOI = "https://doi.org/10.2307/1266548", ISSN = "0040-1706 (print), 1537-2723 (electronic)", ISSN-L = "0040-1706", MRclass = "62.00", MRnumber = "0112225 (22 \#3079)", MRreviewer = "S. S. Wilks", bibdate = "Sat Jun 21 13:17:27 MDT 2014", bibsource = "http://www.jstor.org/journals/00401706.html; http://www.jstor.org/stable/i254222; https://www.math.utah.edu/pub/bibnet/authors/f/fisher-ronald-aylmer.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/technometrics1960.bib", note = "See errata \cite{Anonymous:1960:EFG}.", URL = "http://www.jstor.org/stable/1266548", acknowledgement = ack-nhfb, fjournal = "Technometrics", journal-URL = "http://www.jstor.org/journals/00401706.html", } @Article{Haimovici:1960:MRE, author = "Corina Haimovici", title = "A method of representing elementary functions in algebras of finite order. ({Romanian})", journal = "An. {\c{S}}ti. Univ. ``Al. I. Cuza'' Ia{\c{s}}i Sec{\c{t}} I. (N.S.)", volume = "6", pages = "507--515", year = "1960", MRclass = "26.00", MRnumber = "24 \#A1342", bibdate = "Sat Jan 11 17:44:01 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, } @Article{Harumi:1960:VTN, author = "Kasabur{\^o} Harumi and Shigetoshi Katsura and John W. {Wrench, Jr.}", title = "Values of {$ \frac {2}{\pi } \int^\infty_0 \Big (\frac {\sin t}{t} \Big)^n d t $} (in {Technical Notes and Short Papers})", journal = j-MATH-COMPUT, volume = "14", number = "72", pages = "379--379", month = oct, year = "1960", CODEN = "MCMPAF", DOI = "https://doi.org/10.1090/S0025-5718-1960-0122010-7", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", remark = "Fullerton: 10-digit table for $ n = 1, 2, \ldots {}, 30 $.", } @InCollection{Kogbetliantz:1960:GEF, author = "E. G. (Ervand George) Kogbetliantz", title = "Generation of elementary functions", crossref = "Ralston:1960:MMD", pages = "7--35", year = "1960", MRclass = "68.00 (65.00)", MRnumber = "22 \#8681", MRreviewer = "J. C. P. Miller", bibdate = "Sat Jan 11 17:44:01 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, } @Article{Lang:1960:ECC, author = "H. A. Lang", title = "On the Evaluation of Certain Complex Elliptic Integrals (in {Technical Notes and Short Papers})", journal = j-MATH-COMPUT, volume = "14", number = "70", pages = "195--199", month = apr, year = "1960", CODEN = "MCMPAF", DOI = "https://doi.org/10.1090/S0025-5718-1960-0112241-4", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Book{Lebedev:1960:GMT, author = "A. V. (Aleksandr Vasil'evich) Lebedev and R. M. (Rimma Maksimova) Fedorova and Nina Mikhollovna Burunova", title = "A Guide to Mathematical Tables", publisher = pub-PERGAMON, address = pub-PERGAMON:adr, pages = "xlvi + 586", year = "1960", LCCN = "Z6654.T3 L42", bibdate = "Mon Feb 13 17:12:14 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; z3950.loc.gov:7090/Voyager", note = "English edition prepared by D. G. Fry from the Russian original.", acknowledgement = ack-nhfb, subject = "Mathematics; Tables; Indexes", tableofcontents = "Front Cover \\ A Guide to Mathematical Tables \\ Copyright Page \\ Translator's Preface \\ Preface \\ Index to the Section Headings in the Table of Contents of the Guide and the Supplement \\ Table of Contents \\ Part One: Description of the Tables \\ 1. Powers, Rational and Algebraic Functions \\ Positive Whole Powers \\ Fractional Positive Powers \\ Reciprocals (Negative Whole and Fractional Powers) \\ Rational Functions \\ Algebraic Functions \\ Complex Numbers and Their Powers \\ 2. Trigonometric Functions. Various Values Connected With the Circle and the Sphere \\ Natural Values of Trigonometric Functions \\ Various Expressions Containing Trigonometric Functions \\ Derivatives and Powers of Trigonometric Functions \\ Reciprocal Trigonometric Functions of A Complex Variable \\ Geometric Quantities \\ Tables For Converting From One Angular Measure to Another \\ 3. Exponential and Hyperbolic Functions \\ Exponential Functions \\ Hyperbolic Functions \\ Expressions Containing Trigonometric and Hyperbolic Functions \\ Inverse Hyperbolic Functions \\ 4. Logarithms \\ Common Logarithms and Antilogarithms \\ Natural Logarithms \\ Logarithms to Base 2 \\ 5. Factorials, Euler Integrals and Related Functions \\ Factorials \\ the Gamma Function \\ the Psi Function and Its Derivatives \\ the Beta Function \\ Certain Constants \\ 6. Sine and Cosine Integrals, Exponential and Logarithmic Integrals and Related Functions \\ Sine and Cosine Integrals \\ Hyperbolic Sine and Cosine Integrals \\ Exponential Integral \\ Logarithmic Integral \\ Generalised and Composite Integral Functions \\ Integral Functions of Complex Argument \\ 7. Probability Integrals and Related Functions \\ Functions of the Distribution of Probability and Related Functions \\ Probability Integrals and Expressions Containing Probability Integrals \\ Tables in Which the Integral Serves As the Argument \\ Probability Integrals of Complex Argument \\ Derivatives of Various Orders of Probability Distribution Functions \\ Zeros of Probability Integrals \\ Logarithms of Probability Integrals \\ Fresnel Integrals and Related Functions \\ 8. Elliptic Integrals and Elliptic Functions \\ Moduli of the Integrals \\ Complete Elliptic Integrals \\ Incomplete Elliptic Integrals \\ Elliptic Functions \\ Theta Functions \\ 9. Legendre Functions and Polynomials \\ Legendre Polynomials and Legendre Functions of the First Kind \\ Associated Legendre Functions of the First Kind \\ Roots of Legendre Functions \\ Various Expressions Containing Legendre Functions \\ 10. Cylinder Functions \\ Cylinder Functions of the First and Second Kinds of Real Argument \\ Riccati--Bessel Functions \\ Lommel Functions of Two Real Variables \\ Cylinder Functions of the Third Kind (Hankel Functions) \\ Cylinder Functions of the First and Second Kinds of Imaginary Argument \\ Lommel Functions of Two Imaginary Variables \\ Cylinder Functions of Complex Argument \\ Thomson Functions", } @Manual{Maehly:1960:ACD, author = "Hans J. Maehly", title = "Approximations for the {Control Data 1604}", organization = inst-INST-ADV-STUDY, address = inst-INST-ADV-STUDY:adr, pages = "ii + 44", day = "15", month = jan, year = "1960", bibdate = "Tue Nov 06 00:39:07 2007", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.bitsavers.org/pdf/cdc/1604/Approximations_For_The_Control_Data_1604_Mar60.pdf", acknowledgement = ack-nhfb, keywords = "$\arctan(x)$; $\cos(x)$; $\exp(x)$; $\log(x)$; $\sin(x)$; $\sqrt(x)$; $\tan(x)$; CDC 1604", } @InProceedings{Maehly:1960:RAT, author = "H. J. Maehly", editor = "????", booktitle = "Information Processing, Proceedings of the {International Conference on Information Processing, UNESCO, Paris, 15--20 June 1959}", title = "Rational approximations for transcendental functions", publisher = "UNESCO / R. Oldenbourg / Butterworths", address = "Paris, France / Munich, West Germany / London, UK", pages = "57--62", year = "1960", bibdate = "Mon Nov 10 12:12:52 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "https://www.vldb.org/dblp/db/conf/ifip/ifip1959.html", acknowledgement = ack-nhfb, } @Article{Philip:1960:FI, author = "J. R. Philip", title = "The function $ \operatorname {inverfc} \theta $", journal = j-AUSTRALIAN-J-PHYS, volume = "13", number = "1", pages = "13--20", month = mar, year = "1960", CODEN = "AUJPAS", DOI = "https://doi.org/10.1071/PH600013", ISSN = "0004-9506 (print), 1446-5582 (electronic)", ISSN-L = "0004-9506", MRnumber = "22 9626", bibdate = "Tue Sep 11 20:53:59 2007", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", ZMnumber = "135.28302", abstract = "The function inverfc $ \theta $ arises in certain differential equations when concentration is taken as an independent variable. It enters into a general method of exact solution of the concentration-dependent diffusion equation. An account is given of the properties of this function, and of its derivatives and integrals. The function\par $$ B(\theta) = (2 / \pi^{1 / 2}) \exp [ - (\operatorname {inverfc} \theta)^2] $$ \noindent is intimately connected with the first integral of inverfc $ \theta $ and with its derivatives. Tables of $ \operatorname {inverfc} \theta $ and $ B(\theta) $ are given.", acknowledgement = ack-nhfb, fjournal = "Australian Journal of Physics", journal-URL = "http://www.publish.csiro.au/ph/content/allissues", } @Book{Rainville:1960:SF, author = "Earl David Rainville", title = "Special Functions", publisher = pub-MACMILLAN, address = pub-MACMILLAN:adr, pages = "xii + 365", year = "1960", ISBN = "0-8284-0258-2", ISBN-13 = "978-0-8284-0258-3", LCCN = "QA331 R15; QA351 .R3 1971", bibdate = "Mon Sep 3 16:04:28 MDT 2007", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; z3950.loc.gov:7090/Voyager", note = "Reprinted in 1971.", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", remark = "Based upon the lectures on special functions which \ldots{} [the author has] been giving at the University of Michigan since 1946. Fullerton: All traditional functions and polynomials are discussed in some detail.", shorttableofcontents = "1: Infinite Products \\ 2: The Gamma and Beta Functions \\ 3: Asymptotic Series \\ 4: The Hypergeometric Function \\ 5: Generalized Hypergeometric Functions \\ 6: Bessel Functions \\ 7: The Confluent Hypergeometric Function \\ 8: Generating Functions \\ 9: Orthogonal Polynomials \\ 10: Legendre Polynomials \\ 11: Hermite Polynomials \\ 12: Laguerre Polynomials \\ 13: The Sheffer Classification and Related Topics \\ 14: Pure Recurrence Relations \\ 15: Symbolic Relations \\ 16: Jacobi Polynomials \\ 17: Ultraspherical and Gegenbauer Polynomials \\ 18: Other Polynomial Sets \\ 19: Elliptic Functions \\ 20: Theta Functions \\ 21: Jacobian Elliptic Functions \\ Bibliography \\ Index", subject = "Functions, Special", tableofcontents = "1: Infinite Products \\ 1. Introduction / 1 \\ 2. Definition of an infinite product / 1 \\ 3. A necessary condition for convergence . / 2 \\ 4. The associated series of logarithms / 2 \\ 5. Absolute convergence / 3 \\ 6. Uniform convergence / 5 \\ \\ 2: The Gamma and Beta Functions \\ 7. The Euler or Mascheroni constant $\gamma$ / 8 \\ 8. The Gamma function / 9 \\ 9. A series for $\Gamma'(z) / \Gamma(z)$ / 10 \\ 10. Evaluation of $\Gamma(1)$ and $\Gamma'(1)$ / 10 \\ 11. The Euler product for $\Gamma(z)$ / 11 \\ 12. The difference equation $\Gamma(z + 1) = z \Gamma(z)$ / 12 \\ 13. The order symbols $o$ and $O$ / 12 \\ 14. Evaluation of certain infinite products / 13 \\ 15. Euler's integral for $\Gamma(z)$ / 15 \\ 16. The Beta function / 18 \\ 17. The value of $\Gamma(z) \Gamma(1 - z)$ / 19 \\ 18. The factorial function / 22 \\ 19. Legendre's duplication formula / 23 \\ 20. Gauss' multiplication theorem / 24 \\ 21. A summation formula due to Euler / 26 \\ 22. The behavior of $\log \Gamma(z)$ for large $|z|$ / 29 \\ \\ 3: Asymptotic Series \\ 23. Definition of an asymptotic expansion / 33 \\ 24. Asymptotic expansions about infinity / 36 \\ 25. Algebraic properties / 38 \\ 26. Term-by-term integration / 39 \\ 27. Uniqueness / 40 \\ 28. Watson's lemma / 41 \\ \\ 4: The Hypergeometric Function \\ 29. The function $F(a, b; c; z)$ / 45 \\ 30. A simple integral form / 47 \\ 31. $F(a, b; c; 1)$ as a function of the parameters / 48 \\ 32. Evaluation of $F(a, b; c; 1)$ / 48 \\ 33. The contiguous function relations / 50 \\ 34. The hypergeometric differential equation / 53 \\ 35. Logarithmic solutions of the hypergeometric equation / 54 \\ 36. $F(a, b; c; 2)$ as a function of its parameters / 55 \\ 37. Elementary series manipulations / 56 \\ 38. Simple transformations / 58 \\ 39. Relation between functions of $z$ and $1 - z$ / 61 \\ 40. A quadratic transformation / 63 \\ 41. Other quadratic transformations / 65 \\ 42. A theorem due to Kummer / 68 \\ 43. Additional properties / 68 \\ \\ 5: Generalized Hypergeometric Functions \\ 44. The function $_pF_q$ / 73 \\ 45. The exponential and binomial functions / 74 \\ 46. A differential equation / 74 \\ 47. Other solutions of the differential equation / 76 \\ 48. The contiguous function relations / 80 \\ 49. A simple integral / 85 \\ 50. The $_pF_q$ with unit argument / 85 \\ 51. Saalsch{\"u}tz' theorem / 86 \\ 52. Whipple's theorem / 88 \\ 53. Dixon's theorem / 92 \\ 54. Contour integrals of Barnes' type / 94 \\ 55. The Barnes integrals and the function $_pF_q$ / 98 \\ 56. A useful integral / 102 \\ \\ 6: Bessel Functions \\ 57. Remarks / 108 \\ 58. Definition of $J_n(z)$ / 108 \\ 59. Bessel's differential equation / 109 \\ 60. Differential recurrence relations / 110 \\ 61. A pure recurrence relation / 111 \\ 62. A generating function / 112 \\ 63. Bessel's integral / 114 \\ 64. Index half an odd integer 114 , \\ 65. Modified Bessel functions / 116 \\ 66. Neumann polynomials / 116 \\ 67. Neumann series / 119 \\ \\ 7: The Confluent Hypergeometric Function \\ 68. Basic properties of the $_1F_1$ / 123 \\ 69. Kummer's first formula / 124 \\ 70. Kummer's second formula / 125 \\ \\ 8: Generating Functions \\ 71. The generating function concept / 129 \\ 72. Generating functions of the form $G(2 x t - t^2)$ / 131 \\ 73. Sets generated by $e^t \psi(x t)$ / 132 \\ 74. The generating functions $A(t) \exp[-x t / (1 - t)]$ / 135 \\ 75. Another class of generating functions / 137 \\ 76. Boas and Buck generating functions / 140 \\ 77. An extension / 143 \\ \\ 9: Orthogonal Polynomials \\ 78. Simple sets of polynomials / 147 \\ 79. Orthogonality / 147 \\ 80. An equivalent condition for orthogonality / 148 \\ 81. Zeros of orthogonal polynomials / 149 \\ 82. Expansion of polynomials / 150 \\ 83. The three-term recurrence relation / 151 \\ 84. The Christoffel--Darboux formula / 153 \\ 85. Normalization; Bessel's inequality / 155 \\ \\ 10: Legendre Polynomials \\ 86. A generating function / 157 \\ 87. Differential recurrence relations / 158 \\ 88. The pure recurrence relation / 159 \\ 89. Legendre's differential equation / 160 \\ 90. The Rodrigues formula / 161 \\ 91. Bateman's generating function / 162 \\ 92. Additional generating functions / 163 \\ 93. Hypergeometric forms of $P_n(x)$ / 163 \\ 94. Brafman's generating functions / 167 \\ 95. Special properties of $P_n(x)$ / 168 \\ 96. More generating functions / 169 \\ 97. Laplace's first integral form / 171 \\ 98. Some bounds on $P_n(z)$ / 172 \\ 99. Orthogonality / 173 \\ 100. An expansion theorem / 176 \\ 101. Expansion of $x^n$ / 179 \\ 102. Expansion of analytic functions / 181 \\ \\ 11: Hermite Polynomials \\ 103. Definition of $H_n(x)$ / 187 \\ 104. Recurrence relations / 188 \\ 105. The Rodrigues formula / 189 \\ 106. Other generating functions / 190 \\ 107. Integrals / 190 \\ 108. The Hermite polynomial as a $_2F_0$ / 191 \\ 109. Orthogonality / 191 \\ 110. Expansion of polynomials / 193 \\ 111. More generating functions / 196 \\ \\ 12: Laguerre Polynomials \\ 112. The polynomial $L_n^{(\alpha)}(x)$ / 200 \\ 113. Generating functions / 201 \\ 114. Recurrence relations / 202 \\ 115. The Rodrigues formula / 203 \\ 116. The differential equation / 204 \\ 117. Orthogonality / 204 \\ 118. Expansion of polynomials / 206 \\ 119. Special properties / 209 \\ 120. Other generating functions / 211 \\ 121. The simple Laguerre polynomials / 213 \\ \\ 13: The Sheffer Classification and Related Topics \\ 122. Differential operators and polynomial sets / 218 \\ 123. Sheffer's $A$-type classification / 221 \\ 124. Polynomials of Sheffer $A$-type zero / 222 \\ 195. An extension of Sheffer's classification / 226 \\ 126. Polynomials of $\sigma$-type zero / 228 \\ \\ 14: Pure Recurrence Relations \\ 127. Sister Celine's technique / 233 \\ 128. A mild extension / 240 \\ \\ 15: Symbolic Relations \\ 129. Notation / 246 \\ 130. Symbolic relations among classical polynomials / 247 \\ 131. Polynomials of symbolic form $L_n(y(x))$ / 249 \\ \\ 16: Jacobi Polynomials \\ 132. The Jacobi polynomials / 254 \\ 133. Bateman's generating function / 256 \\ 134. The Rodrigues formula / 257 \\ 135. Orthogonality / 258 \\ 136. Differential recurrence relations / 261 \\ 137. The pure recurrence relation / 263 \\ 138. Mixed relations / 263 \\ 139. Appell's functions of two variables / 265 \\ 140. An elementary generating function / 269 \\ 141. Brafman's generating functions / 271 \\ 142. Expansion in series of polynomials / 272 \\ \\ 17: Ultraspherical and Gegenbauer Polynomials \\ 143. Definitions / 276 \\ 144. The Gegenbauer polynomials / 277 \\ 145. The ultraspherical polynomials / 283 \\ \\ 18: Other Polynomial Sets \\ 146. Bateman's $Z_n(x)$ / 285 \\ 147. Rice's $H_n(\zeta, p, v)$ / 287 \\ 148. Bateman's $F_n(z)$ / 289 \\ 149. Sister Celine's polynomials / 290 \\ 150. Bessel polynomials / 293 \\ 151. Bedient's polynomials / 297 \\ 152. Shively's pseudo-Laguerre and other polynomials / 298 \\ 153. Bernoulli polynomials / 299 \\ 154. Euler polynomials / 300 \\ 155. Tchebicheff polynomials / 301 \\ \\ 19: Elliptic Functions \\ 156. Doubly periodic functions / 305 \\ 157. Elliptic functions / 306 \\ 158. Elementary properties / 306 \\ 159. Order of an elliptic function / 308 \\ 160. The Weierstrass function $P(z)$ / 309 \\ 161. Other elliptic functions / 311 \\ 162. A differential equation for $P(z)$ / 311 \\ 163. Connection with elliptic integrals / 313 \\ \\ 20: Theta Functions \\ 164. Definitions / 314 \\ 165. Elementary properties / 315 \\ 166. The basic property table / 316 \\ 167. Location of zeros / 319 \\ 168. Relations among squares of theta functions / 322 \\ 169. Pseudo addition theorems / 325 \\ 170. Relation to the heat equation / 328 \\ 171. The relation $\theta_1' = \theta_2 \theta_3 \theta_4$ / 329 \\ 172. Infinite products / 332 \\ 173. The value of $G$ / 334 \\ \\ 21: Jacobian Elliptic Functions \\ 174. A differential equation involving theta functions / 339 \\ 175. The function $\sn(u)$ / 342 \\ 176. The functions $\cn(u)$ and $\dn(u)$ / 343 \\ 177. Relations involving squares / 344 \\ 178. Relations involving derivatives / 345 \\ 179. Addition theorems / 347 \\ \\ Bibliography / 349 \\ \\ Index / 359", } @Article{Sarafyan:1960:DCS, author = "Diran Sarafyan", title = "Divisionless computation of square roots through continued squaring", journal = j-CACM, volume = "3", number = "5", pages = "319--321", month = may, year = "1960", CODEN = "CACMA2", DOI = "https://doi.org/10.1145/367236.367267", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", MRclass = "65.00", MRnumber = "22\#8639", bibdate = "Fri Nov 25 18:19:26 MST 2005", bibsource = "http://www.acm.org/pubs/contents/journals/cacm/; https://www.math.utah.edu/pub/tex/bib/cacm1960.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "$\sqrt(x)$; elementary functions", ZMreviewer = "M. Lotkin", } @Article{Sholander:1960:AEE, author = "Marlow Sholander", title = "Analytical expressions and elementary functions", journal = j-AMER-MATH-MONTHLY, volume = "67", number = "3", pages = "213--214", month = mar, year = "1960", CODEN = "AMMYAE", ISSN = "0002-9890 (print), 1930-0972 (electronic)", ISSN-L = "0002-9890", MRclass = "26.00", MRnumber = "22 \#9553", bibdate = "Sat Jan 11 17:44:01 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.jstor.org/stable/2309678", acknowledgement = ack-nhfb, fjournal = "American Mathematical Monthly", journal-URL = "https://www.jstor.org/journals/00029890.htm", } @Book{Slater:1960:CHF, author = "Lucy Joan Slater", title = "Confluent hypergeometric functions", publisher = pub-CAMBRIDGE, address = pub-CAMBRIDGE:adr, pages = "247", year = "1960", LCCN = "QA351 .S56", bibdate = "Sat Oct 30 21:01:55 MDT 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; z3950.loc.gov:7090/Voyager", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", remark = "Fullerton: Discussion of properties and 7 decimal place tables of $_1 F_1 (a; b; x)$.", subject = "Hypergeometric functions", } @Article{Traub:1960:CNM, author = "J. F. Traub", title = "Comments on a recent paper [{``A New Method of Computation of Square Roots Without Using Division''}]", journal = j-CACM, volume = "3", number = "2", pages = "86--86", month = feb, year = "1960", CODEN = "CACMA2", DOI = "https://doi.org/10.1145/366959.366989", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Fri Nov 25 18:19:25 MST 2005", bibsource = "http://dblp.uni-trier.de/db/journals/cacm/cacm3.html#Traub60; http://www.acm.org/pubs/contents/journals/cacm/; https://www.math.utah.edu/pub/tex/bib/cacm1960.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "See \cite{Sarafyan:1959:NMC}.", abstract = "Mr. Diran Sarafyan, in his paper \booktitle{A New Method of Computation of Square Roots Without Using Divisions} (Communications, Nov. 1959) gave a way of computing square roots which converges faster than the standard Newton method. His technique can be generalized as follows.", acknowledgement = ack-nhfb, fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", oldlabel = "Traub60", XMLdata = "ftp://ftp.informatik.uni-trier.de/pub/users/Ley/bib/records.tar.gz#journals/cacm/Traub60", } @Article{Ward:1960:CCE, author = "Morgan Ward", title = "The Calculation of the Complete Elliptic Integral of the Third Kind", journal = j-AMER-MATH-MONTHLY, volume = "67", number = "3", pages = "205--213", month = mar, year = "1960", CODEN = "AMMYAE", DOI = "https://doi.org/10.2307/2309677", ISSN = "0002-9890 (print), 1930-0972 (electronic)", ISSN-L = "0002-9890", bibdate = "Tue Feb 6 16:32:27 2018", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.jstor.org/stable/2309677", acknowledgement = ack-nhfb, fjournal = "American Mathematical Monthly", journal-URL = "http://www.jstor.org/journals/00029890.html", } @Book{Warmus:1960:TEF, author = "Mieczys{\l}aw Warmus", title = "Tables of elementary functions", publisher = pub-PERGAMON, address = pub-PERGAMON:adr, pages = "vii + 567", year = "1960", LCCN = "QA55 .W3 1960", MRclass = "65.05", MRnumber = "23 \#B1141", MRreviewer = "J. C. P. Miller", bibdate = "Sat Jan 11 17:44:01 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "Pa{\'n}stwowe Wydawnictwo Naukowe, Warsaw. Separately bound table of proportional parts, 30 pp.", acknowledgement = ack-nhfb, } @Article{Wynn:1960:RAF, author = "Peter Wynn", title = "The Rational Approximation of Functions which are Formally Defined by a Power Series Expansion", journal = j-MATH-COMPUT, volume = "14", number = "70", pages = "147--186", month = apr, year = "1960", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp1960.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Barakat:1961:EIG, author = "Richard Barakat", title = "Evaluation of the Incomplete Gamma Function of Imaginary Argument by {Chebyshev} Polynomials", journal = j-MATH-COMPUT, volume = "15", number = "73", pages = "7--11", month = jan, year = "1961", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Boersma:1961:TFR, author = "J. Boersma", title = "Two Formulas Relating to Elliptic Integrals of the Third Kind (in {Technical Notes and Short Papers})", journal = j-MATH-COMPUT, volume = "15", number = "75", pages = "296--298", month = jul, year = "1961", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Book{Bowman:1961:IEF, author = "Frank Bowman", title = "Introduction to Elliptic Functions with Applications", volume = "922", publisher = pub-DOVER, address = pub-DOVER:adr, pages = "115", year = "1961", LCCN = "QA343 .B76 1961", bibdate = "Wed Mar 15 06:50:49 MDT 2017", bibsource = "fsz3950.oclc.org:210/WorldCat; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, remark = "Unabridged and corrected republication of \cite{Bowman:1953:IEF}.", } @Article{Cheney:1961:TNA, author = "E. W. Cheney and H. L. Loeb", title = "Two new algorithms for rational approximation", journal = j-NUM-MATH, volume = "3", number = "1", pages = "72--75", month = dec, year = "1961", CODEN = "NUMMA7", DOI = "https://doi.org/10.1007/BF01386002", ISSN = "0029-599X (print), 0945-3245 (electronic)", ISSN-L = "0029-599X", bibdate = "Sun Oct 17 19:01:15 MDT 1999", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/nummath.bib", acknowledgement = ack-nhfb, fjournal = "Numerische Mathematik", journal-URL = "http://link.springer.com/journal/211", } @Article{Corrington:1961:ACE, author = "Murlan S. Corrington", title = "Applications of the Complex Exponential Integral", journal = j-MATH-COMPUT, volume = "15", number = "73", pages = "1--6", month = jan, year = "1961", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Ehrling:1961:NCI, author = "G. Ehrling", title = "On the Numerical Computation of Incomplete Elliptic Integrals", journal = j-NORDISK-TIDSKR-INFORM-BEHAND, volume = "1", number = "1", pages = "8--14", month = mar, year = "1961", CODEN = "BITTEL, NBITAB", DOI = "https://doi.org/10.1007/BF01961946", ISSN = "0006-3835 (print), 1572-9125 (electronic)", ISSN-L = "0006-3835", bibdate = "Wed Jan 4 18:52:06 MST 2006", bibsource = "http://springerlink.metapress.com/openurl.asp?genre=issue&issn=0006-3835&volume=1&issue=1; https://www.math.utah.edu/pub/tex/bib/bit.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.springerlink.com/openurl.asp?genre=article&issn=0006-3835&volume=1&issue=1&spage=8", acknowledgement = ack-nhfb, journal-URL = "http://link.springer.com/journal/10543", } @Article{Fields:1961:EHF, author = "Jerry L. Fields and Jet Wimp", title = "Expansions of Hypergeometric Functions in Hypergeometric Functions", journal = j-MATH-COMPUT, volume = "15", number = "76", pages = "390--395", month = oct, year = "1961", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp1960.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Floyd:1961:ACE, author = "Robert W. Floyd", title = "An algorithm for coding efficient arithmetic operations", journal = j-CACM, volume = "4", number = "1", pages = "42--51", month = jan, year = "1961", CODEN = "CACMA2", DOI = "https://doi.org/10.1145/366062.366082", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Fri Nov 25 18:19:30 MST 2005", bibsource = "ftp://ftp.ira.uka.de/pub/bibliography/Compiler/Compiler.Lins.bib; http://www.acm.org/pubs/contents/journals/cacm/; https://www.math.utah.edu/pub/tex/bib/cacm1960.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, author-dates = "Robert W. Floyd (8 June 1936--25 September 2001)", fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", remark = "Description of binary subdivision method for computing $n$-th powers in $ O(\log_2 (n))$ operations, as referenced in \cite{Knuth:1962:EPC}.", } @Article{Froberg:1961:RCA, author = "Carl-Erik Fr{\"o}berg", title = "Rational {Chebyshev} Approximations of Elementary Functions", journal = j-NORDISK-TIDSKR-INFORM-BEHAND, volume = "1", number = "4", pages = "256--262", month = dec, year = "1961", CODEN = "BITTEL, NBITAB", DOI = "https://doi.org/10.1007/BF01933243", ISSN = "0006-3835 (print), 1572-9125 (electronic)", ISSN-L = "0006-3835", bibdate = "Wed Jan 4 18:52:07 MST 2006", bibsource = "http://springerlink.metapress.com/openurl.asp?genre=issue&issn=0006-3835&volume=1&issue=4; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "See erratum \cite{Froberg:1963:ERC}.", URL = "http://www.springerlink.com/openurl.asp?genre=article&issn=0006-3835&volume=1&issue=4&spage=256", acknowledgement = ack-nhfb, journal-URL = "http://link.springer.com/journal/10543", keywords = "elementary functions", } @Article{Gautschi:1961:RCR, author = "Walter Gautschi", title = "Recursive Computation of the Repeated Integrals of the Error Function", journal = j-MATH-COMPUT, volume = "15", number = "75", pages = "227--232", month = jul, year = "1961", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Gray:1961:BFI, author = "Marion C. Gray", title = "{Bessel} functions of integral order and complex argument", journal = j-CACM, volume = "4", number = "4", pages = "169--169", month = apr, year = "1961", CODEN = "CACMA2", DOI = "https://doi.org/10.1145/355578.366318", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", MRclass = "33.25", MRnumber = "27\#1636", bibdate = "Fri Nov 25 18:19:32 MST 2005", bibsource = "http://www.acm.org/pubs/contents/journals/cacm/; https://www.math.utah.edu/pub/tex/bib/cacm1960.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", abstract = "The FORTRAN II source language [1, 2] places rather severe restrictions on the form a subscript may take, primarily because of the manner in which indices are incremented in iterative loops. In the process of constructing a compiler for a medium-sized (8008-word memory) computer which will accept the FORTRAN II source language, it became clear that the ``recursive address calculation'' scheme, as used in the FORTRAN compilers to minimize object-program running time, was probably not the best one to use. This system, described in some detail by Samelson and Bauer [3], requires that the subscript expression be a linear function of the subscripting variable. The alternative, which requires complete evaluation of the ``storage mapping function'', is usually rejected because of the time required for the object program to perform the necessary address calculation.", acknowledgement = ack-nhfb, fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "Bessel functions; special functions", } @Article{Herndon:1961:ABF, author = "John R. Herndon", title = "{Algorithm 57}: {Ber} or {Bei} Function", journal = j-CACM, volume = "4", number = "4", pages = "181--181", month = apr, year = "1961", CODEN = "CACMA2", DOI = "https://doi.org/10.1145/355578.366476", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Fri Nov 25 18:19:32 MST 2005", bibsource = "http://www.acm.org/pubs/contents/journals/cacm/; https://www.math.utah.edu/pub/tex/bib/cacm1960.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "bei functions; ber functions; special functions", remark = "Fullerton: 20-line Algol procedure that only sums series.", } @Article{Herndon:1961:ACEa, author = "John R. Herndon", title = "{Algorithm 55}: {Complete} elliptic integral of the first kind", journal = j-CACM, volume = "4", number = "4", pages = "180--180", month = apr, year = "1961", CODEN = "CACMA2", DOI = "https://doi.org/10.1145/355578.366454", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Fri Nov 25 18:19:32 MST 2005", bibsource = "http://www.acm.org/pubs/contents/journals/cacm/; https://www.math.utah.edu/pub/tex/bib/cacm1960.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "special functions", } @Article{Herndon:1961:ACEb, author = "John R. Herndon", title = "{Algorithm 56}: {Complete} elliptic integral of the second kind", journal = j-CACM, volume = "4", number = "4", pages = "180--181", month = apr, year = "1961", CODEN = "CACMA2", DOI = "https://doi.org/10.1145/355578.366474", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Fri Nov 25 18:19:32 MST 2005", bibsource = "http://www.acm.org/pubs/contents/journals/cacm/; https://www.math.utah.edu/pub/tex/bib/cacm1960.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "special functions", } @Article{Herndon:1961:AEC, author = "John R. Herndon", title = "{Algorithm 46}: {Exponential} of a Complex Number", journal = j-CACM, volume = "4", number = "4", pages = "178--178", month = apr, year = "1961", CODEN = "CACMA2", DOI = "https://doi.org/10.1145/355578.366356", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Fri Nov 25 18:19:32 MST 2005", bibsource = "http://www.acm.org/pubs/contents/journals/cacm/; https://www.math.utah.edu/pub/tex/bib/cacm1960.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "$\exp(z)$; $e^z$; elementary functions", } @Article{Herndon:1961:AGF, author = "John R. Herndon", title = "{Algorithm 54}: {Gamma} function for range $1$ to $2$", journal = j-CACM, volume = "4", number = "4", pages = "180--180", month = apr, year = "1961", CODEN = "CACMA2", DOI = "https://doi.org/10.1145/355578.366453", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Fri Nov 25 18:19:32 MST 2005", bibsource = "http://www.acm.org/pubs/contents/journals/cacm/; https://www.math.utah.edu/pub/tex/bib/cacm1960.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "$\Gamma(x)$; special functions", } @Article{Herndon:1961:ASN, author = "J. R. Herndon", title = "{Algorithm 49}: {Spherical} {Neumann} Function", journal = j-CACM, volume = "4", number = "4", pages = "179--179", month = apr, year = "1961", CODEN = "CACMA2", DOI = "https://doi.org/10.1145/355578.355579", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Fri Nov 25 18:19:32 MST 2005", bibsource = "http://www.acm.org/pubs/contents/journals/cacm/; https://www.math.utah.edu/pub/tex/bib/cacm1960.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "See also \cite{Coleman:1978:RSN}.", acknowledgement = ack-nhfb, fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "Neumann functions; special functions", } @Book{Hochstadt:1961:SFM, author = "Harry Hochstadt", title = "Special functions of mathematical physics", publisher = pub-HRW, address = pub-HRW:adr, pages = "viii + 81", year = "1961", ISBN = "0-236-73011-8", ISBN-13 = "978-0-236-73011-7", LCCN = "QA351 H65 1961", bibdate = "Sat Oct 30 18:01:08 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, remark = "Reprinted 1966.", } @Article{L:1961:BRF, author = "Y. L. L.", title = "Book Review: {L. Fox, \booktitle{Tables of Weber Parabolic Cylinder Functions and Other Functions for Large Arguments}, National Physical Laboratory Mathematical Tables Volume 4, Her Majesty's Stationery Office, London, 1960, iii + 40 p., 28 cm. (Paperback)}", journal = j-MATH-COMPUT, volume = "15", number = "75", pages = "310--311", month = jul, year = "1961", CODEN = "MCMPAF", DOI = "https://doi.org/10.1090/S0025-5718-61-99215-8", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Wed Nov 15 11:07:36 2023", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/f/fox-leslie.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.jstor.org/stable/2002928; https://www.ams.org/journals/mcom/1961-15-075/S0025-5718-61-99215-8/S0025-5718-61-99215-8.pdf", acknowledgement = ack-nhfb, fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", remark = "Y. L. L. == Yudell L. Luke", subject-dates = "Leslie Fox (30 September 1918--1 August 1992)", } @Article{Landau:1961:PSW, author = "H. J. Landau and H. O. Pollak", title = "Prolate spheroidal wave functions, {Fourier} analysis and uncertainty. {II}", journal = j-BELL-SYST-TECH-J, volume = "40", number = "1", pages = "65--84", month = jan, year = "1961", CODEN = "BSTJAN", ISSN = "0005-8580", MRclass = "33.27", MRnumber = "0140733 (25 \#4147)", MRreviewer = "I. Marx", bibdate = "Tue Nov 9 11:15:54 MST 2010", bibsource = "http://bstj.bell-labs.com/oldfiles/year.1961/BSTJ.1961.4001.html; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://bstj.bell-labs.com/BSTJ/images/Vol40/bstj40-1-65.pdf", acknowledgement = ack-nhfb, fjournal = "The Bell System Technical Journal", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)1538-7305/issues/", } @Article{Luke:1961:EHF, author = "Yudell L. Luke and Richard L. Coleman", title = "Expansion of Hypergeometric Functions in Series of Other Hypergeometric Functions", journal = j-MATH-COMPUT, volume = "15", number = "75", pages = "233--237", month = jul, year = "1961", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp1960.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Rader:1961:CAC, author = "P. J. Rader and Henry C. {Thacher, Jr.}", title = "Certification of {Algorithm 14 [not 13]}: {Complex} exponential integral", journal = j-CACM, volume = "4", number = "2", pages = "105--105", month = feb, year = "1961", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Fri Nov 25 18:19:30 MST 2005", bibsource = "http://www.acm.org/pubs/contents/journals/cacm/; https://www.math.utah.edu/pub/tex/bib/cacm1960.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "$\Ei(z)$; special functions", remark = "Fullerton: It's accurate but sometimes slow.", } @Article{Slepian:1961:PSW, author = "D. Slepian and H. O. Pollak", title = "Prolate spheroidal wave functions, {Fourier} analysis and uncertainty. {I}", journal = j-BELL-SYST-TECH-J, volume = "40", number = "1", pages = "43--63", month = jan, year = "1961", CODEN = "BSTJAN", ISSN = "0005-8580", MRclass = "33.27", MRnumber = "0140732 (25 \#4146)", MRreviewer = "I. Marx", bibdate = "Tue Nov 9 11:15:54 MST 2010", bibsource = "http://bstj.bell-labs.com/oldfiles/year.1961/BSTJ.1961.4001.html; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://bstj.bell-labs.com/BSTJ/images/Vol40/bstj40-1-43.pdf", acknowledgement = ack-nhfb, fjournal = "The Bell System Technical Journal", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)1538-7305/issues/", } @Book{Sneddon:1961:SFM, author = "Ian Naismith Sneddon", title = "Special Functions of Mathematical Physics and Chemistry", publisher = "Oliver and Boyd", address = "Edinburgh, UK", edition = "Ssecond", pages = "184", year = "1961", LCCN = "QA331 .S65 1961", bibdate = "Sat Oct 30 21:22:03 MDT 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; z3950.loc.gov:7090/Voyager", series = "University mathematical texts", acknowledgement = ack-nhfb, remark = "See first edition \cite{Sneddon:1956:SFM} and third edition {Sneddon:1980:SFM}", } @Article{Spielberg:1961:ECF, author = "Kurt Spielberg", title = "Efficient Continued Fraction Approximations To Elementary Functions", journal = j-MATH-COMPUT, volume = "15", number = "76", pages = "409--417", month = oct, year = "1961", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "65.20", MRnumber = "MR0134842 (24 \#B894)", MRreviewer = "M. E. Rose", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "garbo.uwasa.fi:/pc/doc-soft/fpbiblio.txt; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nj # " and " # ack-nhfb, fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Thacher:1961:ISR, author = "Henry C. {Thacher, Jr.}", title = "Iterated Square Root Expansions for the Inverse Cosine and Inverse Hyperbolic Cosine", journal = j-MATH-COMPUT, volume = "15", number = "76", pages = "399--403", month = oct, year = "1961", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Weingarten:1961:TGC, author = "Harry Weingarten and A. R. {Di Donato}", title = "A Table of Generalized Circular Error", journal = j-MATH-COMPUT, volume = "15", number = "74", pages = "169--173", month = apr, year = "1961", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp1960.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Werner:1961:CAG, author = "Helmut Werner and Robert Collinge", title = "{Chebyshev} Approximations to the Gamma Function (in {Technical Notes and Short Papers})", journal = j-MATH-COMPUT, volume = "15", number = "74", pages = "195--197", month = apr, year = "1961", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Wojcicki:1961:ABF, author = "Maria E. Wojcicki", title = "{Algorithm 44}: {Bessel} Functions Computed Recursively", journal = j-CACM, volume = "4", number = "4", pages = "177--178", month = apr, year = "1961", CODEN = "CACMA2", DOI = "https://doi.org/10.1145/355578.366341", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Fri Nov 25 18:19:32 MST 2005", bibsource = "http://www.acm.org/pubs/contents/journals/cacm/; https://www.math.utah.edu/pub/tex/bib/cacm1960.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "Bessel functions; special functions", } @Article{Albrecht:1962:FKM, author = "J. Albrecht", title = "{Fehlerschranken und Konvergenzbeschleunigung bei einer monotonen oder alternierenden Iterationsfolge}. ({German}) [{Error} Bounds and Convergence Acceleration with a Monotone or Alternating Iteration Sequence]", journal = j-NUM-MATH, volume = "4", pages = "196--208", month = dec, year = "1962", CODEN = "NUMMA7", ISSN = "0029-599X (print), 0945-3245 (electronic)", ISSN-L = "0029-599X", bibdate = "Mon Oct 18 01:28:20 MDT 1999", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Numerische Mathematik", journal-URL = "http://link.springer.com/journal/211", keywords = "convergence acceleration", language = "German", } @Article{Arscott:1962:BRI, author = "F. M. Arscott", title = "Book Review: {{\booktitle{Introduction to Elliptic Functions with Applications}} (F. Bowman)}", journal = j-SIAM-REVIEW, volume = "4", number = "4", pages = "408--408", month = "????", year = "1962", CODEN = "SIREAD", DOI = "https://doi.org/10.1137/1004109", ISSN = "0036-1445 (print), 1095-7200 (electronic)", ISSN-L = "0036-1445", bibdate = "Thu Mar 27 09:04:56 MDT 2014", bibsource = "http://epubs.siam.org/toc/siread/4/4; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/siamreview.bib", acknowledgement = ack-nhfb, fjournal = "SIAM Review", journal-URL = "http://epubs.siam.org/sirev", onlinedate = "October 1962", } @TechReport{Blanch:1962:TRR, author = "Gertrude Blanch and Donald S. Clemm", title = "Tables Relating to the Radial {Mathieu} Functions, Vols. 1 \& 2", type = "Report", institution = "Aeronautical Research Labs. U. S. Government Printing Office", address = "Washington, DC, USA", year = "1962", bibdate = "Fri Oct 29 21:37:53 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, author-dates = "Gertrude Blanch (1897--1996)", citedby = "Fullerton:1980:BEM", remark = "Fullerton: 7-place tables.", } @Article{Boersma:1962:CLF, author = "J. Boersma", title = "On the Computation of {Lommel}'s Functions of Two Variables (in {Technical Notes and Short Papers})", journal = j-MATH-COMPUT, volume = "16", number = "78", pages = "232--238", month = apr, year = "1962", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", remark = "Fullerton: Lommel's functions are infinite sums involving Bessel functions $ J_{\nu - 2m}(x) $.", } @Article{Cantor:1962:LEF, author = "D. Cantor and G. Estrin and R. Turn", title = "Logarithmic and Exponential Function Evaluation in a Variable Structure Digital Computer", journal = j-IRE-TRANS-ELEC-COMPUT, volume = "EC-11", number = "2", pages = "155--164", month = apr, year = "1962", CODEN = "IRELAO", DOI = "https://doi.org/10.1109/TEC.1962.5219348", ISSN = "0367-9950", ISSN-L = "0367-9950", bibdate = "Thu Jul 14 09:11:49 MDT 2011", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", URL = "https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5219348", acknowledgement = ack-nj # "\slash " # ack-nhfb, fjournal = "IRE Transactions on Electronic Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=5407885", } @Article{Christiansen:1962:APC, author = "S{\o}ren Christiansen", title = "{Algol} Programming: Contribution no. 3: Calculation of complementary {Fresnel} integrals", journal = j-NORDISK-TIDSKR-INFORM-BEHAND, volume = "2", number = "3", pages = "192--194", year = "1962", CODEN = "BITTEL, NBITAB", ISSN = "0006-3835 (print), 1572-9125 (electronic)", ISSN-L = "0006-3835", bibdate = "Mon Nov 16 14:34:20 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/bit.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", journal-URL = "http://link.springer.com/journal/10543", remark = "Fullerton: A 50-line Algol procedure is given. It calculates $ \int_x^\infty \cos (t) / \sqrt {2 \pi t} \, d t $ and $ \int_x^\infty \sin (t) / \sqrt {2 \pi t} \, d t $.", } @Article{Cundiff:1962:AEA, author = "John L. Cundiff", title = "{Algorithm 88}: {Evaluation} of Asymptotic Expression for the {Fresnel} Sine and Cosine Integrals", journal = j-CACM, volume = "5", number = "5", pages = "280--280", month = may, year = "1962", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Fri Nov 25 18:19:38 MST 2005", bibsource = "http://www.acm.org/pubs/contents/journals/cacm/; https://www.math.utah.edu/pub/tex/bib/cacm1960.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "C(x); S(x); special functions", remark = "Fullerton: Can be used with algorithms 89 and 90. 30-line Algol procedure.", } @Article{Cundiff:1962:AEFa, author = "John L. Cundiff", title = "{Algorithm 89}: {Evaluation} of the {Fresnel} Sine Integral", journal = j-CACM, volume = "5", number = "5", pages = "280--280", month = may, year = "1962", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Fri Nov 25 18:19:38 MST 2005", bibsource = "http://www.acm.org/pubs/contents/journals/cacm/; https://www.math.utah.edu/pub/tex/bib/cacm1960.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "S(x); special functions", remark = "Fullerton: 20-line Algol procedure that must be used with algorithm 88.", } @Article{Cundiff:1962:AEFb, author = "John L. Cundiff", title = "{Algorithm 90}: {Evaluation} of the {Fresnel} Cosine Integral", journal = j-CACM, volume = "5", number = "5", pages = "281--281", month = may, year = "1962", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Fri Nov 25 18:19:38 MST 2005", bibsource = "http://www.acm.org/pubs/contents/journals/cacm/; https://www.math.utah.edu/pub/tex/bib/cacm1960.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "C(x); special functions", remark = "Fullerton: 20-line Algol procedure that must be used with algorithm 88.", } @Article{DiDonato:1962:MCC, author = "A. R. DiDonato and M. P. Jarnagin", title = "A Method for Computing the Circular Coverage Function", journal = j-MATH-COMPUT, volume = "16", number = "79", pages = "347--355", month = jul, year = "1962", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp1960.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @TechReport{DiDonato:1962:MCG, author = "A. R. DiDonato and M. P. Jamagin", title = "A Method for Computing the Generalized Circular Error Function and the Circular Coverage Function", type = "NWL Report", number = "1768", institution = "Naval Surface Weapons Center", address = "Dahlgren, VA 22448, USA", day = "23", month = jan, year = "1962", bibdate = "Wed Nov 12 15:56:35 2014", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, } @Article{Dorn:1962:GHR, author = "William S. Dorn", title = "Generalizations of {Horner}'s rule for polynomial evaluation", journal = j-IBM-JRD, volume = "6", number = "2", pages = "239--245", month = apr, year = "1962", CODEN = "IBMJAE", DOI = "https://doi.org/10.1147/rd.62.0239", ISSN = "0018-8646 (print), 2151-8556 (electronic)", ISSN-L = "0018-8646", MRclass = "65.50", MRnumber = "24 \#B2541", bibdate = "Tue Sep 11 16:10:28 MDT 2012", bibsource = "http://www.research.ibm.com/journal/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/ibmjrd.bib", URL = "https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5392358", ZMnumber = "128.37202", acknowledgement = ack-nhfb, book-URL = "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=5288520", fjournal = "IBM Journal of Research and Development", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=5288520", reviewer = "D. H. Lehmer", } @Article{Fraser:1962:CRA, author = "W. Fraser and J. F. Hart", title = "On the computation of rational approximations to continuous functions", journal = j-CACM, volume = "5", number = "7", pages = "401--403", month = jul, year = "1962", CODEN = "CACMA2", DOI = "https://doi.org/10.1145/368273.368578", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Fri Nov 25 18:19:39 MST 2005", bibsource = "http://www.acm.org/pubs/contents/journals/cacm/; https://www.math.utah.edu/pub/tex/bib/cacm1960.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "$\abs(x)$; $\cos(x)$; $\Gamma(1+x)$; $\sin(x)$; elementary functions; Remes algorithm; special functions", remark = "This paper outlines the Remes algorithm for ``for finding polynomial approximations to the determination of `best' rational approximations.''. It also gives approximations for starting values of Newton--Raphson iterations for $ \abs (x) $, $ \cos (x) $, $ \Gamma (1 + x) $, and $ \sin (x) $.", } @Article{Hansen:1962:SRV, author = "Eldon R. Hansen and Merrell L. Patrick", title = "Some Relations and Values for the Generalized {Riemann} Zeta Function", journal = j-MATH-COMPUT, volume = "16", number = "79", pages = "265--274", month = jul, year = "1962", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Jefferson:1962:RAI, author = "David K. Jefferson", title = "Remark on {Algorithm 73}: {Incomplete} elliptic integrals", journal = j-CACM, volume = "5", number = "10", pages = "514--514", month = oct, year = "1962", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Fri Nov 25 18:19:41 MST 2005", bibsource = "http://www.acm.org/pubs/contents/journals/cacm/; https://www.math.utah.edu/pub/tex/bib/cacm1960.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "special functions", } @Article{Knuth:1962:ECP, author = "Donald E. Knuth", title = "{Euler}'s Constant to $ 1271 $ Places", journal = j-MATH-COMPUT, volume = "16", number = "79", pages = "275--281", month = jul, year = "1962", CODEN = "MCMPAF", DOI = "https://doi.org/10.1090/S0025-5718-1962-0148255-X", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "10.41", MRnumber = "26 #5763", bibdate = "Fri Mar 22 18:03:29 MST 2002", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp1960.bib; JSTOR database; MathSciNet database", acknowledgement = ack-nhfb, fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Knuth:1962:EPC, author = "Donald E. Knuth", title = "Evaluation of polynomials by computer", journal = j-CACM, volume = "5", number = "12", pages = "595--599", month = dec, year = "1962", CODEN = "CACMA2", DOI = "https://doi.org/10.1145/355580.369074", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", MRclass = "68.00 (12.00)", MRnumber = "27 #970", bibdate = "Thu Dec 08 11:11:03 2005", bibsource = "http://www.acm.org/pubs/contents/journals/cacm/; https://www.math.utah.edu/pub/bibnet/subjects/acc-stab-num-alg.bib; https://www.math.utah.edu/pub/tex/bib/cacm1960.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; MathSciNet database", note = "See letter \cite{Knuth:1963:LEE}.", acknowledgement = ack-nhfb, fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", remark = "The author reports that Motzkin (1962) showed that Horner's rule for polynomial evaluation may not be optimal, and develops the idea further for arbitrary polynomials, but also observes that the coefficients of the revised polynomials may be difficult to find. He also asks about, but does not answer, the question of error analysis of the various methods.", } @Article{Landau:1962:PSW, author = "H. J. Landau and H. O. Pollak", title = "Prolate spheroidal wave functions, {Fourier} analysis and uncertainty. {III}. {The} dimension of the space of essentially time- and band-limited signals", journal = j-BELL-SYST-TECH-J, volume = "41", number = "4", pages = "1295--1336", month = jul, year = "1962", CODEN = "BSTJAN", ISSN = "0005-8580", MRclass = "33.28 (94.10)", MRnumber = "0147686 (26 \#5200)", MRreviewer = "I. Marx", bibdate = "Tue Nov 9 11:15:54 MST 2010", bibsource = "http://bstj.bell-labs.com/oldfiles/year.1962/BSTJ.1962.4104.html; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://bstj.bell-labs.com/BSTJ/images/Vol41/bstj41-4-1295.pdf", acknowledgement = ack-nhfb, fjournal = "The Bell System Technical Journal", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)1538-7305/issues/", } @Article{Meggitt:1962:PDP, author = "J. E. Meggitt", title = "Pseudo Division and Pseudo Multiplication Processes", journal = j-IBM-JRD, volume = "6", number = "2", pages = "210--226", month = apr, year = "1962", CODEN = "IBMJAE", DOI = "https://doi.org/10.1147/rd.62.0210", ISSN = "0018-8646 (print), 2151-8556 (electronic)", ISSN-L = "0018-8646", bibdate = "Thu Sep 1 10:15:31 1994", bibsource = "http://www.research.ibm.com/journal/; https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ibmjrd.bib", URL = "https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5392356", ZMnumber = "201.48709", acknowledgement = ack-nhfb, ajournal = "IBM J. Res. Develop.", book-URL = "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=5288520", fjournal = "IBM Journal of Research and Development", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=5288520", keywords = "binary arithmetic; decimal arithmetic; divide; exponential; inverse square root; inverse tangent; logarithm; multiply; square root; tangent", remark = "This paper shows how common digit-at-a-time hardware circuitry can be used to compute several different functions, with the only difference among the functions being initial conditions and stored tables of particular constants, with an error than does not exceed 3 units in the lowest-order decimal digit of the result (assuming base-10 arithmetic). The steps are similar to those in CORDIC algorithms. If M is the time for one multiplication, then other functions are found in these times: cosine (7M), division (3M?), exponential (3M), inverse tangent (3M), logarithm (3M), sine (7M), square root (3M?), and tangent (4M). For $n$-digit numbers, the calculation requires two registers, one of length $ n + 2 $ digits, and the other of length $ 2 n + 2 $ digits.", } @Article{Merner:1962:AAC, author = "J. N. Merner", title = "{ACM Algorithm 149}: Complete Elliptic Integral", journal = j-CACM, volume = "5", number = "12", pages = "605", month = dec, year = "1962", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Thu Sep 08 09:47:50 1994", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "See also \cite{Skovgaard:1978:RCE}.", acknowledgement = ack-nhfb, fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", } @TechReport{Senzig:1962:DDG, author = "D. N. Senzig", title = "Digit-By-Digit Generation of the Trigonometric and Hyperbolic Functions", type = "IBM Research Report", number = "RC-860", institution = pub-IBM, address = pub-IBM:adr, day = "17", month = dec, year = "1962", bibdate = "Mon Nov 10 10:33:33 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, } @Article{Thacher:1962:CAB, author = "Henry C. {Thacher, Jr.}", title = "Certification of {Algorithm 57}: {$ \operatorname {ber} $} or {$ \operatorname {bei} $} function", journal = j-CACM, volume = "5", number = "8", pages = "438--438", month = aug, year = "1962", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Fri Nov 25 18:19:40 MST 2005", bibsource = "http://www.acm.org/pubs/contents/journals/cacm/; https://www.math.utah.edu/pub/tex/bib/cacm1960.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "bei functions; ber functions; special functions", remark = "Fullerton: The algorithm is inaccurate for large $x$.", } @Article{Thacher:1962:CAH, author = "Henry C. {Thacher, Jr.}", title = "Certification of {Algorithms 191 and 192, Hypergeometric and Confluent Hypergeometric Functions}", journal = j-CACM, volume = "7", number = "4", pages = "244--244", month = apr, year = "1962", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Sat Oct 30 11:27:28 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", remark = "Fullerton: An error in a comment is noted.", } @Article{Wimp:1962:PEB, author = "Jet Wimp", title = "Polynomial Expansions of {Bessel} Functions and Some Associated Functions", journal = j-MATH-COMPUT, volume = "16", number = "80", pages = "446--458", month = oct, year = "1962", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp1960.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp1970.bib; JSTOR database", note = "See corrigendum \cite{Wimp:1972:CPE}.", acknowledgement = ack-nhfb, fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Wynn:1962:AAP, author = "P. Wynn", title = "An Arsenal of {ALGOL} Procedures for Complex Arithmetic", journal = j-NORDISK-TIDSKR-INFORM-BEHAND, volume = "2", number = "4", pages = "232--255", month = dec, year = "1962", CODEN = "BITTEL, NBITAB", DOI = "https://doi.org/10.1007/BF01940171", ISSN = "0006-3835 (print), 1572-9125 (electronic)", ISSN-L = "0006-3835", bibdate = "Wed Jan 4 18:52:07 MST 2006", bibsource = "http://springerlink.metapress.com/openurl.asp?genre=issue&issn=0006-3835&volume=2&issue=4; https://www.math.utah.edu/pub/tex/bib/bit.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "http://www.springerlink.com/openurl.asp?genre=article&issn=0006-3835&volume=2&issue=4&spage=232", acknowledgement = ack-nhfb, journal-URL = "http://link.springer.com/journal/10543", keywords = "ALGOL; complex arithmetic; confluence hypergeometric function; continued fractions; incomplete beta function; incomplete gamma function; Stieltjes $S$-fractions; Weber parabolic cylinder function", remark = "Cited in \cite{Sterbenz:1974:FPC}.", } @Article{Barakat:1963:CES, author = "Richard Barakat and Agnes Houston", title = "{Chebyschev} Expansion of the Sine and Cosine Integrals", journal = j-J-MATH-PHYS-MIT, volume = "42", number = "1--4", pages = "331--333", month = apr, year = "1963", CODEN = "JMPHA9", DOI = "https://doi.org/10.1002/sapm1963421331", ISSN = "0097-1421", ISSN-L = "0097-1421", bibdate = "Sat Aug 19 13:36:07 MDT 2023", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jmathphysmit.bib", URL = "https://onlinelibrary.wiley.com/doi/epdf/10.1002/sapm1963421331", acknowledgement = ack-nhfb, ajournal = "J. Math. Phys. (MIT)", fjournal = "Journal of Mathematics and Physics (MIT)", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1111/(ISSN)1467-9590", onlinedate = "April 1963", } @Article{Burgoyne:1963:AKF, author = "F. D. Burgoyne", title = "Approximations to {Kelvin} Functions (in {Technical Notes and Short Papers})", journal = j-MATH-COMPUT, volume = "17", number = "83", pages = "295--298", month = jul, year = "1963", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", remark = "Fullerton: 9-digit approximations.", } @Article{Carlitz:1963:IEF, author = "L. Carlitz", title = "The Inverse of the Error Function", journal = j-PAC-J-MATH, volume = "13", number = "2", pages = "459--470", year = "1963", CODEN = "PJMAAI", ISSN = "0030-8730 (print), 1945-5844 (electronic)", ISSN-L = "0030-8730", bibdate = "Thu Sep 13 21:30:05 2007", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://pjm.math.berkeley.edu/pjm; http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?handle=euclid.pjm/1103035736&view=body&content-type=pdf_1", acknowledgement = ack-nhfb, fjournal = "Pacific Journal of Mathematics", journal-URL = "http://msp.org/pjm", } @Article{Clenshaw:1963:ASF, author = "C. W. Clenshaw and G. F. Miller and M. Woodger", title = "Algorithms for Special Functions {I}", journal = j-NUM-MATH, volume = "4", pages = "403--419", month = dec, year = "1963", CODEN = "NUMMA7", ISSN = "0029-599X (print), 0945-3245 (electronic)", ISSN-L = "0029-599X", bibdate = "Fri Sep 16 10:21:31 1994", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nj, citedby = "Fullerton:1980:BEM", fjournal = "Numerische Mathematik", journal-URL = "http://link.springer.com/journal/211", remark = "Fullerton: ALGOL 60 procedures with accuracy $ \approx 10^{-15} $ for $ \exp $, $ \ln $, $ \sin $, $ \cos $, $ \tan $, $ \arcsin $, $ \arctan $, $ \gamma $, $ \Ei $ and $ \erf $. See G. F. Miller (1965) for corrections.", } @Article{Eisman:1963:PER, author = "S. H. Eisman", title = "Polynomial Evaluation Revisited", journal = j-CACM, volume = "6", number = "7", pages = "384--385", month = jul, year = "1963", CODEN = "CACMA2", DOI = "https://doi.org/10.1145/366663.366668", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Fri Nov 25 18:19:47 MST 2005", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; http://www.acm.org/pubs/contents/journals/cacm/; https://www.math.utah.edu/pub/tex/bib/cacm1960.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", } @Article{Erdelyi:1963:AEC, author = "A. Erd{\'e}lyi and M. Wyman", title = "The asymptotic evaluation of certain integrals", journal = j-ARCH-RAT-MECH-ANAL, volume = "14", number = "1", pages = "217--260", month = jan, year = "1963", CODEN = "AVRMAW", DOI = "https://doi.org/10.1007/BF00250704", ISSN = "0003-9527 (print), 1432-0673 (electronic)", ISSN-L = "0003-9527", bibdate = "Sat Feb 18 14:53:08 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://link.springer.com/article/10.1007/BF00250704", acknowledgement = ack-nhfb, fjournal = "Archive for rational mechanics and analysis", journal-URL = "http://link.springer.com/journal/205", keywords = "incomplete gamma function", } @Article{Eve:1963:SAI, author = "J. Eve", title = "Starting Approximations for the Iterative Calculation of Square Roots", journal = j-COMP-J, volume = "6", number = "3", pages = "274--276", month = nov, year = "1963", CODEN = "CMPJA6", DOI = "https://doi.org/10.1093/comjnl/6.3.274", ISSN = "0010-4620 (print), 1460-2067 (electronic)", ISSN-L = "0010-4620", bibdate = "Tue Dec 4 14:47:30 MST 2012", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; http://comjnl.oxfordjournals.org/content/6/3.toc; http://www3.oup.co.uk/computer_journal/hdb/Volume_06/Issue_03/; https://www.math.utah.edu/pub/tex/bib/compj1960.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", abstract = "Several starting approximations are given which, in conjunction with a well-known iterative process, lead to square root approximations, with a relative error in the range $ (2^{-55}, 2^{-45}) $, at the expense of three divisions. More accurate approximations are given which require in addition a single multiplication.", acknowledgement = ack-nhfb # " and " # ack-nj, fjournal = "The Computer Journal", journal-URL = "http://comjnl.oxfordjournals.org/", } @Article{Fettis:1963:AMH, author = "Henry E. Fettis", title = "{Algorithm 163}: {Modified} {Hankel} function", journal = j-CACM, volume = "6", number = "4", pages = "161--162", month = apr, year = "1963", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Fri Nov 25 18:19:46 MST 2005", bibsource = "http://www.acm.org/pubs/contents/journals/cacm/; https://www.math.utah.edu/pub/tex/bib/cacm1960.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "Hankel functions; special functions", remark = "Fullerton: A 25-line Algol procedure for $ e^x K_p(x) $.", } @Article{Froberg:1963:APC, author = "Carl-Erik Fr{\"o}berg", title = "{Algol} Programming: Contribution no. 5: Computation of the {Fermi} function", journal = j-NORDISK-TIDSKR-INFORM-BEHAND, volume = "3", number = "2", pages = "141--142", year = "1963", CODEN = "BITTEL, NBITAB", ISSN = "0006-3835 (print), 1572-9125 (electronic)", ISSN-L = "0006-3835", bibdate = "Mon Nov 16 14:36:22 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/bit.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", journal-URL = "http://link.springer.com/journal/10543", remark = "Fullerton: The Fermi function depends on several physical parameters of the atomic nucleus.", } @Article{Froberg:1963:ERC, author = "C.-E. Fr{\"o}berg", title = "Erratum: {``Rational Chebyshev Approximations of Elementary Functions'' [BIT {\bf 1}(4), 1961, p. 261, line 12]}", journal = j-NORDISK-TIDSKR-INFORM-BEHAND, volume = "3", number = "1", pages = "68--68", month = mar, year = "1963", CODEN = "BITTEL, NBITAB", DOI = "https://doi.org/10.1007/BF01963538", ISSN = "0006-3835 (print), 1572-9125 (electronic)", ISSN-L = "0006-3835", bibdate = "Wed Jan 4 18:52:07 MST 2006", bibsource = "http://springerlink.metapress.com/openurl.asp?genre=issue&issn=0006-3835&volume=3&issue=1; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "See \cite{Froberg:1961:RCA}.", URL = "http://www.springerlink.com/openurl.asp?genre=article&issn=0006-3835&volume=3&issue=1&spage=68", acknowledgement = ack-nhfb, journal-URL = "http://link.springer.com/journal/10543", keywords = "elementary functions", } @InProceedings{Gautschi:1963:RCS, author = "Walter Gautschi", editor = "????", booktitle = "{The University of Michigan Engineering Summer Conferences, Numerical Analysis, Summer 1963}", title = "Recursive computation of special functions", publisher = "????", address = "????", pages = "??--??", year = "1963", bibdate = "Fri Aug 21 11:08:35 2020", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/g/gautschi-walter.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, } @Article{Gray:1963:AFI, author = "Malcolm D. Gray", title = "{Algorithm 213}: {Fresnel} Integrals", journal = j-CACM, volume = "6", number = "10", pages = "617--617", month = oct, year = "1963", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Fri Nov 25 18:19:49 MST 2005", bibsource = "http://www.acm.org/pubs/contents/journals/cacm/; https://www.math.utah.edu/pub/tex/bib/cacm1960.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "See certification \cite{Gray:1964:CAF} and related remark \cite{Gray:1963:RAE}.", acknowledgement = ack-nhfb, fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "C(x); S(x); special functions", } @Article{Gray:1963:RAE, author = "Malcolm D. Gray", title = "Remark on Algorithms 88, 89, and 90 evaluation of the {Fresnel} integrals", journal = j-CACM, volume = "6", number = "10", pages = "618--618", month = oct, year = "1963", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Fri Nov 25 18:19:49 MST 2005", bibsource = "http://www.acm.org/pubs/contents/journals/cacm/; https://www.math.utah.edu/pub/tex/bib/cacm1960.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "C(x); S(x); special functions", remark = "Fullerton: An error is noted.", } @Article{Hofsommer:1963:NCE, author = "D. J. Hofsommer and R. van de Riet", title = "On the numerical calculation of elliptic integrals of the first and second kind and the elliptic functions of {Jacobi}", journal = j-NUM-MATH, volume = "5", pages = "291--302", month = dec, year = "1963", CODEN = "NUMMA7", ISSN = "0029-599X (print), 0945-3245 (electronic)", ISSN-L = "0029-599X", bibdate = "Mon Oct 18 01:28:20 MDT 1999", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Numerische Mathematik", journal-URL = "http://link.springer.com/journal/211", } @Article{Ibbetson:1963:AG, author = "D. Ibbetson", title = "{Algorithm 209}: {Gauss}", journal = j-CACM, volume = "6", number = "10", pages = "616--616", month = oct, year = "1963", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Fri Nov 25 18:19:49 MST 2005", bibsource = "http://www.acm.org/pubs/contents/journals/cacm/; https://www.math.utah.edu/pub/tex/bib/cacm1960.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", } @Article{Knuth:1963:LEE, author = "Donald E. Knuth", title = "Letter to the {Editor}: {Evaluation} of polynomials by computer", journal = j-CACM, volume = "6", number = "2", pages = "51--51", month = feb, year = "1963", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Tue Dec 26 16:31:38 1995", bibsource = "https://www.math.utah.edu/pub/tex/bib/cacm1960.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "See \cite{Knuth:1962:EPC}.", acknowledgement = ack-nhfb, fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", } @Article{Lee-Whiting:1963:EFC, author = "G. E. Lee-Whiting", title = "Erratum: ``{Formulas} for Computing Incomplete Elliptic Integrals of the First and Second Kinds''", journal = j-J-ACM, volume = "10", pages = "412--412", year = "1963", CODEN = "JACOAH", ISSN = "0004-5411 (print), 1557-735X (electronic)", ISSN-L = "0004-5411", bibdate = "Sat Dec 10 15:59:26 1994", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "See \cite{Lee-Whiting:1963:FCI}.", acknowledgement = ack-nhfb, ajournal = "J. Assoc. Comput. Mach.", fjournal = "Journal of the ACM", journal-URL = "https://dl.acm.org/loi/jacm", xxmonth = "none", xxnumber = "none", } @Article{Lee-Whiting:1963:FCI, author = "G. E. Lee-Whiting", title = "Formulas for Computing Incomplete Elliptic Integrals of the First and Second Kinds", journal = j-J-ACM, volume = "10", number = "2", pages = "126--130", month = apr, year = "1963", CODEN = "JACOAH", ISSN = "0004-5411 (print), 1557-735X (electronic)", ISSN-L = "0004-5411", bibdate = "Sat Nov 05 22:55:28 1994", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "See also \cite{Lee-Whiting:1963:EFC}.", acknowledgement = ack-nhfb, ajournal = "J. Assoc. Comput. Mach.", fjournal = "Journal of the ACM", journal-URL = "https://dl.acm.org/loi/jacm", } @Book{Ljusternik:1963:MVF, author = "L. A. Ljusternik and O. A. {\v{C}}ervonenkis and A. R. Janpol{\'s}ki{{\u{\i}}}", title = "{{\cyr Matematicheski{\u{\i}}analiz. Vychislenie {\`e}lementarnykh funktsi{\u{\i}}}}. [Mathematical analysis. {Computation} of the elementary functions]", publisher = "Gosudarstv. Izdat. Fiz-Mat. Lit.", address = "Moscow, USSR", pages = "247", year = "1963", MRclass = "65.25", MRnumber = "28 \#1733", MRreviewer = "John Todd", bibdate = "Sat Jan 11 17:44:01 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, } @Article{Ludwig:1963:AIB, author = "Oliver G. Ludwig", title = "{Algorithm 179}: {Incomplete} beta ratio", journal = j-CACM, volume = "6", number = "6", pages = "314--314", month = jun, year = "1963", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Fri Nov 25 18:19:47 MST 2005", bibsource = "http://www.acm.org/pubs/contents/journals/cacm/; https://www.math.utah.edu/pub/tex/bib/cacm1960.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "See remark \cite{Pike:1976:RIB,Bosten:1974:RAI}.", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", remark = "Fullerton: This algorithm is the basis for a modern treatment by Bosten and Battiste.", } @Article{Meyer:1963:CAI, author = "Noelle A. Meyer", title = "Certification of {Algorithm 73}: {Incomplete} elliptic integrals", journal = j-CACM, volume = "6", number = "2", pages = "69--69", month = feb, year = "1963", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Fri Nov 25 18:19:44 MST 2005", bibsource = "http://www.acm.org/pubs/contents/journals/cacm/; https://www.math.utah.edu/pub/tex/bib/cacm1960.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "special functions", } @Article{Newman:1963:ICS, author = "J. N. Newman and W. Frank", title = "An Integral Containing the Square of a {Bessel} Function", journal = j-MATH-COMPUT, volume = "17", number = "81", pages = "64--70", month = jan, year = "1963", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", remark = "Fullerton: The integral $ I_n^m(x) = \int_0^{\pi / 2} \frac {J_n^2(x \cos (\theta))}{(x \cos (\theta))^{2m}} \, d \theta $, where $m$ and $n$ are either integers or half integers, is considered.", } @Article{Peuizulaev:1963:AEF, author = "{\v{S}}. I. Pe{\u{\i}}zulaev", title = "An approximation by elementary functions. ({Russian})", journal = "{\v{Z}}. Vy{\v{c}}isl. Mat. i Mat. Fiz.", volume = "3", pages = "769--770", year = "1963", MRclass = "41.30", MRnumber = "28 \#398", MRreviewer = "D. D. Stancu", bibdate = "Sat Jan 11 17:44:01 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, language = "Russian", } @Article{Relph:1963:AAH, author = "A. P. Relph", title = "{ACM Algorithm 191}: Hypergeometric", journal = j-CACM, volume = "6", number = "7", pages = "388--389", month = jul, year = "1963", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Thu Sep 08 09:32:02 1994", bibsource = "https://www.math.utah.edu/pub/tex/bib/cacm1970.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "See certification \cite{Koppelaar:1974:CRA}.", acknowledgement = ack-nhfb, fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", } @Article{Relph:1963:ACH, author = "A. P. Relph", title = "{Algorithm 192}: {Confluent} hypergeometric", journal = j-CACM, volume = "6", number = "7", pages = "388--388", month = jul, year = "1963", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Fri Nov 25 18:19:47 MST 2005", bibsource = "http://www.acm.org/pubs/contents/journals/cacm/; https://www.math.utah.edu/pub/tex/bib/cacm1960.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", remark = "Fullerton: 30-line Algol procedure for complex args. The work of Luke supersedes this.", } @Article{Relph:1963:AH, author = "A. P. Relph", title = "{Algorithm 191}: {Hypergeometric}", journal = j-CACM, volume = "6", number = "7", pages = "388--388", month = jul, year = "1963", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Fri Nov 25 18:19:47 MST 2005", bibsource = "http://www.acm.org/pubs/contents/journals/cacm/; https://www.math.utah.edu/pub/tex/bib/cacm1960.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "See certification \cite{Koppelaar:1974:CRA}.", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", remark = "Fullerton: 30-line Algol procedure. The work of Luke is better.", } @Article{Rosenberg:1963:FTP, author = "R. M. Rosenberg", title = "The {$ {\rm Ateb}(h) $}-functions and their properties", journal = j-QUART-APPL-MATH, volume = "21", pages = "37--47", year = "1963", CODEN = "QAMAAY", DOI = "https://doi.org/10.1090/qam/143948", ISSN = "0033-569X (print), 1552-4485 (electronic)", ISSN-L = "0033-569X", MRclass = "33.15", MRnumber = "143948", bibdate = "Fri Mar 7 10:54:26 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, ajournal = "Q. Appl. Math.", fjournal = "Quarterly of Applied Mathematics", journal-URL = "http://dl.acm.org/citation.cfm?id=J641; http://www.ams.org/journals/qam", remark = "See \cite{Kowalenko:2024:AVT,Dronyuk:2025:ACG}.", } @Article{Rutishauser:1963:BQG, author = "H. Rutishauser", title = "{Betrachtungen zur Quadratwurzeliteration}. ({German}) [{Considerations} on square root iteration]", journal = j-MONAT-MATH, volume = "67", pages = "452--464", year = "1963", CODEN = "MNMTA2", DOI = "https://doi.org/10.1007/BF01295091", ISSN = "0026-9255 (print), 1436-5081 (electronic)", ISSN-L = "0026-9255", MRclass = "65.50", MRnumber = "158532", MRreviewer = "A. S. Householder", bibdate = "Mon Aug 24 21:56:15 2020", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/r/rutishauser-heinz.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, author-dates = "Heinz Rutishauser (30 January 1918--10 November 1970)", fjournal = "Monatshefte f{\"u}r Mathematik", journal-URL = "http://link.springer.com/journal/605", language = "German", } @Book{Sneddon:1963:SFM, author = "Ian Naismith Sneddon", title = "{Spezielle Funktionen der mathematischen Physik und Chemie. Mathematische Formelsammlung II}. ({German}) [Special {functions} of mathematical physics and chemistry. {Mathematical} formula collection {II}]", volume = "54", publisher = pub-BIBLIO-INST, address = pub-BIBLIO-INST:adr, pages = "166", year = "1963", LCCN = "QA351 .S6415", bibdate = "Sat Oct 30 21:22:03 MDT 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; z3950.loc.gov:7090/Voyager", series = "B. I.-Hochschultaschenb{\"u}cher", acknowledgement = ack-nhfb, language = "German", remark = "German translation of \cite{Sneddon:1961:SFM}.", subject = "Functions; Mathematical physics", } @Article{Stern:1963:CSR, author = "T. E. Stern and R. M. Lerner", title = "A circuit for the square root of the sum of the squares", journal = j-PROC-IEEE, volume = "51", number = "4", pages = "593--596", month = apr, year = "1963", CODEN = "IEEPAD", ISSN = "0018-9219 (print), 1558-2256 (electronic)", ISSN-L = "0018-9219", bibdate = "Sat Jul 16 15:28:13 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, fjournal = "Proceedings of the IEEE", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=5", summary = "A piecewise-linear network can produce an output proportional to the square root of the sum of the squares of a set of input voltages, using resistors and diodes alone. The required relationship between voltages can be represented by a multi- \ldots{}", } @Article{Sweeney:1963:CEC, author = "Dura W. Sweeney", title = "On the Computation of {Euler}'s Constant", journal = j-MATH-COMPUT, volume = "17", number = "82", pages = "170--178", month = apr, year = "1963", CODEN = "MCMPAF", DOI = "https://doi.org/10.1090/S0025-5718-1963-0160308-X", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp1960.bib; JSTOR database", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", remark = "Fullerton: 3566 digits should be enough.", } @Article{Thacher:1963:ACEa, author = "Henry C. {Thacher, Jr.}", title = "{Algorithm 165}: {Complete} elliptic integrals", journal = j-CACM, volume = "6", number = "4", pages = "163--164", month = apr, year = "1963", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Fri Nov 25 18:19:46 MST 2005", bibsource = "http://www.acm.org/pubs/contents/journals/cacm/; https://www.math.utah.edu/pub/tex/bib/cacm1960.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "special functions", } @Article{Thacher:1963:CACa, author = "Henry C. {Thacher, Jr.}", title = "Certification of {Algorithm 55}: {Complete} elliptic integral of the first kind", journal = j-CACM, volume = "6", number = "4", pages = "166--167", month = apr, year = "1963", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Fri Nov 25 18:19:46 MST 2005", bibsource = "http://www.acm.org/pubs/contents/journals/cacm/; https://www.math.utah.edu/pub/tex/bib/cacm1960.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "special functions", } @Article{Thacher:1963:CACb, author = "Henry C. {Thacher, Jr.}", title = "Certification of {Algorithm 149}: {Complete} elliptic integral", journal = j-CACM, volume = "6", number = "4", pages = "166--167", month = apr, year = "1963", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Fri Nov 25 18:19:46 MST 2005", bibsource = "http://www.acm.org/pubs/contents/journals/cacm/; https://www.math.utah.edu/pub/tex/bib/cacm1960.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "special functions", } @Book{Tocher:1963:AS, author = "K. D. Tocher", title = "The Art of Simulation", publisher = "Van Nostrand", address = "Princeton, NJ, USA", pages = "viii + 184", year = "1963", LCCN = "TA177 .T6", bibdate = "Sat Dec 16 17:47:12 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; z3950.loc.gov:7090/Voyager", acknowledgement = ack-nhfb, subject = "Simulation methods", } @Article{vandeRiet:1963:CAI, author = "R. P. van de Riet", title = "Certification of {Algorithm 73}: {Incomplete} elliptic integrals", journal = j-CACM, volume = "6", number = "4", pages = "167--167", month = apr, year = "1963", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Fri Nov 25 18:19:46 MST 2005", bibsource = "http://www.acm.org/pubs/contents/journals/cacm/; https://www.math.utah.edu/pub/tex/bib/cacm1960.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "special functions", } @Article{Werner:1963:AFI, author = "H. Werner and G. Raymann", title = "An Approximation to the {Fermi} Integral {$ F_{1 / 2}(x) $}", journal = j-MATH-COMPUT, volume = "17", number = "82", pages = "193--194", month = apr, year = "1963", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", URL = "http://www.jstor.org/stable/2003641", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", remark = "Fullerton: Relative errors of $ 5 \times 10^{-4} $.", } @Article{Whittlesey:1963:IGF, author = "John R. B. Whittlesey", title = "Incomplete Gamma Functions for Evaluating {Erlang} Process Probabilities", journal = j-MATH-COMPUT, volume = "17", number = "81", pages = "11--17", month = jan, year = "1963", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @InCollection{Abramowitz:1964:CWF, author = "Milton Abramowitz", title = "{Coulomb} Wave Functions", crossref = "Abramowitz:1964:HMF", pages = "537--554", year = "1964", bibdate = "Sat Oct 30 19:37:56 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", } @InCollection{Abramowitz:1964:SFR, author = "Milton Abramowitz", title = "{Struve} Functions and Related Functions", crossref = "Abramowitz:1964:HMF", pages = "495--502", year = "1964", bibdate = "Sat Oct 30 19:37:56 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", } @Article{Aiken:1964:PAC, author = "H. H. Aiken and A. G. Oettinger and T. C. Bartee", title = "Proposed automatic calculating machine", journal = j-IEEE-SPECTRUM, volume = "1", number = "8", pages = "62--69", month = aug, year = "1964", CODEN = "IEESAM", DOI = "https://doi.org/10.1109/MSPEC.1964.6500770", ISSN = "0018-9235 (print), 1939-9340 (electronic)", ISSN-L = "0018-9235", bibdate = "Tue Jan 14 11:14:17 2020", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeespectrum1960.bib; https://www.math.utah.edu/pub/tex/bib/master.bib", abstract = "Here presented is the memorandum that 20 years ago initiated a series of events whose revolutionary implications are only beginning to manifest themselves a description of the first large-scale general-purpose automatic digital computer. Twenty years ago, on August 7, 1944, Mark I, the first large-scale general-purpose automatic digital computer ever to be put in operation was dedicated at Harvard University by James B. Conant, then president of Harvard, and the late Thomas J. Watson, founder of IBM.", acknowledgement = ack-nhfb, fjournal = "IEEE Spectrum", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=6", remark = "Pages 66--69 discuss computation of the elementary functions with minimal intermediate storage: recipes are given for integral and fractional power, log, exponential, trigonometric, inverse trigonometric, hyperbolic, and inverse hyperbolic. Mention is also made of the probability integral, elliptic functions, and Bessel functions, but the text says they will be discussed later (meaning, in a future publication). The methods involve recurrences and series summations, and thus, can be regarded as precision independent.", xxnote = "Previously unpublished memorandum written by Aiken and dated by an unknown recipient as 4 November 1937. Reprinted in \cite[\S 5.1]{Randell:1982:ODC}.", } @InCollection{Antosiewicz:1964:BFF, author = "H. A. Antosiewicz", title = "{Bessel} Functions of Fractional Order", crossref = "Abramowitz:1964:HMF", pages = "435--478", year = "1964", bibdate = "Sat Oct 30 19:37:56 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", } @InCollection{Blanch:1964:MF, author = "Gertrude Blanch", title = "{Mathieu} Functions", crossref = "Abramowitz:1964:HMF", pages = "721--750", year = "1964", bibdate = "Sat Oct 30 19:37:56 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, author-dates = "1897--1996", citedby = "Fullerton:1980:BEM", } @Article{Bray:1964:CAGa, author = "T. A. Bray", title = "Certification of {Algorithm 225}: {Gamma} function with controlled accuracy", journal = j-CACM, volume = "7", number = "10", pages = "586--586", month = oct, year = "1964", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Fri Nov 25 18:19:56 MST 2005", bibsource = "http://portal.acm.org/; http://www.acm.org/pubs/contents/journals/cacm/; https://www.math.utah.edu/pub/tex/bib/cacm1960.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "$\Gamma(x)$; special functions", remark = "Fullerton: No corrections were necessary.", } @Article{Burgoyne:1964:GTF, author = "F. D. Burgoyne", title = "Generalized Trigonometric Functions (in {Technical Notes and Short Papers})", journal = j-MATH-COMPUT, volume = "18", number = "86", pages = "314--316", month = apr, year = "1964", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Cody:1964:DPS, author = "William J. {Cody, Jr.}", title = "Double-Precision Square Root for the {CDC-3600}", journal = j-CACM, volume = "7", number = "12", pages = "715--718", month = dec, year = "1964", CODEN = "CACMA2", DOI = "https://doi.org/10.1145/355588.365122", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Fri Nov 25 18:19:57 MST 2005", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; garbo.uwasa.fi:/pc/doc-soft/fpbiblio.txt; http://www.acm.org/pubs/contents/journals/cacm/; https://www.math.utah.edu/pub/bibnet/authors/c/cody-william-j.bib; https://www.math.utah.edu/pub/tex/bib/cacm1960.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", abstract = "In January of 1960, the late Hans J. Maehly completed a summary of approximations to the elementary functions for the CDC-1604 computer. The approximations and techniques suggested by Maehly are equally applicable to the second large computer in the CDC line, the 3600. Unlike the 1604, however, the 3600 has built-in double-precision floating-point arithmetic. The present work, largely inspired by the successes of Maehly and his associates, concerns the extension of one of Maehly's ideas to a double-precision subroutine for the 3600.", acknowledgement = ack-nhfb # "\slash " # ack-nj, fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "$sqrt(x)$; CDC 3600; elementary functions; floating-point arithmetic", } @Article{Cowgill:1964:LEB, author = "D. Cowgill", title = "Logic Equations for a Built-In Square Root Method", journal = j-IEEE-TRANS-ELEC-COMPUT, volume = "EC-13", number = "2", pages = "156--157", month = apr, year = "1964", CODEN = "IEECA8", DOI = "https://doi.org/10.1109/PGEC.1964.263791", ISSN = "0367-7508", bibdate = "Thu Jul 14 06:56:59 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", URL = "https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4038119", acknowledgement = ack-nhfb, fjournal = "IEEE Transactions on Electronic Computers", } @Article{Curtiss:1964:EIB, author = "C. F. Curtiss", title = "Expansions of Integrals of {Bessel} Functions of Large Order", journal = j-J-MATH-PHYS, volume = "5", number = "4", pages = "561--564", month = apr, year = "1964", CODEN = "JMAPAQ", DOI = "https://doi.org/10.1063/1.1704149", ISSN = "0022-2488 (print), 1089-7658 (electronic), 1527-2427", ISSN-L = "0022-2488", bibdate = "Fri Oct 28 08:40:12 MDT 2011", bibsource = "http://www.aip.org/ojs/jmp.html; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jmathphys1960.bib", URL = "http://jmp.aip.org/resource/1/jmapaq/v5/i4/p561_s1", acknowledgement = ack-nhfb, fjournal = "Journal of Mathematical Physics", journal-URL = "http://jmp.aip.org/", onlinedate = "22 December 2004", pagecount = "4", } @Article{Cyvin:1964:AGF, author = "S. J. Cyvin and B. N. Cyvin", title = "{Algorithm 225}: {Gamma} function with controlled accuracy", journal = j-CACM, volume = "7", number = "5", pages = "295--295", month = may, year = "1964", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Fri Nov 25 18:19:53 MST 2005", bibsource = "http://portal.acm.org/; http://www.acm.org/pubs/contents/journals/cacm/; https://www.math.utah.edu/pub/tex/bib/cacm1960.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "$\Gamma(x)$; special functions", remark = "Fullerton: 30-line Algol procedure based on out-of-date method.", } @Article{Cyvin:1964:AND, author = "S. J. Cyvin", title = "{Algorithm 226}: {Normal} distribution function", journal = j-CACM, volume = "7", number = "5", pages = "295--295", month = may, year = "1964", CODEN = "CACMA2", DOI = "https://doi.org/10.1145/364099.364315", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Fri Nov 25 18:19:53 MST 2005", bibsource = "http://portal.acm.org/; http://www.acm.org/pubs/contents/journals/cacm/; https://www.math.utah.edu/pub/tex/bib/cacm1960.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "probability functions", } @InCollection{Davis:1964:GFR, author = "Philip J. Davis", title = "Gamma Function and Related Functions", crossref = "Abramowitz:1964:HMF", pages = "253--294", year = "1964", bibdate = "Sat Oct 30 19:37:56 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", } @Article{Eve:1964:EP, author = "J. Eve", title = "The evaluation of polynomials", journal = j-NUM-MATH, volume = "6", number = "1", pages = "17--21", month = dec, year = "1964", CODEN = "NUMMA7", DOI = "https://doi.org/10.1007/BF01386049", ISSN = "0029-599X (print), 0945-3245 (electronic)", ISSN-L = "0029-599X", bibdate = "Mon Oct 18 20:10:40 MDT 1999", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/nummath.bib", acknowledgement = ack-nhfb, fjournal = "Numerische Mathematik", journal-URL = "http://link.springer.com/journal/211", keywords = "number of multiplications to evaluate a polynomial", remark = "From the first two paragraphs: ``Ostrowski [5] has shown that the $ 2 n $ operations required by this algorithm [Horner's] are minimal for $ n \leq 4 $. Motzkin [4] (see also Todd [8]) and Knuth [3] have given methods whereby polynomials with $ 4 \leq n \leq 6 $ can be evaluated in $ [(1 / 2)(n + 3)] $ multiplications and not more than $ n + 1 $ additions. Similar methods effecting a reduction in the number of multiplications have been described by Pan [6] for $ n \leq 12 $. Each of these methods is valid only for a particular value of $n$.\par A general method due to Pan [7] applicable to all polynomials with $ n \geq 5 $ results in an evaluation involving $ [(1 / 2) (n + 4)] $ multiplications and $ n + 1 $ additions. Knuth has also given a method applicable to all polynomials with $ n \geq 3 $ in which $ n + 1 $ additions are required while the number of multiplications varies between $ [(1 / 2) (n + 3)] $ and approximately $ (3 / 4)n $.''", } @TechReport{Fisherkeller:1964:TCE, author = "M. A. Fisherkeller and W. J. {Cody, Jr.}", title = "Tables of the Complete Elliptic Integrals $ {K} $, $ {K}' $, $ {E} $, and $ {E}' $", type = "Technical Memo", number = "ANL AMD 71", institution = inst-ANL, address = inst-ANL:adr, pages = "14", year = "1964", bibdate = "Thu Nov 17 10:44:21 1994", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/c/cody-william-j.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "See review by John W. Wrench in Mathematics of Computation, {\bf 19}(89--92), 342, 1965.", acknowledgement = ack-nhfb, } @Article{Gargantini:1964:RCA, author = "I. Gargantini and T. Pomentale", title = "Rational {Chebyshev} approximations to the {Bessel} function integrals {$ K i_s(x) $}", journal = j-CACM, volume = "7", number = "12", pages = "727--730", month = dec, year = "1964", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", MRclass = "65.25", MRnumber = "31\#863", bibdate = "Fri Nov 25 18:19:57 MST 2005", bibsource = "http://portal.acm.org/; http://www.acm.org/pubs/contents/journals/cacm/; https://www.math.utah.edu/pub/tex/bib/cacm1960.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", abstract = "The second Remes algorithm is used to approximate the integrals $ K i_s $ by rational functions. The related coefficients for the approximations of $ K i_1, K i_2, K i_3 $ are given for different precisions.", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "Bessel functions; Kis(x); special functions", remark = "Fullerton: Approximations for repeated integrals $ \operatorname {Ki}_s(x) $ of $ K(x) $ for $ s = 1, 2, 3 $ are given for accuracies down to $ 10^{-5} $ for $ s = 1 $ and to $ 10^{-7} $ for $ s = 2, 3 $.", } @Article{Gautschi:1964:AAB, author = "W. Gautschi", title = "{ACM Algorithm 236}: {Bessel} Functions of the First Kind [{S17}]", journal = j-CACM, volume = "7", number = "8", pages = "479--480", month = aug, year = "1964", CODEN = "CACMA2", DOI = "https://doi.org/10.1145/355586.355587", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Fri Nov 25 18:19:55 MST 2005", bibsource = "http://portal.acm.org/; http://www.acm.org/pubs/contents/journals/cacm/; https://www.math.utah.edu/pub/tex/bib/cacm1960.bib; https://www.math.utah.edu/pub/tex/bib/cacm1970.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib", note = "See remark \cite{Skovgaard:1975:RBF}.", acknowledgement = ack-nhfb, fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "$J_n(x)$; Bessel functions of the first kind; special functions", } @Article{Gautschi:1964:AGF, author = "Walter Gautschi", title = "{Algorithm 221}: {Gamma} functions", journal = j-CACM, volume = "7", number = "3", pages = "143--143", month = mar, year = "1964", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Fri Nov 25 18:19:52 MST 2005", bibsource = "http://portal.acm.org/; http://www.acm.org/pubs/contents/journals/cacm/; https://www.math.utah.edu/pub/tex/bib/cacm1960.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "$\Gamma(x)$; special functions", } @Article{Gautschi:1964:AIB, author = "Walter Gautschi", title = "{Algorithm 222}: {Incomplete} beta functions ratios", journal = j-CACM, volume = "7", number = "3", pages = "143--143", month = mar, year = "1964", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Fri Nov 25 18:19:52 MST 2005", bibsource = "http://portal.acm.org/; http://www.acm.org/pubs/contents/journals/cacm/; https://www.math.utah.edu/pub/tex/bib/cacm1960.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "beta functions; special functions", remark = "Fullerton: 200-line Algol procedure.", } @Article{Gautschi:1964:CAI, author = "Walter Gautschi", title = "Certification of {Algorithm 222}: {Incomplete} beta function ratios", journal = j-CACM, volume = "7", number = "4", pages = "244--244", month = apr, year = "1964", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Fri Nov 25 18:19:53 MST 2005", bibsource = "http://www.acm.org/pubs/contents/journals/cacm/; https://www.math.utah.edu/pub/tex/bib/cacm1960.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "beta functions; special functions", remark = "Fullerton: A typographical error is noted.", } @InCollection{Gautschi:1964:EFF, author = "Walter Gautschi", title = "Error Function and {Fresnel} Integrals", crossref = "Abramowitz:1964:HMF", pages = "295--330", year = "1964", bibdate = "Sat Oct 30 19:37:56 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", } @InCollection{Gautschi:1964:EIR, author = "Walter Gautschi and William F. Cahill", title = "Exponential Integral and Related Functions", crossref = "Abramowitz:1964:HMF", pages = "227--252", year = "1964", bibdate = "Sat Oct 30 19:37:56 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", } @Article{Gray:1964:CAF, author = "Malcolm Gray", title = "Certification of {Algorithm 213}: {Fresnel} integrals", journal = j-CACM, volume = "7", number = "11", pages = "661--661", month = nov, year = "1964", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Fri Nov 25 18:19:56 MST 2005", bibsource = "http://portal.acm.org/; http://www.acm.org/pubs/contents/journals/cacm/; https://www.math.utah.edu/pub/tex/bib/cacm1960.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "See \cite{Gray:1963:AFI,Gray:1963:RAE}.", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "C(x); S(x); special functions", remark = "Fullerton: Several corrections are given.", } @InCollection{Haynsworth:1964:BEP, author = "Emilie V. Haynsworth and Karl Goldberg", title = "{Bernoulli} and {Euler} Polynomials, {Riemann Zeta} Function", crossref = "Abramowitz:1964:HMF", pages = "803--820", year = "1964", bibdate = "Sat Oct 30 19:37:56 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", } @InCollection{Hochstrasser:1964:OP, author = "Urs W. Hochstrasser", title = "Orthogonal Polynomials", crossref = "Abramowitz:1964:HMF", pages = "771--802", year = "1964", bibdate = "Sat Oct 30 19:37:56 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", } @Article{Hummer:1964:EDF, author = "David G. Hummer", title = "Expansions of {Dawson}'s Function in a Series of {Chebyshev} Polynomials (in {Technical Notes and Short Papers})", journal = j-MATH-COMPUT, volume = "18", number = "86", pages = "317--319", month = apr, year = "1964", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp1960.bib; JSTOR database", acknowledgement = ack-nhfb, ajournal = "Math. Comput.", citedby = "Fullerton:1980:BEM", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", remark = "Fullerton: Almost l5-digit approximations.", } @Article{Lanczos:1964:PAG, author = "Cornelius Lanczos", title = "A Precision Approximation of the Gamma Function", journal = j-SIAM-J-NUM-ANALYSIS-B, volume = "1", number = "1", pages = "86--96", month = "????", year = "1964", DOI = "https://doi.org/10.1137/0701008", ISSN = "0887-459X (print), 1095-7170 (electronic)", ISSN-L = "0887-459X", MRclass = "33.15", MRnumber = "0176115 (31 \#390)", MRreviewer = "S. C. van Veen", bibdate = "Fri Oct 16 06:57:22 MDT 1998", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/l/lanczos-cornelius.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/siamjnumeranal.bib; JSTOR database", URL = "http://www.jstor.org/stable/2949767", ZMnumber = "Zbl 0136.05201", acknowledgement = ack-nhfb, fjournal = "Journal of the Society for Industrial and Applied Mathematics: Series B, Numerical Analysis", journal-URL = "http://epubs.siam.org/loi/sjnaam.1", } @Article{Lotsch:1964:AFI, author = "Helmut Lotsch and Malcolm Gray", title = "{Algorithm 244}: {Fresnel} Integrals [{S20}]", journal = j-CACM, volume = "7", number = "11", pages = "660--661", month = nov, year = "1964", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Fri Nov 25 18:19:56 MST 2005", bibsource = "http://portal.acm.org/; http://www.acm.org/pubs/contents/journals/cacm/; https://www.math.utah.edu/pub/tex/bib/cacm1960.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", abstract = "This procedure computes the Fresnel sine and cosine integrals $ C(w) = \int_0^\infty \cos [(\pi / 2)t^2] \, d t $ and $ S(w) = \int_0^w \sin [(\pi / 2)t^2] \, d t $. It is a modification of Algorithm 213 (Comm. ACM, 6 (Oct. 1963), 617) such that the accuracy, expressed by \textit{eps}, is improved. eps can arbitrarily be chosen up to $ \textit {eps} = 10^{-6} $ for a computer with sufficient word length as, for example, the Burroughs B5000 which has 11--12 significant digits. Referring to the formulas of Algorithm 213: if $ |w| < \sqrt {(26.20 / \pi)} $ the series expansions $ C(w) $ and $ S(w) $ are terminated when the absolute value of the relative change in two successive terms is $ \leq \textit {eps} $. If $ |w| \geq \sqrt {(26.20 / \pi)} $ the series $ Q(x) $ and $ P(x) $ are terminated when the absolute value of the terms is $ \leq \textit {eps} / 2 $. However, this truncation point is not necessarily valid for the range $ \sqrt {(26.20 / \pi)} \leq |w| < \sqrt {(28.50 / \pi)} $ when $ \textit {eps} = 10^{-6} $, since the asymptotic series must be terminated before arriving at the minimum. In this range the ignored terms of the series expansions are $ < 3 \times 10^6 $, and for larger arguments $ < 10^{-6} $. This accuracy may be improved if desired: the switch-over point from the regular to the asymptotic series expansions has to be displaced to larger arguments.", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "C(x); Fresnel integrals; S(x); special functions", remark = "Fullerton: 100-line Algol procedure.", } @InCollection{Lowan:1964:SWF, author = "Arnold N. Lowan", title = "Spheroidal Wave Functions", crossref = "Abramowitz:1964:HMF", pages = "751--770", year = "1964", bibdate = "Sat Oct 30 19:37:56 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", } @InCollection{Luke:1964:IBF, author = "Yudell L. Luke", title = "Integrals of {Bessel} Functions", crossref = "Abramowitz:1964:HMF", pages = "479--494", year = "1964", bibdate = "Sat Oct 30 19:37:56 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", } @Book{Meinardus:1964:AFI, author = "Gunter Meinardus", title = "{Approximation von Funktionen und ihre numerische Behandlung}. ({German}) [{Approximation} of functions and their numerical treatment]", volume = "4", publisher = pub-SV, address = pub-SV:adr, pages = "viii + 180", year = "1964", DOI = "https://doi.org/10.1007/978-3-642-85646-4", ISBN = "3-540-03219-3, 3-642-85646-2, 3-642-85647-0 (print)", ISBN-13 = "978-3-540-03219-9, 978-3-642-85646-4, 978-3-642-85647-1 (print)", LCCN = "QA320 .M4", bibdate = "Thu Oct 19 17:00:27 MDT 2023", bibsource = "fsz3950.oclc.org:210/WorldCat; https://www.math.utah.edu/pub/bibnet/authors/r/rice-john-r.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", series = "Springer tracts in natural philosophy", abstract = "[numerous OCR errors to be corrected] Erst in den letzten Jahren hat sich derjenige Tell der Approximations theorie, der sich auf numerische Fragestellungen anwenden l{\"a}{\ss}t, starker entwickelt. Das Prinzip der in einem gewissen Sinne besten Ann{\"a}herung von Funktionen gewann insbesondere durch die Verwendung elektronischer Rechenmaschinen an Bedeutung. Einige der theoretischen Grundlagen, die zur Behandlung der auftretenden Probleme herange zogen werden mussen, finden sich verstreut in wenigen Buchern. Der weitaus gro{\ss}te Teil der theoretischen und praktischen Untersuchungen ist jedoch nur in den Originalarbeiten nachzulesen. Hieraus ergab sich die Zielsetzung des vorliegenden Buches: Es sollte eine Zusammen stellung der wesentlichen Ergebnisse der Approximationstheorie gegeben werden, die einerseits ein rasches Eindringen in die modernen Entwicklungen dieses Gebietes ermoglicht und andererseits eine gewisse Vollst{\"a}ndigkeit auf dem Problemkreis der Tschebyscheff-Approximationen bietet, womit keineswegs gemeint ist, da{\ss} eine vollst{\"a}ndige Literatur {\"u}bersicht angestrebt wurde. Die Auswahl erfolgte stets nach dem immer noch subjektiven Gesichtspunkt der Bedeutung f{\"u}r die Anwendungen. Dies gilt z. B. auch f{\"u}r die asymptotischen Untersuchungen des {\S} 3, denn ich bin der Meinung, da{\ss} man sich auch beinumerischen Approximationen {\"u}ber die, wenigstens asymptotisch zu erwartende Genauigkeit Gedanken machen sollte. Fast ausschlie{\ss}lich habe ich mich auf die Theorie der gleich m{\"a}{\ss}igen Approximation beschr{\"a}nkt, da diese die weitaus gro{\ss}te praktische Bedeutung besitzt. Das erste Kapitel behandelt lineare Approximationen. Der {\S} 3 enth{\"a}lt wohl den heute k{\"u}rzesten Zugang zur linearen Theorie.", acknowledgement = ack-nhfb, author-dates = "1926--", language = "German", subject = "Aproximaciones", tableofcontents = "I Lineare Approximationen \\ I.1. Das allgemeine lineare Approximationsproblem \\ I.2. Dichte Systeme \\ I.3. Allgemeine Theorie linearer Tschebyscheff-Approximationen \\ I.4. Spezielle Tschebyscheff-Approximationen \\ I.5. Absch{\"a}tzungen der Gr{\"o}{\ss}enordnung des Fehlers bei trigonometrischer und bei polynomialer Approximation \\ I.6. Polynomapproximationen \\ I.7. Numerische Verfahren bei linearen Tschebyscheff-Approximationen \\ II Nicht-lineare Approximationen \\ II.8. Allgemeine Theorie nicht-linearer Tschebyscheff-Approximationen \\ II.9. Rationale Approximationen \\ II.10. Exponentialapproximationen", } @InCollection{Miller:1964:PCF, author = "J. C. P. Miller", title = "Parabolic Cylinder Functions", crossref = "Abramowitz:1964:HMF", pages = "685--720", year = "1964", bibdate = "Sat Oct 30 19:37:56 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", } @InCollection{Milne-Thomson:1964:EI, author = "L. M. Milne-Thomson", title = "Elliptic Integrals", crossref = "Abramowitz:1964:HMF", pages = "587--626", year = "1964", bibdate = "Sat Oct 30 19:37:56 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", } @InCollection{Milne-Thomson:1964:JEF, author = "L. M. Milne-Thomson", title = "{Jacobian} Elliptic Functions and Theta Functions", crossref = "Abramowitz:1964:HMF", pages = "567--586", year = "1964", bibdate = "Sat Oct 30 19:37:56 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", } @Article{Morelock:1964:AAE, author = "J. C. Morelock", title = "{ACM} Algorithm 229: Elementary Functions by Continued Fractions", journal = j-CACM, volume = "7", number = "5", pages = "296", month = may, year = "1964", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Thu Sep 08 09:32:21 1994", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", } @InCollection{Oberhettinger:1964:HF, author = "Frtiz Oberhettinger", title = "Hypergeometric Functions", crossref = "Abramowitz:1964:HMF", pages = "555--566", year = "1964", bibdate = "Sat Oct 30 19:37:56 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", } @InCollection{Olver:1964:BFI, author = "F. W. J. Olver", title = "{Bessel} Functions of Integer Order", crossref = "Abramowitz:1964:HMF", pages = "355--434", year = "1964", bibdate = "Sat Oct 30 19:37:56 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", } @Article{Schmidt:1964:AEC, author = "Paul W. Schmidt", title = "Asymptotic Expansion of Certain Integrals Containing the {Bessel} Function {$ J_0 (x) $}", journal = j-J-MATH-PHYS, volume = "5", number = "8", pages = "1183--1184", month = aug, year = "1964", CODEN = "JMAPAQ", DOI = "https://doi.org/10.1063/1.1704223", ISSN = "0022-2488 (print), 1089-7658 (electronic), 1527-2427", ISSN-L = "0022-2488", bibdate = "Fri Oct 28 08:40:15 MDT 2011", bibsource = "http://www.aip.org/ojs/jmp.html; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jmathphys1960.bib", URL = "http://jmp.aip.org/resource/1/jmapaq/v5/i8/p1183_s1", acknowledgement = ack-nhfb, fjournal = "Journal of Mathematical Physics", journal-URL = "http://jmp.aip.org/", onlinedate = "22 December 2004", pagecount = "2", } @Article{Simauti:1964:AFS, author = "Takakazu Simauti", title = "Approximation formulas for some elementary functions", journal = "Comment. Math. Univ. St. Paul.", volume = "12", pages = "23--35", year = "1964", CODEN = "COMAAC", ISSN = "0010-258X", MRclass = "65.25", MRnumber = "28 \#5552", MRreviewer = "John Todd", bibdate = "Sat Jan 11 17:44:01 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, } @InCollection{Slater:1964:CHF, author = "Lucy Joan Slater", title = "Confluent Hvpergeometric Functions", crossref = "Abramowitz:1964:HMF", pages = "503--536", year = "1964", bibdate = "Sat Oct 30 19:37:56 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", } @Article{Slepian:1964:PSW, author = "David Slepian", title = "Prolate Spheroidal Wave Functions, {Fourier} Analysis and Uncertainty --- {IV}: Extensions to Many Dimensions; Generalized Prolate Spheroidal Functions", journal = j-BELL-SYST-TECH-J, volume = "43", number = "6", pages = "3009--3057", month = nov, year = "1964", CODEN = "BSTJAN", ISSN = "0005-8580", MRclass = "33.28", MRnumber = "0181766 (31 \#5993)", MRreviewer = "J. Meixner", bibdate = "Tue Nov 9 11:15:55 MST 2010", bibsource = "http://bstj.bell-labs.com/oldfiles/year.1964/BSTJ.1964.4306.html; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://bstj.bell-labs.com/BSTJ/images/Vol43/bstj43-6-3009.pdf", acknowledgement = ack-nhfb, fjournal = "The Bell System Technical Journal", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)1538-7305/issues/", } @InCollection{Southard:1964:WER, author = "Thomas H. Southard", title = "{Weierstrass} Elliptic and Related Functions", crossref = "Abramowitz:1964:HMF", pages = "627--684", year = "1964", bibdate = "Sat Oct 30 19:37:56 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", } @InCollection{Stegun:1964:LF, author = "Irene A. Stegun", title = "{Legendre} Functions", crossref = "Abramowitz:1964:HMF", pages = "331--354", year = "1964", bibdate = "Sat Oct 30 19:37:56 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", } @InCollection{Stegun:1964:MF, author = "Irene A. Stegun", title = "Miscellaneous Functions", crossref = "Abramowitz:1964:HMF", pages = "997--1010", year = "1964", bibdate = "Sat Oct 30 19:37:56 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", keywords = "Clausen's integral; Clebsch--Gordan coefficients; Debye function; Dilogarithm (Spence's integral); Einstein function; Planck function; Sievert and related integrals", remark = "Fullerton: Debye, Planck and Einstein functions. Sievert and related integrals. Dilogarithm (Spence's integral). Clausen's integral. Clebsch--Gordan coefficients.", } @Article{Wengert:1964:SAD, author = "R. E. Wengert", title = "A simple automatic derivative evaluation program", journal = j-CACM, volume = "7", number = "8", pages = "463--464", year = "1964", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Mon May 19 13:30:58 1997", bibsource = "ftp://ftp.ira.uka.de/pub/bibliography/Math/auto.diff.bib.gz; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", abstract = "A procedure for automatic evaluation of total and partial derivatives of arbitrary algebraic functions is presented. The numerical values of derivatives are computed without developing analytic expressions for the derivatives. The function is decomposed into a sequence of elementary expressions A library is provided for differentiating of elementary functions.", acknowledgement = ack-nhfb, fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "computer program.; differentiation arithmetic; point algorithm", referred = "[Bell65a]; [Carl86a]; [Corl88a]; [Garc91a]; [Irim91a]; [Kala83b]; [Laws88a]; [Laws91a]; [Neid87a]; [Neid89a]; [Ostr71a]; [Pfei87a]; [Tesf91a]; [Voli85a]; [Wexl87a]; [Wilk64a].", } @InCollection{Zelen:1964:PF, author = "Marvin Zelen and Norman C. Severo", title = "Probability Functions", crossref = "Abramowitz:1964:HMF", pages = "925--996", year = "1964", bibdate = "Sat Oct 30 19:37:56 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", } @InCollection{Zucker:1964:ETF, author = "Ruth Zucker", title = "Elementary Transcendental Functions. {Logarithmic}, Exponential, Circular and Hyperbolic Functions", crossref = "Abramowitz:1964:HMF", pages = "65--226", year = "1964", bibdate = "Sat Oct 30 19:37:56 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", } @Article{Bingulac:1965:ACG, author = "S. P. Bingulac and E. A. Humo", title = "Analog Computer Generation of {Bessel} Functions of Arbitrary Order", journal = j-IEEE-TRANS-ELEC-COMPUT, volume = "EC-14", number = "6", pages = "886--889", month = dec, year = "1965", CODEN = "IEECA8", DOI = "https://doi.org/10.1109/PGEC.1965.264084", ISSN = "0367-7508", bibdate = "Thu Jul 14 06:26:41 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", URL = "https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4038609", acknowledgement = ack-nhfb, fjournal = "IEEE Transactions on Electronic Computers", } @Article{Braess:1965:MIG, author = "Dietrich Braess", title = "{Monotone Iterationsfolgen bei Gleichungssystemen mit fehlerhaften Koeffizienten und Iterationsbeschleunigung}. ({German}) [{Monotone} Iteration Sequences for Equation Systems with Coefficients Having Errors, and Iteration Acceleration]", journal = j-NUM-MATH, volume = "7", number = "1", pages = "32--41", month = feb, year = "1965", CODEN = "NUMMA7", ISSN = "0029-599X (print), 0945-3245 (electronic)", ISSN-L = "0029-599X", bibdate = "Mon Oct 18 01:28:20 MDT 1999", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Numerische Mathematik", journal-URL = "http://link.springer.com/journal/211", keywords = "convergence acceleration", language = "German", } @Article{Bulirsch:1965:NCEa, author = "R. Bulirsch", title = "Numerical calculation of elliptic integrals and elliptic functions", journal = j-NUM-MATH, volume = "7", number = "1", pages = "78--90", month = feb, year = "1965", CODEN = "NUMMA7", ISSN = "0029-599X (print), 0945-3245 (electronic)", ISSN-L = "0029-599X", bibdate = "Mon Oct 18 20:10:40 MDT 1999", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", series = "Handbook Series Special functions", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Numerische Mathematik", journal-URL = "http://link.springer.com/journal/211", } @Article{Bulirsch:1965:NCEb, author = "R. Bulirsch", title = "Numerical calculation of elliptic integrals and elliptic functions. {II}", journal = j-NUM-MATH, volume = "7", number = "4", pages = "353--354", month = aug, year = "1965", CODEN = "NUMMA7", ISSN = "0029-599X (print), 0945-3245 (electronic)", ISSN-L = "0029-599X", bibdate = "Sun Oct 17 16:12:48 MDT 1999", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Numerische Mathematik", journal-URL = "http://link.springer.com/journal/211", } @Article{Christiansen:1965:APE, author = "S. Christiansen", title = "{Algol} programming: Error Integral with Complex Argument", journal = j-NORDISK-TIDSKR-INFORM-BEHAND, volume = "5", number = "4", pages = "287--293", month = dec, year = "1965", CODEN = "BITTEL, NBITAB", DOI = "https://doi.org/10.1007/BF01937509", ISSN = "0006-3835 (print), 1572-9125 (electronic)", ISSN-L = "0006-3835", bibdate = "Wed Jan 4 18:52:09 MST 2006", bibsource = "http://springerlink.metapress.com/openurl.asp?genre=issue&issn=0006-3835&volume=5&issue=4; https://www.math.utah.edu/pub/tex/bib/bit.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.springerlink.com/openurl.asp?genre=article&issn=0006-3835&volume=5&issue=4&spage=287", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", journal-URL = "http://link.springer.com/journal/10543", remark = "Fullerton: A 75-line Algol procedure with maximum absolute error about $ 2 \times 10^{-6} $ is given for $ w(z) = e^{-z^2} \erfc ( - i z) $.", } @Article{Cochran:1965:ZHF, author = "J. A. Cochran", title = "The zeros of {Hankel} functions as functions of their order", journal = j-NUM-MATH, volume = "7", number = "3", pages = "238--250", month = jun, year = "1965", CODEN = "NUMMA7", ISSN = "0029-599X (print), 0945-3245 (electronic)", ISSN-L = "0029-599X", bibdate = "Mon Oct 18 10:06:00 MDT 1999", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Numerische Mathematik", journal-URL = "http://link.springer.com/journal/211", } @Article{Cody:1965:CAC, author = "W. J. {Cody, Jr.}", title = "{Chebyshev} Approximations for the Complete Elliptic Integrals {$K$} and {$E$}", journal = j-MATH-COMPUT, volume = "19", number = "89--92", pages = "105--112", month = apr, year = "1965", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "65.05", MRnumber = "30\#1601", bibdate = "Fri Oct 23 11:10:16 1998", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/c/cody-william-j.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "See also \cite{Cody:1966:CCA}.", URL = "http://www.jstor.org/stable/2004103", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", remark = "Fullerton: Relative errors down to $ 4 \times 10^{-18} $.", } @Article{Cody:1965:CPE, author = "W. J. {Cody, Jr.}", title = "{Chebyshev} Polynomial Expansions of Complete Elliptic Integrals", journal = j-MATH-COMPUT, volume = "19", number = "89--92", pages = "249--259", month = apr, year = "1965", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "65.25", MRnumber = "31\#2820", bibdate = "Fri Oct 23 11:10:33 1998", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/c/cody-william-j.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.jstor.org/stable/2003350", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", remark = "Fullerton: 25-digit approximations.", } @Article{Combet:1965:CBT, author = "M. Combet and H. {Van Zonneveld} and L. Verbeek", title = "Computation of the Base Two Logarithm of Binary Numbers", journal = j-IEEE-TRANS-ELEC-COMPUT, volume = "EC-14", number = "6", pages = "863--867", month = dec, year = "1965", CODEN = "IEECA8", DOI = "https://doi.org/10.1109/PGEC.1965.264080", ISSN = "0367-7508", bibdate = "Thu Jul 14 06:26:41 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", URL = "https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4038605", acknowledgement = ack-nhfb, fjournal = "IEEE Transactions on Electronic Computers", } @Article{Dahlquist:1965:CAP, author = "Germund Dahlquist and Sven-{\AA}ke Gustafson and K{\'a}roly Sikl{\'o}si", title = "Convergence Acceleration from the Point of View of Linear Programming", journal = j-NORDISK-TIDSKR-INFORM-BEHAND, volume = "5", number = "1", pages = "1--16", month = mar, year = "1965", CODEN = "BITTEL, NBITAB", DOI = "https://doi.org/10.1007/BF01975719", ISSN = "0006-3835 (print), 1572-9125 (electronic)", bibdate = "Wed Jan 4 18:52:08 MST 2006", bibsource = "http://springerlink.metapress.com/openurl.asp?genre=issue&issn=0006-3835&volume=5&issue=1; https://www.math.utah.edu/pub/tex/bib/bit.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.springerlink.com/openurl.asp?genre=article&issn=0006-3835&volume=5&issue=1&spage=1", acknowledgement = ack-nhfb, journal-URL = "http://link.springer.com/journal/10543", keywords = "convergence acceleration", } @Article{Fettis:1965:CEI, author = "Henry E. Fettis", title = "Calculation of Elliptic Integrals of the Third Kind by Means of {Gauss}' Transformation", journal = j-MATH-COMPUT, volume = "19", number = "89", pages = "97--104", month = apr, year = "1965", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", URL = "http://www.jstor.org/stable/2004102", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Fields:1965:RAG, author = "Jerry L. Fields", title = "Rational Approximations to Generalized Hypergeometric Functions", journal = j-MATH-COMPUT, volume = "19", number = "92", pages = "606--624", month = oct, year = "1965", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp1960.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Franke:1965:NEE, author = "Charles H. Franke", title = "Numerical Evaluation of the Elliptic Integral of the Third Kind (in {Technical Notes and Short Papers})", journal = j-MATH-COMPUT, volume = "19", number = "91", pages = "494--496", month = jul, year = "1965", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Fraser:1965:SMC, author = "W. Fraser", title = "A Survey of Methods for Computing Minimax and Near-Minimax Polynomial Approximations for Functions of a Single Independent Variable", journal = j-J-ACM, volume = "12", number = "3", pages = "295--314", month = jul, year = "1965", CODEN = "JACOAH", DOI = "https://doi.org/10.1145/321281.321282", ISSN = "0004-5411 (print), 1557-735X (electronic)", ISSN-L = "0004-5411", bibdate = "Thu Nov 03 08:47:50 1994", bibsource = "http://portal.acm.org/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/jacm.bib", abstract = "Methods are described for the derivation of minimax and near-minimax polynomial approximations. For minimax approximations techniques are considered for both analytically defined functions and functions defined by a table of values. For near-minimax approximations methods of determining the coefficients of the Fourier--Chebyshev expansion are first described. These consist of the rearrangement of the coefficients of a power polynomial, and also direct determination of the coefficients from the integral which defines them, or the differential equation which defines the function. Finally there is given a convenient modification of an interpolation scheme which finds coefficients of a near-minimax approximation without requiring numerical integration or the numerical solution of a system of equations.", acknowledgement = ack-nhfb, ajournal = "J. Assoc. Comput. Mach.", fjournal = "Journal of the ACM", journal-URL = "https://dl.acm.org/loi/jacm", } @Article{Gautschi:1965:ALF, author = "W. Gautschi", title = "{Algorithm 259}: {Legendre} Functions for Arguments Larger than One [{S16}]", journal = j-CACM, volume = "8", number = "8", pages = "488--492", month = aug, year = "1965", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Fri Nov 25 18:20:01 MST 2005", bibsource = "http://www.acm.org/pubs/contents/journals/cacm/; https://www.math.utah.edu/pub/tex/bib/cacm1960.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "See also \cite{Jansen:1977:RLF}.", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "Legendre functions; special functions", remark = "Fullerton: Long Algol procedures for the associated Legendre functions of the first and second kinds: $ P_a^n(x) $ and $ Q_n^m $.", } @Article{Gautschi:1965:CAS, author = "Walter Gautschi", title = "Certification of {Algorithm 236} [{S17}]: {Bessel} functions of the first kind", journal = j-CACM, volume = "8", number = "2", pages = "105--106", month = feb, year = "1965", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Fri Nov 25 18:19:58 MST 2005", bibsource = "http://www.acm.org/pubs/contents/journals/cacm/; https://www.math.utah.edu/pub/tex/bib/cacm1960.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "$J_n(x)$; Bessel functions of the first kind; special functions", } @Article{Gunn:1965:ASa, author = "J. H. Gunn", title = "{Algorithm 260}: {6-$J$} symbols", journal = j-CACM, volume = "8", number = "8", pages = "492--492", month = aug, year = "1965", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Fri Nov 25 18:20:01 MST 2005", bibsource = "http://www.acm.org/pubs/contents/journals/cacm/; https://www.math.utah.edu/pub/tex/bib/cacm1960.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", remark = "Fullerton: Short Algol procedure.", } @Article{Gunn:1965:ASb, author = "J. H. Gunn", title = "{Algorithm 261}: {9-$J$} symbols", journal = j-CACM, volume = "8", number = "8", pages = "492--493", month = aug, year = "1965", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Fri Nov 25 18:20:01 MST 2005", bibsource = "http://www.acm.org/pubs/contents/journals/cacm/; https://www.math.utah.edu/pub/tex/bib/cacm1960.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", remark = "Fullerton: Short Algol procedure.", } @Article{Gunn:1965:AZV, author = "J. H. Gunn", title = "{Algorithm 252} [{Z}]: {Vector} coupling or {Clebsch--Gordan} coefficients", journal = j-CACM, volume = "8", number = "4", pages = "217--217", month = apr, year = "1965", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Fri Nov 25 18:19:59 MST 2005", bibsource = "http://www.acm.org/pubs/contents/journals/cacm/; https://www.math.utah.edu/pub/tex/bib/cacm1960.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", remark = "Fullerton: Short Algol procedure.", } @Article{Heatley:1965:ETT, author = "A. H. Heatley", title = "An Extension of the Table of the {Toronto} Function (in {Technical Notes and Short Papers})", journal = j-MATH-COMPUT, volume = "19", number = "89", pages = "118--123", month = apr, year = "1965", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", remark = "Fullerton: Some 5 and 7-digit values.", } @Article{James:1965:GSR, author = "Wendy James and P. Jarratt", title = "The Generation of Square Roots on a Computer with Rapid Multiplication Compared with Division (in {Technical Notes and Short Papers})", journal = j-MATH-COMPUT, volume = "19", number = "91", pages = "497--500", month = jul, year = "1965", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; garbo.uwasa.fi:/pc/doc-soft/fpbiblio.txt; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp1960.bib; JSTOR database", acknowledgement = ack-nj # " and " # ack-nhfb, fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Kazangapov:1965:REF, author = "A. N. Kazangapov", title = "Representation of elementary function in the system of residual classes. ({Russian})", journal = "Izv. Akad. Nauk Kazah. SSR Ser. Fiz.-Mat. Nauk", volume = "3", pages = "79--84", year = "1965", MRclass = "65.25", MRnumber = "33 \#5090", bibdate = "Sat Jan 11 17:44:01 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, language = "Russian", } @Article{King:1965:LED, author = "R. King", title = "Letter to the {Editor}: On the Double-Precision Square Root Routine", journal = j-CACM, volume = "8", number = "4", pages = "202", month = apr, year = "1965", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Thu Sep 1 10:15:43 1994", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; https://www.math.utah.edu/pub/tex/bib/cacm1960.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nj, fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "$\sqrt(x)$; elementary functions; floating-point arithmetic", } @Book{Lebedev:1965:SFT, author = "N. N. (Nikola{\u\i}i Nikolaevich) Lebedev", title = "Special Functions and Their Applications", publisher = pub-PH, address = pub-PH:adr, pages = "xii + 308", year = "1965", LCCN = "QA351 .L3613", bibdate = "Sat Apr 1 14:42:46 MDT 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; z3950.loc.gov:7090/Voyager", note = "Revised English edition translated and edited by Richard A. Silverman.", series = "Selected Russian publications in the mathematical sciences", acknowledgement = ack-nhfb, subject = "Functions, Special; Mathematical physics", } @Book{Lyusternik:1965:HCE, author = "L. A. Lyusternik and O. A. Chervonenkis and A. R. Yanpol{\'s}kii", title = "Handbook for Computing Elementary Functions", volume = "76", publisher = pub-PERGAMON, address = pub-PERGAMON:adr, pages = "xiii + 251", year = "1965", LCCN = "QA221.L513", MRclass = "65.25", MRnumber = "32 \#584", bibdate = "Sat Jan 11 17:44:01 MST 1997", bibsource = "garbo.uwasa.fi:/pc/doc-soft/fpbiblio.txt; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "Translated from the Russian by G. J. Tee. Translation edited by K. L. Stewart.", series = "International series of monographs on pure and applied mathematics", acknowledgement = ack-nhfb, } @Article{MacLaren:1965:APN, author = "M. D. MacLaren", title = "{Algorithm 272}: {Procedure} for the Normal Distribution Functions [{S15}]", journal = j-CACM, volume = "8", number = "12", pages = "789--790", month = dec, year = "1965", CODEN = "CACMA2", DOI = "https://doi.org/10.1145/365691.365957", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Fri Nov 25 18:20:03 MST 2005", bibsource = "http://www.acm.org/pubs/contents/journals/cacm/; https://www.math.utah.edu/pub/tex/bib/cacm1960.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "See remarks \cite{Hill:1967:RAS,MacLaren:1968:RAP}.", abstract = "The procedure gives $ \Phi (a) = \sqrt {1 / (2 \pi)} \int_{- \infty }^a \exp ( - t^2 / 2) \, d t $ and $ \Phi *(a) = 2 (\Phi (|a|) - 0.5) = \sqrt {2 / \pi } \int_0^{|a|} \exp ( - t^2 / 2) \, d t $.", acknowledgement = ack-nhfb, fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "probability functions", } @Article{Maklovic:1965:IIC, author = "S. T. Maklovi{\v{c}}", title = "Investigation of integrals containing {Bessel} and elementary functions. ({Russian})", journal = "Ki{\v{s}}inev. Gos. Univ. U{\v{c}}en. Zap.", volume = "82", pages = "75--81", year = "1965", MRclass = "33.25", MRnumber = "34 \#387", MRreviewer = "H. A. Lauwerier", bibdate = "Sat Jan 11 17:44:01 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, language = "Russian", } @Article{Markman:1965:RZF, author = "B. Markman", title = "The {Riemann} Zeta Function", journal = j-NORDISK-TIDSKR-INFORM-BEHAND, volume = "5", number = "2", pages = "138--141", year = "1965", bibdate = "Sat Oct 30 08:53:17 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", journal-URL = "http://link.springer.com/journal/10543", remark = "Fullerton: A 25-line Algol procedure for evaluating $ \zeta (s) $ for all $ s \neq 1 $ is given.", } @Article{Medhurst:1965:EI, author = "R. G. Medhurst and J. H. Roberts", title = "Evaluation of the Integral $ {I}_n(b) = \frac {2}{\pi } \int^\infty_0 \bigg (\frac {\sin x}{x} \bigg)^n \cos (b x) d x $", journal = j-MATH-COMPUT, volume = "19", number = "89", pages = "113--117", month = apr, year = "1965", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Metze:1965:MSR, author = "Gernot Metze", title = "Minimal Square Rooting", journal = j-IEEE-TRANS-ELEC-COMPUT, volume = "EC-14", number = "2", pages = "181--185", month = apr, year = "1965", CODEN = "IEECA8", DOI = "https://doi.org/10.1109/PGEC.1965.263963", ISSN = "0367-7508", bibdate = "Thu Jul 14 06:26:22 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", URL = "https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4038397", acknowledgement = ack-nhfb, fjournal = "IEEE Transactions on Electronic Computers", } @Article{Miller:1965:ASF, author = "G. F. Miller", title = "Algorithms for Special Functions {II}", journal = j-NUM-MATH, volume = "7", pages = "194--196", year = "1965", CODEN = "NUMMA7", ISSN = "0029-599X (print), 0945-3245 (electronic)", ISSN-L = "0029-599X", bibdate = "Fri Sep 16 10:22:10 1994", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nj, citedby = "Fullerton:1980:BEM", fjournal = "Numerische Mathematik", journal-URL = "http://link.springer.com/journal/211", remark = "Fullerton: Corrections and simplifications of the $ \sin $, $ \cos $ and $ \tan $ routines given in paper I. See Clenshaw (1963).", xxmonth = "(none)", xxnumber = "(none)", } @Article{Nemeth:1965:CEF, author = "G. N{\'e}meth", title = "{Chebyshev} expansions for {Fresnel} integrals", journal = j-NUM-MATH, volume = "7", number = "4", pages = "310--312", month = aug, year = "1965", CODEN = "NUMMA7", DOI = "https://doi.org/10.1007/BF01436524", ISSN = "0029-599X (print), 0945-3245 (electronic)", ISSN-L = "0029-599X", bibdate = "Sun Oct 17 20:47:18 MDT 1999", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/nummath.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Numerische Mathematik", journal-URL = "http://link.springer.com/journal/211", remark = "Fullerton: Two series of 12-digit coefficients are given to cover the range $ 0 \leq x < \infty $.", } @Article{Rice:1965:CPR, author = "John R. Rice", title = "On the Conditioning of Polynomial and Rational Forms", journal = j-NUM-MATH, volume = "7", number = "5", pages = "426--435", month = oct, year = "1965", CODEN = "NUMMA7", DOI = "https://doi.org/10.1007/BF01436257", ISSN = "0029-599X (print), 0945-3245 (electronic)", ISSN-L = "0029-599X", MRclass = "65.99", MRnumber = "MR0189283 (32 \#6710)", MRreviewer = "James H. Wilkinson", bibdate = "Sun Oct 16 17:22:04 GMT 1994", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/r/rice-john-r.bib; https://www.math.utah.edu/pub/bibnet/subjects/acc-stab-num-alg.bib; https://www.math.utah.edu/pub/mirrors/ftp.ira.uka.de/bibliography/Math/nummath.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/nummath.bib", acknowledgement = ack-nhfb, fjournal = "Numerische Mathematik", journal-URL = "http://link.springer.com/journal/211", keywords = "number of multiplications to evaluate a polynomial", } @Article{Slepian:1965:EAP, author = "David Slepian and Estelle Sonnenblick", title = "Eigenvalues associated with prolate spheroidal wave functions of zero order", journal = j-BELL-SYST-TECH-J, volume = "44", number = "8", pages = "1745--1759", month = oct, year = "1965", CODEN = "BSTJAN", ISSN = "0005-8580", MRclass = "65.25", MRnumber = "0183103 (32 \#585)", MRreviewer = "R. Nicolovius", bibdate = "Tue Nov 9 11:15:55 MST 2010", bibsource = "http://bstj.bell-labs.com/oldfiles/year.1965/BSTJ.1965.4408.html; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://bstj.bell-labs.com/BSTJ/images/Vol44/bstj44-8-1745.pdf", acknowledgement = ack-nhfb, fjournal = "The Bell System Technical Journal", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)1538-7305/issues/", } @Article{Slepian:1965:SAE, author = "David Slepian", title = "Some Asymptotic Expansions for Prolate Spheroidal Wave Functions", journal = "J. Math. and Physics {XLIV(2)}", volume = "??", number = "??", pages = "99--140", month = jun, year = "1965", bibdate = "Sat Oct 30 10:46:38 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.ams.org/mathscinet/search/publications.html?pg1=IID&s1=189661", acknowledgement = ack-nhfb, author-dates = "1923--2007", citedby = "Fullerton:1980:BEM", remark = "Fullerton: Several complicated expansions are derived and presented.", } @Article{Swarztrauber:1965:LED, author = "P. N. Swarztrauber", title = "Letter to the {Editor}: On the Double-Precision Square Root Routine", journal = j-CACM, volume = "8", number = "4", pages = "202", month = apr, year = "1965", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Wed Aug 31 14:02:19 1994", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; https://www.math.utah.edu/pub/tex/bib/cacm1960.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "$\sqrt(x)$; elementary functions; floating-point arithmetic", } @Article{Thompson:1965:AEI, author = "G. T. Thompson", title = "The Asymptotic Expansion of the Integrals Psi and Chi in Terms of {Tchebycheff} Polynomials (in {Technical Notes and Short Papers})", journal = j-MATH-COMPUT, volume = "19", number = "92", pages = "661--663", month = oct, year = "1965", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "Doppler Broadening", } @Book{Clenshaw:1966:CSB, author = "C. W. Clenshaw and Susan M. Picken", title = "{Chebyshev} series for {Bessel} functions of fractional order", volume = "8", publisher = pub-HMSO, address = pub-HMSO:adr, pages = "iii + 53", year = "1966", MRclass = "33.25", MRnumber = "203095", MRreviewer = "L. J. Slater", bibdate = "Sun Nov 12 06:18:24 2023", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/c/clenshaw-charles-w.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", series = "National Physical Laboratory. Mathematical tables", acknowledgement = ack-nhfb, author-dates = "Charles William Clenshaw (15 March 1926--23 September 2004)", xxpages = "51", xxpages = "iii + 54", } @Article{Cody:1966:CCA, author = "W. J. {Cody, Jr.}", title = "Corrigenda: ``{Chebyshev} Approximations for the Complete Elliptic Integrals $ {K} $ and $ {E} $''", journal = j-MATH-COMPUT, volume = "20", number = "93", pages = "207--207", month = jan, year = "1966", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Fri Oct 23 11:13:58 1998", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/c/cody-william-j.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "See \cite{Cody:1965:CAC}.", acknowledgement = ack-nhfb, fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Doring:1966:CZC, author = "Boro Doring", title = "Complex Zeros of Cylinder Functions", journal = j-MATH-COMPUT, volume = "20", number = "94", pages = "215--222", month = apr, year = "1966", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Fike:1966:SAS, author = "C. T. Fike", title = "Starting Approximations for Square Root Calculation on {IBM System\slash 360}", journal = j-CACM, volume = "9", number = "4", pages = "297--299", month = apr, year = "1966", CODEN = "CACMA2", DOI = "https://doi.org/10.1145/365278.365556", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Thu Sep 1 10:15:43 1994", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; http://www.acm.org/pubs/contents/journals/cacm/; https://www.math.utah.edu/pub/tex/bib/cacm1960.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", note = "See letter \cite{Fike:1967:LER}.", abstract = "Several starting approximations for square root calculation by Newton's method are presented in a form to facilitate their use in IBM System/360 square root routines. These approximations include several for the range [1/16, 1], which is the interval of primary interest on IBM System/360.", acknowledgement = ack-nj # " and " # ack-nhfb, fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "$\sqrt(x)$; elementary functions; IBM S/360", } @Article{Filippi:1966:BEE, author = "S. Filippi", title = "{Die Berechnung einiger elementarer transzendenter Funktionen mit Hilfe des Richardson-Algorithmus} \toenglish {The Computation of Some Elementary Transcendental Functions by Means of the Richardson Algorithm} \endtoenglish", journal = j-COMPUTING, volume = "1", number = "2", pages = "127--132", month = jun, year = "1966", CODEN = "CMPTA2", DOI = "https://doi.org/10.1007/BF02342622", ISSN = "0010-485X (print), 1436-5057 (electronic)", ISSN-L = "0010-485X", bibdate = "Fri Sep 16 16:30:40 1994", bibsource = "garbo.uwasa.fi:/pc/doc-soft/fpbiblio.txt; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nj, fjournal = "Computing", journal-URL = "http://link.springer.com/journal/607", } @Article{Gautschi:1966:AD, author = "Walter Gautschi", title = "{Algorithm 282}: {Derivatives} of $ e^x / x $, $ \cos (x) / x $, and $ \sin (x) / x $", journal = j-CACM, volume = "9", number = "4", pages = "272--272", month = apr, year = "1966", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Fri Nov 25 18:20:05 MST 2005", bibsource = "http://www.acm.org/pubs/contents/journals/cacm/; https://www.math.utah.edu/pub/tex/bib/cacm1960.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "See remark \cite{Gautschi:1970:RAD}.", acknowledgement = ack-nhfb, fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "$\cos(x)/x$; $\sin(x)/x$; $e^x/x$; elementary functions", } @Article{Gautschi:1966:ARC, author = "Walter Gautschi", title = "{Algorithm 292}: {Regular} {Coulomb} Wave Functions", journal = j-CACM, volume = "9", number = "11", pages = "793--795", month = nov, year = "1966", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Fri Nov 25 18:20:10 MST 2005", bibsource = "http://www.acm.org/pubs/contents/journals/cacm/; https://www.math.utah.edu/pub/tex/bib/cacm1960.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "Coulomb wave functions; special functions", } @Article{Glasser:1966:ESI, author = "M. L. Glasser", title = "Evaluation of Some Integrals Involving the $ \psi $-Function (in {Technical Notes and Short Papers})", journal = j-MATH-COMPUT, volume = "20", number = "94", pages = "332--333", month = apr, year = "1966", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Gustafson:1966:CAM, author = "Sven-{\AA}ke Gustafson", title = "Convergence Acceleration by Means of Numerical Quadrature", journal = j-NORDISK-TIDSKR-INFORM-BEHAND, volume = "6", number = "2", pages = "117--128", month = jun, year = "1966", CODEN = "BITTEL, NBITAB", DOI = "https://doi.org/10.1007/BF01933103", ISSN = "0006-3835 (print), 1572-9125 (electronic)", bibdate = "Wed Jan 4 18:52:09 MST 2006", bibsource = "http://springerlink.metapress.com/openurl.asp?genre=issue&issn=0006-3835&volume=6&issue=2; https://www.math.utah.edu/pub/tex/bib/bit.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.springerlink.com/openurl.asp?genre=article&issn=0006-3835&volume=6&issue=2&spage=117", acknowledgement = ack-nhfb, journal-URL = "http://link.springer.com/journal/10543", keywords = "convergence acceleration", } @Article{Hart:1966:CAR, author = "Roger G. Hart", title = "A Close Approximation Related to the Error Function (in {Technical Notes and Short Papers})", journal = j-MATH-COMPUT, volume = "20", number = "96", pages = "600--602", month = oct, year = "1966", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp1960.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Hastings:1966:RCB, author = "C. W. Hastings", title = "{R66-78} Computation of the Base Two Logarithm of Binary Number", journal = j-IEEE-TRANS-ELEC-COMPUT, volume = "EC-15", number = "6", pages = "956--957", month = dec, year = "1966", CODEN = "IEECA8", DOI = "https://doi.org/10.1109/PGEC.1966.264517", ISSN = "0367-7508", bibdate = "Thu Jul 14 05:46:46 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", URL = "https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4038956", acknowledgement = ack-nhfb, fjournal = "IEEE Transactions on Electronic Computers", } @Book{Jurimae:1966:KFT, author = "E. J{\"u}rim{\"a}e", title = "Kompleksmuutuja funktsioonide teooria. {I}: Elementaarsed funktsioonid. ({Estonian}) [Theory of functions of a complex variable. {I}: Elementary functions]", publisher = "Tartu Riiklik {\"U}likool", address = "Tartu, Estonia", pages = "131", year = "1966", MRclass = "30.00", MRnumber = "40 \#5827a", bibdate = "Sat Jan 11 17:44:01 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, } @InCollection{Kogbetliantz:1966:GEF, author = "E. G. (Ervand George) Kogbetliantz", title = "Generation of Elementary Functions", crossref = "Ralston:1960:MMD", pages = "7--35", year = "1966", bibdate = "Sat Dec 09 14:09:27 1995", bibsource = "garbo.uwasa.fi:/pc/doc-soft/fpbiblio.txt; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nj, } @TechReport{Kuki:1966:CAE, author = "H. Kuki", title = "Comments on the {ANL} Evaluation of the {OS\slash 360 FORTRAN} Math Function Library", type = "????", number = "SSD 169, C4773", institution = "SHARE Secretary Distribution", address = "????", pages = "47--53", year = "1966", bibdate = "Wed Feb 14 19:13:50 2007", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, } @Article{Larssen:1966:CAC, author = "Gerhard Meidell Larssen", title = "Certification of {Algorithm 56}: {Complete} elliptic integral of the second kind", journal = j-CACM, volume = "9", number = "1", pages = "12--12", month = jan, year = "1966", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Fri Nov 25 18:20:04 MST 2005", bibsource = "http://www.acm.org/pubs/contents/journals/cacm/; https://www.math.utah.edu/pub/tex/bib/cacm1960.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "special functions", } @Article{Mechel:1966:CMB, author = "Fr. Mechel", title = "Calculation of the Modified {Bessel} Functions of the Second Kind with Complex Argument", journal = j-MATH-COMPUT, volume = "20", number = "95", pages = "407--412", month = jul, year = "1966", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Nellis:1966:REE, author = "W. J. Nellis and B. C. Carlson", title = "Reduction and Evaluation of Elliptic Integrals", journal = j-MATH-COMPUT, volume = "20", number = "94", pages = "223--231", month = apr, year = "1966", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Pike:1966:ALG, author = "M. C. Pike and I. D. Hill", title = "{Algorithm 291}: {Logarithm} of Gamma Function", journal = j-CACM, volume = "9", number = "9", pages = "684--684", month = sep, year = "1966", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Fri Nov 25 18:20:09 MST 2005", bibsource = "http://www.acm.org/pubs/contents/journals/cacm/; https://www.math.utah.edu/pub/tex/bib/cacm1960.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "$\log(\Gamma(x))$; special functions", remark = "Fullerton: Short Algol procedure valid only for $ x > 0 $. Accurate to 10 digits.", } @TechReport{Price:1966:NAR, author = "James F. Price", title = "Numerical Analysis and Related Literature for Scientific Computer Users", type = "Mathematical Note", number = "456 (D1-82-0517)", institution = "Mathematics Research Laboratory, Boeing Scientific Research Laboratories", address = "Seattle, WA, USA", edition = "Second", pages = "ix + 191", month = mar, year = "1966", bibdate = "Mon Jun 18 06:55:22 2018", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.dtic.mil/dtic/tr/fulltext/u2/632244.pdf", abstract = "The Second Edition of this annotated bibliography lists the contents of over 150 books in English on numerical analysis and related literature. It is meant for the general scientific computer user and not for the research numerical analyst; the descriptions and suggestions are given with this in mind. It is expected that the most useful section will be the 27-page index which tells in which books various topics may be found. There is also a section describing how to look up further information on such topics which may be found in the literature.", acknowledgement = ack-nhfb, tableofcontents = "Introduction / iv \\ I. Numerical Procedures in Books / 1 \\ II. How to Find What You Want / 157 \\ A. Bibliographies, lists of books, abstracting journals / 157 \\ B. Looking for Tables of Various Functions / 161 \\ C. Keeping Up with Some of the New Literature / 163 \\ III. Subject Index / 165", } @Book{Slater:1966:GHF, author = "Lucy Joan Slater", title = "Generalized Hypergeometric Functions", publisher = pub-CAMBRIDGE, address = pub-CAMBRIDGE:adr, pages = "xiii + 273", year = "1966", LCCN = "QA351 .S565", bibdate = "Sat Oct 30 21:01:55 MDT 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; z3950.loc.gov:7090/Voyager", acknowledgement = ack-nhfb, subject = "Hypergeometric functions", } @Article{Takenaga:1966:EIG, author = "Roy Takenaga", title = "On the Evaluation of the Incomplete Gamma Function (in {Technical Notes and Short Papers})", journal = j-MATH-COMPUT, volume = "20", number = "96", pages = "606--610", month = oct, year = "1966", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Thompson:1966:ESI, author = "Rory Thompson", title = "Evaluation of $ {I}_n(b) = 2 \pi^{-1} \int^\infty_0 \big (\frac {sin x}{x} \big)^n \cos (b x) d x $ and of Similar Integrals (in {Technical Notes and Short Papers})", journal = j-MATH-COMPUT, volume = "20", number = "94", pages = "330--332", month = apr, year = "1966", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Book{Tolke:1966:PFT, author = "Friedrich T{\"o}lke", title = "{Praktische Funktionenlehre. 2. Theta-Funktionen und spezielle Weierstrasssche Funktionen}. ({German}) [{Practical} functional theory. 2. {Theta} functions and special {Weierstrass} functions]", publisher = pub-SV, address = pub-SV:adr, pages = "vii + 248", year = "1966", ISBN = "", ISBN-13 = "", LCCN = "????", bibdate = "Mon Feb 13 19:01:10 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, language = "German", } @TechReport{Tricomi:1966:RUS, author = "F. G. Tricomi", title = "Lectures on the use of special functions by calculations with electronic computers", type = "Lecture Series", number = "47", institution = "The Institute for Fluid Dynamics and Applied Mathematics, University of Maryland, College Park", address = "College Park, MD, USA", year = "1966", bibdate = "Tue Mar 14 18:48:58 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, } @Book{Watson:1966:TTB, author = "G. N. Watson", title = "A Treatise on the Theory of {Bessel} Functions", publisher = pub-CAMBRIDGE, address = pub-CAMBRIDGE:adr, edition = "Second", pages = "vi + 804", year = "1966", ISBN = "0-521-09382-1", ISBN-13 = "978-0-521-09382-8", LCCN = "QA 408 W33t 1966", bibdate = "Fri Nov 24 13:53:35 MST 1995", bibsource = "https://www.math.utah.edu/pub/bibnet/subjects/matched-field-proc.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", alias = "Watson 66a", sthbib = "M2 Wat 82 55", } @Article{Wood:1966:DBI, author = "Van E. Wood and R. P. Kenan and M. L. Glasser", title = "{Doppler} Broadening Integrals (in {Technical Notes and Short Papers})", journal = j-MATH-COMPUT, volume = "20", number = "96", pages = "610--611", month = oct, year = "1966", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Anonymous:1967:CAP, author = "Anonymous", title = "Convergence acceleration from the point of view of linear programming", journal = j-BIT, volume = "7", number = "3", pages = "256--256", month = sep, year = "1967", CODEN = "BITTEL, NBITAB", DOI = "https://doi.org/10.1007/BF01939269", ISSN = "0006-3835 (print), 1572-9125 (electronic)", ISSN-L = "0006-3835", bibdate = "Wed Jan 4 18:52:10 MST 2006", bibsource = "http://springerlink.metapress.com/openurl.asp?genre=issue&issn=0006-3835&volume=7&issue=3; https://www.math.utah.edu/pub/tex/bib/bit.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.springerlink.com/openurl.asp?genre=article&issn=0006-3835&volume=7&issue=3&spage=256", acknowledgement = ack-nhfb, fjournal = "BIT (Nordisk tidskrift for informationsbehandling)", journal-URL = "http://link.springer.com/journal/10543", keywords = "convergence acceleration", } @Article{Bond:1967:AAF, author = "Gillian Bond and M. L. V. Pitteway", title = "{Algorithm 301}: {Airy} Function", journal = j-CACM, volume = "10", number = "5", pages = "291--292", month = may, year = "1967", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Fri Nov 25 18:20:13 MST 2005", bibsource = "http://www.acm.org/pubs/contents/journals/cacm/; https://www.math.utah.edu/pub/tex/bib/cacm1960.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "Airy functions; special functions", remark = "Fullerton: 100-line Algol program for $ \operatorname {Ai} $, $ \operatorname {Bi} $. and their derivatives.", } @Article{Cody:1967:CAN, author = "W. J. Cody and K. E. Hillstrom", title = "{Chebyshev} Approximations for the Natural Logarithm of the Gamma Function", journal = j-MATH-COMPUT, volume = "21", number = "98", pages = "198--203", month = apr, year = "1967", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", remark = "Fullerton: Relative errors down to $ 10^{-17} $.", } @Article{Cody:1967:CRC, author = "W. J. Cody and Henry C. {Thacher, Jr.}", title = "Corrigendum: ``{Rational Chebyshev approximations for Fermi--Dirac integrals of orders $ - 1 / 2 $, $ 1 / 2 $, and $ 3 / 2 $''}", journal = j-MATH-COMPUT, volume = "21", number = "99", pages = "525--525", month = jul, year = "1967", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Mon Sep 26 19:36:03 1994", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/c/cody-william-j.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "See \cite{Cody:1967:RCA}.", URL = "http://www.jstor.org/stable/2003289", acknowledgement = ack-nhfb, fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", xxmonth = "(none)", } @Article{Cody:1967:LEA, author = "William J. {Cody, Jr.}", title = "Letter to the {Editor}: Another Aspect of Economical Polynomials", journal = j-CACM, volume = "10", number = "9", pages = "531--531", month = sep, year = "1967", CODEN = "CACMA2", DOI = "https://doi.org/10.1145/363566.363577", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Thu Nov 17 10:20:03 1994", bibsource = "http://www.acm.org/pubs/contents/journals/cacm/; https://www.math.utah.edu/pub/bibnet/authors/c/cody-william-j.bib; https://www.math.utah.edu/pub/tex/bib/cacm1960.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", note = "See \cite{Fike:1967:MEP}.", abstract = "In his paper ``Methods of Evaluating Polynomial Approximations in Function Evaluation Routines'' [Comm. ACM 10, (March 1967)], C. T. Fike fails to discuss one very important aspect of the ``economical'' methods for polynomials. Since these evaluation methods involve a decreased number of arithmetic operations over the usual Horner's method (or at least replace a multiplication by an addition) the implication is that they are faster to execute. Dr. Fike points out that these methods can be poorly conditioned for particular polynomials, thus requiring extended precision or fixed-point arithmetic to maintain accuracy and costing more in time than Horner's method. But even if we assume the methods are well conditioned, the need to store away and retrieve intermediate results in some machines with only one floating-point arithmetic register can wipe out the time savings effected by a reduction in the number of arithmetic operations. On many of today's high-performance computers the time required to store away and retrieve a result is about the same as the time required for a floating-point addition. It is no longer sufficient to estimate the efficiency of a method by a count of arithmetic operations alone.", acknowledgement = ack-wjc # " and " # ack-nhfb, fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "floating-point arithmetic", } @Article{Cody:1967:RCA, author = "W. J. Cody and Henry C. {Thacher, Jr.}", title = "Rational {Chebyshev} approximations for {Fermi--Dirac} integrals of orders $ - 1 / 2 $, $ 1 / 2 $, and $ 3 / 2 $", journal = j-MATH-COMPUT, volume = "21", number = "97", pages = "30--40", month = jan, year = "1967", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Mon Sep 26 19:23:19 1994", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/c/cody-william-j.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "See also \cite{Cody:1967:CRC}.", URL = "http://www.jstor.org/stable/2003468", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", remark = "Fullerton: Relative errors down to $ 10^{-9} $.", } @Article{DiDonato:1967:ECI, author = "A. R. DiDonato and M. P. Jarnagin", title = "The Efficient Calculation of the Incomplete Beta-Function Ratio for Half-Integer Values of the Parameters $ a, b $", journal = j-MATH-COMPUT, volume = "21", number = "100", pages = "652--662", month = oct, year = "1967", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Fair:1967:RAI, author = "Wyman G. Fair and Yudell L. Luke", title = "Rational Approximations to the Incomplete Elliptic Integrals of the First and Second Kinds", journal = j-MATH-COMPUT, volume = "21", number = "99", pages = "418--422", month = jul, year = "1967", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Fettis:1967:MCI, author = "Henry E. Fettis", title = "More on the Calculation of the Integral $ {I}_n(b) = \frac {2}{\pi } \int^\infty_0 \big (\frac {\sin x}{x} \big)^n \cos b x \, d x $ (in {Technical Notes and Short Papers})", journal = j-MATH-COMPUT, volume = "21", number = "100", pages = "727--730", month = oct, year = "1967", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Fike:1967:LER, author = "C. T. Fike", title = "Letter to the {Editor}: {A} rational approximation optimal by {Moursund}'s criterion", journal = j-CACM, volume = "10", number = "11", pages = "683--684", month = nov, year = "1967", CODEN = "CACMA2", DOI = "https://doi.org/10.1145/363790.363795", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Fri Nov 25 18:20:16 MST 2005", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; http://www.acm.org/pubs/contents/journals/cacm/; https://www.math.utah.edu/pub/tex/bib/cacm1960.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", note = "See \cite{Moursund:1967:OSV,Fike:1966:SAS}", acknowledgement = ack-nhfb, fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "elementary function; square root", remark = "Gives a starting value for $ \sqrt {x} $ ($x$ on $ [1 / 16, 1]$) of $ R*(x) = 1.68212586 - 1.28977371 / (x + 0.84106293)$, with an error of $ 2^{-12.496}$.", } @Article{Fike:1967:MEP, author = "C. T. Fike", title = "Methods of Evaluating Polynomial Approximations in Function Evaluation Routines", journal = j-CACM, volume = "10", number = "3", pages = "175--178", month = mar, year = "1967", CODEN = "CACMA2", DOI = "https://doi.org/10.1145/363162.363200", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Fri Nov 25 18:20:12 MST 2005", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; http://www.acm.org/pubs/contents/journals/cacm/; https://www.math.utah.edu/pub/bibnet/authors/c/cody-william-j.bib; https://www.math.utah.edu/pub/tex/bib/cacm1960.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", note = "See remark on efficiency \cite{Cody:1967:LEA}.", abstract = "The method of nested multiplication is commonly used in function evaluation routines to evaluate approximation polynomials. New polynomial evaluation methods have been developed in recent years which require fewer multiplications than nested multiplication and may therefore be preferable for use in function evaluation routines. Although some of these methods do not appear to be practically useful because of rounding-error difficulties, several methods of evaluating low-degree polynomials have been found to be satisfactory. Three such methods are described and illustrated.", acknowledgement = ack-nhfb, fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", received = "August 1966 (revised December 1966)", } @Article{Friedland:1967:AAV, author = "Paul Friedland", title = "{Algorithm 312}: {Absolute} Value and Square Root of a Complex Number", journal = j-CACM, volume = "10", number = "10", pages = "665--665", month = oct, year = "1967", CODEN = "CACMA2", DOI = "https://doi.org/10.1145/363717.363780", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Fri Nov 25 18:20:15 MST 2005", bibsource = "http://www.acm.org/pubs/contents/journals/cacm/; https://www.math.utah.edu/pub/tex/bib/cacm1960.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "$\abs(z)$; $\sqrt(z)$; elementary functions", } @Article{Gautschi:1967:CAT, author = "Walter Gautschi", title = "Computational Aspects of Three-Term Recurrence Relations", journal = j-SIAM-REVIEW, volume = "9", number = "1", pages = "24--82", month = jan, year = "1967", CODEN = "SIREAD", DOI = "https://doi.org/10.1137/1009002", ISSN = "0036-1445 (print), 1095-7200 (electronic)", ISSN-L = "0036-1445", bibdate = "Thu Mar 27 09:05:42 MDT 2014", bibsource = "http://epubs.siam.org/toc/siread/9/1; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/siamreview.bib", URL = "http://link.aip.org/link/?SIR/9/24/1", acknowledgement = ack-nhfb, fjournal = "SIAM Review", journal-URL = "http://epubs.siam.org/sirev", keywords = "Bessel functions; continued fractions; Coulomb wave functions; Fourier coefficients; incomplete beta functions; incomplete gamma functions; Legendre functions; Sturm--Liouville boundary value problems; three-term recurrence relations", onlinedate = "January 1967", remark = "This paper is frequently cited in later work on continued fractions, three-term recurrence relations, and special functions.", } @Article{Goldstein:1967:CSB, author = "Max Goldstein and C. W. Clenshaw and Susan M. Picken", title = "{Chebyshev} Series for {Bessel} Functions of Fractional Order", journal = j-MATH-COMPUT, volume = "21", number = "99", pages = "509--??", month = jul, year = "1967", CODEN = "MCMPAF", DOI = "https://doi.org/10.2307/2003271", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Sun Nov 12 09:25:35 2023", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/c/clenshaw-charles-w.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, author-dates = "Charles William Clenshaw (15 March 1926--23 September 2004)", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", remark = "TO DO: Why is this missing from journal bibliography file, mathcomp1970.bib?", } @Article{Gunn:1967:ACW, author = "J. H. Gunn", title = "{Algorithm 300}: {Coulomb} Wave Functions", journal = j-CACM, volume = "10", number = "4", pages = "244--245", month = apr, year = "1967", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Fri Nov 25 18:20:12 MST 2005", bibsource = "http://www.acm.org/pubs/contents/journals/cacm/; https://www.math.utah.edu/pub/tex/bib/cacm1960.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "See remark \cite{Vos:1973:RAC}.", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "Coulomb wave functions; special functions", remark = "Fullerton: 150-line Algol procedure that is superseded by other routines in the physics literature.", } @Article{Hill:1967:ACS, author = "I. D. Hill and M. C. Pike", title = "{Algorithm 299}: {Chi}-Squared Integral", journal = j-CACM, volume = "10", number = "4", pages = "243--244", month = apr, year = "1967", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Fri Nov 25 18:20:12 MST 2005", bibsource = "http://www.acm.org/pubs/contents/journals/cacm/; https://www.math.utah.edu/pub/tex/bib/cacm1960.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "See also \cite{Hill:1985:RCS,elLozy:1976:RAC}.", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "chi-squared; probability functions", remark = "Fullerton: Short Algol procedure.", } @Article{Hill:1967:ANCa, author = "I. D. Hill and S. A. Joyce", title = "{Algorithm 304}: {Normal} Curve Integral", journal = j-CACM, volume = "10", number = "6", pages = "374--375", month = jun, year = "1967", CODEN = "CACMA2", DOI = "https://doi.org/10.1145/363332.363411", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Fri Nov 25 18:20:13 MST 2005", bibsource = "http://www.acm.org/pubs/contents/journals/cacm/; https://www.math.utah.edu/pub/tex/bib/cacm1960.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "See remarks \cite{Hill:1967:RAS,Bergson:1968:ACR}.", abstract = "{\tt normal(x,upper)} calculates the curve, i.e., tail area of the standardized normal curve, i.e., $ (1 / \sqrt {2 \pi }) \int \exp ( - t^2 / 2) \, d t $. If {\tt upper} is {\tt true}, the limits of integration are $x$ and $ \infty $. If {\tt upper} is {\tt false}, the limits of integration are $ - \infty $ and $x$.", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "probability functions", remark = "Fullerton: 75-line Algol procedure that is superseded by numerous $ \erf $ routines.", } @Article{Hill:1967:RAS, author = "I. D. Hill and S. A. Joyce", title = "Remarks on {Algorithm 123} [{S15}]: {Real} error function, {{\tt ERF(x)}}; {Algorithm 180} [{S15}]: {Error} Function --- Large $ {X} $; {Algorithm 181} [{S15}]: {Complementary} Error Function --- Large $ {X} $; {Algorithm 209} [{S15}]: {Gauss}; {Algorithm 226} [{S15}]: {Normal} Distribution Function; {Algorithm 272} [{S15}]: {Procedure} for the Normal Distribution Functions; {Algorithm 304} [{S15}]: {Normal} Curve Integral", journal = j-CACM, volume = "10", number = "6", pages = "377--378", month = jun, year = "1967", CODEN = "CACMA2", DOI = "https://doi.org/10.1145/363332.365433", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Fri Nov 25 18:20:13 MST 2005", bibsource = "http://www.acm.org/pubs/contents/journals/cacm/; https://www.math.utah.edu/pub/tex/bib/cacm1960.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "See \cite{Cyvin:1964:AND,MacLaren:1965:APN,Hill:1967:ANCa}.", acknowledgement = ack-nhfb, fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "$\erf(x)$; $\erfc(x)$; probability functions; special functions", } @Article{Kilpatrick:1967:CIP, author = "J. E. Kilpatrick and Shigetoshi Katsura and Yuji Inoue", title = "Calculations of Integrals of Products of {Bessel} Functions", journal = j-MATH-COMPUT, volume = "21", number = "99", pages = "407--412", month = jul, year = "1967", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Knuth:1967:CTE, author = "Donald E. Knuth and Thomas J. Buckholtz", title = "Computation of Tangent, {Euler}, and {Bernoulli} Numbers", journal = j-MATH-COMPUT, volume = "21", number = "100", pages = "663--688", month = oct, year = "1967", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "65.25", MRnumber = "36 #4787", bibdate = "Fri Mar 22 18:03:29 MST 2002", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database; MathSciNet database", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", remark = "Fullerton: The first 119. 120 and 250 tangent, Euler and Bernoulli numbers, respectively.", } @InCollection{Korn:1967:SFA, author = "Granino A. Korn", title = "A Survey of Function-approximation Techniques", crossref = "Klerer:1967:DCU", pages = "2:3--2:17", year = "1967", acknowledgement = ack-nhfb, bibdate = "Thu Feb 12 15:33:11 2026", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", } @Book{MacRobert:1967:SHE, author = "Thomas Murray MacRobert and Ian Naismith Sneddon", title = "Spherical harmonics; an elementary treatise on harmonic functions, with applications", volume = "98", publisher = pub-PERGAMON, address = pub-PERGAMON:adr, edition = "Third", pages = "xviii + 349", year = "1967", LCCN = "QA406 .M3 1967", bibdate = "Sat Oct 30 21:22:03 MDT 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; z3950.loc.gov:7090/Voyager", series = "International series of monographs in pure and applied mathematics", acknowledgement = ack-nhfb, author-dates = "1884--1962", subject = "Spherical harmonics", } @Book{Meinardus:1967:AFT, author = "G{\"u}nter Meinardus", title = "Approximation of functions: Theory and numerical methods", volume = "13", publisher = pub-SV, address = pub-SV:adr, pages = "viii + 198", year = "1967", ISBN = "0-387-03985-6, 3-540-03985-6", ISBN-13 = "978-0-387-03985-5, 978-3-540-03985-3", ISSN = "0081-3877", LCCN = "QA221 .M3813", bibdate = "Thu Oct 19 17:07:54 MDT 2023", bibsource = "fsz3950.oclc.org:210/WorldCat; https://www.math.utah.edu/pub/bibnet/authors/r/rice-john-r.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", series = "Springer Tracts in Natural Philosophy", URL = "http://catalogue.bnf.fr/ark:/12148/cb37378349d", acknowledgement = ack-nhfb, author-dates = "(1926--2007)", remark = "Translation by Larry L. Schumaker of \cite{Meinardus:1964:AFI}.", subject = "Approximation theory; Numerical analysis; Th{\'e}orie de l'approximation; Analyse num{\'e}rique; Fonctions (math{\'e}matiques); Approximation, Th{\'e}orie de l'; Analyse num{\'e}rique; Approximation theory; Numerical analysis; Approximation; Funktion; Mathematik", tableofcontents = "Part I. Linear Approximation \\ 1. The General Linear Approximation Problem / 1 \\ 1.1. Statement of the Problem. Existence Theorem / 1 \\ 1.2. Strictly Convex Spaces. Hilbert Space / 2 \\ 1.3. Maximal Linear Functionals / 4 \\ 2. Dense Systems / 5 \\ 2.1. A General Criterion of Banach / 5 \\ 2.2. Approximation Theorems of Weierstrass and Muntz / 6 \\ 2.3. Approximation Theorems in the Complex Plane / 10 \\ 3. General Theory of Linear Tchebycheff Approximation / 13 \\ 3.1. Fundamentals. The Theorem of Kolmogoroff / 13 \\ 3.2. The Haar Uniqueness Theorem. Linear Functionals and Alternants / 16 \\ 3.3. Further Uniqueness Results / 24 \\ 3.4. Invariants / 26 \\ 3.5. Vector-valued Functions / 28 \\ 4. Special Tchebycheff Approximations / 28 \\ 4.1. Tchebycheff Systems / 28 \\ 4.2. Tchebycheff Polynomials / 31 \\ 4.3. The Function ?? / 33 \\ 4.4. A Problem of Bernstein and Achieser / 36 \\ 4.5 Zolotareff's Problem / 41 \\ 5. Estimating the Magnitude of Error in Trigonometric and Polynomial Approximation / 45 \\ 5.1. Projection Operators. Linear Polynomial Operators / 45 \\ 5.2. The Connection between Trigonometric and Polynomial Approximation / 45 \\ 5.3. The Fej{\'e}r Operator / 47 \\ 5.4. The Korovkin Operators / 50 \\ 5.5. The Theorems of D. Jackson / 52 \\ 5.6. The Theorems of Bernstein and Zygmund / 57 \\ 5.7. Supplements / 65 \\ 6. Approximation by Polynomials and Related Functions / 72 \\ 6.1. Foundations / 72 \\ 6.2. Upper Bounds for En (??) / 77 \\ 6.3. Lower Bounds for En (??) / 82 \\ 6.4. Dependence of the Approximation on the Interval / 85 \\ 6.5. Regular Haar Systems / 87 \\ 6.6. Asymptotic Results / 90 \\ 6.7. Results for tho Alternants / 101 \\ 7. Numerical Methods for Linear Tchebycheff Approximation / 105 \\ 7.1. The Iterative Methods of Remez / 105 \\ 7.2. Initial Approximations / 116 \\ 7.3. Direct Methods / 122 \\ 7.4. Discretization. Other Methods / 124 \\ Part II. Non-linear Approximation \\ 8. General Theory of Non-linear Tchebycheff Approximation / 131 \\ 8.1. Survey of the Problem. A Generalization of the Kolmogoroff Theorem / 131 \\ 8.2. The Haar Uniqueness Theorem. Alternants / 141 \\ 8.3. The Investigations of Rice / 148 \\ 8.4. The Newton Iteration Method / 149 \\ 8.5. H-Sets / 153 \\ 9. Rational Approximation / 154 \\ 9.1. Existence. Invariants. A Theorem of Walsh / 154 \\ 9.2. Theorems on Alternants. Anomalies. Continuity. Examples / 160 \\ 9.3. Asymptotic Results. Small Intervals / 167 \\ 9.4. Numerical Methods / 170 \\ 10. Exponential Approximation / 176 \\ 10.1. The Results of Rice / 176 \\ 10.2. An Anomaly Theorem. Constructive Methods / 179 \\ 11. Segment Approximation / 183 \\ 11.1. Statement of the Problem. Hypotheses / 183 \\ 11.2. The principle of Lawson / 184 \\ H.3. Equidegree Polynomial Approximation / 188 \\ Bibliography / 189 \\ Subject Index / 197", } @Article{Moody:1967:ADF, author = "William T. Moody", title = "Approximations for the Psi (Digamma) Function (in Technical Notes and Short Notices)", journal = j-MATH-COMPUT, volume = "21", number = "97", pages = "112--112", month = jan, year = "1967", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Moursund:1967:OSV, author = "David G. Moursund", title = "Optimal starting values for {Newton--Raphson} calculation of $ \sqrt {x} $", journal = j-CACM, volume = "10", number = "7", pages = "430--432", month = jul, year = "1967", CODEN = "CACMA2", DOI = "https://doi.org/10.1145/363427.363454", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", MRclass = "65.25", MRnumber = "39\#2297", bibdate = "Thu Sep 1 10:15:43 1994", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; http://www.acm.org/pubs/contents/journals/cacm/; https://www.math.utah.edu/pub/tex/bib/cacm1960.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", note = "See letter \cite{Fike:1967:LER}.", abstract = "The problem of obtaining starting values for the Newton-Raphson calculation of $ \sqrt {x} $ on a digital computer is considered. It is shown that the conventionally used best uniform approximations to $ \sqrt {x} $ do not provide optimal starting values. The problem of obtaining optimal starting values is stated, and several basic results are proved. A table of optimal polynomial starting values is given.", acknowledgement = ack-nj # " and " # ack-nhfb, fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "$\sqrt(x)$; elementary functions", remark = "Title of article has incorrect $ \sqrt (x^{1 / 2}) $: the article discusses computation of {\tt sqrt(x)}.", } @Article{Olver:1967:BSS, author = "F. W. J. Olver", title = "Bounds for the Solutions of Second-Order Linear Difference Equations [{Anger--Weber} and {Struve} Functions]", journal = j-J-RES-NATL-BUR-STAND-1934, volume = "71B", number = "4", pages = "161--166", month = oct, year = "1967", ISSN = "0091-0635", bibdate = "Sat Oct 30 09:37:44 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Journal of Research of the National Bureau of Standards", journal-URL = "https://www.nist.gov/nist-research-library/journal-research-nist/past-papers", remark = "Fullerton: Truncation error estimates of an earlier published algorithm.", } @Article{Olver:1967:NSS, author = "F. W. J. Olver", title = "Numerical solution of second-order linear difference equations", journal = j-J-RES-NATL-BUR-STAND-B, volume = "71B", number = "2--3", pages = "111--129", month = apr, year = "1967", CODEN = "JNBBAU", DOI = "https://doi.org/10.6028/jres.071B.018", ISSN = "0022-4340", MRclass = "65.70", MRnumber = "221789", MRreviewer = "G. N. Lance", bibdate = "Sun Nov 5 09:03:34 2023", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/o/olver-frank-w-j.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "https://nvlpubs.nist.gov/nistpubs/jres/71B/jresv71Bn2-3p111_A1b.pdf", abstract = "A new algorithm is given for computing the solution of any second-order linear difference equation which is applicable when simple recurrence procedures cannot be used because of instability. Compared with the well-known Miller algorithm the new method has the advantages of (i) automatically determining the correct number of recurrence steps. (ii) applying to inhomogeneous difference equations, (iii) enabling more powerful error analyses to be constructed.\par The method is illustrated by numerical computations, including error analyses of Anger--Weber, Struve, and Bessel functions, and the solution of a differential equation in Chebyshev series", acknowledgement = ack-nhfb, author-dates = "Frank William John Olver (15 December 1924--23 April 2013)", fjournal = "Journal of Research of the National Bureau of Standards. Section B. Mathematics and Mathematical Physics", journal-URL = "http://www.nist.gov/nvl/jrespastpapers.cfm", keywords = "Chebyshev series; difference equations; error analysis; Miller algorithm. recurrence methods; special functions", } @Article{Pike:1967:RAI, author = "M. C. Pike and I. D. Hill", title = "Remark on {Algorithm 179}: {Incomplete} {Beta} ratio", journal = j-CACM, volume = "10", number = "6", pages = "375--376", month = jun, year = "1967", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Fri Nov 25 18:20:13 MST 2005", bibsource = "http://www.acm.org/pubs/contents/journals/cacm/; https://www.math.utah.edu/pub/tex/bib/cacm1960.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "$B(z,w)$; beta function; special functions", remark = "Fullerton: Corrections to an Algol procedure.", } @Article{Pitteway:1967:RAA, author = "M. L. V. Pitteway", title = "Remark on {Algorithm 301}: {Airy} function", journal = j-CACM, volume = "10", number = "7", pages = "453--453", month = jul, year = "1967", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Fri Nov 25 18:20:14 MST 2005", bibsource = "http://www.acm.org/pubs/contents/journals/cacm/; https://www.math.utah.edu/pub/tex/bib/cacm1960.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "Airy functions; special functions", remark = "Fullerton: Corrections to an Algol procedure.", } @Book{Sen:1967:TSF, author = "Bibhutibhusan Sen", title = "A treatise on special functions, for scientists and engineers", publisher = "Allied Publishers", address = "Bombay, India", pages = "164", year = "1967", LCCN = "QA351 .S45", bibdate = "Fri Oct 29 21:30:38 MDT 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; z3950.loc.gov:7090/Voyager", acknowledgement = ack-nhfb, subject = "Functions, Special; Spherical harmonics", } @Book{Tolke:1967:PFE, author = "Friedrich T{\"o}lke", title = "{Praktische Funktionenlehre. 4. Elliptische Integralgruppen und Jacobische elliptische Funktionen im Komplexen}. ({German}) [{Practical} functional theory. 4. {Elliptical} integral groups and {Jacobian} elliptic functions in the complex plane]", publisher = pub-SV, address = pub-SV:adr, pages = "viii + 191", year = "1967", DOI = "https://doi.org/10.1007/978-3-662-36381-2", ISBN = "3-662-36381-X, 3-662-35552-3 (print), 3-662-36381-X (e-book)", ISBN-13 = "978-3-662-36381-2, 978-3-662-35552-7 (print), 978-3-662-36381-2 (e-book)", LCCN = "????", bibdate = "Mon Feb 13 19:01:10 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.springerlink.com/content/978-3-662-36381-2", acknowledgement = ack-nhfb, language = "German", } @Book{Tolke:1967:PFJ, author = "Friedrich T{\"o}lke", title = "{Praktische Funktionenlehre. 3. Jacobische elliptische Funktionen, Legendresche elliptische Normalintegrale und spezielle Weierstrasssche Zeta- und Sigma-Funktionen}. ({German}) [{Practical} functional theory. 3. {Jacobian} elliptic functions, Legendre elliptical normal Integrals and special {Weierstrass} zeta- and sigma functions]", publisher = pub-SV, address = pub-SV:adr, pages = "viii + 180", year = "1967", DOI = "https://doi.org/10.1007/978-3-662-36379-9", ISBN = "3-642-50264-4, 3-662-36379-8, 3-662-35550-7 (print), 3-662-36379-8 (e-book)", ISBN-13 = "978-3-642-50264-4, 978-3-662-36379-9, 978-3-662-35550-3 (print), 978-3-662-36379-9 (e-book)", LCCN = "????", bibdate = "Mon Feb 13 19:01:10 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.springerlink.com/content/978-3-662-36379-9", acknowledgement = ack-nhfb, language = "German", tableofcontents = "5 Jacobische elliptische Funktionen und zugeh{\"o}rige logarithmische Ableitungen \\ 108. Definitionen \\ 109. Funktionalgleichungen \\ 110. Periodenverhalten und Substitutionen \\ 111. Funktionswerte an den Stellen $ 0, \pm \frac{1}{2}, \pm \frac{ix}{2}, \pm \frac{1}{2}, \pm \frac{ix}{2} $ bzw. $ 0, \pm K, \pm iK', \pm K \pm iK' $ \\ 112. Trigonometrische und hyperbolische Reihenentwicklungen \\ 113. Potenzreihen-Entwicklungen \\ 114. Imagin{\"a}re Argumenttransformation, reziproke Modultransformation und imagin{\"a}re Modultransformation \\ 115. Ableitungen \\ 116. Gausssche und Landensche Transformation. Substitutionen f{\"u}r $ \zeta \pm \frac{1}{4}$ und $\zeta \pm \frac{ix}{4} $ \\ 117. Additionstheoreme. Transformationsgleichungen f{\"u}r doppeltes und halbes Argument. Weitere Substitutionen f{\"u}r $ \zeta \pm \frac{1}{4}$ und $\zeta \pm \frac{ix}{4}$ sowie f{\"u}r $\zeta \pm \frac{1}{4} \pm \frac{ix}{4} $ \\ 118. Die Logarithmen der logarithmischen Ableitungen der Jacobischen elliptischen Funktionen \\ 119. {\"U}berg{\"a}nge vom (?,?)-System auf das $(z, k)$-System \\ 120. Funktionsverlauf der Jacobischen elliptischen Funktionen und der zugeh{\"o}rigen Ableitungen und logarithmischen Ableitungen im Reellen. Ausartungen \\ 121. Differentialgleichungen erster und zweiter Ordnung \\ 122. Die Integrale der Jacobischen elliptischen Funktionen \\ 123. Die Integrale der logarithmischen Ableitungen der Jacobischen elliptischen Funktionen \\ 6 Umkehrfunktionen der Jacobischen elliptischen Funktionen und elliptische Normalintegrale erster Gattung. Elliptische Amplitudenfunktion sowie Legendresche $F$- und $E$-Funktion. Elliptische Normalintegrale zweiter Gattung. Jacobische Zeta- und Heumansche Lambda-Funktion \\ 124. Die 18 Umkehrfunktionen der Jacobischen elliptischen Funktionen und ihrer logarithmischen Ableitungen. (Elliptische Normalintegrale erster Gattung.) Additionstheoreme der Umkehrfunktionen \\ 125. Elliptische Normalintegrale erster Gattung in hyperbolischer Form \\ 126. Potenzreihen-Entwicklungen der Umkehrfunktionen \\ 127. Die elliptische Amplitudenfunktion $? = \am(z, k)$ und ihre Umkehrfunktion $z = F(?, k)$. Die vier trigonometrischen Legendreschen Normalintegrale erster Gattung \\ 128. Darstellung der 18 Umkehrfunktionen und der elliptischen Normalintegrale erster Gattung durch die Funktion F. Die vier hyperbolischen Legendreschen Normalintegrale erster Gattung und die Funktion $F$ f{\"u}r imagin{\"a}res Argument \\ 129. Die Legendresche $E$-Funktion f{\"u}r reelles und imagin{\"a}res Argument \\ 130. Die 18 Integrale der Quadrate der Jacobischen elliptischen Funktionen und ihrer logarithmischen Ableitungen, die 12 durch Umformung der letzteren entstehenden hyperbolischen Integrale, die 24 Normalintegrale zweiter Gattung und die acht trigonometrischen und hyperbolischen Legendreschen Normalintegrale zweiter Gattung \\ 131. Die 46 Normalintegrale erster und zweiter Gattung mit linearen trigonometrischen und hyperbolischen Funktionen \\ 132. Jacobische Zeta-Funktion und Heumansche Lambda-Funktion \\ 7 Normalintegrale dritter Gattung. Legendresche $\Pi$-Funktion. Zur{\"u}ckf{\"u}hrung des allgemeinen elliptischen Integrals auf Normalintegrale erster, zweiter und dritter Gattung \\ 133. Die 96 Normalintegrale dritter Gattung in Jacobischer Form \\ 134. Die acht zu den logarithmischen Ableitungen der Jacobischen elliptischen Funktionen geh{\"o}rigen Normalintegrale dritter Gattung \\ 135. 48 Quotientenintegrale und 48 spezielle Normalintegrale dritter Gattung in der Jacobischen Form \\ 136. Algebraische Form der elliptischen Normalintegrale dritter Gattung \\ 137. Darstellung der vollst{\"a}ndigen Normalintegrale dritter Gattung durch Jacobische Zeta- und Heumansche Lambda-Funktionen \\ 138. Die $\Pi$-Funktion und die Integrale dritter Gattung in trigonometrischer Form \\ 139. Die 48 speziellen Normalintegrale dritter Gattung in algebraischer Form \\ 140. Weitere sechs spezielle Normalintegrale dritter Gattung \\ 141. Zur{\"u}ckf{\"u}hrung des allgemeinen elliptischen Integrals in der Legendreschen Form auf Normalintegrale erster, zweiter und dritter Gattung \\ 8 Spezielle Weierstra{\ss}sche Zeta-Funktionen \\ 142. Definitions- und Funktionalgleichungen \\ 143. Substitutionen \\ 144. Relatives Periodenverhalten. Spezielle Funktionswerte. Funktionsverlauf \\ 145. Lineare Beziehungen zu den logarithmischen Ableitungen der Jacobischen elliptischen Funktionen und deren Ableitungen \\ 146. Integrale der $\Pi$-Funktionen als Weierstrasssche Zeta-Funktionen und Ableitungen der Zeta-Funktionen \\ 147. Differentialtransformationen f{\"u}r doppelte und halbe Parameter \\ 148. Gausssche und Landensche Transformation \\ 149. Additionstheoreme und Transformationsgleichungen f{\"u}r doppeltes und halbes Argument \\ 150. Trigonometrische, hyperbolische und Potenzreihen-Entwicklungen \\ 151. Homogenit{\"a}tstransformation der Funktionen", } @Article{Verma:1967:NSG, author = "Arun Verma", title = "A Note on the Summation of the Generalised Hypergeometric Functions (in {Technical Notes and Short Papers})", journal = j-MATH-COMPUT, volume = "21", number = "98", pages = "232--236", month = apr, year = "1967", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp1960.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Wigner:1967:BRW, author = "Eugene P. Wigner", title = "Book Review: {Wilhelm Magnus, Fritz Oberhettinger, and R. P. Soni, \booktitle{Formulas and Theorems for the Special Functions of Mathematical Physics}}", journal = j-PHYS-TODAY, volume = "20", number = "12", pages = "81--81", month = dec, year = "1967", CODEN = "PHTOAD", DOI = "https://doi.org/10.1063/1.3034082", ISSN = "0031-9228 (print), 1945-0699 (electronic)", ISSN-L = "0031-9228", bibdate = "Sat Jul 28 07:53:52 MDT 2012", bibsource = "http://www.physicstoday.org/search; https://www.math.utah.edu/pub/bibnet/authors/w/wigner-eugene.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://link.aip.org/link/phtoad/v20/i12/p81/s1", acknowledgement = ack-nhfb, fjournal = "Physics Today", journal-URL = "http://www.physicstoday.org/", } @Article{Wood:1967:CEI, author = "Van E. Wood", title = "{Chebyshev} Expansions for Integrals of the Error Function (in {Technical Notes and Short Papers})", journal = j-MATH-COMPUT, volume = "21", number = "99", pages = "494--496", month = jul, year = "1967", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", remark = "Fullerton: 7-digit approximations for $ i^n \erfc (x), n = 1, 2 $.", } @Article{Yarbrough:1967:PCC, author = "Lynn Yarbrough", title = "Precision calculations of $e$ and $ \pi $ constants", journal = j-CACM, volume = "10", number = "9", pages = "537--537", month = sep, year = "1967", CODEN = "CACMA2", DOI = "https://doi.org/10.1145/363566.363578", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Fri Nov 25 18:20:15 MST 2005", bibsource = "http://www.acm.org/pubs/contents/journals/cacm/; https://www.math.utah.edu/pub/tex/bib/cacm1960.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "floating-point arithmetic; number base conversion", remark = "Gives decimal, octal, and hexadecimal values of $e$ and $ \pi $ to 100 digits, and notes ``The difficulty arises because assemblers and compilers are hardly ever designed to convert decimal constants to a precision of more than a dozen or so digits. Thus, if calculations to greater precision are to be done, constants usually must be input in octal or other binary-derived representation.''. Cited in \cite{Sterbenz:1974:FPC}.", } @Article{Anonymous:1968:ISA, author = "Anonymous", title = "Index by Subject to Algorithms, 1960--1968", journal = j-CACM, volume = "11", number = "12", pages = "827--830", month = dec, year = "1968", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Sat Oct 30 09:29:34 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", } @Book{Arsenin:1968:BES, author = "V. Ja. (Vasilii Jakovlevich) Arsenin", title = "Basic equations and special functions of mathematical physics", publisher = "Iliffe", address = "London, UK", pages = "7 + 361", year = "1968", ISBN = "0-592-05035-1", ISBN-13 = "978-0-592-05035-5", LCCN = "QC20 .A693", bibdate = "Sat Oct 30 18:25:22 MDT 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; z3950.loc.gov:7090/Voyager", note = "Translation by S. Chomet, Kings College, London of Matematicheska{\"e}i{\`\i}a fizika.", acknowledgement = ack-nhfb, subject = "Mathematical physics", } @Article{Ascari:1968:LRG, author = "A. Ascari and P. G. Novario", title = "L'algoritmo {$ Q D $} di {Rutishauser} e la generazione di funzioni speciali nel calcolo automatico. ({Italian}) [{The} {$ Q D $} algorithm of {Rutishauser} and the generation of special functions in automatic calculation]", journal = j-CALCOLO, volume = "5", number = "1", pages = "162--173", month = "????", year = "1968", CODEN = "CALOBK", DOI = "https://doi.org/10.1007/BF02576063", ISSN = "0008-0624 (print), 1126-5434 (electronic)", ISSN-L = "0008-0624", bibdate = "Mon Aug 24 21:37:24 MDT 2020", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/r/rutishauser-heinz.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://link.springer.com/article/10.1007/BF02576063", acknowledgement = ack-nhfb, fjournal = "Calcolo: a quarterly on numerical analysis and theory of computation", journal-URL = "http://link.springer.com/journal/10092", language = "Italian", subject-dates = "Heinz Rutishauser (30 January 1918--10 November 1970)", } @Book{Bell:1968:SFS, author = "W. W. (William Wallace) Bell", title = "Special functions for scientists and engineers", publisher = "Van Nostrand", address = "London, UK", pages = "xiv + 247", year = "1968", LCCN = "QA351 .B4", bibdate = "Fri Oct 29 21:30:38 MDT 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; z3950.loc.gov:7090/Voyager", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", remark = "Fullerton: Gamma, Beta, Legendre, Bessel, and Hypergeometric functions as well as orthogonal polynomials. Reprinted in \cite{Bell:2004:SFS}.", subject = "Functions, Special", } @Article{Bergson:1968:ACR, author = "A. Bergson", title = "Certification of and remark on {Algorithm 304} [{S15}]: {Normal} curve integral", journal = j-CACM, volume = "11", number = "4", pages = "271--271", month = apr, year = "1968", CODEN = "CACMA2", DOI = "https://doi.org/10.1145/362991.363048", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Fri Nov 25 18:20:19 MST 2005", bibsource = "http://www.acm.org/pubs/contents/journals/cacm/; https://www.math.utah.edu/pub/tex/bib/cacm1960.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "See \cite{Hill:1967:ANCa,Hill:1967:RAS}.", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "probability functions", remark = "Fullerton: 50-line Algol Program that is superceded by numerous {\tt erf} routines.", } @Article{Bingulac:1968:RAA, author = "S. P. Bingulac", title = "{R68-38} Accurate Analog Computer Generation of {Bessel} Functions for Large Ranges", journal = j-IEEE-TRANS-COMPUT, volume = "C-17", number = "8", pages = "819--819", month = aug, year = "1968", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/TC.1968.229133", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Wed Jul 13 17:40:50 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", URL = "https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=1687462", acknowledgement = ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", } @Article{Chiarella:1968:EIR, author = "C. Chiarella and A. Reichel", title = "On the Evaluation of Integrals Related to the Error Function", journal = j-MATH-COMPUT, volume = "22", number = "101", pages = "137--143", month = jan, year = "1968", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Cody:1968:RCAa, author = "W. J. Cody and H. C. {Thacher, Jr.}", title = "Rational {Chebyshev} approximations for the exponential integral {$ E_1 (x) $}", journal = j-MATH-COMPUT, volume = "22", number = "103", pages = "641--649", month = jul, year = "1968", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "65.25", MRnumber = "38\#6745", bibdate = "Wed Jan 17 08:57:34 1996", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/c/cody-william-j.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", remark = "Fullerton: Relative errors down to $ 10^{-21} $.", } @Article{Dean:1968:GRB, author = "K. J. Dean", title = "Generator for the reciprocals of binary numbers", journal = j-PROC-IEE, volume = "115", number = "6", pages = "787", month = jun, year = "1968", CODEN = "PIEEAH", DOI = "https://doi.org/10.1049/piee.1968.0142", ISSN = "0020-3270 (print), 2053-7891 (electronic)", ISSN-L = "0020-3270", bibdate = "Thu Apr 10 13:07:02 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, fjournal = "Proceedings of the Institution of Electrical Engineers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=5247218", } @Article{Dorrer:1968:ADS, author = "Egon Dorrer", title = "{Algorithm 322}: {$F$}-Distribution [{S14}]", journal = j-CACM, volume = "11", number = "2", pages = "116--117", month = feb, year = "1968", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Fri Nov 25 18:20:18 MST 2005", bibsource = "http://www.acm.org/pubs/contents/journals/cacm/; https://www.math.utah.edu/pub/tex/bib/cacm1960.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", remark = "Fullerton: 50-line Algol Program.", } @Book{Fike:1968:CEM, author = "C. T. Fike", title = "Computer Evaluation of Mathematical Functions", publisher = pub-PH, address = pub-PH:adr, pages = "xii + 227", year = "1968", LCCN = "QA297 .F5", bibdate = "Thu Sep 1 10:12:51 1994", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nj, } @Article{Fleckner:1968:MCF, author = "Oscar L. Fleckner", title = "A Method for the Computation of the {Fresnel} Integrals and Related Functions", journal = j-MATH-COMPUT, volume = "22", number = "103", pages = "635--640", month = jul, year = "1968", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Book{Fox:1968:CPN, author = "L. Fox and I. B. Parker", title = "{Chebyshev} Polynomials in Numerical Analysis", publisher = pub-OXFORD, address = pub-OXFORD:adr, pages = "ix + 205", year = "1968", ISBN = "0-19-859614-6", ISBN-13 = "978-0-19-859614-1", LCCN = "QA297 .F65", MRclass = "65.10", MRnumber = "228149", MRreviewer = "G. N. Lance", bibdate = "Mon Nov 13 14:02:18 2023", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/f/fox-leslie.bib; https://www.math.utah.edu/pub/bibnet/subjects/acc-stab-num-alg.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcw.bib; z3950.loc.gov:7090/Voyager", note = "Reprinted in 1972 with corrections, but same ISBN.", series = "Oxford mathematical handbooks", acknowledgement = ack-nhfb, author-dates = "Leslie Fox (30 September 1918--1 August 1992)", mynote = "JRUL: 517", series-editor = "John Crank and C. C. Ritchie", subject = "Chebyshev polynomials; Numerical analysis", tableofcontents = "TO DO: find this!", xxauthor = "L. (Leslie) Fox and I. B. (Ian Bax) Parker", } @Article{Galant:1968:HAG, author = "D. C. Galant and P. F. Byrd", title = "High Accuracy Gamma Function Values for Some Rational Arguments (in {Technical Notes and Short Papers})", journal = j-MATH-COMPUT, volume = "22", number = "104", pages = "885--887", month = oct, year = "1968", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Book{Hart:1968:CAa, author = "John F. Hart and E. W. Cheney and Charles L. Lawson and Hans J. Maehly and Charles K. Mesztenyi and John R. Rice and Henry G. {Thatcher, Jr.} and Christoph Witzgall", title = "Computer Approximations", publisher = pub-R-E-KRIEGER, address = pub-R-E-KRIEGER:adr, pages = "x + 343", year = "1968", ISBN = "0-88275-642-7", ISBN-13 = "978-0-88275-642-4", LCCN = "QA 297 C64 1978", bibdate = "Tue Dec 14 22:55:11 1993", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/r/rice-john-r.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/master.bib", note = "Reprinted 1978 with corrections.", acknowledgement = ack-nhfb, shorttableofcontents = "1: The Design of a Function Subroutine / 1 \\ 2: General Methods of Computing Functions / 10 \\ 3: Least Maximum Approximations / 42 \\ 4: The Choice and Application of Approximations / 58 \\ 5: Description and Use of the Tables / 82 \\ 6: Function Notes / 89 \\ 7: Tables of Coefficients / 155 \\ Appendix A: Conversion Algorithms / 307 \\ Appendix B: Bibliography of Approximations / 313 \\ Appendix C: Decimal and Octal Constants / 333 \\ References / 336 \\ Index / 341", tableofcontents = "1: The Design of a Function Subroutine / 1 \\ 1.1 Introduction / 1 \\ 1.2 General Considerations in Writing a Function Subroutine / 2 \\ 1.3 Relation of the Function Subroutine to the Computer System / 3 \\ 1.4 The Three Main Types of Function Subroutine / 4 \\ 1.5 Special Programming Techniques / 7 \\ 1.6 Subroutine Errors / 7 \\ 1.7 Final Steps / 9 \\ 2: General Methods of Computing Functions / 10 \\ 2.1 Introduction / 10 \\ 2.2 Application of Infinite Expansions / 11 \\ 2.3 Recurrence and Difference Relations / 23 \\ 2.4 Iterative Techniques / 27 \\ 2.5 Integral Representations / 28 \\ 2.6 Differential Equations / 29 \\ 2.7 Tabular Data / 32 \\ 2.8 Convergence Acceleration / 33 \\ 3: Least Maximum Approximations / 42 \\ 3.1 Introduction / 42 \\ 3.2 Properties of Least Maximum Approximations / 43 \\ 3.3 Nearly Least Maximum Approximations / 46 \\ 3.4 Rational Approximation / 51 \\ 3.5 Segmented Approximation / 54 \\ 3.6 Computation of the Tables / 55 \\ 4: The Choice and Application of Approximations / 58 \\ 4.1 Introduction / 5 8 \\ 4.2 Domain Considerations / 58 \\ 4.3 Machine Considerations / 62 \\ 4.4 Conditioning of Approximations / 65 \\ 4.5 Polynomial Forms / 67 \\ 4.6 Rational Forms / 73 \\ 4.7 Transformation Algorithms / 78 \\ 5: Description and Use of the Tables / 82 \\ 5.1 Introduction / 22 \\ 5.2 Function Notes / 82 \\ 5.3 Accuracy of the Coefficients / 83 \\ 5.4 How to Use the Tables / 86 \\ 5.5 Preparation of the Tables / 88 \\ 6: Function Notes / 89 \\ 6.1 Square Root, Cube Root / 89 \\ 6.2 Exponential and Hyperbolic Functions / 96 \\ 6.3 The Logarithm Function / 105 \\ 6.4 Trigonometric Functions / 112 \\ 6.5 The Inverse Trigonometric Functions / 120 \\ 6.6 The Gamma Function and Its Logarithm / 130 \\ 6.7 The Error Function / 136 \\ 6.8 Bessel Functions / 141 \\ 6.9 Complete Elliptic Integrals / 150 \\ 7: Tables of Coefficients / 155 \\ Appendix A Conversion Algorithms / 307 \\ Appendix B Bibliography of Approximations / 313 \\ Appendix C Decimal and Octal Constants / 333 \\ References / 336 \\ Index / 341", } @Book{Hart:1968:CAb, author = "John F. Hart and E. W. Cheney and Charles L. Lawson and Hans J. Maehly and Charles K. Mesztenyi and John R. Rice and Henry G. {Thatcher, Jr.} and Christoph Witzgall", title = "Computer Approximations", publisher = pub-WILEY, address = pub-WILEY:adr, pages = "x + 343", year = "1968", ISBN = "0-471-35630-1", ISBN-13 = "978-0-471-35630-1", LCCN = "QA297 .C64", bibdate = "Sat Jan 14 14:53:06 2006", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/r/rice-john-r.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", series = "The SIAM series in applied mathematics", acknowledgement = ack-nhfb, } @Article{Kolbig:1968:ADS, author = "K. S. K{\"o}lbig", title = "{Algorithm 327}: {Dilogarithm} [{S22}]", journal = j-CACM, volume = "11", number = "4", pages = "270--271", month = apr, year = "1968", CODEN = "CACMA2", DOI = "https://doi.org/10.1145/362991.363043", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Fri Nov 25 18:20:19 MST 2005", bibsource = "http://www.acm.org/pubs/contents/journals/cacm/; https://www.math.utah.edu/pub/tex/bib/cacm1960.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "$d(x) = \int_0^x (\ln|1-y|/y)\,dy$; dilogarithm; special functions", } @Article{Luke:1968:AEI, author = "Yudell L. Luke", title = "Approximations for Elliptic Integrals", journal = j-MATH-COMPUT, volume = "22", number = "103", pages = "627--634", month = jul, year = "1968", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{MacLaren:1968:RAP, author = "M. D. MacLaren", title = "Remark on {Algorithm 272}: {Procedure} for the normal distribution functions", journal = j-CACM, volume = "11", number = "7", pages = "498--498", month = jul, year = "1968", CODEN = "CACMA2", DOI = "https://doi.org/10.1145/363397.363553", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Fri Nov 25 18:20:20 MST 2005", bibsource = "http://www.acm.org/pubs/contents/journals/cacm/; https://www.math.utah.edu/pub/tex/bib/cacm1960.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "See \cite{MacLaren:1965:APN}.", acknowledgement = ack-nhfb, fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "probability functions", } @Article{Mavromatis:1968:IFP, author = "H. A. Mavromatis and K. Schilcher", title = "Inverse Functions of the Products of Two {Bessel} Functions and Applications to Potential Scattering", journal = j-J-MATH-PHYS, volume = "9", number = "10", pages = "1627--1632", month = oct, year = "1968", CODEN = "JMAPAQ", DOI = "https://doi.org/10.1063/1.1664492", ISSN = "0022-2488 (print), 1089-7658 (electronic), 1527-2427", ISSN-L = "0022-2488", bibdate = "Fri Oct 28 11:55:17 MDT 2011", bibsource = "http://www.aip.org/ojs/jmp.html; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jmathphys1965.bib", URL = "http://jmp.aip.org/resource/1/jmapaq/v9/i10/p1627_s1", acknowledgement = ack-nhfb, classification = "A0380 (General theory of scattering)", corpsource = "Dept. Physics, American Univ. Beirut, Lebanon", fjournal = "Journal of Mathematical Physics", journal-URL = "http://jmp.aip.org/", keywords = "functions; mathematics; scattering", onlinedate = "28 October 2003", pagecount = "6", } @Article{Mechel:1968:IRT, author = "Fr. Mechel", title = "Improvement in Recurrence Techniques for the Computation of {Bessel} Functions of Integral Order (in {Technical Notes and Short Papers})", journal = j-MATH-COMPUT, volume = "22", number = "101", pages = "202--205", month = jan, year = "1968", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Book{Miller:1968:LTS, author = "Willard Miller", title = "{Lie} theory and special functions", volume = "43", publisher = pub-ACADEMIC, address = pub-ACADEMIC:adr, pages = "xv + 338", year = "1968", ISBN = "0-12-497450-3", ISBN-13 = "978-0-12-497450-0", LCCN = "QA387 .M55 1968eb", bibdate = "Sat Oct 30 19:07:04 MDT 2010", bibsource = "catalog.library.cornell.edu:7090/voyager; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", series = "Mathematics in science and engineering", acknowledgement = ack-nhfb, remark = "Reprinted 1979 with same ISBN.", subject = "Lie groups; Functions, Special", } @Article{Nagashima:1968:EFN, author = "Takashi Nagashima", title = "On elementary functions of natural numbers", journal = "Hitotsubashi J. Arts Sci.", volume = "9", pages = "50--58", year = "1968", ISSN = "0073-2788", MRclass = "02.72", MRnumber = "MR0232678 (38 \#1001)", MRreviewer = "R. L. Goodstein", bibdate = "Mon Oct 24 11:33:08 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Hitotsubashi Journal of Arts \& Sciences", } @Article{Ng:1968:DSS, author = "Edward W. Ng", title = "On the direct summation of series involving higher transcendental functions", journal = j-J-COMPUT-PHYS, volume = "3", number = "2", pages = "334--338", month = oct, year = "1968", CODEN = "JCTPAH", DOI = "https://doi.org/10.1016/0021-9991(68)90029-6", ISSN = "0021-9991 (print), 1090-2716 (electronic)", ISSN-L = "0021-9991", bibdate = "Sun Jan 1 08:28:02 MST 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputphys1960.bib", URL = "http://www.sciencedirect.com/science/article/pii/0021999168900296", acknowledgement = ack-nhfb, fjournal = "Journal of Computational Physics", journal-URL = "http://www.sciencedirect.com/science/journal/00219991/", } @Article{OBrien:1968:CAC, author = "William M. O'Brien and Joan Wood", title = "Certification of {Algorithm 299} [{S15}]: {Chi-squared} integral", journal = j-CACM, volume = "11", number = "4", pages = "271--271", month = apr, year = "1968", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Fri Nov 25 18:20:19 MST 2005", bibsource = "http://www.acm.org/pubs/contents/journals/cacm/; https://www.math.utah.edu/pub/tex/bib/cacm1960.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "chi-squared; probability functions", remark = "Fullerton: Corrections to an Algol procedure.", } @Article{Osborn:1968:IBF, author = "David Osborn and Richard Madey", title = "The Incomplete Beta Function and its Ratio to the Complete Beta Function", journal = j-MATH-COMPUT, volume = "22", number = "101", pages = "159--162", month = jan, year = "1968", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Richardson:1968:SUP, author = "Daniel Richardson", title = "Some Undecidable Problems Involving Elementary Functions of a Real Variable", journal = j-J-SYMBOLIC-LOGIC, volume = "33", number = "4", pages = "514--520", month = dec, year = "1968", CODEN = "JSYLA6", ISSN = "0022-4812 (print), 1943-5886 (electronic)", ISSN-L = "0022-4812", MRclass = "02.75", MRnumber = "39 #1330", bibdate = "Mon May 19 13:04:20 1997", bibsource = "ftp://ftp.ira.uka.de/pub/bibliography/Math/hilbert10.bib.gz; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Journal of Symbolic Logic", journal-URL = "http://projecteuclid.org/euclid.jsl; http://www.jstor.org/journal/jsymboliclogic", } @Article{Schmidt:1968:AEK, author = "Jochen W. Schmidt", title = "{Asymptotische Einschlie{\ss}ung bei konvergenzbeschleunigenden Verfahren. II}. ({German}) [{Asymptotic enclosure with convergence acceleration method. II}]", journal = j-NUM-MATH, volume = "11", number = "1", pages = "53--56", month = jan, year = "1968", CODEN = "NUMMA7", ISSN = "0029-599X (print), 0945-3245 (electronic)", ISSN-L = "0029-599X", bibdate = "Sun Oct 17 16:12:48 MDT 1999", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Numerische Mathematik", journal-URL = "http://link.springer.com/journal/211", keywords = "convergence acceleration", language = "German", } @Article{Strecok:1968:CIE, author = "Anthony J. Strecok", title = "On the Calculation of the Inverse of the Error Function", journal = j-MATH-COMPUT, volume = "22", number = "101", pages = "144--158", month = jan, year = "1968", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", URL = "http://www.jstor.org/stable/2004772", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", remark = "Fullerton: 18-digit approximations.", } @Book{Talman:1968:SFG, author = "James D. Talman", title = "Special Functions: a Group Theoretic Approach Based on Lectures by {Eugene P. Wigner}", publisher = pub-BENJAMIN, address = pub-BENJAMIN:adr, pages = "xii + 260", year = "1968", LCCN = "????", bibdate = "Sat Oct 30 16:57:02 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "With an introduction by Eugene P. Wigner.", series = "The mathematical physics monograph series", acknowledgement = ack-nhfb, keywords = "group theory; mathematical function; special functions", } @Book{Tolke:1968:PFA, author = "Friedrich T{\"o}lke", title = "{Praktische Funktionenlehre. 5. Allgemeine Weierstrasssche Funktionen und Ableitungen nach dem Parameter: Integrale der Theta-Funktionen und Bilinear-Entwicklungen}. ({German}) [{Practical} functional theory. 5. {General} information on {Weierstrass} functions and derivatives according to the parameters: integrals of theta functions and bilinear developments]", publisher = pub-SV, address = pub-SV:adr, pages = "viii + 158", year = "1968", ISBN = "3-662-11121-7, 3-662-11120-9", ISBN-13 = "978-3-662-11121-5, 978-3-662-11120-8", LCCN = "????", bibdate = "Mon Feb 13 19:01:10 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, language = "German", tableofcontents = "12 Allgemeine Weierstra{\ss}sche Funktionen. Doppelreihen-Entwicklungen \\ 13 Die Ableitungen nach dem Parameter und dem Modul \\ 14 Integrale von Theta-Funktionen (D-Funktionen) \\ 15 Mehrdimensionale Theta- und D-Funktionen \\ 16 Theta- und D-Funktionen mit imagin{\"a}ren Parametern \\ 17 Greensche Funktionen und Bilinear-Entwicklungen", xxISBN = "3-662-11031-8", xxISBN-13 = "978-3-662-11031-7", } @Article{Tooper:1968:SCP, author = "Robert F. Tooper and John Mark", title = "Simplified Calculation of {$ \operatorname {Ei}(x) $} for Positive Arguments, and a Short Table of $ \operatorname {Shi}(x) $ (in {Technical Notes and Short Papers})", journal = j-MATH-COMPUT, volume = "22", number = "102", pages = "448--449", month = apr, year = "1968", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "exponential integral ($\operatorname{Ei}(x)$); hyperbolic sine integral ($\operatorname{Shi}(x)$)", } @PhdThesis{Tung:1968:CAF, author = "Chin Tung", title = "A Combinational Arithmetic Function Generation System", type = "{Ph.D.} thesis", school = "University of California, Los Angeles", address = "Los Angeles, CA, USA", pages = "230", year = "1968", bibdate = "Mon Nov 10 13:18:11 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "https://www.proquest.com/pqdtglobal/docview/302294704", acknowledgement = ack-nhfb, remark = "Text not available online.", } @Article{Wilcox:1968:ZTN, author = "Peter H. Wilcox", title = "The Zeros of $ {P}^1_\nu (\cos \theta) $ and $ \frac {\partial }{\partial \theta } {P}^1_\mu (\cos \theta) $ (in {Technical Notes and Short Papers})", journal = j-MATH-COMPUT, volume = "22", number = "101", pages = "205--208", month = jan, year = "1968", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", URL = "http://www.jstor.org/stable/2004783", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", remark = "Fullerton: $ P_\nu^1 $ are Legendre functions.", } @Article{Wimp:1968:RFH, author = "Jet Wimp", title = "Recursion Formulae of Hypergeometric Functions", journal = j-MATH-COMPUT, volume = "22", number = "102", pages = "363--373", month = apr, year = "1968", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp1960.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Witte:1968:AAJ, author = "B. F. W. Witte", title = "{ACM Algorithm 332}: {Jacobi} Polynomials", journal = j-CACM, volume = "11", number = "6", pages = "436--437", month = jun, year = "1968", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Thu Sep 08 09:33:08 1994", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "See remark \cite{Skovgaard:1975:RAJ}.", acknowledgement = ack-nhfb, fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", } @Article{Wrench:1968:CTS, author = "John W. {Wrench, Jr.}", title = "Concerning Two Series for the Gamma Function", journal = j-MATH-COMPUT, volume = "22", number = "103", pages = "617--626", month = jul, year = "1968", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRnumber = "MR 0237078 (38:5371)", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Aharoni:1969:C, author = "Amikam Aharoni", title = "Computation of {$ K_p(x) $}", journal = j-J-COMPUT-PHYS, volume = "4", number = "2", pages = "270--271", month = aug, year = "1969", CODEN = "JCTPAH", DOI = "https://doi.org/10.1016/0021-9991(69)90072-2", ISSN = "0021-9991 (print), 1090-2716 (electronic)", ISSN-L = "0021-9991", bibdate = "Sun Jan 1 08:28:03 MST 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputphys1960.bib", URL = "http://www.sciencedirect.com/science/article/pii/0021999169900722", acknowledgement = ack-nhfb, fjournal = "Journal of Computational Physics", journal-URL = "http://www.sciencedirect.com/science/journal/00219991/", } @Article{Boersma:1969:ECW, author = "J. Boersma", title = "Expansions for {Coulomb} Wave Functions", journal = j-MATH-COMPUT, volume = "23", number = "105", pages = "51--59", month = jan, year = "1969", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Book{Buchholz:1969:CHF, author = "Herbert Buchholz", title = "The confluent hypergeometric function with special emphasis on its applications", volume = "15", publisher = pub-SV, address = pub-SV:adr, pages = "xviii + 238", year = "1969", LCCN = "QA351 .B813", bibdate = "Sat Oct 30 21:06:31 MDT 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; z3950.loc.gov:7090/Voyager", note = "Translation by H. Lichtblau and K. Wetzel to English from German ``Die konfluente hypergeometrische Funktion.''", series = "Springer tracts in natural philosophy", acknowledgement = ack-nhfb, subject = "Hypergeometric functions", } @Article{Bulirsch:1969:EBT, author = "R. Bulirsch", title = "An extension of the {Bartky}-transformation to incomplete elliptic integrals of the third kind", journal = j-NUM-MATH, volume = "13", number = "3", pages = "266--284", month = jul, year = "1969", CODEN = "NUMMA7", ISSN = "0029-599X (print), 0945-3245 (electronic)", ISSN-L = "0029-599X", bibdate = "Sun Oct 17 19:01:15 MDT 1999", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Numerische Mathematik", journal-URL = "http://link.springer.com/journal/211", } @Article{Bulirsch:1969:NCE, author = "R. Bulirsch", title = "Numerical calculation of elliptic integrals and elliptic functions. {III}", journal = j-NUM-MATH, volume = "13", number = "4", pages = "305--315", month = aug, year = "1969", CODEN = "NUMMA7", ISSN = "0029-599X (print), 0945-3245 (electronic)", ISSN-L = "0029-599X", bibdate = "Sun Oct 17 19:01:15 MDT 1999", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Numerische Mathematik", journal-URL = "http://link.springer.com/journal/211", } @Article{Carlson:1969:CBE, author = "B. C. Carlson", title = "A connection between elementary functions and higher transcendental functions", journal = j-SIAM-J-APPL-MATH, volume = "17", pages = "116--148", year = "1969", CODEN = "SMJMAP", ISSN = "0036-1399 (print), 1095-712X (electronic)", ISSN-L = "0036-1399", MRclass = "33.20 (30.00)", MRnumber = "40 \#408", MRreviewer = "S. K. Bose", bibdate = "Sat Jan 11 17:44:01 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "SIAM Journal on Applied Mathematics", journal-URL = "http://epubs.siam.org/siap", } @InProceedings{Clark:1969:SCE, author = "N. W. Clark and W. J. Cody", title = "Self-contained exponentiation", crossref = "AFIPS:1969:ACPb", pages = "701--706", year = "1969", bibdate = "Wed Sep 07 10:49:33 1994", bibsource = "garbo.uwasa.fi:/pc/doc-soft/fpbiblio.txt; https://www.math.utah.edu/pub/bibnet/authors/c/cody-william-j.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nj, } @Article{Clemm:1969:ACV, author = "Donald S. Clemm", title = "{Algorithm 352}: {Characteristic} Values and Associated Solutions of {Mathieu}'s Differential Equation [{S22}]", journal = j-CACM, volume = "12", number = "7", pages = "399--407", month = jul, year = "1969", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Fri Nov 25 18:20:27 MST 2005", bibsource = "http://www.acm.org/pubs/contents/journals/cacm/; https://www.math.utah.edu/pub/tex/bib/cacm1960.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "See remark \cite{Frisch:1972:RAR}.", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", classcodes = "C4170 (Differential equations); C7300 (Natural sciences computing)", corpsource = "Wright-Patterson Air Force Base, USA", fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "differential equations; function evaluation; subroutines", remark = "Fullerton: Long set of FORTRAN routines to evaluate Mathieu functions as well as several Bessel functions.", } @Article{Cobb:1969:CAS, author = "S. M. Cobb", title = "Certification of {Algorithm 47} [{S16}]: {Associated} {Legendre} functions of the first kind for real or imaginary arguments", journal = j-CACM, volume = "12", number = "11", pages = "635--636", month = nov, year = "1969", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Fri Nov 25 18:20:28 MST 2005", bibsource = "http://www.acm.org/pubs/contents/journals/cacm/; https://www.math.utah.edu/pub/tex/bib/cacm1960.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "associated Legendre functions of the first kind; special functions", remark = "Fullerton: Numerous additional corrections and changes to an Algol procedure.", } @Article{Cody:1969:CAE, author = "W. J. Cody and Henry C. {Thacher, Jr.}", title = "{Chebyshev} Approximations for the Exponential Integral {$ \hbox {Ei}(x) $}", journal = j-MATH-COMPUT, volume = "23", number = "106", pages = "289--303", month = apr, year = "1969", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "65.25", MRnumber = "39\#3680", bibdate = "Wed Jan 17 08:57:33 1996", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/c/cody-william-j.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Cody:1969:CRA, author = "W. J. Cody and G. Meinardus and R. S. Varga", title = "{Chebyshev} rational approximations to $ e^{-x} $ on $ [0, \infty) $ and applications to heat conduction problems", journal = j-J-APPROX-THEORY, volume = "2", number = "??", pages = "50--65", month = "??", year = "1969", CODEN = "JAXTAZ", ISSN = "0021-9045 (print), 1096-0430 (electronic)", ISSN-L = "0021-9045", MRclass = "65.67 (41.00)", MRnumber = "40\#999", bibdate = "Wed Jan 17 08:57:33 1996", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/c/cody-william-j.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-wjc, fjournal = "Journal of Approximation Theory", journal-URL = "http://www.sciencedirect.com/science/journal/00219045", } @Article{Cody:1969:RCA, author = "W. J. Cody", title = "Rational {Chebyshev} Approximations for the Error Function", journal = j-MATH-COMPUT, volume = "23", number = "107", pages = "631--637", month = jul, year = "1969", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp1960.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Farkas:1969:CAS, author = "I. Farkas", title = "Certification of {Algorithm 165} [{S21}]: {Complete} elliptic integrals", journal = j-CACM, volume = "12", number = "1", pages = "38--38", month = jan, year = "1969", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Fri Nov 25 18:20:24 MST 2005", bibsource = "http://www.acm.org/pubs/contents/journals/cacm/; https://www.math.utah.edu/pub/tex/bib/cacm1960.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "special functions", } @Article{Gautschi:1969:ACE, author = "Walter Gautschi", title = "{Algorithm 363}: {Complex} Error Function [{S15}]", journal = j-CACM, volume = "12", number = "11", pages = "635--635", month = nov, year = "1969", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Fri Nov 25 18:20:28 MST 2005", bibsource = "http://www.acm.org/pubs/contents/journals/cacm/; https://www.math.utah.edu/pub/tex/bib/cacm1960.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "See certification \cite{Kolbig:1972:CAC}.", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "$\erf(z)$; special functions", remark = "Fullerton: 50-line Algol procedure with accuracy to 10 decimal places.", } @Article{Gautschi:1969:RAS, author = "Walter Gautschi", title = "Remark on {Algorithm 292} [{S22}]: {Regular} {Coulomb} wave functions", journal = j-CACM, volume = "12", number = "5", pages = "280--280", month = may, year = "1969", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Fri Nov 25 18:20:26 MST 2005", bibsource = "http://www.acm.org/pubs/contents/journals/cacm/; https://www.math.utah.edu/pub/tex/bib/cacm1960.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "Coulomb wave functions; special functions", remark = "Fullerton: The first of many remarks.", } @Article{Herman:1969:NHE, author = "G. T. Herman", title = "A new hierarchy of elementary functions", journal = j-PROC-AM-MATH-SOC, volume = "20", pages = "557--562", year = "1969", CODEN = "PAMYAR", ISSN = "0002-9939 (print), 1088-6826 (electronic)", ISSN-L = "0002-9939", MRclass = "02.77", MRnumber = "40 \#4110", MRreviewer = "G. E. Sacks", bibdate = "Sat Jan 11 17:44:01 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Proceedings of the American Mathematical Society", journal-URL = "http://www.ams.org/journals/proc", } @Article{Holzwarth:1969:VBB, author = "A. Holzwarth", title = "{Ein Verfahren zur Bestimmung bester Tscheb\-y\-scheff-Ap\-prox\-i\-ma\-tion\-en der Quadratwurzelfunktion}. ({German}) {A Method for Determination of Best Chebyshev Approximations to the Square Root Function}", journal = j-COMPUTING, volume = "4", number = "2", pages = "168--177", month = jun, year = "1969", CODEN = "CMPTA2", ISSN = "0010-485X (print), 1436-5057 (electronic)", ISSN-L = "0010-485X", bibdate = "Tue Jan 2 17:40:51 MST 2001", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; garbo.uwasa.fi:/pc/doc-soft/fpbiblio.txt; https://www.math.utah.edu/pub/tex/bib/computing.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; INSPEC Axiom database (1968--date)", acknowledgement = ack-nj # " and " # ack-nhfb, affiliation = "T{\"u}bingen, West Germany", classification = "C4130", description = "Chebyshev approximation; numerical analysis", fjournal = "Computing", journal-URL = "http://link.springer.com/journal/607", language = "German", } @Article{King:1969:LEN, author = "Richard F. King and David L. Phillips", title = "The Logarithmic Error and {Newton}'s Method for the Square Root", journal = j-CACM, volume = "12", number = "2", pages = "87--88", month = feb, year = "1969", CODEN = "CACMA2", DOI = "https://doi.org/10.1145/362848.362861", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", MRclass = "65.50", MRnumber = "44\#2333", bibdate = "Fri Nov 25 18:20:24 MST 2005", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; http://www.acm.org/pubs/contents/journals/cacm/; https://www.math.utah.edu/pub/tex/bib/cacm1960.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", abstract = "The problem of obtaining optimal starting values for the calculation of the square root using Newton's method is considered. It has been pointed out elsewhere that if relative error is used as the measure of goodness of fit, optimal results are not obtained when the initial approximation is a best fit. It is shown here that if, instead, the so-called logarithmic error is used, then a best initial fit is optimal for both types of error. Moreover, use of the logarithmic error appears to simplify the problem of determining the optimal initial approximation.", acknowledgement = ack-nj # " and " # ack-nhfb, classcodes = "C4120 (Functional analysis)", corpsource = "Argonne Nat. Lab., Argonne, IL, USA", fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "$\sqrt(x)$; elementary functions; function evaluation; iterative methods", } @Article{Kolbig:1969:CASa, author = "K. S. K{\"o}lbig", title = "Certification of {Algorithm 292} [{S22}]: {Regular} {Coulomb} wave functions and of remark on {Algorithm 292} [{S22}]: {Regular} {Coulomb} wave functions", journal = j-CACM, volume = "12", number = "5", pages = "278--279", month = may, year = "1969", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Fri Nov 25 18:20:26 MST 2005", bibsource = "http://www.acm.org/pubs/contents/journals/cacm/; https://www.math.utah.edu/pub/tex/bib/cacm1960.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "Coulomb wave functions; special functions", remark = "Fullerton: Tests of an Algol procedure.", } @Article{Kolbig:1969:CASb, author = "K. S. K{\"o}lbig", title = "Certification of {Algorithm 300} [{S22}]: {Coulomb} wave functions", journal = j-CACM, volume = "12", number = "5", pages = "279--280", month = may, year = "1969", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Fri Nov 25 18:20:26 MST 2005", bibsource = "http://www.acm.org/pubs/contents/journals/cacm/; https://www.math.utah.edu/pub/tex/bib/cacm1960.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "Coulomb wave functions; special functions", remark = "Fullerton: The first of many remarks.", } @Article{Kolbig:1969:RAS, author = "K. S. K{\"o}lbig", title = "Remark on {Algorithm 300} [{S22}]: {Coulomb} wave functions", journal = j-CACM, volume = "12", number = "12", pages = "692--692", month = dec, year = "1969", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Fri Nov 25 18:20:29 MST 2005", bibsource = "http://www.acm.org/pubs/contents/journals/cacm/; https://www.math.utah.edu/pub/tex/bib/cacm1960.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "Coulomb wave functions; special functions", remark = "Fullerton: One of many remarks.", } @Article{Lardner:1969:RBB, author = "Thomas J. Lardner", title = "Relations Between {${}_0 F_3 $} and {Bessel} Functions", journal = j-SIAM-REVIEW, volume = "11", number = "1", pages = "69--72", month = "????", year = "1969", CODEN = "SIREAD", DOI = "https://doi.org/10.1137/1011007", ISSN = "0036-1445 (print), 1095-7200 (electronic)", ISSN-L = "0036-1445", bibdate = "Thu Mar 27 09:06:04 MDT 2014", bibsource = "http://epubs.siam.org/toc/siread/11/1; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/siamreview.bib", acknowledgement = ack-nhfb, fjournal = "SIAM Review", journal-URL = "http://epubs.siam.org/sirev", onlinedate = "January 1969", } @Article{Lewis:1969:FII, author = "Richard L. Lewis", title = "On Finite Integrals Involving Trigonometric, {Bessel}, and {Legendre} Functions", journal = j-MATH-COMPUT, volume = "23", number = "106", pages = "259--273", month = apr, year = "1969", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Book{Luke:1969:SFTa, author = "Yudell L. Luke", title = "The Special Functions and Their Approximations", volume = "I", publisher = pub-ACADEMIC, address = pub-ACADEMIC:adr, pages = "xx + 349", year = "1969", ISBN = "0-12-459901-X", ISBN-13 = "978-0-12-459901-7", LCCN = "QA351 .L94 1969", bibdate = "Wed Dec 15 17:55:35 1993", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", series = "Mathematics in Science and Engineering, Volume 53-I, Editor: Richard Bellman", acknowledgement = ack-nhfb, } @Book{Luke:1969:SFTb, author = "Yudell L. Luke", title = "The Special Functions and Their Approximations", volume = "II", publisher = pub-ACADEMIC, address = pub-ACADEMIC:adr, pages = "xx + 485", year = "1969", ISBN = "0-12-459902-8", ISBN-13 = "978-0-12-459902-4", LCCN = "QA351 .L797", bibdate = "Wed Dec 15 17:55:38 1993", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/master.bib", series = "Mathematics in Science and Engineering, Volume 53-II, Editor: Richard Bellman", URL = "http://www.sciencedirect.com/science/book/9780124599024", acknowledgement = ack-nhfb, tableofcontents = "Dedication / v \\ Preface / vii--ix \\ Contents of Volume I / xv \\ Introduction / xvii--xx \\ IX: Expansions of Generalized Hypergeometric Functions in Series of Functions of the Same Kind / 1--65 \\ X: The $\tau$-Method / 66--91 \\ XI: Polynomial and Rational Approximations to Generalized Hypergeometric Functions / 92--132 \\ XII: Recursion Formulas for Polynomials and Functions which Occur in Infinite Series and Rational Approximations to Generalized Hypergeometric Functions / 133--166 \\ XIII: Polynomial and Rational Approximations for $E(z) = _2F_1(1, \sigma; \rho + 1; 1/z)$ / 167--185 \\ XIV: Polynomial and Rational Approximations for the Incomplete Gamma Function / 186--213 \\ XV: Trapezoidal Rule Integration Formulas / 214--226 \\ XVI: Applications / 227--281 \\ XVII: Tables of Coefficients / 282--452 \\ Bibliography / 453--461 \\ Notation Index / 463--467 \\ Subject Index to Volumes I and II / 468--485", } @TechReport{Moses:1969:ICS, author = "Joel Moses", title = "The integration of a class of special functions with the {Risch} algorithm", type = "{AI} Memo (180)", number = "MAC-M-421", institution = "Artificial Intelligence Laboratory, Massachusetts Institute of Technology", address = "Cambridge, MA, USA", pages = "13 + 1", year = "1969", LCCN = "Q334 M533 no. 180", bibdate = "Sat Oct 30 18:37:28 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, } @Article{Ng:1969:CPE, author = "Edward W. Ng and C. J. Devine and R. F. Tooper", title = "{Chebyshev} Polynomial Expansion of {Bose--Einstein} Functions of Orders $1$ to $ 10$", journal = j-MATH-COMPUT, volume = "23", number = "107", pages = "639--643", month = jul, year = "1969", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", remark = "Fullerton: $ B_p(\eta) = \frac {1}{\Gamma (p - 1)} \int_0^\infty \frac {x^p}{e^{x - \eta } - 1} \, d t $. Relative errors down to $ 3 \times 10^{-19} $.", } @Article{Reichel:1969:IPV, author = "Alex Reichel", title = "The Integral of the $n$ th Power of the {Voigt} Function", journal = j-MATH-COMPUT, volume = "23", number = "107", pages = "645--649", month = jul, year = "1969", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", remark = "Fullerton: The function $ \chi_n(t) = \int_{- \infty }^\infty \{ U_0 (x, t) \}^n \, d x $, where $ U_0 (x, t) = \frac {1}{\sqrt {4 \pi t}} \frac {\exp [ - (x - y)^2 / 4 t]}{1 + y^2} \, d y $ is considered.", } @Article{Robertson:1969:CNC, author = "G. H. Robertson", title = "Computation of the Noncentral Chi-Square Distribution", journal = j-BELL-SYST-TECH-J, volume = "48", number = "1", pages = "201--207", month = jan, year = "1969", CODEN = "BSTJAN", ISSN = "0005-8580", bibdate = "Tue Nov 9 11:15:55 MST 2010", bibsource = "http://bstj.bell-labs.com/oldfiles/year.1969/BSTJ.1969.4801.html; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://bstj.bell-labs.com/BSTJ/images/Vol48/bstj48-1-201.pdf", acknowledgement = ack-nhfb, fjournal = "The Bell System Technical Journal", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)1538-7305/issues/", } @Article{Sollfrey:1969:IFP, author = "William Sollfrey", title = "Inverse Functions of the Products of Two {Bessel} Functions", journal = j-J-MATH-PHYS, volume = "10", number = "8", pages = "1429--1430", month = aug, year = "1969", CODEN = "JMAPAQ", DOI = "https://doi.org/10.1063/1.1664985", ISSN = "0022-2488 (print), 1089-7658 (electronic), 1527-2427", ISSN-L = "0022-2488", bibdate = "Fri Oct 28 11:55:26 MDT 2011", bibsource = "http://www.aip.org/ojs/jmp.html; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jmathphys1965.bib", URL = "http://jmp.aip.org/resource/1/jmapaq/v10/i8/p1429_s1", acknowledgement = ack-nhfb, classification = "A0200 (Mathematical methods in physics)", corpsource = "RAND Corp., Santa Monica, CA, USA", fjournal = "Journal of Mathematical Physics", journal-URL = "http://jmp.aip.org/", keywords = "functions", onlinedate = "4 November 2003", pagecount = "2", } @Article{Sterbenz:1969:OSA, author = "P. H. Sterbenz and C. T. Fike", title = "Optimal Starting Approximations for {Newton}'s Method", journal = j-MATH-COMPUT, volume = "23", number = "106", pages = "313--318", month = apr, year = "1969", CODEN = "MCMPAF", DOI = "", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp1960.bib; JSTOR database", abstract = "Various writers have dealt with the subject of optimal starting approximations for square-root calculation by Newton's method. Three optimality criteria that have been used can be shown to lead to closely related approximations. This fact makes it surprisingly easy to choose a starting approximation of some prescribed form so that the maximum relative error after any number of Newton iterations is as small as possible.", acknowledgement = ack-nj # " and " # ack-nhfb, ajournal = "Math. Comput.", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{TadeudeMedeiros:1969:APF, author = "Adilson {Tadeu de Medeiros} and Georges Schwachheim", title = "{Algorithm 349}: {Polygamma} Functions with Arbitrary Precision [{S14}]", journal = j-CACM, volume = "12", number = "4", pages = "213--214", month = apr, year = "1969", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Fri Nov 25 18:20:25 MST 2005", bibsource = "http://www.acm.org/pubs/contents/journals/cacm/; https://www.math.utah.edu/pub/tex/bib/cacm1960.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "See certification \cite{Lewis:1975:CPF}.", abstract = "This procedure assigns to polygam the value of the polygamma function of order n for any real argument $z$. For $ n = 0$, we have the psi or digamma function, for $ n = 1$ the trigamma function, for $ n = 2$ the tetragamma function, and so on.", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", classcodes = "C7300 (Natural sciences computing)", corpsource = "Centro Brasileiro de Pesquisas Fisicas, Rio de Janeiro, Brazil", fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "digamma function; mathematics; polygamma function; psi function; special functions; subroutines; tetragamma function; trigamma function", remark = "Fullerton: 150-line Algol procedure.", } @InProceedings{Tesler:1969:AEF, author = "G. S. Tesler", booktitle = "Mathematical provisioning of electronic digital computers and effective organization of the computing process (Proc. Sem., Kiev, 1969) ({Russian}), No. 2", title = "The approximation of elementary functions by means of polynomials of degree zero and one. ({Russian})", publisher = "Akad. Nauk Ukrain. SSR", address = "Kiev, USSR", pages = "75--88", year = "1969", MRclass = "65D15", MRnumber = "45 \#4600", MRreviewer = "I. Selihova", bibdate = "Sat Jan 11 17:44:01 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, language = "Russian", } @Book{Tolke:1969:PFT, author = "Friedrich T{\"o}lke", title = "{Praktische Funktionenlehre. 6. Tafeln aus dem Gebiet der Theta-Funktionen und der elliptischen Funtionen}. ({German}) [{Practical} functional theory. 6. {Tables} from the field of theta functions and elliptic functions]", publisher = pub-SV, address = pub-SV:adr, pages = "lxxxii + 449 (vol. 1)", year = "1969", ISBN = "", ISBN-13 = "", LCCN = "????", bibdate = "Mon Feb 13 19:01:10 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "Two volumes", acknowledgement = ack-nhfb, language = "German", } @Article{Turner:1969:DSC, author = "L. R. Turner", title = "Difficulty in {$ \sin $ \slash$ \cos $} Routine", journal = j-SIGNUM, volume = "4", number = "3", pages = "13--13", year = "1969", CODEN = "SNEWD6", ISSN = "0163-5778 (print), 1558-0237 (electronic)", ISSN-L = "0163-5778", bibdate = "Thu Feb 15 15:23:23 2007", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "ACM SIGNUM Newsletter", journal-URL = "https://dl.acm.org/loi/signum", } @Article{VandeVel:1969:SEM, author = "H. {Van de Vel}", title = "On the Series Expansion Method for Computing Incomplete Elliptic Integrals of the First and Second Kinds", journal = j-MATH-COMPUT, volume = "23", number = "105", pages = "61--69", month = jan, year = "1969", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Zaker:1969:CCE, author = "T. A. Zaker", title = "Calculation of the complementary error function of complex argument", journal = j-J-COMPUT-PHYS, volume = "4", number = "3", pages = "427--430", month = oct, year = "1969", CODEN = "JCTPAH", DOI = "https://doi.org/10.1016/0021-9991(69)90011-4", ISSN = "0021-9991 (print), 1090-2716 (electronic)", ISSN-L = "0021-9991", bibdate = "Thu Dec 04 16:20:39 2008", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputphys1960.bib", acknowledgement = ack-nhfb, fjournal = "Journal of Computational Physics", journal-URL = "http://www.sciencedirect.com/science/journal/00219991/", } @Article{Bray:1970:CAR, author = "T. Bray", title = "Certification of {Algorithm 22, Ricatti--Bessel Functions of First and Second Kind}", journal = j-CACM, volume = "13", number = "7", pages = "448--448", month = jul, year = "1970", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Fri Oct 29 21:49:15 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", remark = "Fullerton: An error in an Algol procedure is reported.", } @Article{Carlitz:1970:SRF, author = "L. Carlitz", title = "Some Reduction Formulas for Generalized Hypergeometric Functions", journal = j-SIAM-J-MATH-ANA, volume = "1", number = "2", pages = "243--245", month = may, year = "1970", CODEN = "SJMAAH", ISSN = "0036-1410 (print), 1095-7154 (electronic)", ISSN-L = "0036-1410", bibdate = "Sun Nov 28 19:21:58 MST 2010", bibsource = "http://epubs.siam.org/sam-bin/dbq/toc/SIMA/1/2; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/siamjmathana.bib", acknowledgement = ack-nhfb, fjournal = "SIAM Journal on Mathematical Analysis", journal-URL = "http://epubs.siam.org/sima", } @Article{Carlson:1970:HSS, author = "B. C. Carlson", title = "Hidden Symmetries of Special Functions", journal = j-SIAM-REVIEW, volume = "12", number = "3", pages = "332--345", month = jul, year = "1970", CODEN = "SIREAD", DOI = "https://doi.org/10.1137/1012078", ISSN = "0036-1445 (print), 1095-7200 (electronic)", ISSN-L = "0036-1445", bibdate = "Thu Mar 27 09:06:20 MDT 2014", bibsource = "http://epubs.siam.org/toc/siread/12/3; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/siamreview.bib", URL = "http://www.jstor.org/stable/2028552", acknowledgement = ack-nhfb, fjournal = "SIAM Review", journal-URL = "http://epubs.siam.org/sirev", onlinedate = "July 1970", } @Article{Chen:1970:CGI, author = "Reuven Chen", title = "On the Computation of the Generalized Integral in Glow Curve Theory", journal = j-J-COMPUT-PHYS, volume = "6", number = "2", pages = "314--316", month = oct, year = "1970", CODEN = "JCTPAH", DOI = "https://doi.org/10.1016/0021-9991(70)90027-6", ISSN = "0021-9991 (print), 1090-2716 (electronic)", ISSN-L = "0021-9991", bibdate = "Fri Oct 29 22:09:19 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputphys1970.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Journal of Computational Physics", journal-URL = "http://www.sciencedirect.com/science/journal/00219991/", } @Article{Cochran:1970:NTF, author = "James Alan Cochran and Judith N. Hoffspiegel", title = "Numerical Techniques for Finding $ \nu $-Zeros of {Hankel} Functions", journal = j-MATH-COMPUT, volume = "24", number = "110", pages = "413--422", month = apr, year = "1970", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Cody:1970:CAC, author = "W. J. Cody and K. E. Hillstrom", title = "{Chebyshev} Approximations for the {Coulomb} Phase Shift", journal = j-MATH-COMPUT, volume = "24", number = "111", pages = "671--677", month = jul, year = "1970", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "65.25", MRnumber = "42\#8661", bibdate = "Wed Jan 17 08:57:04 1996", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/c/cody-william-j.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-wjc, citedby = "Fullerton:1980:BEM", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", remark = "Fullerton: Relative errors down to $ 4 \times 10^{-19} $.", } @Article{Cody:1970:CAD, author = "W. J. Cody and Kathleen A. Paciorek and Henry C. {Thacher, Jr.}", title = "{Chebyshev} approximations for {Dawson}'s integral", journal = j-MATH-COMPUT, volume = "24", number = "109", pages = "171--178", month = jan, year = "1970", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "65.20", MRnumber = "41\#2883", bibdate = "Wed Jan 17 08:57:30 1996", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/c/cody-william-j.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Cody:1970:RAC, author = "W. J. Cody and Kathleen A. Paciorek", title = "Remark on {Algorithm} 292 [{S22}]: Regular {Coulomb} Wave Functions", journal = j-CACM, volume = "13", number = "9", pages = "573", month = sep, year = "1970", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Wed Nov 16 23:58:51 1994", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/c/cody-william-j.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-wjc, citedby = "Fullerton:1980:BEM", fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", remark = "Fullerton: More modifications to an Algol procedure.", } @Article{Darlington:1970:AAA, author = "Sidney Darlington", title = "Analytical Approximations to Approximations in the {Chebyshev} Sense", journal = j-BELL-SYST-TECH-J, volume = "49", number = "1", pages = "1--32", month = jan, year = "1970", CODEN = "BSTJAN", ISSN = "0005-8580", bibdate = "Tue Nov 9 11:15:55 MST 2010", bibsource = "http://bstj.bell-labs.com/oldfiles/year.1970/BSTJ.1970.4901.html; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://bstj.bell-labs.com/BSTJ/images/Vol49/bstj49-1-1.pdf", acknowledgement = ack-nhfb, fjournal = "The Bell System Technical Journal", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)1538-7305/issues/", } @TechReport{DeLugish:1970:CAAa, author = "Bruce Gene {De Lugish}", title = "A Class of Algorithms for Automatic Evaluation of Certain Elementary Functions in a Binary Computer", number = "399", institution = "Department of Computer Science, University of Illinois at Urbana-Champaign", address = "Urbana, Illinois", pages = "191", year = "1970", bibdate = "Mon May 19 13:30:58 1997", bibsource = "ftp://ftp.ira.uka.de/pub/bibliography/Techreports/Uiuc.Tr.bib.gz; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, } @PhdThesis{DeLugish:1970:CAAb, author = "Bruce Gene {De Lugish}", title = "A Class of Algorithms for Automatic Evaluation of Certain Elementary Functions in a Binary Computer", type = "{Ph.D.} thesis", school = "Department of Electrical Engineering, University of Illinois at Urbana--Champaign", address = "Urbana, IL, USA", pages = "ix + 244", month = jun, year = "1970", bibdate = "Mon Nov 10 10:24:34 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "https://www.proquest.com/pqdtglobal/docview/302409094/", acknowledgement = ack-nhfb, advisor = "James E. Robertson", } @Article{Flynn:1970:DFI, author = "M. J. Flynn", title = "On Division by Functional Iteration", journal = j-IEEE-TRANS-COMPUT, volume = "C-19", number = "8", pages = "702--706", month = aug, year = "1970", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/T-C.1970.223019", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Thu Nov 15 07:59:25 2007", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", note = "Reprinted in \cite{Swartzlander:1990:CAa}.", URL = "http://www.acsel-lab.com/arithmetic/arith1/papers/ARITH1_Flynn.pdf", abstract = "In order to avoid the time delays associated with linearly convergent division based on subtraction, other iterative schemes can be used. These are based on (1) series expansion of the reciprocal, (2) multiplicative sequence, or (3) additive sequence convergent to the quotient. These latter techniques are based on finding the root of an arbitrary function at either the quotient or reciprocal value. A Newton--Raphson iteration or root finding iteration can be used. The most useful techniques are quadratically convergent (i.e., $ \mathrm {error}_{i + 1} = O((\mathrm {error}_i)^2) $). These techniques generally require two arithmetic operations (add or multiply) to double the precision of the quotient.", acknowledgement = ack-sfo # " and " # ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", keywords = "ARITH-1", } @Article{Gautschi:1970:RAD, author = "Walter Gautschi and Bruce J. Klein", title = "Remark on {Algorithm 282, Derivatives of $ e^x / x $, $ \cos (x) / x $, and $ \sin (x) / x $}", journal = j-CACM, volume = "13", number = "1", pages = "53--54", month = jan, year = "1970", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Sat Oct 30 07:27:17 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "See \cite{Gautschi:1966:AD}.", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", remark = "Fullerton: Corrections are given for several Algol procedures.", } @Article{Gautschi:1970:RCC, author = "Walter Gautschi and Bruce J. Klein", title = "Recursive computation of certain derivatives --- a study of error propagation", journal = j-CACM, volume = "13", number = "1", pages = "7--9", month = jan, year = "1970", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", MRclass = "65Q05", MRnumber = "46 1115", MRreviewer = "D. F. Mayers", bibdate = "Tue Mar 25 13:26:09 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", abstract = "A brief study is made of the propagation of errors in linear first-order difference equations. The recursive computation of successive derivatives of $ (e^x) / x $ and $ (\cos x) / x $ is considered as an illustration.", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", classcodes = "C4110 (Error analysis in numerical methods)", corpsource = "Purdue Univ., Lafayette, IN, USA", fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "difference equations; error analysis; error propagation; recursive computation; successive derivatives", remark = "Fullerton: Recursive calculation of derivatives of $ e^x / x $ and $ \cos (x) / x $ is considered.", } @Article{Hill:1970:AASa, author = "G. W. Hill", title = "{ACM Algorithm 395}: {Student}'s $t$-Distribution", journal = j-CACM, volume = "13", number = "10", pages = "617--619", month = oct, year = "1970", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Tue Mar 25 13:26:09 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "See also \cite{elLozy:1979:RAS,Hill:1981:RSD}.", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", classcodes = "C7310 (Mathematics computing)", corpsource = "CSIRO, Glen Osmond, Australia", fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "statistics; subroutines", remark = "Fullerton: Description of a 50-line Algol procedure.", } @Article{Hill:1970:AASb, author = "G. W. Hill", title = "{ACM Algorithm 396}: {Student}'s $t$-Quantiles", journal = j-CACM, volume = "13", number = "10", pages = "619--620", month = oct, year = "1970", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Tue Mar 25 13:26:09 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "See also \cite{Hill:1981:RSD,Hill:1981:RSQ,elLozy:1979:RAS}.", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", classcodes = "C4120 (Functional analysis); C7310 (Mathematics computing)", corpsource = "CSIRO, Glen Osmond, Australia", fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "function evaluation; statistics; subroutines", remark = "Fullerton: Description of a 50-line Algol procedure.", } @Article{Holmgren:1970:RAN, author = "Bo Holmgren", title = "Remark on {Algorithm 304, Normal Curve Integral}", journal = j-CACM, volume = "13", number = "10", pages = "624--624", month = oct, year = "1970", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Sat Oct 30 08:18:38 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", } @Article{Jones:1970:GHF, author = "Alan L. Jones", title = "The generalized hypergeometric function", journal = j-SIGPLAN, volume = "5", number = "3", pages = "26--27", month = mar, year = "1970", CODEN = "SINODQ", ISSN = "0362-1340 (print), 1523-2867 (print), 1558-1160 (electronic)", ISSN-L = "0362-1340", bibdate = "Thu May 25 06:40:57 MDT 2006", bibsource = "http://portal.acm.org/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "ACM SIGPLAN Notices", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J706", } @Article{Kolbig:1970:CZI, author = "K. S. Kolbig", title = "Complex Zeros of an Incomplete {Riemann} Zeta Function and of the Incomplete Gamma Function", journal = j-MATH-COMPUT, volume = "24", number = "111", pages = "679--696", month = jul, year = "1970", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Lehman:1970:DZR, author = "R. S. Lehman", title = "On the distribution of zeros of the {Riemann} zeta-function", journal = j-PROC-LONDON-MATH-SOC-1, volume = "3", number = "20", pages = "303--320", month = "????", year = "1970", ISSN = "0024-6115 (print), 1460-244X (electronic)", ISSN-L = "0024-6115", MRnumber = "MR0258768 (41:3414)", bibdate = "Mon Oct 24 12:42:07 2011", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/t/turing-alan-mathison.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "This paper corrects several errors in \cite{Turing:1953:SCR}. See also \cite{Trudgian:2011:ITM}.", acknowledgement = ack-nhfb, fjournal = "Proceedings of the London Mathematical Society. First Series", journal-URL = "http://plms.oxfordjournals.org/content/by/year", } @TechReport{Lugish:1970:CAA, author = "B. G. de Lugish", title = "A Class of Algorithms for Automatic Evaluation of Certain Elementary Function in a Binary Computer", type = "Report", number = "399", institution = "Department of Computer Science, University of Illinois", pages = "????", month = jun, year = "1970", bibdate = "Fri Sep 02 22:49:20 1994", bibsource = "garbo.uwasa.fi:/pc/doc-soft/fpbiblio.txt; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nj, } @Article{Luke:1970:FAE, author = "Yudell L. Luke", title = "Further Approximations for Elliptic Integrals", journal = j-MATH-COMPUT, volume = "24", number = "109", pages = "191--198", month = jan, year = "1970", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Minton:1970:GHF, author = "Barry M. Minton", title = "Generalized Hypergeometric Function of Unit Argument", journal = j-J-MATH-PHYS, volume = "11", number = "4", pages = "1375--1376", month = apr, year = "1970", CODEN = "JMAPAQ", DOI = "https://doi.org/10.1063/1.1665270", ISSN = "0022-2488 (print), 1089-7658 (electronic), 1527-2427", ISSN-L = "0022-2488", bibdate = "Fri Oct 28 16:39:25 MDT 2011", bibsource = "http://jmp.aip.org/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jmathphys1970.bib", URL = "http://jmp.aip.org/resource/1/jmapaq/v11/i4/p1375_s1", acknowledgement = ack-nhfb, classification = "A0200 (Mathematical methods in physics)", corpsource = "Univ. Calgary, Alta., Canada", fjournal = "Journal of Mathematical Physics", journal-URL = "http://jmp.aip.org/", keywords = "functions", onlinedate = "28 October 2003", pagecount = "2", } @Article{Ng:1970:CAE, author = "E. N. Ng", title = "Certification of {Algorithm 385, Exponential Integral $ \operatorname {Ei}(x) $}", journal = j-CACM, volume = "13", number = "7", pages = "448--449", month = jul, year = "1970", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Sat Oct 30 09:18:14 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", remark = "Fullerton: Comments on a FORTRAN routine.", } @Article{Ng:1970:CDF, author = "E. W. Ng and C. J. Devine", title = "On the Computation of {Debye} Functions of Integer Orders", journal = j-MATH-COMPUT, volume = "24", number = "110", pages = "405--407", month = apr, year = "1970", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", remark = "Fullerton: Debye functions are incomplete Riemann zeta functions. $ \overbar {D}_p(x) = \frac {1}{\Gamma (p - 1)} \int_0^x \frac {t^p}{e^t - 1} \, d t $ and the complementary integral are calculated to 20 digits.", } @Article{Ninomiya:1970:BRS, author = "Ichizo Ninomiya", title = "Best Rational Starting Approximations and Improved {Newton} Iteration for the Square Root", journal = j-MATH-COMPUT, volume = "24", number = "110", pages = "391--404", month = apr, year = "1970", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "garbo.uwasa.fi:/pc/doc-soft/fpbiblio.txt; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb # " and " # ack-nj, classcodes = "C4130 (Interpolation and function approximation)", corpsource = "Nagoya Univ., Chikua ku, Japan", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "computing procedure; function approximation; iterative methods; Newton iteration; rational approximation; square root", treatment = "T Theoretical or Mathematical", } @Article{Paciorek:1970:AEI, author = "K. A. Paciorek", title = "{Algorithm 385}: {Exponential} Integral {$ \operatorname {Ei}(x) $}", journal = j-CACM, volume = "13", number = "7", pages = "446--447", month = jul, year = "1970", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Tue Mar 25 13:26:09 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/cacm1960.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "See remark \cite{Redish:1970:RAE,Frisch:1972:RAR}.", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", classcodes = "C4160 (Numerical integration and differentiation); C7300 (Natural sciences computing)", corpsource = "Argonne Nat. Lab., IL, USA", fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "integration; subroutines", remark = "Fullerton: A 100-line FORTRAN routine for both $ \operatorname {E1}(x) $ and $ \operatorname {Ei}(x) $.", } @Article{Phillips:1970:GLE, author = "David L. Phillips", title = "Generalized Logarithmic Error and {Newton}'s Method for the $m$-th Root", journal = j-MATH-COMPUT, volume = "24", number = "110", pages = "383--389", month = apr, year = "1970", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "garbo.uwasa.fi:/pc/doc-soft/fpbiblio.txt; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp1970.bib; JSTOR database", acknowledgement = ack-nj # " and " # ack-nhfb, ajournal = "Math. Comput.", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Raff:1970:CGF, author = "Morton S. Raff", title = "On Calculating the Gamma Function of Non-Integral Arguments", journal = j-AMER-STAT, volume = "24", number = "2", pages = "22--24", month = apr, year = "1970", CODEN = "ASTAAJ", ISSN = "0003-1305 (print), 1537-2731 (electronic)", ISSN-L = "0003-1305", bibdate = "Fri Jan 27 10:52:18 MST 2012", bibsource = "http://www.jstor.org/journals/00031305.html; http://www.jstor.org/stable/i326364; https://www.math.utah.edu/pub/tex/bib/amstat1970.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.jstor.org/stable/2681926", acknowledgement = ack-nhfb, fjournal = "The American Statistician", journal-URL = "http://www.tandfonline.com/loi/utas20", } @Article{Redish:1970:RAE, author = "K. A. Redish", title = "Remark on {Algorithm 385, Exponential Integral $ \operatorname {Ei}(x) $}", journal = j-CACM, volume = "13", number = "12", pages = "750--750", month = dec, year = "1970", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Sat Oct 30 09:56:59 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "See \cite{Paciorek:1970:AEI}", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", remark = "Fullerton: Minor corrections to a FORTRAN routine.", } @TechReport{Rothmaier:1970:BQN, author = "B. Rothmaier", title = "{Die Berechnung der Quadratwurzel nebst Schranken auf Dualmaschinen} \toenglish {The Computation of the Square Root together with [Interval] Bounds on Binary Machines} \endtoenglish", type = "{Interner Bericht}", number = "Nr. 70/17", institution = "Institut f{\"u}r Informatik, Universit{\"a}t Karlsruhe", pages = "??", year = "1970", bibdate = "Fri Sep 16 16:30:41 1994", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nj, } @TechReport{Rothmaier:1970:DSB, author = "B. Rothmaier", title = "{Dokumentation der Standardfunktionen des Betriebssystems Hydra X8} \toenglish {Documentation} of the Elementary Functions of the Operating System {Hydra X8} \endtoenglish", type = "Interner {Bericht}", number = "Nr. 70/8", institution = "Institut f{\"u}r Informatik, Universit{\"a}t Karlsruhe", address = "Karlsruhe, Germany", pages = "????", year = "1970", bibdate = "Fri Jun 11 12:37:53 1999", bibsource = "garbo.uwasa.fi:/pc/doc-soft/fpbiblio.txt; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nj, } @Article{Smith:1970:ASH, author = "Robert R. Smith and Dennis McCall", title = "{Algorithm 392}: {Systems} of Hyperbolic {P.D.E.}", journal = j-CACM, volume = "13", number = "9", pages = "567--570", month = sep, year = "1970", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Tue Mar 25 13:26:09 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/cacm1960.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "See remark \cite{Frisch:1972:RAR}.", acknowledgement = ack-nhfb, classcodes = "C4170 (Differential equations); C7310 (Mathematics computing)", corpsource = "US Naval Electronics Lab. Center, San Diego, CA, USA", fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "boundary-value problems; partial differential equations", } @Book{Spain:1970:FMP, author = "Barry Spain and M. G. (Michael Gambier) Smith", title = "Functions of Mathematical Physics", publisher = pub-VAN-NOSTRAND-REINHOLD, address = pub-VAN-NOSTRAND-REINHOLD:adr, pages = "xi + 208", year = "1970", ISBN = "0-442-07871-4, 0-442-07876-5 (hardcover)", ISBN-13 = "978-0-442-07871-3, 978-0-442-07876-8 (hardcover)", LCCN = "QA351 .S69", bibdate = "Tue Dec 5 10:54:45 MST 2023", bibsource = "fsz3950.oclc.org:210/WorldCat; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", series = "The New university mathematics series", URL = "http://books.google.com/books?id=kYgZAQAAIAAJ", acknowledgement = ack-nhfb, subject = "Functions, Special; Mathematical physics; Fonctions sp{\'e}ciales; Physique math{\'e}matique; Functions, Special; Mathematical physics; Functions, Special; Fonctions", tableofcontents = "1: Series solution of second-order linear homogeneous equations \\ 2: Contour integral solutions of an ordinary linear differential equation \\ 3: Oscillation Theorems and Sturm--Liouville Theory \\ 4: Asymptotics \\ 5: The Gamma function \\ 6: The Hypergeometric Equation \\ 7: The confluent hypergeometric function \\ 8: The Legendre Functions \\ 9: Bessel Functions \\ 10: Laguerre Polynomials \\ 11: Hermite Polynomials \\ Appendix 1: The Laplace and Helmholtz Equations \\ Appendix 2: The Schr{\"o}dinger Equation \\ References \\ Index", } @Article{Squire:1970:RAI, author = "William Squire", title = "A Rational Approximation to an Integral Appearing in Glow Curve Theory", journal = j-J-COMPUT-PHYS, volume = "6", number = "1", pages = "152--253", month = aug, year = "1970", CODEN = "JCTPAH", DOI = "https://doi.org/10.1016/0021-9991(70)90016-1", ISSN = "0021-9991 (print), 1090-2716 (electronic)", ISSN-L = "0021-9991", bibdate = "Sat Oct 30 10:57:00 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputphys1970.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Journal of Computational Physics", journal-URL = "http://www.sciencedirect.com/science/journal/00219991/", } @Article{Srivastava:1970:CRI, author = "H. M. Srivastava", title = "Certain Results Involving Generalized Hypergeometric Functions", journal = j-SIAM-J-MATH-ANA, volume = "1", number = "1", pages = "75--81", month = feb, year = "1970", CODEN = "SJMAAH", ISSN = "0036-1410 (print), 1095-7154 (electronic)", ISSN-L = "0036-1410", bibdate = "Sun Nov 28 19:21:56 MST 2010", bibsource = "http://epubs.siam.org/sam-bin/dbq/toc/SIMA/1/1; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/siamjmathana.bib", acknowledgement = ack-nhfb, fjournal = "SIAM Journal on Mathematical Analysis", journal-URL = "http://epubs.siam.org/sima", } @Article{Stegun:1970:ACM, author = "I. A. Stegun and R. Zucker", title = "Automatic Computing Methods for Special Functions. Part 1. {Error}, Probability, and Related Functions", journal = j-J-RES-NATL-BUR-STAND-1934, volume = "74B", number = "3", pages = "211--224", month = jul, year = "1970", ISSN = "0091-0635", bibdate = "Sat Oct 30 10:58:39 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Journal of Research of the National Bureau of Standards (1934)", journal-URL = "https://www.nist.gov/nist-research-library/journal-research-nist/past-papers", remark = "Fullerton: Adjustable double precision FORTRAN routines for $ \erf $ and $ \erfc $.", } @Article{Taylor:1970:OSA, author = "G. D. Taylor", title = "Optimal starting approximations for {Newton}'s method", journal = j-J-APPROX-THEORY, volume = "3", number = "2", pages = "156--163", month = jun, year = "1970", CODEN = "JAXTAZ", DOI = "https://doi.org/10.1016/0021-9045(70)90024-9", ISSN = "0021-9045 (print), 1096-0430 (electronic)", ISSN-L = "0021-9045", bibdate = "Mon Nov 10 09:44:35 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Journal of Approximation Theory", journal-URL = "http://www.sciencedirect.com/science/journal/00219045", } @Book{Tolke:1970:PFT, author = "Friedrich T{\"o}lke", title = "{Praktische Funktionenlehre. 6. Tafeln aus dem Gebiet der Theta-Funktionen und der elliptischen Funtionen}. ({German}) [{Practical} functional theory. 6. {Tables} from the field of theta functions and elliptic functions]", publisher = pub-SV, address = pub-SV:adr, pages = "452--1047 (vol. 2)", year = "1970", ISBN = "3-662-13079-3 (print), 3-662-13078-5", ISBN-13 = "978-3-662-13079-7 (print), 978-3-662-13078-0", LCCN = "????", bibdate = "Mon Feb 13 19:01:10 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "Two volumes", acknowledgement = ack-nhfb, language = "German", tableofcontents = "Sechsstellige Tafel III der Theta-Funktionen und ihrer logarithmischen Ableitungen, der Jacobischen elliptischen Funktionen und ihrer logarithmischen Ableitungen sowie der Weierstrassschen $z$, $?$, und $?$-Funktionen einschlie{\ss}lich einiger Parameterfunktionen f{\"u}r $? = z/2K$ als Argument und bzw. $1/?$ als Parameter $2$ H{\"a}lfte, Parameterbereich $1 ? 1 / ? 0$ \\ Neunstellige Tafel IV der Legendreschen Normalintegrale erster und zweiter Gattung sowie der Jacobischen Zeta-Funktion und der abgewandelten Heumanschen Lambda-Funktion \\ Sechsstellige Tafel V der $D$-Funktionen erster bis vierter Ordnung f{\"u}r die Charakteristiken 1 bis 4 \\ Sechsstellige Tafel VI der Legendreschen Normalintegrale erster und zweiter Gattung sowie der Funktion \\ \ldots{}", } @Article{Wilson:1970:OSA, author = "M. Wayne Wilson", title = "Optimal Starting Approximations for Generating Square Root for Slow or No Divide", journal = j-CACM, volume = "13", number = "9", pages = "559--560", month = sep, year = "1970", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", MRclass = "65.50", MRnumber = "44\#2338", MRreviewer = "J. E. {Dennis, Jr.}", bibdate = "Tue Apr 08 20:38:30 1997", bibsource = "Compendex database; ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; https://www.math.utah.edu/pub/tex/bib/cacm1970.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", abstract = "On computing machines with slow or no division, it is preferable to use an iterative scheme for the square root different from the classical Heron scheme. The problem of optimal initial approximants is considered, and some optimal polynomial initial approximations are tabulated.", acknowledgement = ack-nj # " and " # ack-nhfb, classcodes = "C5230 (Digital arithmetic methods)", corpsource = "IBM, Houston, TX, USA", fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", journalabr = "Commun ACM", keywords = "CACMA; digital arithmetic; ele; iterative methods; mathematics; numerical methods; optimisation", } @Article{Winograd:1970:NMN, author = "Shmuel Winograd", title = "On the number of multiplications necessary to compute certain functions", journal = j-COMM-PURE-APPL-MATH, volume = "23", number = "2", pages = "165--179", month = mar, year = "1970", CODEN = "CPAMAT, CPMAMV", DOI = "https://doi.org/10.1002/cpa.3160230204", ISSN = "0010-3640 (print), 1097-0312 (electronic)", ISSN-L = "0010-3640", bibdate = "Sat Oct 21 12:05:50 2023", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, ajournal = "Comm. Pure Appl. Math.", fjournal = "Communications on Pure and Applied Mathematics (New York)", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)1097-0312", keywords = "number of multiplications to evaluate a polynomial", remark = "From the second paragraph: ``Motzkin [3] introduced the notion of preconditioning of the coefficients. Motzkin showed that if, in the course of computing $ P_n(x) = \sum_{i = 0}^n a_i x^i $, operations which depend only the $ a_i $ are not counted, then only about $ n / 2 $ multiplications are necessary to evaluate $ P_n(x) $. The obvious application of this result is when the same polynomial $ P_n(x) $ has to be evaluated at many different points.''.", } @TechReport{Yohe:1970:RBC, author = "J. M. Yohe", title = "Rigorous Bounds on Computed Approximations to Square Roots and Cube Roots", type = "MRC Technical Summary", number = "1088", institution = "University of Wisconsin, Madison", year = "1970", bibdate = "Fri Jan 12 11:37:56 1996", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-jr, } @Article{Yong:1970:GBA, author = "Lam Lay Yong", title = "The Geometrical Basis of the {Ancient Chinese} Square-Root Method", journal = j-ISIS, volume = "61", number = "1", pages = "92--102", month = "Spring", year = "1970", CODEN = "ISISA4", ISSN = "0021-1753 (print), 1545-6994 (electronic)", ISSN-L = "0021-1753", bibdate = "Tue Jul 30 21:28:39 MDT 2013", bibsource = "http://www.jstor.org/action/showPublication?journalCode=isis; http://www.jstor.org/stable/i302287; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/isis1970.bib", URL = "http://www.jstor.org/stable/229151", acknowledgement = ack-nhfb, fjournal = "Isis", journal-URL = "http://www.jstor.org/journal/isis", } @Article{Zill:1970:SEI, author = "D. G. Zill and B. C. Carlson", title = "Symmetric Elliptic Integrals of the Third Kind", journal = j-MATH-COMPUT, volume = "24", number = "109", pages = "199--214", month = jan, year = "1970", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Bohan:1971:ADC, author = "K. A. Bohan and K. V. La{\v{s}}{\v{c}}enov", title = "The analytic definition of certain elementary functions. ({Russian}) Questions of modern mathematics and methods of teaching it at institutions of higher learning", journal = "Leningrad. Gos. Ped. Inst. U{\v{c}}en. Zap.", volume = "404", pages = "59--78", year = "1971", MRclass = "26A09", MRnumber = "55 \#10612", bibdate = "Sat Jan 11 17:44:01 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, language = "Russian", } @Book{Byrd:1971:HEI, author = "Paul F. Byrd and Morris D. Friedman", title = "Handbook of Elliptic Integrals for Engineers and Scientists", volume = "67", publisher = pub-SV, address = pub-SV:adr, edition = "Second", pages = "xvi + 358", year = "1971", DOI = "https://doi.org/10.1007/978-3-642-65138-0", ISBN = "0-387-05318-2 (New York)", ISBN-13 = "978-0-387-05318-9 (New York)", LCCN = "QA343 .B95 1971", bibdate = "Mon Oct 15 16:40:14 MDT 2007", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/master.bib; https://www.math.utah.edu/pub/tex/bib/mathcw.bib; z3950.loc.gov:7090/Voyager", series = "Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen", acknowledgement = ack-nhfb, subject = "Elliptic functions", tableofcontents = "Introduction / 1--7 \\ Definitions and Fundamental Relations / 8--41 \\ Reduction of Algebraic Integrands to Jacobian Elliptic Functions / 42--161 \\ Reduction of Trigonometric Integrands to Jacobian Elliptic Functions / 162--181 \\ Reduction of Hyperbolic Integrands to Jacobian Elliptic Functions / 182--190 \\ Table of Integrals of Jacobian Elliptic Functions / 191--222 \\ Elliptic Integrals of the Third Kind / 223--239 \\ Miscellaneous Elliptic Integrals Involving Trigonometric and Hyperbolic Integrands / 240--248 \\ Elliptic Integrals Resulting from Laplace Transformations / 249--251 \\ Hyperelliptic Integrals / 252--271 \\ Integrals of the Elliptic Integrals / 272--281 \\ Derivatives / 282--287 \\ Miscellaneous Integrals and Formulas / 288--297 \\ Expansions in Series / 298--307 \\ Appendix / 308 \\ Bibliography / 351 \\ Supplemental Bibliography / 353 \\ Index / 355", } @Misc{Chen:1971:BAU, author = "Tien Chi Chen", title = "Binary arithmetic unit implementing a multiplicative iteration for the exponential, logarithm, quotient and square root functions", howpublished = "United States Patent 3,631,230", day = "28", month = dec, year = "1971", bibdate = "Tue Jan 08 21:54:11 2008", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "http://www.freepatentsonline.com/3631230.html", abstract = "Apparatus and a method is described for efficiently achieving arithmetic evaluations for functions such as exponential, logarithm, quotient, and square root with a minimum use of multiplications or divisions. Basically, use is made of the fact that $ x(1 \pm 2^{-m}) $ can be evaluated by a shift followed by an add. A pair of numbers $ (x_k, y_k) $ can represent a function $ x : f(x) = g(x_k, y_k) $, such that $ g(l, y_n) = y_n $ for logarithm, quotient and square root. Then, multiplication by shifting is applied to $ x_k $ with suitable adjustments on $ y_k $, until $ x_k $ is close to unity, at which time $ y_k $ represents the desired answer. The exponential is computed by essentially reversing the logarithm procedure. A termination algorithm further improves accuracy. The apparatus involves two registers for $ x_k $ and $ y_k $, a local memory, an adder and a shift register.", acknowledgement = ack-nhfb, } @Article{Choong:1971:RA, author = "K. Y. Choong and D. E. Daykin and C. R. Rathbone", title = "Rational Approximations to $ \pi $", journal = j-MATH-COMPUT, volume = "25", number = "114", pages = "387--392", month = apr, year = "1971", CODEN = "MCMPAF", DOI = "https://doi.org/10.1090/S0025-5718-1971-0300981-0", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp1970.bib; https://www.math.utah.edu/pub/tex/bib/pi.bib; JSTOR database", note = "See errata \cite{Shanks:1976:TER}.", URL = "http://www.ams.org/journals/mcom/1971-25-114/S0025-5718-1971-0300981-0", acknowledgement = ack-nhfb, fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Cody:1971:CAR, author = "W. J. Cody and K. E. Hillstrom and Henry C. {Thatcher, Jr.}", title = "{Chebyshev} approximations for the {Riemann} zeta function", journal = j-MATH-COMPUT, volume = "25", number = "115", pages = "537--547", month = jul, year = "1971", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "65D20", MRnumber = "47 2785", bibdate = "Wed Jan 17 08:57:00 1996", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/c/cody-william-j.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-wjc, citedby = "Fullerton:1980:BEM", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", remark = "Fullerton: 20-digit approximations for either $ \zeta (s) $ or $ \zeta (s) - 1 $.", } @InCollection{Cody:1971:SEF, author = "W. J. Cody", title = "Software for the Elementary Functions", crossref = "Rice:1971:MS", pages = "171--186", year = "1971", bibdate = "Thu Sep 15 18:56:47 1994", bibsource = "garbo.uwasa.fi:/pc/doc-soft/fpbiblio.txt; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nj, } @Article{Deverell:1971:GSC, author = "J. Deverell", title = "Generation of sines and cosines using iterative arrays", journal = j-ELECT-LETTERS, volume = "7", number = "20", pages = "616--618", day = "1", month = oct, year = "1971", CODEN = "ELLEAK", DOI = "https://doi.org/10.1049/el:19710416", ISSN = "0013-5194 (print), 1350-911X (electronic)", ISSN-L = "0013-5194", bibdate = "Mon Nov 10 12:38:32 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "https://digital-library.theiet.org/doi/abs/10.1049/el%3A19710416", acknowledgement = ack-nhfb, fjournal = "Electronics Letters", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=2220; https://digital-library.theiet.org/journal/el", remark = "Specialized circuits that sum up to six terms of the Taylor series.", } @Article{Glasser:1971:CFE, author = "M. L. Glasser and V. E. Wood", title = "A Closed Form Evaluation of the Elliptic Integral", journal = j-MATH-COMPUT, volume = "25", number = "115", pages = "535--536", month = jul, year = "1971", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Glasser:1971:EII, author = "M. L. Glasser", title = "An Elliptic Integral Identity", journal = j-MATH-COMPUT, volume = "25", number = "115", pages = "533--534", month = jul, year = "1971", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Herman:1971:EDH, author = "G. T. Herman", title = "The equivalence of different hierarchies of elementary functions", journal = "Z. Math. Logik Grundlagen Math.", volume = "17", pages = "219--224", year = "1971", MRclass = "02.77 (68.00)", MRnumber = "44 \#6494", MRreviewer = "D. A. Clarke", bibdate = "Sat Jan 11 17:44:01 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, } @Book{Hochstadt:1971:FMP, author = "Harry Hochstadt", title = "The Functions of Mathematical Physics", volume = "XXIII", publisher = pub-WI, address = pub-WI:adr, pages = "xi + 322", year = "1971", ISBN = "0-471-40170-6 (hardcover)", ISBN-13 = "978-0-471-40170-4 (hardcover)", LCCN = "QA351 .H68", bibdate = "Tue Dec 5 10:48:41 MST 2023", bibsource = "fsz3950.oclc.org:210/WorldCat; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", series = "Pure and applied mathematics: a series of texts and monographs", acknowledgement = ack-nhfb, subject = "Functions, Special; Fonctions sp{\'e}ciales; Fonctions (math{\'e}matiques)", tableofcontents = "1: Orthogonal Polynomials \\ 1 Linear Spaces / 1 \\ 2 Orthogonal Polynomials / 6 \\ 3 The Recurrence Formula / 8 \\ 4 The Christoffel--Darboux Formula / 9 \\ 5 The Weierstrass Approximation Theorem / 11 \\ 6 The Zeros of the Orthogonal Polynomials / 14 \\ 7 Approximation Theory / 16 \\ 8 More about the Zeros of the Orthonormal Polynomials / 23 \\ 9 The completeness of the Orthonormal Polynomials in the Space of Square-Integrable Functions / 27 \\ 10 Generalizations and an Application to Conformal Mappings / 32 \\ \\ 2: The Classical Orthogonal Polynomials 1 Rodrigues' Formula and the Classical Orthogonal Polynomials / 39 \\ 2 The Differential Equations Satisfied by the Classical Orthogonal Polynomials / 43 \\ 3 On the Zeros of the Jacobi Polynomials / 45 \\ 4 An Alternative Approach to the Tchebicheff Polynomials / 46 \\ 5 An Application of the Hermite Polynomials to Quantum Mechanics / 49 \\ 6 The Completeness of the Hermite and Laguerre Polynomials / 53 \\ 7 Generating Functions / 57 \\ \\ 3: The Gamma Function 1 Definitions and Basic Properties / 61 \\ 2 Analytic Continuation and Integral Representations / 65 \\ 3 Asymptotic Expansions / 69 \\ 4 Beta Functions / 75 \\ 5 The Logarithmic Derivative of the Gamma Function / 77 \\ 6 Mellin--Barnes Integrals / 78 \\ 7 Mellin Transforms / 80 \\ 8 Applications to Algebraic Equations / 81 \\ \\ 4: Hypergeometric Functions 1 Review of Linear Differential Equations with Regular Singular Points / 88 \\ 2 The Hypergeometric Differential Equation / 90 \\ 3 The Hypergeometric Function / 93 \\ 4 A General Method for Finding Integral Representations / 100 \\ 5 Integral Representations for the Hypergeometric Function / 105 \\ 6 The Twenty-four Solutions of the Hypergeometric / Equation / 106 \\ 7 The Schwarz--Christoffel Transformation / 112 \\ 8 Mappings of Curvilinear Triangles / 119 \\ 9 Group Theoretic Discussion of the Case $ \pi(\alpha_1 + \alpha_2 + \alpha_3) > \pi$ / 130 \\ 10 Nonlinear Transformations of Hypergeometric Functions / 132 \\ \\ 5: The Legendre Functions 1 Laplace's Differential Equation / 138 \\ 2 Maxwell's Theory of Poles / 140 \\ 3 Relationship to the Hypergeometric Functions / 141 \\ 4 Expansion Formulas / 147 \\ 5 The Addition Theorem / 149 \\ 6 Green's Functions / 153 \\ 7 The Complete Solution of Legendre's Differential Equation / 156 \\ 8 Asymptotic Formulas / 161 \\ \\ 6: Spherical Harmonics in $p$ Dimensions 1 Homogeneous Polynomials / 168 \\ 2 Orthogonality of Spherical Harmonics / 171 \\ 3 Legendre Polynomials / 175 \\ 4 Applications to Boundary Value Problems / 183 \\ \\ 7: Confluent Hypergeometric Functions 1 Relationship to the Hypergeometric Functions / 189 \\ 2 Applications of These Functions in Mathematical Physics / 191 \\ 3 Integral Representations / 195 \\ 4 Asymptotic Representations / 198 \\ \\ 8: Bessel Functions 1 Basic Definitions / 200 \\ 2 Integral Representations / 203 \\ 3 Relationship to the Legendre Functions / 205 \\ 4 The Generating Function of the Bessel Function / 207 \\ 5 More Integral Representations / 210 \\ 6 Addition Theorems / 216 \\ 7 The Complete Solution of Bessel's Equation / 223 \\ 8 Asymptotic Expansions for Large Argument / 225 \\ 9 Airy Functions / 230 \\ 10 Asymptotic Expansions for Large Indices and Large Arguments / 235 \\ 11 Some Applications of Bessel Functions in Physical Optics / 241 \\ 12 The Zeros of Bessel Functions / 249 \\ 13 Fourier--Bessel Expansions / 257 \\ 14 Applications in Mathematical Physics / 266 \\ 15 Discontinuous Integrals / 269 \\ \\ 9: Hill's Equation 1 Mathieu's Equation / 281 \\ 2 Hill's Equation / 282 \\ 3 The Discriminant / 287 \\ 4 Expansion Theorems / 299 \\ 5 Inverse Problems / 305 \\ 6 Hill's Equations with Even Coefficients / 309 \\ 7 Mathieu's Equation Revisited / 310 \\ 8 Energy Bands in Crystals / 313 \\ Appendix / 314 \\ \\ Bibliography / 318 \\ \\ Index / 321", } @Article{Honey:1971:CCD, author = "D. W. Honey", title = "Correspondence: Calculation of a double-length square root from a double length number using single precision techniques", journal = j-COMP-J, volume = "14", number = "4", pages = "443--443", month = nov, year = "1971", CODEN = "CMPJA6", ISSN = "0010-4620 (print), 1460-2067 (electronic)", ISSN-L = "0010-4620", bibdate = "Fri Sep 29 08:51:58 MDT 2000", bibsource = "http://www3.oup.co.uk/computer_journal/hdb/Volume_14/Issue_04/; https://www.math.utah.edu/pub/tex/bib/compj1970.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "http://www3.oup.co.uk/computer_journal/hdb/Volume_14/Issue_04/140443.sgm.abs.html; http://www3.oup.co.uk/computer_journal/hdb/Volume_14/Issue_04/tiff/443.tif", acknowledgement = ack-nhfb, fjournal = "The Computer Journal", journal-URL = "http://comjnl.oxfordjournals.org/", } @Article{Kuki:1971:FEP, author = "H. Kuki and J. Ascoly", title = "{FORTRAN} extended-precision library", journal = j-IBM-SYS-J, volume = "10", number = "1", pages = "39--61", year = "1971", CODEN = "IBMSA7", ISSN = "0018-8670", bibdate = "Thu Sep 15 18:51:32 1994", bibsource = "garbo.uwasa.fi:/pc/doc-soft/fpbiblio.txt; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ibmsysj.bib", acknowledgement = ack-nj, fjournal = "IBM Systems Journal", xxmonth = "(none)", } @InCollection{Kuki:1971:MFS, author = "H. Kuki", title = "Mathematical Function Subprograms for Basic System Libraries{}\emdash Objectives, Constraints, and Trade-Off", crossref = "Rice:1971:MS", pages = "187--199", year = "1971", bibdate = "Fri Sep 16 16:27:40 1994", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nj, } @Article{Lehmer:1971:CCM, author = "D. H. Lehmer", title = "On the compounding of certain means", journal = j-J-MATH-ANAL-APPL, volume = "36", number = "1", pages = "183--200", month = oct, year = "1971", CODEN = "JMANAK", DOI = "https://doi.org/10.1016/0022-247x(71)90029-1", ISSN = "0022-247X (print), 1096-0813 (electronic)", ISSN-L = "0022-247X", bibdate = "Tue Mar 14 18:52:11 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Journal of Mathematical Analysis and Applications", journal-URL = "http://www.sciencedirect.com/science/journal/0022247X", keywords = "arithmetic--geometric mean (AGM) iteration; complete elliptic integrals of the first and second kinds. Landen s transformation", } @Article{Liron:1971:ISR, author = "N. Liron", title = "Infinite Sums of Roots for a Class of Transcendental Equations and {Bessel} Functions of Order One-Half", journal = j-MATH-COMPUT, volume = "25", number = "116", pages = "769--781", month = oct, year = "1971", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Lucas:1971:AAC, author = "C. W. {Lucas, Jr.} and C. W. Terrill", title = "{ACM Algorithm 404}: Complex Gamma Function [{S14}]", journal = j-CACM, volume = "14", number = "1", pages = "48--49", month = jan, year = "1971", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Mon Jan 22 07:00:03 MST 2001", bibsource = "http://dblp.uni-trier.de/db/journals/cacm/cacm14.html#LucasT71; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", classcodes = "C4120 (Functional analysis); C7310 (Mathematics computing)", corpsource = "Coll. William and Mary, Williamsburg, VA, USA", fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "algorithm; CGAMMA; complex gamma function evaluation; formula; function evaluation; poles of gamma function; recursion formula; reflection; Stirling's asymptotic series; subroutine in ALGOL; subroutines", oldlabel = "LucasT71", remark = "Fullerton: Fortran routine with machine-dependent constants.", treatment = "T Theoretical or Mathematical", XMLdata = "ftp://ftp.informatik.uni-trier.de/pub/users/Ley/bib/records.tar.gz#journals/cacm/LucasT71", } @Article{Luke:1971:MTBa, author = "Yudell L. Luke", title = "Miniaturized Tables of {Bessel} Functions", journal = j-MATH-COMPUT, volume = "25", number = "114", pages = "323--330", month = apr, year = "1971", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Luke:1971:MTBb, author = "Yudell L. Luke", title = "Miniaturized Tables of {Bessel} Functions, {II}", journal = j-MATH-COMPUT, volume = "25", number = "116", pages = "789--795", month = oct, year = "1971", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Majithia:1971:CAN, author = "J. C. Majithia and R. Kitai", title = "A Cellular Array for the Nonrestoring Extraction of Square Roots", journal = j-IEEE-TRANS-COMPUT, volume = "C-20", number = "12", pages = "1617--1618", month = dec, year = "1971", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/T-C.1971.223191", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Wed Jul 13 06:38:22 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput1970.bib", URL = "https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=1671784", acknowledgement = ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", } @Article{Matta:1971:UCE, author = "F. Matta and A. Reichel", title = "Uniform Computation of the Error Function and Other Related Functions", journal = j-MATH-COMPUT, volume = "25", number = "114", pages = "339--344", month = apr, year = "1971", CODEN = "MCMPAF", DOI = "https://doi.org/10.1090/S0025-5718-1971-0295538-4", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp1970.bib; JSTOR database", URL = "http://www.ams.org/journals/mcom/1971-25-114/S0025-5718-1971-0295538-4", acknowledgement = ack-nhfb, fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @PhdThesis{Rothmaier:1971:BEF, author = "B. Rothmaier", title = "{Die Berechnung der elementaren Funktionen mit beliebiger Genauigkeit} \toenglish {The Computation of Elementary Functions with Arbitrary Accuracy} \endtoenglish", type = "Dissertation", school = "Universit{\"a}t Karlsruhe", address = "Karlsruhe, Germany", pages = "????", year = "1971", bibdate = "Fri Sep 16 16:30:40 1994", bibsource = "garbo.uwasa.fi:/pc/doc-soft/fpbiblio.txt; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nj, } @Article{Sarkar:1971:EPP, author = "B. P. Sarkar and E. V. Krishnamurthy", title = "Economic Pseudodivision Processes for Obtaining Square Root, Logarithm, and Arctan", journal = j-IEEE-TRANS-COMPUT, volume = "C-20", number = "12", pages = "1589--1593", month = dec, year = "1971", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/T-C.1971.223178", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Thu Sep 01 10:32:36 1994", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", acknowledgement = ack-nj, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", } @Article{Scarton:1971:DPF, author = "Henry A. Scarton", title = "Double precision {FORTRAN} subroutines to compute both ordinary and modified {Bessel} functions of the first kind and of integer order with arbitrary complex argument: {$ J_n(x + j y) $} and {$ I_n(x + j y) $}", journal = j-J-COMPUT-PHYS, volume = "8", number = "2", pages = "295--299", month = oct, year = "1971", CODEN = "JCTPAH", DOI = "https://doi.org/10.1016/0021-9991(71)90010-6", ISSN = "0021-9991 (print), 1090-2716 (electronic)", ISSN-L = "0021-9991", bibdate = "Sun Jan 1 09:15:04 MST 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fortran1.bib; https://www.math.utah.edu/pub/tex/bib/jcomputphys1970.bib", URL = "http://www.sciencedirect.com/science/article/pii/0021999171900106", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Journal of Computational Physics", journal-URL = "http://www.sciencedirect.com/science/journal/00219991/", } @Article{Shenton:1971:CFP, author = "L. R. Shenton and K. O. Bowman", title = "Continued Fractions for the Psi Function and its Derivatives", journal = j-SIAM-J-APPL-MATH, volume = "20", number = "4", pages = "547--554", month = jun, year = "1971", CODEN = "SMJMAP", ISSN = "0036-1399 (print), 1095-712X (electronic)", ISSN-L = "0036-1399", bibdate = "Thu Oct 15 18:16:06 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", URL = "http://www.jstor.org/stable/2099856", acknowledgement = ack-nhfb, fjournal = "SIAM Journal on Applied Mathematics", journal-URL = "http://epubs.siam.org/siap", } @Article{Shipman:1971:HSE, author = "L. L. Shipman and R. E. Christoffersen", title = "High Speed Evaluation of {$ F_0 (x) $}", journal = j-COMP-PHYS-COMM, volume = "2", number = "4", pages = "201--206", month = may # "\slash " # jun, year = "1971", CODEN = "CPHCBZ", DOI = "https://doi.org/10.1016/0010-4655(71)90053-1", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", bibdate = "Sat Oct 30 10:40:08 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/compphyscomm1970.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/0010465571900531", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Computer Physics Communications", journal-URL = "http://www.sciencedirect.com/science/journal/00104655", remark = "Fullerton: $ F_0 (x) = \int_0^1 \exp ( - x u^2) \, d u $, which is simply related to $ \erf $ for $ x > 0 $ and to Dawson's function for $ x < 0 $.", } @Article{Spellucci:1971:DPA, author = "P. Spellucci", title = "Double precision approximations to the elementary functions using {Jacobi-fractions}", journal = j-NUM-MATH, volume = "18", pages = "127--143", year = "1971/1972", CODEN = "NUMMA7", ISSN = "0029-599X (print), 0945-3245 (electronic)", ISSN-L = "0029-599X", MRclass = "65D20", MRnumber = "45 \#7938", bibdate = "Sat Jan 11 17:44:01 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Numerische Mathematik", journal-URL = "http://link.springer.com/journal/211", } @Article{Spira:1971:CGF, author = "Robert Spira", title = "Calculation of the Gamma Function by {Stirling}'s Formula", journal = j-MATH-COMPUT, volume = "25", number = "114", pages = "317--322", month = apr, year = "1971", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Sundblad:1971:AFT, author = "Y. Sundblad", title = "The {Ackermann} Function. {A} Theoretical, Computational, and Formula Manipulative Study", journal = j-BIT, volume = "11", number = "1", pages = "107--119", month = mar, year = "1971", CODEN = "BITTEL, NBITAB", DOI = "https://doi.org/10.1007/BF01935330", ISSN = "0006-3835 (print), 1572-9125 (electronic)", ISSN-L = "0006-3835", bibdate = "Wed Jan 4 18:52:11 MST 2006", bibsource = "http://springerlink.metapress.com/openurl.asp?genre=issue&issn=0006-3835&volume=11&issue=1; https://www.math.utah.edu/pub/tex/bib/bit.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.springerlink.com/openurl.asp?genre=article&issn=0006-3835&volume=11&issue=1&spage=107", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "BIT (Nordisk tidskrift for informationsbehandling)", journal-URL = "http://link.springer.com/journal/10543", remark = "Fullerton: Ackermann's function is a recursively defined function of importance to computer science theorists.", } @InProceedings{Walther:1971:UAE, author = "J. S. Walther", title = "A unified algorithm for elementary functions", crossref = "Macon:1971:SJC", pages = "379--385", year = "1971", DOI = "https://doi.org/10.1145/1478786.1478840", bibdate = "Thu Sep 1 10:15:31 1994", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; https://www.math.utah.edu/pub/tex/bib/cordic.bib https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", note = "Reprinted in \cite{Walther:1972:UAE,Hwang:1979:CAP}.", abstract = "This paper describes a single unified algorithm for the calculation of elementary functions including multiplication, division, sin, cos, tan, arctan, sinh, cosh, tanh, arctanh, ln, exp and square-root. The basis for the algorithm is coordinate rotation in a linear, circular, or hyperbolic coordinate system depending on which function is to be calculated. The only operations required are shifting, adding, subtracting and the recall of prestored constants. The limited domain of convergence of the algorithm is calculated, leading to a discussion of the modifications required to extend the domain for floating-point calculations", acknowledgement = ack-nj, } @Article{Wills:1971:URR, author = "John G. Wills", title = "On the use of recursion relations in the numerical evaluation of spherical {Bessel} functions and {Coulomb} functions", journal = j-J-COMPUT-PHYS, volume = "8", number = "1", pages = "162--166", month = aug, year = "1971", CODEN = "JCTPAH", DOI = "https://doi.org/10.1016/0021-9991(71)90043-X", ISSN = "0021-9991 (print), 1090-2716 (electronic)", ISSN-L = "0021-9991", bibdate = "Sun Jan 1 09:15:04 MST 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputphys1970.bib", URL = "http://www.sciencedirect.com/science/article/pii/002199917190043X", acknowledgement = ack-nhfb, fjournal = "Journal of Computational Physics", journal-URL = "http://www.sciencedirect.com/science/journal/00219991/", } @Article{Wong:1971:SEI, author = "R. Wong and E. Rosenbloom", title = "Series Expansions of $ {W}_{k, m}(z) $ Involving Parabolic Cylinder Functions", journal = j-MATH-COMPUT, volume = "25", number = "116", pages = "783--787", month = oct, year = "1971", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Bardin:1972:CFE, author = "C. Bardin and Y. Dandeu and L. Gauthier and J. Guillermin. T. Lena. J.-M. Pernet and H. H. Wolter and T. Tamura", title = "{Coulomb} Functions in Entire $ (\eta, \rho) $ Plane", journal = j-COMP-PHYS-COMM, volume = "3", number = "2", pages = "73--87", month = mar, year = "1972", CODEN = "CPHCBZ", DOI = "https://doi.org/10.1016/0010-4655(72)90057-4", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", bibdate = "Fri Oct 29 21:09:33 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/compphyscomm1970.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Computer Physics Communications", journal-URL = "http://www.sciencedirect.com/science/journal/00104655", } @Book{Bark:1972:MTF, author = "L. S. Bark", title = "{{\cyr Mnogoznachnye tablitsy {\`e}lementarnykh funktsi{\u\i}}} ($ \operatorname {sin} x $, $ \operatorname {cos} x $, $ e^x $ {\cyr i} $ e^{-x} $). ({Russian}) [Multiplace tables of the elementary functions ($ \operatorname {sin} \ x $, $ \operatorname {cos} \ x $, $ e^x $ and $ e^{-x} $ )]", publisher = "Vy{\v{c}}isl. Centr Akad. Nauk SSSR", address = "Moscow, USSR", edition = "Second, unrevised", pages = "134", year = "1972", MRclass = "65A05", MRnumber = "50 \#6100", bibdate = "Sat Jan 11 17:44:01 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "Tables processed and text translated from the English by L. S. Bark. Library of Mathematical Tables, No. 9. 1972", acknowledgement = ack-nhfb, language = "Russian", } @Article{Carlson:1972:ACL, author = "B. C. Carlson", title = "An algorithm for computing logarithms and arctangents", journal = j-MATH-COMPUT, volume = "26", number = "118", pages = "543--549", month = apr, year = "1972", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, classcodes = "C4130 (Interpolation and function approximation)", corpsource = "Iowa State Univ., Ames, IA, USA", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "acceleration; arctangents; auxiliary; Borchardt's algorithm; convergence; fast; function approximation; functions; inverse circular functions; inverse hyperbolic; iterative algorithm; logarithms; numerical methods; rational operations; recurrence relation; square roots", treatment = "T Theoretical or Mathematical", } @Article{Chen:1972:ACE, author = "Tien Chi Chen", title = "Automatic Computation of Exponentials, Logarithms, Ratios and Square Roots", journal = j-IBM-JRD, volume = "16", number = "4", pages = "380--388", month = jul, year = "1972", CODEN = "IBMJAE", ISSN = "0018-8646 (print), 2151-8556 (electronic)", ISSN-L = "0018-8646", MRclass = "65D20", MRnumber = "49 \#1738", bibdate = "Tue Mar 25 14:26:59 MST 1997", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "http://www.research.ibm.com/journal/rd/164/chen.pdf", abstract = "It is shown how a relatively simple device can evaluate exponentials, logarithms, ratios and square roots for fraction arguments, employing only shifts, adds, high-speed table lookups, and bit counting. The scheme is based on the cotransformation of a number pair $ (x, y) $ such that the $ F(x, y) = f(x_0) $ is invariant; when $x$ is driven towards a known value $ x_w $, $y$ is driven towards the result. For an $N$-bit fraction about $ N / 4 $ iterations are required, each involving two or three adds; then a termination algorithm, based on an add and an abbreviated multiply, completes the process, for a total cost of about one conventional multiply time. Convergence, errors and simulation using APL are discussed.", acknowledgement = ack-nhfb # " and " # ack-nj, classcodes = "C5230 (Digital arithmetic methods)", corpsource = "IBM, San Jose, CA, USA", fjournal = "IBM Journal of Research and Development", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=5288520", keywords = "adds; APL; bit counting; convergence; cotransformation; digital arithmetic; errors; exponentials; high speed table; iteration; logarithms; lookups; ratios; shifts; simulation; square roots; termination algorithm", reviewer = "F. Gotze", treatment = "P Practical", } @TechReport{Ercegovac:1972:RES, author = "Milos D. Ercegovac", title = "Radix 16 Evaluation of Some Elementary Functions", number = "540", institution = "Department of Computer Science, University of Illinois at Urbana-Champaign", address = "Urbana, Illinois", pages = "30", year = "1972", bibdate = "Mon May 19 13:30:58 1997", bibsource = "ftp://ftp.ira.uka.de/pub/bibliography/Techreports/Uiuc.Tr.bib.gz; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, } @Article{Frisch:1972:RAR, author = "Michael J. Frisch", title = "Remark on ``{Algorithms 352, 385, 392}: Remarks on Characteristic Values and Associated Solutions of {Mathieu}'s Differential Equation, Exponential Integral, and Systems of Hyperbolic {P.D.E.}''", journal = j-CACM, volume = "15", number = "12", pages = "1074--??", year = "1972", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Mon Jan 22 06:42:24 MST 2001", bibsource = "http://dblp.uni-trier.de/db/journals/cacm/cacm15.html#Frisch72; https://www.math.utah.edu/pub/tex/bib/cacm1960.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "See \cite{Clemm:1969:ACV,Paciorek:1970:AEI,Smith:1970:ASH}.", acknowledgement = ack-nhfb, fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", oldlabel = "Frisch72", XMLdata = "ftp://ftp.informatik.uni-trier.de/pub/users/Ley/bib/records.tar.gz#journals/cacm/Frisch72", } @Article{Fullerton:1972:MIG, author = "W. Fullerton", title = "{ACM Algorithm 435}: Modified Incomplete Gamma Function", journal = j-CACM, volume = "15", number = "11", pages = "993--995", month = nov, year = "1972", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Thu Sep 08 09:47:55 1994", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "See also \cite{Schoene:1978:RMI}.", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", remark = "Fullerton: Fortran subprogram for evaluating $ e^{x_1} \int_{x_1}^{x_2} |y|^{a - 1} e^{-y} \, d y $ for $a$ roughly between $1$ and $2$.", } @Article{Hunter:1972:NEC, author = "D. B. Hunter and T. Regan", title = "A note on the evaluation of the complementary error function", journal = j-MATH-COMPUT, volume = "26", number = "118", pages = "539--541", month = apr, year = "1972", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", classcodes = "C4160 (Numerical integration and differentiation)", corpsource = "Univ. Bradford, UK", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "complementary error function; complex variable; evaluation; integration; method of Matta and Reichel; modification; numerical; stability", treatment = "T Theoretical or Mathematical", } @Article{Kim:1972:AEH, author = "Shoon K. Kim", title = "The Asymptotic Expansion of a Hypergeometric Function $_2 {F}_2 (1, \alpha; \rho_1, \rho_2; z)$", journal = j-MATH-COMPUT, volume = "26", number = "120", pages = "963--963", month = oct, year = "1972", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp1970.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Kolbig:1972:CAC, author = "K. S. K{\"o}lbig", title = "Certification of ``{Algorithm 363}: {Complex} error function''", journal = j-CACM, volume = "15", number = "6", pages = "465--466", month = jun, year = "1972", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Mon Jan 22 06:55:38 MST 2001", bibsource = "http://dblp.uni-trier.de/db/journals/cacm/cacm15.html#Kolbig72; https://www.math.utah.edu/pub/tex/bib/cacm1960.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "See \cite{Gautschi:1969:ACE}.", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", classcodes = "C4120 (Functional analysis); C7310 (Mathematics computing)", corpsource = "CERN, Geneva, Switzerland", fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "$\erf(z)$; complex error function; function evaluation; special functions; subroutines; Voigt function", oldlabel = "Kolbig72", remark = "Fullerton: Corrections and tests of an Algol procedure.", treatment = "T Theoretical or Mathematical", XMLdata = "ftp://ftp.informatik.uni-trier.de/pub/users/Ley/bib/records.tar.gz#journals/cacm/Kolbig72", } @Article{Kolbig:1972:PCL, author = "K. S. K{\"o}lbig", title = "Programs for Computing the Logarithm of the Gamma Function, and the Digamma Function, for Complex Argument", journal = j-COMP-PHYS-COMM, volume = "4", number = "2", pages = "221--226", month = nov, year = "1972", CODEN = "CPHCBZ", DOI = "https://doi.org/10.1016/0010-4655(72)90012-4", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", bibdate = "Sat Oct 30 08:33:42 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/compphyscomm1970.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Computer Physics Communications", journal-URL = "http://www.sciencedirect.com/science/journal/00104655", } @Article{Kolbig:1972:RCC, author = "K. S. K{\"o}lbig", title = "Remarks on the Computation of {Coulomb} Wavefunctions", journal = j-COMP-PHYS-COMM, volume = "4", number = "2", pages = "214--220", month = nov, year = "1972", CODEN = "CPHCBZ", DOI = "https://doi.org/10.1016/0010-4655(72)90011-2", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", bibdate = "Sat Oct 30 08:35:38 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/compphyscomm1970.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Computer Physics Communications", journal-URL = "http://www.sciencedirect.com/science/journal/00104655", } @Article{Kolbig:1972:ZIG, author = "K. S. Kolbig", title = "On the Zeros of the Incomplete Gamma Function", journal = j-MATH-COMPUT, volume = "26", number = "119", pages = "751--755", month = jul, year = "1972", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, classcodes = "B0290F (Interpolation and function approximation); C4130 (Interpolation and function approximation)", corpsource = "CERN, Geneva, Switzerland", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "asymptotic formulae; complex w plane; function approximation; incomplete gamma function; poles and zeros; zeros", treatment = "T Theoretical or Mathematical", } @Article{Kuki:1972:AAC, author = "Hirondo Kuki", title = "{ACM Algorithm 421}: Complex Gamma Function with Error Control [{S14}]", journal = j-CACM, volume = "15", number = "4", pages = "271--272", month = apr, year = "1972", CODEN = "CACMA2", DOI = "https://doi.org/10.1145/361284.361296", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", MRclass = "65D20", MRnumber = "47 1249", MRreviewer = "L. Fox", bibdate = "Mon Jan 22 06:56:30 MST 2001", bibsource = "http://dblp.uni-trier.de/db/journals/cacm/cacm15.html#Kuki72a; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", classcodes = "C4130 (Interpolation and function approximation); C7310 (Mathematics computing)", corpsource = "Univ. Chicago, IL, USA", fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "algorithm; complex; complex gamma function; complex loggamma; error control; FORTRAN; function; function approximation; loggamma function; programme; subroutines", oldlabel = "Kuki72a", remark = "Fullerton: 100-line FORTRAN routine with double complex accuracy to $ 10^{-14} $.", treatment = "T Theoretical or Mathematical", XMLdata = "ftp://ftp.informatik.uni-trier.de/pub/users/Ley/bib/records.tar.gz#journals/cacm/Kuki72a", } @Article{Kuki:1972:CGF, author = "Hirondo Kuki", title = "Complex Gamma Function with Error Control [{S14}]", journal = j-CACM, volume = "15", number = "4", pages = "262--267", month = apr, year = "1972", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", MRclass = "65D20", MRnumber = "47 1249", MRreviewer = "L. Fox", bibdate = "Mon Jan 22 06:56:30 MST 2001", bibsource = "http://dblp.uni-trier.de/db/journals/cacm/cacm15.html#Kuki72a; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", classcodes = "C4130 (Interpolation and function approximation); C7310 (Mathematics computing)", corpsource = "Univ. Chicago, IL, USA", fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "algorithm; complex; complex gamma function; complex loggamma; error control; FORTRAN; function; function approximation; loggamma function; programme; subroutines", oldlabel = "Kuki72a", remark = "Fullerton: Description of a FORTRAN routine with some math details.", treatment = "T Theoretical or Mathematical", XMLdata = "ftp://ftp.informatik.uni-trier.de/pub/users/Ley/bib/records.tar.gz#journals/cacm/Kuki72a", } @Book{Lebedev:1972:SFT, author = "N. N. (Nikolai Nikolaevich) Lebedev", title = "Special functions and their applications", publisher = pub-DOVER, address = pub-DOVER:adr, pages = "xii + 308", year = "1972", ISBN = "0-486-60624-4 (paperback)", ISBN-13 = "978-0-486-60624-8 (paperback)", LCCN = "QA351 .L3613 1972", bibdate = "Sat Oct 30 16:25:05 MDT 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/master.bib; z3950.loc.gov:7090/Voyager", note = "Translated to English and edited by Richard A. Silverman.", URL = "http://www.loc.gov/catdir/description/dover031/72086228.html", acknowledgement = ack-nhfb, remark = "Translation of Spe{\"e}t{\`\i}sial\S{}nye funk{\"e}t{\`\i}sii i ikh prilozheni{\"e}i{\`\i}a.", subject = "Functions, Special; Mathematical physics", tableofcontents = "1 The Gamma Function \\ \\ 1.1. Definition of the Gamma Function \\ 1.2. Some Relations Satisfied by the Gamma Function \\ 1.3. The Logarithmic Derivative of the Gamma Function \\ 1.4. Asymptotic Representation of the Gamma Function for Large $|z|$ \\ 1.5. Definite Integrals Related to the Gamma Function \\ Problems \\ \\ 2 The Probability Integral and Related Functions \\ \\ 2.1. The Probability Integral and Its Basic Properties \\ 2.2. Asymptotic Representation of the Probability Integral for Large $|z|$ \\ 2.3. The Probability Integral of Imaginary Argument. The Function $F(z)$ \\ 2.4. The Probability Integral of Argument $\sqrt{i} x$. The Fresnel Integrals, 21. \\ 2.5. Application to Probability Theory \\ 2.6. Application to the Theory of Heat Conduction. Cooling of the Surface of a Heated Object \\ 2.7. Application to the Theory of Vibrations. Transverse Vibrations of an Infinite Rod under the Action of a Suddenly Applied Concentrated Force \\ Problems \\ \\ 3 The Exponential Integral and Related Functions \\ \\ 3.1. The Exponential Integral and its Basic Properties \\ 3.2. Asymptotic Representation of the Exponential Integral for Large $|z|$ \\ 3.3. The Exponential Integral of Imaginary Argument. The Sine and Cosine Integrals \\ 3.4. The Logarithmic Integral \\ 3.5. Application to Electromagnetic Theory, Radiation of a Linear Half-Wave Oscillator Problems \\ \\ 4 Orthogonal Polynomials \\ \\ 4.1. Introductory Remarks \\ 4.2. Definition and Generating Function of the Legendre Polynomials \\ 4.3. Recurrence Relations and Differential Equation for the Legendre Polynomials \\ 4.4. Integral Representations of the Legendre Polynomials \\ 4.5. Orthogonality of the Legendre Polynomials \\ 4.6. Asymptotic Representation of the Legendre Polynomials for Large $n$ \\ 4.7. Expansion of Functions in Series of Legendre Polynomials \\ 4.8. Examples of Expansions in Series of Legendre Polynomials \\ 4.9. Definition and Generating Function of the Hermite Polynomials \\ 4.10. Recurrence Relations and Differential Equation for the Hermite Polynomials \\ 4.11. Integral Representations of the Hermite Polynomials \\ 4.12. Integral Equations Satisfied by the Hermite Polynomials \\ 4.13. Orthogonality of the Hermite Polynomials \\ 4.14. Asymptotic Representation of the Hermite Polynomials for Large n \\ 4.15. Expansion of Functions in Series of Hermite Polynomials \\ 4.16. Examples of Expansions in Series of Hermite Polynomials \\ 4.17. Definition and Generating Function of the Laguerre Polynomials \\ 4.18. Recurrence Relations and Differential Equation for the Laguerre Polynomials \\ 4.19. An Integral Representation of the Laguerre Polynomials. Relation between the Laguerre and Hermite Polynomials \\ 4.20. An Integral Equation Satisfied by the Laguerre Polynomials \\ 4.21. Orthogonality of the Laguerre Polynomials \\ 4.22. Asymptotic Representation of the Laguerre Polynomials for Large $n$ \\ 4.23. Expansion of Functions in Series of Laguerre Polynomials \\ 4.24. Examples of Expansions in Series of Laguerre Polynomials \\ 4.25. Application to the Theory of Propagation of Electromagnetic Waves. Reflection from the End of a Long Transmission Line Terminated by a Lumped Inductance \\ Problems \\ \\ 5 Cylinder Functions: Theory \\ \\ 5.1. Introductory Remarks \\ 5.2. Bessel Functions of Nonnegative Integral Order \\ 5.3. Bessel Functions of Arbitrary Order \\ 5.4. General Cylinder Functions. Bessel Functions of the Second Kind \\ 5.5. Series Expansion of the Function $Y_n(z)$ \\ 5.6. Bessel Functions of the Third Kind \\ 5.7. Bessel Functions of Imaginary Argument \\ 5.8. Cylinder Functions of Half-Integral Order \\ 5.9. Wronskians of Pairs of Solutions of Bessel s Equation \\ 5.10. Integral Representations of the Cylinder Functions \\ 5.11. Asymptotic Representations of the Cylinder Functions for Large $|z|$ \\ 5.12. Addition Theorems for the Cylinder Functions, 124.Zeros of the Cylinder Functions \\ 5.13. Expansions in Series and Integrals Involving Cylinder Functions \\ 5.14. Definite Integrals Involving Cylinder Functions \\ 5.15. Cylinder Functions of Nonnegative Argument and Order \\ 5.16. Airy Functions \\ Problems \\ \\ 6 Cylinder Functions: Applications \\ \\ 6.1. Introductory Remarks \\ 6.2. Separation of Variables in Cylindrical Coordinates \\ 6.3. The Boundary Value Problems of Potential Theory. The Dirichlet Problem for a Cylinder \\ 6.4. The Dirichlet Problem for a Domain Bounded by Two Parallel Planes \\ 6.5. The Dirichlet Problem for a Wedge \\ 6.6. The Field of a Point Charge near the Edge of a Conducting Sheet \\ 6.7. Cooling of a Heated Cylinder \\ 6.8. Diffraction by a Cylinder \\ Problems \\ \\ 7 Spherical Harmonics: Theory \\ \\ 7.1. Introductory Remarks \\ 7.2. The Hypergeometric Equation and Its Series Solution \\ 7.3. Legendre Functions \\ 7.4. Integral Representations of the Legendre Functions \\ 7.5. Some Relations Satisfied by the Legendre Functions \\ 7.6. Series Representations of the Legendre Functions \\ 7.7. Wronskians of Pairs of Solutions of Legend-re s Equation \\ 7.8. Recurrence Relations for the Legendre Functions \\ 7.9. Legendre Functions of Nonnegative Integral Degree and Their Relation to Legendre Polynomials \\ 7.10. Legendre Functions of Half-Integral Degree \\ 7.11. Asymptotic Representations of the Legendre Functions for Large $|v|$ \\ 7.12. Associated Legendre Functions \\ Problems \\ \\ 8 Spherical Harmonics: Applications \\ \\ 8.1. Introductory Remarks \\ 8.2. Solution of Laplace s Equation in Spherical Coordinates \\ 8.3. The Dirichlet Problem for a Sphere \\ 8.4. The Field of a Point Charge Inside a Hollow Conducting Sphere \\ 8.5. The Dirichlet Problem for a Cone \\ 8.6. Solution of Laplace s Equation in Spheroidal Coordinates \\ 8.7. The Dirichlet Problem for a Spheroid \\ 8.8. The Gravitational Attraction of a Homogeneous Solid Spheroid \\ 8.9. The Dirichlet Problem for a Hyperboloid of Revolution \\ 8.10. Solution of Laplace s Equation in Toroidal Coordinates \\ 8.11. The Dirichlet Problem for a Torus \\ 8.12. The Dirichlet Problem for a Domain Bounded by Two Intersecting Spheres \\ 8.13. Solution of Laplace s Equation in Bipolar Coordinates \\ 8.14. Solution of Helmholtz s Equation in Spherical Coordinates \\ Problems \\ \\ 9 Hypergeometric Functions \\ \\ 9.1. The Hypergeometric Series and Its Analytic Continuation \\ 9.2. Elementary Properties of the Hypergeometric Function \\ 9.3. Evaluation of $F(\alpha, \beta; \gamma; z)$ for $\Re(\gamma \alpha \beta) > 0$, 243. \\ 9.4. $F(\alpha, \beta; \gamma; z)$ as a Function of its Parameters \\ 9.5. Linear Transformations of the Hypergeometric Function \\ 9.6. Quadratic Transformations of the Hypergeometric Function \\ 9.7. Formulas for Analytic Continuation of $F(\alpha, \beta; \gamma; z)$ in Exceptional Cases \\ 9.8. Representation of Various Functions in Terms of the Hypergeometric Function \\ 9.9. The Confluent Hypergeometric Function \\ 9.10. The Differential Equation for the Confluent Hypergeometric Function and Its Solution. The Confluent Hypergeometric Function of the Second Kind \\ 9.11. Integral Representations of the Confluent Hypergeometric Functions \\ 9.12. Asymptotic Representations of the Confluent Hypergeometric Functions for Large $|z|$ \\ 9.13. Representation of Various Functions in Terms of the Confluent Hypergeometric Functions \\ 9.14. Generalized Hypergeometric Functions \\ Problems \\ \\ 10 Parabolic Cylinder Functions \\ \\ 10.1. Separation of Variables in Laplace s Equation in Parabolic Coordinates \\ 10.2. Hermite Functions \\ 10.3. Some Relations Satisfied by the Hermite Functions \\ 10.4. Recurrence Relations for the Hermite Functions \\ 10.5. Integral Representations of the Hermite Functions \\ 10.6. Asymptotic Representations of the Hermite Functions for Large $|z|$ \\ 10.7. The Dirichlet Problem for a Parabolic Cylinder \\ 10.8. Application to Quantum Mechanics \\ Problems \\ \\ Bibliography \\ \\ Index", } @Article{Ling:1972:EM, author = "Chih-Bing Ling and Jung Lin", title = "On Evaluation of Moments of $ {K}_\nu (t) / {I}_\nu (t) $", journal = j-MATH-COMPUT, volume = "26", number = "118", pages = "529--537", month = apr, year = "1972", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, classcodes = "C4190 (Other numerical methods)", corpsource = "Virginia Politech. Inst., Blacksburg, VA, USA", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "evaluation; K/sub nu/(t)/I/sub; moments; nu/(t); numerical methods; Watson's method", treatment = "T Theoretical or Mathematical", } @Article{Linz:1972:MCB, author = "Peter Linz", title = "A Method for Computing {Bessel} Function Integrals", journal = j-MATH-COMPUT, volume = "26", number = "118", pages = "509--513", month = apr, year = "1972", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", classcodes = "C4160 (Numerical integration and differentiation)", corpsource = "Univ. California, Davis, CA, USA", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "Abel; Bessel function integrals; Fourier integrals; infinite integrals; integration; numerical computation; transform", treatment = "T Theoretical or Mathematical", } @Article{Luke:1972:MTB, author = "Yudell L. Luke", title = "Miniaturized Tables of {Bessel} Functions. {III}", journal = j-MATH-COMPUT, volume = "26", number = "117", pages = "237--240", month = jan, year = "1972", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, classcodes = "C4190 (Other numerical methods)", corpsource = "Univ. Missouri, Kansas City, KS, USA", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "Bessel functions; miniaturized tables; numerical methods", treatment = "T Theoretical or Mathematical", } @Article{MacKinnon:1972:AEH, author = "Robert F. MacKinnon", title = "The asymptotic expansions of {Hankel} transforms and related integrals", journal = j-MATH-COMPUT, volume = "26", number = "118", pages = "515--527", month = apr, year = "1972", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", classcodes = "B0230 (Integral transforms); C1130 (Integral transforms)", corpsource = "Defence Res. Establ., Pacific, Victoria, BC, Canada", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "asymptotic expansions; Bessel; functions; Hankel transforms; integrals; transforms", treatment = "T Theoretical or Mathematical", } @Article{Majithia:1972:CAE, author = "J. C. Majithia", title = "Cellular Array for Extraction of Squares and Square Roots of Binary Numbers", journal = j-IEEE-TRANS-COMPUT, volume = "C-21", number = "9", pages = "1023--1024", month = sep, year = "1972", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/TC.1972.5009084", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Tue Jul 12 18:58:46 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput1970.bib", URL = "https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5009084", acknowledgement = ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", } @TechReport{Manos:1972:CCA, author = "Paul Manos and L. Richard Turner", title = "Constrained {Chebyshev} approximations to some elementary functions suitable for evaluation with floating-point arithmetic", type = "{NASA} Technical Note", number = "TN D-6698", institution = pub-NASA, address = pub-NASA:adr, pages = "iii + 68", month = mar, year = "1972", bibdate = "Mon May 22 11:27:24 2006", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19720010958_1972010958.pdf", acknowledgement = ack-nhfb, } @Article{Marino:1972:NAA, author = "D. Marino", title = "New Algorithms for the Approximate Evaluation in Hardware of Binary Logarithms and Elementary Functions", journal = j-IEEE-TRANS-COMPUT, volume = "C-21", number = "12", pages = "1416--1421", month = dec, year = "1972", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/T-C.1972.223516", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Thu Sep 08 08:05:51 1994", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nj, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", } @Article{Morita:1972:CAG, author = "Tohru Morita and Tsuyoshi Horiguchi", title = "Convergence of the arithmetic-geometric mean procedure for the complex variables and the calculation of the complete elliptic integrals with complex modulus", journal = j-NUM-MATH, volume = "20", number = "5", pages = "425--430", month = oct, year = "1972", CODEN = "NUMMA7", DOI = "https://doi.org/10.1007/BF01402565", ISSN = "0029-599X (print), 0945-3245 (electronic)", ISSN-L = "0029-599X", bibdate = "Thu Jan 13 10:01:46 MST 2011", bibsource = "http://springerlink.metapress.com/openurl.asp?genre=issue&issn=0029-599X&volume=20&issue=5; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.springerlink.com/openurl.asp?genre=article&issn=0029-599X&volume=20&issue=5&spage=425", abstract = "The convergence of the arithmetic-geometric mean procedure is checked for complex variables. The procedure is shown to be useful for the evaluation of the complete elliptic integrals of the first and second kinds with complex modulus. It is suggested that the procedure will be useful also for the numerical calculation of the elliptic integrals and the Jacobian elliptic functions with complex modulus in general.", acknowledgement = ack-nhfb, fjournal = "Numerische Mathematik", journal-URL = "http://link.springer.com/journal/211", } @Article{Moses:1972:TGT, author = "Joel Moses", title = "Toward a General Theory of Special Functions", journal = j-CACM, volume = "15", number = "7", pages = "550--554", month = jul, year = "1972", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", MRclass = "34-02 12H05", MRnumber = "53 3384", MRreviewer = "K. Okugawa", bibdate = "Mon Jan 22 07:06:21 MST 2001", bibsource = "http://dblp.uni-trier.de/db/journals/cacm/cacm15.html#Moses72; https://www.math.utah.edu/pub/tex/bib/cacm1970.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/macsyma.bib", note = "Twenty-fifth anniversary of the Association for Computing Machinery.", acknowledgement = ack-nhfb, classcodes = "C1100 (Mathematical techniques)", fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "general theory; mathematics; special functions", oldlabel = "Moses72", treatment = "T Theoretical or Mathematical", XMLdata = "ftp://ftp.informatik.uni-trier.de/pub/users/Ley/bib/records.tar.gz#journals/cacm/Moses72", } @Article{Olver:1972:NBR, author = "F. W. J. Olver and D. J. Sookne", title = "Note on Backward Recurrence Algorithms", journal = j-MATH-COMPUT, volume = "26", number = "120", pages = "941--947", month = oct, year = "1972", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, classcodes = "B0290Z (Other numerical methods); C4190 (Other numerical methods)", corpsource = "Univ. Maryland, College Park, MD, USA", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "backward recurrence algorithms; Bessel; Bessel functions; difference equations; functions; recessive solution; second order linear difference equation", treatment = "T Theoretical or Mathematical", } @Article{Parnes:1972:CZM, author = "R. Parnes", title = "Complex zeros of the modified {Bessel} function {$ K_n(Z) $}", journal = j-MATH-COMPUT, volume = "26", number = "120", pages = "949--953", month = oct, year = "1972", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, classcodes = "B0290F (Interpolation and function approximation); C4130 (Interpolation and function approximation)", corpsource = "City Univ., New York, NY, USA", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "and zeros; Bessel functions; complex zeros; interpolation; interpolation scheme; iterative; iterative methods; modified Bessel function; poles", treatment = "T Theoretical or Mathematical", } @Article{Rabin:1972:FEP, author = "Michael O. Rabin and Shmuel Winograd", title = "Fast evaluation of polynomials by rational preparation", journal = j-COMM-PURE-APPL-MATH, volume = "25", number = "4", pages = "433--458", month = jul, year = "1972", CODEN = "CPAMAT, CPMAMV", DOI = "https://doi.org/10.1002/cpa.3160250405", ISSN = "0010-3640 (print), 1097-0312 (electronic)", ISSN-L = "0010-3640", bibdate = "Fri Oct 20 09:03:28 2023", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, ajournal = "Comm. Pure Appl. Math.", fjournal = "Communications on Pure and Applied Mathematics (New York)", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)1097-0312", keywords = "number of multiplications to evaluate a polynomial", } @Article{Ramamoorthy:1972:SPI, author = "C. V. Ramamoorthy and James R. Goodman and K. H. Kim", title = "Some Properties of Iterative Square-Rooting Methods Using High-Speed Multiplication", journal = j-IEEE-TRANS-COMPUT, volume = "C-21", number = "8", pages = "837--847", month = aug, year = "1972", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/TC.1972.5009039", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Tue Jul 12 18:58:45 MDT 2011", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput1970.bib", URL = "https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5009039", acknowledgement = ack-nj # " and " # ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", } @Article{Samet:1972:CDL, author = "P. A. Samet and D. W. Honey", title = "Calculation of a Double-Length Square Root from Double-Length Number using Single Precision Techniques", journal = j-COMP-J, volume = "15", number = "2", pages = "116--116", month = may, year = "1972", CODEN = "CMPJA6", DOI = "https://doi.org/10.1093/comjnl/15.2.116", ISSN = "0010-4620 (print), 1460-2067 (electronic)", ISSN-L = "0010-4620", bibdate = "Tue Dec 4 14:47:49 MST 2012", bibsource = "http://comjnl.oxfordjournals.org/content/15/2.toc; http://www3.oup.co.uk/computer_journal/hdb/Volume_15/Issue_02/; https://www.math.utah.edu/pub/tex/bib/compj1970.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "http://comjnl.oxfordjournals.org/content/15/2/116.full.pdf+html; http://www3.oup.co.uk/computer_journal/hdb/Volume_15/Issue_02/tiff/116.tif", acknowledgement = ack-nhfb, classcodes = "C5230 (Digital arithmetic methods)", corpsource = "Univ. Coll., London, UK", fjournal = "The Computer Journal", journal-URL = "http://comjnl.oxfordjournals.org/", keywords = "digital arithmetic; double length; precision techniques; single; square root", treatment = "T Theoretical or Mathematical", } @Article{Strecok:1972:HPE, author = "A. J. Strecok and J. A. Gregory", title = "High Precision Evaluation of the Irregular {Coulomb} Wave Functions", journal = j-MATH-COMPUT, volume = "26", number = "120", pages = "955--961 + s1--s10", month = oct, year = "1972", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", classcodes = "A0365G (Solutions of wave equations: bound state in quantum theory); B0290F (Interpolation and function approximation); C4130 (Interpolation and function approximation)", corpsource = "Argonne Nat. Lab., IL, USA", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "function approximation; high precision evaluation; irregular Coulomb wave functions; numerical methods; wave functions", treatment = "T Theoretical or Mathematical", } @Article{Turunov:1972:EFD, author = "M. Turunov", title = "Elementary functions of a discrete real and complex argument. ({Russian})", journal = "Ta{\v{s}}kent. Gos. Univ. Nau{\v{c}}n. Trudy", volume = "418 Voprosy Mat.", pages = "263--271, 386", year = "1972", MRclass = "30A95", MRnumber = "50 \#13556", MRreviewer = "G. Berzsenyi", bibdate = "Sat Jan 11 17:44:01 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, language = "Russian", } @Article{Wimp:1972:CPE, author = "Jet Wimp", title = "Corrigendum: {{\booktitle{Polynomial expansions of Bessel functions and some associated functions}} (Math. Comp. {\bf 16} (1962), 446--458)}", journal = j-MATH-COMPUT, volume = "26", number = "117", pages = "309--309", month = jan, year = "1972", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Sat Dec 22 06:54:10 MST 2018", bibsource = "http://www.ams.org/mcom/1972-26-117; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp1960.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp1970.bib", note = "See \cite{Wimp:1962:PEB}.", URL = "http://www.ams.org/journals/mcom/1972-26-117/S0025-5718-1972-0400659-X; http://www.ams.org/journals/mcom/1972-26-117/S0025-5718-1972-0400659-X/S0025-5718-1972-0400659-X.pdf; https://www.ams.org/mathscinet-getitem?mr=400659; https://www.ams.org/mathscinet/search/authors.html?authorName=Wimp%2C%20Jet", acknowledgement = ack-nhfb, fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Wynn:1972:CAM, author = "Peter Wynn", title = "Convergence Acceleration by a Method of Intercalation", journal = j-COMPUTING, volume = "9", number = "4", pages = "267--273", year = "1972", CODEN = "CMPTA2", ISSN = "0010-485X (print), 1436-5057 (electronic)", ISSN-L = "0010-485X", bibdate = "Tue Jan 2 17:40:51 MST 2001", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; INSPEC Axiom database (1968--date)", acknowledgement = ack-ec # " and " # ack-nhfb, affiliation = "Louisiana State Univ., New Orleans, LA, USA", classification = "B0290Z; C4190", description = "convergence; series (mathematics)", fjournal = "Computing", journal-URL = "http://link.springer.com/journal/607", keywords = "convergence acceleration; method of intercalation; series of real terms", } @Article{Amos:1973:BIC, author = "D. E. Amos", title = "Bounds on Iterated Coerror Functions and Their Ratios", journal = j-MATH-COMPUT, volume = "27", number = "122", pages = "413--427", month = apr, year = "1973", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Baker:1973:PAS, author = "P. W. Baker", title = "Predictive algorithms for some elementary functions in radix 2", journal = j-ELECT-LETTERS, volume = "9", number = "21", pages = "493--494", month = oct, year = "1973", CODEN = "ELLEAK", DOI = "https://doi.org/10.1049/el:19730363", ISSN = "0013-5194 (print), 1350-911X (electronic)", ISSN-L = "0013-5194", MRclass = "68A10", MRnumber = "57 \#18203", bibdate = "Sat Jan 11 17:44:01 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Electronics Letters", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=2220; https://digital-library.theiet.org/journal/el", } @Article{Besslich:1973:MDS, author = "P. W. Besslich and S. Raman", title = "Multiplication, Division and Square Root Extraction Methods for Electronic Desk Calculators", journal = "Journal of the Institution of Telecommunication Engineers (India)", volume = "19", number = "4", month = apr, year = "1973", bibdate = "Thu Sep 1 10:16:11 1994", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nj, } @Article{Braithwaite:1973:ALP, author = "W. J. Braithwaite", title = "Associated {Legendre} Polynomials, Ordinary and Modified Spherical Harmonics", journal = j-COMP-PHYS-COMM, volume = "5", number = "5", pages = "390--394", month = may, year = "1973", CODEN = "CPHCBZ", DOI = "https://doi.org/10.1016/0010-4655(73)90065-9", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", bibdate = "Fri Oct 29 21:45:41 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/compphyscomm1970.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Computer Physics Communications", journal-URL = "http://www.sciencedirect.com/science/journal/00104655", } @Article{Cody:1973:CAP, author = "W. J. Cody and Anthony J. Strecok and Henry C. {Thacher, Jr.}", title = "{Chebyshev} Approximations for the Psi Function", journal = j-MATH-COMPUT, volume = "27", number = "121", pages = "123--127", month = jan, year = "1973", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "65-06 (68-06)", MRnumber = "50 6095", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", classcodes = "B0290F (Interpolation and function approximation); C4130 (Interpolation and function approximation)", corpsource = "Argonne Nat. Lab., IL, USA", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "Chebyshev approximation; Chebyshev approximations; digamma function; psi function", remark = "Fullerton: Relative errors down to $ 10^{-20} $", treatment = "T Theoretical or Mathematical", } @Book{Dingle:1973:AET, author = "Robert B. Dingle", title = "Asymptotic expansions: their derivation and interpretation", publisher = pub-ACADEMIC, address = pub-ACADEMIC:adr, pages = "xv + 521", year = "1973", ISBN = "0-12-216550-0", ISBN-13 = "978-0-12-216550-4", LCCN = "QA295 .D45", bibdate = "Sat Feb 18 14:52:17 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; z3950.loc.gov:7090/Voyager", acknowledgement = ack-nhfb, subject = "Asymptotic expansions; Power series", } @Article{Ercegovac:1973:REC, author = "Milo{\v{s}} D. Ercegovac", title = "Radix-16 Evaluation of Certain Elementary Functions", journal = j-IEEE-TRANS-COMPUT, volume = "C-22", number = "6", pages = "561--566", month = jun, year = "1973", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/TC.1973.5009107", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Thu Sep 1 10:15:39 1994", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; garbo.uwasa.fi:/pc/doc-soft/fpbiblio.txt; https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput1970.bib", URL = "http://www.acsel-lab.com/arithmetic/arith2/papers/ARITH2_Ercegovac.pdf; https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5009107", acknowledgement = ack-nj # " and " # ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", keywords = "ARITH-2", } @Article{Fettis:1973:CZE, author = "Henry E. Fettis and James C. Caslin and Kenneth R. Cramer", title = "Complex zeros of the error function and of the complementary error function", journal = j-MATH-COMPUT, volume = "27", number = "122", pages = "401--407", month = apr, year = "1973", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, classcodes = "B0290D (Functional analysis); C4120 (Functional analysis)", corpsource = "Wright-Patterson Air Force Base, OH, USA", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "asymptotic formula; complementary error function; complex zeros; error function; errors; first one hundred zeros; function evaluation; poles and zeros", treatment = "T Theoretical or Mathematical", } @Article{Fettis:1973:SPC, author = "Henry E. Fettis and James C. Caslin and Kenneth R. Cramer", title = "Saddle Points of the Complementary Error Function", journal = j-MATH-COMPUT, volume = "27", number = "122", pages = "409--412", month = apr, year = "1973", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, classcodes = "B0290D (Functional analysis); C4120 (Functional analysis)", corpsource = "Sandia Labs., Albuquerque, NM, USA", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "asymptotic; complementary error function; errors; formula; function evaluation; poles and zeros; saddle points", treatment = "T Theoretical or Mathematical", } @Article{Fisher:1973:NEI, author = "N. I. Fisher", title = "A note on the evaluation of the incomplete gamma function", journal = j-J-STAT-COMPUT-SIMUL, volume = "2", number = "4", pages = "325--332", year = "1973", CODEN = "JSCSAJ", DOI = "https://doi.org/10.1080/00949657308810058", ISSN = "0094-9655 (print), 1026-7778 (electronic), 1563-5163", ISSN-L = "0094-9655", bibdate = "Tue Apr 22 09:10:34 MDT 2014", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jstatcomputsimul.bib", acknowledgement = ack-nhfb, fjournal = "Journal of Statistical Computation and Simulation", journal-URL = "http://www.tandfonline.com/loi/gscs20", } @TechReport{Franke:1973:AAH, author = "R. Franke", title = "An analysis of algorithms for hardware evaluation of elementary functions", type = "Report", number = "NPS-53FE73051A", institution = "Naval Post-Graduate School", address = "Monterey, CA, USA", month = may, year = "1973", bibdate = "Mon Nov 10 14:43:43 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, xxnumber = "AD-761519", } @Article{Gautschi:1973:AAE, author = "Walter Gautschi", title = "{ACM Algorithm 471}: Exponential Integrals [{S13}]", journal = j-CACM, volume = "16", number = "12", pages = "761--763", month = dec, year = "1973", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Mon Jan 22 06:43:23 MST 2001", bibsource = "http://dblp.uni-trier.de/db/journals/cacm/cacm16.html#Gautschi73; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", classcodes = "B0290M (Numerical integration and differentiation); C4160 (Numerical integration and differentiation); C7310 (Mathematics computing)", corpsource = "Purdue Univ., Lafayette, IN, USA", fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "ALGOL; computation; continued fractions; exponential integrals; integration; recurrence relations; recursive; subroutine; subroutines", oldlabel = "Gautschi73", remark = "Fullerton: Algol-language routine for $ E_n(x) = \int_1^\infty e^{-x t} t^n \, d t, x > 0 $.", treatment = "A Application; T Theoretical or Mathematical", XMLdata = "ftp://ftp.informatik.uni-trier.de/pub/users/Ley/bib/records.tar.gz#journals/cacm/Gautschi73", } @TechReport{Hemker:1973:SDL, author = "P. W. Hemker and W. Hoffmann and S. P. N. {van Kampen} and H. L. Oudshoorn and D. T. Winter", title = "Single- and double-length computation of elementary functions", number = "NW 7", institution = "Mathematical Centre", address = "Amsterdam", year = "1973", bibdate = "Mon May 19 13:30:58 1997", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/h/hemker-pieter-w.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, } @Article{Hill:1973:AAN, author = "G. W. Hill and A. W. Davis", title = "{ACM Algorithm 442}: Normal Deviate [{S14}]", journal = j-CACM, volume = "16", number = "1", pages = "51--52", month = jan, year = "1973", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Mon Jan 22 06:49:54 MST 2001", bibsource = "ftp://ftp.ira.uka.de/pub/bibliography/Distributed/QLD/1973.bib; http://dblp.uni-trier.de/db/journals/cacm/cacm16.html#HillD73; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", classcodes = "C7310 (Mathematics computing)", corpsource = "CSIRO, Glen Osmond, Australia", country = "USA", descriptors = "RVG", enum = "7393", fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "ALGOL; normal deviate; normal distribution inverse; probit; statistics; subroutines; Taylor series approximation; transform", oldlabel = "HillD73", references = "0", remark = "Fullerton: Short Algol-language procedure with accuracy to 24 digits.", treatment = "P Practical", XMLdata = "ftp://ftp.informatik.uni-trier.de/pub/users/Ley/bib/records.tar.gz#journals/cacm/HillD73", } @Article{Hill:1973:AAS, author = "G. W. Hill", title = "{ACM Algorithm 465}: {Student}'s $t$ Frequency [{S14}]", journal = j-CACM, volume = "16", number = "11", pages = "690--690", month = nov, year = "1973", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Mon Jan 22 06:49:52 MST 2001", bibsource = "http://dblp.uni-trier.de/db/journals/cacm/cacm16.html#Hill73a; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", classcodes = "C7310 (Mathematics computing)", corpsource = "CSIRO, Glen Osmond, SA, Australia", fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "ALGOL; approximation; density function; series; statistics; student's t statistic; subroutine; subroutines", oldlabel = "Hill73a", remark = "Fullerton: Algol-language routine for $ f(t | n) = \frac {\Gamma (n / 2 + 1 / 2)}{(\pi n)^{1 / 2} \Gamma (n / 2)} (1 + t^2 / n)^{n / 2 + 1 / 2} $.", treatment = "P Practical", XMLdata = "ftp://ftp.informatik.uni-trier.de/pub/users/Ley/bib/records.tar.gz#journals/cacm/Hill73a", } @Article{Hill:1973:SAA, author = "I. D. Hill", title = "Statistical Algorithms: {Algorithm AS 66}: The Normal Integral", journal = j-APPL-STAT, volume = "22", number = "3", pages = "424--427", month = sep, year = "1973", CODEN = "APSTAG", ISSN = "0035-9254 (print), 1467-9876 (electronic)", ISSN-L = "0035-9254", bibdate = "Sat Apr 21 10:20:49 MDT 2001", bibsource = "https://www.math.utah.edu/pub/tex/bib/as1970.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", URL = "http://lib.stat.cmu.edu/apstat/66", acknowledgement = ack-nhfb, fjournal = "Applied Statistics", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1111/(ISSN)1467-9876/issues", } @Article{Hwang:1973:RRS, author = "W. G. Hwang and John Todd", title = "A recurrence relation for the square root", journal = j-J-APPROX-THEORY, volume = "9", pages = "299--306", year = "1973", CODEN = "JAXTAZ", DOI = "https://doi.org/10.1016/0021-9045(73)90075-0", ISSN = "0021-9045,1096-0430", ISSN-L = "0021-9045", MRclass = "65H05", MRnumber = "373270", MRreviewer = "L. Fox", bibdate = "Sat Oct 21 14:25:01 2023", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/t/todd-john.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", ZMnumber = "0271.65032", acknowledgement = ack-nhfb, author-dates = "John Todd (16 May 1911--21 June 2007)", fjournal = "Journal of Approximation Theory", journal-URL = "http://www.sciencedirect.com/science/journal/00219045", received = "19 April 1971", ZBmath = "3426800", } @Article{Laurenzi:1973:DWF, author = "Bernard J. Laurenzi", title = "Derivatives of {Whittaker} Functions $ {W}_{k, 1 / 2} $ and $ {M}_{k, 1 / 2} $ with Respect to Order $ {K} $", journal = j-MATH-COMPUT, volume = "27", number = "121", pages = "129--132", month = jan, year = "1973", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Levin:1973:DNL, author = "D. Levin", title = "Development of non-linear transformations for improving convergence of sequences", journal = j-INT-J-COMPUT-MATH, volume = "3", number = "1--4", pages = "371--388", month = "????", year = "1973", CODEN = "IJCMAT", DOI = "https://doi.org/10.1080/00207167308803075", ISSN = "0020-7160", ISSN-L = "0020-7160", bibdate = "Thu Dec 01 10:27:34 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", ZMnumber = "0274.65004", acknowledgement = ack-nhfb, fjournal = "International Journal of Computer Mathematics", journal-URL = "http://www.tandfonline.com/loi/gcom20", keywords = "convergence acceleration", } @Article{Linz:1973:NCI, author = "Peter Linz and T. E. Kropp", title = "A note on the computation of integrals involving products of trigonometric and {Bessel} functions", journal = j-MATH-COMPUT, volume = "27", number = "124", pages = "871--872", month = oct, year = "1973", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, classcodes = "B0290M (Numerical integration and differentiation); C4160 (Numerical integration and differentiation)", corpsource = "Univ. California, Davies, CA, USA", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "Bessel functions; computation; integration; numerical; numerical methods; products; trigonometric", treatment = "T Theoretical or Mathematical", } @Article{Lozier:1973:BCP, author = "D. W. Lozier and L. C. Maximon and W. L. Sadowski", title = "A bit comparison program for algorithm testing", journal = j-COMP-J, volume = "16", number = "2", pages = "111--117", month = may, year = "1973", CODEN = "CMPJA6", ISSN = "0010-4620 (print), 1460-2067 (electronic)", ISSN-L = "0010-4620", bibdate = "Fri Sep 29 08:52:11 MDT 2000", bibsource = "http://www3.oup.co.uk/computer_journal/hdb/Volume_16/Issue_02/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www3.oup.co.uk/computer_journal/hdb/Volume_16/Issue_02/160111.sgm.abs.html; http://www3.oup.co.uk/computer_journal/hdb/Volume_16/Issue_02/tiff/111.tif; http://www3.oup.co.uk/computer_journal/hdb/Volume_16/Issue_02/tiff/112.tif; http://www3.oup.co.uk/computer_journal/hdb/Volume_16/Issue_02/tiff/113.tif; http://www3.oup.co.uk/computer_journal/hdb/Volume_16/Issue_02/tiff/114.tif; http://www3.oup.co.uk/computer_journal/hdb/Volume_16/Issue_02/tiff/115.tif; http://www3.oup.co.uk/computer_journal/hdb/Volume_16/Issue_02/tiff/116.tif; http://www3.oup.co.uk/computer_journal/hdb/Volume_16/Issue_02/tiff/117.tif", acknowledgement = ack-nhfb, classcodes = "C6150G (Diagnostic, testing, debugging and evaluating systems)", corpsource = "Nat. Bur. Stand., Washington, DC, USA", fjournal = "The Computer Journal", journal-URL = "http://comjnl.oxfordjournals.org/", keywords = "accuracy; algorithm testing; bit comparison program; computer algorithms; program debugging", treatment = "P Practical", } @Article{McNolty:1973:SPD, author = "Frank McNolty", title = "Some probability density functions and their characteristic functions", journal = j-MATH-COMPUT, volume = "27", number = "123", pages = "495--504", month = jul, year = "1973", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, classcodes = "B0240 (Probability and statistics); C1140 (Probability and statistics)", corpsource = "Lockheed Palo Alto Res. Lab., CA, USA", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "Bessel function; characteristic functions; functions; hypergeometric function; probability; probability density functions", treatment = "T Theoretical or Mathematical", } @Article{Meinardus:1973:OSA, author = "G{\"u}nter Meinardus and G. D. Taylor", title = "Optimal starting approximations for iterative schemes", journal = j-J-APPROX-THEORY, volume = "9", number = "1", pages = "1--19", month = sep, year = "1973", CODEN = "JAXTAZ", DOI = "https://doi.org/10.1016/0021-9045(73)90108-1", ISSN = "0021-9045 (print), 1096-0430 (electronic)", ISSN-L = "0021-9045", bibdate = "Mon Nov 10 09:49:33 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Journal of Approximation Theory", journal-URL = "http://www.sciencedirect.com/science/journal/00219045", } @Article{Rice:1973:EEI, author = "S. O. Rice", title = "Efficient Evaluation of Integrals of Analytic Functions by the Trapezoidal Rule", journal = j-BELL-SYST-TECH-J, volume = "52", number = "5", pages = "707--722", month = may # "--" # jun, year = "1973", CODEN = "BSTJAN", ISSN = "0005-8580", bibdate = "Tue Nov 9 11:15:55 MST 2010", bibsource = "http://bstj.bell-labs.com/oldfiles/year.1973/BSTJ.1973.5205.html; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://bstj.bell-labs.com/BSTJ/images/Vol52/bstj52-5-707.pdf", acknowledgement = ack-nhfb, fjournal = "The Bell System Technical Journal", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)1538-7305/issues/", } @Article{Schmid:1973:BLVa, author = "H. Schmid", title = "{BCD} logic {V}: {BCD} square root", journal = j-ELECTRONIC-DESIGN, volume = "21", number = "17", pages = "62--69", month = aug, year = "1973", CODEN = "ELODAW", ISSN = "0013-4872 (print), 1944-9550 (electronic)", ISSN-L = "0013-4872", bibdate = "Thu Sep 1 10:16:11 1994", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nj, fjournal = "Electronic Design", keywords = "decimal fixed-point arithmetic", } @Article{Sookne:1973:BFC, author = "D. J. Sookne", title = "{Bessel} Functions {$I$} and {$J$} of Complex Argument and Integer Order", journal = j-J-RES-NATL-BUR-STAND-1934, volume = "77B", number = "3--4", pages = "111--114", month = jul, year = "1973", ISSN = "0091-0635", bibdate = "Sat Oct 30 10:49:13 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Journal of Research of the National Bureau of Standards (1934)", journal-URL = "https://www.nist.gov/nist-research-library/journal-research-nist/past-papers", remark = "Fullerton: A program is described but not published.", } @Article{Sookne:1973:BFR, author = "D. J. Sookne", title = "{Bessel} Functions of Real Argument and Integer Order", journal = j-J-RES-NATL-BUR-STAND-1934, volume = "77A", number = "3--4", pages = "125--132", month = jul, year = "1973", ISSN = "0091-0635", bibdate = "Sat Oct 30 10:51:18 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Journal of Research of the National Bureau of Standards (1934)", journal-URL = "https://www.nist.gov/nist-research-library/journal-research-nist/past-papers", remark = "Fullerton: Sequences of $ I_n(x) $ and $ J_n(x) $ are computed with a FORTRAN routine.", } @Article{Sookne:1973:CABa, author = "D. J. Sookne", title = "Certification of an Algorithm for {Bessel} Functions of Real Argument", journal = j-J-RES-NATL-BUR-STAND-1934, volume = "77B", number = "3--4", pages = "115--124", month = jul, year = "1973", ISSN = "0091-0635", bibdate = "Sat Oct 30 10:55:03 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Journal of Research of the National Bureau of Standards (1934)", journal-URL = "https://www.nist.gov/nist-research-library/journal-research-nist/past-papers", remark = "Fullerton: An algorithm is shown to lose at most about 3 bits of precision.", } @Article{Sookne:1973:CABb, author = "D. J. Sookne", title = "Certification of an Algorithm for {Bessel} Functions of Complex Argument", journal = j-J-RES-NATL-BUR-STAND-1934, volume = "77B", number = "3", pages = "133--136", month = jul, year = "1973", ISSN = "0091-0635", bibdate = "Sat Oct 30 10:53:39 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Journal of Research of the National Bureau of Standards (1934)", journal-URL = "https://www.nist.gov/nist-research-library/journal-research-nist/past-papers", remark = "Fullerton: The algorithm is shown to lose at most about 3 bits of precision.", } @Article{Vos:1973:RAC, author = "H. Vos", title = "Remark on ``{Algorithm 300}: {Coulomb} Wave Functions''", journal = j-CACM, volume = "16", number = "5", pages = "308--309", month = may, year = "1973", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Mon Jan 22 07:27:34 MST 2001", bibsource = "http://dblp.uni-trier.de/db/journals/cacm/cacm16.html#Vos73; https://www.math.utah.edu/pub/tex/bib/cacm1960.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "See \cite{Gunn:1967:ACW}.", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", classcodes = "B0290D (Functional analysis); C4120 (Functional analysis); C7310 (Mathematics computing)", corpsource = "Vrije Univ., Amsterdam, Netherlands", fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "Coulomb wave functions; function evaluation; mathematics; wave functions", oldlabel = "Vos73", remark = "Fullerton: Algol-language accuracy monitor for Algorithm 300, which is generally accurate only to 3 digits.", treatment = "A Application; T Theoretical or Mathematical", XMLdata = "ftp://ftp.informatik.uni-trier.de/pub/users/Ley/bib/records.tar.gz#journals/cacm/Vos73", } @Article{Wong:1973:AEL, author = "R. Wong", title = "An Asymptotic Expansion of {$ W_{k, m}(z) $} with Large Variable and Parameters", journal = j-MATH-COMPUT, volume = "27", number = "122", pages = "429--436", month = apr, year = "1973", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Wong:1973:UAE, author = "R. Wong", title = "On uniform asymptotic expansion of definite integrals", journal = j-J-APPROX-THEORY, volume = "7", number = "1", pages = "76--86", month = jan, year = "1973", CODEN = "JAXTAZ", DOI = "https://doi.org/10.1016/0021-9045(73)90055-5", ISSN = "0021-9045 (print), 1096-0430 (electronic)", ISSN-L = "0021-9045", MRclass = "41A60", MRnumber = "0340910", MRreviewer = "L. Berg", bibdate = "Sat Feb 18 15:20:40 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/0021904573900555", acknowledgement = ack-nhfb, fjournal = "Journal of Approximation Theory", journal-URL = "http://www.sciencedirect.com/science/journal/00219045", keywords = "incomplete gamma functions", } @Article{Wrench:1973:ECT, author = "John W. Wrench", title = "Erratum: {Concerning Two Series for the Gamma Function}", journal = j-MATH-COMPUT, volume = "27", number = "123", pages = "681--682", month = jul, year = "1973", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", remark = "Fullerton: Minor last-digit rounding errors are reported.", } @Article{Wrigge:1973:EII, author = "H. S. Wrigge", title = "An Elliptic Integral Identity", journal = j-MATH-COMPUT, volume = "27", number = "124", pages = "839--840", month = oct, year = "1973", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Yohe:1973:IBS, author = "J. M. Yohe", title = "Interval Bounds for Square Roots and Cube Roots", journal = j-COMPUTING, volume = "11", number = "1", pages = "51--57", month = mar, year = "1973", CODEN = "CMPTA2", ISSN = "0010-485X (print), 1436-5057 (electronic)", ISSN-L = "0010-485X", bibdate = "Tue Jan 2 17:40:51 MST 2001", bibsource = "https://www.math.utah.edu/pub/tex/bib/computing.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; INSPEC Axiom database (1968--date)", acknowledgement = ack-jr # " and " # ack-nhfb, affiliation = "Univ. Wisconsin, Madison, WI, USA", classification = "C5230", description = "digital arithmetic; error analysis", fjournal = "Computing", journal-URL = "http://link.springer.com/journal/607", keywords = "binary computers; cube roots; error analysis; interval bounds; machine representable number; optimal upward directed rounding; smallest machine representable interval; square roots", } @Article{Acton:1974:RRF, author = "Forman S. Acton", title = "Recurrence relations for the {Fresnel} integral $ \int_0^{\infty } \exp ( - c t) \, d t / \sqrt {t (1 + t^2)} $ and similar integrals", journal = j-CACM, volume = "17", number = "8", pages = "480--481", month = aug, year = "1974", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", MRclass = "65D20 (33A70)", MRnumber = "49 6554", bibdate = "Mon Jan 22 06:20:27 MST 2001", bibsource = "Compendex database; http://dblp.uni-trier.de/db/journals/cacm/cacm17.html#Acton74; https://www.math.utah.edu/pub/bibnet/subjects/acc-stab-num-alg.bib; https://www.math.utah.edu/pub/tex/bib/cacm1970.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", abstract = "The class of functions defined by $ \int_0^\infty [\exp ( - c X)d t / (1 + Y)(t^{1 / 2})^k] $ where $X$ and $Y$ are either $t$ or $ t^2 $ and $k$ is $ - 1 $, $0$, or $1$ can be evaluated by recurrences for all but small values of the parameter $c$. These recurrences, given here, are more efficient than the usual asymptotic series.", acknowledgement = ack-nhfb, classification = "921", fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", journalabr = "Commun ACM", keywords = "exponential integral; Fresnel integral; mathematical techniques; recurrence relations", oldlabel = "Acton74", XMLdata = "ftp://ftp.informatik.uni-trier.de/pub/users/Ley/bib/records.tar.gz#journals/cacm/Acton74", } @Article{Amos:1974:CMB, author = "D. E. Amos", title = "Computation of modified {Bessel} functions and their ratios", journal = j-MATH-COMPUT, volume = "28", number = "125", pages = "239--251", month = jan, year = "1974", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", classcodes = "B0290D (Functional analysis); C4120 (Functional analysis)", corpsource = "Sandia Labs., Albuquerque, NM, USA", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "Bessel functions; computation; low order Bessel functions; modified Bessel functions; monotonicity; properties; ratios; recursion; relation", remark = "Fullerton: Ratios $ I_{\nu + 1}(x) / I_\nu (x) $ are considered.", treatment = "T Theoretical or Mathematical", } @Book{Anonymous:1974:TCH, editor = "Anonymous", title = "Tables of Complex Hyperbolic and Circular Functions", volume = "23", publisher = "Corona Pub. Co.", address = "Tokyo, Japan", pages = "621", year = "1974", LCCN = "QA55 .T172", bibdate = "Sat Apr 1 14:49:41 MDT 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; z3950.loc.gov:7090/Voyager", series = "Advanced series of mathematical and engineering tables", acknowledgement = ack-nhfb, remark = "Title and introduction also in Japanese: Fukuso s{\aa}okyokusen kans{\aa}u hy{\aa}o.", subject = "Mathematics; Tables; Exponential functions; Trigonometrical functions", } @Article{Barlow:1974:CCF, author = "R. H. Barlow", title = "Convergent Continued Fraction Approximants to Generalised Polylogarithms", journal = j-BIT, volume = "14", number = "1", pages = "112--116", month = mar, year = "1974", CODEN = "BITTEL, NBITAB", DOI = "https://doi.org/10.1007/BF01933124", ISSN = "0006-3835 (print), 1572-9125 (electronic)", ISSN-L = "0006-3835", bibdate = "Wed Jan 4 18:52:13 MST 2006", bibsource = "http://springerlink.metapress.com/openurl.asp?genre=issue&issn=0006-3835&volume=14&issue=1; https://www.math.utah.edu/pub/tex/bib/bit.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.springerlink.com/openurl.asp?genre=article&issn=0006-3835&volume=14&issue=1&spage=112", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "BIT (Nordisk tidskrift for informationsbehandling)", journal-URL = "http://link.springer.com/journal/10543", remark = "Fullerton: Nielsen's generalization $ S_{n, p}(z) = \frac {( - 1)^{n + p - 1}(n - 1)! p!} \int_0^1 \ln^{n - 1}(t) \ln^p(1 - z t) / t \, d t $ is evaluated in the complex plane.", } @Article{Barnett:1974:CWF, author = "A. R. Barnett and D. H. Feng and J. W. Steed and L. J. B. Goldfarb", title = "{Coulomb} wave functions for all real $ \eta $ and $ \rho $", journal = j-COMP-PHYS-COMM, volume = "8", number = "5", pages = "377--395", month = dec, year = "1974", CODEN = "CPHCBZ", DOI = "https://doi.org/10.1016/0010-4655(74)90013-7", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", bibdate = "Mon Jul 14 09:47:42 2008", bibsource = "https://www.math.utah.edu/pub/tex/bib/compphyscomm1970.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Computer Physics Communications", journal-URL = "http://www.sciencedirect.com/science/journal/00104655", remark = "This paper may contain the first presentation of Steed's algorithm for computing continued fractions.", } @Article{Blair:1974:RCA, author = "J. M. Blair", title = "Rational {Chebyshev} approximations for the modified {Bessel} functions {$ I_0 (x) $} and {$ I_1 (x) $}", journal = j-MATH-COMPUT, volume = "28", number = "126", pages = "581--583", month = apr, year = "1974", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", classcodes = "B0290F (Interpolation and function approximation); C4130 (Interpolation and function approximation)", corpsource = "Atomic Energy Canada Ltd., Chalk River, Ont., Canada", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "Chebyshev approximation; functions; modified Bessel; nearly best rational approximations; rational Chebyshev approximations; series (mathematics)", remark = "Fullerton: With microfiche supplement. Relative errors down to $ 10^{-23} $.", treatment = "T Theoretical or Mathematical", } @Article{Bosten:1974:RAI, author = "Nancy E. Bosten and E. L. Battiste", title = "Remark on ``{Algorithm 179}: {Incomplete} Beta Ratio''", journal = j-CACM, volume = "17", number = "3", pages = "156--157", month = mar, year = "1974", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Mon Jan 22 06:27:36 MST 2001", bibsource = "http://dblp.uni-trier.de/db/journals/cacm/cacm17.html#BostenB74; https://www.math.utah.edu/pub/tex/bib/cacm1960.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "See \cite{Ludwig:1963:AIB,Pike:1976:RIB}.", acknowledgement = ack-nhfb, bdate = "Mon Jan 22 06:27:36 MST 2001", citedby = "Fullerton:1980:BEM", classcodes = "B0290F (Interpolation and function approximation); C4130 (Interpolation and function approximation); C7310 (Mathematics computing)", corpsource = "IMSL, Houston, TX, USA", fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "Algorithm 179; computer aided analysis; function approximation; incomplete beta ratio; subroutines", oldlabel = "BostenB74", remark = "Fullerton: FORTRAN routine with accuracy about $ 10^{-6} $. See M. C. Pike (1976) for a Remark.", treatment = "T Theoretical or Mathematical", XMLdata = "ftp://ftp.informatik.uni-trier.de/pub/users/Ley/bib/records.tar.gz#journals/cacm/BostenB74", } @Article{Brent:1974:AAG, author = "Richard P. Brent", title = "{ACM Algorithm 488}: a {Gaussian} pseudo-random number generator [{G5}]", journal = j-CACM, volume = "17", number = "12", pages = "704--706", month = dec, year = "1974", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Mon Jan 22 06:28:05 MST 2001", bibsource = "ftp://ftp.ira.uka.de/pub/bibliography/Distributed/QLD.bib; http://dblp.uni-trier.de/db/journals/cacm/cacm17.html#Brent74; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", classcodes = "C7890 (Other special applications of computing)", corpsource = "Australian Nat. Univ., Canberra, Australia", country = "USA", descriptors = "RVG", enum = "7061", fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "distribution; FORTRAN; Gaussian; generator; GRAND; normal distribution; pseudo random numbers; random number generation; random numbers; subroutines", location = "SEL: Wi", oldlabel = "Brent74", references = "0", remark = "Fullerton: A FORTRAN routine that returns normally distributed numbers with zero mean and unit standard deviation.", revision = "16/01/94", treatment = "A Application; T Theoretical or Mathematical", XMLdata = "ftp://ftp.informatik.uni-trier.de/pub/users/Ley/bib/records.tar.gz#journals/cacm/Brent74", } @Article{Burrell:1974:AAE, author = "Keith H. Burrell", title = "{ACM Algorithm 484}: Evaluation of the Modified {Bessel} Functions {$ K_0 (z) $} and {$ K_1 (z) $} for Complex Arguments [{S17}]", journal = j-CACM, volume = "17", number = "9", pages = "524--526", month = sep, year = "1974", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Mon Jan 22 06:28:58 MST 2001", bibsource = "http://dblp.uni-trier.de/db/journals/cacm/cacm17.html#Burrell74; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", classcodes = "B0290D (Functional analysis); C4120 (Functional analysis); C7310 (Mathematics computing)", corpsource = "California Inst. Technol., Pasadena, CA, USA", fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "algorithm; applications of computers; Bessel functions; complex arguments; function evaluation; Gauss-Hermite quadrature; Hankel functions; modified Bessel functions; natural sciences; subroutines", oldlabel = "Burrell74", remark = "Fullerton: 10-digit accuracy FORTRAN program.", treatment = "A Application; T Theoretical or Mathematical", XMLdata = "ftp://ftp.informatik.uni-trier.de/pub/users/Ley/bib/records.tar.gz#journals/cacm/Burrell74", } @Article{Buschman:1974:FSR, author = "R. G. Buschman", title = "Finite sum representations for partial derivatives of special functions with respect to parameters", journal = j-MATH-COMPUT, volume = "28", number = "127", pages = "817--824", month = jul, year = "1974", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", classcodes = "B0290M (Numerical integration and differentiation); C4160 (Numerical integration and differentiation)", corpsource = "Univ. Wyoming, Laramie, WY, USA", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "Bessel functions; differentiation; finite sum representations; finite sums; functions; G function; Gegenbauer functions; hypergeometric; Legendre; Mellin transformation; partial derivatives; special functions; Whittaker functions", remark = "Fullerton: Whittaker and Bessel functions are considered.", treatment = "T Theoretical or Mathematical", } @Article{Carta:1974:HLR, author = "David G. Carta", title = "Help!!: {The} Lost Reference: ({A} Modified {Newton} Method for Square Roots)", journal = j-SIGNUM, volume = "9", number = "4", pages = "9--9", month = oct, year = "1974", CODEN = "SNEWD6", DOI = "https://doi.org/10.1145/1206085.1206086", ISSN = "0163-5778 (print), 1558-0237 (electronic)", ISSN-L = "0163-5778", bibdate = "Tue Jun 17 18:47:00 MDT 2008", bibsource = "http://portal.acm.org/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/signum.bib", abstract = "Around 1970 I saw a journal article describing a modified Newton iteration for square roots. It involved changing the usual factor of 0.5 in $ x_{n + 1} = 0.5 (x_n + a / x_n) $ to $ c_n $ where $ c_n \rightarrow 0.5 $, thereby increasing the asymptotic rate of convergence from $ e_{n + 1} = 0.5 e_n^2 $ to $ e_{n + 1} = 0.25 e_n^2 $.", acknowledgement = ack-nhfb, fjournal = "ACM SIGNUM Newsletter", journal-URL = "https://dl.acm.org/loi/signum", } @Article{Fettis:1974:SAC, author = "Henry E. Fettis", title = "A stable algorithm for computing the inverse error function in the `tail-end' region", journal = j-MATH-COMPUT, volume = "28", number = "126", pages = "585--587", month = apr, year = "1974", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, classcodes = "B0290D (Functional analysis); C4120 (Functional analysis)", corpsource = "Wright-Patterson Air Force Base, OH, USA", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "function evaluation; inverse error function; iterative algorithm; stability; stable algorithm; tail end region", treatment = "T Theoretical or Mathematical", } @Article{Glasser:1974:SDI, author = "M. L. Glasser", title = "Some definite integrals of the product of two {Bessel} functions of the second kind: (order zero)", journal = j-MATH-COMPUT, volume = "28", number = "126", pages = "613--615", month = apr, year = "1974", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, classcodes = "B0290D (Functional analysis); C4120 (Functional analysis)", corpsource = "Battelle Memorial Inst., Columbus, OH, USA", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "Bessel functions; definite integrals; evaluate; function evaluation; integral representation; order; second kind; zero", treatment = "T Theoretical or Mathematical", } @Article{Hitotumatu:1974:NMC, author = "Sin Hitotumatu", title = "A new method for the computation of square root, exponential and logarithmic functions through hyperbolic {CORDIC}", journal = "Revue d'Analyse Num{\'e}rique et de la Th{\'e}orie de l'Approximation", volume = "3", number = "2", pages = "173--180", year = "1974", ISSN = "1010-3376 (print), 2457-8118 (electronic)", ISSN-L = "1010-3376", MRclass = "65D20", MRnumber = "381249", MRreviewer = "L. Fox", bibdate = "Tue Nov 14 17:19:58 2023", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/f/fox-leslie.bib; https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "https://ictp.acad.ro/jnaat/journal/article/view/1974-vol3-no2-art7", acknowledgement = ack-nhfb, ajournal = "Rev. Anal. Num{\'e}r. Th{\'e}orie Approximation", fjournal = "Revue d'Analyse Num{\'e}rique et de la Th{\'e}orie de l'Approximation", journal-URL = "https://ictp.acad.ro/jnaat/journal", reviewer-dates = "Leslie Fox (30 September 1918--1 August 1992)", } @Article{Koppelaar:1974:CRA, author = "Henk Koppelaar", title = "Certification and Remark on ``{Algorithm 191}: Hypergeometric''", journal = j-CACM, volume = "17", number = "10", pages = "589--590", month = oct, year = "1974", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Mon Jan 22 06:55:45 MST 2001", bibsource = "http://dblp.uni-trier.de/db/journals/cacm/cacm17.html#Kopelaar74; https://www.math.utah.edu/pub/tex/bib/cacm1960.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "See \cite{Relph:1963:AH}.", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", classcodes = "C7310 (Mathematics computing)", corpsource = "Utrecht State Univ., Netherlands", fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "algorithm; hypergeometric; improvements; inefficiency; natural sciences applications of computers; subroutines", oldlabel = "Kopelaar74", remark = "Fullerton: Algol-language modifications for Algorithm 191, which does not appear to be accurate far from the origin.", treatment = "G General Review; T Theoretical or Mathematical", XMLdata = "ftp://ftp.informatik.uni-trier.de/pub/users/Ley/bib/records.tar.gz#journals/cacm/Kopelaar74", } @Article{Kyriakopoulos:1974:GFH, author = "E. Kyriakopoulos", title = "Generating functions of the hypergeometric functions", journal = j-J-MATH-PHYS, volume = "15", number = "6", pages = "753--759", month = jun, year = "1974", CODEN = "JMAPAQ", DOI = "https://doi.org/10.1063/1.1666724", ISSN = "0022-2488 (print), 1089-7658 (electronic), 1527-2427", ISSN-L = "0022-2488", bibdate = "Fri Oct 28 16:40:13 MDT 2011", bibsource = "http://jmp.aip.org/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jmathphys1970.bib", URL = "http://jmp.aip.org/resource/1/jmapaq/v15/i6/p753_s1", acknowledgement = ack-nhfb, classification = "A0210 (Algebra, set theory, and graph theory); A0220 (Group theory); A0230 (Function theory, analysis)", corpsource = "Nuclear Res. Center 'Democritos', Athens, Greece", fjournal = "Journal of Mathematical Physics", journal-URL = "http://jmp.aip.org/", keywords = "functions; generating functions; hypergeometric functions; Lie groups; Lie theory; multiplier representation theory; Weisner's technique", onlinedate = "4 November 2003", pagecount = "7", treatment = "T Theoretical or Mathematical", } @Article{Latham:1974:CPC, author = "W. P. Latham and Rogers W. Redding", title = "On the calculation of the parabolic cylinder functions", journal = j-J-COMPUT-PHYS, volume = "16", number = "1", pages = "66--75", month = sep, year = "1974", CODEN = "JCTPAH", DOI = "https://doi.org/10.1016/0021-9991(74)90104-1", ISSN = "0021-9991 (print), 1090-2716 (electronic)", ISSN-L = "0021-9991", bibdate = "Sun Jan 1 09:15:15 MST 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputphys1970.bib", URL = "http://www.sciencedirect.com/science/article/pii/0021999174901041", acknowledgement = ack-nhfb, fjournal = "Journal of Computational Physics", journal-URL = "http://www.sciencedirect.com/science/journal/00219991/", } @Article{McCabe:1974:CFE, author = "J. H. McCabe", title = "A continued fraction expansion, with a truncation error estimate, for {Dawson}'s integral", journal = j-MATH-COMPUT, volume = "28", number = "127", pages = "811--816", month = jul, year = "1974", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", classcodes = "B0290B (Error analysis in numerical methods); B0290F (Interpolation and function approximation); C4110 (Error analysis in numerical methods); C4130 (Interpolation and function approximation)", corpsource = "Univ. St. Andrews, Fife, UK", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "continued fraction expansion; convergents; Dawson's integral; error analysis; function approximation; rational approximations; truncation error estimate", treatment = "T Theoretical or Mathematical", } @Article{Nasell:1974:IMB, author = "Ingemar Nasell", title = "Inequalities for Modified {Bessel} Functions", journal = j-MATH-COMPUT, volume = "28", number = "125", pages = "253--256", month = jan, year = "1974", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, classcodes = "B0290D (Functional analysis); C4120 (Functional analysis)", corpsource = "Bell Labs., Holmdel, NJ, USA", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "Bessel functions; inequalities; modified Bessel functions; sharp versions", treatment = "T Theoretical or Mathematical", } @Book{Olver:1974:ASF, author = "F. W. J. Olver", title = "Asymptotics and Special Functions", publisher = pub-ACADEMIC, address = pub-ACADEMIC:adr, pages = "xvi + 572", year = "1974", ISBN = "0-12-525850-X", ISBN-13 = "978-0-12-525850-0", LCCN = "QA351 .O481 1974", bibdate = "Wed Dec 15 10:40:06 1993", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, } @Article{Pisarskii:1974:QCD, author = "A. V. Pisarski{\u\i} and A. F. Kurgaev and A. V. Palagin", title = "On the question of the convergence of the {``digit by digit''} methods of computing the elementary functions", journal = "Kibernetika (Kiev)", volume = "4", pages = "147--149", year = "1974", CODEN = "KBRNA5", ISSN = "0023-1274", MRclass = "65D20", MRnumber = "53 \#1915", MRreviewer = "V. V. Ivanov", bibdate = "Sat Jan 11 17:44:01 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, } @Article{Pomeranz:1974:AAE, author = "John Pomeranz", title = "{ACM Algorithm 487}: Exact Cumulative Distribution of the {Kolmogorov--Smirnov} Statistic for Small Samples", journal = j-CACM, volume = "17", number = "12", pages = "703--704", month = dec, year = "1974", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Mon Jan 22 07:12:56 MST 2001", bibsource = "http://dblp.uni-trier.de/db/journals/cacm/cacm17.html#Pomeranz74; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "See also \cite{Pomeranz:1976:REC}.", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", classcodes = "C7310 (Mathematics computing)", corpsource = "Purdue Univ., West Lafayette, IN, USA", fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "algorithm; exact cumulative distribution; FORTRAN; Kolmogorov Smirnov test; natural sciences applications of computers; small samples; statistic; statistics; subroutines", oldlabel = "Pomeranz74", remark = "Fullerton: FORTRAN routine accurate apparently to 5 digits.", treatment = "A Application; T Theoretical or Mathematical", XMLdata = "ftp://ftp.informatik.uni-trier.de/pub/users/Ley/bib/records.tar.gz#journals/cacm/Pomeranz74", } @Article{Shaw:1974:NME, author = "Mary Shaw and J. F. Traub", title = "On the Number of Multiplications for the Evaluation of a Polynomial and Some of Its Derivatives", journal = j-J-ACM, volume = "21", number = "1", pages = "161--167", month = jan, year = "1974", CODEN = "JACOAH", DOI = "https://doi.org/10.1145/321796.321810", ISSN = "0004-5411 (print), 1557-735X (electronic)", ISSN-L = "0004-5411", bibdate = "Wed Jan 15 18:12:53 MST 1997", bibsource = "Compendex database; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jacm.bib", abstract = "A family of new algorithms is given for evaluating the first $m$ derivatives of a polynomial. In particular, it is shown that all derivatives may be evaluated in $ 3 n - 2 $ multiplications. The best previous result required $ (1 / 2) n (n + 1) $ multiplications. Some optimality results are presented.", acknowledgement = ack-nhfb, ajournal = "J. Assoc. Comput. Mach.", classification = "921", fjournal = "Journal of the Association for Computing Machinery", journal-URL = "https://dl.acm.org/loi/jacm", keywords = "mathematical techniques; number of multiplications to evaluate a polynomial", } @Article{Sheorey:1974:CEW, author = "V. B. Sheorey", title = "{Chebyshev} Expansions for Wavefunctions", journal = j-COMP-PHYS-COMM, volume = "7", number = "1", pages = "1--12", month = jan, year = "1974", CODEN = "CPHCBZ", DOI = "https://doi.org/10.1016/0010-4655(74)90053-8", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", bibdate = "Sat Oct 30 10:36:29 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/compphyscomm1970.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Computer Physics Communications", journal-URL = "http://www.sciencedirect.com/science/journal/00104655", } @Article{Stegun:1974:ACM, author = "I. A. Stegun and R. Zucker", title = "Automatic Computing Methods for Special Functions. {Part II}. {The} Exponential Integral {$ E_n(x) $}", journal = j-J-RES-NATL-BUR-STAND-1934, volume = "78B", number = "4", pages = "199--216", month = oct, year = "1974", ISSN = "0091-0635", bibdate = "Sat Oct 30 11:00:40 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Journal of Research of the National Bureau of Standards (1934)", journal-URL = "https://www.nist.gov/nist-research-library/journal-research-nist/past-papers", remark = "Fullerton: Adjustable double precision FORTRAN routines for $ E_n(x) $ and $ e^x E_n(x) $.", } @Article{Wang:1974:UEZ, author = "Paul S. Wang", title = "The Undecidability of the Existence of Zeros of Real Elementary Functions", journal = j-J-ACM, volume = "21", number = "4", pages = "586--589", month = oct, year = "1974", CODEN = "JACOAH", ISSN = "0004-5411 (print), 1557-735X (electronic)", ISSN-L = "0004-5411", bibdate = "Wed Jan 15 18:12:53 MST 1997", bibsource = "Compendex database; ftp://ftp.ira.uka.de/pub/bibliography/Math/hilbert10.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", abstract = "From Richardson's undecidability results, it is shown that the predicate ``there exists a real number such that G(r) equals 0'' is recursively undecidable for G(x) in a class of functions which involves polynomials and the sine function. The deduction follows that the convergence of a class of improper integrals is recursively undecidable.", acknowledgement = ack-nhfb, ajournal = "J. Assoc. Comput. Mach.", classification = "921", fjournal = "Journal of the ACM", journal-URL = "https://dl.acm.org/loi/jacm", keywords = "mathematical techniques", } @Article{Wimp:1974:CTP, author = "J. Wimp", title = "On the computation of {Tricomi}'s {Psi} function", journal = j-COMPUTING, volume = "13", number = "3--4", pages = "195--203", year = "1974", CODEN = "CMPTA2", ISSN = "0010-485X (print), 1436-5057 (electronic)", ISSN-L = "0010-485X", bibdate = "Tue Jan 2 17:40:52 MST 2001", bibsource = "Compendex database; http://springerlink.metapress.com/openurl.asp?genre=journal&issn=0010-485X; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; INSPEC Axiom database (1968--date)", acknowledgement = ack-nhfb, affiliation = "Drexel Univ., Philadelphia, PA, USA", citedby = "Fullerton:1980:BEM", classification = "723; 921; B0290D; C4120", description = "convergence of numerical methods; function evaluation", fjournal = "Computing", journal-URL = "http://link.springer.com/journal/607", journalabr = "Comput (Vienna/NY)", keywords = "computation; computer programming --- Subroutines; confluent hypergeometric function; convergence; mathematical techniques; recurrence relations; Tricomi's", remark = "Fullerton: Backward recursion methods are discussed.", } @Article{Baker:1975:MER, author = "P. W. Baker", title = "More efficient radix-2 algorithms for some elementary functions", journal = j-IEEE-TRANS-COMPUT, volume = "C-24", number = "11", pages = "1049--1054", month = nov, year = "1975", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/T-C.1975.224132", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", MRclass = "68A10", MRnumber = "52 \#7193", MRreviewer = "I. Kaufmann", bibdate = "Tue Jul 12 07:57:58 MDT 2011", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; garbo.uwasa.fi:/pc/doc-soft/fpbiblio.txt; https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput1970.bib", URL = "https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=1672725", acknowledgement = ack-nhfb # "\slash " # ack-nj, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", } @Article{Baker:1975:PMA, author = "P. W. Baker", title = "Parallel Multiplicative Algorithms for Some Elementary Functions", journal = j-IEEE-TRANS-COMPUT, volume = "C-24", number = "3", pages = "322--325", month = mar, year = "1975", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/T-C.1975.224215", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Tue Jul 12 07:57:51 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput1970.bib", URL = "https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=1672808", abstract = "This correspondence presents generalized higher radix algorithms for some elementary functions which use fast parallel $m$-bit multipliers where $ \mathrm {radix} = 2^m $. These algorithms are extensions of those iterative schemes which are based on multiplications by $ (1 + 2^{-i}) $ and the use of prestored values of $ \ln (1 + 2^{-i}) $ and $ \tan^{-1}(2^{-i}) $. The particular functions under consideration are $ y / x $, $ y / x^{1 / 2} $, $ y \exp (x) $, $ y + \ln (x) $, $ \sin (x) $ and $ \cos (x) $ [and hence $ \tan (x) $ ]. The extended algorithms rely on multiplication by $ (1 + \mathrm {dir}^{-k}) $ where $ \mathrm {dir} $, $ 0 \leq \mathrm {dir} $, is an $m$-bit integer. Using a simple selection procedure for $ \mathrm {dir} $, simulations show that $p$ (radix $r$) digits of a function may be generated, on the average, in less than $ p + 1 $ iterations.", acknowledgement = ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", received = "27 March 1973", remark = "The final paragraph says: ``In order to obtain an answer with $p$ correct digits in radix $ 2^m$, an extra $ \log_2 ((p + 1) m)$ guard digits ought to be used and $k$ should be taken to $ p + 2$ with the final result rounded to $p$ digits.'' That corresponds to 5 to 7 extra bits in the four IEEE 754 binary formats.", revised = "7 October 1974", } @Article{Barinka:1975:SEC, author = "Lawrence L. Barinka", title = "Some Experience with Constructing, Testing, and Certifying a Standard Mathematical Subroutine Library", journal = j-TOMS, volume = "1", number = "2", pages = "165--177", month = jun, year = "1975", CODEN = "ACMSCU", DOI = "https://doi.org/10.1145/355637.355642", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Fri Aug 26 23:44:16 1994", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib", acknowledgement = ack-nhfb, ajournal = "ACM Trans. Math. Softw.", fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", } @Article{Boris:1975:NEO, author = "Jay P. Boris and Elaine S. Oran", title = "Numerical evaluation of oscillatory integrals such as the modified {Bessel} function {$ K_{i \zeta }(x) $}", journal = j-J-COMPUT-PHYS, volume = "17", number = "4", pages = "425--433", month = apr, year = "1975", CODEN = "JCTPAH", DOI = "https://doi.org/10.1016/0021-9991(75)90045-5", ISSN = "0021-9991 (print), 1090-2716 (electronic)", ISSN-L = "0021-9991", bibdate = "Sun Jan 1 09:15:17 MST 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputphys1970.bib", URL = "http://www.sciencedirect.com/science/article/pii/0021999175900455", acknowledgement = ack-nhfb, fjournal = "Journal of Computational Physics", journal-URL = "http://www.sciencedirect.com/science/journal/00219991/", } @TechReport{Brent:1975:FMP, author = "R. P. Brent", title = "Fast Multiple-precision Evaluation of Elementary Functions", type = "Technical Report", number = "STAN-CS-75-515", institution = inst-STAN-CS, address = inst-STAN-CS:adr, pages = "i + 22", month = aug, year = "1975", bibdate = "Thu Jan 11 16:47:21 2024", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "https://bitsavers.org/pdf/stanford/cs_techReports/STAN-CS-75_515_Brent_Fast_Multiple-Precision_Evaluation_Of_Elementary_Functions_Aug75.pdf", abstract = "Let $ f(x) $ be one of the usual elementary functions ($ \exp $, $ \log $, $ \arctan $, $ \sin $, $ \cosh $, etc.), and let $ M(n) $ be the number of single-precision operations required to multiply n-bit integers. We show that f(x) can be evaluated, with relative error $ O(2^{-n}) $, in $ O(M(n) \log (n)) $ operations as $ n \to \infty $, for any floating-point number $x$ (with an $n$-bit fraction) in a suitable finite interval. From the Sch{\"o}nhage--Strassen bound on $ M(n)$, it follows that an $n$-bit approximation to $ f(x)$ may be evaluated in $ O(n \log^2 (n) \log \log (n))$ operations. Special cases include the evaluation of constants such as $ \pi $, $e$, and $ e^p i$. The algorithms depend on the theory of elliptic integrals, using the arithmetic--geometric mean iteration and ascending Landen transformations.", acknowledgement = ack-nhfb, } @TechReport{Brent:1975:MZM, author = "R. P. (Richard P.) Brent", title = "Multiple-precision zero-finding methods and the complexity of elementary function evaluation", institution = "Department of Computer Science, Carnegie-Mellon University", address = "Pittsburgh, PA, USA", pages = "26", year = "1975", bibdate = "Sat Jan 11 10:14:06 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, keywords = "Iterative methods (Mathematics)", searchkey = "ti:elementary n1 function", } @Article{Carta:1975:LOA, author = "David G. Carta", title = "Low-Order Approximations for the Normal Probability Integral and the Error Function", journal = j-MATH-COMPUT, volume = "29", number = "131", pages = "856--862", month = jul, year = "1975", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp1970.bib; JSTOR database", acknowledgement = ack-nhfb, classcodes = "B0260 (Optimisation techniques); B0290R (Integral equations); C1180 (Optimisation techniques); C4180 (Integral equations)", corpsource = "Jet Propulsion Lab., California Inst. of Technol., Pasadena, CA, USA", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "error function; fraction; integral equations; linear minimax problems; linear programming; low; normal probability integral; order approximations; polynomials; rational", treatment = "T Theoretical or Mathematical", } @Article{Cody:1975:FPS, author = "W. J. Cody", title = "The {FUNPACK} Package of Special Function Subroutines", journal = j-TOMS, volume = "1", number = "1", pages = "13--25", month = mar, year = "1975", CODEN = "ACMSCU", DOI = "https://doi.org/10.1145/355626.355631", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Fri Aug 26 23:44:16 1994", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", } @PhdThesis{Epstein:1975:AET, author = "Harvey Irwin Epstein", title = "Algorithms for elementary transcendental function arithmetic", school = "University of Wisconsin", address = "Madison, WI, USA", pages = "409", year = "1975", bibdate = "Sat Jan 11 10:14:06 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, annote = "Typescript. Vita. Thesis (Ph. D.)--University of Wisconsin--Madison, 1975.", keywords = "Algorithms.; Functions -- Data processing.; Functions, Transcendental.", searchkey = "ti:elementary n1 function", } @PhdThesis{Ercegovac:1975:GMEa, author = "Milo{\v{s}} Dragutin Ercegovac", title = "A General Method for Evaluation of Functions and Computations in a Digital Computer", type = "{Ph.D.} Thesis", school = "Department of Computer Science, University of Illinois at Urbana-Champaign", address = "Urbana-Champaign, IL, USA", pages = "viii + 109", month = jul, year = "1975", bibdate = "Mon Feb 10 07:18:12 2020", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "https://search.proquest.com/pqdtglobal/docview/302756306", acknowledgement = ack-nhfb, advisor = "James E. Robertson", } @Article{Ferguson:1975:PFI, author = "Helaman Rolfe Pratt Ferguson and Dale E. Nielsen and Grant Cook", title = "A partition formula for the integer coefficients of the theta function nome", journal = j-MATH-COMPUT, volume = "29", number = "131", pages = "851--855", month = jul, year = "1975", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", classcodes = "B0290R (Integral equations); C4180 (Integral equations)", corpsource = "Dept. of Math., Brigham Young Univ., Provo, UT, USA", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "coefficients; complete elliptic integrals; elliptic function theory; elliptic integrals; incomplete; integral equations; partition formula; theta function nome integer", remark = "Fullerton: Incomplete elliptic integrals can be expressed as a series of theta functions.", treatment = "T Theoretical or Mathematical", } @Article{Fornberg:1975:CZJ, author = "B. Fornberg and K. S. Kolbig", title = "Complex zeros of the {Jonquiere} or polylogarithm function", journal = j-MATH-COMPUT, volume = "29", number = "130", pages = "582--599", month = apr, year = "1975", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp1970.bib; JSTOR database", acknowledgement = ack-nhfb, classcodes = "B0290F (Interpolation and function approximation); C4130 (Interpolation and function approximation)", corpsource = "CERN, Geneva, Switzerland", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "asymptotic; behaviour; complex zero trajectories; Jonquiere function; poles and zeros; polylogarithm function; polynomials; Riemann zeta function", treatment = "T Theoretical or Mathematical", } @InProceedings{Gargantini:1975:PSR, author = "I. Gargantini", editor = "K. Nickel", booktitle = "Interval Mathematics", title = "Parallel Square Root Iterations", volume = "29", publisher = pub-SV, address = pub-SV:adr, pages = "196--204", year = "1975", bibdate = "Fri Jan 12 11:37:56 1996", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", series = "Lecture Notes In Computer Science", acknowledgement = ack-jr, } @InProceedings{Gautschi:1975:CMS, author = "W. Gautschi", title = "Computational Methods in Special Functions --- a Survey", crossref = "Askey:1975:TAS", pages = "1--98", year = "1975", bibdate = "Sat Oct 30 07:42:41 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", remark = "Fullerton: Extensive list of references.", } @Article{Ginsberg:1975:AAD, author = "E. S. Ginsberg and Dorothy Zaborowski", title = "{ACM Algorithm 490}: The Dilogarithm Function of a Real Argument [{S22}]", journal = j-CACM, volume = "18", number = "4", pages = "200--202", month = apr, year = "1975", CODEN = "CACMA2", DOI = "https://doi.org/10.1145/360715.360722", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Mon Jan 22 06:44:28 MST 2001", bibsource = "http://dblp.uni-trier.de/db/journals/cacm/cacm18.html#GinsbergZ75; https://www.math.utah.edu/pub/tex/bib/cacm1970.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "See also \cite{Morris:1976:RDF}.", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", classcodes = "B0290D (Functional analysis); C4120 (Functional analysis); C7310 (Mathematics computing)", corpsource = "Dept. of Phys., Univ. of Massachusetts, Boston, MA, USA", fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "dilogarithm function; electrodynamics; ferromagnets; function evaluation; function subroutine; ideal; library; network analysis; polymers; quantum; real argument; subprograms; subroutines; thermodynamics", oldlabel = "GinsbergZ75", remark = "Fullerton: FORTRAN routine accurate to 15 digits for evaluating $ \operatorname {Li}_2 (x) = - \int_0^\infty \frac {\ln (1 z)z} \, d z $.", treatment = "A Application; T Theoretical or Mathematical", XMLdata = "ftp://ftp.informatik.uni-trier.de/pub/users/Ley/bib/records.tar.gz#journals/cacm/GinsbergZ75", } @Article{Headley:1975:DZG, author = "V. B. Headley and V. K. Barwell", title = "On the distribution of the zeros of generalized {Airy} functions", journal = j-MATH-COMPUT, volume = "29", number = "131", pages = "863--877", month = jul, year = "1975", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, classcodes = "B0290P (Differential equations); C4170 (Differential equations)", corpsource = "Dept. of Math., Brock Univ., St. Catherines, Ont., Canada", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "Bessel functions; boundary rays; differential equations; generalized Airy functions; nonreal zeros; zeros distribution", treatment = "T Theoretical or Mathematical", } @InProceedings{Hitotumatu:1975:SRU, author = "Sin Hitotumatu", title = "Some remarks on the unified treatment of elementary functions by microprogramming", crossref = "Miller:1975:TNA", pages = "51--56 (vol. 2)", year = "1975", MRclass = "65D15", MRnumber = "54 \#11733", MRreviewer = "Luciano Biasini", bibdate = "Sat Jan 11 17:44:01 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, } @Article{Ikebe:1975:ZRC, author = "Yasuhiko Ikebe", title = "The Zeros of Regular {Coulomb} Wave Functions and of Their Derivatives", journal = j-MATH-COMPUT, volume = "29", number = "131", pages = "878--887", month = jul, year = "1975", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, classcodes = "B0290H (Linear algebra); C4140 (Linear algebra)", corpsource = "Centre Numerical Analysis, Univ. of Texas, Austin, TX, USA", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "Bessel function; compact matrix; eigenvalues; eigenvalues and eigenfunctions; function zeros characterisation; matrix algebra; methods; numerical; operators; regular Coulomb wave functions; wave functions; zeros", treatment = "T Theoretical or Mathematical", } @Article{Kaufman:1975:URA, author = "E. H. Kaufman and G. D. Taylor", title = "Uniform rational approximation of functions of several variables", journal = j-INT-J-NUMER-METHODS-ENG, volume = "9", number = "2", pages = "297--323", month = jan, year = "1975", CODEN = "IJNMBH", DOI = "https://doi.org/10.1002/nme.1620090204", ISSN = "0029-5981 (print), 1097-0207 (electronic)", ISSN-L = "0029-5981", bibdate = "Mon Nov 10 09:55:39 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, ajournal = "Int. J. Numer. Methods Eng.", fjournal = "International Journal for Numerical Methods in Engineering", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)1097-0207", } @Article{Kioustelidis:1975:PLA, author = "J. B. Kioustelidis and J. K. Petrou", title = "A Piecewise Linear Approximation of $ \log_2 x $ with Equal Maximum Errors in All Intervals", journal = j-IEEE-TRANS-COMPUT, volume = "C-24", number = "9", pages = "858--861", month = sep, year = "1975", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/T-C.1975.224330", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Tue Jul 12 07:57:56 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", URL = "https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=1672923", abstract = "In this paper it is shown how to divide the interval $ [1, 2] $ into $n$ parts so that the uniform linear approximation of $ \log_2 x $ in each subinterval has the same maximum error. This error is, in the case $ n = 4 $, smaller by a factor of $ 2.3 $ than the error of the linear mean-square approximation given by Hall et al. [1]. The final products of the mathematical analysis are explicit formulas which allow the direct determination of all parameters and the maximum error for any desired number $n$ of subdivisions of $ [1, 2] $.", acknowledgement = ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", keywords = "$\log_2(x)$; elementary function", } @Article{Lewis:1975:CPF, author = "John Gregg Lewis", title = "Certification of ``{Algorithm} 349: Polygamma Functions with Arbitrary Precision''", journal = j-TOMS, volume = "1", number = "4", pages = "380--382", month = dec, year = "1975", CODEN = "ACMSCU", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Fri Jun 16 10:31:40 2000", bibsource = "https://www.math.utah.edu/pub/tex/bib/cacm1960.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "See \cite{TadeudeMedeiros:1969:APF}.", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", keywords = "polygamma functions; special functions", } @Book{Luke:1975:MFT, author = "Yudell L. Luke", title = "Mathematical Functions and Their Approximations", publisher = pub-ACADEMIC, address = pub-ACADEMIC:adr, pages = "xvii + 568", year = "1975", ISBN = "0-12-459950-8, 1-4832-6245-6 (e-book)", ISBN-13 = "978-0-12-459950-5, 978-1-4832-6245-1 (e-book)", LCCN = "QA55 .L96 1975", bibdate = "Fri Jun 30 05:58:16 MDT 2023", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/master.bib", URL = "https://shop.elsevier.com/books/mathematical-functions-and-their-approximations/luke/978-0-12-459950-5", acknowledgement = ack-nhfb, libnote = "Not in my library.", remark = "An updated version of part of Handbook of mathematical functions with formulas, graphs, and mathematical tables, edited by M. Abramowitz and I.A. Stegun. Includes indexes.", subject = "Mathematics; Tables; Fonctions (Math{\'e}ematiques); Math{\'e}ematiques; Calculus; Mathematical Analysis; Mathematics; Approximation; Funktion; Mathematik; Spezielle Funktion", tableofcontents = "Preface / xv \\ \\ I. The Gamma Function and Related Functions \\ \\ 1.1 Definitions and Elementary Properties / 1 \\ 1.2 Power Series and Other Series Expansions / 1 \\ 1.3 Asymptotic Expansions / 7 \\ 1.4 Rational Approximations for y (z) / 13 \\ 1.5 Inequalities / 17 \\ 1.6 Bibliographic and Numerical Data / 20 \\ 1.6.1 General References / 20 \\ 1.6.2 Description of and References to Tables / 21 \\ 1.6.3 Description of and References to Other Approximations and Expansions / 22 \\ \\ II. The Binomial Function \\ \\ 2.1 Power Series / 24 \\ 2.2 Expansions in Series of Jacobi and Chebyshev Polynomials / 24 \\ 2.3 Expansions in Series of Bessel Functions / 26 \\ 2.4 Pad{\'e} Approximations / 27 \\ 24.1 $(1 + 1 / z)^{-c}$ / 27 \\ 2.4.2 The Square Root / 28 \\ 2.4.3 Pad{\'e} Coefficients / 30 \\ 2.4.4 The Function $e^{-w}$ / 31 \\ 2.5 Inequalities / 34 \\ \\ III. Elementary Functions \\ \\ 3.1 Logarithmic Functions / 36 \\ 3.1.1 Power Series / 36 \\ 3.1.2 Expansion in Series of Chebyshev Polynomials / 38 \\ 3.1.3 Pad{\'e} Approximations / 39 \\ 3.1.4 Inequalities / 41 \\ 3.2 Exponential Function / 42 \\ 3.2.1 Series Expansions / 42 \\ 3.2.2 Expansions in Series of Jacobi and Chebyshev Polynomials and Bessel Functions / 42 \\ 3.2.3 Pad{\'e} Approximations / 46 \\ 3.2.4 Inequalities / 51 \\ 3.3 Circular and Hyperbolic Functions / 52 \\ 3.3.1 Power Series / 52 \\ 3.3.2 Expansions in Series of Jacobi and Chebyshev Polynomials and Bessel Functions / 52 \\ 3.3.3 Rational and Pad{\'e} Approximations / 57 \\ 3.3.4 Inequalities / 60 \\ 3.4 Inverse Circular and Hyperbolic Functions / 61 \\ 3.4.1 Power Series / 61 \\ 3.4.2 Expansions in Series of Chebyshev Polynomials / 63 \\ 3.4.3 Pad{\'e} Approximations / 68 \\ 3.4.4 Inequalities / 72 \\ 3.5 Bibliographic and Numerical Data / 74 \\ 3.5.1 Description of and References to Tables / 74 \\ 3.5.2 Description of and References to Other Approximations and Expansions / 74 \\ \\ IV. Incomplete Gamma Functions \\ \\ 4.1 Definitions and Series Expansions / 77 \\ 4.2 Differential Equations and Difference Equations / 78 \\ 4.3 Pad{\'e} Approximations / 79 \\ 4.3.1 $_1F_1(1; \nu + 1; -z)$ / 79 \\ 4.3.2 $z^{1 - \nu} e^z \Gamma(\nu, z)$ / 82 \\ 4.3.3 The Error $T_n(\nu, z)$ for $|{\rm arg} z/k| \leq \pi$ / 84 \\ 4.3.4 The Negative Real Axis and the Zeros of $F_n(\nu, z)$ / 89 \\ 4.4 Inequalities / 95 \\ 4.4.1 $H(\nu, z)$ / 95 \\ 4.4.2 $\Gamma(\nu, z)$ / 96 \\ 4.5 Notes on the Computation of the Incomplete Gamma Function / 97 \\ 4.6 Exponential Integrals / 103 \\ 4.6.1 Relation to Incomplete Gamma Function and Other Properties / 103 \\ 4.6.2 Expansions in Series of Chebyshev Polynomials / 104 \\ 4.6.3 Rational and Pad Approximations / 106 \\ 4.7 Cosine and Sine Integrals / 115 \\ 4.7.1 Relation to Exponential Integral and Other Properties / 115 \\ 4.7.2 Expansions in Series of Chebyshev Polynomials / 116 \\ 4.8 Error Functions / 119 \\ 4.8.1 Relation to Incomplete Gamma Function and Other Properties / 119 \\ 4.8.2 Expansions in Series of Chebyshev Polynomials and Bessel Functions / 122 \\ 4.8.3 Pad{\'e} Approximations / 124 \\ 4.8.4 Trapezoidal Rule Approximations / 134 \\ 4.8.5 Inequalities / 137 \\ 4.9 Fresnel Integrals / 139 \\ 4.9.1 Relation to Error Functions and Other Properties / 139 \\ 4.9.2 Expansions in Series of Chebyshev Polynomials / 140 \\ 4.10 Bibliographic and Numerical Data / 143 \\ 4.10.1 References / 143 \\ 4.10.2 Description of and References to Tables / 143 \\ 4.10.3 Description of and References to Other Approximations and Expansions / 149 \\ \\ V. The Generalized Hypergeometric Function $_pF_g$ and the $G$-Function \\ \\ 5.1 Introduction / 154 \\ 5.2 The $_pF_q$ / 155 \\ 5.2.1 Power Series / 155 \\ 5.2.2 Derivatives and Contiguous Relations / 159 \\ 5.2.3 Integral Representations and Integrals Involving the $_pF_q$ / 160 \\ 5.2.4 Evaluation for Special Values of the Variable and Parameters / 163 \\ 5.3 The $G$-Function / 170 \\ 5.3.1 Definition and Relation to the $_pF_q$ / 170 \\ 5.3.2 Elementary Properties / 176 \\ 5.3.3 Analytic Continuation of $G_{p, p}^{m, n}(z)$ / 178 \\ 5.4 The Confluence Principle / 179 \\ 5.5 Multiplication Theorems / 184 \\ 5.6 Integrals Involving $G$-Functions / 186 \\ 5.7 Differential Equations / 190 \\ 5.7.1 The $_pF_q$ / 190 \\ 5.7.2 The $G$-Function / 192 \\ 5.8 Series of $G$-Functions / 194 \\ 5.8.1 Introduction / 194 \\ 5.8.2 Notation / 194 \\ 5.8.3 Expansion Theorems / 197 \\ 5.9 Asymptotic Expansions / 199 \\ 5.9.1 $G_{p, q}^{q, n}(z)$, $n = 0, 1$ / 199 \\ 5.9.2 $G_{p, q}^{m, n}(z)$ / 201 \\ 5.9.3 $_pF_q(z)$ / 206 \\ 5.10 Expansions in Series of Generalized Jacobi, Generalized Laguerre and Chebyshev Polynomials / 213 \\ 5.10.1 Expansions for $G$-Functions / 213 \\ 5.10.2 Expansions for $_pF_q$ / 220 \\ 5.11 Expansions in Series of Bessel Functions / 223 \\ 5.12 Polynomial and Rational Approximations / 224 \\ 5.13 Recurrence Formulas for Polynomials and Functions Occurring in Approximations to Generalized Hypergeometric Functions / 234 \\ 5.13.1 Introduction / 234 \\ 5.13.2 Recursion Formulas for Extended Jacobi and Laguerre Functions / 235 \\ 5.13.3 Recursion Formulas for the Numerator and Denominator Polynomials in the Rational Approximations for the Generalized Hypergeometric Function / 244 \\ 5.13.4 Recursion Formula for Coefficients in the Expansion of the $G$-Function in Series of Extended Jacobi Polynomials / 247 \\ 5.14 Inequalities / 252 \\ \\ VI. The Gaussian Hypergeometric Function $_2F_1$ \\ \\ 6.1 Introduction / 257 \\ 6.2 Elementary Properties / 257 \\ 6.2.1 Derivatives / 257 \\ 6.2.2 Contiguous Relations / 258 \\ 6.2.3 Integral Representations / 259 \\ 6.3 Differential Equations / 260 \\ 6.4 Kummer Solutions and Transformation Formulae / 262 \\ 6.5 Analytic Continuation / 263 \\ 6.6 The Complete Solution and Wronskians / 265 \\ 6.7 Quadratic Transformations / 270 \\ 6.8 The $_2F_1$ for Special Values of the Argument / 271 \\ 6.9 Expansion in Series of Chebyshev Polynomials / 274 \\ 6.10 Pad{\'e} Approximations for $_2F_1(1, \sigma;\rho + 1;-1/z)$ / 274 \\ 6.11 Inequalities / 278 \\ 6.12 Bibliographic and Numerical Data / 279 \\ 6.12.1 References / 279 \\ 6.12.2 Description of and References to Tables / 279 \\ \\ VII. The Confluent Hypergeometric Function \\ \\ 7.1 Introduction / 284 \\ 7.2 Integral Representations / 284 \\ 7.3 Elementary Relations / 285 \\ 7.3.1 Derivatives / 285 \\ 7.3.2 Contiguous Relations / 285 \\ 7.3.3 Products of Confluent Functions / 286 \\ 7.4 Differential Equations / 287 \\ 7.5 The Complete Solution and Wronskians / 288 \\ 7.6 Asymptotic Expansions / 291 \\ 7.7 Expansions in Series of Chebyshev Polynomials / 293 \\ 7.8 Expansions in Series of Besse! Functions / 294 \\ 7.9 Inequalities / 295 \\ 7.10 Other Notations and Related Functions / 295 \\ 7.11 Bibliographic and Numerical Data / 296 \\ 7.11.1 References / 296 \\ 7.11.2 Description of and References to Tables and Other Approximations / 296 \\ \\ VIII. Identification of the $_pF_q$, and $G$-Functions with the Special Functions \\ \\ 8.1 Introduction / 298 \\ 8.2 Named Special Functions Expressed as $_pF_q$'s / 298 \\ 8.2.1 Elementary Functions / 298 \\ 8.2.2 The Incomplete Gamma Function and Related Functions / 298 \\ 8.2.3 The Gaussian Hypergeometric Function / 298 \\ 8.2.4 Legendre Functions / 299 \\ 8.2.5 Orthogonal Polynomials / 299 \\ 8.2.6 Complete Elliptic Integrals / 299 \\ 8.2.7 Confluent Hypergeometric Functions, Whittaker Functions and Bessel Functions / 300 \\ 8.3 Named Functions Expressed in Terms of the $G$-Function / 300 \\ 8.4 The $G$-Function Expressed as a Named Function / 306 \\ \\ IX. Bessel Functions and Their Integrals \\ \\ 9.1 Introduction / 311 \\ 9.2 Definitions, Connecting Relations and Power Series / 311 \\ 9.3 Difference--Differential Formulas / 313 \\ 9.4 Products of Bessel Functions / 314 \\ 9.5 Asymptotic Expansions for Large Variable / 315 \\ 9.6 Integrals of Bessel Functions / 315 \\ 9.7 Expansions in Series of Chebyshev Polynomials / 316 \\ 9.8 Expansions in Series of Bessel Functions / 360 \\ 9.9 Rational Approximations / 361 \\ 9.9.1 Introduction / 361 \\ 9.9.2 $I_\nu(z)$, $z$ Small / 361 \\ 9.9.3 $K_\nu(z)$, $z$ Large / 366 \\ 9.10 Computation of Bessel Functions by Use of Recurrence Formulas / 380 \\ 9.10.1 Introduction / 380 \\ 9.10.2 Backward Recurrence Schemata for Generating $I_\nu(z)$ / 380 \\ 9.10.3 Closed Form Expressions / 382 \\ 9.10.4 Expressions for $J_\nu(z)$ / 389 \\ 9.10.5 Numerical Examples / 392 \\ 9.11 Evaluation of Bessel Functions by Application of Trapezoidal Type Integration Formulas / 395 \\ 9.12 Inequalities / 399 \\ 9.13 Bibliographic and Numerical Data / 403 \\ 9.13.1 References / 403 \\ 9.13.2 Description of and References to Tables / 404 \\ 9.13.3 Description of and References to Other Approximations and Expansions / 410 \\ \\ X. Lommel Functions, Struve Functions, and Associated Bessel Functions \\ \\ 10.1 Definitions, Connecting Relations and Power Series / 413 \\ 10.2 Asymptotic Expansions / 415 \\ 10.3 Expansions in Series of Chebyshev Polynomials and Bessel Functions / 415 \\ 10.4 Rational Approximations for $H_\nu(z) - Y_\nu(z)$ and the Errors in These Approximations / 422 \\ 10.5 Bibliographic and Numerical Data / 426 \\ 10.5.1 References / 426 \\ 10.5.2 Description of and References to Tables / 426 \\ \\ XI. Orthogonal Polynomials \\ \\ 11.1 Introduction / 428 \\ 11.2 Orthogonal Properties / 428 \\ 11.3 Jacobi Polynomials / 436 \\ 11.3.1 Expansion Formulae / 436 \\ 11.3.2 Difference--Differential Formulae / 439 \\ 11.3.3 Integrals / 439 \\ 11.3.4 Expansion of $x^\rho$ in Series of Jacobi Polynomials / 440 \\ 11.3.5 Convergence Theorems for the Expansion of Arbitrary Functions in Series of Jacobi Polynomials / 442 \\ 11.3.6 Evaluation and Estimation of the Coefficients in the Expansion of a Given Function $f(x)$ in Series of Jacobi Polynomials / 443 \\ 11.4 The Chebyshev Polynomials $T_n(x)$ and $U_n(x)$ / 453 \\ 11.5 The Chebyshev Polynomials $T_n^*(x)$ and $U_n^*(x)$ / 459 \\ 11.6 Coefficients for Expansion of Integrals of Functions in Series of Chebyshev Polynomials of the First Kind / 464 \\ 11.6.1 Introduction / 464 \\ 11.6.2 Series of Shifted Chebyshev Polynomials / 464 \\ 11.6.3 Series of Chebyshev Polynomials of Even Order / 468 \\ 11.6.4 Series of Chebyshev Polynomials of Odd Order / 468 \\ 11.7 Orthogonality Properties of Chebyshev Polynomials with Respect to Summation / 469 \\ 11.8 A Nesting Procedure for the Computation of Expansions in Series of Functions Where the Functions Satisfy a Linear Finite Difference Equation / 475 \\ \\ XII. Computation by Use of Recurrence Formulas \\ \\ 12.1 Introduction / 483 \\ 12.2 Homogeneous Difference Equations / 483 \\ 12.3 Inhomogeneous Difference Equations / 487 \\ \\ XIII. Some Aspects of Rational and Polynomial Approximations \\ \\ 13.1 Introduction / 490 \\ 13.2 Approximations in Series of Chebyshev Polynomials of the First Kind / 490 \\ 13.3 The Pad{\'e} Table / 493 \\ 13.4 Approximation of Functions Defined by a Differential Equation --- The $\tau$-Method / 495 \\ 13.5 Approximations of Functions Defined by a Series / 499 \\ 13.6 Solution of Differential Equations in Series of Chebyshev Polynomials of the First Kind / 500 \\ \\ XIV. Miscellaneous Topics \\ \\ 14.1 Introduction / 505 \\ 14.2 Bernoulli Polynomials and Numbers / 505 \\ 14.3 $D$ and $\delta$ Operators / 507 \\ 14.4 Computation and Check of the Tables / 509 \\ 14.5 Mathematical Constants / 512 \\ 14.6 Late Bibliography / 516 \\ \\ Bibliography / 517 \\ \\ Notation Index / 545 \\ \\ Subject Index / 551", } @Article{Luke:1975:SRU, author = "Y. L. Luke", title = "Some remarks on uniform asymptotic expansions for {Bessel} functions", journal = j-COMPUT-MATH-APPL, volume = "1", number = "3--4", pages = "285--290", month = "????", year = "1975", CODEN = "CMAPDK", ISSN = "0898-1221 (print), 1873-7668 (electronic)", ISSN-L = "0898-1221", bibdate = "Wed Mar 1 18:51:12 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/computmathappl1970.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/0898122175900279", acknowledgement = ack-nhfb, fjournal = "Computers and Mathematics with Applications", journal-URL = "http://www.sciencedirect.com/science/journal/08981221", } @Book{Masser:1975:EFT, author = "D. W. Masser", title = "Elliptic Functions and Transcendence", volume = "437", publisher = pub-SV, address = pub-SV:adr, pages = "112 (est.)", year = "1975", CODEN = "LNMAA2", DOI = "https://doi.org/10.1007/BFb0069432", ISBN = "3-540-07136-9 (print), 3-540-37410-8 (e-book)", ISBN-13 = "978-3-540-07136-5 (print), 978-3-540-37410-7 (e-book)", ISSN = "0075-8434 (print), 1617-9692 (electronic)", ISSN-L = "0075-8434", LCCN = "QA3 .L28 no. 437", MRclass = "11J81 (14K22; 33E05; 11G15; 11J17; 11-02)", bibdate = "Tue May 6 14:52:13 MDT 2014", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/lnm1975.bib", series = ser-LECT-NOTES-MATH, URL = "http://link.springer.com/book/10.1007/BFb0069432; http://www.springerlink.com/content/978-3-540-37410-7", ZMnumber = "0312.10023", acknowledgement = ack-nhfb, series-URL = "http://link.springer.com/bookseries/304", } @Article{Midy:1975:ICG, author = "P. Midy", title = "An improved calculation of the general elliptic integral of the second kind in the neighbourhood of $ x = 0 $", journal = j-NUM-MATH, volume = "25", number = "1", pages = "99--101", month = mar, year = "1975", CODEN = "NUMMA7", ISSN = "0029-599X (print), 0945-3245 (electronic)", ISSN-L = "0029-599X", bibdate = "Sun Oct 17 16:12:48 MDT 1999", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, classification = "B0290M (Numerical integration and differentiation); C4160 (Numerical integration and differentiation)", corpsource = "Centre de Calcul Paris Sud Informatique, Univ. Paris XI, Orsay, France", fjournal = "Numerische Mathematik", journal-URL = "http://link.springer.com/journal/211", keywords = "elliptic integral; integration; Landen transformation; second kind", treatment = "T Theoretical or Mathematical", } @Article{Miller:1975:CCN, author = "Webb Miller", key = "Miller", title = "Computational Complexity and Numerical Stability", journal = j-SIAM-J-COMPUT, volume = "4", number = "2", pages = "97--107", month = jun, year = "1975", CODEN = "SMJCAT", DOI = "https://doi.org/10.1137/0204009", ISSN = "0097-5397 (print), 1095-7111 (electronic)", ISSN-L = "0097-5397", bibdate = "Mon Nov 29 10:58:08 MST 2010", bibsource = "http://epubs.siam.org/sam-bin/dbq/toclist/SICOMP/4/2; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/siamjcomput.bib; Parallel/Multi.bib; Theory/Matrix.bib", acknowledgement = ack-nhfb, fjournal = "SIAM Journal on Computing", journal-URL = "http://epubs.siam.org/sicomp", keywords = "complexity; number of multiplications to evaluate a polynomial; numerical analysis; rounding error", } @TechReport{Morris:1975:LRS, author = "Robert Morris", title = "A Library of Reference Standard Mathematical Subroutines", type = "Technical Memorandum", number = "1074 (TM 75-1271-6)", institution = inst-ATT-BELL, address = inst-ATT-BELL:adr, pages = "??", day = "1", month = may, year = "1975", bibdate = "Tue Jun 06 08:07:45 2023", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/unix.bib", abstract = "This memo describes a set of mathematical library functions to use arbitrary accuracy. Relevant error analysis and subroutines listings are given.", acknowledgement = ack-nhfb, author-dates = "Robert Morris (25 July 1932--26 June 2011)", } @Article{Ng:1975:CCM, author = "Edward W. Ng", title = "A Comparison of Computational Methods and Algorithms for the Complex Gamma Function", journal = j-TOMS, volume = "1", number = "1", pages = "56--70", month = mar, year = "1975", CODEN = "ACMSCU", DOI = "https://doi.org/10.1145/355626.355635", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", MRclass = "65D20", MRnumber = "52 #2148", bibdate = "Fri Aug 26 23:44:16 1994", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", reviewer = "R. H. Bartels", } @Article{Oliver:1975:SME, author = "J. Oliver", title = "Stable methods for evaluating the points $ \cos (i \pi / n) $", journal = j-J-INST-MATH-APPL, volume = "16", number = "2", pages = "247--257", month = oct, year = "1975", CODEN = "JMTAA8", DOI = "https://doi.org/10.1093/imamat/16.2.247", ISSN = "0020-2932", ISSN-L = "0020-2932", MRclass = "65D05", MRnumber = "52 #12292 (391471)", MRreviewer = "C. W. Clenshaw", bibdate = "Mon Nov 13 08:14:42 2023", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/c/clenshaw-charles-w.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jinstmathappl.bib", ZMnumber = "0308.65011", acknowledgement = ack-nhfb, fjournal = "Journal of the Institute of Mathematics and its Applications", journal-URL = "http://imamat.oxfordjournals.org/content/by/year", reviewer-dates = "Charles William Clenshaw (15 March 1926--23 September 2004)", } @Article{Prince:1975:AAF, author = "P. J. Prince", title = "{Algorithm 498}: {Airy} Functions Using {Chebyshev} Series Approximations", journal = j-TOMS, volume = "1", number = "4", pages = "372--379", month = dec, year = "1975", CODEN = "ACMSCU", DOI = "https://doi.org/10.1145/355656.355663", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Sat Aug 27 00:24:33 1994", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "See also \cite{Razaz:1981:RAF}.", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", remark = "Fullerton: FORTRAN routines of fixed 10D precision for the two Airy functions and their derivatives are given.", } @Article{Rudnicki-Bujnowski:1975:EFC, author = "Georges Rudnicki-Bujnowski", title = "Explicit Formulas for {Clebsch--Gordan} Coefficients", journal = j-COMP-PHYS-COMM, volume = "10", number = "4", pages = "245--250", month = oct, year = "1975", CODEN = "CPHCBZ", DOI = "https://doi.org/10.1016/0010-4655(75)90069-7", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", bibdate = "Sun Feb 12 14:24:38 MST 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/compphyscomm1970.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/0010465575900697", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Computer Physics Communications", journal-URL = "http://www.sciencedirect.com/science/journal/00104655", remark = "Fullerton: A PL/I-FORMAC procedure is discussed.", } @Article{Skovgaard:1975:RAJ, author = "Ove Skovgaard", title = "Remark on ``{Algorithm 332}: {Jacobi} Polynomials''", journal = j-CACM, volume = "18", number = "2", pages = "116--117", year = "1975", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Mon Jan 22 07:22:18 MST 2001", bibsource = "http://dblp.uni-trier.de/db/journals/cacm/cacm18.html#Skovgaard75; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "See \cite{Witte:1968:AAJ}.", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", oldlabel = "Skovgaard75", remark = "Fullerton: Modifications to an adjustable-precision FORTRAN routine.", XMLdata = "ftp://ftp.informatik.uni-trier.de/pub/users/Ley/bib/records.tar.gz#journals/cacm/Skovgaard75", } @Article{Skovgaard:1975:RBF, author = "Ove Skovgaard", title = "Remark on ``{Algorithm 236: Bessel Functions of the First Kind [S17]}''", journal = j-TOMS, volume = "1", number = "3", pages = "282--284", month = sep, year = "1975", CODEN = "ACMSCU", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Tue Feb 06 05:26:43 1996", bibsource = "https://www.math.utah.edu/pub/tex/bib/cacm1970.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "See \cite{Gautschi:1964:AAB}.", acknowledgement = ack-nhfb, fjournal = "ACM Transactions on Mathematical Software", journal-URL = "https://dl.acm.org/loi/toms", } @Article{Temme:1975:NEM, author = "N. M. Temme", title = "On the Numerical Evaluation of the Modified {Bessel} Function of the Third Kind", journal = j-J-COMPUT-PHYS, volume = "19", number = "3", pages = "324--337", month = nov, year = "1975", CODEN = "JCTPAH", DOI = "https://doi.org/10.1016/0021-9991(75)90082-0", ISSN = "0021-9991 (print), 1090-2716 (electronic)", ISSN-L = "0021-9991", bibdate = "Sat Oct 30 11:20:31 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputphys1970.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Journal of Computational Physics", journal-URL = "http://www.sciencedirect.com/science/journal/00219991/", } @Article{Temme:1975:UAE, author = "N. M. Temme", title = "Uniform asymptotic expansions of the incomplete gamma functions and the incomplete beta function", journal = j-MATH-COMPUT, volume = "29", number = "132", pages = "1109--1114", month = oct, year = "1975", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", classcodes = "B0290F (Interpolation and function approximation); C4130 (Interpolation and function approximation)", corpsource = "Dept. of Appl. Math., Amsterdam, Netherlands", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "asymptotic expansions; asymptotic series; complementary error function; function approximation; incomplete beta function; incomplete gamma functions; uniform", treatment = "T Theoretical or Mathematical", } @Book{VanBuren:1975:TAS, author = "A. L. (Arnie Lee) {Van Buren} and others", title = "Tables of Angular Spheroidal Wave Functions", publisher = "Naval Research Laboratory", address = "Washington, DC, USA", pages = "????", year = "1975", LCCN = "QC174.26.W3 T27", bibdate = "Sat Apr 1 14:32:29 MDT 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; z3950.loc.gov:7090/Voyager", acknowledgement = ack-nhfb, subject = "Wave functions; Tables; Spheroidal functions", tableofcontents = "v. 1. Prolate, m = O \\ v. 2. Oblate, m = O", } @TechReport{Warner:1975:PDG, author = "D. D. Warner", title = "A partial derivative generator", type = "Computing Science Technical Report", number = "28", institution = inst-ATT-BELL, address = inst-ATT-BELL:adr, year = "1975", bibdate = "Mon May 19 13:30:58 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; Theory/auto.diff.bib", abstract = "A precompiler is described which takes a specification of a function as input and produces a Fortran subroutine which will evaluate the component functions and the corresponding Jacobian. Many of the Fortran elementary functions are provided, as well as a facility which allows the user to specify their own differentiation rules.", acknowledgement = ack-nhfb, keywords = "differentiation arithmetic; precompiler.", referred = "[Carl86a]; [Hali83a]; [Hill82a]; [Spee80a].", } @InProceedings{Andrews:1976:ESR, author = "M. Andrews and S. F. McCormick and G. D. Taylor", editor = "John Gosden", booktitle = "{ACM '76: Proceedings of the 1976 annual conference, Houston Texas USA October 20--22, 1976}", title = "Evaluation of the square root function on microprocessors", publisher = pub-ACM, address = pub-ACM:adr, bookpages = "576", year = "1976", DOI = "https://doi.org/10.1145/800191.805571", ISBN = "1-4503-7489-1", ISBN-13 = "978-1-4503-7489-7", LCCN = "????", bibdate = "Mon Nov 10 10:11:48 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", series = "ACM 76", acknowledgement = ack-nhfb, collection = "ACM 76", } @Article{Assmus:1976:NFS, author = "E. F. {Assmus, Jr.} and H. F. {Mattson, Jr.} and Howard E. Sachar", title = "A New Form of the Square Root Bound", journal = j-SIAM-J-APPL-MATH, volume = "30", number = "2", pages = "352--354", month = mar, year = "1976", CODEN = "SMJMAP", ISSN = "0036-1399 (print), 1095-712X (electronic)", ISSN-L = "0036-1399", bibdate = "Thu Oct 15 18:16:06 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/siamjapplmath.bib; JSTOR database", acknowledgement = ack-nhfb, classification = "B0250 (Combinatorial mathematics); C1160 (Combinatorial mathematics)", corpsource = "Dept. of Math., Lehigh Univ., Bethlehem, PA, USA", fjournal = "SIAM Journal on Applied Mathematics", journal-URL = "http://epubs.siam.org/siap", keywords = "combinatorial mathematics; linear codes; square root bound; sufficient combinatorial conditions", treatment = "T Theoretical or Mathematical", } @Article{Badhe:1976:NAN, author = "Sahadeo K. Badhe", title = "New approximation of the normal distribution function", journal = j-COMMUN-STAT-SIMUL-COMPUT, volume = "5", number = "4", pages = "173--176", year = "1976", CODEN = "CSSCDB", DOI = "https://doi.org/10.1080/03610917608812017", ISSN = "0361-0918", ISSN-L = "0361-0918", bibdate = "Sat Jan 30 06:32:08 MST 2016", bibsource = "https://www.math.utah.edu/pub/tex/bib/communstatsimulcomput1970.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Communications in Statistics: Simulation and Computation", journal-URL = "http://www.tandfonline.com/loi/lssp20", } @Article{Baker:1976:SFB, author = "P. W. Baker", title = "Suggestion for a fast binary sine\slash cosine generator", journal = j-IEEE-TRANS-COMPUT, volume = "C-25", number = "11", pages = "1134--1136", month = nov, year = "1976", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/TC.1976.1674566", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Tue Jul 12 06:24:55 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput1970.bib", URL = "https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=1674566", acknowledgement = ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", keywords = "$\cos(x)$; $\sin(x)$; elementary function", } @Article{Barnett:1976:MRC, author = "A. R. Barnett", title = "{RCWFF} --- Modification of the Real {Coulomb} Wavefunction Program {RCWFN}", journal = j-COMP-PHYS-COMM, volume = "11", number = "1", pages = "141--142", month = jan # "\slash " # feb, year = "1976", CODEN = "CPHCBZ", DOI = "https://doi.org/10.1016/0010-4655(76)90045-X", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", bibdate = "Fri Oct 29 21:24:02 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/compphyscomm1970.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Computer Physics Communications", journal-URL = "http://www.sciencedirect.com/science/journal/00104655", } @Article{Barnett:1976:RMR, author = "A. R. Barnett", title = "{RCWFF} --- Modification of the Real {Coulomb} Wavefunction Program {RCWFN}", journal = j-COMP-PHYS-COMM, volume = "11", number = "1", pages = "141--142", month = jan # "\slash " # feb, year = "1976", CODEN = "CPHCBZ", DOI = "https://doi.org/10.1016/0010-4655(76)90045-X", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", bibdate = "Mon Feb 13 06:01:19 MST 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/compphyscomm1970.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/001046557690045X", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Computer Physics Communications", journal-URL = "http://www.sciencedirect.com/science/journal/00104655", xxtitle = "{RCWFF} --- a modification of the real {Coulomb} wavefunction program {RCWFN}", } @Article{Blagoveshchenskii:1976:MCM, author = "Yu V. Blagoveshchenskii and B. A. Popov and G. S. Tesler", title = "Methods for computing mutually inverse functions", journal = j-CYBER, volume = "11", number = "2", pages = "252--256", month = mar, year = "1976", CODEN = "CYBNAW", DOI = "https://doi.org/10.1007/BF01069867", ISSN = "0011-4235 (print), 2375-0189 (electronic)", ISSN-L = "0011-4235", bibdate = "Tue Jan 24 08:29:23 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.springerlink.com/index/10.1007/BF01069867", acknowledgement = ack-nhfb, fjournal = "Cybernetics", journal-URL = "http://link.springer.com/journal/10559", remark = "Translated from \booktitle{Kibernetika}, No. 2, pp. 69--72, March--April, 1975.", } @Article{Blair:1976:RCA, author = "J. M. Blair and C. A. Edwards and J. H. Johnson", title = "Rational {Chebyshev} approximations for the inverse of the error function", journal = j-MATH-COMPUT, volume = "30", number = "136", pages = "827--830", month = oct, year = "1976", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", classcodes = "B0290F (Interpolation and function approximation); C4130 (Interpolation and function approximation)", corpsource = "Atomic Energy of Canada Ltd., Chalk River Nuclear Lab., Chalk River, Ont., Canada", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "Chebyshev approximation; Chebyshev approximations; error function; inverse; rational", remark = "Fullerton: With microfiche supplement.", treatment = "T Theoretical or Mathematical", } @Article{Brent:1976:FMP, author = "Richard P. Brent", title = "Fast Multiple-Precision Evaluation of Elementary Functions", journal = j-J-ACM, volume = "23", number = "2", pages = "242--251", month = apr, year = "1976", CODEN = "JACOAH", DOI = "https://doi.org/10.1145/321941.321944", ISSN = "0004-5411 (print), 1557-735X (electronic)", ISSN-L = "0004-5411", MRclass = "68A20 (68A10)", MRnumber = "52 \#16111", MRreviewer = "Amnon Barak", bibdate = "Wed Jan 15 18:12:53 MST 1997", bibsource = "Compendex database; garbo.uwasa.fi:/pc/doc-soft/fpbiblio.txt; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", abstract = "Let $ f(x) $ be one of the usual elementary functions ($ \exp $, $ \log $, $ \artan $, $ \sin $, $ \cosh $, etc.), and let $ M(n) $ be the number of single-precision operations required to multiply $n$-bit integers. It is shown that $ f(x) $ can be evaluated, with relative error $ O(2 - n) $, in $ O(M(n)l o g (n)) $ operations as $ n \rightarrow \infty $, for any floating-point number $x$ (with an $n$-bit fraction) in a suitable finite interval. From the Sch{\"o}nhage--Strassen bound on $ M(n) $, it follows that an $n$-bit approximation to $ f(x) $ may be evaluated in $ O(n \log_(n) \log \log (n)) $ operations. Special cases include the evaluation of constants such as $ \pi $ $e$, and $ e^\pi $. The algorithms depend on the theory of elliptic integrals, using the arithmetic-geometric mean iteration and ascending Landen transformations.", acknowledgement = ack-nhfb, ajournal = "J. Assoc. Comput. Mach.", classification = "723", fjournal = "Journal of the ACM", journal-URL = "https://dl.acm.org/loi/jacm", keywords = "computational complexity; computer arithmetic; computer programming", } @InProceedings{Brent:1976:MPZ, author = "Richard P. Brent", title = "Multiple-precision zero-finding methods and the complexity of elementary function evaluation", crossref = "Traub:1976:ACC", pages = "151--176", year = "1976", MRclass = "68A20", MRnumber = "54 \#11843", MRreviewer = "Claus-Peter Schnorr", bibdate = "Sat Jan 11 17:44:01 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, } @Article{Buschman:1976:IHF, author = "R. G. Buschman", title = "Inequalities for Hypergeometric Functions", journal = j-MATH-COMPUT, volume = "30", number = "134", pages = "303--305", month = apr, year = "1976", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", classcodes = "B0220 (Mathematical analysis); C1120 (Mathematical analysis)", corpsource = "Dept. of Math. and Statistics, Univ. of Guelph, Guelph, Ont., Canada", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "Bessel functions; classical orthogonal polynomials; confluence; dominant diagonal matrix; function; functions; Gauss' hypergeometric; hypergeometric functions; Kummer's hypergeometric function; modified Bessel function; principle", treatment = "T Theoretical or Mathematical", } @Article{Davies:1976:IPS, author = "M. Davies and B. Dawson", title = "The incrementation parameter in square root iteration", journal = j-J-INST-MATH-APPL, volume = "17", number = "2", pages = "219--223", year = "1976", CODEN = "JMTAA8", ISSN = "0020-2932", MRclass = "65H05", MRnumber = "55 #9514", MRreviewer = "Luciano Biasini", bibdate = "Fri Apr 5 07:38:01 MST 2002", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", ZMnumber = "0319.65039", acknowledgement = ack-nhfb, fjournal = "Journal of the Institute of Mathematics and its Applications", journal-URL = "http://imamat.oxfordjournals.org/content/by/year", } @Article{Deuflhard:1976:ASC, author = "P. Deuflhard", title = "On Algorithms for the Summation of Certain Special Functions", journal = j-COMPUTING, volume = "17", number = "1", pages = "37--48", month = mar, year = "1976", CODEN = "CMPTA2", DOI = "https://doi.org/10.1007/BF02252258", ISSN = "0010-485X (print), 1436-5057 (electronic)", ISSN-L = "0010-485X", bibdate = "Tue Jan 2 17:40:52 MST 2001", bibsource = "Compendex database; http://springerlink.metapress.com/openurl.asp?genre=journal&issn=0010-485X; https://www.math.utah.edu/pub/tex/bib/computing.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; INSPEC Axiom database (1968--date)", acknowledgement = ack-nhfb, affiliation = "Inst. f{\"u}r Math., Tech. Univ. M{\"u}nchen, M{\"u}nchen, West Germany", citedby = "Fullerton:1980:BEM", classification = "723; 921; B0290B; B0290H; C4110; C4140", description = "error analysis; linear algebra", fjournal = "Computing", journal-URL = "http://link.springer.com/journal/607", journalabr = "Comput (Vienna/NY)", keywords = "algorithms; backward error analysis; computer programming; graph representation; mathematical techniques; special functions; stability; summation", remark = "Fullerton: An extension of Clenshaw's summation method is discussed. Spherical harmonic sums are considered as a special case.", } @TechReport{DiDonato:1976:CIG, author = "Armido R. DiDonato and R. K. Hageman", title = "Computation of the incomplete gamma function ratios", type = "Report", number = "NSWC/DL TR-3482", institution = "Naval Surface Weapons Center", address = "Dahlgren, VA, USA", pages = "86", month = apr, year = "1976", bibdate = "Mon Jun 03 12:24:32 2024", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "https://apps.dtic.mil/sti/citations/tr/ADA031812", acknowledgement = ack-nhfb, } @Article{Eckhardt:1976:RAW, author = "Ulrich Eckhardt", title = "A Rational Approximation to {Weierstrass}' $ \wp $-Function", journal = j-MATH-COMPUT, volume = "30", number = "136", pages = "818--826", month = oct, year = "1976", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, classcodes = "A0260 (Numerical approximation and analysis); C4130 (Interpolation and function approximation)", corpsource = "Nuclear Res. Center, Central Inst. for Appl. Math., Julich, West Germany", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "equianharmonic; function approximation; rational approximation; unit period parallelogram; Weierstrass' elliptic function; Weierstrass' p function", remark = "Fullerton: A complex FORTRAN algorithm with accuracy down to $ 10^{-18} $ is given.", treatment = "T Theoretical or Mathematical", } @Article{Ellacott:1976:RCA, author = "S. Ellacott and Jack Williams", title = "Rational {Chebyshev} Approximation in the Complex Plane", journal = j-SIAM-J-NUMER-ANAL, volume = "13", number = "3", pages = "310--323", month = jun, year = "1976", CODEN = "SJNAAM", ISSN = "0036-1429 (print), 1095-7170 (electronic)", ISSN-L = "0036-1429", bibdate = "Fri Oct 16 06:57:22 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "SIAM Journal on Numerical Analysis", journal-URL = "http://epubs.siam.org/sinum", } @Article{elLozy:1976:RAC, author = "Mohamed el Lozy", title = "Remark on {``Algorithm 299: Chi-Squared Integral [S15]''}", journal = j-TOMS, volume = "2", number = "4", pages = "393--395", month = dec, year = "1976", CODEN = "ACMSCU", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Sat Jul 05 16:47:38 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/cacm1960.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "See \cite{Hill:1967:ACS,Hill:1985:RCS}.", acknowledgement = ack-nhfb, fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", } @Article{Fettis:1976:CR, author = "Henry E. Fettis", title = "Complex Roots of $ \sin z = a z, \cos z = a z $, and $ \cosh z = a z $", journal = j-MATH-COMPUT, volume = "30", number = "135", pages = "541--545", month = jul, year = "1976", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, classcodes = "B0220 (Mathematical analysis); C1120 (Mathematical analysis)", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "complex roots; cos z=az; cosh z=az; functional equations; sin z=az", treatment = "T Theoretical or Mathematical", } @Article{Fullerton:1976:AEM, author = "L. W. Fullerton and G. A. {Rinker, Jr.}", title = "Accurate and Efficient Methods for the Evaluation of Vacuum-Polarization Potentials of Order {$ Z \alpha $} and {$ Z \alpha^2 $}", journal = j-PHYS-REV-A, volume = "13", number = "3", pages = "1283--1287", month = mar, year = "1976", CODEN = "PLRAAN", ISSN = "1050-2947 (print), 1094-1622, 1538-4446, 1538-4519", ISSN-L = "1050-2947", bibdate = "Sat Oct 30 06:32:26 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Physical Review A (Atomic, Molecular, and Optical Physics)", journal-URL = "http://pra.aps.org/browse", remark = "Fullerton: Nine-figure approximations to $ K_n(x) = \int_1^\infty e^{-x t} t^n (1 / r^3 + 1 / (2 r^5)) (r^2 - 1)^{1 / 2} \, d t $ for $ n = 0, 1, 3 $, and $5$.", } @Article{Ikebe:1976:CZB, author = "Y. Ikebe", title = "Computing Zeros of {Bessel} and Regular {Coulomb} Wave Functions and of Their Derivatives by Matrix Theoretic Approach --- Practical Accuracy Criteria", journal = j-SIAM-REVIEW, volume = "18", number = "4", pages = "810--810", month = "????", year = "1976", CODEN = "SIREAD", ISSN = "0036-1445 (print), 1095-7200 (electronic)", ISSN-L = "0036-1445", bibdate = "Fri Jun 21 11:25:02 MDT 2013", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/siamreview.bib", acknowledgement = ack-nhfb, fjournal = "SIAM Review", journal-URL = "http://epubs.siam.org/sirev", } @Article{Kerridge:1976:YAS, author = "D. F. Kerridge and G. W. Cook", title = "Yet Another Series for the Normal Integral", journal = j-BIOMETRIKA, volume = "63", number = "2", pages = "401--403", month = aug, year = "1976", CODEN = "BIOKAX", DOI = "https://doi.org/10.2307/2335636", ISSN = "0006-3444 (print), 1464-3510 (electronic)", ISSN-L = "0006-3444", bibdate = "Sat Jun 21 14:34:06 MDT 2014", bibsource = "http://www.jstor.org/journals/00063444.html; http://www.jstor.org/stable/i315483; https://www.math.utah.edu/pub/tex/bib/biometrika1970.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.jstor.org/stable/2335636", acknowledgement = ack-nhfb, fjournal = "Biometrika", journal-URL = "http://www.jstor.org/journals/00063444.html", } @Article{Kononova:1976:CEF, author = "N. F. Kononova", title = "The computation of elementary functions by means of polynomial approximations by the method of {V. K. Dzjadik}. ({Russian})", journal = "Vy{\v{c}}isl. Prikl. Mat. (Kiev)", volume = "29", pages = "27--39", year = "1976", ISSN = "0321-4117", MRclass = "151. 65D15", MRnumber = "57 \#18026", bibdate = "Sat Jan 11 17:44:01 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, language = "Russian", } @Article{Lentz:1976:GBF, author = "William J. Lentz", title = "Generating {Bessel} Functions in {Mie} Scattering Calculations Using Continued Fractions", journal = j-APPL-OPTICS, volume = "15", number = "3", pages = "668--671", month = mar, year = "1976", CODEN = "APOPAI", DOI = "https://doi.org/10.1364/AO.15.000668", ISSN = "0003-6935", bibdate = "Sat Oct 30 08:41:40 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", abstract = "A new method of generating the Bessel functions and ratios of Bessel functions necessary for Mie calculations is presented. Accuracy is improved while eliminating the need for extended precision word lengths or large storage capability. The algorithm uses a new technique of evaluating continued fractions that starts at the beginning rather than the tail and has a built-in error check. The continued fraction representations for both spherical Bessel functions and ratios of Bessel functions of consecutive order are presented.", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Applied Optics", journal-URL = "http://www.osapublishing.org/ao/browse.cfm", } @Article{Luke:1976:CER, author = "Yudell L. Luke", title = "{Chebyshev} expansions and rational approximations", journal = j-J-COMPUT-APPL-MATH, volume = "2", number = "2", pages = "85--93", month = jun, year = "1976", CODEN = "JCAMDI", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Sat Feb 25 11:59:14 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath1970.bib", URL = "http://www.sciencedirect.com/science/article/pii/0771050X76900139", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @Article{Morris:1976:RDF, author = "Robert Morris", title = "Remark on ``{Algorithm 490}: The Dilogarithm Function of a Real Argument [{S22}]''", journal = j-TOMS, volume = "2", number = "1", pages = "112--112", month = mar, year = "1976", CODEN = "ACMSCU", DOI = "https://doi.org/10.1145/355666.355680", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Tue Aug 30 00:27:18 1994", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "See \cite{Ginsberg:1975:AAD}.", acknowledgement = ack-nhfb, fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", } @Article{Overington:1976:DSI, author = "William J. G. Overington", title = "A design study for an integer order {Bessel} function of the first kind function generator for an analogue computer", journal = j-MATH-COMPUT-SIMUL, volume = "18", number = "1", pages = "63--64", month = jan, year = "1976", CODEN = "MCSIDR", DOI = "https://doi.org/10.1016/0378-4754(76)90032-X", ISSN = "0378-4754 (print), 1872-7166 (electronic)", ISSN-L = "0378-4754", bibdate = "Fri Aug 15 13:24:09 MDT 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcomputsimul1970.bib", URL = "https://www.sciencedirect.com/science/article/pii/037847547690032X", acknowledgement = ack-nhfb, ajournal = "Math. Comput. Simul.", fjournal = "Mathematics and Computers in Simulation", journal-URL = "https://www.sciencedirect.com/science/journal/03784754", } @Article{Paul:1976:SEF, author = "George Paul and M. Wayne Wilson", title = "Should the Elementary Function Library Be Incorporated Into Computer Instruction Sets?", journal = j-TOMS, volume = "2", number = "2", pages = "132--142", month = jun, year = "1976", CODEN = "ACMSCU", DOI = "https://doi.org/10.1145/355681.355684", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Sat Aug 27 00:30:21 1994", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib", abstract = "The question whether elementary mathematical function library routines should be incorporated into a computer system as part of the machine instruction set is discussed. The prime issues affecting such a decision are accuracy, substitution, and user expectations, Support of the COMPLEX arithmetic data type is also a complicating factor. These issues are discussed and conclusions are drawn.", acknowledgement = ack-nhfb, ajournal = "ACM Trans. Math. Softw.", fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", keywords = "computer instruction sets; elementary mathematical functions; hardware function evaluation; mathematical function libraries; programming languages", } @Article{Pike:1976:RIB, author = "Malcolm C. Pike and Jennie SooHoo and N. E. Bosten", title = "Remark on {``Algorithm 179: Incomplete Beta Ratio [S14]''}", journal = j-TOMS, volume = "2", number = "2", pages = "207--208", month = jun, year = "1976", CODEN = "ACMSCU", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Sat Jul 05 16:45:39 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/cacm1960.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "See remark \cite{Ludwig:1963:AIB,Bosten:1974:RAI}.", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", } @Article{Pomeranz:1976:REC, author = "J. Pomeranz", title = "Remark on ``{Algorithm 487: Exact Cumulative Distribution of the Kolmogorov--Smirnov Statistic for Small Samples [S14]}''", journal = j-TOMS, volume = "2", number = "1", pages = "111--111", month = mar, year = "1976", CODEN = "ACMSCU", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Tue Feb 06 05:28:05 1996", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "See \cite{Pomeranz:1974:AAE}.", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", } @Article{Redding:1976:CPC, author = "R. W. Redding and W. P. Latham", title = "On the calculation of the parabolic cylinder functions. {II}. {The} function {$ V(a, x) $}", journal = j-J-COMPUT-PHYS, volume = "20", number = "2", pages = "256--258", month = feb, year = "1976", CODEN = "JCTPAH", DOI = "https://doi.org/10.1016/0021-9991(76)90071-1", ISSN = "0021-9991 (print), 1090-2716 (electronic)", ISSN-L = "0021-9991", bibdate = "Sun Jan 1 09:15:20 MST 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputphys1970.bib", URL = "http://www.sciencedirect.com/science/article/pii/0021999176900711", acknowledgement = ack-nhfb, fjournal = "Journal of Computational Physics", journal-URL = "http://www.sciencedirect.com/science/journal/00219991/", } @Article{Salamin:1976:CUA, author = "Eugene Salamin", title = "Computation of $ \pi $ Using Arithmetic-Geometric Mean", journal = j-MATH-COMPUT, volume = "30", number = "135", pages = "565--570", month = jul, year = "1976", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", classcodes = "B0290D (Functional analysis); C4120 (Functional analysis)", corpsource = "Charles Stark Draper Lab., Cambridge, MA, USA", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "arithmetic geometric mean; convergence; elliptic integrals; error analysis; fast Fourier transform multiplication; function evaluation; Landen's; Legendre's relation; numerical computation of pi; transformation", remark = "Fullerton: A quadratically convergent algorithm.", treatment = "A Application; T Theoretical or Mathematical", } @InCollection{Salimov:1976:OCE, author = "F. I. Salimov", booktitle = "Probabilistic methods and cybernetics", title = "The organization of calculations of elementary functions into tables. ({Russian})", volume = "12--13", publisher = "Kazan University", address = "Kazan, USSR", pages = "77--90", year = "1976", MRclass = "65A05", MRnumber = "58 \#31706", bibdate = "Sat Jan 11 17:44:01 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, language = "Russian", } @Article{Schett:1976:PTS, author = "Alois Schett", title = "Properties of the {Taylor} series expansion coefficients of the {Jacobian} elliptic functions", journal = j-MATH-COMPUT, volume = "30", number = "133", pages = "143--147", month = jan, year = "1976", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", classcodes = "B0290P (Differential equations); C4170 (Differential equations)", corpsource = "CENS, Gif-sur-Yvette, France", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "differential equations; Jacobian elliptic functions; randomisation distributions; series (mathematics); Taylor series expansion", remark = "Fullerton: The first several coefficients are tabulated.", treatment = "T Theoretical or Mathematical", } @Article{Schonfelder:1976:PSF, author = "J. L. Schonfelder", title = "The Production of Special Function Routines for a Multi-Machine Library", journal = j-SPE, volume = "6", number = "1", pages = "71--82", month = jan # "\slash " # mar, year = "1976", CODEN = "SPEXBL", ISSN = "0038-0644 (print), 1097-024X (electronic)", ISSN-L = "0038-0644", bibdate = "Sat May 31 13:36:16 MDT 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Software---Practice and Experience", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)1097-024X", remark = "Fullerton: The design of transportable routines for the NAG library is discussed.", } @Article{Schulten:1976:REC, author = "K. Schulten and R. G. Gordon", title = "Recursive Evaluation of $ 3 j $ and $ 6 j $ Coefficients", journal = j-COMP-PHYS-COMM, volume = "11", number = "2", pages = "269--278", month = mar # "\slash " # may, year = "1976", CODEN = "CPHCBZ", DOI = "https://doi.org/10.1016/0010-4655(76)90058-8", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", bibdate = "Sat Oct 30 10:32:44 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/compphyscomm1970.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Computer Physics Communications", journal-URL = "http://www.sciencedirect.com/science/journal/00104655", } @Article{Shanks:1976:TER, author = "D. Shanks", title = "Table errata: {``Regular continued fractions for $ \pi $ and $ \gamma $'', (Math. Comp. {\bf 25} (1971), 403); ``Rational approximations to $ \pi $'' (ibid. {\bf 25} (1971), 387--392) by K. Y. Choong, D. E. Daykin and C. R. Rathbone}", journal = j-MATH-COMPUT, volume = "30", number = "134", pages = "381--381", year = "1976", CODEN = "MCMPAF", DOI = "https://doi.org/10.1090/S0025-5718-1976-0386215-4", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "65A05 (10-04 10F20)", MRnumber = "0386215 (52 \#7073)", bibdate = "Wed Jan 14 13:22:34 2015", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/pi.bib", URL = "http://www.ams.org/journals/mcom/1976-30-134/S0025-5718-1976-0386215-4", acknowledgement = ack-nhfb, fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", remark = "The second paper in the title is actually a review of a report containing table of partial quotients for a simple continued fraction for $ \pi $.", } @Article{Sheorey:1976:DCE, author = "V. B. Sheorey", title = "Double {Chebyshev} Expansions for Wave Functions", journal = j-COMP-PHYS-COMM, volume = "12", number = "2", pages = "125--134", month = nov, year = "1976", CODEN = "CPHCBZ", DOI = "https://doi.org/10.1016/0010-4655(76)90061-8", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", bibdate = "Sat Oct 30 10:38:43 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/compphyscomm1970.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/0010465576900618", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Computer Physics Communications", journal-URL = "http://www.sciencedirect.com/science/journal/00104655", } @Article{Siemieniuch:1976:PCR, author = "J. L. Siemieniuch", title = "Properties of certain rational approximations to $ e^{-z} $", journal = j-BIT, volume = "16", number = "2", pages = "172--191", month = jun, year = "1976", CODEN = "BITTEL, NBITAB", DOI = "https://doi.org/10.1007/BF01931369", ISSN = "0006-3835 (print), 1572-9125 (electronic)", ISSN-L = "0006-3835", bibdate = "Wed Jan 4 18:52:14 MST 2006", bibsource = "http://springerlink.metapress.com/openurl.asp?genre=issue&issn=0006-3835&volume=16&issue=2; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.springerlink.com/openurl.asp?genre=article&issn=0006-3835&volume=16&issue=2&spage=172", acknowledgement = ack-nhfb, fjournal = "BIT (Nordisk tidskrift for informationsbehandling)", journal-URL = "http://link.springer.com/journal/10543", keywords = "elefunt; elementary functions", } @Article{Stegun:1976:ACM, author = "I. A. Stegun and R. Zucker", title = "Automatic Computing Methods for Special Functions. {Part III}. {The} Sine, Cosine, Exponential Integrals, and Related Functions", journal = j-J-RES-NATL-BUR-STAND-1934, volume = "80B", number = "2", pages = "291--311", month = apr, year = "1976", ISSN = "0091-0635", bibdate = "Sat Oct 30 11:07:19 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Journal of Research of the National Bureau of Standards (1934)", journal-URL = "https://www.nist.gov/nist-research-library/journal-research-nist/past-papers", remark = "Fullerton: Adjustable double precision FORTRAN. routines for $ \operatorname {Si} $, $ \operatorname {Ci} $, $ \operatorname {Ei} $, $ \operatorname {Shi} $, and $ \operatorname {Chi} $.", } @Article{Temme:1976:NEO, author = "Nico M. Temme", title = "On the numerical evaluation of the ordinary {Bessel} function of the second kind", journal = j-J-COMPUT-PHYS, volume = "21", number = "3", pages = "343--350", month = jul, year = "1976", CODEN = "JCTPAH", DOI = "https://doi.org/10.1016/0021-9991(76)90032-2", ISSN = "0021-9991 (print), 1090-2716 (electronic)", ISSN-L = "0021-9991", bibdate = "Sun Jan 1 09:15:21 MST 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputphys1970.bib", URL = "http://www.sciencedirect.com/science/article/pii/0021999176900322", acknowledgement = ack-nhfb, fjournal = "Journal of Computational Physics", journal-URL = "http://www.sciencedirect.com/science/journal/00219991/", } @Article{Tugov:1976:MCA, author = "I. I. Tugov and Yu. L. Shitkov", title = "A Method of Calculating the {Appell} Functions $ {F}_a(\alpha; \beta, \beta '; \gamma; x, y) $", journal = j-USSR-COMP-MATH-MATH-PHYS, volume = "16", number = "6", pages = "1587--1590", year = "1976", CODEN = "CMMPA9", ISSN = "0041-5553, 0502-9902", ISSN-L = "0041-5553", bibdate = "Sat Oct 30 11:38:05 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "U.S.S.R. Computational Mathematics and Mathematical Physics", journal-URL = "http://www.sciencedirect.com/science/journal/00415553", remark = "Fullerton: [English] translation of Russian-language Zhurnal Vychislitel'noi Matematikii Matemancheskoi Fiziki (1976).", } @InProceedings{Aird:1977:IFC, author = "T. J. Aird", title = "The {IMSL Fortran} converter: an approach to solving portability problems", crossref = "Cowell:1977:PNS", pages = "368--388", year = "1977", DOI = "https://doi.org/10.1007/3-540-08446-0_49", bibdate = "Thu Dec 11 15:15:52 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fortran1.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, } @Article{Alexander:1977:SRR, author = "V. L. Alexander", title = "Square Root Routine", journal = j-IBM-TDB, volume = "20", number = "3", pages = "1222", month = aug, year = "1977", CODEN = "IBMTAA", ISSN = "0018-8689", bibdate = "Thu Sep 1 10:15:41 1994", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nj, fjournal = "IBM Technical Disclosure Bulletin", } @Article{Amos:1977:ACS, author = "D. E. Amos and S. L. Daniel and M. K. Weston", title = "{Algorithm 511}: {CDC} 6600 Subroutines {IBESS} and {JBESS} for {Bessel} Functions {$ I_\nu (x) $} and {$ J_\nu (x) $}, {$ x \ge 0, \nu \ge 0 $} [{S18}]", journal = j-TOMS, volume = "3", number = "1", pages = "93--95", month = mar, year = "1977", CODEN = "ACMSCU", DOI = "https://doi.org/10.1145/355719.355727", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Thu Apr 29 15:14:12 1999", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "See erratum \cite{Amos:1978:ECS}.", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", } @Article{Amos:1977:CSI, author = "D. E. Amos and S. L. Daniel and M. K. Weston", title = "{CDC} 6600 Subroutines {IBESS} and {JBESS} for {Bessel} Functions {$ I_\nu (x) $} and {$ J_\nu (x) $}, {$ x \ge 0, \nu \ge 0 $}", journal = j-TOMS, volume = "3", number = "1", pages = "76--92", month = mar, year = "1977", CODEN = "ACMSCU", DOI = "https://doi.org/10.1145/355719.355726", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", MRclass = "65D20", MRnumber = "55 #6781", bibdate = "Tue Sep 06 19:20:02 1994", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", reviewer = "Sven-{\AA}ke Gustafson", } @Article{Ardill:1977:BFC, author = "R. W. B. Ardill and K. J. M. Moriarty", title = "The {Bessel} Functions {$ J_0 $} and {$ J_1 $} of Complex Argument", journal = j-COMP-PHYS-COMM, volume = "13", number = "1", pages = "17--24", month = may, year = "1977", CODEN = "CPHCBZ", DOI = "https://doi.org/10.1016/0010-4655(77)90023-6", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", bibdate = "Fri Oct 29 21:19:09 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/compphyscomm1970.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/0010465577900236", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Computer Physics Communications", journal-URL = "http://www.sciencedirect.com/science/journal/00104655", } @Manual{Arnold:1977:SMF, author = "Mark G. Arnold", title = "{SCELBAL} Mathematical Functions Supplement (8008\slash 8080)", organization = "Scelbi Computer Consulting. Inc.", address = "1322 Rear --- Boston Post Road, Milford, CT 0646, USA", pages = "31", year = "1977", bibdate = "Fri Dec 01 16:04:29 2023", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.scelbi.com/files/docs/scelbal/SCELBAL%20Mathematical%20Functions%20Supplement.pdf", acknowledgement = ack-nhfb, remark = "Brief description of implementations of cos, sin, exp, log, and atn functions.", } @InProceedings{Bentley:1977:EPW, author = "J. Bentley and B. Ford", title = "On the enhancement of portability within the {NAG} project --- a statistical survey", crossref = "Cowell:1977:PNS", pages = "505--528", year = "1977", DOI = "https://doi.org/10.1007/3-540-08446-0_57", bibdate = "Thu Dec 11 15:15:52 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, } @InProceedings{Boyle:1977:MST, author = "James M. Boyle", title = "Mathematical software transportability systems --- have the variations a theme?", crossref = "Cowell:1977:PNS", pages = "305--360", year = "1977", DOI = "https://doi.org/10.1007/3-540-08446-0_47", bibdate = "Thu Dec 11 15:15:52 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, } @Book{Brezinski:1977:ACA, author = "Claude Brezinski", title = "Acc{\'e}l{\'e}ration de la convergence en analyse num{\'e}rique. ({French}) [{Convergence} acceleration in numerical analysis]", publisher = pub-SV, address = pub-SV:adr, pages = "313", year = "1977", ISBN = "0-387-08241-7, 3-540-08241-7", ISBN-13 = "978-0-387-08241-7, 978-3-540-08241-5", LCCN = "????", bibdate = "Thu Dec 1 10:17:17 MST 2011", bibsource = "carmin.sudoc.abes.fr:210/ABES-Z39-PUBLIC; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, keywords = "convergence acceleration", language = "French", } @InProceedings{Brown:1977:FPM, author = "W. S. Brown and A. D. Hall", title = "{Fortran} portability via models and tools", crossref = "Cowell:1977:PNS", pages = "158--164", year = "1977", DOI = "https://doi.org/10.1007/3-540-08446-0_41", bibdate = "Thu Dec 11 15:15:52 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fortran1.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, } @Article{Carlson:1977:EIF, author = "B. C. Carlson", title = "Elliptic Integrals of the First Kind", journal = j-SIAM-J-MATH-ANA, volume = "8", number = "2", pages = "231--242", month = "????", year = "1977", CODEN = "SJMAAH", ISSN = "0036-1410 (print), 1095-7154 (electronic)", ISSN-L = "0036-1410", MRnumber = "MR 0430341 (55:3346)", bibdate = "Fri Oct 29 22:03:28 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "SIAM Journal on Mathematical Analysis", journal-URL = "http://epubs.siam.org/sima", } @Book{Carlson:1977:SFA, author = "Bille Chandler Carlson", title = "Special Functions of Applied Mathematics", publisher = pub-ACADEMIC, address = pub-ACADEMIC:adr, pages = "xv + 335", year = "1977", ISBN = "0-12-160150-1", ISBN-13 = "978-0-12-160150-8", LCCN = "QA351 .C32", bibdate = "Fri Jan 22 10:33:57 MST 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; z3950.loc.gov:7090/Voyager", acknowledgement = ack-nhfb, subject = "Functions, Special", } @Article{Cody:1977:CRF, author = "W. J. Cody and Rose M. Motley and L. Wayne Fullerton", title = "The Computation of Real Fractional Order {Bessel} Functions of the Second Kind", journal = j-TOMS, volume = "3", number = "3", pages = "232--239", month = sep, year = "1977", CODEN = "ACMSCU", DOI = "https://doi.org/10.1145/355744.355747", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Tue Sep 20 18:24:22 1994", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/c/cody-william-j.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", } @InProceedings{Cody:1977:MPN, author = "William J. {Cody, Jr.}", title = "Machine parameters for numerical analysis", crossref = "Cowell:1977:PNS", pages = "49--67", year = "1977", DOI = "https://doi.org/10.1007/3-540-08446-0_35", bibdate = "Thu Dec 11 15:15:52 2025", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/c/cody-william-j.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, } @InProceedings{Dahlstrand:1977:SPT, author = "Ingemar Dahlstrand", title = "A study of portability in technical and scientific computing", crossref = "Cowell:1977:PNS", pages = "529--539", year = "1977", DOI = "https://doi.org/10.1007/3-540-08446-0_58", bibdate = "Thu Dec 11 15:15:52 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, } @Article{Danilcenko:1977:ETC, author = "L. S. Danil'{\v{c}}enko", title = "An efficient technique for the construction of rational approximations of elementary functions. ({Russian}) Optimization of computations (approximation and minimization of functions)", journal = "Akad. Nauk Ukrain. SSR Inst. Kibernet. Preprint", volume = "18", pages = "17--21", year = "1977", MRclass = "65D15", MRnumber = "80b:65016", bibdate = "Sat Jan 11 17:44:01 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, language = "Russian", } @Article{deAPMartins:1977:DSB, author = "Pedro {de A.P.Martins}", title = "Determination of spherical {Bessel} functions of an order larger than the argument", journal = j-J-COMPUT-PHYS, volume = "25", number = "2", pages = "194--198", month = oct, year = "1977", CODEN = "JCTPAH", DOI = "https://doi.org/10.1016/0021-9991(77)90021-3", ISSN = "0021-9991 (print), 1090-2716 (electronic)", ISSN-L = "0021-9991", bibdate = "Sun Jan 1 09:15:26 MST 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputphys1970.bib", URL = "http://www.sciencedirect.com/science/article/pii/0021999177900213", acknowledgement = ack-nhfb, fjournal = "Journal of Computational Physics", journal-URL = "http://www.sciencedirect.com/science/journal/00219991/", xxtitle = "Determination of spherical {Bessel}'s functions of an order larger than the argument", } @InProceedings{Delves:1977:ALN, author = "L. M. Delves", title = "{Algol 68} as a language for numerical software", crossref = "Cowell:1977:PNS", pages = "95--126", year = "1977", DOI = "https://doi.org/10.1007/3-540-08446-0_38", bibdate = "Thu Dec 11 15:15:52 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, } @Article{Derenzo:1977:AHC, author = "Stephen E. Derenzo", title = "Approximations for Hand Calculators Using Small Integer Coefficients", journal = j-MATH-COMPUT, volume = "31", number = "137", pages = "214--222", month = jan, year = "1977", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "garbo.uwasa.fi:/pc/doc-soft/fpbiblio.txt; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp1970.bib; JSTOR database", acknowledgement = ack-nhfb # " and " # ack-nj, classcodes = "B0290D (Functional analysis); B0290F (Interpolation and function approximation); C4120 (Functional analysis); C4130 (Interpolation and function approximation); C7310 (Mathematics computing)", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "approximations; function approximation; function evaluation; hand calculators; programmable calculators; small integer coefficients", treatment = "A Application; T Theoretical or Mathematical", } @TechReport{DiDonato:1977:CPP, author = "Armido R. DiDonato and R. K. Hageman", title = "Computation of the percentage points of the chi-square distribution", type = "Report", number = "NSWC/DL TR-3569", institution = "Naval Surface Weapons Center", address = "Dahlgren, VA, USA", pages = "200", year = "1977", bibdate = "Mon Jun 03 12:24:32 2024", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "https://apps.dtic.mil/sti/pdfs/ADA043997.pdf", acknowledgement = ack-nhfb, } @Article{Dijkstra:1977:CFE, author = "D. Dijkstra", title = "A continued fraction expansion for a generalization of {Dawson}'s integral", journal = j-MATH-COMPUT, volume = "31", number = "138", pages = "503--510", month = apr, year = "1977", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", classcodes = "C4110 (Error analysis in numerical methods); C4120 (Functional analysis); C4160 (Numerical integration and differentiation)", corpsource = "Dept. of Math., Tech. Univ. Twente, Enschede, Netherlands", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "confluent hypergeometric; continued fraction expansion; Dawson's; error analysis; function; function evaluation; generalization; integral; integration; truncation error", remark = "Fullerton: An expansion for $ F(p, x) = e^{-x^2} \int_0^x e^{t^2} \, d t $ is given.", treatment = "T Theoretical or Mathematical", } @InProceedings{Dritz:1977:MPR, author = "Kenneth W. Dritz", title = "Multiple program realizations using the {TAMPR} system", crossref = "Cowell:1977:PNS", pages = "405--423", year = "1977", DOI = "https://doi.org/10.1007/3-540-08446-0_51", bibdate = "Thu Dec 11 15:15:52 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, } @InProceedings{DuCroz:1977:APW, author = "J. J. {Du Croz} and S. J. Hague and J. L. Siemieniuch", title = "Aids to portability within the {NAG} project", crossref = "Cowell:1977:PNS", pages = "389--404", year = "1977", DOI = "https://doi.org/10.1007/3-540-08446-0_50", bibdate = "Thu Dec 11 15:15:52 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, } @TechReport{Dzjadyk:1977:TPA, author = "V. K. Dzjadyk and S. F. Karpenko", title = "Tables of polynomials for the approximate solution of elementary functions. ({Russian})", number = "28", institution = "Akad. Nauk Ukrain. SSR Inst. Mat. Preprint", pages = "28", year = "1977", MRclass = "65A05 (65D20)", MRnumber = "58 \#19016", bibdate = "Sat Jan 11 17:44:01 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, language = "Russian", } @TechReport{Dzyadyk:1977:OPM, author = "V. K. Dzjadyk and Z. V. Zarickaja and S. F. Karpenko and N. F. Kononova", title = "{{\cyr Ob {\`e}ffektivnom priblizhenii mnogochlenami {\`e}lementarnykh funktsi{\u\i}.}} ({Russian}) [Efficient approximation by polynomials of elementary functions]", type = "Preprint", number = "IM-77-21", institution = "Akad. Nauk Ukrain. SSR Inst. Mat.", address = "Kiev, USSR", pages = "42", year = "1977", MRclass = "65L99", MRnumber = "57 \#11075", MRreviewer = "B. D. Donevski", bibdate = "Sat Jan 11 17:44:01 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, language = "Russian", } @Article{Eckhardt:1977:RAW, author = "Ulrich Eckhardt", title = "A rational approximation to {Weierstrass}' elliptic function. {II}. {The} lemniscatic case", journal = j-COMPUTING, volume = "18", number = "4", pages = "341--349", year = "1977", CODEN = "CMPTA2", ISSN = "0010-485X (print), 1436-5057 (electronic)", ISSN-L = "0010-485X", bibdate = "Tue Jan 2 17:40:53 MST 2001", bibsource = "http://springerlink.metapress.com/openurl.asp?genre=journal&issn=0010-485X; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; INSPEC Axiom database (1968--date)", acknowledgement = ack-nhfb, affiliation = "Zentralinst. f{\"u}r Angewandte Math., Julich, West Germany", citedby = "Fullerton:1980:BEM", classification = "C4130", description = "function approximation", fjournal = "Computing", journal-URL = "http://link.springer.com/journal/607", keywords = "elliptic functions; lemniscatic case; rational approximation; Weierstrass elliptic function", remark = "Fullerton: Weierstrass' function and its derivative are approximated in the complex plane to 16 S.", } @Article{Egbert:1977:PCAa, author = "W. E. Egbert", title = "Personal calculator algorithms. {I}. Square roots", journal = j-HEWLETT-PACKARD-J, volume = "28", number = "9", pages = "22--23", month = may, year = "1977", CODEN = "HPJOAX", ISSN = "0018-1153", bibdate = "Tue Mar 25 14:12:15 MST 1997", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nj # " and " # ack-nhfb, classcodes = "C5420 (Mainframes and minicomputers); C7310 (Mathematics computing)", fjournal = "Hewlett-Packard Journal: technical information from the laboratories of Hewlett-Packard Company", keywords = "electronic calculators; HP personal calculator; square root algorithm", treatment = "A Application; T Theoretical or Mathematical", xxpages = "22--24", } @Article{Ercegovac:1977:GHO, author = "Milo{\v{s}} D. Ercegovac", title = "A General Hardware-Oriented Method for Evaluation of Functions and Computations in a Digital Computer", journal = j-IEEE-TRANS-COMPUT, volume = "C-26", number = "7", pages = "667--680", month = jul, year = "1977", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/TC.1977.1674900", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Mon Jul 11 21:56:56 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput1970.bib", URL = "https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=1674900", abstract = "A parallel computational method, amenable for efficient hardware-level implementation, is described. It provides a simple and fast algorithm for the evaluation of polynomials, certain rational functions and arithmetic expressions, solving a class of systems of linear equations, or performing the basic arithmetic operations in a fixed-point number representation system. The time required to perform the computation is of the order of $m$ carry-free addition operations, $m$ being the number of digits in the solution. In particular, the method is suitable for fast evaluation of mathematical functions in hardware.", acknowledgement = ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", keywords = "Arithmetic expressions; digital computer arithmetic; E-method; evaluation of real-valued functions; fixed-point representation; hardware-level implementation; integral powers; linear systems; on-line algorithms; parallel computation; polynomials; rational functions; redundant number systems", } @InProceedings{Ford:1977:PCP, author = "Brian Ford", title = "Preparing conventions for parameters for transportable numerical software", crossref = "Cowell:1977:PNS", pages = "68--91", year = "1977", DOI = "https://doi.org/10.1007/3-540-08446-0_36", bibdate = "Thu Dec 11 15:15:52 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, } @TechReport{Fox:1977:PMS, author = "P. A. Fox and A. D. Hall and N. L. Schryer", title = "The {PORT} Mathematical Subroutine Library", type = "Computing Science Technical Report", number = "47", institution = inst-ATT-BELL, address = inst-ATT-BELL:adr, pages = "ii + 50", day = "22", month = mar, year = "1977", bibdate = "Fri Sep 01 09:08:27 2023", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fortran1.bib; https://www.math.utah.edu/pub/tex/bib/unix.bib", URL = "http://history.siam.org/%5C/sup/Fox_bell_subroutine.pdf", abstract = "The development at Bell Laboratories of PORT, a library of portable Fortran programs for numerical computation, is discussed.\par Portability is achieved by careful language specification, together with the key technique of specifying computer classes by means of pre-defined machine constants.\par The library is built around an automatic error-handling facility and a dynamic storage allocation scheme, both of which are implemented portably. These, together with the modular structure of the library, lead to simplified calling sequences and ease of use.", acknowledgement = ack-nhfb, remark = "May 1977 revision of version of September 1976.", tableofcontents = "Part 1: Description \\ Part 2: Utility program listings: \\ Machine constants \\ Error handling \\ Stack allocation", } @InProceedings{Fox:1977:PPM, author = "Phyllis A. Fox", title = "{Port} --- A portable mathematical subroutine library", crossref = "Cowell:1977:PNS", pages = "165--177", year = "1977", DOI = "https://doi.org/10.1007/3-540-08446-0_42", bibdate = "Thu Dec 11 15:15:52 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fortran1.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, } @InProceedings{Fullerton:1977:PSF, author = "L. W. Fullerton", title = "Portable Special Function Routines", crossref = "Cowell:1977:PNS", pages = "452--483", year = "1977", bibdate = "Sat Oct 30 06:40:00 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", } @Article{Gautschi:1977:ACC, author = "Walter Gautschi", title = "Anomalous convergence of a continued fraction for ratios of {Kummer} functions", journal = j-MATH-COMPUT, volume = "31", number = "140", pages = "994--999", month = oct, year = "1977", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", classcodes = "C1120 (Mathematical analysis); C4170 (Differential equations)", corpsource = "Dept. of Computer Sci., Purdue Univ., Lafayette, IN, USA", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "apparent; Bessel functions; continued fraction; convergence; differential equations; gamma functions; Kummer functions; wrong limit", treatment = "T Theoretical or Mathematical", } @Article{Gautschi:1977:ERI, author = "Walter Gautschi", title = "Evaluation of Repeated Integrals of the Coerror Function", journal = j-TOMS, volume = "3", number = "3", pages = "240--252", month = sep, year = "1977", CODEN = "ACMSCU", DOI = "https://doi.org/10.1145/355744.355748", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Sat Aug 27 22:26:34 1994", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", remark = "Fullerton: An arbitrary-accuracy method for evaluating $ i^n \erfc (x) $ is given.", } @Article{Harris:1977:CAT, author = "F. E. Harris", title = "Convergence acceleration technique for lattice sums arising in electronic-structure studies of crystalline solids", journal = j-J-MATH-PHYS, volume = "18", number = "12", pages = "2377--2381", month = dec, year = "1977", CODEN = "JMAPAQ", ISSN = "0022-2488 (print), 1089-7658 (electronic), 1527-2427", ISSN-L = "0022-2488", bibdate = "Fri Jan 2 14:59:17 MST 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, classification = "A0260 (Numerical approximation and analysis); A6150L (Crystal binding)", corpsource = "Dept. of Chem., Univ. of Hawaii, Honolulu, HI, USA", fjournal = "Journal of Mathematical Physics", journal-URL = "http://jmp.aip.org/", keywords = "convergence acceleration; convergence of numerical methods; crystalline solids; electronic structure; Laplace transform; Laplace transforms; lattice energy; lattice sums; Poisson's summation formula", pubcountry = "USA", treatment = "T Theoretical or Mathematical", } @InProceedings{Hemker:1977:CTA, author = "Pieter W. Hemker", title = "Criteria for transportable {Algol} libraries", crossref = "Cowell:1977:PNS", pages = "145--157", year = "1977", DOI = "https://doi.org/10.1007/3-540-08446-0_40", bibdate = "Thu Dec 11 15:15:52 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, } @Book{Henrici:1977:ACC, author = "Peter Henrici", title = "Applied and Computational Complex Analysis. Volume 2: Special Functions, Integral Transforms, Asymptotics, Continued Fractions", publisher = pub-WILEY, address = pub-WILEY:adr, pages = "ix + 662", year = "1977", ISBN = "0-471-01525-3", ISBN-13 = "978-0-471-01525-3", LCCN = "QA331 .H453 1974", MRclass = "30-02 (33-02 44A10)", MRnumber = "0453984", MRreviewer = "A. E. Heins", bibdate = "Mon Jan 28 07:08:10 2019", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/h/henrici-peter.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, author-dates = "Peter Karl Henrici (13 September 1923--13 March 1987)", tableofcontents = "8: Infinite Products / 1--74 \\ 9: Ordinary Differential Equations / 75--194 \\ 10: Integral Transforms / 195--350 \\ 11: Asymptotic Methods / 351--472 \\ 12: Continued Fractions / 473--641 \\ Bibliography / 642--649 \\ Appendix [Some additional problems for Volume 1] / 650--651 \\ Index / 653--662", } @Book{Henrici:1977:ART, author = "Peter Henrici", title = "{Analytische Rechenverfahren f{\"u}r den Taschenrechner HP-25}. ({German}) [{Analytical} Calculations for the {HP-25} Calculator]", publisher = "Oldenbourg", address = "M{\"u}nchen, West Germany", pages = "280", year = "1977", ISBN = "0-471-02938-6", ISBN-13 = "978-0-471-02938-0", LCCN = "????", bibdate = "Mon Jan 28 08:25:06 MST 2019", bibsource = "fsz3950.oclc.org:210/WorldCat; https://www.math.utah.edu/pub/bibnet/authors/h/henrici-peter.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, author-dates = "1923--1987", language = "German", subject = "Numerische Mathematik; Programmierung ; Kleinrechner", tableofcontents = "Einleitung / 9 \\ Teil I: Zahlentheorie / 13 \\ Primfaktorzerlegung / 14 \\ Euklidscher Algorithmus / 18 \\ Rationale Binomialkoeffizienten / 22 \\ Kettenbruchdarstellungen reeller Zahlen / 28 \\ Genaue Kettenbruchdarstellungen von quadratischen Irrationalit{\"a}ten / 33 \\ Teil II: Iteration / 39 \\ Iteration / 40 \\ Iteration mit Aitken-Beschleunigung / 45 \\ Aitken--Steffensen-Iteration / 50 \\ Newton-Iteration f{\"u}r Wurzeln komplexer Zahlen / 55 \\ Teil III: Polynome / 61 \\ Der Horner-Algorithmus 6 / 2 \\ Das Newton-Verfahren bei Polynomen / 66 \\ Bernoulli-Verfahren: Eine reelle dominante Nullstelle / 71 \\ Bernoulli-Verfahren: Zwei konjugiert komplexe dominante Nullstellen / 76 \\ Der Quotienten--Differenzen-Algorithmus / 82 \\ Der Routh-Algorithmus / 87 \\ Der Schur--Cohn-Algorithmus I / 92 \\ Der Schur--Cohn-Algorithmus II / 97 \\ Teil IV: Potenzreihen / 101 \\ Reziproke Potenzreihe / 102 \\ Potenz einer Potenzreihe / 111 \\ Exponentiation einer Potenzreihe / 118 \\ Teil V: Integration / 123 \\ Numerische Integration mit Schrittverfeinerung / 124 \\ Der Romberg-Algorithmus / 129 \\ Die Planasche Summationsformel / 136 \\ Eine Differentialgleichung erster Ordnung: Trapezregel / 143 \\ Autonome Differentialgleichung zweiter Ordnung ohne erste Ableitung / 150 \\ Lineare Differentialgleichung zweiter Ordnung / 155 \\ Teil VI: Spezielle Konstanten und Funktionen / 161 \\ Log-Arcsinus-Algorithmus / 162 \\ Die Gamma-Funktion / 167 \\ Unvollst{\"a}ndige Gamma-Funktion / 174 \\ Die Fehlerfunktion / 181 \\ Vollst{\"a}ndige elliptische Integrale / 187 \\ Besselfunktionen ganzzahliger Ordnung / 191 \\ Besselfunktionen beliebiger Ordnung / 196 \\ Besselfunktionen: Asymptotische Reihe / 201 \\ Die Riemannsche Zetafunktion auf der kritischen Geraden / 207 \\ Stichwortverzeichnis / 213", } @Book{Higgins:1977:CBP, author = "John Rowland Higgins", title = "Completeness and basis properties of sets of special functions", publisher = pub-CAMBRIDGE, address = pub-CAMBRIDGE:adr, pages = "x + 134", year = "1977", ISBN = "0-521-21376-2 (hardcover), 0-521-60488-5 (paperback)", ISBN-13 = "978-0-521-21376-9 (hardcover), 978-0-521-60488-8 (paperback)", LCCN = "????", bibdate = "Sat Oct 30 16:52:55 MDT 2010", bibsource = "carmin.sudoc.abes.fr:210/ABES-Z39-PUBLIC; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, } @Article{Hill:1977:AIB, author = "G. W. Hill", title = "{Algorithm 518}: Incomplete {Bessel} Function {$ I_0 $}. {The von Mises} Distribution [{S14}]", journal = j-TOMS, volume = "3", number = "3", pages = "279--284", month = sep, year = "1977", CODEN = "ACMSCU", DOI = "https://doi.org/10.1145/355744.355753", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Sat Oct 24 15:46:06 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", remark = "Fullerton: Adjustable-accuracy 50-statement FORTRAN subprogram.", } @Article{Hinden:1977:PAR, author = "Harvey J. Hinden", title = "Phi Again: a Relationship Between the Golden Ratio and the Limit of a Ratio of Modified {Bessel} Functions", journal = j-FIB-QUART, volume = "15", number = "2", pages = "112, 152", month = apr, year = "1977", CODEN = "FIBQAU", ISSN = "0015-0517", ISSN-L = "0015-0517", bibdate = "Thu Oct 20 17:59:17 MDT 2011", bibsource = "http://www.fq.math.ca/15-2.html; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fibquart.bib", URL = "http://www.fq.math.ca/Scanned/15-2/hinden-a.pdf", acknowledgement = ack-nhfb, ajournal = "Fib. Quart", fjournal = "The Fibonacci Quarterly. Official Organ of the Fibonacci Association", journal-URL = "http://www.fq.math.ca/", } @InProceedings{Hull:1977:SFP, author = "T. E. Hull", title = "Semantics of floating point arithmetic and elementary functions", crossref = "Cowell:1977:PNS", pages = "37--48", year = "1977", DOI = "https://doi.org/10.1007/3-540-08446-0_34", bibdate = "Thu Dec 11 15:15:52 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, } @Article{Ismail:1977:IRC, author = "Mourad E. H. Ismail", title = "Integral representations and complete monotonicity of various quotients of {Bessel} functions", journal = j-CAN-J-MATH, volume = "29", number = "??", pages = "1198--1207", month = "????", year = "1977", CODEN = "CJMAAB", DOI = "https://doi.org/10.4153/CJM-1977-119-5", ISSN = "0008-414X (print), 1496-4279 (electronic)", ISSN-L = "0008-414X", bibdate = "Sat Sep 10 15:38:50 MDT 2011", bibsource = "http://cms.math.ca/cjm/v29/; https://www.math.utah.edu/pub/tex/bib/canjmath1970.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Canadian Journal of Mathematics = Journal canadien de math{\'e}matiques", journal-URL = "http://cms.math.ca/cjm/", } @Article{Jansen:1977:RLF, author = "J. K. M. Jansen", title = "Remark on ``{Algorithm 259: Legendre Functions for Arguments Larger than One}''", journal = j-TOMS, volume = "3", number = "2", pages = "204--250", month = jun, year = "1977", CODEN = "ACMSCU", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Tue Feb 06 05:28:08 1996", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "See \cite{Gautschi:1965:ALF}.", acknowledgement = ack-nhfb, fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", } @InProceedings{Kemp:1977:WEF, author = "P. Kemp", title = "Writing the elementary function procedures for the {ALGOL68C} compiler", crossref = "Cowell:1977:PNS", pages = "127--144", year = "1977", DOI = "https://doi.org/10.1007/3-540-08446-0_39", bibdate = "Thu Dec 11 15:15:52 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, } @InProceedings{Krogh:1977:FFP, author = "Fred T. Krogh", title = "Features for {Fortran} portability", crossref = "Cowell:1977:PNS", pages = "361--367", year = "1977", DOI = "https://doi.org/10.1007/3-540-08446-0_48", bibdate = "Thu Dec 11 15:15:52 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fortran1.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, } @InProceedings{Lawson:1977:TNA, author = "C. L. Lawson and J. K. Reid", title = "Two numerical analysts' views on the {Draft Proposed ANS Fortran}", crossref = "Cowell:1977:PNS", pages = "257--268", year = "1977", DOI = "https://doi.org/10.1007/3-540-08446-0_44", bibdate = "Thu Dec 11 15:15:52 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fortran1.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, } @Book{Luke:1977:ACM, author = "Yudell L. Luke", title = "Algorithms for the Computation of Mathematical Functions", publisher = pub-ACADEMIC, address = pub-ACADEMIC:adr, pages = "xiii + 284", year = "1977", ISBN = "0-12-459940-0", ISBN-13 = "978-0-12-459940-6", LCCN = "QA351 .L7961", bibdate = "Wed Dec 15 10:38:19 1993", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/master.bib; https://www.math.utah.edu/pub/tex/bib/mathcw.bib", acknowledgement = ack-nhfb, tableofcontents = "Preface / xi \\ 1: Basic Formulas / 1 \\ 1.1 Introduction / 1 \\ 1.2 The Generalized Hypergeometric Function and the $G$-Function / 1 \\ 1.3 Expansion of $_pF_q(z)$ and $G^{q - r, 1}_{p + 1, q}(z)$, $r = 0$ or $r = 1$, in Series of Chebyshev Polynomials of the First Kind / 4 \\ 1.4 Efficient Evaluation of Series of Chebyshev Polynomials / 17 \\ 1.5 Rational Approximations for Generalized Hypergeometric Functions / 20 \\ 1.6 The Pad{\'e} Table / 27 \\ 1.7 Computations of and Checks on Coefficients and Tables / 29 \\ 1.8 Tables of the Functions $e^{-\zeta}$, and $e^{-\xi}$ / 35 \\ 2: Identification of Functions / 41 \\ 2.1 Introduction / 41 \\ 2.2 The Generalized Hypergeometric Function $_pF_q(z)$ / 41 \\ 2.3 The G-Function / 47 \\ 2.4 Miscellaneous Functions / 48 \\ 3: General Remarks on the Algorithms and Programs / 49 \\ 3.1 Introduction / 49 \\ 3.2 Precision and Complex Arithmetic / 49 \\ 4: Chebyshev Coefficients for $_2F_1(a.b;c;z)$ / 52 \\ 5: Coefficients for the Expansion of the Confluent Hypergeometric Function $_1F_1(a;c;z)$ in Ascending Series of Chebyshev Polynomials / 70 \\ 6: Chebyshev Coefficients for $_0F_1(c;z)$ / 77 \\ 7: Coefficients for the Expansion of $_1F_2(a;b,c;z)$ in Ascending Series of Chebyshev Polynomials / 82 \\ 8: Coefficients for the Expansion of the Confluent Hypergeometric Functions $U(a;c;z)$ and $_1F_1(a;c;-z)$ in Descending Series of Chebyshev Polynomials / 88 \\ 9: Coefficients for the Expansion of the Functions $G^{m,1}_{1,3}(z^2/4|^1_{a,b,c})$, $m = 3$ or $m = 2$, in Descending Series of Chebyshev Polynomials / 101 \\ 10: Differential and Integral Properties of Expansions in Series of Chebyshev Polynomials of the First Kind / 116 \\ 11: Expansion of Exponential Type Integrals in Series of Chebyshev Polynomials of the First Kind / 126 \\ 11.1 Introduction / 126 \\ 11.2 The Representation for $g(x)$ / 127 \\ 11.3 The Representation for $G(x)$ / 129 \\ 11.4 Exponential Type Integrals Involving Logarithms / 133 \\ 11.5 Numerical Examples / 135 \\ 11.6 Errata / 139 \\ 12: Conversion of a Power Series into a Series of Chebyshev Polynomials of the First Kind / 154 \\ 13: Rational Approximations for $_2F_1(a,b;c;-z)$ / 159 \\ 14: Pad{\'e} Approximations for $_2F_1(1,b;c;-z)$ / 174 \\ 15: Rational Approximations for $_1F_1(a;c;-z)$ / 182 \\ 16: Pad{\'e} Approximations for $_1F_1(1;c;-z)$ / 192 \\ 17: Rational Approximations for Bessel Functions of the First Kind / 203 \\ 18: Pad{\'e} Approximations for $I_{\nu + 1}(z)/I_\nu(z)$ / 220 \\ 19: Evaluation of Bessel Functions of the First Kind by Use of the Backward Recurrence Formula \\ 19.1 Introduction / 230 \\ 19.2 Backward Recurrence Schemata for $I_\nu(z)$ and $J_\nu(z)$ / 230 \\ 19.3 Numerical Examples / 240 \\ 19.4 Mathematical Description of Programs / 243 \\ 19.4.1 Evaluation of Functions Related to $I_{m + \nu}(z)$ and $J_{m + \nu}(z)$ / 243 \\ 19.4.2 Evaluation of Functions Related to $e^{-l}I_{m + \nu}(z)$ / 245 \\ 20: Rational Approximations for $z^aU(a;1 + a - b;z)$ / 252 \\ 21: Pad{\'e} Approximations for $z U(1;2-b;z)$ / 265 \\ Appendices \\ Bibliography / 280 \\ Notation Index / 281 \\ Subject Index / 283", wrongisbn = "0-12-459940-6", } @Article{Marsaglia:1977:SMG, author = "George Marsaglia", title = "The squeeze method for generating gamma variates", journal = j-COMPUT-MATH-APPL, volume = "3", number = "4", pages = "321--325", year = "1977", CODEN = "CMAPDK", DOI = "https://doi.org/10.1016/0898-1221(77)90089-X", ISSN = "0898-1221 (print), 1873-7668 (electronic)", ISSN-L = "0898-1221", MRclass = "65C10", MRnumber = "58 \#13613", bibdate = "Mon Oct 24 11:37:20 2011", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; MathSciNet database", ZMnumber = "0384.65005", abstract = "This paper describes an exact method for computer generation of random variables with a gamma distribution. The method is based on the Wilson--Hilferty transformation and an improvement on the rejection technique. The idea is to ``squeeze'' a target density between two functions, the top one easy to sample from, the bottom one easy to evaluate.", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Computers and Mathematics with Applications", journal-URL = "http://www.sciencedirect.com/science/journal/08981221", ZMclass = "*65C10 Random number generation 60E05 General theory of probability distributions", } @Article{McCarthy:1977:OAE, author = "D. P. McCarthy", title = "The optimal algorithm to evaluate $ x^n $ using elementary multiplication methods", journal = j-MATH-COMPUT, volume = "31", number = "137", pages = "251--256", month = jan, year = "1977", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, classcodes = "C4120 (Functional analysis); C4240 (Programming and algorithm theory)", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "computational complexity; function evaluation; optimal multiplication chains; symbolic algebraic manipulation", treatment = "T Theoretical or Mathematical", } @Article{Ng:1977:CAL, author = "E. W. Ng", title = "Computations and Applications of Linear Hypergeometric Transformations", journal = j-COMPUT-MATH-APPL, volume = "3", number = "1", pages = "65--70", year = "1977", CODEN = "CMAPDK", DOI = "https://doi.org/10.1016/0898-1221(77)90115-8", ISSN = "0898-1221 (print), 1873-7668 (electronic)", ISSN-L = "0898-1221", bibdate = "Sat Oct 30 09:27:55 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", abstract = "Linear transformations are well-known in the theory of hypergeometric functions. In this note, it is indicated, both by analyses and by supporting numerical experiments, how these transformations can be applied to the computation of Legendre's functions, the incomplete Beta function, and the variance-ratio probability distribution function. It is shown that a simple transformation can in many cases cause dramatic improvement in computation.", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Computers and Mathematics with Applications", journal-URL = "http://www.sciencedirect.com/science/journal/08981221", } @Article{Page:1977:MAC, author = "E. Page", title = "Miscellanea: Approximations to the Cumulative Normal Function and its Inverse for Use on a Pocket Calculator", journal = j-APPL-STAT, volume = "26", number = "1", pages = "75--76", year = "1977", CODEN = "APSTAG", ISSN = "0035-9254 (print), 1467-9876 (electronic)", ISSN-L = "0035-9254", bibdate = "Sat Apr 21 10:21:55 MDT 2001", bibsource = "https://www.math.utah.edu/pub/tex/bib/as1970.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "Applied Statistics", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1111/(ISSN)1467-9876/issues", } @Article{Pexton:1977:RTTa, author = "Robert L. Pexton and Arno D. Steiger", title = "Roots of two transcendental equations involving spherical {Bessel} functions", journal = j-MATH-COMPUT, volume = "31", number = "139", pages = "752--753", month = jul, year = "1977", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, classcodes = "C1110 (Algebra)", corpsource = "Univ. of California, Livermore, CA, USA", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "Bessel functions; root; spherical Bessel functions; transcendental equations", treatment = "T Theoretical or Mathematical", } @InProceedings{Randazzo:1977:DFE, author = "D. J. Randazzo", booktitle = "Numerical analysis ({Proceedings of the Colloquium, Lausanne, 1976})", title = "Data fits with exponential functions", volume = "37", publisher = pub-BIRKHAUSER, address = pub-BIRKHAUSER:adr, pages = "77--94", year = "1977", ISBN = "3-7643-0939-3", ISBN-13 = "978-3-7643-0939-8", MRclass = "65D15", MRnumber = "468111", MRreviewer = "C. W. Clenshaw", bibdate = "Mon Nov 13 08:14:42 2023", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/c/clenshaw-charles-w.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", series = "Internat. Ser. Numer. Math.", acknowledgement = ack-nhfb, reviewer-dates = "Charles William Clenshaw (15 March 1926--23 September 2004)", } @Article{Schett:1977:RFT, author = "Alois Schett", title = "Recurrence formula of the {Taylor} series expansion coefficients of the {Jacobian} elliptic functions", journal = j-MATH-COMPUT, volume = "31", number = "140", pages = "1003--1005", month = oct, year = "1977", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", classcodes = "C1110 (Algebra)", corpsource = "CENS, Gif-sur-Yvette, France", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "elliptic functions; functions; Jacobian elliptic; peak; recurrence formula; run up; Taylor series expansion coefficients", treatment = "T Theoretical or Mathematical", } @Article{Schindler:1977:CCS, author = "Susan Schindler and R. Mirman", title = "The {Clebsch--Gordan} coefficients of {$ S_n $}", journal = j-J-MATH-PHYS, volume = "18", number = "8", pages = "1697--1704", month = aug, year = "1977", CODEN = "JMAPAQ", DOI = "https://doi.org/10.1063/1.523470", ISSN = "0022-2488 (print), 1089-7658 (electronic), 1527-2427", ISSN-L = "0022-2488", bibdate = "Fri Jan 2 14:59:17 MST 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", abstract = "Ordering schemes for the frames and tableaux of $ S_n $ are presented, and some results, expressible in terms of these, are developed. A formula is derived for the sign function on tableaux. A table of the nonzero Clebsch--Gordon coefficients for the ''working triplets'' is given. Methods are described, and a needed table supplied, for finding the other coefficients from the tabulated ones. The values are for $ n = 2, \ldots {}, 6 $, with the coefficients for $ n = 6 $ relegated to PAPS.", acknowledgement = ack-nhfb, classification = "A0365F (Algebraic methods in quantum theory); A1130L (Other internal and higher symmetries in particle physics)", corpsource = "Baruch Coll., City Univ. of New York, NY, USA", fjournal = "Journal of Mathematical Physics", journal-URL = "http://jmp.aip.org/", keywords = "$S_n$ group; Clebsch Gordan coefficients; Clebsch--Gordan coefficients; working triplets", pubcountry = "USA", treatment = "T Theoretical or Mathematical", } @InProceedings{Schonfelder:1977:PTS, author = "J. L. Schonfelder", title = "The Production and Testing of Special Function Software in the {NAG} Library", crossref = "Cowell:1977:PNS", pages = "425--451", year = "1977", bibdate = "Sat Oct 30 10:24:29 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", } @Article{Sharma:1977:GMA, author = "R. R. Sharma and Bahman Zohuri", title = "A general method for an accurate evaluation of exponential integrals {$ E_1 (x), x > 0 $}", journal = j-J-COMPUT-PHYS, volume = "25", number = "2", pages = "199--204", month = oct, year = "1977", CODEN = "JCTPAH", DOI = "https://doi.org/10.1016/0021-9991(77)90022-5", ISSN = "0021-9991 (print), 1090-2716 (electronic)", ISSN-L = "0021-9991", bibdate = "Sun Jan 1 09:15:26 MST 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputphys1970.bib", URL = "http://www.sciencedirect.com/science/article/pii/0021999177900225", acknowledgement = ack-nhfb, fjournal = "Journal of Computational Physics", journal-URL = "http://www.sciencedirect.com/science/journal/00219991/", } @InProceedings{Smith:1977:FPA, author = "Brian T. Smith", title = "{Fortran} poisoning and antidotes", crossref = "Cowell:1977:PNS", pages = "178--256", year = "1977", DOI = "https://doi.org/10.1007/3-540-08446-0_43", bibdate = "Thu Dec 11 15:15:52 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fortran1.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, } @Article{Verma:1977:CSF, author = "Arun Verma", title = "Certain summation formulae for basic hypergeometric series", journal = j-CAN-MATH-BULL, volume = "20", number = "??", pages = "369--376", month = "????", year = "1977", CODEN = "CMBUA3", DOI = "https://doi.org/10.4153/CMB-1977-055-8", ISSN = "0008-4395 (print), 1496-4287 (electronic)", ISSN-L = "0008-4395", bibdate = "Thu Sep 8 10:04:41 MDT 2011", bibsource = "http://cms.math.ca/cmb/v20/; https://www.math.utah.edu/pub/tex/bib/canmathbull.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Canadian mathematical bulletin = Bulletin canadien de math{\'e}matiques", journal-URL = "http://cms.math.ca/cmb/", } @InProceedings{Victor:1977:ISI, author = "N. Victor and M. Sund", title = "The importance of standardized interfaces for portable statistical software", crossref = "Cowell:1977:PNS", pages = "484--503", year = "1977", DOI = "https://doi.org/10.1007/3-540-08446-0_55", bibdate = "Thu Dec 11 15:15:52 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, } @Article{Amos:1978:ECS, author = "Donald E. Amos", title = "Erratum: ``{Algorithm 511}: {CDC} 6600 Subroutines {IBESS} and {JBESS} for {Bessel} Functions {$ I_\nu (x) $} and {$ J_\nu (x) $}, {$ x \ge 0, \nu \ge 0 $} [{S18}]''", journal = j-TOMS, volume = "4", number = "4", pages = "411--411", month = dec, year = "1978", CODEN = "ACMSCU", DOI = "https://doi.org/10.1145/356502.356501", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Tue Aug 30 00:28:02 1994", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "See \cite{Amos:1977:ACS}.", acknowledgement = ack-nhfb, fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", } @Article{Andrews:1978:EFM, author = "M. Andrews and S. F. McCormick and G. D. Taylor", title = "Evaluation of Functions on Microcomputers: Square Root", journal = j-COMPUT-MATH-APPL, volume = "4", number = "4", pages = "359--367", year = "1978", CODEN = "CMAPDK", ISSN = "0898-1221 (print), 1873-7668 (electronic)", ISSN-L = "0898-1221", bibdate = "Thu Sep 15 18:40:29 1994", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nj, fjournal = "Computers and Mathematics with Applications", journal-URL = "http://www.sciencedirect.com/science/journal/08981221", xxmonth = "(none)", } @Article{Andrews:1978:UEF, author = "M. Andrews and T. Mraz", title = "Unified elementary function generator", journal = j-MICROPROC-MICROSYS, volume = "2", number = "5", pages = "270--273", month = oct, year = "1978", CODEN = "MIMID5", ISSN = "0141-9331 (print), 1872-9436 (electronic)", ISSN-L = "0141-9331", bibdate = "Thu Sep 1 10:15:39 1994", bibsource = "garbo.uwasa.fi:/pc/doc-soft/fpbiblio.txt; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nj, fjournal = "Microprocessors and Microsystems", } @Article{Ardill:1978:SBF, author = "R. W. B. Ardill and K. J. M. Moriarty", title = "Spherical {Bessel} functions $ j_n $ and $ y_n $ of integer order and real argument", journal = j-COMP-PHYS-COMM, volume = "14", number = "3--4", pages = "261--265", month = may # "\slash " # jun, year = "1978", CODEN = "CPHCBZ", DOI = "https://doi.org/10.1016/0010-4655(78)90019-X", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", bibdate = "Thu Apr 24 10:35:27 2008", bibsource = "https://www.math.utah.edu/pub/tex/bib/compphyscomm1970.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/001046557890019X", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Computer Physics Communications", journal-URL = "http://www.sciencedirect.com/science/journal/00104655", } @Article{Benton:1978:CZT, author = "T. C. Benton and H. D. Knoble", title = "Common Zeros of Two {Bessel} Functions", journal = j-MATH-COMPUT, volume = "32", number = "142", pages = "533--535", month = apr, year = "1978", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, classcodes = "C1110 (Algebra)", corpsource = "Pennsylvania State Univ., University Park, PA, USA", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "Bessel functions; common zeros; poles and zeros; positive zeros; zeros", treatment = "T Theoretical or Mathematical", } @Article{Blair:1978:RCA, author = "J. M. Blair and C. A. Edwards and J. H. Johnson", title = "Rational {Chebyshev} approximations for the {Bickley} functions {$ K i_n(x) $}", journal = j-MATH-COMPUT, volume = "32", number = "143", pages = "876--886", month = jul, year = "1978", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", classcodes = "C4130 (Interpolation and function approximation)", corpsource = "Atomic Energy of Canada Ltd., Chalk River, Ont., Canada", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "approximation; Bessel functions; Bickley functions; Chebyshev approximation; Chebyshev approximations; function; recurrence; relation", remark = "Fullerton: Approximations accurate to 23 digits of repeated integrals of the Bessel function $ K_0 (x) $ for $ n = 1, 2, \ldots {}, 10 $.", treatment = "T Theoretical or Mathematical", } @Article{Bowman:1978:ASS, author = "K. O. Bowman and L. R. Shenton", title = "Asymptotic series and {Stieltjes} continued fractions for a gamma function ratio", journal = j-J-COMPUT-APPL-MATH, volume = "4", number = "2", pages = "105--111", month = jun, year = "1978", CODEN = "JCAMDI", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Sat Feb 25 11:59:17 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath1970.bib", URL = "http://www.sciencedirect.com/science/article/pii/0771050X78900347", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @Book{Brezinski:1978:AAC, author = "Claude Brezinski", title = "Algorithmes d'acc{\'e}l{\'e}ration de la convergence: {\'e}tude num{\'e}rique. ({French}) [Algorithms for convergence acceleration: numerical study]", publisher = "{\'E}ditions Technip", address = "Paris, France", pages = "xi + 392", year = "1978", ISBN = "2-7108-0341-0 (paperback)", ISBN-13 = "978-2-7108-0341-6 (paperback)", LCCN = "????", bibdate = "Thu Dec 1 10:20:23 MST 2011", bibsource = "carmin.sudoc.abes.fr:210/ABES-Z39-PUBLIC; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, keywords = "convergence acceleration", language = "French", } @Article{Brezinski:1978:CAS, author = "C. Brezinski", title = "Convergence acceleration of some sequences by the $ \epsilon $-algorithm", journal = j-NUM-MATH, volume = "29", number = "2", pages = "173--177", month = jan, year = "1978", CODEN = "NUMMA7", ISSN = "0029-599X (print), 0945-3245 (electronic)", ISSN-L = "0029-599X", bibdate = "Mon May 26 11:49:34 MDT 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/nummath.bib", acknowledgement = ack-nhfb, classification = "C4140 (Linear algebra)", corpsource = "UER d'IEEA-Informatique, Univ. de Lille I, Villeneuve d'Ascq, France", fjournal = "Numerische Mathematik", journal-URL = "http://link.springer.com/journal/211", keywords = "acceleration; converge; convergence acceleration; convergence of numerical methods; epsilon-algorithm; sequences", treatment = "T Theoretical or Mathematical", } @Article{Brezinski:1978:SCA, author = "C. Brezinski", title = "Survey on convergence acceleration methods in numerical analysis", journal = j-MATH-STUDENT, volume = "46", number = "1", pages = "28--41 (1979)", year = "1978", CODEN = "MTHSBH", ISSN = "0025-5742", MRclass = "65B99", MRnumber = "698176 (84d:65003)", bibdate = "Thu Dec 1 17:36:45 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "The Mathematics Student", keywords = "convergence acceleration", } @Article{Chin:1978:DAD, author = "R. C. Y. Chin and G. W. Hedstrom", title = "A dispersion analysis for difference schemes: tables of generalized {Airy} functions", journal = j-MATH-COMPUT, volume = "32", number = "144", pages = "1163--1170", month = oct, year = "1978", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, classcodes = "C4130 (Interpolation and function approximation); C4170 (Differential equations)", corpsource = "Univ. of California, Lawrence Livermore Lab., Livermore, CA, USA", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "artificial viscosity; difference schemes; dispersion analysis; function approximation; functions; generalized Airy; linear hyperbolic equation", treatment = "T Theoretical or Mathematical", } @Article{Coleman:1978:RSN, author = "John P. Coleman", title = "Remark on {``Algorithm 49: Spherical Neumann Function''}", journal = j-TOMS, volume = "4", number = "3", pages = "295--295", month = sep, year = "1978", CODEN = "ACMSCU", ISSN = "0098-3500", bibdate = "Sat Jul 05 16:48:40 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/cacm1960.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "See \cite{Herndon:1961:ASN}.", acknowledgement = ack-nhfb, keywords = "Neumann functions; special functions", } @Article{DiDonato:1978:AR, author = "A. R. DiDonato", title = "An Approximation for $ \int^\infty_x e^{-t^2 / 2} t^p d t, x > 0, p $ Real", journal = j-MATH-COMPUT, volume = "32", number = "141", pages = "271--275", month = jan, year = "1978", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", classcodes = "C4160 (Numerical integration and differentiation)", corpsource = "Naval Surface Weapons Center, Dahlgren, VA, USA", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "integral approximation; integration; numerical analysis", remark = "Fullerton: Closely related to the complementary incomplete gamma function.", treatment = "A Application; N New Development; T Theoretical or Mathematical", } @Article{Drezner:1978:CBN, author = "Z. Drezner", title = "Computation of the Bivariate Normal Integral", journal = j-MATH-COMPUT, volume = "32", number = "141", pages = "277--279", month = jan, year = "1978", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", classcodes = "B0290M (Numerical integration and differentiation); C4160 (Numerical integration and differentiation)", corpsource = "Faculty of Business, McMaster Univ., Hamilton, Ont., Canada", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "bivariate normal integral; computation; Gauss quadrature method; integration; numerical analysis", treatment = "A Application; T Theoretical or Mathematical", } @Article{Dzjadyk:1978:CRP, author = "V. K. Dzjadyk and L. {\=I}. F{\'\i}lozof", title = "The convergence rate of {Pad{\'e}} approximants for some elementary functions. ({Russian})", journal = "Mat. Sb. (N.S.)", volume = "107(149)", number = "3", pages = "347--363, 463", year = "1978", ISSN = "0368-8666", MRclass = "41A21 (30E10 41A25)", MRnumber = "81b:41043", MRreviewer = "B. D. Donevski", bibdate = "Sat Jan 11 17:44:01 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, language = "Russian", } @InProceedings{Ercegovac:1978:LSR, author = "Milo{\v{s}} D. Ercegovac", title = "An On-Line Square Rooting Algorithm", crossref = "IEEE:1978:PSC", pages = "183--189", year = "1978", bibdate = "Thu Nov 15 10:49:40 2007", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "http://www.acsel-lab.com/arithmetic/arith4/papers/ARITH4_Ercegovac.pdf", abstract = "An on-line algorithm for computing square roots in a radix 2, normalized floating-point number system with the redundant digit set $ \{ - 1, 0, 1 \} $ is described. The algorithm has on-line delay of one and it is amenable for modular implementation. A systematic approach, used in deriving this algorithm, is presented in detail.", acknowledgement = ack-nhfb, keywords = "ARITH-4", } @Book{Feinsilver:1978:SFP, author = "Philip J. (Philip Joel) Feinsilver", title = "Special functions, probability semigroups, and {Hamiltonian} flows", volume = "696", publisher = pub-SV, address = pub-SV:adr, pages = "vi + 112", year = "1978", ISBN = "0-387-09100-9", ISBN-13 = "978-0-387-09100-6", LCCN = "QA3 .L28 no. 696; QA273", bibdate = "Sat Oct 30 19:22:05 MDT 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; z3950.loc.gov:7090/Voyager", series = "Lecture notes in mathematics", acknowledgement = ack-nhfb, subject = "probabilities; semigroups; functions, special; Hamiltonian systems", } @InProceedings{Frankowski:1978:RME, author = "Krzysztof S. Frankowski", title = "A Realistic Model for Error Estimates in the Evaluation of Elementary Functions", crossref = "IEEE:1978:PSC", pages = "70--74", year = "1978", MRclass = "65G05 (65D20)", MRnumber = "80g:65050", bibdate = "Sat Jan 11 17:44:01 MST 1997", bibsource = "garbo.uwasa.fi:/pc/doc-soft/fpbiblio.txt; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, } @Article{Gautschi:1978:CMB, author = "Walter Gautschi and Josef Slavik", title = "On the computation of modified {Bessel} function ratios", journal = j-MATH-COMPUT, volume = "32", number = "143", pages = "865--875", month = jul, year = "1978", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", classcodes = "C1120 (Mathematical analysis); C4130 (Interpolation and function approximation)", corpsource = "Dept. of Computer Sci., Purdue Univ., Lafayette, IN, USA", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "Bessel; Bessel functions; continued fraction; fraction; function approximation; functions; Gauss' continued; modified Bessel function ratios; Perron's continued fraction", treatment = "T Theoretical or Mathematical", } @Article{Gustafson:1978:ATC, author = "S.-{\AA}. Gustafson", title = "Algorithm $ 38 $. {Two} computer codes for convergence acceleration", journal = j-COMPUTING, volume = "21", number = "1", pages = "87--91", year = "1978", CODEN = "CMPTA2", ISSN = "0010-485X (print), 1436-5057 (electronic)", ISSN-L = "0010-485X", MRclass = "65B10", MRnumber = "83a:65005", bibdate = "Tue Jan 2 17:40:53 MST 2001", bibsource = "Compendex database; http://springerlink.metapress.com/openurl.asp?genre=journal&issn=0010-485X; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; INSPEC Axiom database (1968--date); MathSciNet database", acknowledgement = ack-nhfb, affiliation = "Australian Nat. Univ., Canberra, ACT, Australia", classification = "723; C4140", description = "convergence of numerical methods", fjournal = "Computing", journal-URL = "http://link.springer.com/journal/607", journalabr = "Computing (Vienna/New York)", keywords = "codes, symbolic; convergence acceleration; power series sum", } @Article{Gustafson:1978:CAG, author = "S.-{\AA}. Gustafson", title = "Convergence acceleration on a general class of power series", journal = j-COMPUTING, volume = "21", number = "1", pages = "53--69", year = "1978", CODEN = "CMPTA2", ISSN = "0010-485X (print), 1436-5057 (electronic)", ISSN-L = "0010-485X", MRclass = "65B10 (40A05)", MRnumber = "83m:65005", MRreviewer = "A. M. Cohen", bibdate = "Tue Jan 2 17:40:53 MST 2001", bibsource = "http://springerlink.metapress.com/openurl.asp?genre=journal&issn=0010-485X; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; INSPEC Axiom database (1968--date); MathSciNet database", acknowledgement = ack-nhfb, affiliation = "Australian Nat. Univ., Canberra, ACT, Australia", classification = "C4120", description = "convergence of numerical methods; function evaluation", fjournal = "Computing", journal-URL = "http://link.springer.com/journal/607", keywords = "algorithms; convergence acceleration; power series", } @Article{Hamaker:1978:MAC, author = "Hugo C. Hamaker", title = "Miscellanea: Approximating the Cumulative Normal Distribution and its Inverse", journal = j-APPL-STAT, volume = "27", number = "1", pages = "76--77", month = jan, year = "1978", CODEN = "APSTAG", ISSN = "0035-9254 (print), 1467-9876 (electronic)", ISSN-L = "0035-9254", bibdate = "Sat Apr 21 10:22:12 MDT 2001", bibsource = "https://www.math.utah.edu/pub/tex/bib/as1970.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "Applied Statistics", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1111/(ISSN)1467-9876/issues", } @InProceedings{Hull:1978:DFP, author = "T. E. Hull", title = "Desirable Floating-Point Arithmetic and Elementary Functions for Numerical Computation", crossref = "IEEE:1978:PSC", pages = "63--69", year = "1978", bibdate = "Thu Sep 01 12:14:34 1994", bibsource = "garbo.uwasa.fi:/pc/doc-soft/fpbiblio.txt; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nj, } @Article{Katholi:1978:CVP, author = "Charles R. Katholi", title = "On the computation of values of the psi function from rapidly converging power series expansions", journal = j-J-STAT-COMPUT-SIMUL, volume = "8", number = "1", pages = "25--42", year = "1978", CODEN = "JSCSAJ", DOI = "https://doi.org/10.1080/00949657808810245", ISSN = "0094-9655 (print), 1026-7778 (electronic), 1563-5163", ISSN-L = "0094-9655", bibdate = "Tue Apr 22 09:10:43 MDT 2014", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jstatcomputsimul.bib", acknowledgement = ack-nhfb, fjournal = "Journal of Statistical Computation and Simulation", journal-URL = "http://www.tandfonline.com/loi/gscs20", } @Article{Kobayashi:1978:FPA, author = "Yasuhiro Kobayashi and Masaaki Ohkita and Michio Inoue", title = "Fractional power approximations of elliptic integrals and {Bessel} functions", journal = j-MATH-COMPUT-SIMUL, volume = "20", number = "4", pages = "285--290", month = dec, year = "1978", CODEN = "MCSIDR", DOI = "https://doi.org/10.1016/0378-4754(78)90020-4", ISSN = "0378-4754 (print), 1872-7166 (electronic)", ISSN-L = "0378-4754", bibdate = "Fri Aug 15 13:24:17 MDT 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcomputsimul1970.bib", URL = "https://www.sciencedirect.com/science/article/pii/0378475478900204", acknowledgement = ack-nhfb, ajournal = "Math. Comput. Simul.", fjournal = "Mathematics and Computers in Simulation", journal-URL = "https://www.sciencedirect.com/science/journal/03784754", } @Article{Ling:1978:EWZ, author = "Chin-Bing Ling", title = "Evaluation of {Weierstrass} Zeta Functions", journal = j-SIAM-REVIEW, volume = "20", number = "1", pages = "183--183", month = "????", year = "1978", CODEN = "SIREAD", DOI = "https://doi.org/10.1137/1020017", ISSN = "0036-1445 (print), 1095-7200 (electronic)", ISSN-L = "0036-1445", bibdate = "Sat Mar 29 09:52:48 MDT 2014", bibsource = "http://epubs.siam.org/toc/siread/20/1; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/siamreview.bib", acknowledgement = ack-nhfb, fjournal = "SIAM Review", journal-URL = "http://epubs.siam.org/sirev", onlinedate = "January 1978", } @Article{Morita:1978:CCE, author = "T. Morita", title = "Calculation of the complete elliptic integrals with complex modulus", journal = j-NUM-MATH, volume = "29", number = "2", pages = "233--236", month = jan, year = "1978", CODEN = "NUMMA7", ISSN = "0029-599X (print), 0945-3245 (electronic)", ISSN-L = "0029-599X", bibdate = "Mon May 26 11:49:34 MDT 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", classification = "C4160 (Numerical integration and differentiation)", corpsource = "Dept. of Appl. Sci., Faculty of Engng., Tohoku Univ., Sendai, Japan", fjournal = "Numerische Mathematik", journal-URL = "http://link.springer.com/journal/211", keywords = "complete elliptic integrals; complex modulus; integration", remark = "Fullerton: Addendum to a paper by Morita and Horiguchi.", treatment = "T Theoretical or Mathematical", } @Article{Pexton:1978:RTT, author = "Robert L. Pexton and Arno D. Steiger", title = "Roots of two transcendental equations as functions of a continuous real parameter", journal = j-MATH-COMPUT, volume = "32", number = "142", pages = "511--518", month = apr, year = "1978", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, classcodes = "C1110 (Algebra)", corpsource = "Lawrence Livermore Lab., Univ. of California, Livermore, CA, USA", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "continuous real parameter; equations; roots; spherical Bessel functions; transcendental equations", treatment = "T Theoretical or Mathematical", } @Article{Preston:1978:NAT, author = "F. S. Preston", title = "A New Algorithm for the Tangent", journal = j-IEEE-TRANS-COMPUT, volume = "C-27", number = "2", pages = "167--167", month = feb, year = "1978", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/TC.1978.1675052", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Mon Jul 11 08:13:26 MDT 2011", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", URL = "https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=1675052", abstract = "A new mathematical algorithm has been developed for the tangent. The form of the equation guarantees that the error is zero at $ 0^\circ $, $ 45^\circ $, and $ 90^\circ $ corresponding to tangents of $0$, $1$, and infinity. With only one constant the error is brought to zero at two more points and the maximum error is less than one part in $ 3000 $. By adding a second constant, the error is reduced to less than one in $ 720 \, 000 $. Further terms improve the accuracy geometrically.", acknowledgement = ack-nj # "\slash " # ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", } @Article{Schindler:1978:GCG, author = "Susan Schindler and R. Mirman", title = "Generation of the {Clebsch-Gordan} coefficients for {$ S_n $}", journal = j-COMP-PHYS-COMM, volume = "15", number = "1--2", pages = "131--145", month = sep, year = "1978", CODEN = "CPHCBZ", DOI = "https://doi.org/10.1016/0010-4655(78)90087-5", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", bibdate = "Sun Oct 31 09:20:58 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/compphyscomm1970.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Computer Physics Communications", journal-URL = "http://www.sciencedirect.com/science/journal/00104655", } @Article{Schmidt:1978:EI, author = "Paul W. Schmidt", title = "Evaluation of the Integral $ \int^\infty_0 \frac {t^{2 \alpha - 1} J_\nu (x \sqrt {1 + t^2})(1 + t^2)^{\alpha + \beta - 1}} d t $", journal = j-MATH-COMPUT, volume = "32", number = "141", pages = "265--269", month = jan, year = "1978", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", classcodes = "B0290M (Numerical integration and differentiation); C4160 (Numerical integration and differentiation)", corpsource = "Phys. Dept., Univ. of Missouri, Columbia, MO, USA", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "cylinder diameter distribution; first kind Bessel function; integral evaluation; integration; long circular; neutron scattering data; numerical analysis; power series expansions; recurrence; relations", treatment = "A Application; T Theoretical or Mathematical", } @Article{Schoene:1978:RMI, author = "Andrew Y. Schoene", title = "Remark on ``{Algorithm 435}: Modified Incomplete Gamma Function [{S14}]''", journal = j-TOMS, volume = "4", number = "3", pages = "296--304", month = sep, year = "1978", CODEN = "ACMSCU", DOI = "https://doi.org/10.1145/355791.355803", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Tue Aug 30 00:28:02 1994", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "See \cite{Fullerton:1972:MIG}.", acknowledgement = ack-nhfb, fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", } @Article{Schonfelder:1978:CEE, author = "J. L. Schonfelder", title = "{Chebyshev} expansions for the error and related functions", journal = j-MATH-COMPUT, volume = "32", number = "144", pages = "1232--1240", month = oct, year = "1978", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", classcodes = "C4110 (Error analysis in numerical methods); C4130 (Interpolation and function approximation)", corpsource = "Computer Centre, Univ. of Birmingham, Birmingham, UK", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "approximations; Chebyshev approximation; Chebyshev expansions; error analysis; error function; mappings; NAG library", remark = "Fullerton: Approximations for $ \erf (x) $, $ \erfc (x) $, and probability functions $ P(x) $, $ Q(x) $ with accuracy down to $ 10^{-30} $.", treatment = "T Theoretical or Mathematical", } @Article{Skovgaard:1978:RCE, author = "Ove Skovgaard", title = "Remark on ``{Algorithm 149: Complete Elliptic Integral [S21]}''", journal = j-TOMS, volume = "4", number = "1", pages = "95--95", month = mar, year = "1978", CODEN = "ACMSCU", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Tue Feb 06 05:28:13 1996", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "See \cite{Merner:1962:AAC}.", acknowledgement = ack-nhfb, fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", } @Article{Slepian:1978:PSW, author = "D. Slepian", title = "Prolate Spheroidal Wave Functions, {Fourier} Analysis, and Uncertainty --- {V}: The Discrete Case", journal = j-BELL-SYST-TECH-J, volume = "57", number = "5", pages = "1371--1430", month = may # "--" # jun, year = "1978", CODEN = "BSTJAN", ISSN = "0005-8580", bibdate = "Tue Nov 9 11:15:56 MST 2010", bibsource = "http://bstj.bell-labs.com/oldfiles/year.1978/BSTJ.1978.5705.html; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://bstj.bell-labs.com/BSTJ/images/Vol57/bstj57-5-1371.pdf", acknowledgement = ack-nhfb, fjournal = "The Bell System Technical Journal", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)1538-7305/issues/", } @Article{Temme:1978:UAE, author = "N. M. Temme", title = "Uniform asymptotic expansions of confluent hypergeometric functions", journal = j-J-INST-MATH-APPL, volume = "22", number = "2", pages = "215--223", year = "1978", CODEN = "JMTAA8", ISSN = "0020-2932", MRclass = "33A30", MRnumber = "80a:33004", MRreviewer = "F. W. J. Olver", bibdate = "Fri Apr 5 05:48:27 MST 2002", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; MathSciNet database", acknowledgement = ack-nhfb, fjournal = "Journal of the Institute of Mathematics and its Applications", journal-URL = "http://imamat.oxfordjournals.org/content/by/year", } @Article{Wang:1978:EPF, author = "J. Y. Wang", title = "The Evaluation of Periodic Functions with Large Input Arguments", journal = j-SIGNUM, volume = "13", number = "4", pages = "7--8", month = dec, year = "1978", CODEN = "SNEWD6", DOI = "https://doi.org/10.1145/1053412.1053413", ISSN = "0163-5778 (print), 1558-0237 (electronic)", ISSN-L = "0163-5778", bibdate = "Tue Apr 12 07:50:05 MDT 2005", bibsource = "garbo.uwasa.fi:/pc/doc-soft/fpbiblio.txt; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/signum.bib", abstract = "The argument reduction scheme plays an important role in the evaluation of periodic functions. In this report we discuss the criteria for determining the domain of computer routines that evaluate periodic functions, and apply the results to the widely used sine and cosine functions.", acknowledgement = ack-nhfb # " and " # ack-nj, classcodes = "C4120 (Functional analysis); C4240 (Programming and algorithm theory)", corpsource = "Appl. Math. Div., Argonne Nat. Lab., Argonne, IL, USA", journal-URL = "https://dl.acm.org/loi/signum", keywords = "argument; computational complexity; computer routines; cosine functions; FORTRAN IV; function evaluation; large input arguments; periodic functions; reduction scheme", remark-1 = "From page 7: ``In this report, we will examine bounds on the domain that guarantees that the computed function values retain half of the significant digits.''", remark-2 = "From page 8: ``Improvement: \ldots{} this can he accomplished by storing PI/4 in two words, i.e., PI/4 = C1 + C2.", treatment = "T Theoretical or Mathematical", } @TechReport{Wang:1978:SMR, author = "J. Y. Wang and J. Boyer", title = "A Studv of the Mathematical Routines in the {IBM System/360 FORTRAN IV} and {FORTRAN IV (Mod II)} Libraries", type = "Report", number = "AMD-TM 304", institution = "Applied Mathematics Division, Argonne National Laboratory", address = "Argonne, IL, USA", month = jan, year = "1978", bibdate = "Fri Sep 20 14:26:48 2024", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "Cited in \cite[Reference 8]{Agarwal:1986:NSV} in elefunt.bib and fparith.bib.", acknowledgement = ack-nhfb, } @Article{Alt:1979:SRD, author = "H. Alt", title = "Square Rooting Is as Difficult as Multiplication", journal = j-COMPUTING, volume = "21", number = "3", pages = "221--232", month = sep, year = "1979", CODEN = "CMPTA2", ISSN = "0010-485X (print), 1436-5057 (electronic)", ISSN-L = "0010-485X", MRclass = "68C25", MRnumber = "82m:68081", bibdate = "Tue Jan 2 17:40:54 MST 2001", bibsource = "Compendex database; garbo.uwasa.fi:/pc/doc-soft/fpbiblio.txt; https://www.math.utah.edu/pub/tex/bib/computing.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; INSPEC Axiom database (1968--date); MathSciNet database", acknowledgement = ack-nj # " and " # ack-nhfb, affiliation = "Math. \& Information, Univ. of Saarlandes, Saarbrucken, West Germany", classification = "723; C5230", description = "digital arithmetic", fjournal = "Computing", journal-URL = "http://link.springer.com/journal/607", journalabr = "Computing (Vienna/New York)", keywords = "algorithm; computer programming; square rooting", } @Article{Ardill:1979:ABF, author = "R. W. B. Ardill and K. J. M. Moriarty", title = "Accurate {Bessel} functions {$ J_n(z) $}, {$ Y_n(z) $}, {$ H_n^{(1)}(z) $} and {$ H_n^{(2)}(z) $} of integer order and complex argument", journal = j-COMP-PHYS-COMM, volume = "17", number = "3", pages = "321--336", month = jun, year = "1979", CODEN = "CPHCBZ", DOI = "https://doi.org/10.1016/0010-4655(79)90060-2", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", bibdate = "Thu Apr 24 10:35:27 2008", bibsource = "https://www.math.utah.edu/pub/tex/bib/compphyscomm1970.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Computer Physics Communications", journal-URL = "http://www.sciencedirect.com/science/journal/00104655", remark = "Fullerton: Description of a program with accuracy about $ 10^{-10} $.", } @Article{Atkins:1979:FSC, author = "D. E. Atkins", title = "{Fourth Symposium on Computer Arithmetic}: crunching with quality and {LSI}", journal = j-COMPUTER, volume = "12", number = "4", pages = "94--97", month = apr, year = "1979", CODEN = "CPTRB4", ISSN = "0018-9162 (print), 1558-0814 (electronic)", ISSN-L = "0018-9162", bibdate = "Thu Dec 12 07:20:54 MST 1996", bibsource = "Compendex database; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", abstract = "Computer arithmetic problems --- faster computation rates and more efficient representations of real numbers --- are considered in the paper. Floating-point arithmetic standardization, novel implementation of basic arithmetic operators, evaluation of elementary functions --- these are the main considerations of the conference review.", acknowledgement = ack-nhfb, classification = "722; 723", fjournal = "Computer", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=2", journalabr = "Computer", keywords = "computer arithmetic; computer systems, digital; data processing --- data description; mathematical techniques --- digital arithmetic", } @Article{Babb:1979:OEC, author = "Stanley E. {Babb, Jr.} and James W. Cafky", title = "Operational evaluation of certain infinite {Bessel} function integrals", journal = j-MATH-COMPUT, volume = "33", number = "147", pages = "1033--1039", month = jul, year = "1979", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "44A99", MRnumber = "80e:44002", MRreviewer = "L. Arteaga", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, classcodes = "C4180 (Integral equations)", corpsource = "Dept. of Phys. and Astron., Univ. of Oklahoma, Norman, OK, USA", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "Bessel function integral; integration; numerical methods; Schafheitlin method; trigonometric function; Weber", treatment = "A Application; T Theoretical or Mathematical", } @Article{Borjesson:1979:SAE, author = "P. B{\"o}rjesson and C.-E. Sundberg", title = "Simple Approximations of the Error Function {$ Q(x) $} for Communications Applications", journal = j-IEEE-TRANS-COMM, volume = "27", number = "3", pages = "639--643", month = mar, year = "1979", CODEN = "IECMBT", DOI = "https://doi.org/10.1109/tcom.1979.1094433", ISSN = "0090-6778 (print), 1558-0857 (electronic)", ISSN-L = "0090-6778", bibdate = "Sat Dec 16 15:29:00 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "IEEE Transactions on Communications", } @Article{Campbell:1979:BFR, author = "J. B. Campbell", title = "{Bessel} functions {$ J_\nu (x) $} and {$ Y_\nu (x) $} of real order and real argument", journal = j-COMP-PHYS-COMM, volume = "18", number = "1", pages = "133--142", month = sep, year = "1979", CODEN = "CPHCBZ", DOI = "https://doi.org/10.1016/0010-4655(79)90030-4", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", bibdate = "Mon Feb 13 06:01:26 MST 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/compphyscomm1970.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/0010465579900304", acknowledgement = ack-nhfb, fjournal = "Computer Physics Communications", journal-URL = "http://www.sciencedirect.com/science/journal/00104655", } @Article{Carlson:1979:CEI, author = "B. C. Carlson", title = "Computing elliptic integrals by duplication", journal = j-NUM-MATH, volume = "33", number = "1", pages = "1--16", month = mar, year = "1979", CODEN = "NUMMA7", ISSN = "0029-599X (print), 0945-3245 (electronic)", ISSN-L = "0029-599X", MRclass = "65D20", MRnumber = "80h:65008", bibdate = "Mon May 26 11:49:34 MDT 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", abstract = "Logarithms, arctangents, and elliptic integrals of all three kinds (including complete integrals) are evaluated numerically by successive applications of the duplication theorem. When the convergence is improved by including a fixed number of terms of Taylor's series, the error ultimately decreases by a factor of 4096 in each cycle of iteration. Except for Cauchy principal values there is no separation of cases according to the values of the variables, and no serious cancellations occur if the variables are real and nonnegative. Only rational operations and square roots are required. An appendix contains a recurrence relation and two new representations (in terms of elementary symmetric functions and power sums) for $R$-polynomials, as well as an upper bound for the error made in truncating the Taylor series of an $R$-function.", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", classification = "C4180 (Integral equations)", corpsource = "Dept. of Math. and Phys., Iowa Univ., Ames, IA, USA", fjournal = "Numerische Mathematik", journal-URL = "http://link.springer.com/journal/211", keywords = "convergence; duplication theory; elliptic integral; integration; numerical methods; R-polynomial; Taylor series", treatment = "A Application; T Theoretical or Mathematical", } @Article{Cole:1979:EI, author = "R. J. Cole and C. Pescatore", title = "Evaluation of the Integral $ \int_0^\infty t^n \exp ( - t^2 - x / t) \, d t $", journal = j-J-COMPUT-PHYS, volume = "32", number = "2", pages = "280--287", month = aug, year = "1979", CODEN = "JCTPAH", DOI = "https://doi.org/10.1016/0021-9991(79)90135-9", ISSN = "0021-9991 (print), 1090-2716 (electronic)", ISSN-L = "0021-9991", bibdate = "Sun Jan 1 09:15:34 MST 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputphys1970.bib", URL = "http://www.sciencedirect.com/science/article/pii/0021999179901359", abstract = "The main purpose of this paper is to provide a unified approach to the treatment of linear recurrence relations for single or pairs of order statistics. Suppose such a relation has been proved in the simplest case when $ X_1, \ldots {}, X_n $ are independent variates having an arbitrary absolutely continuous distribution. It is pointed out that the same relation continues to hold when the $X$'s are exchangeable, whether continuous or not. As has recently become well known, further generalizations are possible when the $X$'s have any joint distribution. Attention is also drawn to a useful nonlinear recurrence relation due to Boncelet (1987).", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Journal of Computational Physics", journal-URL = "http://www.sciencedirect.com/science/journal/00219991/", } @Article{Conde:1979:Z, author = "S. Conde and Shyam L. Kalla", title = "The $ \nu $-zeros of {$ J_{- \nu }(x) $}", journal = j-MATH-COMPUT, volume = "33", number = "145", pages = "423--426", month = jan, year = "1979", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "65D20", MRnumber = "80b:65021", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", classcodes = "C4170 (Differential equations); C4190 (Other numerical methods)", corpsource = "Facultad de Ingenieria, Univ. del Zulia, Maracaibo, Venezuela", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "Bessel function; boundary value; boundary-value problems; J/sub -v/(x); partial differential equations; poles and zeros; problems; transforms; v-zeros", remark = "Fullerton: With a microfiche supplement.", treatment = "T Theoretical or Mathematical", } @Article{Delic:1979:CSS, author = "G. Delic", title = "{Chebyshev} series for the spherical {Bessel} function $ j_l(r) $", journal = j-COMP-PHYS-COMM, volume = "18", number = "1", pages = "73--86", month = sep, year = "1979", CODEN = "CPHCBZ", DOI = "https://doi.org/10.1016/0010-4655(79)90025-0", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", bibdate = "Mon Feb 13 06:01:26 MST 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/compphyscomm1970.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/0010465579900250", acknowledgement = ack-nhfb, fjournal = "Computer Physics Communications", journal-URL = "http://www.sciencedirect.com/science/journal/00104655", } @Article{Divgi:1979:CUB, author = "D. R. Divgi", title = "Calculation of Univariate and Bivariate Normal Probability Functions", journal = j-ANN-STAT, volume = "7", number = "4", pages = "903--910", month = jul, year = "1979", CODEN = "ASTSC7", DOI = "https://doi.org/10.1214/aos/1176344739", ISSN = "0090-5364 (print), 2168-8966 (electronic)", ISSN-L = "0090-5364", bibdate = "Wed Jun 4 06:39:50 MDT 2014", bibsource = "https://www.math.utah.edu/pub/tex/bib/annstat1970.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://projecteuclid.org/euclid.aos/1176344739", acknowledgement = ack-nhfb, fjournal = "Annals of Statistics", journal-URL = "http://projecteuclid.org/all/euclid.aos/", } @Article{Einarsson:1979:BEN, author = "Bo Einarsson", title = "Bibliography on the evaluation of numerical software", journal = j-J-COMPUT-APPL-MATH, volume = "5", number = "2", pages = "145--159", month = jun, year = "1979", CODEN = "JCAMDI", DOI = "https://doi.org/10.1016/0771-050X(79)90011-1", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Thu Oct 28 17:14:49 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", abstract = "This bibliography on the evaluation of numerical software has been written at the request of the IFIP Working Group on Numerical Software (IFIP WG 2.5), and is divided into nine different sections. Within each section the references are given in alphabetical order by the first author.\par The aim of the bibliography is to be useful in the production and evaluation of good software for numerical mathematics.", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", tableofcontents = "1. General Articles \\ 2. Miscellaneous evaluations \\ 3. Linear algebra \\ 4. Optimization and nonlinear equations \\ 5. Functions \\ 6. Quadrature \\ 7. Integral equations \\ 8. Ordinary differential equations \\ 9. Partial differential equations", } @Article{elLozy:1979:RAS, author = "Mohamed el Lozy", title = "Remark on ``{Algorithm 395: Student's $t$-Distribution}'' and Remark on ``{Algorithm 396: Student's Quantiles [S14]}''", journal = j-TOMS, volume = "5", number = "2", pages = "238--239", month = jun, year = "1979", CODEN = "ACMSCU", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Tue Feb 06 05:28:16 1996", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "See \cite{Hill:1970:AASa,Hill:1970:AASb,Hill:1981:RSD,Hill:1985:RCS}.", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", remark = "Fullerton: The algorithms are corrected for computers with anomalously small word lengths (e.g., IBM and Interdata).", } @Article{Epstein:1979:STE, author = "H. I. Epstein and B. F. Caviness", title = "A structure theorem for the elementary functions and its application to the identity problem", journal = j-INT-J-COMPUT-INF-SCI, volume = "8", number = "1", pages = "9--37", year = "1979", CODEN = "IJCIAH", ISSN = "0091-7036", MRclass = "12H05 (68C05)", MRnumber = "80k:12032", MRreviewer = "Michael F. Singer", bibdate = "Sat Apr 26 12:45:53 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "International Journal of Computer and Information Sciences", journal-URL = "http://link.springer.com/journal/10766", } @Article{Fettis:1979:AEU, author = "Henry E. Fettis", title = "An asymptotic expansion for the upper percentage points of the $ \chi^2 $-distribution", journal = j-MATH-COMPUT, volume = "33", number = "147", pages = "1059--1064", month = jul, year = "1979", CODEN = "MCMPAF", DOI = "https://doi.org/10.2307/2006079", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "62E20", MRnumber = "80h:62014", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp1970.bib; JSTOR database", acknowledgement = ack-nhfb, ajournal = "Math. Comput.", citedby = "Fullerton:1980:BEM", classcodes = "C4130 (Interpolation and function approximation)", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "approximation theory; asymptotic expansion; upper percentage point", treatment = "A Application; T Theoretical or Mathematical", } @Article{Gabutti:1979:HPM, author = "Bruno Gabutti", title = "On high precision methods for computing integrals involving {Bessel} functions", journal = j-MATH-COMPUT, volume = "33", number = "147", pages = "1049--1057", month = jul, year = "1979", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "65D30", MRnumber = "80c:65048", MRreviewer = "K. Jetter", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, classcodes = "C4180 (Integral equations); C4240 (Programming and algorithm theory)", corpsource = "Istituto di Calcoli Numerici, Univ. di Torino, Torino, Italy", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "Bessel function integral; computational complexity; exponential function integral; integration", treatment = "T Theoretical or Mathematical", } @Article{Gargantini:1979:NSS, author = "Irene Gargantini", title = "The Numerical Stability of Simultaneous Iterations Via Square-Rooting", journal = j-COMPUT-MATH-APPL, volume = "5", number = "1", pages = "25--31", month = "????", year = "1979", CODEN = "CMAPDK", DOI = "https://doi.org/10.1016/0898-1221(81)90136-X", ISSN = "0898-1221 (print), 1873-7668 (electronic)", ISSN-L = "0898-1221", bibdate = "Wed Mar 1 18:51:16 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/computmathappl1970.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/089812218190136X", acknowledgement = ack-jr # " and " # ack-nhfb, fjournal = "Computers and Mathematics with Applications", journal-URL = "http://www.sciencedirect.com/science/journal/08981221", } @Article{Gautschi:1979:AIG, author = "W. Gautschi", title = "{Algorithm 542}: Incomplete Gamma Functions [{S14}]", journal = j-TOMS, volume = "5", number = "4", pages = "482--489", month = dec, year = "1979", CODEN = "ACMSCU", DOI = "https://doi.org/10.1145/355853.355864", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Sun Aug 28 00:39:50 1994", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", remark = "Fullerton: FORTRAN routines of adjustable accuracy.", } @Article{Gautschi:1979:CPI, author = "Walter Gautschi", title = "A Computational Procedure for Incomplete Gamma Functions", journal = j-TOMS, volume = "5", number = "4", pages = "466--481", month = dec, year = "1979", CODEN = "ACMSCU", DOI = "https://doi.org/10.1145/355853.355863", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Sun Aug 28 00:32:50 1994", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", remark = "Fullerton: Algorithms for the incomplete gamma function, $ \gamma (a, x) $, the complementary function, $ \Gamma (a, x) $, and Tricomi's form, $ \gamma '(a, x) $, are given.", } @Article{Glasser:1979:NI, author = "M. L. Glasser", title = "A Note on the Integral $ \int^\infty_0 t^{2 \alpha - 1}(1 + t^2)^{1 - \alpha - \beta } {J}_\nu (x \sqrt {1 + t^2}) d t $", journal = j-MATH-COMPUT, volume = "33", number = "146", pages = "792--793", month = apr, year = "1979", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "33A40", MRnumber = "80d:33004", MRreviewer = "T. Erber", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", classcodes = "C1120 (Mathematical analysis); C4180 (Integral equations)", corpsource = "Math. and Computer Sci. Dept., Clarkson Coll. of Technol., Potsdam, NY, USA", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "Bessel functions; hypergeometric series; integral; integral equations", treatment = "T Theoretical or Mathematical", } @Article{Haavie:1979:GNT, author = "Tore H{\aa}vie", title = "Generalized {Neville} type extrapolation schemes", journal = j-BIT, volume = "19", number = "2", pages = "204--213", month = jun, year = "1979", CODEN = "BITTEL, NBITAB", DOI = "https://doi.org/10.1007/BF01930850", ISSN = "0006-3835 (print), 1572-9125 (electronic)", ISSN-L = "0006-3835", MRclass = "65B05 (65D05 65D30)", MRnumber = "80f:65005", MRreviewer = "Siegfried Filippi", bibdate = "Wed Jan 4 18:52:16 MST 2006", bibsource = "http://springerlink.metapress.com/openurl.asp?genre=issue&issn=0006-3835&volume=19&issue=2; https://www.math.utah.edu/pub/tex/bib/bit.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.springerlink.com/openurl.asp?genre=article&issn=0006-3835&volume=19&issue=2&spage=204", acknowledgement = ack-nhfb, fjournal = "BIT (Nordisk tidskrift for informationsbehandling)", journal-URL = "http://link.springer.com/journal/10543", } @Article{Johnson:1979:RAF, author = "Donald B. Johnson and Webb Miller and Brian Minnihan and Celia Wrathall", title = "Reducibility Among Floating-Point Graphs", journal = j-J-ACM, volume = "26", number = "4", pages = "739--760", month = oct, year = "1979", CODEN = "JACOAH", ISSN = "0004-5411 (print), 1557-735X (electronic)", ISSN-L = "0004-5411", MRclass = "65G05", MRnumber = "80i:65045", bibdate = "Fri Dec 08 11:55:10 1995", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", abstract = "The graph-theoretic models of this paper can be used to compare the rounding-error behavior of numerical programs. The models follow the approach, popularized by Wilkinson, of assuming independent rounding errors in each arithmetic operation. Models constructed on this assumption are more tractable than would be the case under more realistic assumptions. There are identified two easily tested conditions on programs which guarantee that error analyses are relatively insensitive to the particular graph model employed. The development has the additional benefit of sometimes providing an elementary proof that one program is comparable in stability to another. Examples of such results are given.", acknowledgement = ack-nhfb, ajournal = "J. Assoc. Comput. Mach.", fjournal = "Journal of the ACM", journal-URL = "https://dl.acm.org/loi/jacm", } @Article{Kobayashi:1979:UEF, author = "Y. Kobayashi and M. Ohkita and M. Inoue", title = "On the use of an exponential function in approximation of elliptic integrals", journal = j-MATH-COMPUT-SIMUL, volume = "21", number = "2", pages = "226--230", month = aug, year = "1979", CODEN = "MCSIDR", DOI = "https://doi.org/10.1016/0378-4754(79)90138-1", ISSN = "0378-4754 (print), 1872-7166 (electronic)", ISSN-L = "0378-4754", bibdate = "Fri Aug 15 13:24:19 MDT 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcomputsimul1970.bib", URL = "https://www.sciencedirect.com/science/article/pii/0378475479901381", acknowledgement = ack-nhfb, ajournal = "Math. Comput. Simul.", fjournal = "Mathematics and Computers in Simulation", journal-URL = "https://www.sciencedirect.com/science/journal/03784754", } @Article{Kusterer:1979:SEP, author = "Roland Kusterer and Manfred Reimer", title = "Stable Evaluation of Polynomials in Time $ \log n $", journal = j-MATH-COMPUT, volume = "33", number = "147", pages = "1019--1031", month = jul, year = "1979", CODEN = "MCMPAF", DOI = "https://doi.org/10.1090/S0025-5718-1979-0528054-X; https://doi.org/10.2307/2006075", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "65G05 (68C25)", MRnumber = "80d:65050 (528054)", MRreviewer = "C. W. Clenshaw", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; garbo.uwasa.fi:/pc/doc-soft/fpbiblio.txt; https://www.math.utah.edu/pub/bibnet/authors/c/clenshaw-charles-w.bib; https://www.math.utah.edu/pub/bibnet/authors/t/todd-john.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp1970.bib; JSTOR database", acknowledgement = ack-nhfb, classcodes = "C4130 (Interpolation and function approximation)", corpsource = "Math. Inst., University of Dortmund, Dortmund, West Germany", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "algorithm; approximation theory; number of multiplications to evaluate a polynomial; polynomials", reviewer-dates = "Charles William Clenshaw (15 March 1926--23 September 2004)", treatment = "A Application; T Theoretical or Mathematical", } @Article{Leung:1979:AFE, author = "K. V. Leung and S. S. Ghaderpanah", title = "An application of the finite element approximation method to find the complex zeros of the modified {Bessel} function $ {K}_n(z) $", journal = j-MATH-COMPUT, volume = "33", number = "148", pages = "1299--1306", month = oct, year = "1979", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "65D20 (33-04)", MRnumber = "80e:65024", MRreviewer = "R. P. Boas, Jr.", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, classcodes = "C4140 (Linear algebra)", corpsource = "Dept. of Computer Sci., Concordia Univ., Montreal, Que., Canada", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "Bessel function; complex zeros; finite element analysis; finite element approximation method; iterative optimisation scheme; modified; poles and zeros", treatment = "T Theoretical or Mathematical", } @Article{Lindstrom:1979:MSM, author = "F. T. Lindstrom", title = "A Modified $3$-Spline Method for Evaluating the {Euler} Digamma Function", journal = j-TECHNOMETRICS, volume = "21", number = "3", pages = "307--311", month = aug, year = "1979", CODEN = "TCMTA2", DOI = "https://doi.org/10.2307/1267752", ISSN = "0040-1706 (print), 1537-2723 (electronic)", ISSN-L = "0040-1706", bibdate = "Sat Jun 21 13:18:50 MDT 2014", bibsource = "http://www.jstor.org/journals/00401706.html; http://www.jstor.org/stable/i254300; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/technometrics1970.bib", URL = "http://www.jstor.org/stable/1267752", acknowledgement = ack-nhfb, fjournal = "Technometrics", journal-URL = "http://www.jstor.org/journals/00401706.html", } @Article{Ling:1979:EWZ, author = "C. B. Ling", title = "Evaluation of {Weierstrass} Zeta Functions", journal = j-SIAM-REVIEW, volume = "21", number = "1", pages = "146--147", month = "????", year = "1979", CODEN = "SIREAD", DOI = "https://doi.org/10.1137/1021020", ISSN = "0036-1445 (print), 1095-7200 (electronic)", ISSN-L = "0036-1445", bibdate = "Fri Jun 21 11:25:02 MDT 2013", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/siamreview.bib", acknowledgement = ack-nhfb, fjournal = "SIAM Review", journal-URL = "http://epubs.siam.org/sirev", } @Article{Maurone:1979:ABD, author = "Philip A. Maurone and Alain J. Phares", title = "On the asymptotic behavior of the derivatives of {Airy} functions", journal = j-J-MATH-PHYS, volume = "20", number = "11", pages = "2191--2191", month = nov, year = "1979", CODEN = "JMAPAQ", DOI = "https://doi.org/10.1063/1.523997", ISSN = "0022-2488 (print), 1089-7658 (electronic), 1527-2427", ISSN-L = "0022-2488", MRclass = "33A60", MRnumber = "80j:33014", bibdate = "Sat Oct 29 11:28:40 MDT 2011", bibsource = "http://jmp.aip.org/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jmathphys1975.bib", URL = "http://jmp.aip.org/resource/1/jmapaq/v20/i11/p2191_s1", acknowledgement = ack-nhfb, classification = "A0230 (Function theory, analysis); A0365D (Functional analytical methods in quantum theory); A1235 (Composite models of particles)", corpsource = "Dept. of Phys., Villanova Univ., Villanova, PA, USA", fjournal = "Journal of Mathematical Physics", journal-URL = "http://jmp.aip.org/", keywords = "Airy functions; angular momentum theory; asymptotic behavior; derivatives; functional analysis; noniterative functional solution; quantum theory; quark confinement; recursion relation", onlinedate = "29 July 2008", pagecount = "1", treatment = "T Theoretical or Mathematical", } @Article{Morris:1979:DFR, author = "Robert Morris", title = "The Dilogarithm Function of a Real Argument", journal = j-MATH-COMPUT, volume = "33", number = "146", pages = "778--787", month = apr, year = "1979", CODEN = "MCMPAF", DOI = "https://doi.org/10.1090/S0025-5718-1979-0521291-X", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "65D20 (33A70)", MRnumber = "80e:65025", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp1970.bib; JSTOR database", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", classcodes = "C4240 (Programming and algorithm theory)", corpsource = "Bell Labs., Murray Hill, NJ, USA", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "algorithm theory; dilogarithm function; real argument", received = "22 May 1978", remark = "Fullerton: Relative errors down to $ 10^{-24} $ for $ \operatorname {Li}_2 (z) = - \int_0^z \frac {\log (1 - z)z} \, d z $. $ \operatorname {Li}_2 (z) $ is a form of Spence's integral.", treatment = "T Theoretical or Mathematical", } @TechReport{Morris:1979:DMS, author = "Alfred H. {Morris, Jr.}", title = "Development of Mathematical Software and Mathematical Software Libraries", type = "Technical Report", number = "TR 79-102 (AD-A068023)", institution = "Naval Surface Warfare Center", address = "Dahlgren, VA 22448-5000, USA; Silver Spring, MD 20903-5000, USA", pages = "iii + 20", month = apr, year = "1979", bibdate = "Wed Dec 17 11:01:53 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "https://apps.dtic.mil/sti/tr/pdf/ADA068023.pdf", abstract = "The purpose of this paper is to briefly examine several of the major issues concerning the development of numerical mathematical software and numerical mathematical software libraries. The paper begins with a brief summary of the evolution of general-purpose mathematical software libraries. This is followed by an introductory discussion on software reliability. It is often tacitly assumed that nest of the basic numerical mathematics problems have satisfactorily been solved. This is shown not to be the case. Indeed, it is noted that many of the problems encounter not only deep theoretical difficulties, but also numerous software engineering problems.\par The next major topic is software portability. Here the emphasis is on portability difficulties that arise from design deficiencies in the programming languages. It is noted that FORTRAN permits an arithmetic expression to be altered when it is known that the modification can produce different results.\par The final issues considered are those involved in forming a library. If the purpose of the library is to serve as broad an audience as possible, then it is recognized that the subroutines in the library should be as simple to use and as comprehensive as is practical. Thus formation of the library can be characterized as a packaging problem, the objective being to package mathematical formulae and theory into comprehensive, simple-to-use subroutines.", acknowledgement = ack-nhfb, pdfpages = "31", } @Article{Pexton:1979:DRT, author = "Robert L. Pexton and Arno D. Steiger", title = "Degenerate roots of three transcendental equations involving spherical {Bessel} functions", journal = j-MATH-COMPUT, volume = "33", number = "147", pages = "1041--1048", month = jul, year = "1979", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "loose microfiche suppl. 65H10 (65D20)", MRnumber = "80g:65057", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, classcodes = "C4130 (Interpolation and function approximation)", corpsource = "Lawrence Livermore Lab., Univ. of California, Livermore, CA, USA", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "function approximation; spherical Bessel function; transcendental equation", treatment = "A Application; T Theoretical or Mathematical", } @Article{Phillips:1979:FAC, author = "G. Phillips", title = "A fast approximation to the complementary error function for use in fitting gamma-ray peaks", journal = j-NUCL-INSTR-METH, volume = "164", number = "??", pages = "561--563", month = sep, year = "1979", CODEN = "NUIMAL", DOI = "https://doi.org/10.1016/0029-554X(79)90094-6", ISSN = "0029-554x (print), 1878-3759 (electronic)", ISSN-L = "0029-554X", bibdate = "Mon Oct 24 11:37:20 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://adsabs.harvard.edu/abs/1979NucIM.164..561P", abstract = "A fast approximation to the complementary error function has been programmed and tested for use in the peak-shape function for fitting peaks in gamma-ray spectra. The function was compared for speed and accuracy on the NRL ASC 7 computer to the mathematical library version of the complementary error function. The approximation has resulted in a 50\% time savings in the computer program HYPERMET which was developed at NRL for automatic analysis of gamma-ray spectra from germanium detectors.", acknowledgement = ack-nhfb, fjournal = "Nuclear Instruments and Methods", } @Article{Reeves:1979:AB, author = "C. M. Reeves", title = "Algorithm 9: {Boyserf}", journal = j-COMP-J, volume = "22", number = "1", pages = "89--90", month = feb, year = "1979", CODEN = "CMPJA6", DOI = "https://doi.org/10.1093/comjnl/22.1.86; https://doi.org/10.1093/comjnl/22.1.89", ISSN = "0010-4620 (print), 1460-2067 (electronic)", ISSN-L = "0010-4620", bibdate = "Thu Oct 05 15:40:50 2000", bibsource = "http://www3.oup.co.uk/computer_journal/hdb/Volume_22/Issue_01/; https://www.math.utah.edu/pub/tex/bib/compj1970.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "Reprinted from \booktitle{The Computer Bulletin}, September 1964, p. 67.", URL = "http://www3.oup.co.uk/computer_journal/hdb/Volume_22/Issue_01/tiff/89.tif; http://www3.oup.co.uk/computer_journal/hdb/Volume_22/Issue_01/tiff/90.tif", acknowledgement = ack-nhfb, fjournal = "The Computer Journal", journal-URL = "http://comjnl.oxfordjournals.org/", remark = "The boyserf() function computes $ F(x) = \int_0^1 \exp ( - x * t**2) \, d t $. A Maple simplification says that $ F(x) = \sqrt {\pi } \erf (\sqrt {x}) / (2 \sqrt {x}) $.", } @Article{Risch:1979:APE, author = "Robert H. Risch", title = "Algebraic properties of the elementary functions of analysis", journal = j-AM-J-MATH, volume = "101", number = "4", pages = "743--759", year = "1979", CODEN = "AJMAAN", ISSN = "0002-9327 (print), 1080-6377 (electronic)", ISSN-L = "0002-9327", MRclass = "12H05", MRnumber = "81b:12029", MRreviewer = "J. L. Johnson", bibdate = "Sat Jan 11 17:44:01 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "American Journal of Mathematics", } @Article{Schulten:1979:AEC, author = "Z. (or K. ??) Schulten and D. G. M. Anderson and R. G. Gordon", title = "An Algorithm for the Evaluation of the Complex {Airy} Functions", journal = j-J-COMPUT-PHYS, volume = "31", number = "1", pages = "60--75", month = apr, year = "1979", CODEN = "JCTPAH", DOI = "https://doi.org/10.1016/0021-9991(79)90062-7", ISSN = "0021-9991 (print), 1090-2716 (electronic)", ISSN-L = "0021-9991", bibdate = "Sat Oct 30 10:34:34 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputphys1970.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Journal of Computational Physics", journal-URL = "http://www.sciencedirect.com/science/journal/00219991/", remark = "Fullerton: Simple formulae over the whole complex plane are presented.", } @Article{Smith:1979:ALL, author = "David A. Smith and William F. Ford", title = "Acceleration of Linear and Logarithmic Convergence", journal = j-SIAM-J-NUMER-ANAL, volume = "16", number = "2", pages = "223--240", month = apr, year = "1979", CODEN = "SJNAAM", ISSN = "0036-1429 (print), 1095-7170 (electronic)", ISSN-L = "0036-1429", MRclass = "65B10", MRnumber = "82a:65012", bibdate = "Fri Oct 16 06:57:22 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/siamjnumeranal.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "SIAM Journal on Numerical Analysis", journal-URL = "http://epubs.siam.org/sinum", } @Article{Takemasa:1979:CFC, author = "T. Takemasa and T. Tamura and H. H. Wolter", title = "{Coulomb} Functions with Complex Angular Momenta", journal = j-COMP-PHYS-COMM, volume = "17", number = "4", pages = "351--355", month = jul # "\slash " # aug, year = "1979", CODEN = "CPHCBZ", DOI = "https://doi.org/10.1016/0010-4655(79)90097-3", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", bibdate = "Sat Oct 30 11:14:02 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/compphyscomm1970.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Computer Physics Communications", journal-URL = "http://www.sciencedirect.com/science/journal/00104655", remark = "Fullerton: Description of program CCOULOM, which works only in the part of the $ (\rho, \eta) $-plane where $ \eta^2 \ll \rho $ and $ | \ell |^2 \ll \rho $.", } @Article{Temme:1979:AAP, author = "N. M. Temme", title = "An Algorithm with {ALGOL 60} Program for the Computation of the Zeros of Ordinary {Bessel} Functions and those of their Derivatives", journal = j-J-COMPUT-PHYS, volume = "32", number = "2", pages = "270--279", month = aug, year = "1979", CODEN = "JCTPAH", DOI = "https://doi.org/10.1016/0021-9991(79)90134-7", ISSN = "0021-9991 (print), 1090-2716 (electronic)", ISSN-L = "0021-9991", bibdate = "Sat Oct 30 11:23:13 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputphys1970.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Journal of Computational Physics", journal-URL = "http://www.sciencedirect.com/science/journal/00219991/", remark = "Fullerton: An adjustable-accuracy 100 line Algol procedure is discussed.", } @Article{Temme:1979:AEI, author = "N. M. Temme", title = "The asymptotic expansion of the incomplete gamma functions", journal = j-SIAM-J-MATH-ANA, volume = "10", number = "4", pages = "757--766", month = jul, year = "1979", CODEN = "SJMAAH", ISSN = "0036-1410 (print), 1095-7154 (electronic)", ISSN-L = "0036-1410", MRclass = "33A15", MRnumber = "80i:33002", MRreviewer = "E. Rieksti\lfhook n{\v{s}}", bibdate = "Sun Nov 28 19:22:16 MST 2010", bibsource = "http://epubs.siam.org/sam-bin/dbq/toc/SIMA/10/4; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/siamjmathana.bib", acknowledgement = ack-nhfb, fjournal = "SIAM Journal on Mathematical Analysis", journal-URL = "http://epubs.siam.org/sima", } @Article{Terras:1979:DIG, author = "Riho Terras", title = "The determination of incomplete gamma functions through analytic integration", journal = j-J-COMPUT-PHYS, volume = "31", number = "1", pages = "146--151", month = apr, year = "1979", CODEN = "JCTPAH", DOI = "https://doi.org/10.1016/0021-9991(79)90066-4", ISSN = "0021-9991 (print), 1090-2716 (electronic)", ISSN-L = "0021-9991", bibdate = "Sun Jan 1 09:15:33 MST 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputphys1970.bib", URL = "http://www.sciencedirect.com/science/article/pii/0021999179900664", acknowledgement = ack-nhfb, fjournal = "Journal of Computational Physics", journal-URL = "http://www.sciencedirect.com/science/journal/00219991/", } @Article{Thacher:1979:NBR, author = "Henry C. {Thacher, Jr.}", title = "New Backward Recurrences for {Bessel} Functions", journal = j-MATH-COMPUT, volume = "33", number = "146", pages = "744--764", month = apr, year = "1979", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "65D20 (33A40)", MRnumber = "81b:65019", MRreviewer = "R. G. Langebartel", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", classcodes = "C1120 (Mathematical analysis)", corpsource = "Dept. of Computer Sci., Univ. of Kentucky, Lexington, KY, USA", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "backward recurrences; Bessel functions; converges", treatment = "N New Development; T Theoretical or Mathematical", } @Article{Amos:1980:CEI, author = "Donald E. Amos", title = "Computation of Exponential Integrals", journal = j-TOMS, volume = "6", number = "3", pages = "365--377", month = sep, year = "1980", CODEN = "ACMSCU", DOI = "https://doi.org/10.1145/355900.355908", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", MRclass = "65D30 (68-04)", MRnumber = "82b:65011", bibdate = "Mon Aug 29 10:57:24 1994", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib", acknowledgement = ack-nhfb, fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", keywords = "exponential integral; Miller algorithm; recursion; Taylor series", reviewer = "M. M. Chawla", } @Article{Brent:1980:SNA, author = "Richard P. Brent and Edwin M. McMillan", title = "Some new algorithms for high-precision computation of {Euler}'s constant", journal = j-MATH-COMPUT, volume = "34", number = "149", pages = "305--312", month = jan, year = "1980", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "10-04 (10A40 68C05)", MRnumber = "82g:10002", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", classcodes = "B0290D (Functional analysis); C4120 (Functional analysis)", corpsource = "Univ. of California, Berkeley, CA, USA", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "Bessel functions; computation; Euler's constant; function evaluation; high precision", remark = "Fullerton: Calculation to 30,100 places is discussed, but only a 3-digit value appears in the paper.", treatment = "N New Development; T Theoretical or Mathematical", } @InProceedings{Brent:1980:UAE, author = "R. P. Brent", title = "Unrestricted Algorithms for Elementary and Special Functions", crossref = "Lavington:1980:IPP", pages = "613--619", year = "1980", bibdate = "Thu Sep 01 11:55:31 1994", bibsource = "garbo.uwasa.fi:/pc/doc-soft/fpbiblio.txt; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://maths-people.anu.edu.au/~brent/pub/pub052.html", acknowledgement = ack-nj, remark = "From the author's Web site: Errata for original version:\par Page 615, remove the absolute value signs in equation (13) and the following paragraph (x should be large and positive here).\par Page 616, second half of equation (26): insert a minus sign after the equals sign.\par Page 617, equation (39): delete `` / j ! ''.\par Page 617, equation (42): the assumption ``j < k'' should be added. Also, the contour C needs to be enlarged slightly.\par Page 617, left-hand-side of equation (44): replace ``Sj,k'' by ``S2j,k''.\par Page 617, ten lines after equation (44): replace ``O(jn2)'' by ``O(j2n)''.", } @Article{Brezinski:1980:GEA, author = "C. Brezinski", title = "A general extrapolation algorithm", journal = j-NUM-MATH, volume = "35", number = "2", pages = "175--187", month = jun, year = "1980", CODEN = "NUMMA7", ISSN = "0029-599X (print), 0945-3245 (electronic)", ISSN-L = "0029-599X", MRclass = "65B05", MRnumber = "81j:65015", MRreviewer = "L. Fox", bibdate = "Mon May 26 11:49:34 MDT 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/nummath.bib", acknowledgement = ack-nhfb, classification = "C4130 (Interpolation and function approximation)", corpsource = "UER IEEA, Univ. de Lille 1, Villeneuve d'Ascq, France", fjournal = "Numerische Mathematik", journal-URL = "http://link.springer.com/journal/211", keywords = "convergence acceleration; extrapolation; extrapolation algorithm; linear extrapolation; rational extrapolation; recursive algorithm; sequence transformations", treatment = "T Theoretical or Mathematical", } @Article{Char:1980:SCF, author = "Bruce W. Char", title = "On {Stieltjes}' continued fraction for the gamma function", journal = j-MATH-COMPUT, volume = "34", number = "150", pages = "547--551", month = apr, year = "1980", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "65A05 (65D20)", MRnumber = "81b:65008", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", remark = "Fullerton: The first 41 coefficients to 40 digits are given.", } @Article{Clenshaw:1980:UAE, author = "C. W. Clenshaw and Frank W. J. Olver", title = "An unrestricted algorithm for the exponential function", journal = j-SIAM-J-NUMER-ANAL, volume = "17", number = "2", pages = "310--331", month = apr, year = "1980", CODEN = "SJNAAM", DOI = "https://doi.org/10.1137/0717026", ISSN = "0036-1429 (print), 1095-7170 (electronic)", ISSN-L = "0036-1429", MRclass = "65D20", MRnumber = "567276", MRreviewer = "A. M. Cohen", bibdate = "Sun Nov 12 06:18:24 2023", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/c/clenshaw-charles-w.bib; https://www.math.utah.edu/pub/bibnet/authors/o/olver-frank-w-j.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", abstract = "An algorithm is presented for the computation of the exponential function of real argument. There are no restrictions on the range of the argument or on the precision that may be demanded in the results.", acknowledgement = ack-nhfb, author-dates = "Charles William Clenshaw (15 March 1926--23 September 2004); Frank William John Olver (15 December 1924--23 April 2013)", fjournal = "SIAM Journal on Numerical Analysis", journal-URL = "http://epubs.siam.org/sinum", } @Book{Cody:1980:SME, author = "William J. {Cody, Jr.} and William Waite", title = "Software Manual for the Elementary Functions", publisher = pub-PH, address = pub-PH:adr, pages = "x + 269", year = "1980", ISBN = "0-13-822064-6", ISBN-13 = "978-0-13-822064-8", LCCN = "QA331 .C635 1980", bibdate = "Tue Dec 14 23:28:38 1993", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/master.bib", acknowledgement = ack-nhfb, shorttableofcontents = "Preface / ix \\ 1. Introduction / 1 \\ 2. Preliminaries / 3 \\ 3. Performance Testing / 11 \\ 4. SQRT / 17 \\ 5. ALOG/ALOG10 / 35 \\ 6. EXP / 60 \\ 7. POWER (**) / 84 \\ 8. SIN/COS / 125 \\ 9. TAN/COT / 150 \\ 10. ASIN/ACOS / 174 \\ 11. ATAN/ATAN2 / 194 \\ 12. SINH/COSH / 217", tableofcontents = "Preface / ix \\ 1. Introduction / 1 \\ 2. Preliminaries / 3 \\ 3. Performance Testing / 11 \\ 4. SQRT / 17 \\ a. General Discussion / 17 \\ b. Flow Chart for SQRT(X) / 18 \\ c. Implementation Notes, Non-Decimal Fixed-Point Machines / 19 \\ d. Implementation Notes, Binary Floating-Point Machines / 23 \\ e. Implementation Notes, Non-Binary Floating-Point Machines / 25 \\ f. Testing / 28 \\ 5. ALOG/ALOG10 / 35 \\ a. General Discussion / 35 \\ b. Flow Chart for ALOG(X)/ALOG10(X) / 37 \\ c. Implementation Notes, Non-Decimal Fixed-Point Machines / 38 \\ d. Implementation Notes, Non-Decimal Floating-Point Machines / 42 \\ e. Implementation Notes, Decimal Floating-Point Machines / 46 \\ f. Testing / 49 \\ 6. EXP / 60 \\ a. General Discussion / 60 \\ b. Flow Chart for EXP(X) / 62 \\ c. Implementation Notes, Non-Decimal Fixed-Point Machines / 63 \\ d. Implementation Notes, Non-Decimal Floating-Point Machines / 67 \\ e. Implementation Notes, Decimal Floating-Point Machines / 71 \\ f. Testing / 75 \\ 7. POWER (**) / 84 \\ a. General Discussion / 84 \\ b. Flow Chart for POWER(X,Y) / 88 \\ c. Implementation Notes, Non-Decimal Fixed-Point Machines / 90 \\ d. Implementation Notes, Non-Decimal Floating-Point Machines / 97 \\ e. Implementation Notes, Decimal Floating-Point Machines / 106 \\ f. Testing / 113 \\ 8. SIN/COS / 125 \\ a. General Discussion / 125 \\ b. Flow Chart for SIN(X)/COS(X) / 127 \\ c. Implementation Notes, Non-Decimal Fixed-Point Machines / 129 \\ d. Implementation Notes, All Floating-Point Machines / 134 \\ e. Testing / 139 \\ 9. TAN/COT / 150 \\ a. General Discussion / 150 \\ b. Flow Chart for TAN(X)/COTAN(X) / 152 \\ c. Implementation Notes, Non-Decimal Fixed-Point Machines / 154 \\ d. Implementation Notes, All Floating-Point Machines / 159 \\ e. Testing / 164 \\ 10. ASIN/ACOS / 174 \\ a. General Discuss i on / 174 \\ b. Flow Chart for AS IN(X)/ACOS(X) / 176 \\ c. Implementation Not es, Non-Decimal Fixed-Point Machines / 177 \\ d. Implementation Notes, All Floating-Point Machines / 181 \\ e. Testing / 185 \\ 11. ATAN/ATAN2 / 194 \\ a. General Discussion / 194 \\ b. Flow Chart for ATAN(X)/ATAN2(V,U) / 196 \\ c. Implementation Notes, Non-Decimal Fixed-Point Machines / 198 \\ d. Implementation Notes, All Floating-Point Machines / 203 \\ e. Testing / 207 \\ 12. SINH/COSH / 217 \\ a. General Discussion / 217 \\ b. Flow Chart for SINH(X)/COSH(X) / 220 \\ c. Implementation Notes, Non-Decimal Fixed-Point Machines / 221 \\ d. Implementation Notes, All Floating-Point Machines / 225 \\ e. Testing / 229", } @Article{Coleman:1980:FSB, author = "J. P. Coleman", title = "A {Fortran} subroutine for the {Bessel} function {$ J_n(x) $} of order $0$ to $ 10 $", journal = j-COMP-PHYS-COMM, volume = "21", number = "1", pages = "109--118", day = "1", month = dec, year = "1980", CODEN = "CPHCBZ", DOI = "https://doi.org/10.1016/0010-4655(80)90080-6", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", bibdate = "Mon Feb 13 06:01:19 MST 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/compphyscomm1980.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fortran1.bib", URL = "http://www.sciencedirect.com/science/article/pii/0010465580900806", acknowledgement = ack-nhfb, fjournal = "Computer Physics Communications", journal-URL = "http://www.sciencedirect.com/science/journal/00104655", } @Article{Delahaye:1980:RNA, author = "J. P. Delahaye and B. Germain-Bonne", title = "{R}{\'e}sultats n{\'e}gatifs en acc{\'e}l{\'e}ration de la convergence. ({French}) [{Negative} results in convergence acceleration]", journal = j-NUM-MATH, volume = "35", number = "4", pages = "443--457", month = nov, year = "1980", CODEN = "NUMMA7", ISSN = "0029-599X (print), 0945-3245 (electronic)", ISSN-L = "0029-599X", MRclass = "65B99 (40A99)", MRnumber = "81k:65007", MRreviewer = "Claude Brezinski", bibdate = "Mon May 26 11:49:34 MDT 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, classification = "C4130 (Interpolation and function approximation); C4240 (Programming and algorithm theory)", corpsource = "Univ. des Sci. et Tech. de Lille I, Villeneuve d'Ascq, France", fjournal = "Numerische Mathematik", journal-URL = "http://link.springer.com/journal/211", keywords = "acceleration; algorithm; approximation theory; computational complexity; convergence; convergence acceleration; convergence of numerical methods", language = "French", treatment = "T Theoretical or Mathematical", } @Article{Ditkin:1980:CSF, author = "V. A. Ditkin and K. A. Karpov and M. K. Kerimov", title = "The computation of special functions", journal = j-USSR-COMP-MATH-MATH-PHYS, volume = "20", number = "5", pages = "3--12", year = "1980", CODEN = "CMMPA9", ISSN = "0041-5553, 0502-9902", ISSN-L = "0041-5553", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, catcode = "G.2.1; G.1.2", CRclass = "G.2.1 Combinatorics; G.2.1 Generating functions; G.1.2 Approximation; G.1.2 Elementary function approximation", descriptor = "Mathematics of Computing, DISCRETE MATHEMATICS, Combinatorics, Generating functions; Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation", fjournal = "U.S.S.R. Computational Mathematics and Mathematical Physics", genterm = "algorithms; design", guideno = "09092", journal-URL = "http://www.sciencedirect.com/science/journal/00415553", journalabr = "USSR Comput. Math. Math. Phys", jrldate = "1980", subject = "G. Mathematics of Computing; G.2 DISCRETE MATHEMATICS; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS", } @Article{Eckhardt:1980:AWE, author = "Ulrich Eckhardt", title = "{Algorithm 549}: {Weierstrass}' Elliptic Functions [{S21}]", journal = j-TOMS, volume = "6", number = "1", pages = "112--120", month = mar, year = "1980", CODEN = "ACMSCU", DOI = "https://doi.org/10.1145/355873.355884", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Mon Aug 29 10:31:24 1994", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", keywords = "Weierstrass' elliptic functions", remark = "Fullerton: A complex FORTRAN algorithm with accuracy down to $ 10^{-18} $ is given", } @Article{Fransen:1980:HPV, author = "Arne Frans{\'e}n and Staffan Wrigge", title = "High-precision values of the gamma function and of some related coefficients", journal = j-MATH-COMPUT, volume = "34", number = "150", pages = "553--566", month = apr, year = "1980", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "65A05 (65D20)", MRnumber = "81f:65004", MRreviewer = "F. W. J. Olver", bibdate = "Sat Apr 01 10:12:58 2000", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", note = "See addendum and corrigendum \cite{Fransen:1981:ACH}.", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", remark = "Fullerton: 80D values of coefficients in the Taylor series for $ \Gamma^m(s + x) $ are given.", } @TechReport{Fullerton:1980:BEM, author = "L. W. Fullerton", title = "A Bibliography on the Evaluation of Mathematical Functions", type = "Technical report", number = "TM 80-1274-4 and CSTR 86", institution = inst-ATT-BELL, address = inst-ATT-BELL:adr, month = sep, year = "1980", bibdate = "Sat Feb 05 17:39:14 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, } @Article{Gargantini:1980:PSR, author = "Irene Gargantini", title = "Parallel Square-Root Iterations for Multiple Roots", journal = j-COMPUT-MATH-APPL, volume = "6", number = "3", pages = "279--288", month = "????", year = "1980", CODEN = "CMAPDK", ISSN = "0898-1221 (print), 1873-7668 (electronic)", ISSN-L = "0898-1221", bibdate = "Wed Mar 1 18:51:19 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/computmathappl1980.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/0898122180900358", acknowledgement = ack-jr # " and " # ack-nhfb, fjournal = "Computers and Mathematics with Applications", journal-URL = "http://www.sciencedirect.com/science/journal/08981221/", } @Unpublished{Kahan:1980:SPI, author = "W. Kahan", title = "Software $ \sqrt x $ for the Proposed {IEEE Floating-Point Standard}", institution = inst-BERKELEY-CS, address = inst-BERKELEY-CS:adr, pages = "????", day = "25", month = aug, year = "1980", bibdate = "Mon Apr 25 18:24:02 2005", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/k/kahan-william-m.bib; https://www.math.utah.edu/pub/bibnet/subjects/acc-stab-num-alg.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", note = "Manuscript", acknowledgement = ack-nhfb, mynote = "Dated August 25 1980", } @Article{Kasperkovitz:1980:AAM, author = "P. Kasperkovitz", title = "Asymptotic approximations for modified {Bessel} functions", journal = j-J-MATH-PHYS, volume = "21", number = "1", pages = "6--13", month = jan, year = "1980", CODEN = "JMAPAQ", DOI = "https://doi.org/10.1063/1.524310", ISSN = "0022-2488 (print), 1089-7658 (electronic), 1527-2427", ISSN-L = "0022-2488", MRclass = "33A40", MRnumber = "81a:33012", MRreviewer = "N. Hayek Calil", bibdate = "Sat Oct 29 18:18:22 MDT 2011", bibsource = "http://jmp.aip.org/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jmathphys1980.bib", URL = "http://jmp.aip.org/resource/1/jmapaq/v21/i1/p6_s1", acknowledgement = ack-nhfb, fjournal = "Journal of Mathematical Physics", journal-URL = "http://jmp.aip.org/", onlinedate = "21 July 2008", pagecount = "8", } @Article{Langebartel:1980:FER, author = "R. G. Langebartel", title = "{Fourier} expansions of rational fractions of elliptic integrals and {Jacobian} elliptic functions", journal = j-SIAM-J-MATH-ANA, volume = "11", number = "3", pages = "506--513", month = may, year = "1980", CODEN = "SJMAAH", ISSN = "0036-1410 (print), 1095-7154 (electronic)", ISSN-L = "0036-1410", MRclass = "42A16 (33A25)", MRnumber = "81e:42008", MRreviewer = "R. C. Varma", bibdate = "Sat Dec 5 18:14:13 MST 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "SIAM Journal on Mathematical Analysis", journal-URL = "http://epubs.siam.org/sima", } @InProceedings{Maksymiv:1980:APT, author = "E. M. Maksymiv", booktitle = "Differentsialnye Uravneniya i ikh Prilozhen", title = "Approximate properties of {Thiele}'s formula in a class of elementary functions. ({Russian})", volume = "141", publisher = "Vestnik L'vov. Politekhn. Inst.", address = "L'vov, USSR", pages = "55--56", year = "1980", MRclass = "119.65D20", MRnumber = "81i:65021", bibdate = "Sat Jan 11 17:44:01 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, language = "Russian", } @Article{Meinardus:1980:OPN, author = "G{\"u}nter Meinardus and G. D. Taylor", title = "Optimal Partitioning of {Newton}'s Method for Calculating Roots", journal = j-MATH-COMPUT, volume = "35", number = "152", pages = "1221--1230", month = oct, year = "1980", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "65H05 (41A30)", MRnumber = "81j:65069", MRreviewer = "Derek W. Arthur", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; JSTOR database", acknowledgement = ack-nhfb, ajournal = "Math. Comput.", classcodes = "C4130 (Interpolation and function approximation)", corpsource = "Fachbereich Math., University of Siegen, Siegen, West Germany", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "cube root; domain interval; function approximation; iterative methods; method; Newton; optimal partitioning; reciprocal square root; square root; subinterval", treatment = "A Application; T Theoretical or Mathematical", } @Article{Moran:1980:CND, author = "P. A. P. Moran", title = "Calculation of the Normal Distribution Function", journal = j-BIOMETRIKA, volume = "67", number = "3", pages = "675--676", month = dec, year = "1980", CODEN = "BIOKAX", DOI = "https://doi.org/10.1093/biomet/67.3.675; https://doi.org/10.2307/2335138", ISSN = "0006-3444 (print), 1464-3510 (electronic)", ISSN-L = "0006-3444", MRclass = "62E30", MRnumber = "601106 (82d:62044)", MRreviewer = "G. P. Bhattacharjee", bibdate = "Sat Jun 21 14:34:26 MDT 2014", bibsource = "http://www.jstor.org/journals/00063444.html; http://www.jstor.org/stable/i315495; https://www.math.utah.edu/pub/tex/bib/biometrika1980.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.jstor.org/stable/2335138", acknowledgement = ack-nhfb, fjournal = "Biometrika", journal-URL = "http://biomet.oxfordjournals.org/content/by/year; http://www.jstor.org/journals/00063444.html", } @Article{OBrien:1980:SBF, author = "D. M. O'Brien", title = "Spherical {Bessel} functions of large order", journal = j-J-COMPUT-PHYS, volume = "36", number = "1", pages = "128--132", month = jun, year = "1980", CODEN = "JCTPAH", DOI = "https://doi.org/10.1016/0021-9991(80)90177-1", ISSN = "0021-9991 (print), 1090-2716 (electronic)", ISSN-L = "0021-9991", bibdate = "Sun Jan 1 15:59:01 MST 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputphys1980.bib", URL = "http://www.sciencedirect.com/science/article/pii/0021999180901771", abstract = "This note introduces functions $ b_n(x) $, related to spherical Bessel functions $ j_n(x) $ and $ y_n(x) $. They are scaled so that they are bounded functions of $n$ and polynomially bounded functions of $x$, and therefore avoid the problems of underflow and overflow which are so common with Bessel functions. They can be generated from a stable recurrence relation for which starting values are readily computable.", acknowledgement = ack-nhfb, fjournal = "Journal of Computational Physics", journal-URL = "http://www.sciencedirect.com/science/journal/00219991/", } @InProceedings{Olver:1980:UAG, author = "F. W. J. Olver", title = "Unrestricted algorithms for generating elementary functions", crossref = "Alefeld:1980:PSE", pages = "131--140", year = "1980", MRclass = "65G05 (65D15)", MRnumber = "82b:65034", MRreviewer = "John Todd", bibdate = "Sat Jan 11 17:44:01 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, } @Article{Pedersen:1980:HBM, author = "P. W. Pedersen", title = "Hvordan beregner man kvadratroden? \toenglish {How do you calculate the square root?} \endtoenglish", journal = "Elektronik (Denmark)", volume = "??", number = "4", pages = "18--21", month = apr, year = "1980", bibdate = "Fri Sep 16 16:30:41 1994", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nj, } @Book{Popov:1980:PFD, author = "B. A. Popov and G. S. Tesler", title = "Priblizhenie funktsii dlya tekhnicheskikh prilozhenii. ({Russian}) [{Approximation} of functions for technical applications]", publisher = "Naukova Dumka", address = "Kiev, USSR", pages = "351", year = "1980", MRclass = "65D15 (41-02 65D07)", MRnumber = "602955", bibdate = "Tue Jan 24 08:23:12 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "Nauka i tekhnicheskii progress. [Science and technical progress]", acknowledgement = ack-nhfb # " and " # ack-mv, } @Article{Rengarajan:1980:MFI, author = "S. R. Rengarajan and J. E. Lewis", title = "{Mathieu} Functions of Integral Order and Real Arguments", journal = j-IEEE-TRANS-MICROWAVE-THEORY-TECH, volume = "28", number = "3", pages = "276--277", month = mar, year = "1980", CODEN = "IETMAB", DOI = "https://doi.org/10.1109/TMTT.1980.1130060", ISSN = "0018-9480 (print), 1557-9670 (electronic)", ISSN-L = "0018-9480", bibdate = "Sat Oct 30 10:09:01 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", abstract = "To compute Mathieu functions, modified Mathieu functions and related parameters for integral orders and real arguments, encountered in wave propagation involving elliptic geometries.", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "IEEE transactions on microwave theory and techniques", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=22", remark = "Fullerton: A long FORTRAN program is very briefly described. This work is superior to that of Clemm (1969), because $q$ may be negative.", } @Article{Schell:1980:AEU, author = "Hans-Joachim Schell", title = "{Asymptotische Entwicklungen f{\"u}r die unvollst{\"a}ndige Gammafunktion}. ({German}) [{Asymptotic} developments for the incomplete gamma function]", journal = "{Wissenschaftliche Zeitschrift der Technischen Hochschule Karl-Marx-Stadt}", volume = "22", number = "5", pages = "477 485", month = "????", year = "1980", bibdate = "Sat Feb 18 15:11:46 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, ajournal = "{Wiss. Schr. Tech. Univ. Karl-Marx-Stadt}", keywords = "incomplete gamma function; uniform asymptotic expansions", language = "German", } @Article{Schonfelder:1980:VHA, author = "J. L. Schonfelder", title = "Very high accuracy {Chebyshev} expansions for the basic trigonometric functions", journal = j-MATH-COMPUT, volume = "34", number = "149", pages = "237--244", month = jan, year = "1980", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "65D20", MRnumber = "81f:65016", MRreviewer = "Claude Carasso", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", remark = "Fullerton: 40D coefficients for the sine, cosine and tangent are given.", } @Book{Sneddon:1980:SFM, author = "Ian Naismith Sneddon", title = "Special Functions of Mathematical Physics and Chemistry", publisher = "Longman", address = "London, UK", edition = "Third", pages = "ix + 182", year = "1980", ISBN = "0-582-44396-2 (paperback)", ISBN-13 = "978-0-582-44396-9 (paperback)", LCCN = "QA351 .S64 1980", bibdate = "Sat Oct 30 18:25:01 MDT 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; z3950.loc.gov:7090/Voyager", series = "Longman mathematical texts", acknowledgement = ack-nhfb, remark = "See first edition \cite{Sneddon:1956:SFM} and second edition \cite{Sneddon:1961:SFM}.", subject = "Functions, Special", } @InCollection{Tretjakov:1980:PSE, author = "V. A. Tret'jakov", booktitle = "Mathematical analysis and the theory of functions ({Russian})", title = "On the properties of some elementary functions that are defined on the algebra of bicomplex numbers. ({Russian})", publisher = "Moskov. Oblast. Ped. Inst.", address = "Moscow, USSR", pages = "99--106", year = "1980", MRclass = "30G35 (78A35)", MRnumber = "82i:30069", MRreviewer = "Toma V. Tonev", bibdate = "Sat Jan 11 17:44:01 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, language = "Russian", } @Article{Tretter:1980:ASA, author = "Marietta J. Tretter and G. W. Walster", title = "Analytic subtraction applied to the incomplete gamma and beta functions", journal = j-SIAM-J-SCI-STAT-COMP, volume = "1", number = "3", pages = "321--326", month = sep, year = "1980", CODEN = "SIJCD4", DOI = "https://doi.org/10.1137/0901022", ISSN = "0196-5204", ISSN-L = "0196-5204", MRclass = "65D20 (33A15)", MRnumber = "81m:65029", MRreviewer = "Anton Hut'a", bibdate = "Mon Mar 31 09:58:49 MDT 2014", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/siamjscistatcomp.bib", acknowledgement = ack-nhfb, fjournal = "SIAM Journal on Scientific and Statistical Computing", journal-URL = "http://epubs.siam.org/loi/sjoce3", keywords = "analytic subtraction; continued fraction; incomplete beta function; incomplete gamma function", onlinedate = "September 1980", } @PhdThesis{vonGudenberg:1980:EAR, author = "J. Wolff {von Gudenberg}", title = "{Einbettung allgemeiner Rechnerarithmetik in Pascal mittels eines Operatorkonzepts und Implementierung der Standardfunktionen mit optimaler Genauigkeit} \toenglish {Embedding a General Computer Arithmetic in Pascal by Means of an Operator Concept and the Implementation of Elementary Functions with Optimal Accuracy} \endtoenglish", type = "Dissertation", school = "Universit{\"a}t Karlsruhe", address = "Karlsruhe, Germany", pages = "????", year = "1980", bibdate = "Sun Oct 25 10:29:29 1998", bibsource = "garbo.uwasa.fi:/pc/doc-soft/fpbiblio.txt; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nj, } @Article{Waldecker:1980:NSR, author = "D. E. Waldecker", title = "Nonrestoring Square Root with Simplified Answer Generation", journal = j-IBM-TDB, volume = "22", number = "11", pages = "4807--4808", month = apr, year = "1980", CODEN = "IBMTAA", ISSN = "0018-8689", bibdate = "Thu Sep 1 10:15:41 1994", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nj, fjournal = "IBM Technical Disclosure Bulletin", } @InCollection{Wang:1980:ISM, author = "J. Y. Wang", booktitle = "{SHARE-54}, Anaheim, {CA}, March 3, 1980", title = "On the Improvement of Some Mathematical Subroutines in the {IBM S/360 FORTRAN IV} Libraries", volume = "1", publisher = "????", address = "????", pages = "75--77", year = "1980", DOI = "", ISBN = "", ISBN-13 = "", LCCN = "", bibdate = "Fri Sep 20 14:23:46 2024", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "", acknowledgement = ack-nhfb, remark = "Cited in \cite[Reference 7]{Agarwal:1986:NSV} in elefunt.bib and fparith.bib.", } @InProceedings{Ahmed:1981:VSA, author = "H. M. Ahmed and M. Morf and D. T. L. Lee and P. H. Ang", editor = "????", booktitle = "{Proceedings of 1981 ICASSP. Atlanta. GA, 1981}", title = "A {VLSI} speech analysis chip set based on square-root normalized ladder forms", publisher = "????", address = "????", pages = "648--653", year = "1981", DOI = "", ISBN = "", ISBN-13 = "", LCCN = "", bibdate = "Wed Oct 29 10:23:36 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "", acknowledgement = ack-nhfb, } @Article{Andrews:1981:EFM, author = "M. Andrews and D. Jaeger and S. F. McCormick and G. D. Taylor", title = "Evaluation of Functions on Microcomputers: $ \exp (x) $", journal = j-COMPUT-MATH-APPL, volume = "7", number = "6", pages = "503--508", year = "1981", CODEN = "CMAPDK", DOI = "https://doi.org/10.1016/0898-1221(81)90034-1", ISSN = "0898-1221 (print), 1873-7668 (electronic)", ISSN-L = "0898-1221", bibdate = "Thu Sep 15 18:40:45 1994", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; https://www.math.utah.edu/pub/tex/bib/computmathappl1980.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "http://www.sciencedirect.com/science/article/pii/0898122181900341", acknowledgement = ack-nj # " and " # ack-nhfb, fjournal = "Computers and Mathematics with Applications", journal-URL = "http://www.sciencedirect.com/science/journal/08981221", } @Article{Arscott:1981:LBB, author = "F. M. Arscott", title = "The land beyond {Bessel}: a survey of higher special functions", journal = j-LECT-NOTES-MATH, volume = "846", pages = "26--45", year = "1981", CODEN = "LNMAA2", DOI = "https://doi.org/10.1007/BFb0089822", ISBN = "3-540-10569-7 (print), 3-540-38538-X (e-book)", ISBN-13 = "978-3-540-10569-5 (print), 978-3-540-38538-7 (e-book)", ISSN = "0075-8434 (print), 1617-9692 (electronic)", ISSN-L = "0075-8434", bibdate = "Fri May 9 19:07:31 MDT 2014", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/lnm1980.bib", URL = "http://link.springer.com/chapter/10.1007/BFb0089822/", acknowledgement = ack-nhfb, book-DOI = "https://doi.org/10.1007/BFb0089819", book-URL = "http://www.springerlink.com/content/978-3-540-38538-7", fjournal = "Lecture Notes in Mathematics", journal-URL = "http://link.springer.com/bookseries/304", } @Article{Banuelos:1981:PCF, author = "Alicia Ba{\~n}uelos and Ricardo Angel Depine and Roberto Claudio Mancini", title = "A program for computing the {Fermi--Dirac} functions", journal = j-COMP-PHYS-COMM, volume = "21", number = "3", pages = "315--322", month = jan, year = "1981", CODEN = "CPHCBZ", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", bibdate = "Wed Feb 5 09:02:18 MST 2014", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/d/dirac-p-a-m.bib; https://www.math.utah.edu/pub/bibnet/authors/f/fermi-enrico.bib; https://www.math.utah.edu/pub/tex/bib/compphyscomm1980.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/0010465581900126", acknowledgement = ack-nhfb, fjournal = "Computer Physics Communications", journal-URL = "http://www.sciencedirect.com/science/journal/00104655/", } @Article{Baratella:1981:ABF, author = "P. Baratella and M. Garetto and G. Vinardi", title = "Approximation of the {Bessel} function {$ J_\nu (x) $} by numerical integration", journal = j-J-COMPUT-APPL-MATH, volume = "7", number = "2", pages = "87--91", month = jun, year = "1981", CODEN = "JCAMDI", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Sat Feb 25 11:59:21 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath1980.bib", URL = "http://www.sciencedirect.com/science/article/pii/0771050X81900401", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @Article{Barnett:1981:ARI, author = "A. R. Barnett", title = "An algorithm for regular and irregular {Coulomb} and {Bessel} functions of real order to machine accuracy", journal = j-COMP-PHYS-COMM, volume = "21", number = "3", pages = "297--314", month = jan, year = "1981", CODEN = "CPHCBZ", DOI = "https://doi.org/10.1016/0010-4655(81)90011-4", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", bibdate = "Thu Apr 24 10:35:27 2008", bibsource = "https://www.math.utah.edu/pub/tex/bib/compphyscomm1980.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", abstract = "We describe an algorithm to evaluate a wide class of functions and their derivatives, to extreme precision (25--30S) if required, which does not use any function calls other than square root. The functions are the Coulomb functions of positive argument ($ F_\lambda (x, \eta) $, $ G_\lambda (x, \eta) $, $ x > 0 $, $ \eta $, $ \lambda $ real) and hence, as special cases with $ \eta = 0 $, the cylindrical Bessel functions ($ J_\mu (x) $, $ Y_\mu (x) $, $ x > 0 $, $ \mu $ real), the spherical Bessel functions ($ i_\lambda (x) $, $ y_\lambda (x) $, $ x > 0 $, $ \lambda $ real), Airy functions of negative argument $ \textrm {Ai}( - x) $, $ \textrm {Bi}( - x) $ and others. The present method has a number of attractive features: both the regular and irregular solution are calculated, all others of the functions can be produced from a specified minimum (not necessarily zero) to a specified maximum, functions of a single order can be found without all of the orders from zero, the derivatives of the functions arise naturally in the solution and are readily available, the results are available to different precisions from the same subroutine (in contrast to rational approximation techniques) and the methods can be used for estimating final accuracies. In addition, the sole constant required in the algorithm is $ \pi $, no precalculated arrays of coefficients are needed, and the final accuracy is not dependent on that of other subroutines. The method works most efficiently in the region $ x \approx 0.5 $ to $ x \approx 1000 $ but outside this region the results are still reliable, even though the number of iterations within the subroutine rises. Even in these more asymptotic regions the unchanged algorithm can be used with known accuracy to test other specific subroutines more appropriate to these regions. The algorithm uses the recursion relations satisfied by the Coulomb functions and contains a significant advance over Miller's method for evaluating the ratio of successive minimal solutions ($ F_\lambda + 1 / F_\lambda $ ). It relies on the evaluation of two continued fractions and no infinite series is required for normalisation: instead the Wronskian is used.", acknowledgement = ack-nhfb, fjournal = "Computer Physics Communications", journal-URL = "http://www.sciencedirect.com/science/journal/00104655", } @Article{Bice:1981:AAS, author = "P. K. Bice", title = "Algorithm adds square root to micro's arithmetic capability", journal = j-ELECTRONIC-DESIGN, volume = "29", number = "11", pages = "146", month = may, year = "1981", CODEN = "ELODAW", ISSN = "0013-4872 (print), 1944-9550 (electronic)", ISSN-L = "0013-4872", bibdate = "Thu Sep 1 10:15:42 1994", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nj, fjournal = "Electronic Design", } @Article{Branrers:1981:RAZ, author = "M. Branrers and R. Piessens and M. {De Meue}", title = "Rational approximations for zeros of {Bessel} functions", journal = j-J-COMPUT-PHYS, volume = "42", number = "2", pages = "403--405", month = aug, year = "1981", CODEN = "JCTPAH", DOI = "https://doi.org/10.1016/0021-9991(81)90253-9", ISSN = "0021-9991 (print), 1090-2716 (electronic)", ISSN-L = "0021-9991", bibdate = "Sun Jan 1 15:59:07 MST 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputphys1980.bib", URL = "http://www.sciencedirect.com/science/article/pii/0021999181902539", acknowledgement = ack-nhfb, fjournal = "Journal of Computational Physics", journal-URL = "http://www.sciencedirect.com/science/journal/00219991/", } @Article{Campbell:1981:BFR, author = "J. B. Campbell", title = "{Bessel} functions {$ I_\nu (z) $} and {$ K_\nu (z) $} of real order and complex argument", journal = j-COMP-PHYS-COMM, volume = "24", number = "1", pages = "97--105", month = sep # "\slash " # oct, year = "1981", CODEN = "CPHCBZ", DOI = "https://doi.org/10.1016/0010-4655(81)90109-0", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", bibdate = "Mon Feb 13 10:27:59 MST 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/compphyscomm1980.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/0010465581901090", acknowledgement = ack-nhfb, fjournal = "Computer Physics Communications", journal-URL = "http://www.sciencedirect.com/science/journal/00104655", } @Article{Carlson:1981:AAI, author = "B. C. Carlson and Elaine M. Notis", title = "{Algorithm 577}: Algorithms for Incomplete Elliptic Integrals [{S21}]", journal = j-TOMS, volume = "7", number = "3", pages = "398--403", month = sep, year = "1981", CODEN = "ACMSCU", DOI = "https://doi.org/10.1145/355958.355970", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Mon Aug 29 22:58:27 1994", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", keywords = "$R$-functions; elliptic integrals; inverse circular functions; inverse hyperbolic functions; logarithms", } @Article{Chen:1981:AFD, author = "Gang Chen", title = "An attempt to find the derivatives of elementary functions using the algorithmic language {BCY}. ({Chinese})", journal = "Zhejiang Daxue Xuebao", volume = "3", pages = "141--149", year = "1981", MRclass = "26A09", MRnumber = "714 524", bibdate = "Sat Jan 11 17:44:01 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, } @InProceedings{Davis:1981:EFA, author = "Diane F. Davis", title = "Elementary Functions on an Array Processor", crossref = "IEEE:1981:PIS", pages = "170--178", year = "1981", bibdate = "Mon May 19 13:30:58 1997", bibsource = "ftp://ftp.ira.uka.de/pub/bibliography/Theory/arith.bib.gz; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, } @Article{Delahaye:1981:ACS, author = "J.-P. Delahaye", title = "Acc{\'e}l{\'e}ration de la convergence des suites dont le rapport des erreurs est born{\'e}. ({French}) [{Convergence} acceleration for sequences with bounded error ratios]", journal = j-CALCOLO, volume = "18", number = "2", pages = "1--116", year = "1981", CODEN = "CDABAE", DOI = "https://doi.org/10.1007/BF02576491", ISSN = "0008-0624 (print), 1126-5434 (electronic)", ISSN-L = "0008-0624", MRclass = "65B05", MRnumber = "647821 (83a:65004)", bibdate = "Thu Dec 1 17:36:45 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Calcolo", journal-URL = "http://link.springer.com/journal/10092", keywords = "convergence acceleration", language = "{French}", } @Article{Drachman:1981:TTH, author = "B. Drachman and C. I. Chuang", title = "A table of two hundred zeros of the derivative of the modified {Bessel} function {$ K_n(z) $} and a graph of their distribution", journal = j-J-COMPUT-APPL-MATH, volume = "7", number = "3", pages = "167--171", month = sep, year = "1981", CODEN = "JCAMDI", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Sat Feb 25 11:59:22 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath1980.bib", URL = "http://www.sciencedirect.com/science/article/pii/0771050X81900140", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @InProceedings{Farmwald:1981:HBE, author = "P. Michael Farmwald", title = "High Bandwidth Evaluation of Elementary Functions", crossref = "IEEE:1981:PIS", pages = "139--142", year = "1981", bibdate = "Mon May 19 13:30:58 1997", bibsource = "ftp://ftp.ira.uka.de/pub/bibliography/Theory/arith.bib.gz; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, } @Article{Fettis:1981:TEHb, author = "Henry E. Fettis", title = "Table errata: {{\em Handbook of elliptic integrals for engineers and physicists} [second edition, Springer, New York, 1971 and MR {\bf 43} \#3506] by P. F. Byrd and M. D. Friedman}", journal = j-MATH-COMPUT, volume = "36", number = "153", pages = "317--317", month = jan, year = "1981", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "65A05", MRnumber = "82a:65008a", bibdate = "Sat Apr 12 15:32:35 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Fettis:1981:TETb, author = "Henry E. Fettis", title = "Table errata: {{\em A table of the complete elliptic integral of the first kind for complex values of the modulus, Part I} [Rep. No. ARL 69-0172, Aerospace Res. Lab., Wright--Patterson Air Force Base, Ohio, 1969; MR {\bf 40} \#6725] by Fettis and J. C. Caslin}", journal = j-MATH-COMPUT, volume = "36", number = "153", pages = "318", month = jan, year = "1981", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "318.65A05", MRnumber = "82a:65010", bibdate = "Sat Jan 11 13:29:06 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Fransen:1981:ACH, author = "Arne Frans{\'e}n", title = "Addendum and corrigendum to: {``High-precision values of the gamma function and of some related coefficients''} {[Math. Comp. {\bf 34} (1980), no. 150, 553--566, MR 81f:65004] by Frans{\'e}n and S. Wrigge}", journal = j-MATH-COMPUT, volume = "37", number = "155", pages = "233--235", month = jul, year = "1981", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "65A05 (65D20)", MRnumber = "82m:65002", bibdate = "Sat Apr 01 10:12:58 2000", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", note = "See \cite{Fransen:1980:HPV}.", acknowledgement = ack-nhfb, fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Fredette:1981:RES, author = "G. Fredette", title = "68000 routine extracts square roots", journal = j-EDN, volume = "26", number = "16", pages = "185--194", month = aug, year = "1981", CODEN = "EDNSBH", ISSN = "0012-7515, 0364-6637", bibdate = "Thu Sep 1 10:15:56 1994", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nj, fjournal = "EDN", } @TechReport{Fullerton:1981:FUMa, author = "L. W. Fullerton", title = "{FNLIB} User's Manual Explanatory Table of Contents", type = "Technical report", number = "CSTR 92", institution = inst-ATT-BELL, address = inst-ATT-BELL:adr, month = mar, year = "1981", bibdate = "Sat Feb 05 17:39:14 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, } @TechReport{Fullerton:1981:FUMb, author = "L. W. Fullerton", title = "{FNLIB} User's Manual", type = "Technical report", number = "CSTR 95", institution = inst-ATT-BELL, address = inst-ATT-BELL:adr, month = mar, year = "1981", bibdate = "Sat Feb 05 17:39:14 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, } @Article{Gasper:1981:SFB, author = "George Gasper", title = "Summation Formulas for Basic Hypergeometric Series", journal = j-SIAM-J-MATH-ANA, volume = "12", number = "2", pages = "196--200", month = mar, year = "1981", CODEN = "SJMAAH", ISSN = "0036-1410 (print), 1095-7154 (electronic)", ISSN-L = "0036-1410", MRclass = "33A30", MRnumber = "82a:33005", MRreviewer = "L. J. Slater", bibdate = "Sun Nov 28 19:22:39 MST 2010", bibsource = "http://epubs.siam.org/sam-bin/dbq/toc/SIMA/12/2; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/siamjmathana.bib", acknowledgement = ack-nhfb, fjournal = "SIAM Journal on Mathematical Analysis", journal-URL = "http://epubs.siam.org/sima", } @Article{Gatto:1981:NEM, author = "M. A. Gatto and J. B. Seery", title = "Numerical evaluation of the modified {Bessel} functions {$I$} and {$K$}", journal = j-COMPUT-MATH-APPL, volume = "7", number = "3", pages = "203--209", month = "????", year = "1981", CODEN = "CMAPDK", ISSN = "0898-1221 (print), 1873-7668 (electronic)", ISSN-L = "0898-1221", bibdate = "Wed Mar 1 18:51:20 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/computmathappl1980.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/0898122181900808", acknowledgement = ack-nhfb, fjournal = "Computers and Mathematics with Applications", journal-URL = "http://www.sciencedirect.com/science/journal/08981221/", } @Article{Glasser:1981:CBS, author = "M. L. Glasser", title = "A Class of {Bessel} Summations", journal = j-MATH-COMPUT, volume = "37", number = "156", pages = "499--501", month = oct, year = "1981", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "33A40 (42A16 44A15)", MRnumber = "82j:33015", MRreviewer = "B. D. Agrawal", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, classcodes = "B0220 (Mathematical analysis); C1120 (Mathematical analysis)", corpsource = "Dept. of Math. and Computer Sci., Clarkson Coll., Potsdam, NY, USA", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "Bessel functions; Bessel summations; infinite series", treatment = "T Theoretical or Mathematical", } @Article{Haavie:1981:RUT, author = "Tore H{\aa}vie", title = "Remarks on a unified theory for classical and generalized interpolation and extrapolation", journal = j-BIT, volume = "21", number = "4", pages = "465--474", month = dec, year = "1981", CODEN = "BITTEL, NBITAB", DOI = "https://doi.org/10.1007/BF01932843", ISSN = "0006-3835 (print), 1572-9125 (electronic)", ISSN-L = "0006-3835", MRclass = "41A05 (65D05)", MRnumber = "83d:41004", MRreviewer = "G. M{\"u}hlbach", bibdate = "Wed Jan 4 18:52:17 MST 2006", bibsource = "http://springerlink.metapress.com/openurl.asp?genre=issue&issn=0006-3835&volume=21&issue=4; https://www.math.utah.edu/pub/tex/bib/bit.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.springerlink.com/openurl.asp?genre=article&issn=0006-3835&volume=21&issue=4&spage=465", acknowledgement = ack-nhfb, fjournal = "BIT (Nordisk tidskrift for informationsbehandling)", journal-URL = "http://link.springer.com/journal/10543", } @Article{Hill:1981:RSD, author = "G. W. Hill", title = "Remark on ``{Algorithm 395: Student's $t$-Distribution}''", journal = j-TOMS, volume = "7", number = "2", pages = "247--249", month = jun, year = "1981", CODEN = "ACMSCU", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Tue Feb 06 05:28:18 1996", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "See \cite{Hill:1970:AASa,Hill:1970:AASb,elLozy:1979:RAS}.", acknowledgement = ack-nhfb, fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", } @Article{Hill:1981:RSQ, author = "G. W. Hill", title = "Remark on ``{Algorithm 396: Student's $t$-Quantiles}''", journal = j-TOMS, volume = "7", number = "2", pages = "250--251", month = jun, year = "1981", CODEN = "ACMSCU", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Tue Feb 06 05:28:19 1996", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "See \cite{Hill:1970:AASb}.", acknowledgement = ack-nhfb, fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", } @Article{Hough:1981:API, author = "D. Hough", title = "Application of the proposed {IEEE 754} standard for floating-point arithmetic", journal = j-COMPUTER, volume = "14", number = "3", pages = "70--74", year = "1981", CODEN = "CPTRB4", ISSN = "0018-9162 (print), 1558-0814 (electronic)", ISSN-L = "0018-9162", bibdate = "Mon May 19 13:30:58 1997", bibsource = "ftp://ftp.ira.uka.de/pub/bibliography/Distributed/QLD/1981.bib.gz; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, annote = "Various features of the proposed standard provide an especially convenient environment for programming numerical procedures such as the familiar elementary functions.", bydate = "MB", byrev = "Le", country = "USA", date = "14/06/82", descriptors = "Computer arithmetic; floating point; computation structure; method; application; standard", enum = "1418", fjournal = "Computer", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=2", location = "RWTH-AC-DFV: Bibl.", references = "10", revision = "21/04/91", } @Article{Iserles:1981:ARA, author = "A. Iserles and M. J. D. Powell", title = "On the {$A$}-acceptability of rational approximations that interpolate the exponential function", journal = j-IMA-J-NUMER-ANAL, volume = "1", number = "3", pages = "241--251", month = jul, year = "1981", CODEN = "IJNADH", DOI = "https://doi.org/10.1093/imanum/1.3.241", ISSN = "0272-4979 (print), 1464-3642 (electronic)", ISSN-L = "0272-4979", MRclass = "65D15 (30E10)", MRnumber = "83a:65015 (641308)", bibdate = "Sat Dec 23 17:06:35 MST 2000", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/p/powell-m-j-d.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/imajnumeranal.bib; MathSciNet database", acknowledgement = ack-nhfb, author-dates = "Michael James David Powell (29 July 1936--19 April 2015)", fjournal = "IMA Journal of Numerical Analysis", journal-URL = "http://imajna.oxfordjournals.org/content/by/year", } @Article{James:1981:LTS, author = "D. G. James", title = "Linear transformations of the second elementary function", journal = j-LIN-AND-MULT-ALGEBRA, volume = "10", number = "4", pages = "347--349", year = "1981", CODEN = "LNMLAZ", ISSN = "0308-1087 (print), 1563-5139 (electronic)", ISSN-L = "0308-1087", MRclass = "15A69 (10C15)", MRnumber = "83c:15023", MRreviewer = "E. W. Ellers", bibdate = "Sat Jan 11 17:44:01 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Linear and Multilinear Algebra", journal-URL = "http://www.tandfonline.com/loi/glma20", } @Article{Kunz:1981:QZ, author = "W. Kunz", title = "{Quadratwurzel mit dem $ \mu $P Z80} \toenglish {Square Roots with the Z80 Microprocessor} \endtoenglish", journal = j-ELECTRONIK, volume = "7", pages = "109--110", year = "1981", CODEN = "EKRKAR", ISSN = "0013-5658", bibdate = "Fri Sep 16 16:30:41 1994", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nj, fjournal = "Elektronik", } @Book{Lewin:1981:PAF, author = "Leonard Lewin", title = "Polylogarithms and Associated Functions", publisher = pub-NORTH-HOLLAND, address = pub-NORTH-HOLLAND:adr, pages = "xvii + 359", year = "1981", ISBN = "0-444-00550-1", ISBN-13 = "978-0-444-00550-2", LCCN = "QA342 .L47", bibdate = "Fri Jun 16 13:56:23 MDT 2023", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; z3950.loc.gov:7090/Voyager", acknowledgement = ack-nhfb, author-dates = "22-Jul-1919--13-Aug-2007", author-url = "https://en.wikipedia.org/wiki/Leonard_Lewin_(telecommunications_engineer)", remark = "Lightly revised and retitled edition of \cite{Lewin:1958:DAF}.", subject = "Logarithmic functions", } @InCollection{Longman:1981:DCA, author = "I. M. Longman", booktitle = "{Pad{\'e} approximation and its applications, Amsterdam 1980 (Amsterdam, 1980)}", title = "Difficulties of convergence acceleration", volume = "888", publisher = pub-SV, address = pub-SV:adr, pages = "273--289", year = "1981", MRclass = "65B10", MRnumber = "649102 (83d:65013)", bibdate = "Thu Dec 1 17:36:45 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", series = "Lecture Notes in Mathematics", acknowledgement = ack-nhfb, keywords = "convergence acceleration", } @Article{Maino:1981:CPC, author = "G. Maino and E. Menapace and A. Ventura", title = "Computation of parabolic cylinder functions by means of a {Tricomi} expansion", journal = j-J-COMPUT-PHYS, volume = "40", number = "2", pages = "294--304", month = apr, year = "1981", CODEN = "JCTPAH", DOI = "https://doi.org/10.1016/0021-9991(81)90211-4", ISSN = "0021-9991 (print), 1090-2716 (electronic)", ISSN-L = "0021-9991", bibdate = "Sun Jan 1 15:59:06 MST 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputphys1980.bib", URL = "http://www.sciencedirect.com/science/article/pii/0021999181902114", acknowledgement = ack-nhfb, fjournal = "Journal of Computational Physics", journal-URL = "http://www.sciencedirect.com/science/journal/00219991/", } @Article{Markov:1981:ICE, author = "S. M. Markov", title = "On the interval computation of elementary functions", journal = j-C-R-ACAD-BULGARE-SCI, volume = "34", number = "3", pages = "319--322", year = "1981", CODEN = "DBANAD", ISSN = "0366-8681", MRclass = "65G10", MRnumber = "83e:65084", MRreviewer = "David F. Griffiths", bibdate = "Sat Jan 11 17:44:01 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Comptes rendus de l'Acad{\'e}mie bulgare des sciences", } @Article{McCullagh:1981:RCS, author = "Peter McCullagh", title = "A rapidly convergent series for computing $ \psi (z) $ and its derivatives", journal = j-MATH-COMPUT, volume = "36", number = "153", pages = "247--248", month = jan, year = "1981", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "65D20 (33A15)", MRnumber = "81m:65028", MRreviewer = "J. Gregor", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", URL = "http://www.jstor.org/stable/2007741", acknowledgement = ack-nhfb, classcodes = "C4120 (Functional analysis)", corpsource = "Imperial Coll. of Sci. and Technol., London, UK", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "(mathematics); convergence; function evaluation; log gamma function; poles; poles and zeros; rapidly convergent series; series; series expansion; uniformly convergent", treatment = "T Theoretical or Mathematical", } @Article{Moon:1981:AFC, author = "Wooil Moon", title = "{Airy} function with complex arguments", journal = j-COMP-PHYS-COMM, volume = "22", number = "4", pages = "411--417", month = may, year = "1981", CODEN = "CPHCBZ", DOI = "https://doi.org/10.1016/0010-4655(81)90138-7", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", bibdate = "Mon Feb 13 09:27:06 MST 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/compphyscomm1980.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/0010465581901387", acknowledgement = ack-nhfb, fjournal = "Computer Physics Communications", journal-URL = "http://www.sciencedirect.com/science/journal/00104655", } @TechReport{Morris:1981:NDL, author = "Alfred H. {Morris, Jr.}", title = "{NSWC\slash DL} Library of Mathematics Subroutines", type = "Report", number = "NSWC/TR-79-338", institution = "Naval Surface Warfare Center", address = "Dahlgren, VA 22448-5000, USA; Silver Spring, MD 20903-5000, USA", pages = "235", year = "1981", bibdate = "Tue Jun 13 08:47:19 2023", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fortran2.bib", note = "See also later editions \cite{Morris:1987:NLM,Morris:1990:NLM,Morris:1993:NLM}.", URL = "https://ntrl.ntis.gov/NTRL/dashboard/searchResults/titleDetail/ADA108106.xhtml", abstract = "The NSWC/DL library is a library of general-purpose FORTRAN subroutines that provide a basic computational capability in a variety of mathematical activities. Although intended for use on the CDC 6000 series computers, emphasis has been placed on the transportability of the codes. Subroutines are available in the following areas: Elementary Operations, Geometry, Special Functions, Polynomials, Solutions of Nonlinear Equations, Vectors, Matrices, Sparse Matrices, Eigenvalues and Eigenvectors, Least Squares Solutions of Linear Equations, Optimization, Transforms, Approximation of Functions, Curve Fitting, Surface Fitting over Rectangular Grids, Surface Fitting over Arbitrarily Positioned Data Points, Numerical Integration, Ordinary Differential Equations/Initial Value Problems, and Random Number Generation.", acknowledgement = ack-nhfb, } @InProceedings{Peng:1981:AES, author = "Hong Peng", title = "Algorithms for extracting square roots and cube roots", crossref = "IEEE:1981:PSC", pages = "121--126", year = "1981", bibdate = "Thu Sep 01 11:37:17 1994", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "http://www.acsel-lab.com/arithmetic/arith5/papers/ARITH5_Peng.pdf", abstract = "This paper describes a kind of algorithms for fast extracting square roots and cube roots, their mathematical proofs, their revised algorithm formulae, and hardware implementation of the square root algorithm. These algorithms may be of no significance for large scale computer with fast division. But I am sure that it is effective and economical to apply these algorithms to the circuit designs of some mini- and microcomputers with general multiplication and division, such as nonrestoring division.", acknowledgement = ack-nj, keywords = "ARITH-5", } @Article{Razaz:1981:RAF, author = "M. Razaz and J. L. Schonfelder", title = "Remark on ``{Algorithm} 498: {Airy} Functions Using {Chebyshev} Series Approximations''", journal = j-TOMS, volume = "7", number = "3", pages = "404--405", month = sep, year = "1981", CODEN = "ACMSCU", DOI = "https://doi.org/10.1145/355958.355971", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Tue Aug 30 00:28:07 1994", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "See \cite{Prince:1975:AAF}.", acknowledgement = ack-nhfb, fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", } @Article{Schulten:1981:NAE, author = "Z. Schulten and R. G. Gordon and D. G. M. Anderson", title = "A numerical algorithm for the evaluation of {Weber} parabolic cylinder functions {$ U(a, x) $}, {$ V(a, x) $}, and {$ W(a, \pm x) $}", journal = j-J-COMPUT-PHYS, volume = "42", number = "2", pages = "213--237", month = aug, year = "1981", CODEN = "JCTPAH", DOI = "https://doi.org/10.1016/0021-9991(81)90241-2", ISSN = "0021-9991 (print), 1090-2716 (electronic)", ISSN-L = "0021-9991", bibdate = "Sun Jan 1 15:59:07 MST 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputphys1980.bib", URL = "http://www.sciencedirect.com/science/article/pii/0021999181902412", acknowledgement = ack-nhfb, fjournal = "Journal of Computational Physics", journal-URL = "http://www.sciencedirect.com/science/journal/00219991/", } @Article{Shepherd:1981:CA, author = "M. M. Shepherd and J. G. Laframboise", title = "{Chebyshev} Approximation of $ (1 + 2 x) \exp (x^2) \erfc (x) $ in $ 0 \leq x < \infty $", journal = j-MATH-COMPUT, volume = "36", number = "153", pages = "249--253", month = jan, year = "1981", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "65D20", MRnumber = "83c:65029", MRreviewer = "John P. Coleman", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp1980.bib; JSTOR database", acknowledgement = ack-nhfb, classcodes = "C4130 (Interpolation and function approximation)", corpsource = "York Univ., Toronto, Ont., Canada", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "(1+2x)exp(x/sup 2/)erfc x; Chebyshev approximation; Chebyshev expansion; erfc x; single", treatment = "T Theoretical or Mathematical", } @Article{Smith:1981:ERA, author = "J. M. Smith and F. W. J. Olver and D. W. Lozier", title = "Extended-Range Arithmetic and Normalized {Legendre} Polynomials", journal = j-TOMS, volume = "7", number = "1", pages = "93--105", month = mar, year = "1981", CODEN = "ACMSCU", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", MRclass = "65D20 (65G05)", MRnumber = "83a:65017", bibdate = "Mon Aug 29 22:02:12 1994", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://doi.acm.org/10.1145/355934.355940", acknowledgement = ack-nhfb, fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", keywords = "angular momentum; extended-range arithmetic; Legendre polynomials; overflow; underflow", } @Article{Steinberg:1981:LSE, author = "D. Steinberg and M. Rodeh", title = "A layout for the shuffle-exchange network with {$ O(N^2 / \log^{3 / 2N}) $} area", journal = j-IEEE-TRANS-COMPUT, volume = "C-30", number = "12", pages = "977--982", month = dec, year = "1981", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/TC.1981.1675738", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, ajournal = "IEEE Trans. Comput.", catcode = "C.1.2; G.1.2", CRclass = "C.1.2 Multiple Data Stream Architectures (Multiprocessors); C.1.2 Interconnection architectures; G.1.2 Approximation; G.1.2 Elementary function approximation", descriptor = "Computer Systems Organization, PROCESSOR ARCHITECTURES, Multiple Data Stream Architectures (Multiprocessors), Interconnection architectures; Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation", fjournal = "IEEE Transactions on Computers", genterm = "design", guideno = "06519", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", jrldate = "Dec. 1981", subject = "C. Computer Systems Organization; C.1 PROCESSOR ARCHITECTURES; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS", } @Article{Steinhardt:1981:ASF, author = "Paul J. Steinhardt and P. Chaudhari", title = "{Airy} stress function for atomic models", journal = j-J-COMPUT-PHYS, volume = "42", number = "2", pages = "266--276", month = aug, year = "1981", CODEN = "JCTPAH", DOI = "https://doi.org/10.1016/0021-9991(81)90244-8", ISSN = "0021-9991 (print), 1090-2716 (electronic)", ISSN-L = "0021-9991", bibdate = "Sun Jan 1 15:59:07 MST 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputphys1980.bib", URL = "http://www.sciencedirect.com/science/article/pii/0021999181902448", acknowledgement = ack-nhfb, fjournal = "Journal of Computational Physics", journal-URL = "http://www.sciencedirect.com/science/journal/00219991/", } @InProceedings{Taylor:1981:CHD, author = "George S. Taylor", title = "Compatible hardware for division and square root", crossref = "IEEE:1981:PSC", pages = "127--134", year = "1981", bibdate = "Mon Sep 16 16:30:51 2002", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "http://www.acsel-lab.com/arithmetic/arith5/papers/ARITH5_Taylor.pdf", abstract = "Hardware for radix four division and radix two square root is shared in a processor designed to implement the proposed IEEE floating-point standard. The division hardware looks ahead to find the next quotient digit in parallel with the next partial remainder. An 8-bit ALU estimates the next remainder's leading bits. The quotient digit look-up table is addressed with a truncation of the estimate rather than a truncation of the full partial remainder. The estimation ALU and the look-up table are asymmetric for positive and negative remainders. This asymmetry reduces the width of the ALU and the number of minterms in the logic equations for the look-up table. The square root algorithm obtains the correctly rounded result in about two division times using small extensions to the division hardware.", acknowledgement = ack-nhfb, keywords = "ARITH-5", } @Article{Temme:1981:ECH, author = "N. M. Temme", title = "On the expansion of confluent hypergeometric functions in terms of {Bessel} functions", journal = j-J-COMPUT-APPL-MATH, volume = "7", number = "1", pages = "27--32", month = mar, year = "1981", CODEN = "JCAMDI", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Sat Feb 25 11:59:21 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath1980.bib", URL = "http://www.sciencedirect.com/science/article/pii/0771050X81900048", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @Article{Terras:1981:ASI, author = "Riho Terras", title = "Algorithms for some integrals of {Bessel} functions and multivariate {Gaussian} integrals", journal = j-J-COMPUT-PHYS, volume = "41", number = "1", pages = "192--199", month = may, year = "1981", CODEN = "JCTPAH", DOI = "https://doi.org/10.1016/0021-9991(81)90087-5", ISSN = "0021-9991 (print), 1090-2716 (electronic)", ISSN-L = "0021-9991", bibdate = "Sun Jan 1 15:59:06 MST 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputphys1980.bib", URL = "http://www.sciencedirect.com/science/article/pii/0021999181900875", acknowledgement = ack-nhfb, fjournal = "Journal of Computational Physics", journal-URL = "http://www.sciencedirect.com/science/journal/00219991/", } @Article{Terras:1981:MAI, author = "Riho Terras", title = "A {Miller} algorithm for an incomplete {Bessel} function", journal = j-J-COMPUT-PHYS, volume = "39", number = "1", pages = "233--240", month = jan, year = "1981", CODEN = "JCTPAH", DOI = "https://doi.org/10.1016/0021-9991(81)90147-9", ISSN = "0021-9991 (print), 1090-2716 (electronic)", ISSN-L = "0021-9991", bibdate = "Sun Jan 1 15:59:04 MST 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputphys1980.bib", URL = "http://www.sciencedirect.com/science/article/pii/0021999181901479", acknowledgement = ack-nhfb, fjournal = "Journal of Computational Physics", journal-URL = "http://www.sciencedirect.com/science/journal/00219991/", } @Article{Vogelius:1981:DRM, author = "M. Vogelius and I. Babuska", title = "On a dimensional reduction method. {II}. {Some} approximation-theoretic results", journal = j-MATH-COMPUT, volume = "37", number = "155", pages = "47--68", month = jul, year = "1981", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, catcode = "G.1.2; G.1.7", CRclass = "G.1.2 Approximation; G.1.2 Elementary function approximation; G.1.7 Ordinary Differential Equations; G.1.7 Boundary value problems", descriptor = "Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation; Mathematics of Computing, NUMERICAL ANALYSIS, Ordinary Differential Equations, Boundary value problems", fjournal = "Mathematics of Computation", genterm = "theory", guideno = "09396", journal-URL = "http://www.ams.org/mcom/", jrldate = "July 1981", subject = "G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS", } @Book{Wimp:1981:STT, author = "Jet Wimp", title = "Sequence Transformations and Their Applications", volume = "154", publisher = pub-ACADEMIC, address = pub-ACADEMIC:adr, pages = "xix + 257", year = "1981", ISBN = "0-08-095662-9 (e-book), 0-12-757940-0", ISBN-13 = "978-0-08-095662-6 (e-book), 978-0-12-757940-5", LCCN = "QA292 .W54", bibdate = "Thu Dec 1 11:08:47 MST 2011", bibsource = "fsz3950.oclc.org:210/WorldCat; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", series = "Mathematics in Science and Engineering", URL = "http://public.eblib.com/EBLPublic/PublicView.do?ptiID=453177", acknowledgement = ack-nhfb, subject = "Sequences (Mathematics) Transformations (Mathematics) Numerical analysis; Acceleration of convergence", tableofcontents = "Front Cover \\ Sequence Transformations and Their Applications \\ Copyright Page \\ Contents \\ Preface \\ Acknowledgments \\ Notation \\ Chapter 1. Sequences and Series \\ Chapter 2. Linear Transformations \\ Chapter 3. Linear Lozenge Methods \\ Chapter 4. Optimal Methods and Methods Based on Power Series \\ Chapter 5. Nonlinear Lozenges \\ Iteration Sequences \\ Chapter 6. The Schmidt Transformation \\ The e-Algorithm \\ Chapter 7. Aitken's $d^2$-Process and Related Methods \\ Chapter 8. Lozenge Algorithms and the Theory of Continued Fractions \\ Chapter 9. Other Lozenge Algorithms and Nonlinear MethodsChapter 10. The Brezinski--H{\aa}vie ProtocolChapter 11. The Brezinski--H{\aa}vie Protocol and Numerical Quadrature \\ Chapter 12. Probabilistic Methods \\ Chapter 13. Multiple Sequences \\ Appendix \\ Bibliography \\ Index", } @Article{Andrews:1982:MMS, author = "M. Andrews", title = "Mathematical Microprocessor Software: a $ \sqrt {x} $ Comparison", journal = j-IEEE-MICRO, volume = "2", number = "3", pages = "63--79", month = aug, year = "1982", CODEN = "IEMIDZ", DOI = "https://doi.org/10.1109/MM.1982.290970", ISSN = "0272-1732 (print), 1937-4143 (electronic)", ISSN-L = "0272-1732", bibdate = "Thu Dec 14 06:08:58 MST 2000", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; garbo.uwasa.fi:/pc/doc-soft/fpbiblio.txt; https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeemicro.bib; Science Citation Index database (1980--2000)", acknowledgement = ack-nj # " and " # ack-nhfb, ajournal = "IEEE Micro", classcodes = "C4130 (Interpolation and function approximation); C6150G (Diagnostic, testing, debugging and evaluating systems)", corpsource = "Colorado State Univ., Fort Collins, CO, USA", fjournal = "IEEE Micro", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=40", keywords = "16-bit machines; 8-bit machine; accuracy; Chen method; computer testing; Cordic method; direct method; function approximation; hardware; Intel 8080; Newton method; PDP-11/20; software requirements; speed; square-roots", remark = "This article compares instruction-level implementation on the Intel 8080 and DEC PDP-11/20 of the square root using five methods: direct, CORDIC, Chen's, and two variations of Newton's iteration. The concluding paragraph says: ``The conclusions are fairly obvious: Even with the availability of hardware features most suitable to any of the other methods, Newton's method remains the technique of choice. Although the advent of hardware multiple-bit-shift instructions will alter this comparison somewhat, Newton's method, with optimal initialization, will again prove to be the best when hardware multiply\slash divide becomes generally available.''", treatment = "T Theoretical or Mathematical", } @Article{Armengou:1982:ASQ, author = "Santiago Zarzuela Armengou", title = "About some questions of differential algebra concerning to elementary functions", journal = "Publ. Sec. Mat. Univ. Aut{\`o}noma Barcelona", volume = "26", number = "1", pages = "5--15", year = "1982", MRclass = "12H05", MRnumber = "86i:12009", MRreviewer = "Michael F. Singer", bibdate = "Sat Jan 11 17:44:01 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, } @Article{Barnett:1982:CCB, author = "A. R. Barnett", title = "{COULFG}: {Coulomb} and {Bessel} functions and their derivatives, for real arguments, by {Steed}'s method", journal = j-COMP-PHYS-COMM, volume = "27", number = "2", pages = "147--166", month = aug, year = "1982", CODEN = "CPHCBZ", DOI = "https://doi.org/10.1016/0010-4655(82)90070-4", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", bibdate = "Mon Feb 13 10:28:03 MST 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/compphyscomm1980.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/0010465582900704", acknowledgement = ack-nhfb, fjournal = "Computer Physics Communications", journal-URL = "http://www.sciencedirect.com/science/journal/00104655", } @Article{Barnett:1982:CFE, author = "A. R. Barnett", title = "Continued-fraction evaluation of {Coulomb} functions {$ F_\lambda (\eta, x) $}, {$ G_\lambda (\eta, x) $} and their derivatives", journal = j-J-COMPUT-PHYS, volume = "46", number = "2", pages = "171--188", month = may, year = "1982", CODEN = "JCTPAH", DOI = "https://doi.org/10.1016/0021-9991(82)90012-2", ISSN = "0021-9991 (print), 1090-2716 (electronic)", ISSN-L = "0021-9991", bibdate = "Sun Jan 1 15:59:11 MST 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputphys1980.bib", URL = "http://www.sciencedirect.com/science/article/pii/0021999182900122", acknowledgement = ack-nhfb, fjournal = "Journal of Computational Physics", journal-URL = "http://www.sciencedirect.com/science/journal/00219991/", } @Article{Barnett:1982:HPE, author = "A. R. Barnett", title = "High-precision evaluation of the regular and irregular {Coulomb} wavefunctions", journal = j-J-COMPUT-APPL-MATH, volume = "8", number = "1", pages = "29--33", month = mar, year = "1982", CODEN = "JCAMDI", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Sat Feb 25 11:59:22 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath1980.bib", URL = "http://www.sciencedirect.com/science/article/pii/0771050X82900043", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @TechReport{Bazarov:1982:EEF, author = "M. B. Bazarov and Yu. I. Shokin and Z. Kh. Yuldashev", title = "On the Evaluation of Elementary Functions in Interval Analysis (In {Russian})", institution = "Applied Mathematics and Mechanics, Tashkent State Univ.", address = "Tashkent, USSR", pages = "26--31", year = "1982", bibdate = "Fri Jan 12 11:37:56 1996", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-jr # "\slash " # ack-nhfb, } @Article{Belevitch:1982:SIT, author = "V. Belevitch and J. Boersma", title = "On {Stieltjes} integral transforms involving {$ \Gamma $}-functions", journal = j-MATH-COMPUT, volume = "38", number = "157", pages = "223--226", month = jan, year = "1982", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "44A15 (33A15 33A45)", MRnumber = "83d:44001", MRreviewer = "V. M. Bhise", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, classcodes = "B0230 (Integral transforms); C1130 (Integral transforms)", corpsource = "Philips Res Lab., Brussels, Belgium", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "Gamma functions; integral transforms; Stieltjes transforms; systematic classification; transforms", treatment = "T Theoretical or Mathematical", } @Article{Borwein:1982:MNT, author = "P. B. Borwein", title = "On a method of {Newman} and a theorem of {Bernstein}", journal = j-J-APPROX-THEORY, volume = "34", number = "1", pages = "37--41", month = jan, year = "1982", CODEN = "JAXTAZ", ISSN = "0021-9045 (print), 1096-0430 (electronic)", ISSN-L = "0021-9045", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, catcode = "G.1.2", CRclass = "G.1.2 Approximation; G.1.2 Elementary function approximation", descriptor = "Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation", fjournal = "Journal of Approximation Theory", genterm = "theory", guideno = "06012", journal-URL = "http://www.sciencedirect.com/science/journal/00219045", jrldate = "Jan. 1982", subject = "G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS", } @Article{Brezinski:1982:ASG, author = "C. Brezinski", title = "{Algorithm 585}: a Subroutine for the General Interpolation and Extrapolation Problems", journal = j-TOMS, volume = "8", number = "3", pages = "290--301", month = sep, year = "1982", CODEN = "ACMSCU", DOI = "https://doi.org/10.1145/356004.356008", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Mon Aug 29 23:49:19 1994", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib", acknowledgement = ack-nhfb, fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", keywords = "algorithms; convergence acceleration; extrapolation; interpolation; least squares approximation; Neville--Aitken scheme", } @Article{Brezinski:1982:SNC, author = "Claude Brezinski", title = "Some New Convergence Acceleration Methods", journal = j-MATH-COMPUT, volume = "39", number = "159", pages = "133--145", month = jul, year = "1982", CODEN = "MCMPAF", DOI = "https://doi.org/10.2307/2007624", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "65B05", MRnumber = "658218 (83f:65003)", bibdate = "Thu Dec 1 17:36:45 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "convergence acceleration", } @Article{Burr:1982:CCR, author = "S. A. Burr", title = "Computing cube roots when a fast square root is available", journal = j-COMPUT-MATH-APPL, volume = "8", number = "3", pages = "181--183", month = "????", year = "1982", CODEN = "CMAPDK", DOI = "https://doi.org/10.1016/0898-1221(82)90041-4", ISSN = "0898-1221 (print), 1873-7668 (electronic)", ISSN-L = "0898-1221", bibdate = "Wed Mar 1 18:51:22 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/computmathappl1980.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/0898122182900414", acknowledgement = ack-nhfb, fjournal = "Computers and Mathematics with Applications", journal-URL = "http://www.sciencedirect.com/science/journal/08981221/", } @Book{Carroll:1982:TST, author = "Robert Wayne Carroll", title = "Transmutation, scattering theory, and special functions", volume = "87; 69", publisher = pub-NORTH-HOLLAND, address = pub-NORTH-HOLLAND:adr, pages = "x + 457", year = "1982", ISBN = "0-444-86426-1 (paperback)", ISBN-13 = "978-0-444-86426-0 (paperback)", LCCN = "QA1 .N86 no. 87; QA329", bibdate = "Sat Oct 30 18:29:29 MDT 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; z3950.loc.gov:7090/Voyager", series = "North-Holland mathematics studies", acknowledgement = ack-nhfb, subject = "Transmutation operators; Scattering (Mathematics); Inverse problems (Differential equations); Functions, Special", } @Article{Chambers:1982:UBF, author = "Ll. G. Chambers", title = "An upper bound for the first zero of {Bessel} functions", journal = j-MATH-COMPUT, volume = "38", number = "158", pages = "589--591", month = apr, year = "1982", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "33A65", MRnumber = "83h:33011", MRreviewer = "S. Ahmed", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @InProceedings{Cody:1982:TTP, author = "W. J. Cody", title = "Transportable test procedures for elementary function software", crossref = "Mulvey:1982:EMP", pages = "236--247", year = "1982", bibdate = "Thu Nov 17 06:39:08 1994", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/c/cody-william-j.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-wjc, } @Article{Crick:1982:IRB, author = "S. E. Crick", title = "Inclusion relations for {Bernstein} quasi-analytic classes", journal = j-J-APPROX-THEORY, volume = "34", number = "4", pages = "375--379", month = apr, year = "1982", CODEN = "JAXTAZ", ISSN = "0021-9045 (print), 1096-0430 (electronic)", ISSN-L = "0021-9045", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, catcode = "G.1.2", CRclass = "G.1.2 Approximation; G.1.2 Elementary function approximation", descriptor = "Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation", fjournal = "Journal of Approximation Theory", genterm = "algorithms; theory", guideno = "07841", journal-URL = "http://www.sciencedirect.com/science/journal/00219045", jrldate = "April 1982", subject = "G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS", } @Article{Cruz:1982:ZHF, author = "Andr{\'e}s Cruz and Javier Sesma", title = "Zeros of the {Hankel} function of real order and of its derivative", journal = j-MATH-COMPUT, volume = "39", number = "160", pages = "639--645", month = oct, year = "1982", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "33A40", MRnumber = "83j:33005", MRreviewer = "S. Ahmed", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, classcodes = "B0290D (Functional analysis); B0290F (Interpolation and function approximation); C4120 (Functional analysis); C4130 (Interpolation and function approximation)", corpsource = "Dept. de Fisica Teorica, Univ. de Zaragoza, Zaragoza, Spain", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "approximation; derivative; evaluation; function approximation; function evaluation; Hankel function; poles and zeros; real order; trajectories; zeros", treatment = "T Theoretical or Mathematical", } @Article{Danielopoulos:1982:CEP, author = "S. D. Danielopoulos", title = "On the Cost of Evaluating Polynomials and Their Derivatives", journal = j-COMPUTING, volume = "29", number = "4", pages = "373--380", year = "1982", CODEN = "CMPTA2", ISSN = "0010-485X (print), 1436-5057 (electronic)", ISSN-L = "0010-485X", MRclass = "68C25 (68C20)", MRnumber = "84a:68039", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); Compendex database; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; MathSciNet database", acknowledgement = ack-nhfb, affiliation = "Univ of Ioannina, Greece", catcode = "G.1.2; G.4; G.1.0", CRclass = "G.1.2 Approximation; G.1.2 Elementary function approximation; G.4 Algorithm analysis; G.1.0 General; G.1.0 Computer arithmetic", descriptor = "Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation; Mathematics of Computing, MATHEMATICAL SOFTWARE, Algorithm analysis; Mathematics of Computing, NUMERICAL ANALYSIS, General, Computer arithmetic", fjournal = "Computing", genterm = "economics; theory", guideno = "04049", journal-URL = "http://link.springer.com/journal/607", journalabr = "Computing (Vienna/New York)", jrldate = "1982", keywords = "mathematical techniques", subject = "G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; G. Mathematics of Computing; G.4 MATHEMATICAL SOFTWARE; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS", } @Article{Decker:1982:CAN, author = "D. W. Decker and C. T. Kelley", title = "Convergence acceleration for {Newton}'s method at singular points", journal = j-SIAM-J-NUMER-ANAL, volume = "19", number = "1", pages = "219--229", month = feb, year = "1982", CODEN = "SJNAAM", ISSN = "0036-1429 (print), 1095-7170 (electronic)", ISSN-L = "0036-1429", MRclass = "65H05 (58E07 65J15)", MRnumber = "83e:65090", MRreviewer = "Michael Pr{\"u}fer", bibdate = "Fri Oct 16 06:57:22 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "SIAM Journal on Numerical Analysis", journal-URL = "http://epubs.siam.org/sinum", keywords = "convergence acceleration", } @Article{Delahaye:1982:SLC, author = "J. P. Delahaye and B. Germain-Bonne", title = "The set of logarithmically convergent sequences cannot be accelerated", journal = j-SIAM-J-NUMER-ANAL, volume = "19", number = "4", pages = "840--844", month = aug, year = "1982", CODEN = "SJNAAM", ISSN = "0036-1429 (print), 1095-7170 (electronic)", ISSN-L = "0036-1429", MRclass = "65B99", MRnumber = "83f:65005", bibdate = "Fri Oct 16 06:57:22 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/siamjnumeranal.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "SIAM Journal on Numerical Analysis", journal-URL = "http://epubs.siam.org/sinum", keywords = "convergence acceleration", } @Article{Epstein:1982:UAF, author = "C. Epstein and W. L. Miranker and T. J. Rivlin", title = "Ultra-arithmetic {I}: function data types", journal = j-MATH-COMPUT-SIMUL, volume = "24", number = "1", pages = "1--18", month = feb, year = "1982", CODEN = "MCSIDR", ISSN = "0378-4754 (print), 1872-7166 (electronic)", ISSN-L = "0378-4754", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, catcode = "G.1; G.1.2; G.1.2", CRclass = "G.1.5 Roots of Nonlinear Equations; G.1.2 Approximation; G.1.2 Chebyshev approximation and theory; G.1.2 Approximation; G.1.2 Elementary function approximation", descriptor = "Mathematics of Computing, NUMERICAL ANALYSIS, Roots of Nonlinear Equations; Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Chebyshev approximation and theory; Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation", fjournal = "Mathematics and Computers in Simulation", genterm = "algorithms", guideno = "09324", journal-URL = "http://www.sciencedirect.com/science/journal/03784754", jrldate = "Feb. 1982", subject = "G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS", } @Article{Epstein:1982:UAI, author = "C. Epstein and W. L. Miranker and T. J. Rivlin", title = "Ultra-arithmetic {II}: intervals of polynomials", journal = j-MATH-COMPUT-SIMUL, volume = "24", number = "1", pages = "19--29", month = feb, year = "1982", CODEN = "MCSIDR", ISSN = "0378-4754 (print), 1872-7166 (electronic)", ISSN-L = "0378-4754", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, catcode = "G.1.2; G.1.1; G.1.0", CRclass = "G.1.2 Approximation; G.1.2 Elementary function approximation; G.1.1 Interpolation; G.1.1 Spline and piecewise polynomial interpolation; G.1.0 General; G.1.0 Error analysis", descriptor = "Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation; Mathematics of Computing, NUMERICAL ANALYSIS, Interpolation, Spline and piecewise polynomial interpolation; Mathematics of Computing, NUMERICAL ANALYSIS, General, Error analysis", fjournal = "Mathematics and Computers in Simulation", genterm = "algorithms", guideno = "09325", journal-URL = "http://www.sciencedirect.com/science/journal/03784754", jrldate = "Feb. 1982", subject = "G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS", } @Article{Fenton:1982:RCM, author = "J. D. Fenton and R. S. Gardiner-Garden", title = "Rapidly-convergent methods for evaluating elliptic integrals and theta and elliptic functions", journal = j-J-AUSTRAL-MATH-SOC-SER-B, volume = "24", number = "1", pages = "47--58", month = jul, year = "1982", CODEN = "JAMMDU", DOI = "https://doi.org/10.1017/S0334270000003301", ISSN = "0334-2700", ISSN-L = "0334-2700", bibdate = "Fri Apr 26 16:13:14 MDT 2019", bibsource = "https://www.math.utah.edu/pub/tex/bib/anziamj.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "https://www.cambridge.org/core/journals/anziam-journal/article/rapidlyconvergent-methods-for-evaluating-elliptic-integrals-and-theta-and-elliptic-functions/2D993C9A7C9EB1D4B61B856E22B45A34", acknowledgement = ack-nhfb, ajournal = "J. Austral Math. Soc. Ser. B", fjournal = "Journal of the Australian Mathematical Society. Series B, Applied Mathematics", journal-URL = "http://journals.cambridge.org/action/displayJournal?jid=ANZ", onlinedate = "17 February 2009", } @Article{Fernandez:1982:HCI, author = "F. M. Fern{\'a}ndez and A. Mes{\'o}n and E. A. Castro", title = "Hypervirial calculation of integrals involving {Bessel} functions", journal = j-J-MATH-PHYS, volume = "23", number = "2", pages = "254--255", month = feb, year = "1982", CODEN = "JMAPAQ", DOI = "https://doi.org/10.1063/1.525345", ISSN = "0022-2488 (print), 1089-7658 (electronic), 1527-2427", ISSN-L = "0022-2488", MRclass = "81C05 (33A40 82A51)", MRnumber = "83c:81010", bibdate = "Sat Oct 29 18:19:03 MDT 2011", bibsource = "http://jmp.aip.org/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jmathphys1980.bib", URL = "http://jmp.aip.org/resource/1/jmapaq/v23/i2/p254_s1", acknowledgement = ack-nhfb, fjournal = "Journal of Mathematical Physics", journal-URL = "http://jmp.aip.org/", pagecount = "2", } @Article{Gawronski:1982:ACF, author = "W. Gawronski and U. Stadtm{\"u}ller", title = "Approximation of continuous functions by generalized {Favard} operators", journal = j-J-APPROX-THEORY, volume = "34", number = "4", pages = "384--396", month = apr, year = "1982", CODEN = "JAXTAZ", ISSN = "0021-9045 (print), 1096-0430 (electronic)", ISSN-L = "0021-9045", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, catcode = "G.1.2", CRclass = "G.1.2 Approximation; G.1.2 Elementary function approximation", descriptor = "Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation", fjournal = "Journal of Approximation Theory", genterm = "theory; algorithms", guideno = "06038", journal-URL = "http://www.sciencedirect.com/science/journal/00219045", jrldate = "April 1982", subject = "G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS", } @Article{Gessel:1982:SEH, author = "Ira Gessel and Dennis Stanton", title = "Strange Evaluations of Hypergeometric Series", journal = j-SIAM-J-MATH-ANA, volume = "13", number = "2", pages = "295--308", month = mar, year = "1982", CODEN = "SJMAAH", ISSN = "0036-1410 (print), 1095-7154 (electronic)", ISSN-L = "0036-1410", MRclass = "33A30", MRnumber = "83c:33002", MRreviewer = "C. L. Parihar", bibdate = "Sun Nov 28 19:22:53 MST 2010", bibsource = "http://epubs.siam.org/sam-bin/dbq/toc/SIMA/13/2; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/siamjmathana.bib", acknowledgement = ack-nhfb, fjournal = "SIAM Journal on Mathematical Analysis", journal-URL = "http://epubs.siam.org/sima", } @Article{Gordon:1982:RAN, author = "H. T. Gordon", title = "Rough approximation numerical algorithms", journal = j-DDJ, volume = "7", number = "7", pages = "54--56", month = jul, year = "1982", CODEN = "DDJOEB", ISSN = "1044-789X", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); http://www.ddj.com/index/author/index.htm; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, catcode = "G.1.0; G.1.2", CRclass = "G.1.0 General; G.1.0 Numerical algorithms; G.1.2 Approximation; G.1.2 Elementary function approximation", descriptor = "Mathematics of Computing, NUMERICAL ANALYSIS, General, Numerical algorithms; Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation", fjournal = "Dr. Dobb's Journal of Software Tools", guideno = "04547", jrldate = "July 1982", subject = "G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS", } @Article{Greaves:1982:AHM, author = "G. Greaves", title = "An algorithm for the {Hausdorff} moment problem", journal = j-NUM-MATH, volume = "39", number = "2", pages = "231--238", month = aug, year = "1982", CODEN = "NUMMA7", ISSN = "0029-599X (print), 0945-3245 (electronic)", ISSN-L = "0029-599X", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, catcode = "G.1.2", CRclass = "G.1.2 Approximation; G.1.2 Elementary function approximation", descriptor = "Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation", fjournal = "Numerische Mathematik", genterm = "algorithms", guideno = "07489", journal-URL = "http://link.springer.com/journal/211", jrldate = "Aug. 1982", subject = "G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS", } @Article{Hawkes:1982:ANT, author = "Alan G. Hawkes", title = "Approximating the Normal Tail", journal = j-J-R-STAT-SOC-SER-D-STATISTICIAN, volume = "31", number = "3", pages = "231--236", month = sep, year = "1982", CODEN = "????", DOI = "https://doi.org/10.2307/2987989", ISSN = "0039-0526 (print), 1467-9884 (electronic)", ISSN-L = "0039-0526", bibdate = "Thu Jan 22 18:10:21 MST 2015", bibsource = "http://www.jstor.org/stable/i349970; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jrss-d-1980.bib", URL = "http://www.jstor.org/stable/2987989", acknowledgement = ack-nhfb, fjournal = "Journal of the Royal Statistical Society. Series D (The Statistician)", journal-URL = "http://www.jstor.org/journals/00390526.html", } @Article{Hermann:1982:SAG, author = "Robert Hermann", title = "Some algebraic, geometric, and system-theoretic properties of the {Special Functions} of mathematical physics", journal = j-J-MATH-PHYS, volume = "23", number = "7", pages = "1282--1294", month = jul, year = "1982", CODEN = "JMAPAQ", DOI = "https://doi.org/10.1063/1.525511", ISSN = "0022-2488 (print), 1089-7658 (electronic), 1527-2427", ISSN-L = "0022-2488", bibdate = "Sat Oct 29 18:19:10 MDT 2011", bibsource = "http://jmp.aip.org/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jmathphys1980.bib", URL = "http://jmp.aip.org/resource/1/jmapaq/v23/i7/p1282_s1", abstract = "It is known that many of the Special Functions of mathematical physics appear as matrix elements of Lie group representations. This paper is concerned with a beginning attack on the converse problem, i.e., finding conditions that a given function be a matrix element. The methods used are based on a combination of ideas from system theory, functional analysis, Lie theory, differential algebra, and linear ordinary differential equation theory. A key idea is to attach a symbol as an element of a commutative algebra. In favorable cases, this symbol defines a Riemann surface, and a meromorphic differential form on that surface. The topological and analytical invariants attached to this form play a key role in system theory. The Lie algebras of the groups appear as linear differential operators on this Riemann surface. Finally, it is shown how the Picard--Vessiot--Infeld--Hull theory of factorization of linear differential operators leads to realization of many Special Functions as matrix representations of group representations.", acknowledgement = ack-nhfb, fjournal = "Journal of Mathematical Physics", journal-URL = "http://jmp.aip.org/", pagecount = "13", } @TechReport{Kahan:1982:BCC, author = "W. Kahan", title = "Branch Cuts for Complex Elementary Functions", type = "Technical Report", number = "PAM-105", institution = inst-CPAM-UCB, address = inst-CPAM-UCB:adr, year = "1982", bibdate = "Mon May 19 13:30:58 1997", bibsource = "ftp://ftp.ira.uka.de/pub/bibliography/Math/Matrix.bib.gz; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, keywords = "na, elementary function", } @Article{Keener:1982:CLB, author = "L. L. Keener", title = "Characterizing local best {SAIN} approximations", journal = j-J-APPROX-THEORY, volume = "36", number = "1", pages = "55--63", month = sep, year = "1982", CODEN = "JAXTAZ", ISSN = "0021-9045 (print), 1096-0430 (electronic)", ISSN-L = "0021-9045", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, catcode = "G.1.2; G.1.2", CRclass = "G.1.2 Approximation; G.1.2 Chebyshev approximation and theory; G.1.2 Approximation; G.1.2 Elementary function approximation", descriptor = "Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Chebyshev approximation and theory; Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation", fjournal = "Journal of Approximation Theory", genterm = "theory", guideno = "06074", journal-URL = "http://www.sciencedirect.com/science/journal/00219045", jrldate = "Sept. 1982", subject = "G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS", } @Article{Lassey:1982:CCI, author = "Keith R. Lassey", title = "On the computation of certain integrals containing the modified {Bessel} function $ {I}_0 (\xi) $", journal = j-MATH-COMPUT, volume = "39", number = "160", pages = "625--637", month = oct, year = "1982", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "65D20", MRnumber = "83j:65029", MRreviewer = "Walter Gautschi", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, classcodes = "B0290F (Interpolation and function approximation); B0290M (Numerical integration and differentiation); C4130 (Interpolation and function approximation); C4160 (Numerical integration and differentiation)", corpsource = "Inst. of Nuclear Sci., DSIR, Lower Hutt, New Zealand", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "approximation; approximation theory; convergence of numerical methods; convergent series; function; function approximation; integration; limiting behaviour; modified Bessel; numerical integration; one-dimensional integrals; two-dimensional integrals", treatment = "T Theoretical or Mathematical", } @Article{Ling:1982:EIH, author = "Chih Bing Ling and Ming Jing Wu", title = "Evaluation of integrals of {Howland} type involving a {Bessel} function", journal = j-MATH-COMPUT, volume = "38", number = "157", pages = "215--222", month = jan, year = "1982", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "65A05 (65D20)", MRnumber = "82m:65003", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, classcodes = "B0220 (Mathematical analysis); B0290M (Numerical integration and differentiation); C1120 (Mathematical analysis); C4160 (Numerical integration and differentiation)", corpsource = "Dept. of Maths., Virginia Polytech. Inst. and State Univ., Blacksburg, VA, USA", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "10D; 20D; accuracy; Bessel function; Bessel functions; Howland type; integrals; integration; tabulated values", treatment = "T Theoretical or Mathematical", } @Article{McCormick:1982:EFM, author = "S. F. McCormick and G. D. Taylor and D. V. Pryor", title = "Evaluation of Functions on Microcomputers: $ \ln (x) $", journal = j-COMPUT-MATH-APPL, volume = "8", number = "5", pages = "389--392", month = "????", year = "1982", CODEN = "CMAPDK", ISSN = "0898-1221 (print), 1873-7668 (electronic)", ISSN-L = "0898-1221", bibdate = "Wed Mar 1 18:51:23 MST 2017", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; https://www.math.utah.edu/pub/tex/bib/computmathappl1980.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "http://www.sciencedirect.com/science/article/pii/0898122182900323", acknowledgement = ack-nj # " and " # ack-nhfb, fjournal = "Computers and Mathematics with Applications", journal-URL = "http://www.sciencedirect.com/science/journal/08981221/", xxmonth = "(none)", } @InCollection{Mori:1982:ARS, author = "S. Mori and C. Y. Suen", editor = "Ching Y. Suen and Renato {De Mori}", key = "Scanners", booktitle = "Computer analysis and perception: vol. I, {Visual} signals", title = "Automatic recognition of symbols and architecture of the recognition unit", publisher = pub-CRC, address = pub-CRC:adr, bookpages = "various", pages = "17--40", year = "1982", ISBN = "0-8493-6305-5 (vol. 1), 0-8493-6306-3 (vol. 2)", ISBN-13 = "978-0-8493-6305-4 (vol. 1), 978-0-8493-6306-1 (vol. 2)", LCCN = "TA1650 .C65 1982", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, catcode = "I.5; G.1.2; C.3; B.7; I.5.4", CRclass = "I.5.2 Design Methodology; G.1.2 Approximation; G.1.2 Elementary function approximation; C.3 Signal processing systems; B.7.1 Types and Design Styles; I.5.4 Applications; I.5.4 Signal processing", descriptor = "Computing Methodologies, PATTERN RECOGNITION, Design Methodology; Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation; Computer Systems Organization, SPECIAL-PURPOSE AND APPLICATION-BASED SYSTEMS, Signal processing systems; Hardware, INTEGRATED CIRCUITS, Types and Design Styles; Computing Methodologies, PATTERN RECOGNITION, Applications, Signal processing", genterm = "documentation; theory; design", guideno = "01691", subject = "I. Computing Methodologies; I.5 PATTERN RECOGNITION; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; C. Computer Systems Organization; C.3 SPECIAL-PURPOSE AND APPLICATION-BASED SYSTEMS; B. Hardware; B.7 INTEGRATED CIRCUITS; I. Computing Methodologies; I.5 PATTERN RECOGNITION", } @Article{Oklobdzija:1982:LSR, author = "V. G. Oklobdzija and M. D. Ercegovac", title = "An On-Line Square Root Algorithm", journal = j-IEEE-TRANS-COMPUT, volume = "C-31", number = "1", pages = "70--75", month = jan, year = "1982", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/TC.1982.1675887", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Sun Jul 10 10:33:09 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput1980.bib", URL = "https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=1675887", acknowledgement = ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", } @Article{Ollin:1982:CFE, author = "H. Z. Ollin and I. Gerst", title = "Classes of functions with explicit best uniform approximations", journal = j-J-APPROX-THEORY, volume = "34", number = "3", pages = "264--276", month = mar, year = "1982", CODEN = "JAXTAZ", ISSN = "0021-9045 (print), 1096-0430 (electronic)", ISSN-L = "0021-9045", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, catcode = "G.1.2", CRclass = "G.1.2 Approximation; G.1.2 Elementary function approximation", descriptor = "Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation", fjournal = "Journal of Approximation Theory", genterm = "theory", guideno = "06030", journal-URL = "http://www.sciencedirect.com/science/journal/00219045", jrldate = "March 1982", subject = "G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS", } @Book{Patel:1982:HND, author = "Jagdish K. Patel and Campbell B. Read", title = "Handbook of the Normal Distribution", volume = "40", publisher = pub-DEKKER, address = pub-DEKKER:adr, pages = "ix + 337", year = "1982", ISBN = "0-8247-1541-1", ISBN-13 = "978-0-8247-1541-0", LCCN = "QA273.6 .P373", bibdate = "Sat Dec 16 17:22:16 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; z3950.loc.gov:7090/Voyager", series = "Statistics, textbooks and monographs", acknowledgement = ack-nhfb, subject = "Gaussian distribution", } @Article{Piessens:1982:ABF, author = "R. Piessens and Maria Branders", title = "Approximation for {Bessel} functions and their application in the computation of {Hankel} transforms", journal = j-COMPUT-MATH-APPL, volume = "8", number = "4", pages = "305--311", month = "????", year = "1982", CODEN = "CMAPDK", ISSN = "0898-1221 (print), 1873-7668 (electronic)", ISSN-L = "0898-1221", bibdate = "Wed Mar 1 18:51:22 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/computmathappl1980.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/0898122182900128", acknowledgement = ack-nhfb, fjournal = "Computers and Mathematics with Applications", journal-URL = "http://www.sciencedirect.com/science/journal/08981221/", } @Article{Piessens:1982:ACB, author = "R. Piessens", title = "Automatic computation of {Bessel} function integrals", journal = j-COMP-PHYS-COMM, volume = "25", number = "3", pages = "289--295", month = mar, year = "1982", CODEN = "CPHCBZ", DOI = "https://doi.org/10.1016/0010-4655(82)90024-8", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", bibdate = "Mon Feb 13 10:28:01 MST 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/compphyscomm1980.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/0010465582900248", acknowledgement = ack-nhfb, fjournal = "Computer Physics Communications", journal-URL = "http://www.sciencedirect.com/science/journal/00104655", } @Article{Rack:1982:GIV, author = "H.-J Rack", title = "A generalization of an inequality of {V. Markov} to multivariate polynomials", journal = j-J-APPROX-THEORY, volume = "35", number = "1", pages = "94--97", month = may, year = "1982", CODEN = "JAXTAZ", ISSN = "0021-9045 (print), 1096-0430 (electronic)", ISSN-L = "0021-9045", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, catcode = "G.1.2", CRclass = "G.1.2 Approximation; G.1.2 Elementary function approximation", descriptor = "Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation", fjournal = "Journal of Approximation Theory", genterm = "theory", guideno = "06047", journal-URL = "http://www.sciencedirect.com/science/journal/00219045", jrldate = "May 1982", subject = "G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS", } @Article{Rix:1982:UQA, author = "P. Rix", title = "{Universeller Quad\-rat\-wurz\-el-Al\-go\-rith\-mus} \toenglish {Universal Square Root Algorithms} \endtoenglish", journal = j-ELECTRONIK, volume = "23", pages = "81--82", year = "1982", CODEN = "EKRKAR", ISSN = "0013-5658", bibdate = "Fri Sep 16 16:30:41 1994", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nj, fjournal = "Elektronik", } @PhdThesis{Rockey:1982:DMS, author = "S. A. Rockey", title = "Discrete methods of state approximation, parameter identification and optimal control for hereditary systems", type = "{Ph.D} Thesis", school = "Brown University", address = "Providence, RI", pages = "208", year = "1982", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, catcode = "G.1.2; G.1.2; G.1.5", CRclass = "G.1.2 Approximation; G.1.2 Elementary function approximation; G.1.2 Approximation; G.1.2 Linear approximation; G.1.5 Roots of Nonlinear Equations; G.1.5 Convergence", descriptor = "Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation; Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Linear approximation; Mathematics of Computing, NUMERICAL ANALYSIS, Roots of Nonlinear Equations, Convergence", guideno = "15449", source = "UMI order no. DA8228325", subject = "G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS", } @Article{Sommer:1982:EPL, author = "M. Sommer", title = "Existence of pointwise-{Lipschitz}-continuous selections of the metric projection for a class of {$Z$}-spaces", journal = j-J-APPROX-THEORY, volume = "34", number = "2", pages = "115--130", month = feb, year = "1982", CODEN = "JAXTAZ", ISSN = "0021-9045 (print), 1096-0430 (electronic)", ISSN-L = "0021-9045", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, catcode = "G.1.2", CRclass = "G.1.2 Approximation; G.1.2 Elementary function approximation", descriptor = "Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation", fjournal = "Journal of Approximation Theory", genterm = "theory", guideno = "06018", journal-URL = "http://www.sciencedirect.com/science/journal/00219045", jrldate = "Feb. 1982", subject = "G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS", } @PhdThesis{Wang:1982:AME, author = "J.-L Wang", title = "Asymptotically minimax estimators for distributions with increasing failure rate", type = "{Ph.D} Thesis", school = "University of California, Berkeley", address = "Berkeley, CA, USA", pages = "42", year = "1982", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, catcode = "G.1.2", CRclass = "G.1.2 Approximation; G.1.2 Elementary function approximation", descriptor = "Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation", genterm = "design", guideno = "15084", source = "UMI order no. DA8300696", subject = "G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS", } @Article{Whitley:1982:MBI, author = "R. Whitley", title = "{Markov} and {Bernstein}'s inequalities, and compact and strictly singular operators", journal = j-J-APPROX-THEORY, volume = "34", number = "3", pages = "277--285", month = mar, year = "1982", CODEN = "JAXTAZ", ISSN = "0021-9045 (print), 1096-0430 (electronic)", ISSN-L = "0021-9045", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, catcode = "G.1.2", CRclass = "G.1.2 Approximation; G.1.2 Elementary function approximation", descriptor = "Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation", fjournal = "Journal of Approximation Theory", genterm = "theory", guideno = "06031", journal-URL = "http://www.sciencedirect.com/science/journal/00219045", jrldate = "March 1982", subject = "G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS", } @Book{Wilkes:1982:PPE, author = "M. V. (Maurice Vincent) Wilkes and David J. Wheeler and Stanley Gill", title = "The Preparation of Programs for an Electronic Digital Computer: with Special Reference to the {EDSAC} and the Use of a Library of Subroutines", volume = "1", publisher = pub-TOMASH, address = pub-TOMASH:adr, pages = "xxxi + 167", year = "1982", ISBN = "0-262-23118-2 (MIT Press 1984), 0-938228-03-X", ISBN-13 = "978-0-262-23118-3 (MIT Press 1984), 978-0-938228-03-5", LCCN = "QA76.6 .W545 1982", bibdate = "Mon Feb 10 11:33:59 MST 2020", bibsource = "fsz3950.oclc.org:210/WorldCat; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", note = "With a new introduction by Martin Campbell-Kelly.", series = "Charles Babbage Institute reprint series for the history of computing", acknowledgement = ack-nhfb, } @Article{Wills:1982:RCA, author = "C. A. Wills and J. M. Blair and P. L. Ragde", title = "Rational {Chebyshev} approximations for the {Bessel} functions $ {J}_0 (x) $, $ {J}_1 (x) $, $ {Y}_0 (x) $, $ {Y}_1 (x) $", journal = j-MATH-COMPUT, volume = "39", number = "160", pages = "617--623", month = oct, year = "1982", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "65D20 (33A40 41A50)", MRnumber = "83j:65030", MRreviewer = "C. W. Clenshaw", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, classcodes = "B0260 (Optimisation techniques); B0290F (Interpolation and function approximation); C1180 (Optimisation techniques); C4130 (Interpolation and function approximation)", corpsource = "AEG Ltd., Chalk River Nuclear Labs., Ont., Canada", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "approximations; Bessel functions; Chebyshev; Chebyshev approximation; formulae; McMahon asymptotic; minimax techniques; near-minimax rational approximation", treatment = "T Theoretical or Mathematical", } @PhdThesis{Wimp:1982:CMS, author = "Jet (Jesse Jet) Wimp", title = "Computational methods and special functions", type = "{D.Sc.} thesis", school = "University of Edinburgh", address = "Edinburgh, UK", year = "1982", bibdate = "Thu Dec 01 11:15:32 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, } @Article{Amos:1983:APFa, author = "D. E. Amos", title = "{Algorithm 609}: a Portable {FORTRAN} Subroutine for the {Bickley} Functions {$ \hbox {Ki}_n(x) $}", journal = j-TOMS, volume = "9", number = "4", pages = "480--493", month = dec, year = "1983", CODEN = "ACMSCU", DOI = "https://doi.org/10.1145/356056.356064", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", MRclass = "65D20 (33A70 65-04)", MRnumber = "87a:65044", bibdate = "Sun Sep 4 20:00:39 1994", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib", acknowledgement = ack-nhfb, fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", reviewer = "Marietta J. Tretter", } @Article{Amos:1983:APFb, author = "Donald E. Amos", title = "{Algorithm 610}: a Portable {FORTRAN} Subroutine for Derivatives of the Psi Function", journal = j-TOMS, volume = "9", number = "4", pages = "494--502", month = dec, year = "1983", CODEN = "ACMSCU", DOI = "https://doi.org/10.1145/356056.356065", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", MRclass = "65D20", MRnumber = "791 979", bibdate = "Sun Sep 4 20:00:39 1994", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib", acknowledgement = ack-nhfb, catcode = "G.1.0; G.1; G; D.3.2", CRclass = "G.1.0 General; G.1.0 Numerical algorithms; G.1.m Miscellaneous; D.3.2 Language Classifications; D.3.2 FORTRAN", descriptor = "Mathematics of Computing, NUMERICAL ANALYSIS, General, Numerical algorithms; Mathematics of Computing, NUMERICAL ANALYSIS, Miscellaneous; Mathematics of Computing, MISCELLANEOUS; Software, PROGRAMMING LANGUAGES, Language Classifications, FORTRAN", fjournal = "ACM Transactions on Mathematical Software (TOMS)", genterm = "ALGORITHMS", guideno = "02212", journal-URL = "https://dl.acm.org/loi/toms", subject = "G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; G. Mathematics of Computing; G.m MISCELLANEOUS; D. Software; D.3 PROGRAMMING LANGUAGES", } @TechReport{Amos:1983:CBFa, author = "Donald E. Amos", title = "Computation of {Bessel} functions of complex argument", type = "Technical Report", number = "SAND83-0086", institution = "Sandia National Laboratories", address = "Albuquerque, NM, USA", year = "1983", bibdate = "Fri Apr 25 14:41:51 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, } @TechReport{Amos:1983:CBFb, author = "Donald E. Amos", title = "Computation of {Bessel} functions of complex argument and large order", type = "Technical Report", number = "SAND83-0643", institution = "Sandia National Laboratories", address = "Albuquerque, NM, USA", year = "1983", bibdate = "Fri Apr 25 14:41:51 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, } @Article{Benton:1983:CZT, author = "T. C. Benton", title = "Common Zeros of Two {Bessel} Functions. {Part II}. {Approximations} and Tables", journal = j-MATH-COMPUT, volume = "41", number = "163", pages = "203--217", month = jul, year = "1983", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "33A40 (65A05)", MRnumber = "85a:33010", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, classcodes = "B0220 (Mathematical analysis); C1120 (Mathematical analysis)", corpsource = "Dept. of Math., Pennsylvania State Univ., University Park, PA, USA", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "Bessel functions; computer program; poles and zeros", treatment = "T Theoretical or Mathematical", } @Article{Bowman:1983:CFP, author = "K. O. Bowman and L. R. Shenton", title = "Continued fractions and the polygamma functions", journal = j-J-COMPUT-APPL-MATH, volume = "9", number = "1", pages = "29--39", month = mar, year = "1983", CODEN = "JCAMDI", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Sat Feb 25 11:59:24 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath1980.bib", URL = "http://www.sciencedirect.com/science/article/pii/0377042783900262", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @Article{Brezinski:1983:CAE, author = "C. Brezinski and J. P. Delahaye and B. Germain-Bonne", title = "Convergence acceleration by extraction of linear subsequences", journal = j-SIAM-J-NUMER-ANAL, volume = "20", number = "6", pages = "1099--1105", month = dec, year = "1983", CODEN = "SJNAAM", DOI = "https://doi.org/10.1137/0720079", ISSN = "0036-1429 (print), 1095-7170 (electronic)", ISSN-L = "0036-1429", MRclass = "65B99 (40A05)", MRnumber = "723826 (85g:65014)", MRreviewer = "John H. McCabe", bibdate = "Thu Dec 1 17:36:45 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "SIAM Journal on Numerical Analysis", journal-URL = "http://epubs.siam.org/sinum", keywords = "convergence acceleration", } @Article{Brezinski:1983:ECC, author = "Claude Brezinski", title = "Error control in convergence acceleration processes", journal = j-IMA-J-NUMER-ANAL, volume = "3", number = "1", pages = "65--80", year = "1983", CODEN = "IJNADH", DOI = "https://doi.org/10.1093/imanum/3.1.65", ISSN = "0272-4979 (print), 1464-3642 (electronic)", ISSN-L = "0272-4979", MRclass = "65B99 (65D32 65G05)", MRnumber = "705081 (85a:65004)", MRreviewer = "John P. Coleman", bibdate = "Thu Dec 1 17:36:45 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; MathSciNet database", acknowledgement = ack-nhfb, fjournal = "IMA Journal of Numerical Analysis", journal-URL = "http://imajna.oxfordjournals.org/content/by/year", keywords = "convergence acceleration", } @Article{Cash:1983:BRKa, author = "J. R. Cash", title = "Block {Runge--Kutta} Methods for the Numerical Integration of Initial Value Problems in Ordinary Differential Equations. {Part I}. {The} Nonstiff Case", journal = j-MATH-COMPUT, volume = "40", number = "161", pages = "175--191", month = jan, year = "1983", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "65L05", MRnumber = "84d:65044a", MRreviewer = "W. C. Rheinboldt", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, classcodes = "B0290F (Interpolation and function approximation); B0290M (Numerical integration and differentiation); B0290P (Differential equations); C4130 (Interpolation and function approximation); C4160 (Numerical integration and differentiation); C4170 (Differential equations)", corpsource = "Dept. of Math., Imperial Coll., London, UK", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "approximate numerical integration; approximation theory; block implicit formulae; block Runge--Kutta formulae; C. W. Gear; differential equations; equations; first order; formulae; initial value; initial value problems; integration; linear multistep methods; nonstiff problems; order; ordinary differential; problems; Runge--Kutta methods; Runge--Kutta starters; stepsize; stiff problems; systems; variable order; variable order block explicit", treatment = "T Theoretical or Mathematical", } @Article{Cody:1983:ASM, author = "W. J. Cody", title = "Algorithm 597: Sequence of Modified {Bessel} Functions of the First Kind", journal = j-TOMS, volume = "9", number = "2", pages = "242--245", month = jun, year = "1983", CODEN = "ACMSCU", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, catcode = "G.1.2; G", CRclass = "G.1.2 Approximation; G.1.2 Elementary function approximation", descriptor = "Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation; Mathematics of Computing, MISCELLANEOUS", fjournal = "ACM Transactions on Mathematical Software (TOMS)", genterm = "algorithms", guideno = "02186", journal-URL = "https://dl.acm.org/loi/toms", jrldate = "June 1983", keywords = "algorithms", subject = "G.1.2 Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation G Mathematics of Computing, MISCELLANEOUS", } @Article{Coleman:1983:CEB, author = "J. P. Coleman and A. J. Monaghan", title = "{Chebyshev} expansions for the {Bessel} function $ {J}_n(z) $ in the complex plane", journal = j-MATH-COMPUT, volume = "40", number = "161", pages = "343--366", month = jan, year = "1983", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "65A05 (30E10 33A40 65D20)", MRnumber = "84c:65013", MRreviewer = "C. W. Clenshaw", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Cruz:1983:MPR, author = "Andr{\'e}s Cruz and Javier Sesma", title = "Modulus and phase of the reduced logarithmic derivative of the {Hankel} function", journal = j-MATH-COMPUT, volume = "41", number = "164", pages = "597--605", month = oct, year = "1983", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "33A40 (65H05 81F10)", MRnumber = "85b:33006", MRreviewer = "H. E. Fettis", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Cusick:1983:CCL, author = "David Cusick", title = "Computers \& Calculators: a Logarithm Algorithm for Four-Function Calculators", journal = j-TWO-YEAR-COLL-MATH-J, volume = "14", number = "4", pages = "322--324", month = sep, year = "1983", CODEN = "????", DOI = "https://doi.org/10.1080/00494925.1983.11972706", ISSN = "0049-4925 (print), 2325-9116 (electronic)", ISSN-L = "0049-4925", bibdate = "Thu Feb 14 09:49:45 MST 2019", bibsource = "https://www.math.utah.edu/pub/tex/bib/collegemathj.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.tandfonline.com/doi/abs/10.1080/00494925.1983.11972706; https://www.jstor.org/stable/3027283", acknowledgement = ack-nhfb, fjournal = "Two-Year College Mathematics Journal", journal-URL = "https://maa.tandfonline.com/loi/ucmj20; http://www.jstor.org/journals/00494925.html", onlinedate = "30 Jan 2018", } @Article{Demsky:1983:MMC, author = "J. Demsky and M. Schlesinger and R. D. Kent", title = "Micro/mini computer program for calculating the square root of rationals at arbitrary precision", journal = j-COMP-PHYS-COMM, volume = "29", number = "3", pages = "237--244", month = may, year = "1983", CODEN = "CPHCBZ", DOI = "https://doi.org/10.1016/0010-4655(83)90004-8", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", bibdate = "Mon Feb 13 10:28:04 MST 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/compphyscomm1980.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/0010465583900048", acknowledgement = ack-nhfb, fjournal = "Computer Physics Communications", journal-URL = "http://www.sciencedirect.com/science/journal/00104655", } @Article{Dietrich:1983:VQF, author = "D. Dietrich", title = "{Verfahren zur L{\"o}sung von Quadratwurzeln f{\"u}r Mikrorechnerprozeduren} \toenglish {Methods for the Solution of Square Roots for Microprocessor Subroutines} \endtoenglish", journal = j-ELEKTRONIKER, volume = "8", pages = "EL-1--EL-6", year = "1983", CODEN = "ELKRBL", ISSN = "0531-9218", bibdate = "Fri Dec 08 13:05:49 1995", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nj, fjournal = "Elektroniker (Switzerland)", } @Article{Dubrulle:1983:CNM, author = "Augustin A. Dubrulle", title = "Class of Numerical Methods for the Computation of {Pythagorean} Sums", journal = j-IBM-JRD, volume = "27", number = "6", pages = "582--589", month = nov, year = "1983", CODEN = "IBMJAE", ISSN = "0018-8646 (print), 2151-8556 (electronic)", ISSN-L = "0018-8646", bibdate = "Tue Mar 25 14:26:59 MST 1997", bibsource = "Compendex database; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "See \cite{Moler:1983:RSR} and generalization \cite{Jamieson:1989:RCI}.", abstract = "Moler and Morrison have described an iterative algorithm for the computation of the Pythagorean sum (a**2 plus b**2)** one-half of two real numbers a and b. This algorithm is immune to unwarranted floating-point overflows, has a cubic rate of convergence, and is easily transportable. This paper, which shows that the algorithm is essentially Halley's method applied to the computation of square roots, provides a generalization to any order of convergence. Formulas of orders 2 through 9 are illustrated with numerical examples. The generalization keeps the number of floating-point divisions constant and should be particularly useful for computation in high-precision floating-point arithmetic.", acknowledgement = ack-nhfb, classcodes = "C4190 (Other numerical methods); C5230 (Digital arithmetic methods)", classification = "723; 921", corpsource = "IBM Sci. Centre, Palo Alto, CA, USA", fjournal = "IBM Journal of Research and Development", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=5288520", journalabr = "IBM J Res Dev", keywords = "computer programming; digital arithmetic; floating-point divisions; Halley's method; high-precision floating-point arithmetic; iterative algorithm; iterative methods; mathematical techniques --- Numerical Methods; Pythagorean sums; rate of convergence; square roots", treatment = "T Theoretical or Mathematical", } @Article{Ellacott:1983:FTE, author = "S. W. Ellacott", title = "On the {Faber} transform and efficient numerical rational approximation", journal = j-SIAM-J-NUMER-ANAL, volume = "20", number = "5", pages = "989--1000", month = oct, year = "1983", CODEN = "SJNAAM", ISSN = "0036-1429 (print), 1095-7170 (electronic)", ISSN-L = "0036-1429", MRclass = "41A20 (41A21)", MRnumber = "85f:41010", MRreviewer = "Lee L. Keener", bibdate = "Fri Oct 16 06:57:22 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "SIAM Journal on Numerical Analysis", journal-URL = "http://epubs.siam.org/sinum", } @Article{Fessler:1983:HAA, author = "Theodore Fessler and William F. Ford and David A. Smith", title = "{HURRY}: An Acceleration Algorithm for Scalar Sequences and Series", journal = j-TOMS, volume = "9", number = "3", pages = "346--354", month = sep, year = "1983", CODEN = "ACMSCU", DOI = "https://doi.org/10.1145/356044.356051", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", MRclass = "65B10", MRnumber = "791 970", bibdate = "Sun Sep 04 19:50:51 1994", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib", acknowledgement = ack-nhfb, fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", } @Article{Fukushima:1983:OAA, author = "M. Fukushima", title = "An outer approximation algorithm for solving general convex programs", journal = j-OPER-RES, volume = "31", number = "1", pages = "101--113", month = feb, year = "1983", CODEN = "OPREAI", ISSN = "0030-364X (print), 1526-5463 (electronic)", ISSN-L = "0030-364X", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, catcode = "G.1.2; G.4", CRclass = "G.1.2 Approximation; G.1.2 Elementary function approximation; G.4 Efficiency", descriptor = "Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation; Mathematics of Computing, MATHEMATICAL SOFTWARE, Efficiency", fjournal = "Operations Research", genterm = "algorithms; documentation; performance; reliability; theory", guideno = "09992", journal-URL = "http://pubsonline.informs.org/loi/opre", jrldate = "Jan./Feb. 1983", subject = "G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; G. Mathematics of Computing; G.4 MATHEMATICAL SOFTWARE", } @Article{Giordano:1983:EAZ, author = "C. Giordano and A. Laforgia", title = "Elementary approximations for zeros of {Bessel} functions", journal = j-J-COMPUT-APPL-MATH, volume = "9", number = "3", pages = "221--228", month = sep, year = "1983", CODEN = "JCAMDI", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, catcode = "G.1.2", CRclass = "G.1.2 Approximation; G.1.2 Elementary function approximation", descriptor = "Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation", fjournal = "Journal of Computational and Applied Mathematics", genterm = "theory", guideno = "08115", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", jrldate = "Sept. 1983", subject = "G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS", } @Article{Hasson:1983:C, author = "M. Hasson and O. Shisha", title = "On the condition {$ \sum^{\infty }_{n = 1}n^{p - 1}E^*_n(f) < \infty $}", journal = j-J-APPROX-THEORY, volume = "39", number = "4", pages = "389--398", month = dec, year = "1983", CODEN = "JAXTAZ", ISSN = "0021-9045 (print), 1096-0430 (electronic)", ISSN-L = "0021-9045", MRclass = "42A10 (41A10)", MRnumber = "85a:42002", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, catcode = "G.1.2", CRclass = "G.1.2 Approximation; G.1.2 Elementary function approximation", descriptor = "Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation", fjournal = "Journal of Approximation Theory", genterm = "theory", guideno = "07952", journal-URL = "http://www.sciencedirect.com/science/journal/00219045", jrldate = "Dec. 1983", subject = "G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS", } @Article{Johnson:1983:MGC, author = "Gary M. Johnson", title = "Multiple-grid convergence acceleration of viscous and inviscid flow computations", journal = j-APPL-MATH-COMP, volume = "13", number = "3--4", pages = "375--398", month = nov, year = "1983", CODEN = "AMHCBQ", ISSN = "0096-3003 (print), 1873-5649 (electronic)", ISSN-L = "0096-3003", MRclass = "76-08", MRnumber = "84m:76010", bibdate = "Thu Feb 27 09:47:09 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Applied Mathematics and Computation", journal-URL = "http://www.sciencedirect.com/science/journal/00963003", keywords = "convergence acceleration", } @Article{Kershaw:1983:SEW, author = "D. Kershaw", title = "Some extensions of {W. Gautschi}'s inequalities for the gamma function", journal = j-MATH-COMPUT, volume = "41", number = "164", pages = "607--611", month = oct, year = "1983", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "33A15 (26D20 65D20)", MRnumber = "84m:33003", MRreviewer = "P. Anandani", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Lehnhoff:1983:NPT, author = "H.-G Lehnhoff", title = "A new proof of {Teljakowskii}'s theorem", journal = j-J-APPROX-THEORY, volume = "38", number = "2", pages = "177--181", month = jun, year = "1983", CODEN = "JAXTAZ", ISSN = "0021-9045 (print), 1096-0430 (electronic)", ISSN-L = "0021-9045", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, catcode = "G.1.2", CRclass = "G.1.2 Approximation; G.1.2 Elementary function approximation", descriptor = "Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation", fjournal = "Journal of Approximation Theory", genterm = "theory", guideno = "07897", journal-URL = "http://www.sciencedirect.com/science/journal/00219045", jrldate = "June 1983", subject = "G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS", } @Article{Lehnhoff:1983:SPF, author = "H.-G Lehnhoff", title = "A simple proof of a {A. F. Timan}'s theorem", journal = j-J-APPROX-THEORY, volume = "38", number = "2", pages = "172--176", month = jun, year = "1983", CODEN = "JAXTAZ", ISSN = "0021-9045 (print), 1096-0430 (electronic)", ISSN-L = "0021-9045", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, catcode = "G.1.2", CRclass = "G.1.2 Approximation; G.1.2 Elementary function approximation", descriptor = "Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation", fjournal = "Journal of Approximation Theory", genterm = "theory", guideno = "07896", journal-URL = "http://www.sciencedirect.com/science/journal/00219045", jrldate = "June 1983", subject = "G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS", } @InProceedings{Little:1983:CCS, author = "F. Little", editor = "Robert E. Barnhill and Wolfgang Boehm", booktitle = "Surfaces in computer aided geometric design: proceedings of a conference held at Mathematisches Forschungsinstitut Oberwolfach, {F.R.G.}, April 25--30, 1982, organized by Wolfgang Boehm and Josef Hoschek", title = "Convex combination surfaces", publisher = pub-NORTH-HOLLAND, address = pub-NORTH-HOLLAND:adr, bookpages = "xvi + 215", pages = "99--109", year = "1983", ISBN = "0-444-86550-0", ISBN-13 = "978-0-444-86550-2", LCCN = "T385 .S827 1982", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, catcode = "G.1.1; G.1.1; G.1.2", CRclass = "G.1.1 Interpolation; G.1.1 Interpolation formulas; G.1.1 Interpolation; G.1.1 Smoothing; G.1.2 Approximation; G.1.2 Elementary function approximation", descriptor = "Mathematics of Computing, NUMERICAL ANALYSIS, Interpolation, Interpolation formulas; Mathematics of Computing, NUMERICAL ANALYSIS, Interpolation, Smoothing; Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation", genterm = "theory", guideno = "13093", subject = "G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS", } @TechReport{Lutskii:1983:VFT, author = "G. M. Lutski{\u\i} and O. I. Penchev", title = "{{\cyr Vychislenie {\`e}lementarnykh funktsi{\u\i} metodom tsifra za tsifro{\u\i} v izbytochnykh sistemakh schisleniya}}. ({Russian}) [Calculation of elementary functions by the digit-by-digit method in redundant number systems]", type = "Preprint", number = "83-22", institution = "Akad. Nauk Ukrain. SSR, Inst. Kibernet.", address = "Kiev, USSR", pages = "30", year = "1983", MRclass = "65D20", MRnumber = "719 021", bibdate = "Sat Jan 11 17:44:01 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, language = "Russian", } @Article{Mason:1983:CBF, author = "Janet P. Mason", title = "Cylindrical {Bessel} functions for a large range of complex arguments", journal = j-COMP-PHYS-COMM, volume = "30", number = "1", pages = "1--11", month = jul # "\slash " # aug, year = "1983", CODEN = "CPHCBZ", DOI = "https://doi.org/10.1016/0010-4655(83)90116-9", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", bibdate = "Mon Feb 13 10:28:05 MST 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/compphyscomm1980.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/0010465583901169", abstract = "The evaluation of Bessel functions of the first and second kinds, covering a wide range of complex arguments and integer orders, is required in the determination of the intensity of acoustic reflection from absorbing bodies. Numerical problems associated with the calculations are discussed and various means by which these problems have been overcome are explained. The numerical methods used in calculating the Bessel functions of the first, second and third kinds are given, as well as sample results and numerical checks in the form of computer plots and printouts.", acknowledgement = ack-nhfb, fjournal = "Computer Physics Communications", journal-URL = "http://www.sciencedirect.com/science/journal/00104655", } @Article{McCabe:1983:ASC, author = "J. H. McCabe", title = "On an asymptotic series and corresponding continued fraction for a gamma function ratio", journal = j-J-COMPUT-APPL-MATH, volume = "9", number = "2", pages = "125--130", month = jun, year = "1983", CODEN = "JCAMDI", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Sat Feb 25 11:59:24 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath1980.bib", URL = "http://www.sciencedirect.com/science/article/pii/0377042783900353", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @TechReport{McCurdy:1983:ACD, author = "A. McCurdy and K. C. Ng and Beresford N. Parlett", title = "Accurate computation of divided differences of the exponential function", type = "Report", number = "PAM-160", institution = inst-CPAM-UCB, address = inst-CPAM-UCB:adr, month = jun, year = "1983", bibdate = "Fri Nov 11 09:06:19 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, } @Article{Meister:1983:MYF, author = "B. Meister", title = "On {Murphy}'s yield formula", journal = j-IBM-JRD, volume = "27", number = "6", pages = "545--548", month = nov, year = "1983", CODEN = "IBMJAE", ISSN = "0018-8646 (print), 2151-8556 (electronic)", ISSN-L = "0018-8646", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, catcode = "B.7.1; G.1.2", CRclass = "B.7.1 Types and Design Styles; B.7.1 Input/Output circuits; G.1.2 Approximation; G.1.2 Elementary function approximation", descriptor = "Hardware, INTEGRATED CIRCUITS, Types and Design Styles, Input/Output circuits; Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation", fjournal = "IBM Journal of Research and Development", genterm = "theory; design; reliability", guideno = "06316", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=5288520", jrldate = "Nov. 1983", subject = "B. Hardware; B.7 INTEGRATED CIRCUITS; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS", } @Article{Mlodzki:1983:PPC, author = "J. Mlodzki and J. Kuszkowski and M. Suffczynski", title = "A {Pascal} program for calculating the reduced {Coulomb} {Green}'s functions and their partial waves", journal = j-COMP-PHYS-COMM, volume = "29", number = "4", pages = "341--350", month = jun, year = "1983", CODEN = "CPHCBZ", DOI = "https://doi.org/10.1016/0010-4655(83)90013-9", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", bibdate = "Fri Feb 24 18:49:59 MST 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/compphyscomm1980.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/0010465583900139", acknowledgement = ack-nhfb, fjournal = "Computer Physics Communications", journal-URL = "http://www.sciencedirect.com/science/journal/00104655", } @InProceedings{Moler:1983:MSV, author = "C. Moler", editor = "????", booktitle = "{SIAM Conference on Parallel Processing for Scientific Computing, Norfolk, VA, November 10--11, 1983}", title = "Mathematical Software for Vector Computers", publisher = pub-SIAM, address = pub-SIAM:adr, pages = "??--??", year = "1983", DOI = "", ISBN = "", ISBN-13 = "", LCCN = "", bibdate = "Fri Sep 20 14:42:46 2024", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/m/moler-cleve-b.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, remark = "Cited in \cite[Reference 11]{Agarwal:1986:NSV} in elefunt.bib and fparith.bib.", } @Article{Moler:1983:RSR, author = "Cleve B. Moler and Donald Morrison", title = "Replacing Square Roots by {Pythagorean} Sums", journal = j-IBM-JRD, volume = "27", number = "6", pages = "577--581", month = nov, year = "1983", CODEN = "IBMJAE", ISSN = "0018-8646 (print), 2151-8556 (electronic)", ISSN-L = "0018-8646", bibdate = "Thu Sep 1 10:15:41 1994", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; https://www.math.utah.edu/pub/bibnet/authors/m/moler-cleve-b.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "See \cite{Dubrulle:1983:CNM} and generalization \cite{Jamieson:1989:RCI}.", URL = "http://www.research.ibm.com/journal/rd/276/ibmrd2706P.pdf", abstract = "An algorithm is presented for computing a 'Pythagorean sum' a(+)b= square root a/sup 2/+b/sup 2/ directly from a and b without computing their squares or taking a square root. No destructive floating point overflows or underflows are possible. The algorithm can be extended to compute the Euclidean norm of a vector. The resulting subroutine is short, portable, robust, and accurate, but not as efficient as some other possibilities. The algorithm is particularly attractive for computers where space and reliability are more important than speed", acknowledgement = ack-nj # " and " # ack-nhfb, classcodes = "C4190 (Other numerical methods); C5230 (Digital arithmetic methods)", corpsource = "Dept. of Computer Sci., Univ. of New Mexico, Albuquerque, NM, USA", fjournal = "IBM Journal of Research and Development", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=5288520", keywords = "algorithms; digital arithmetic; Euclidean norm; floating-point arithmetic; iterative methods; performance; Pythagorean sums; subroutine; vector", review = "ACM CR 8406-0463", subject = "G.1 Mathematics of Computing, NUMERICAL ANALYSIS, Roots of Nonlinear Equations \\ F.2.1 Theory of Computation, ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY, Numerical Algorithms and Problems, Computations on polynomials \\ F.2.2 Theory of Computation, ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY, Nonnumerical Algorithms and Problems, Geometrical problems and computations", treatment = "T Theoretical or Mathematical", } @PhdThesis{Monk:1983:SFE, author = "P. B. Monk", title = "Some finite element methods for the approximation of the biharmonic equation", type = "{Ph.D} Thesis", school = "Rutgers University, The State University of New Jersey", address = "New Brunswick, NJ, USA", pages = "242", year = "1983", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, catcode = "G.1.8; G.1.2", CRclass = "G.1.8 Partial Differential Equations; G.1.8 Finite element methods; G.1.2 Approximation; G.1.2 Elementary function approximation", descriptor = "Mathematics of Computing, NUMERICAL ANALYSIS, Partial Differential Equations, Finite element methods; Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation", genterm = "design; algorithms; experimentation", guideno = "15929", source = "UMI order no. DA8308441", subject = "G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS", } @Article{Muench:1983:LAF, author = "Donald L. Muench and Gerald Wildenberg", title = "A Logarithm Algorithm for a Five-Function Calculator", journal = j-TWO-YEAR-COLL-MATH-J, volume = "14", number = "4", pages = "324--326", month = sep, year = "1983", CODEN = "????", DOI = "https://doi.org/10.1080/00494925.1983.11972707", ISSN = "0049-4925 (print), 2325-9116 (electronic)", ISSN-L = "0049-4925", bibdate = "Thu Feb 14 09:49:45 MST 2019", bibsource = "https://www.math.utah.edu/pub/tex/bib/collegemathj.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.tandfonline.com/doi/abs/10.1080/00494925.1983.11972707", acknowledgement = ack-nhfb, fjournal = "Two-Year College Mathematics Journal", journal-URL = "https://maa.tandfonline.com/loi/ucmj20; http://www.jstor.org/journals/00494925.html", onlinedate = "30 Jan 2018", } @Article{Nave:1983:ITF, author = "Rafi Nave", key = "Nav83", title = "Implementation of Transcendental Functions on a Numerics Processor", journal = j-MICROPROC-MICROPROG, volume = "11", pages = "221--225", year = "1983", CODEN = "MMICDT", ISSN = "0165-6074 (print), 1878-7061 (electronic)", ISSN-L = "0165-6074", bibdate = "Mon May 19 13:30:58 1997", bibsource = "ftp://ftp.ira.uka.de/pub/bibliography/Math/elefunt.bib.gz; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Microprocessing and Microprogramming", journal-URL = "https://www.sciencedirect.com/journal/microprocessing-and-microprogramming/issues", } @Article{Piessens:1983:MCC, author = "Robert Piessens and Maria Branders", title = "Modified {Clenshaw--Curtis} method for the computation of {Bessel} function integrals", journal = j-BIT, volume = "23", number = "3", pages = "370--381", month = sep, year = "1983", CODEN = "BITTEL, NBITAB", DOI = "https://doi.org/10.1007/BF01934465", ISSN = "0006-3835 (print), 1572-9125 (electronic)", ISSN-L = "0006-3835", MRclass = "65D30 (65R10)", MRnumber = "85b:65019 (705003)", MRreviewer = "H. E. Fettis", bibdate = "Sun Nov 12 06:18:24 2023", bibsource = "http://springerlink.metapress.com/openurl.asp?genre=issue&issn=0006-3835&volume=23&issue=3; https://www.math.utah.edu/pub/bibnet/authors/c/clenshaw-charles-w.bib; https://www.math.utah.edu/pub/tex/bib/bit.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.springerlink.com/openurl.asp?genre=article&issn=0006-3835&volume=23&issue=3&spage=370", acknowledgement = ack-nhfb, fjournal = "BIT. Nordisk Tidskrift for Informationsbehandling (BIT)", journal-URL = "http://link.springer.com/journal/10543", subject-dates = "Charles William Clenshaw (15 March 1926--23 September 2004)", } @Article{Prosser:1983:NCS, author = "C. J. Prosser", title = "A note on computing the square root of an integer", journal = j-COMP-J, volume = "26", number = "2", pages = "187--188", month = may, year = "1983", CODEN = "CMPJA6", ISSN = "0010-4620 (print), 1460-2067 (electronic)", ISSN-L = "0010-4620", bibdate = "Tue Mar 25 13:51:56 MST 1997", bibsource = "http://www3.oup.co.uk/computer_journal/hdb/Volume_26/Issue_02/; https://www.math.utah.edu/pub/tex/bib/compj1980.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "http://www3.oup.co.uk/computer_journal/hdb/Volume_26/Issue_02/tiff/187.tif; http://www3.oup.co.uk/computer_journal/hdb/Volume_26/Issue_02/tiff/188.tif", acknowledgement = ack-nhfb, classcodes = "C4190 (Other numerical methods); C7310 (Mathematics computing)", corpsource = "Rutherford and Appleton Lab., Chilton, Didcot, UK", fjournal = "The Computer Journal", journal-URL = "http://comjnl.oxfordjournals.org/", keywords = "binary; computer; fixed-point number; integer; interactive methods; iterative methods; PASCAL; Pascal implementation; square root; subroutines; successive subtraction", treatment = "P Practical", } @Article{Salzer:1983:NDG, author = "Herbert E. Salzer", title = "Note on the {Do{\v{c}}ev--Grosswald} asymptotic series for generalized {Bessel} polynomials", journal = j-J-COMPUT-APPL-MATH, volume = "9", number = "2", pages = "131--135", month = jun, year = "1983", CODEN = "JCAMDI", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Sat Feb 25 11:59:24 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath1980.bib", note = "See errata \cite{Anonymous:1984:EJCb}.", URL = "http://www.sciencedirect.com/science/article/pii/0377042783900365", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @Article{Sidi:1983:ZSP, author = "Avram Sidi and Doron S. Lubinsky", title = "On the zeros of some polynomials that arise in numerical quadrature and convergence acceleration", journal = j-SIAM-J-NUMER-ANAL, volume = "20", number = "2", pages = "400--405", month = apr, year = "1983", CODEN = "SJNAAM", DOI = "https://doi.org/10.1137/0720028", ISSN = "0036-1429 (print), 1095-7170 (electronic)", ISSN-L = "0036-1429", MRclass = "65H05 (65D30)", MRnumber = "694528 (84f:65046)", MRreviewer = "J. G. Herriot", bibdate = "Thu Dec 1 17:36:45 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "SIAM Journal on Numerical Analysis", journal-URL = "http://epubs.siam.org/sinum", keywords = "convergence acceleration", } @Article{Talman:1983:LSC, author = "James D. Talman", title = "{LSFBTR}: a subroutine for calculating spherical {Bessel} transforms", journal = j-COMP-PHYS-COMM, volume = "30", number = "1", pages = "93--99", month = jul # "\slash " # aug, year = "1983", CODEN = "CPHCBZ", DOI = "https://doi.org/10.1016/0010-4655(83)90126-1", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", bibdate = "Mon Feb 13 10:28:05 MST 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/compphyscomm1980.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/0010465583901261", acknowledgement = ack-nhfb, fjournal = "Computer Physics Communications", journal-URL = "http://www.sciencedirect.com/science/journal/00104655", } @Article{Temme:1983:NCC, author = "N. M. Temme", title = "The numerical computation of the confluent hypergeometric function $ {U}(a, \, b, \, z) $", journal = j-NUM-MATH, volume = "41", number = "1", pages = "63--82", month = apr, year = "1983", CODEN = "NUMMA7", ISSN = "0029-599X (print), 0945-3245 (electronic)", ISSN-L = "0029-599X", MRclass = "65D20 (33A30 65D15)", MRnumber = "84g:65030", MRreviewer = "H. E. Fettis", bibdate = "Mon May 26 11:49:34 MDT 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, classification = "C4120 (Functional analysis); C7310 (Mathematics computing)", corpsource = "Math. Centrum, Amsterdam, Netherlands", fjournal = "Numerische Mathematik", journal-URL = "http://link.springer.com/journal/211", keywords = "ALGOL 60 procedures; asymptotic expansions; confluent hypergeometric function; function computation; function evaluation; Miller algorithm; subroutines", treatment = "P Practical; T Theoretical or Mathematical", } @TechReport{Temme:1983:TTR, author = "N. M. Temme", title = "Traces to {Tricomi} in recent work on special functions and asymptotics of integrals", type = "Report", number = "TW 239/83", institution = "Stichting mathematisch Centrum", address = "Amsterdam, The Netherlands", pages = "15", year = "1983", LCCN = "A1 M462 TW239/83", bibdate = "Sat Oct 30 18:29:48 2010", bibsource = "http://cat.cisti-icist.nrc-cnrc.gc.ca/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, } @Article{Temme:1983:UAE, author = "Nico M. Temme", title = "Uniform asymptotic expansions of {Laplace} integrals", journal = "Analysis", volume = "3", number = "1--4", pages = "221--249", year = "1983", ISSN = "0174-4747 (print), 2196-6753 (electronic)", ISSN-L = "0174-4747", MRclass = "41A60 (44A10)", MRnumber = "756117", MRreviewer = "F. W. J. Olver", bibdate = "Tue Feb 6 11:39:36 2018", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Analysis. International Journal of Analysis and its Application", } @Article{Volz:1983:CAA, author = "H. V{\"o}lz", title = "{CORDIC und {\"a}hnliche Algorithmen der elementaren Funktionen mit besonderer Eignung f{\"u}r Mikrorechner}. ({German}) [{CORDIC} and Similar Algorithms for Elementary Functions with Particular Aptitude for Microcomputers]", journal = j-NACH-ELEK, volume = "33", number = "12", pages = "506--510", month = "????", year = "1983", CODEN = "NTELAP", ISSN = "0323-4657", bibdate = "Fri Sep 16 16:30:40 1994", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nj, fjournal = "Nachrichtentechnik Elektronik", } @Article{Wejntrob:1983:ASR, author = "Leon Wejntrob", title = "Approximation of Square Roots", journal = j-TWO-YEAR-COLL-MATH-J, volume = "14", number = "5", pages = "427--431", month = nov, year = "1983", CODEN = "????", DOI = "https://doi.org/10.1080/00494925.1983.11972733", ISSN = "0049-4925 (print), 2325-9116 (electronic)", ISSN-L = "0049-4925", bibdate = "Thu Feb 14 09:49:48 MST 2019", bibsource = "https://www.math.utah.edu/pub/tex/bib/collegemathj.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.tandfonline.com/doi/abs/10.1080/00494925.1983.11972733", acknowledgement = ack-nhfb, fjournal = "Two-Year College Mathematics Journal", journal-URL = "https://maa.tandfonline.com/loi/ucmj20; http://www.jstor.org/journals/00494925.html", keywords = "rational square roots of rational numbers", onlinedate = "30 Jan 2018", } @Article{Xu:1983:HPG, author = "Xian Yu Xu and Jia Kai Li and Gui Jing Xiong and Guo Liang Xu and Chun Qing Lu", title = "High-precision generation of elementary functions. ({Chinese})", journal = "Appl. Math. Math. Comput.", volume = "6", pages = "24--32", year = "1983", MRclass = "65D20", MRnumber = "86e:65031", bibdate = "Sat Jan 11 17:44:01 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, } @Article{Anonymous:1984:EJCb, author = "Anonymous", title = "Errata: {J. Comput. Appl. Math. {\bf 9}: H. E. Salzer, Note on the Do{\v{c}}ev--Grosswald asymptotic series for generalized Bessel polynomials, (1983) 131--135}", journal = j-J-COMPUT-APPL-MATH, volume = "10", number = "1", pages = "133--133", month = feb, year = "1984", CODEN = "JCAMDI", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Sat Feb 25 11:59:53 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath1980.bib", note = "See \cite{Salzer:1983:NDG}.", URL = "http://www.sciencedirect.com/science/article/pii/0377042784900773", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @Article{Ardill:1984:ABF, author = "R. W. B. Ardill and K. J. M. Moriarty", title = "Accurate {Bessel} functions {$ J_n(z) $}, {$ Y_n(z) $}, {$ H_n^{(1)}(z) $} and {$ H_n^{(2)}(z) $} of integer order and complex argument", journal = j-COMP-PHYS-COMM, volume = "35", number = "1--3", pages = "C-559--C-559", month = "????", year = "1984", CODEN = "CPHCBZ", DOI = "https://doi.org/10.1016/S0010-4655(84)82734-4", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", bibdate = "Sat Feb 25 10:56:33 MST 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/compphyscomm1980.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0010465584827344", acknowledgement = ack-nhfb, fjournal = "Computer Physics Communications", journal-URL = "http://www.sciencedirect.com/science/journal/00104655", } @Article{Ardill:1984:BFC, author = "W. B. Ardill and K. J. M. Moriarty", title = "The {Bessel} functions {$ J_0 $} and {$ J_1 $} of complex argument", journal = j-COMP-PHYS-COMM, volume = "35", pages = "C-409--C-409", year = "1984", CODEN = "CPHCBZ", DOI = "https://doi.org/10.1016/S0010-4655(84)82619-3", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", bibdate = "Thu Apr 24 10:35:27 2008", bibsource = "https://www.math.utah.edu/pub/tex/bib/compphyscomm1980.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Computer Physics Communications", journal-URL = "http://www.sciencedirect.com/science/journal/00104655", } @Article{Ardill:1984:SBF, author = "R. W. B. Ardill and K. J. M. Moriarty", title = "Spherical {Bessel} functions $ j_n $ and $ y_n $ of integer order and real argument", journal = j-COMP-PHYS-COMM, volume = "35", pages = "C-466--C-466", year = "1984", CODEN = "CPHCBZ", DOI = "https://doi.org/10.1016/S0010-4655(84)82666-1", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", bibdate = "Thu Apr 24 10:35:27 2008", bibsource = "https://www.math.utah.edu/pub/tex/bib/compphyscomm1980.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Computer Physics Communications", journal-URL = "http://www.sciencedirect.com/science/journal/00104655", remark = "No code shown, but uses formulas from \cite{Abramowitz:1964:HMF} for evaluations. Function name is {\tt sphbes}.", } @Article{Bardin:1984:CFE, author = "C. Bardin and Y. Dandeu and L. Gauthier and J. Guillermin and T. Lena and J.-M. Pernet and H. H. Wolter and T. Tamura", title = "{Coulomb} functions in entire $ (\eta, \pi) $-plane", journal = j-COMP-PHYS-COMM, volume = "35", number = "1--3", pages = "C-125--C-126", month = "????", year = "1984", CODEN = "CPHCBZ", DOI = "https://doi.org/10.1016/S0010-4655(84)82382-6", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", bibdate = "Sat Feb 25 10:56:06 MST 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/compphyscomm1980.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0010465584823826", acknowledgement = ack-nhfb, fjournal = "Computer Physics Communications", journal-URL = "http://www.sciencedirect.com/science/journal/00104655", } @Article{Barnett:1984:CCB, author = "A. R. Barnett", title = "{Coulfg}: {Coulomb} and {Bessel} functions and their derivatives, for real arguments, by {Steed}'s method", journal = j-COMP-PHYS-COMM, volume = "35", number = "1--3", pages = "C-812--C-813", month = "????", year = "1984", CODEN = "CPHCBZ", DOI = "https://doi.org/10.1016/S0010-4655(84)82930-6", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", bibdate = "Sat Feb 25 10:56:49 MST 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/compphyscomm1980.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0010465584829306", acknowledgement = ack-nhfb, fjournal = "Computer Physics Communications", journal-URL = "http://www.sciencedirect.com/science/journal/00104655", } @Article{Barnett:1984:CWF, author = "A. R. Barnett and D. H. Feng and J. W. Steed and L. J. B. Goldfarb", title = "{Coulomb} wave functions for all real $ \eta $ and $ \rho $", journal = j-COMP-PHYS-COMM, volume = "35", number = "1--3", pages = "C-285", month = "????", year = "1984", CODEN = "CPHCBZ", DOI = "https://doi.org/10.1016/S0010-4655(84)82515-1", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", bibdate = "Sat Feb 25 10:56:15 MST 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/compphyscomm1980.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0010465584825151", acknowledgement = ack-nhfb, fjournal = "Computer Physics Communications", journal-URL = "http://www.sciencedirect.com/science/journal/00104655", } @Article{Barnett:1984:KCF, author = "A. R. Barnett", title = "{Klein}: {Coulomb} functions for real $ \lambda $ and positive energy to high accuracy", journal = j-COMP-PHYS-COMM, volume = "35", number = "1--3", pages = "C-753", month = "????", year = "1984", CODEN = "CPHCBZ", DOI = "https://doi.org/10.1016/S0010-4655(84)82884-2", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", bibdate = "Sat Feb 25 10:56:49 MST 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/compphyscomm1980.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0010465584828842", acknowledgement = ack-nhfb, fjournal = "Computer Physics Communications", journal-URL = "http://www.sciencedirect.com/science/journal/00104655", } @Article{Barnett:1984:RMR, author = "A. R. Barnett", title = "{RCWFF} --- a modification of the real {Coulomb} wavefunction program {RCWFN}", journal = j-COMP-PHYS-COMM, volume = "35", number = "1--3", pages = "C-370", month = "????", year = "1984", CODEN = "CPHCBZ", DOI = "https://doi.org/10.1016/S0010-4655(84)82585-0", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", bibdate = "Sat Feb 25 10:56:24 MST 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/compphyscomm1980.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0010465584825850", acknowledgement = ack-nhfb, fjournal = "Computer Physics Communications", journal-URL = "http://www.sciencedirect.com/science/journal/00104655", } @Article{Bell:1984:CFN, author = "K. L. Bell and N. S. Scott", title = "{Coulomb} functions (negative energies)", journal = j-COMP-PHYS-COMM, volume = "35", number = "1--3", pages = "C-648", month = "????", year = "1984", CODEN = "CPHCBZ", DOI = "https://doi.org/10.1016/S0010-4655(84)82808-8", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", bibdate = "Sat Feb 25 10:56:41 MST 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/compphyscomm1980.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0010465584828088", acknowledgement = ack-nhfb, fjournal = "Computer Physics Communications", journal-URL = "http://www.sciencedirect.com/science/journal/00104655", } @InCollection{Berges:1984:AFE, author = "J. C. Berges", booktitle = "Space mathematics", title = "Arithm{\'e}tique et fonctions {\'e}l{\'e}mentaires sur mini-micro calculateurs. ({French}) [Arithmetic and elementary functions on mini-micro computers]", publisher = "C{\'e}padu{\`e}s", address = "Toulouse, France", pages = "193--229", year = "1984", MRclass = "65-01 (65-04)", MRnumber = "849 200", bibdate = "Sat Jan 11 17:44:01 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, language = "French", } @Article{Berndt:1984:APA, author = "Bruce C. Berndt and Larry A. Goldberg", title = "Analytic properties of arithmetic sums arising in the theory of the classical theta functions", journal = j-SIAM-J-MATH-ANA, volume = "15", number = "1", pages = "143--150", month = jan, year = "1984", CODEN = "SJMAAH", ISSN = "0036-1410 (print), 1095-7154 (electronic)", ISSN-L = "0036-1410", MRclass = "11A15 (11A25 11F27)", MRnumber = "85d:11008", MRreviewer = "T. M. Apostol", bibdate = "Sun Nov 28 19:23:19 MST 2010", bibsource = "http://epubs.siam.org/sam-bin/dbq/toc/SIMA/15/1; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "SIAM Journal on Mathematical Analysis", journal-URL = "http://epubs.siam.org/sima", } @Article{Black:1984:NIS, author = "Cheryl M. Black and Robert P. Burton and Thomas H. Miller", title = "The Need for an Industry Standard of Accuracy for Elementary-Function Programs", journal = j-TOMS, volume = "10", number = "4", pages = "361--366", month = dec, year = "1984", CODEN = "ACMSCU", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", MRclass = "65D20", MRnumber = "792 000", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, catcode = "G.1.0; G.1.2; G.4", CRclass = "G.1.0 General; G.1.0 Numerical algorithms; G.1.2 Approximation; G.1.2 Elementary function approximation; G.4 Efficiency", descriptor = "Mathematics of Computing, NUMERICAL ANALYSIS, General, Numerical algorithms; Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation; Mathematics of Computing, MATHEMATICAL SOFTWARE, Efficiency", fjournal = "ACM Transactions on Mathematical Software (TOMS)", genterm = "theory; algorithms; reliability; standardization", guideno = "02897", journal-URL = "https://dl.acm.org/loi/toms", jrldate = "Dec. 1984", subject = "G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; G. Mathematics of Computing; G.4 MATHEMATICAL SOFTWARE", } @Article{Borwein:1984:AGM, author = "J. M. Borwein and P. B. Borwein", title = "The Arithmetic-Geometric Mean and Fast Computation of Elementary Functions", journal = j-SIAM-REVIEW, volume = "26", number = "3", pages = "351--366", month = jul, year = "1984", CODEN = "SIREAD", DOI = "https://doi.org/10.1137/1026073", ISSN = "0036-1445 (print), 1095-7200 (electronic)", ISSN-L = "0036-1445", MRclass = "65D20 (26A09)", MRnumber = "86d:65029", MRreviewer = "S. Conde", bibdate = "Fri Jun 21 11:25:02 MDT 2013", bibsource = "Compendex database; garbo.uwasa.fi:/pc/doc-soft/fpbiblio.txt; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/siamreview.bib", abstract = "We produce a self contained account of the relationship between the Gaussian arithmetic-geometric mean iteration and the fast computation of elementary functions. A particularly pleasant algorithm for pi is one of the by-products.", acknowledgement = ack-nhfb # " and " # ack-nj, affiliationaddress = "Dalhousie Univ, Halifax, NS, Can", classification = "723; 921", fjournal = "SIAM Review", journal-URL = "http://epubs.siam.org/sirev", journalabr = "SIAM Rev", keywords = "AGM (Arithmetic-Geometric Mean); arithmetic-geometric mean; calculation of pi; computational methods; elliptic functions; Iterative Methods; mathematical techniques; numerical mathematics", } @Article{Braess:1984:RAE, author = "Dietrich Braess", title = "On rational approximation of the exponential and the square root function", journal = j-LECT-NOTES-MATH, volume = "1105", pages = "89--99", year = "1984", CODEN = "LNMAA2", DOI = "https://doi.org/10.1007/BFb0072401", ISBN = "3-540-13899-4 (print), 3-540-39113-4 (e-book)", ISBN-13 = "978-3-540-13899-0 (print), 978-3-540-39113-5 (e-book)", ISSN = "0075-8434 (print), 1617-9692 (electronic)", ISSN-L = "0075-8434", MRclass = "41A20 (41A25 65D15)", MRnumber = "783263 (86g:41025)", MRreviewer = "G. Meinardus", bibdate = "Fri May 9 19:07:44 MDT 2014", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/lnm1980.bib", URL = "http://link.springer.com/chapter/10.1007/BFb0072401/", acknowledgement = ack-nhfb, book-DOI = "https://doi.org/10.1007/BFb0072395", book-URL = "http://www.springerlink.com/content/978-3-540-39113-5", fjournal = "Lecture Notes in Mathematics", journal-URL = "http://link.springer.com/bookseries/304", } @Article{Campbell:1984:BFRa, author = "J. B. Campbell", title = "{Bessel} Functions {$ J_\nu (x) $} of real order and real argument", journal = j-COMP-PHYS-COMM, volume = "35", number = "1--3", pages = "C-583--C-583", month = "????", year = "1984", CODEN = "CPHCBZ", DOI = "https://doi.org/10.1016/S0010-4655(84)82756-3", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", bibdate = "Sat Feb 25 10:56:33 MST 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/compphyscomm1980.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0010465584827563", acknowledgement = ack-nhfb, fjournal = "Computer Physics Communications", journal-URL = "http://www.sciencedirect.com/science/journal/00104655", } @Article{Campbell:1984:BFZ, author = "J. B. Campbell", title = "{Bessel} functions {$ I_\nu (z) $} and {$ K_\nu (z) $} of real order and complex argument", journal = j-COMP-PHYS-COMM, volume = "35", number = "1--3", pages = "C-747--C-748", month = "????", year = "1984", CODEN = "CPHCBZ", DOI = "https://doi.org/10.1016/S0010-4655(84)82880-5", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", bibdate = "Sat Feb 25 10:56:49 MST 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/compphyscomm1980.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0010465584828805", acknowledgement = ack-nhfb, fjournal = "Computer Physics Communications", journal-URL = "http://www.sciencedirect.com/science/journal/00104655", } @Article{Coleman:1984:FSB, author = "J. P. Coleman", title = "A {Fortran} subroutine for the {Bessel} function {$ J_n(x) $} of order $0$ to $ 10 $", journal = j-COMP-PHYS-COMM, volume = "35", number = "1--3", pages = "C-654--C-654", month = "????", year = "1984", CODEN = "CPHCBZ", DOI = "https://doi.org/10.1016/S0010-4655(84)82814-3", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", bibdate = "Sat Feb 25 10:56:41 MST 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/compphyscomm1980.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fortran2.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0010465584828143", acknowledgement = ack-nhfb, fjournal = "Computer Physics Communications", journal-URL = "http://www.sciencedirect.com/science/journal/00104655", remark = "No code shown, but sums two separate Chebyshev series, one for $x$ in $ [0, 8] $, and a second for $x$ in $ (8, \infty) $. Function name is {\tt realjn}.", } @Article{Delic:1984:CSS, author = "G. Delic", title = "{Chebyshev} series for the spherical {Bessel} function $ j_l(r) $", journal = j-COMP-PHYS-COMM, volume = "35", number = "1--3", pages = "C-577--C-577", month = "????", year = "1984", CODEN = "CPHCBZ", DOI = "https://doi.org/10.1016/S0010-4655(84)82751-4", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", bibdate = "Sat Feb 25 10:56:33 MST 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/compphyscomm1980.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0010465584827514", acknowledgement = ack-nhfb, fjournal = "Computer Physics Communications", journal-URL = "http://www.sciencedirect.com/science/journal/00104655", } @Article{Demsky:1984:MMC, author = "J. Demsky and M. Schlesinger and R. D. Kent", title = "Micro/mini computer program for calculating the square root of rationals at arbitrary precision", journal = j-COMP-PHYS-COMM, volume = "35", number = "1--3", pages = "C-877", month = "????", year = "1984", CODEN = "CPHCBZ", DOI = "https://doi.org/10.1016/S0010-4655(84)82981-1", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", bibdate = "Sat Feb 25 10:56:58 MST 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/compphyscomm1980.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0010465584829811", acknowledgement = ack-nhfb, fjournal = "Computer Physics Communications", journal-URL = "http://www.sciencedirect.com/science/journal/00104655", } @Article{Dhanoa:1984:BPE, author = "M. S. Dhanoa and J. France", title = "A {BASIC} program for the evaluation of the gamma functions", journal = j-J-APPL-STAT, volume = "11", number = "2", pages = "225--228", year = "1984", CODEN = "????", DOI = "https://doi.org/10.1080/02664768400000021", ISSN = "0266-4763 (print), 1360-0532 (electronic)", ISSN-L = "0266-4763", bibdate = "Tue Sep 6 11:15:50 MDT 2011", bibsource = "http://www.tandf.co.uk/journals/routledge/02664763.html; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Journal of Applied Statistics", journal-URL = "http://www.tandfonline.com/loi/cjas20", onlinedate = "24 May 2006", } @TechReport{DiDonato:1984:IGF, author = "Armido R. DiDonato", title = "The incomplete gamma function ratios using {Temme}'s asymptotic expansions", type = "Report", number = "NSWC TR 84-79", institution = "Naval Surface Weapons Center", address = "Dahlgren, VA, USA", year = "1984", bibdate = "Mon Jun 03 12:24:32 2024", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, } @Article{Dutka:1984:EHH, author = "Jacques Dutka", title = "The early history of the hypergeometric function", journal = j-ARCH-HIST-EXACT-SCI, volume = "31", number = "1", pages = "15--34", month = mar, year = "1984", CODEN = "AHESAN", DOI = "https://doi.org/10.1007/BF00330241", ISSN = "0003-9519 (print), 1432-0657 (electronic)", ISSN-L = "0003-9519", MRclass = "01A50 (33-03)", MRnumber = "769538 (86d:01010)", MRreviewer = "Willard Parker", bibdate = "Fri Feb 4 21:50:21 MST 2011", bibsource = "http://springerlink.metapress.com/openurl.asp?genre=issue&issn=0003-9519&volume=31&issue=1; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.springerlink.com/openurl.asp?genre=article&issn=0003-9519&volume=31&issue=1&spage=15", acknowledgement = ack-nhfb, fjournal = "Archive for History of Exact Sciences", journal-URL = "http://link.springer.com/journal/407", MRtitle = "The early history of the hypergeometric function", } @Article{Fransen:1984:CMM, author = "Arne Frans{\'e}n and Staffan Wrigge", title = "Calculation of the moments and the moment generating function for the reciprocal gamma distribution", journal = j-MATH-COMPUT, volume = "42", number = "166", pages = "601--616", month = apr, year = "1984", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "65D20 (60E10 62E15 65U05)", MRnumber = "86f:65042a", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, classcodes = "B0290F (Interpolation and function approximation); C1210B (Reliability theory); C4130 (Interpolation and function approximation)", corpsource = "Nat. Defence Res. Inst., Stockholm, Sweden", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "kurtosis; moment generating function; moments; polynomials; reciprocal gamma distribution; reliability theory; skewness; variance", treatment = "T Theoretical or Mathematical", } @Article{Glaeske:1984:LTS, author = "H.-J. Glaeske and O. I. Mari{\v{c}}ev", title = "The {Laguerre} transform of some elementary functions", journal = j-Z-ANAL-ANWEND, volume = "3", number = "3", pages = "237--244", year = "1984", ISSN = "0232-2064 (print), 1661-4534 (electronic)", ISSN-L = "0232-2064", MRclass = "44A15 (34A10)", MRnumber = "86a:44005", MRreviewer = "Ram Kishore Saxena", bibdate = "Sat Jan 11 17:44:01 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "{Zeitschrift f{\"u}r Analysis und ihre Anwendungen}", } @Article{Grodd:1984:REN, author = "Laurence W. Grodd and Charles M. Patton", title = "{ROM} extends numerical function set of handheld computer", journal = j-HEWLETT-PACKARD-J, volume = "35", number = "7", pages = "25--36", month = jul, year = "1984", CODEN = "HPJOAX", ISSN = "0018-1153", bibdate = "Tue Mar 25 14:12:15 MST 1997", bibsource = "Compendex database; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.hpl.hp.com/hpjournal/pdfs/IssuePDFs/1984-07.pdf", abstract = "The plug-in math PAC for HP's new HP-71B Handheld Computer further extends the HP-71B's comprehensive standard numerical function set to provide a mathematical tool of unprecedented capability and power in a personal machine. Full use of complex variables, integration, matrix algebra, and polynomial root finding are some of the capabilities provided by this plug-in module.", acknowledgement = ack-nhfb, affiliation = "Hewlett--Packard Co, Corvallis, OR, USA", affiliationaddress = "Hewlett--Packard Co, Corvallis, OR, USA", classcodes = "C7310 (Mathematics computing)", classification = "723", fjournal = "Hewlett-Packard Journal: technical information from the laboratories of Hewlett-Packard Company", journalabr = "Hewlett Packard J", keywords = "complex; complex variables; computers, miniature; data storage, digital --- Fixed; data type; extended I/O functions; fast Fourier transform; handheld computer; HP-71B hand-held computer; matrix operations; numerical analysis; numerical function set; polynomial; read-only storage; ROM; root finder", remark = "The paper notes: ``Completely support provisions of the proposed IEEE floating-point mathematics standard. \ldots{} an HP-71B REAL variable --- a 12-digit mantissa and a three-digit exponent in the range from $ - 499 $ to $ 499 $. Each part of a COMPLEX SHORT variable or array element has the same precision as an HP-71B SHORT variable --- a five-digit mantissa and a three-digit exponent in the range from $ - 499 $ to $ 499 $. Of course, denormalized numbers, Inf (infinity), and NaNs (not-a-numbers) are also permitted.''", treatment = "P Practical; X Experimental", } @Article{Gustafson:1984:SCC, author = "Sven-{\AA}ke Gustafson", title = "On the stability of a class of convergence acceleration methods for power series", journal = j-BIT, volume = "24", number = "4", pages = "510--519", month = dec, year = "1984", CODEN = "BITTEL, NBITAB", DOI = "https://doi.org/10.1007/BF01934909", ISSN = "0006-3835 (print), 1572-9125 (electronic)", ISSN-L = "0006-3835", MRclass = "65B10", MRnumber = "764823 (86c:65006)", MRreviewer = "D. Levin", bibdate = "Thu Dec 1 17:36:45 2011", bibsource = "http://springerlink.metapress.com/openurl.asp?genre=issue&issn=0006-3835&volume=24&issue=4; https://www.math.utah.edu/pub/tex/bib/bit.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.springerlink.com/openurl.asp?genre=article&issn=0006-3835&volume=24&issue=4&spage=510", acknowledgement = ack-nhfb, fjournal = "BIT (Nordisk tidskrift for informationsbehandling)", journal-URL = "http://link.springer.com/journal/10543", keywords = "convergence acceleration", } @Article{Karp:1984:ELS, author = "A. H. Karp", title = "Exponential and Logarithm by Sequential Squaring", journal = j-IEEE-TRANS-COMPUT, volume = "C-33", number = "5", pages = "462--464", month = may, year = "1984", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/TC.1984.1676464", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Sun Jul 10 09:22:52 MDT 2011", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput1980.bib", URL = "https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=1676464", acknowledgement = ack-nj # "\slash " # ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", } @Article{Kolbig:1984:PCL, author = "K. S. K{\"o}lbig", title = "Programs for computing the logarithm of the gamma function, and the digamma function, for complex argument", journal = j-COMP-PHYS-COMM, volume = "35", number = "1--3", pages = "C-152", month = "????", year = "1984", CODEN = "CPHCBZ", DOI = "https://doi.org/10.1016/S0010-4655(84)82404-2", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", bibdate = "Sat Feb 25 10:56:06 MST 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/compphyscomm1980.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0010465584824042", acknowledgement = ack-nhfb, fjournal = "Computer Physics Communications", journal-URL = "http://www.sciencedirect.com/science/journal/00104655", } @Article{Laforgia:1984:FIG, author = "Andrea Laforgia", title = "Further Inequalities for the Gamma Function", journal = j-MATH-COMPUT, volume = "42", number = "166", pages = "597--600", month = apr, year = "1984", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "33A15", MRnumber = "85i:33001", MRreviewer = "H. E. Fettis", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, classcodes = "B0290F (Interpolation and function approximation); C4130 (Interpolation and function approximation)", corpsource = "Dept. of Math., Univ. of Torino, Torino, Italy", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "gamma function; inequalities; polynomials", treatment = "T Theoretical or Mathematical", } @Article{McCurley:1984:EE, author = "Kevin S. McCurley", title = "Explicit Estimates for $ \theta (x; 3, l) $ and $ \psi (x; 3, l) $", journal = j-MATH-COMPUT, volume = "42", number = "165", pages = "287--296", month = jan, year = "1984", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "11N56", MRnumber = "85g:11085", MRreviewer = "G. J. Rieger", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, classcodes = "B0290F (Interpolation and function approximation); C4130 (Interpolation and function approximation)", corpsource = "Dept. of Maths., Michigan State Univ., East Lansing, MI, USA", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "arithmetic progressions; Chebyshev approximation; Chebyshev functions; Dirichlet L-; explicit estimates; functions; prime number; theorem; zeros", treatment = "T Theoretical or Mathematical", } @Article{Moon:1984:AFC, author = "Wooil Moon", title = "{Airy} function with complex arguments", journal = j-COMP-PHYS-COMM, volume = "35", number = "1--3", pages = "C-692", month = "????", year = "1984", CODEN = "CPHCBZ", DOI = "https://doi.org/10.1016/S0010-4655(84)82842-8", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", bibdate = "Sat Feb 25 10:56:41 MST 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/compphyscomm1980.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0010465584828428", acknowledgement = ack-nhfb, fjournal = "Computer Physics Communications", journal-URL = "http://www.sciencedirect.com/science/journal/00104655", } @TechReport{Morris:1984:NLM, author = "A. H. Morris", title = "{NSWC} library of mathematics subroutines", type = "Report", number = "NSWC TR 84-143", institution = "Naval Surface Weapons Center", address = "Dahlgren, VA, USA", year = "1984", bibdate = "Mon Jun 03 12:24:32 2024", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, } @Article{Nesbet:1984:ARI, author = "R. K. Nesbet", title = "Algorithms for regular and irregular {Coulomb} and {Bessel} functions", journal = j-COMP-PHYS-COMM, volume = "32", number = "4", pages = "341--347", month = jul, year = "1984", CODEN = "CPHCBZ", DOI = "https://doi.org/10.1016/0010-4655(84)90051-1", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", bibdate = "Thu Apr 24 10:35:27 2008", bibsource = "https://www.math.utah.edu/pub/tex/bib/compphyscomm1980.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/0010465584900511", abstract = "Algorithms for computing Coulomb--Bessel functions are considered, with emphasis on obtaining accurate values when the argument $x$ is inside the classical turning point $ x \lambda $. Algorithms of Barnett et al. for the generalized Coulomb functions and their derivatives are discussed in the context of the phase integral formalism. Modified or alternative algorithms are considered that are designed to be valid for all values of argument $x$ and index $ \lambda $ for the functions $ F_\lambda (x) $, $ G_\lambda (x) $. An algorithm for accelerating convergence of a power series by conversion to a continued fraction is presented, and is applied to the evaluation of spherical Bessel functions. An explicit formula for the integrand of the phase integral is presented for spherical Bessel functions. The methods considered need to be augmented by an efficient algorithm for computing the logarithmic derivative of $ G_0 + i F_0 $ for Coulomb functions when $x$ is smaller than the charge parameter $ \eta $.", acknowledgement = ack-nhfb, fjournal = "Computer Physics Communications", journal-URL = "http://www.sciencedirect.com/science/journal/00104655", } @Article{Newman:1984:ABS, author = "J. N. Newman", title = "Approximations for the {Bessel} and {Struve} functions", journal = j-MATH-COMPUT, volume = "43", number = "168", pages = "551--556", month = oct, year = "1984", CODEN = "MCMPAF", DOI = "https://doi.org/10.1090/S0025-5718-1984-0758202-X", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "65D20 (33A40)", MRnumber = "86c:65021", MRreviewer = "S. Conde", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp1980.bib; JSTOR database", acknowledgement = ack-nhfb, classcodes = "B0290F (Interpolation and function approximation); C1120 (Mathematical analysis); C4130 (Interpolation and function approximation)", corpsource = "Dept. of Ocean Eng., MIT, Cambridge, MA, USA", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "accuracy; Bessel functions; function approximation; functions; IBM PC computer; minimax; polynomial approximations; polynomials; rational-fraction approximations; single-precision computations; Struve", treatment = "T Theoretical or Mathematical", } @TechReport{Ng:1984:DAA, author = "K. C. Ng", title = "Contributions to the computation of the matrix exponential", type = "Report", number = "PAM-212", institution = inst-CPAM-UCB, address = inst-CPAM-UCB:adr, month = feb, year = "1984", bibdate = "Fri Nov 11 09:06:19 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "Based on the author's Ph.D. thesis.", acknowledgement = ack-nhfb, keywords = "$\exp(Bt)$", } @Article{Nishimoto:1984:TFD, author = "Katsuyuki Nishimoto", title = "Tables of fractional differintegrations of elementary functions", journal = "J. College Engrg. Nihon Univ. Ser. B", volume = "25", pages = "41--46", year = "1984", ISSN = "0285-6182", MRclass = "30E20 (26A33)", MRnumber = "85f:30065", bibdate = "Sat Jan 11 17:44:01 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, } @Article{Noble:1984:CPE, author = "C. J. Noble and I. J. Thompson", title = "{COULN}, a program for evaluating negative energy {Coulomb} functions", journal = j-COMP-PHYS-COMM, volume = "33", number = "4", pages = "413--419", month = oct, year = "1984", CODEN = "CPHCBZ", DOI = "https://doi.org/10.1016/0010-4655(84)90146-2", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", bibdate = "Fri Feb 24 13:39:14 MST 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/compphyscomm1980.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/0010465584901462", acknowledgement = ack-nhfb, fjournal = "Computer Physics Communications", journal-URL = "http://www.sciencedirect.com/science/journal/00104655", } @Article{Piessens:1984:ACB, author = "R. Piessens", title = "Automatic computation of {Bessel} function integrals", journal = j-COMP-PHYS-COMM, volume = "35", number = "1--3", pages = "C-791", month = "????", year = "1984", CODEN = "CPHCBZ", DOI = "https://doi.org/10.1016/S0010-4655(84)82915-X", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", bibdate = "Sat Feb 25 10:56:49 MST 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/compphyscomm1980.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/S001046558482915X", acknowledgement = ack-nhfb, fjournal = "Computer Physics Communications", journal-URL = "http://www.sciencedirect.com/science/journal/00104655", } @Article{Piessens:1984:CBF, author = "Robert Piessens", title = "The computation of {Bessel} functions on a small computer", journal = j-COMPUT-MATH-APPL, volume = "10", number = "2", pages = "161--166", month = "????", year = "1984", CODEN = "CMAPDK", ISSN = "0898-1221 (print), 1873-7668 (electronic)", ISSN-L = "0898-1221", bibdate = "Wed Mar 1 19:00:50 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/computmathappl1980.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/0898122184900452", acknowledgement = ack-nhfb, fjournal = "Computers and Mathematics with Applications", journal-URL = "http://www.sciencedirect.com/science/journal/08981221", } @Article{Piessens:1984:CSA, author = "R. Piessens", title = "{Chebyshev} series approximations for the zeros of the {Bessel} functions", journal = j-J-COMPUT-PHYS, volume = "53", number = "1", pages = "188--192", month = jan, year = "1984", CODEN = "JCTPAH", DOI = "https://doi.org/10.1016/0021-9991(84)90060-3", ISSN = "0021-9991 (print), 1090-2716 (electronic)", ISSN-L = "0021-9991", bibdate = "Sun Jan 1 15:59:18 MST 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputphys1980.bib", URL = "http://www.sciencedirect.com/science/article/pii/0021999184900603", acknowledgement = ack-nhfb, fjournal = "Journal of Computational Physics", journal-URL = "http://www.sciencedirect.com/science/journal/00219991/", } @Article{Piessens:1984:SEF, author = "R. Piessens", title = "A Series Expansion for the First Positive Zero of the {Bessel} Functions", journal = j-MATH-COMPUT, volume = "42", number = "165", pages = "195--197", month = jan, year = "1984", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "33A40 (65D20)", MRnumber = "84m:33014", MRreviewer = "M. E. Muldoon", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database; Theory/Comp.Alg.1.bib", acknowledgement = ack-nhfb, annote = "Gives explicit series for first positive zero for 4 terms, using REDUCE.", classcodes = "B0220 (Mathematical analysis); C1120 (Mathematical analysis)", corpsource = "Dept. of Computer Sci., Univ. of Leuven, Heverlee, Belgium", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "Bessel functions; poles and zeros; positive zero; Reduce; series (mathematics); series expansion", treatment = "T Theoretical or Mathematical", } @Article{Schmidt:1984:TAI, author = "J. W. Schmidt", title = "Two-Sided Approximations of Inverses, Square Roots and {Cholesky} Factors", journal = "Comput. Math., Banach Center Publ.", volume = "13", pages = "483--497", year = "1984", bibdate = "Fri Jan 12 11:37:56 1996", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-jr, } @Article{Seaton:1984:CFA, author = "M. J. Seaton", title = "{Coulomb} functions analytic in the energy", journal = j-COMP-PHYS-COMM, volume = "35", number = "1--3", pages = "C-771", month = "????", year = "1984", CODEN = "CPHCBZ", DOI = "https://doi.org/10.1016/S0010-4655(84)82899-4", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", bibdate = "Sat Feb 25 10:56:49 MST 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/compphyscomm1980.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0010465584828994", acknowledgement = ack-nhfb, fjournal = "Computer Physics Communications", journal-URL = "http://www.sciencedirect.com/science/journal/00104655", } @Article{Takemasa:1984:CFC, author = "T. Takemasa and T. Tamura and H. H. Wolter", title = "{Coulomb} functions with complex angular momenta", journal = j-COMP-PHYS-COMM, volume = "35", number = "1--3", pages = "C-562", month = "????", year = "1984", CODEN = "CPHCBZ", DOI = "https://doi.org/10.1016/S0010-4655(84)82737-X", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", bibdate = "Sat Feb 25 10:56:33 MST 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/compphyscomm1980.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/S001046558482737X", acknowledgement = ack-nhfb, fjournal = "Computer Physics Communications", journal-URL = "http://www.sciencedirect.com/science/journal/00104655", } @Article{Talman:1984:LSC, author = "James D. Talman", title = "{LSFBTR}: a subroutine for calculating spherical {Bessel} transforms", journal = j-COMP-PHYS-COMM, volume = "35", number = "1--3", pages = "C-903", month = "????", year = "1984", CODEN = "CPHCBZ", DOI = "https://doi.org/10.1016/S0010-4655(84)83002-7", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", bibdate = "Sat Feb 25 10:56:58 MST 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/compphyscomm1980.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0010465584830027", acknowledgement = ack-nhfb, fjournal = "Computer Physics Communications", journal-URL = "http://www.sciencedirect.com/science/journal/00104655", } @Article{Tamura:1984:CFC, author = "Taro Tamura and Frank Rybicki", title = "{Coulomb} functions for complex energies", journal = j-COMP-PHYS-COMM, volume = "35", number = "1--3", pages = "C-5", month = "????", year = "1984", CODEN = "CPHCBZ", DOI = "https://doi.org/10.1016/S0010-4655(84)82276-6", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", bibdate = "Sat Feb 25 10:55:58 MST 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/compphyscomm1980.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0010465584822766", acknowledgement = ack-nhfb, fjournal = "Computer Physics Communications", journal-URL = "http://www.sciencedirect.com/science/journal/00104655", } @Article{Trojan:1984:LBF, author = "George M. Trojan", title = "Lower Bounds and Fast Algorithms for Sequence Acceleration", journal = j-J-ACM, volume = "31", number = "2", pages = "329--335", month = apr, year = "1984", CODEN = "JACOAH", ISSN = "0004-5411 (print), 1557-735X (electronic)", ISSN-L = "0004-5411", bibdate = "Wed Jan 15 18:12:53 MST 1997", bibsource = "Compendex database; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", abstract = "Tight upper and lower bounds are obtained for sequence accelerating. The lower bounds follow from a powerful asymptotic adversary principle. Algorithms are presented and shown to be almost optimal.", acknowledgement = ack-nhfb, affiliationaddress = "Univ of Western Ontario, Dep of Physics, London, Ont, Can", ajournal = "J. Assoc. Comput. Mach.", classification = "723", fjournal = "Journal of the ACM", journal-URL = "https://dl.acm.org/loi/jacm", keywords = "Algorithms; computer programming; convergence acceleration; lower bounds; sequence acceleration; upper bounds", } @Book{vanderLaan:1984:CSF, author = "C. G. van der Laan and N. M. Temme", title = "Calculation of special functions: the gamma function, the exponential integrals and error-like functions", volume = "10", publisher = "Centre for Mathematics and Computer Science", address = "Amsterdam, The Netherlands", pages = "iv + 231", year = "1984", ISBN = "90-6196-277-3", ISBN-13 = "978-90-6196-277-9", LCCN = "QA1 M4591 no. 10", bibdate = "Sat Oct 30 18:43:03 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", series = "CWI tract / Centrum voor Wiskunde en Informatica", acknowledgement = ack-nhfb, } @Article{vonGudenberg:1984:BMG, author = "J. Wolff {von Gudenberg}", title = "{Berechnung maximal genauer Standardfunktionen mit einfacher Mantissenl{\"a}nge} \toenglish {Computation of Maximally Accurate Elementary Functions Using Simple Mantissa Length} \endtoenglish", journal = j-ELEK-RECHENANLAGEN, volume = "26", number = "5", pages = "230--238", month = oct, year = "1984", CODEN = "ELRAA4", ISSN = "0013-5720", bibdate = "Sun Oct 25 10:29:27 1998", bibsource = "garbo.uwasa.fi:/pc/doc-soft/fpbiblio.txt; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nj, fjournal = "Elektronische Rechenanlagen", } @Article{Walmsley:1984:EEM, author = "John L. Walmsley", title = "On the efficient evaluation of modified {Bessel} functions of zeroth and first orders for arguments of the form $ x \exp (i \pi / 4) $", journal = j-J-COMPUT-PHYS, volume = "56", number = "2", pages = "349--355", month = nov, year = "1984", CODEN = "JCTPAH", DOI = "https://doi.org/10.1016/0021-9991(84)90100-1", ISSN = "0021-9991 (print), 1090-2716 (electronic)", ISSN-L = "0021-9991", bibdate = "Sun Jan 1 15:59:21 MST 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputphys1980.bib", URL = "http://www.sciencedirect.com/science/article/pii/0021999184901001", acknowledgement = ack-nhfb, fjournal = "Journal of Computational Physics", journal-URL = "http://www.sciencedirect.com/science/journal/00219991/", } @Book{Wawrzynczyk:1984:GRS, author = "Antoni Wawrzy{\'n}czyk and Aleksander Strasburger", title = "Group representations and special functions", volume = "8", publisher = pub-REIDEL, address = pub-REIDEL:adr, pages = "xvi + 688", year = "1984", ISBN = "90-277-1269-7", ISBN-13 = "978-90-277-1269-1", LCCN = "QA171 .W3513 1984; QA1 M4281 v. 8", bibdate = "Sat Oct 30 18:29:38 MDT 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; z3950.loc.gov:7090/Voyager", series = "Mathematics and its applications. East European series", acknowledgement = ack-nhfb, remark = "Translation of Polish title Wsp{\'o}\pm{}czesna teoria funkcji specjalnych.", subject = "Representations of groups; Functions, Special", } @Article{Wrigge:1984:NMG, author = "Staffan Wrigge", title = "A note on the moment generating function for the reciprocal gamma distribution", journal = j-MATH-COMPUT, volume = "42", number = "166", pages = "617--621", month = apr, year = "1984", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "65D20 (60E10 62E15 65U05)", MRnumber = "86f:65042b", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, classcodes = "B0290F (Interpolation and function approximation); C4130 (Interpolation and function approximation)", corpsource = "Nat. Defence Res. Inst., Stockholm, Sweden", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "Euler--Maclaurin expansion; moment generating function; polynomials; reciprocal gamma distribution", treatment = "T Theoretical or Mathematical", } @Article{Akrivis:1985:ENC, author = "G. Akrivis", title = "The error norm of certain {Gaussian} quadrature formulae", journal = j-MATH-COMPUT, volume = "45", number = "172", pages = "513--519", month = oct, year = "1985", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "65D32", MRnumber = "87a:65051", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, classcodes = "B0290B (Error analysis in numerical methods); B0290F (Interpolation and function approximation); C4110 (Error analysis in numerical methods); C4130 (Interpolation and function approximation)", corpsource = "Math. Inst., Munchen Univ., West Germany", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "error analysis; error functional; error norm; function approximation; Gaussian quadrature formulae; integration; weight functions; wide class", treatment = "T Theoretical or Mathematical", } @Book{Arfken:1985:MMP, author = "George B. (George Brown) Arfken", title = "Mathematical methods for physicists", publisher = pub-ACADEMIC, address = pub-ACADEMIC:adr, edition = "Third", pages = "xxii + 985", year = "1985", ISBN = "0-12-059820-5", ISBN-13 = "978-0-12-059820-5", LCCN = "QA37.2 .A74 1985", bibdate = "Wed Mar 15 06:50:49 MDT 2017", bibsource = "fsz3950.oclc.org:210/WorldCat; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/book/9780120598205", abstract = "Mathematical Methods for Physicists, Third Edition provides an advanced undergraduate and beginning graduate study in physical science, focusing on the mathematics of theoretical physics. This edition includes sections on the non-Cartesian tensors, dispersion theory, first-order differential equations, numerical application of Chebyshev polynomials, the fast Fourier transform, and transfer functions. Many of the physical examples provided in this book, which are used to illustrate the applications of mathematics, are taken from the fields of electromagnetic theory and quantum mechanics. The Hermitian operators, Hilbert space, and concept of completeness are also deliberated. This book is beneficial to students studying graduate level physics, particularly theoretical physics.", acknowledgement = ack-nhfb, author-dates = "1922--", subject = "Mathematics; Mathematical physics; Math{\'e}matiques; Physique math{\'e}matique; Mathematical physics.; Mathematics.; Wiskundige methoden.; Reactoren.; Groepentheorie.; Kwantummechanica.; Elektromechanica.; Vectoren (wiskunde); Elektrodynamica.; Math{\'e}matiques.; Physique math{\'e}matique.; Math{\'e}matiques de l'ing{\'e}nieur.", tableofcontents = "Vector Analysis \\ Rotation of the Coordinate Axes \\ Scalar or Dot Product \\ Vector or Cross Product \\ Triple Scalar Product, Triple Vector Product \\ Gradient, [down triangle, open] \\ Divergence, [down triangle, open] \\ Curl, [down triangle, open] x \\ Successive Applications of [down triangle, open] \\ Vector Integration \\ Gauss's Theorem \\ Stokes's Theorem \\ Potential Theory \\ Gauss's Law, Poisson's Equation \\ Dirac Delta Function \\ Helmholtz's Theorem \\ Curved Coordinates, Tensors \\ Orthogonal Coordinates \\ Differential Vector Operators \\ Special Coordinate Systems: Introduction \\ Circular Cylindrical Coordinates \\ Spherical Polar Coordinates \\ Tensor Analysis \\ Contraction, Direct Product \\ Quotient Rule \\ Pseudotensors, Dual Tensors \\ Non-Cartesian Tensors \\ Tensor Derivative Operators \\ Determinants and Matrices \\ Determinants \\ Matrices \\ Orthogonal Matrices \\ Hermitian Matrices, Unitary Matrices \\ Diagonalization of Matrices \\ Normal Matrices \\ Group Theory \\ Introduction to Group Theory \\ Generators of Continuous Groups \\ Orbital Angular Momentum \\ Angular Momentum Coupling \\ Homogeneous Lorentz Group \\ Lorentz Covariance of Maxwell's Equations \\ Discrete Groups \\ Infinite Series \\ Convergence Tests \\ Alternating Series \\ Algebra of Series \\ Series of Functions \\ Taylor's Expansion \\ Power Series \\ Elliptic Integrals \\ Bernoulli Numbers, Euler-Maclaurin Formula \\ Asymptotic Series \\ Infinite Products \\ Functions of a Complex Variable I \\ Complex Algebra", } @Article{Bank:1985:SEM, author = "Randolph E. Bank and Craig C. Douglas", title = "Sharp estimates for multigrid rates of convergence with general smoothing and acceleration", journal = j-SIAM-J-NUMER-ANAL, volume = "22", number = "4", pages = "617--633", month = aug, year = "1985", CODEN = "SJNAAM", ISSN = "0036-1429 (print), 1095-7170 (electronic)", ISSN-L = "0036-1429", MRclass = "65F10 (65N20)", MRnumber = "86j:65037", MRreviewer = "L. W. Ehrlich", bibdate = "Fri Oct 16 06:57:22 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "SIAM Journal on Numerical Analysis", journal-URL = "http://epubs.siam.org/sinum", keywords = "convergence acceleration", } @InProceedings{Bannur:1985:VIS, author = "J. Bannur and A. Varma", title = "The {VLSI} Implementation of a Square Root Algorithm", crossref = "Hwang:1985:PSC", pages = "159--165", year = "1985", bibdate = "Fri Nov 16 08:47:34 2007", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "http://www.acsel-lab.com/arithmetic/arith7/papers/ARITH7_Bannur_Varma.pdf", abstract = "VLSI implementation of a square root algorithm is studied. Two possible implementations of the basic nonrestoring algorithm are presented --- the second is more area-efficient and modular than the first. The implementations are simple and easy to control, but, at the same time, are more area-time efficient than many existing designs. A hardware algorithm suited to microprogram implementation is also given. Extension of the algorithms to achieve $ 1 / 2 $-bit precision is discussed.", acknowledgement = ack-nhfb, keywords = "ARITH-7", } @Article{Borodin:1985:DND, author = "Allan Borodin and Ronald Fagin and John E. Hopcroft and Martin Tompa", title = "Decreasing the Nesting Depth of Expressions Involving Square Roots", journal = j-J-SYMBOLIC-COMP, volume = "1", number = "2", pages = "169--188", month = jun, year = "1985", CODEN = "JSYCEH", ISSN = "0747-7171 (print), 1095-855X (electronic)", ISSN-L = "0747-7171", MRclass = "68Q40 (12F05)", MRnumber = "87a:68087", bibdate = "Sat May 10 15:54:09 MDT 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Journal of Symbolic Computation", journal-URL = "http://www.sciencedirect.com/science/journal/07477171", keywords = "Simplification", } @Article{Brezinski:1985:CAM, author = "Claude Brezinski", title = "Convergence acceleration methods: the past decade", journal = j-J-COMPUT-APPL-MATH, volume = "12--13", number = "??", pages = "19--36", month = may, year = "1985", CODEN = "JCAMDI", DOI = "https://doi.org/10.1016/0377-0427(85)90005-6", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", MRclass = "65Bxx (65J05)", MRnumber = "793942 (86f:65019)", bibdate = "Thu Dec 01 10:11:33 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", keywords = "convergence acceleration", remark = "Proceedings of the international conference on computational and applied mathematics (Leuven, 1984).", } @Article{Carlson:1985:AEF, author = "B. C. Carlson and John L. Gustafson", title = "Asymptotic expansion of the first elliptic integral", journal = j-SIAM-J-MATH-ANA, volume = "16", number = "5", pages = "1072--1092", month = sep, year = "1985", CODEN = "SJMAAH", ISSN = "0036-1410 (print), 1095-7154 (electronic)", ISSN-L = "0036-1410", MRclass = "33A25 (41A60)", MRnumber = "87d:33002", MRreviewer = "Kusum Soni", bibdate = "Sat Dec 5 18:14:13 MST 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "SIAM Journal on Mathematical Analysis", journal-URL = "http://epubs.siam.org/sima", } @Article{Cathey:1985:ISR, author = "James Cathey", title = "68000 Integer square root routine in {16BST}", journal = j-DDJ, volume = "10", number = "5", pages = "118--??", month = may, year = "1985", CODEN = "DDJOEB", ISSN = "1044-789X", bibdate = "Mon Sep 2 09:09:39 MDT 1996", bibsource = "http://www.ddj.com/index/author/index.htm; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, fjournal = "Dr. Dobb's Journal of Software Tools", } @InProceedings{Conover:1985:AHS, author = "B. Conover and D. L. Gustafson", title = "An Algorithm for High Speed Square Roots", crossref = "IEEE:1985:ERC", pages = "19--21", year = "1985", bibdate = "Fri Jun 11 18:04:41 1999", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nj, } @Article{Frenzen:1985:NAE, author = "C. L. Frenzen and R. Wong", title = "A note on asymptotic evaluation of some {Hankel} transforms", journal = j-MATH-COMPUT, volume = "45", number = "172", pages = "537--548", month = oct, year = "1985", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "41A60 (44A15 65R10)", MRnumber = "87c:41024", MRreviewer = "F. W. J. Olver", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, classcodes = "B0230 (Integral transforms); B0290Z (Other numerical methods)C1130 (Integral transforms); C4190 (Other numerical methods)", corpsource = "Dept. of Math., British Columbia Univ., Vancouver, BC, Canada", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "asymptotic expansion; Bessel function; Bessel functions; growth condition; Hankel transforms; meromorphic function; transforms", treatment = "T Theoretical or Mathematical", } @InProceedings{Gal:1985:CEF, author = "Shmuel Gal", title = "Computing Elementary Functions: a New Approach for Achieving High Accuracy and Good Performance", crossref = "Miranker:1985:ASC", pages = "1--16", year = "1985", DOI = "https://doi.org/10.1007/3-540-16798-6_1", bibdate = "Thu Sep 01 12:27:23 1994", bibsource = "garbo.uwasa.fi:/pc/doc-soft/fpbiblio.txt; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nj, } @Article{Gustafson:1985:SCA, author = "Sven-{\AA}ke Gustafson", title = "Stable convergence acceleration using {Laplace} transforms", journal = j-NUM-MATH, volume = "47", number = "3", pages = "387--394", month = nov, year = "1985", CODEN = "NUMMA7", ISSN = "0029-599X (print), 0945-3245 (electronic)", ISSN-L = "0029-599X", MRclass = "65B10 (65D30)", MRnumber = "86m:65009", bibdate = "Mon May 26 11:49:34 MDT 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, classification = "B0230 (Integral transforms); C1130 (Integral transforms)", corpsource = "Dept. of Numerical Anal. and Comput. Sci., R. Inst. of Technol., Stockholm, Sweden", fjournal = "Numerische Mathematik", journal-URL = "http://link.springer.com/journal/211", keywords = "convergence acceleration; Laplace transforms; power series; quadrature schemes; series (mathematics); stable convergence acceleration", treatment = "T Theoretical or Mathematical", } @Article{Hill:1985:RCS, author = "I. D. Hill and M. C. Pike", title = "Remark on ``{Algorithm 299: Chi-Squared Integral}''", journal = j-TOMS, volume = "11", number = "2", pages = "185--185", month = jun, year = "1985", CODEN = "ACMSCU", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Tue Feb 06 05:28:22 1996", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "See \cite{Hill:1967:ACS,elLozy:1976:RAC,elLozy:1979:RAS}.", acknowledgement = ack-nhfb, fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", } @Article{Hull:1985:PRV, author = "T. E. Hull and A. Abrham", title = "Properly Rounded Variable Precision Square Root", journal = j-TOMS, volume = "11", number = "3", pages = "229--237", month = sep, year = "1985", CODEN = "ACMSCU", DOI = "https://doi.org/10.1145/214408.214413", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", MRclass = "65D15 (65G05)", MRnumber = "87a:65041", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib", URL = "http://www.acm.org/pubs/citations/journals/toms/1985-11-3/p229-hull/; http://www.acm.org/pubs/toc/Abstracts/toms/214413.html", abstract = "The square root function presented here returns a properly rounded approximation to the square root of its argument, or it raises an error condition if the argument is negative. {\em Properly rounded} means rounded to the nearest, or to nearest even in case of a tie. It is {\em variable precision} in that it is designed to return a $p$-digit approximation to a $p$-digit argument, for any $ p > 0 $. (Precision $p$ means $p$ decimal digits.) The program and the analysis are valid for all $ p > 0 $, but current implementations place some restrictions on $p$.", acknowledgement = ack-nhfb, catcode = "G.4; G.4; G.1.0; G.1.2; G.4; G.1.0", CRclass = "G.4 Algorithm analysis; G.4 Verification; G.1.0 General; G.1.0 Numerical algorithms; G.1.2 Approximation; G.1.2 Elementary function approximation; G.4 Certification and testing; G.1.0 General; G.1.0 Error analysis", descriptor = "Mathematics of Computing, MATHEMATICAL SOFTWARE, Algorithm analysis; Mathematics of Computing, MATHEMATICAL SOFTWARE, Verification; Mathematics of Computing, NUMERICAL ANALYSIS, General, Numerical algorithms; Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation; Mathematics of Computing, MATHEMATICAL SOFTWARE, Certification and testing; Mathematics of Computing, NUMERICAL ANALYSIS, General, Error analysis", fjournal = "ACM Transactions on Mathematical Software (TOMS)", genterm = "algorithms; verification", guideno = "02789", journal-URL = "https://dl.acm.org/loi/toms", jrldate = "Sept. 1985", keywords = "algorithms; decimal floating-point arithmetic; verification", subject = "{\bf G.1.2}: Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation. {\bf G.4}: Mathematics of Computing, MATHEMATICAL SOFTWARE, Algorithm analysis. {\bf G.4}: Mathematics of Computing, MATHEMATICAL SOFTWARE, Certification and testing. {\bf G.4}: Mathematics of Computing, MATHEMATICAL SOFTWARE, Verification. {\bf G.1.0}: Mathematics of Computing, NUMERICAL ANALYSIS, General, Error analysis. {\bf G.1.0}: Mathematics of Computing, NUMERICAL ANALYSIS, General, Numerical algorithms.", } @Article{Humblet:1985:BFE, author = "J. Humblet", title = "{Bessel} function expansions of {Coulomb} wave functions", journal = j-J-MATH-PHYS, volume = "26", number = "4", pages = "656--659", month = apr, year = "1985", CODEN = "JMAPAQ", DOI = "https://doi.org/10.1063/1.526602", ISSN = "0022-2488 (print), 1089-7658 (electronic), 1527-2427", ISSN-L = "0022-2488", MRclass = "81C05 (33A40 81G45)", MRnumber = "87c:81034", bibdate = "Mon Oct 31 11:57:19 MDT 2011", bibsource = "http://jmp.aip.org/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jmathphys1985.bib", URL = "http://jmp.aip.org/resource/1/jmapaq/v26/i4/p656_s1", acknowledgement = ack-nhfb, fjournal = "Journal of Mathematical Physics", journal-URL = "http://jmp.aip.org/", pagecount = "4", } @Article{Jones:1985:CIG, author = "William B. Jones and W. J. Thron", title = "On the computation of incomplete gamma functions in the complex domain", journal = j-J-COMPUT-APPL-MATH, volume = "12--13", number = "??", pages = "401--417", month = may, year = "1985", CODEN = "JCAMDI", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Sat Feb 25 12:27:12 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath1980.bib", URL = "http://www.sciencedirect.com/science/article/pii/0377042785900342", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @Unpublished{Kahan:1985:EFK, author = "William Kahan", title = "Elementary Functions from Kernels and Elementary Inequalities among Elementary Functions", institution = inst-BERKELEY-CS, address = inst-BERKELEY-CS:adr, pages = "5", day = "24", month = oct, year = "1985", bibdate = "Sat Aug 23 06:12:44 2025", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/k/kahan-william-m.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "https://ieeemilestones.ethw.org/w/images/1/16/Wk_elems_from_kernels_oct85.pdf", abstract = "Given binary floating-point subprograms to calculate the ``Kernels'' $ \ln (x) $ for $ x \ge 0 $ and $ \ln 1 p(x) = \ln (1 + x) $ for $ x \ge - 1 $, $ \exp (x) $ and $ \expmone (x) = \exp (x) - 1 $ for all $x$, and $ \tan (x)$ for $ |x| < \pi / 8$ and $ \arctan (x)$ for $ |x| < \sqrt {2} - 1$, to nearly full working accuracy, we may calculate all the other elementary transcendental functions almost as accurately, and with no violation of (weak) monotonicity,", acknowledgement = ack-nhfb, } @Article{Kravchuk:1985:ACE, author = "V. R. Kravchuk", title = "Approximation of certain elementary functions by rational functions of order $ (n, 2) $. ({Russian})", journal = "Akad. Nauk Ukrain. SSR Inst. Mat. Preprint", volume = "18", pages = "7--40", year = "1985", MRclass = "41A25 (41A10)", MRnumber = "87b:41017", MRreviewer = "R. Smarzewski", bibdate = "Sat Jan 11 17:44:01 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, language = "Russian", } @Article{Kravchuk:1985:EAE, author = "V. R. Kravchuk", title = "Effective approximation of elementary functions by rational polynomials of order $ (n, 1) $. ({Russian})", journal = j-UKR-MAT-Z, volume = "37", number = "2", pages = "175--180, 270", year = "1985", CODEN = "UMZHAA", ISSN = "0041-6053", MRclass = "41A20 (41A25)", MRnumber = "86h:41013", MRreviewer = "Miguel A. Jim{\'e}nez Pozo", bibdate = "Sat Jan 11 17:44:01 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Ukrainskii matematicheskii zhurnal", language = "Russian", } @Article{Lazard:1985:PFD, author = "Daniel Lazard", title = "Primitives des fonctions {\'e}l{\'e}mentaires (d'apr{\`e}s {Risch} et {Davenport}). ({French}) [Primitives of elementary functions (following {Risch} and {Davenport})] {Seminar Bourbaki, Vol. 1983/84, No. 121-122}", journal = "Ast{\'e}risque", volume = "121--122", pages = "295--308", year = "1985", MRclass = "12H05 (12-04)", MRnumber = "86k:12010", MRreviewer = "J. H. Davenport", bibdate = "Sat Jan 11 17:44:01 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, language = "French", } @Article{Lewanowicz:1985:RRH, author = "Stanis{\l}aw Lewanowicz", title = "Recurrence relations for hypergeometric functions of unit argument", journal = j-MATH-COMPUT, volume = "45", number = "172", pages = "521--535", month = oct, year = "1985", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "33A35 (65Q05)", MRnumber = "86m:33004", MRreviewer = "S. Conde", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp1980.bib; JSTOR database", acknowledgement = ack-nhfb, classcodes = "B0290 (Numerical analysis); C4100 (Numerical analysis)", corpsource = "Inst. of Comput. Sci., Wroclaw Univ., Poland", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "(mathematics); convergence of numerical methods; hypergeometric function; numerical analysis; recurrence relation; series; unit argument", treatment = "T Theoretical or Mathematical", } @Article{Lo:1985:GPB, author = "Hao-Yung Lo and Y. Aoki", title = "Generation of a Precise Binary Logarithm with Difference Grouping Programmable Logic Array", journal = j-IEEE-TRANS-COMPUT, volume = "C-34", number = "8", pages = "681--691", month = aug, year = "1985", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/TC.1985.1676614", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Sun Jul 10 08:33:17 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput1980.bib", URL = "https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=1676614", acknowledgement = ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", } @Article{Majerski:1985:SRA, author = "S. Majerski", title = "Square-Rooting Algorithms for High-Speed Digital Circuits", journal = j-IEEE-TRANS-COMPUT, volume = "C-34", number = "8", pages = "724--733", month = aug, year = "1985", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/TC.1985.1676618", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Sun Jul 10 08:33:17 MDT 2011", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput1980.bib", URL = "https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=1676618", acknowledgement = ack-nj # "\slash " # ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", } @Article{Martin:1985:FAB, author = "Pablo Mart{\'\i}n and Antonio L. Guerrero", title = "Fractional approximations to the {Bessel} function {$ J_0 (x) $}", journal = j-J-MATH-PHYS, volume = "26", number = "4", pages = "705--707", month = apr, year = "1985", CODEN = "JMAPAQ", DOI = "https://doi.org/10.1063/1.526610", ISSN = "0022-2488 (print), 1089-7658 (electronic), 1527-2427", ISSN-L = "0022-2488", MRclass = "41A21 (33A40)", MRnumber = "86g:41031", MRreviewer = "Hans-J{\"u}rgen Albrand", bibdate = "Mon Oct 31 11:57:19 MDT 2011", bibsource = "http://jmp.aip.org/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jmathphys1985.bib", URL = "http://jmp.aip.org/resource/1/jmapaq/v26/i4/p705_s1", acknowledgement = ack-nhfb, fjournal = "Journal of Mathematical Physics", journal-URL = "http://jmp.aip.org/", pagecount = "3", } @Article{Milgram:1985:GIE, author = "M. S. Milgram", title = "The Generalized Integro-Exponential Function", journal = j-MATH-COMPUT, volume = "44", number = "170", pages = "443--458", month = apr, year = "1985", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "33A70 (65D15)", MRnumber = "86c:33024", MRreviewer = "S. Conde", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, classcodes = "B0290D (Functional analysis); B0290F (Interpolation and function approximation); C4120 (Functional analysis); C4130 (Interpolation and function approximation)", corpsource = "AECL, Chalk River, Ont., Canada", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "exponential function; exponential integral; first-order functions; function; function approximation; function evaluation; generalized integro-; incomplete gamma; minimax; rational minimax approximations; techniques", treatment = "T Theoretical or Mathematical", } @Article{Muller:1985:DBC, author = "Jean-Michel Muller", title = "Discrete Basis and Computation of Elementary Functions", journal = j-IEEE-TRANS-COMPUT, volume = "C-34", number = "9", pages = "857--862", month = sep, year = "1985", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/TC.1985.1676643", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", MRclass = "65D20 (65V05)", MRnumber = "87e:65016", MRreviewer = "D. Zwick", bibdate = "Sun Jul 10 08:33:33 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput1980.bib", URL = "https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=1676643", abstract = "We give necessary and sufficient conditions in order that the infinite product or sum of the terms of a positive decreasing sequence generates the reals in a given interval.", acknowledgement = ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", keywords = "Additives; Approximation algorithms; Computers; Convergence; CORDIC-like algorithms, hardware computation of elementary functions; Delays; Electrons; Hardware; Iterative methods; Logic gates; representation of real numbers by infinite series; Sufficient conditions", } @InCollection{Muller:1985:RNR, author = "Jean-Michel Muller", booktitle = "Seminar on number theory, 1984--1985 (Talence, 1984/1985)", title = "Repr{\'e}sentation des nombres r{\'e}els et calcul des fonctions {\'e}l{\'e}mentaires. ({French}) [Representation of real numbers and calculation of elementary functions]", volume = "12", publisher = "Univ. Bordeaux {I}", address = "Talence, France", pages = "22", year = "1985", MRclass = "26-04 (11B13 11B34 26A09)", MRnumber = "87k:26001", MRreviewer = "S. L. Segal", bibdate = "Sat Jan 11 17:44:01 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, language = "French", remark = "From the MRreview: ``About one third of the paper is devoted to algorithms for calculating quantities like square root, exponential, logarithm, or trigonometric funtions, using discrete bases. Indeed, the major motivation for the present paper is obtaining simple algorithms which can easily be realized by hardware.''", } @TechReport{Parlett:1985:DAA, author = "Beresford N. Parlett and K. C. Ng", title = "Development of an accurate algorithm for {$ \exp (B t) $}", type = "Technical Report", number = "PAM-294", institution = inst-CPAM-UCB, address = inst-CPAM-UCB:adr, month = aug, year = "1985", bibdate = "Fri Nov 11 09:06:19 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, } @PhdThesis{Peralta:1985:TRN, author = "Rene Caupolican Peralta", title = "Three results in number theory and cryptography: a new algorithm to compute square roots modulo a prime number; On the bit complexity of the discrete logarithm; a framework for the study of cryptoprotocols", type = "Thesis ({Ph.D.})", school = "Department of Computer Science, University of California, Berkeley", address = "Berkeley, CA, USA", pages = "52", month = dec, year = "1985", LCCN = "????", bibdate = "Sat Oct 17 16:25:07 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/cryptography.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, keywords = "dissertations; dissertations, academic --- UCB --- computer science --- 1981--1990; University of California, Berkeley. computer science division --", } @Article{Pereira:1985:ECF, author = "N. Costa Pereira", title = "Estimates for the {Chebyshev} function $ \psi (x) - \theta (x) $", journal = j-MATH-COMPUT, volume = "44", number = "169", pages = "211--221", month = jan, year = "1985", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "11A25 (11N45 11Y35 33A70)", MRnumber = "86k:11005", bibdate = "Thu Jun 15 07:26:46 2000", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", note = "See corrigendum \cite{Pereira:1987:CEC}.", acknowledgement = ack-nhfb, classcodes = "B0290F (Interpolation and function approximation); C4130 (Interpolation and function approximation)", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "Chebyshev approximation; Chebyshev function", treatment = "T Theoretical or Mathematical", } @Article{Schoof:1985:ECF, author = "Ren{\'e} Schoof", title = "Elliptic Curves Over Finite Fields and the Computation of Square Roots $ \operatorname {mod} p $", journal = j-MATH-COMPUT, volume = "44", number = "170", pages = "483--494", month = apr, year = "1985", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "11Y16 (11G20 14G15)", MRnumber = "86e:11122", MRreviewer = "Karl Rubin", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp1980.bib; JSTOR database", acknowledgement = ack-nhfb, classcodes = "B0250 (Combinatorial mathematics); B0290D (Functional analysis); C1160 (Combinatorial mathematics); C4120 (Functional analysis); C4240 (Programming and algorithm theory)", corpsource = "Amsterdam Univ., Netherlands", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "computational complexity; deterministic algorithm; elliptic curve; F/sub q/-points; finite fields; function evaluation; number theory; square roots mod p; Weierstrass equation", treatment = "T Theoretical or Mathematical", } @Article{Shah:1985:SAA, author = "Arvind K. Shah", title = "A Simpler Approximation for Areas Under the Standard Normal Curve", journal = j-AMER-STAT, volume = "39", number = "1", pages = "80--80", month = feb, year = "1985", CODEN = "ASTAAJ", ISSN = "0003-1305 (print), 1537-2731 (electronic)", ISSN-L = "0003-1305", bibdate = "Fri Jan 27 12:40:28 MST 2012", bibsource = "http://www.jstor.org/journals/00031305.html; http://www.jstor.org/stable/i326426; https://www.math.utah.edu/pub/tex/bib/amstat1980.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.jstor.org/stable/2683918", acknowledgement = ack-nhfb, fjournal = "The American Statistician", journal-URL = "http://www.tandfonline.com/loi/utas20", } @Article{Spijker:1985:SRS, author = "M. N. Spijker", title = "Stepsize restrictions for stability of one-step methods in the numerical solution of initial value problems", journal = j-MATH-COMPUT, volume = "45", number = "172", pages = "377--392", month = oct, year = "1985", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "65L20 (65M10)", MRnumber = "86j:65106", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, classcodes = "A0260 (Numerical approximation and analysis); A0560 (Transport processes: theory); B0290F (Interpolation and function approximation); B0290P (Differential equations); C4130 (Interpolation and function approximation); C4170 (Differential equations); C7320 (Physics and chemistry computing)", corpsource = "Inst. of Appl. Math. and Comput. Sci., Leiden Univ., Netherlands", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "convection; convergence of numerical methods; differential equations; diffusion; diffusion-convection; error growth; initial value problems; iterative methods; numerical solution; partial; problem; stability of one-step methods; stepsize restrictions", treatment = "T Theoretical or Mathematical", } @Article{Sreedharan:1985:ASS, author = "J. Sreedharan and A. Dhurkadas", title = "8086 algorithm solves square roots", journal = j-EDN, volume = "30", number = "7", pages = "272", month = apr, year = "1985", CODEN = "EDNSBH", ISSN = "0012-7515, 0364-6637", bibdate = "Thu Sep 1 10:15:42 1994", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nj, fjournal = "EDN", } @Article{Temme:1985:LTI, author = "N. M. Temme", title = "{Laplace} type integrals: transformation to standard form and uniform asymptotic expansions", journal = j-QUART-APPL-MATH, volume = "43", number = "1", pages = "103--123", year = "1985", CODEN = "QAMAAY", DOI = "https://doi.org/10.1090/qam/782260", ISSN = "0033-569x (print), 1552-4485 (electronic)", ISSN-L = "0033-569X", MRclass = "44A10 (41A60)", MRnumber = "782260", MRreviewer = "I. Feny{\H{o}}", bibdate = "Tue Feb 6 11:42:02 2018", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, ajournal = "Q. Appl. Math.", fjournal = "Quarterly of Applied Mathematics", journal-URL = "http://dl.acm.org/citation.cfm?id=J641; http://www.ams.org/journals/qam", } @Article{Agarwal:1986:NSV, author = "Ramesh C. Agarwal and James W. Cooley and Fred G. Gustavson and James B. Shearer and Gordon Slishman and Bryant Tuckerman", title = "New Scalar and Vector Elementary Functions for the {IBM System\slash 370}", journal = j-IBM-JRD, volume = "30", number = "2", pages = "126--144", month = mar, year = "1986", CODEN = "IBMJAE", DOI = "https://doi.org/10.1147/rd.302.0126", ISSN = "0018-8646 (print), 2151-8556 (electronic)", ISSN-L = "0018-8646", MRclass = "76W05", MRnumber = "840 341", bibdate = "Sat Jan 11 17:44:01 MST 1997", bibsource = "Compendex database; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ibmjrd.bib", note = "See clarification \cite{Agarwal:1987:CNS}.", abstract = "Algorithms have been developed to compute short-and long-precision elementary functions: SIN, COS, TAN, COTAN, LOG, LOG10, EXP, POWER, SQRT, ATAN, ASIN, ACOS, ATAN2, and CABS, in scalar (28 functions) and vector (22 functions) mode. They have been implemented as part of the new VS FORTRAN library recently announced along with the IBM 3090 Vector Facility. These algorithms are essentially table-based algorithms. An important feature of these algorithms is that they produce bitwise-identical results on scalar and vector System\slash 370 machines. Of these, for five functions the computed value result is always the correctly rounded value of the infinite-precision result. For the rest of the functions, the value returned is one of the two floating-point neighbors bordering the infinite-precision result, which implies exact results if they are machine-representable. For the five correctly rounded elementary functions, scalar and vector algorithms are designed independently to optimize performance in each case.", accepted = "2 December 1985", acknowledgement = ack-nhfb, ajournal = "IBM J. Res. Develop.", classcodes = "C6140D (High level languages); C7310 (Mathematics computing)", classification = "723", corpsource = "IBM Res. Div., Yorktown Heights, NY, USA", fjournal = "IBM Journal of Research and Development", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=5288520", journalabr = "IBM J Res Dev", keywords = "ACOS; Algorithms; algorithms; ASIN; ATAN; ATAN2; bitwise-identical results; CABS; computer metatheory; computer programming; computer programming languages --- fortran; COS; COTAN; design; elementary functions; EXP; FORTRAN; fortran library; functions; IBM computers; IBM System/370; infinite-precision result; LOG; LOG10; mainframes; measurement; performance; POWER; scalar elementary functions; SIN; SQRT; subroutines; table-based algorithms; TAN; vector elementary; VS FORTRAN library", received = "5 November 1985", remark-1 = "Numerous figures show errors in ulps, in either linear or logarithmic scales, as dot plots over a range of arguments, an idea that the authors credit to a suggestion by Cleve Moler, then consulting with IBM Palo Alto labs; such plots are used extensively in \cite{Beebe:2017:MFC}.", remark-2 = "From pages 128--129: ``A great deal of satisfaction was obtained from the fact that five of the intrinsic functions reported here always deliver correctly rounded results; these are SQRT, DSQRT, CABS, CDABS, and EXP. One important aspect of this is that correctly rounded results were obtained with surprisingly little sacrifice in performance.''", remark-3 = "From page 132: ``Our CABS and CDABS functions satisfy $\e/u < 0.5$ (this can also be called a half-ulp criterion). They have best-possible rounding, except that unavoidably there are cases when $| e/u | = 0.5$, in which case it would be equally correct to round downward or upward; we choose to round upward. This is consistent with the System/370 definition of rounding.''", remark-4 = "From pages 134--135: ``Tuckerman's condition is of historic significance, as its use allowed us to produce IBM's first elementary function that delivered correctly rounded results for all arguments.''", remark-5 = "From page 137: ``X**2.0 usually produces a correctly rounded value, while X*X always produces the truncated value of $X^2$ .''", remark-6 = "From page 139: ``Generating precise times is difficult, since seemingly inconsequential changes in the timing procedure may have a noticeable effect on the measured times. For example, on the 3081KX the performance of the STM [store multiple] and LM [load multiple] instructions is severely degraded near page boundaries. This means that in the rare event that the save area of a subroutine is near a page boundary, the speed of execution of the subroutine will be substantially decreased.''", subject = "C.4 Computer Systems Organization, PERFORMANCE OF SYSTEMS \\ I.1.2 Computing Methodologies, ALGEBRAIC MANIPULATION, Algorithms \\ F.3.3 Theory of Computation, LOGICS AND MEANINGS OF PROGRAMS, Studies of Program Constructs, Functional constructs \\ C.1.2 Computer Systems Organization, PROCESSOR ARCHITECTURES, Multiple Data Stream Architectures (Multiprocessors), Array and vector processors", treatment = "N New Development; P Practical", } @Article{Amos:1986:APP, author = "D. E. Amos", title = "{Algorithm 644}: a Portable Package for {Bessel} Functions of a Complex Argument and Nonnegative Order", journal = j-TOMS, volume = "12", number = "3", pages = "265--273", month = sep, year = "1986", CODEN = "ACMSCU", DOI = "https://doi.org/10.1145/7921.214331", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", MRclass = "65D20", MRnumber = "889 069", bibdate = "Tue Mar 09 10:26:27 1999", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib", note = "See also \cite{Amos:1990:RPP,Amos:1995:RAP,Kodama:2007:RA}.", URL = "http://www.acm.org/pubs/citations/journals/toms/1986-12-3/p265-amos/", abstract = "This algorithm is a package of subroutines for Computing Bessel functions $ H_v^{(1)}(z) $, $ H_v^{(2)}(z) $, $ I_v(z) $, $ J_v(z) $, $ K_v(z) $, $ Y_v(z) $ and Airy functions $ \mbox {Ai}(z) $, $ \mbox {Ai}'(z) $, $ \mbox {Bi}(z) $, $ \mbox {Bi}'(z) $ for orders $ v \geq 0 $ and complex $z$ in $ - \pi < \mbox {arg} z \leq \pi $. Eight callable subroutines and their double-precision counterparts are provided. Exponential scaling and sequence generation are auxiliary options.", acknowledgement = ack-nhfb, fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", keywords = "algorithms", subject = "{\bf G.1.0}: Mathematics of Computing, NUMERICAL ANALYSIS, General, Numerical algorithms. {\bf G.1.m}: Mathematics of Computing, NUMERICAL ANALYSIS, Miscellaneous. {\bf G.m}: Mathematics of Computing, MISCELLANEOUS.", } @Article{Andrews:1986:SCA, author = "George E. Andrews and Ian P. Goulden and David M. Jackson", title = "{Shanks}' convergence acceleration transform, {Pad{\'e}} approximants and partitions", journal = j-J-COMB-THEORY-A, volume = "43", number = "1", pages = "70--84", year = "1986", CODEN = "JCBTA7", DOI = "https://doi.org/10.1016/0097-3165(86)90024-5", ISSN = "0097-3165 (print), 1096-0899 (electronic)", ISSN-L = "0097-3165", MRclass = "65B99 (11N99 11Y35)", MRnumber = "859298 (88c:65005)", MRreviewer = "Kenneth A. Jukes", bibdate = "Thu Dec 1 17:36:45 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Journal of Combinatorial Theory (Series A)", journal-URL = "http://www.sciencedirect.com/science/journal/00973165", keywords = "convergence acceleration", } @InProceedings{Baxter:1986:PTF, author = "Michael Baxter and Zwie Amitai", editor = "????", booktitle = "Midcon 86, Dallas, {TX}, September 1986, Pap. 17.2, 5p and Northcon 86, Seattle, {WA}, September 1986", title = "Parallel Transcendental-Function Processor Built from {LSI} Building Blocks", publisher = "????", address = "????", pages = "4.2.1--4.2.4", year = "1986", DOI = "", ISBN = "", ISBN-13 = "", LCCN = "", bibdate = "Wed Nov 12 08:41:47 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "", acknowledgement = ack-nhfb, } @Article{Bustoz:1986:GFI, author = "Joaquin Bustoz and Mourad E. H. Ismail", title = "On Gamma Function Inequalities", journal = j-MATH-COMPUT, volume = "47", number = "176", pages = "659--667", month = oct, year = "1986", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "33A15 (26D20)", MRnumber = "87m:33002", MRreviewer = "G. Gasper", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, classcodes = "C1130 (Integral transforms); C1140Z (Other and miscellaneous)", corpsource = "Dept. of Math., Arizona State Univ., Tempe, AZ, USA", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "gamma function inequalities; infinite divisibility; Laplace; Laplace transforms; monotonic functions; probability; probability distributions; quotients; transforms", treatment = "T Theoretical or Mathematical", } @Article{Campbell:1986:NSR, author = "R. A. Campbell", title = "{NS32000} Square Roots", journal = j-DDJ, volume = "11", number = "3", pages = "122--123, 106", month = mar, year = "1986", CODEN = "DDJOEB", ISSN = "1044-789X", bibdate = "Fri Dec 08 13:05:56 1995", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; garbo.uwasa.fi:/pc/doc-soft/fpbiblio.txt; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nj, fjournal = "Dr. Dobb's Journal of Software Tools", } @Article{Cathey:1986:LEI, author = "J. Cathey", title = "Letter to the editor [Integer Square Root]", journal = j-DDJ, volume = "11", number = "8", pages = "14, 82--85", month = aug, year = "1986", CODEN = "DDJOEB", ISSN = "1044-789X", bibdate = "Thu Sep 08 07:59:25 1994", bibsource = "http://www.ddj.com/index/author/index.htm; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nj, fjournal = "Dr. Dobb's Journal of Software Tools", } @Article{Clenshaw:1986:GEL, author = "C. W. Clenshaw and Daniel W. Lozier and F. W. J. Olver and P. R. Turner", title = "Generalized Exponential and Logarithmic Functions", journal = j-COMPUT-MATH-APPL, volume = "12", number = "5--6", pages = "1091--1101", month = sep # "\slash " # dec, year = "1986", CODEN = "CMAPDK", DOI = "https://doi.org/10.1016/0898-1221(86)90233-6", ISSN = "0898-1221 (print), 1873-7668 (electronic)", ISSN-L = "0898-1221", MRclass = "33A70 (39B10 65G05)", MRnumber = "MR0871348 (88a:33027)", bibdate = "Fri Jul 09 06:27:26 2004", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", abstract = "Generalizations of the exponential and logarithmic functions are defined and an investigation is made of two possible versions of these functions. Some applications are described, including computer arithmetic, properties of very large and very small numbers, and the determination of functional roots.", acknowledgement = ack-nhfb, fjournal = "Computers and Mathematics with Applications", journal-URL = "http://www.sciencedirect.com/science/journal/08981221", } @Article{Clenshaw:1986:UAR, author = "C. W. Clenshaw and F. W. J. Olver", title = "Unrestricted algorithms for reciprocals and square roots", journal = j-BIT, volume = "26", number = "4", pages = "475--492", month = dec, year = "1986", CODEN = "BITTEL, NBITAB", DOI = "https://doi.org/10.1007/BF01935054", ISSN = "0006-3835 (print), 1572-9125 (electronic)", ISSN-L = "0006-3835", MRclass = "65D20", MRnumber = "87k:65019", MRreviewer = "Luciano Biasini", bibdate = "Wed Jan 4 18:52:19 MST 2006", bibsource = "http://springerlink.metapress.com/openurl.asp?genre=issue&issn=0006-3835&volume=26&issue=4; https://www.math.utah.edu/pub/tex/bib/bit.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "http://www.springerlink.com/openurl.asp?genre=article&issn=0006-3835&volume=26&issue=4&spage=475", acknowledgement = ack-nhfb, fjournal = "BIT (Nordisk tidskrift for informationsbehandling)", journal-URL = "http://link.springer.com/journal/10543", xxpages = "476--492??", } @Article{DiDonato:1986:CIG, author = "Armido R. DiDonato and Alfred H. {Morris, Jr.}", title = "Computation of the Incomplete Gamma Function Ratios and Their Inverse", journal = j-TOMS, volume = "12", number = "4", pages = "377--393", month = dec, year = "1986", CODEN = "ACMSCU", DOI = "https://doi.org/10.1145/22721.23109", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Sun Sep 04 21:31:03 1994", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib", URL = "http://www.acm.org/pubs/citations/journals/toms/1986-12-4/p377-didonato/", abstract = "An algorithm is given for computing the incomplete gamma function ratios $ P(a, x) $ and $ Q(a, x) $ for $ a \geq 0 $, $ x \geq 0 $, $ a + x \neq 0 $. Temme's uniform asymptotic expansions are used. The algorithm is robust; results accurate to 14 significant digits can be obtained. An extensive set of coefficients for the Temme expansions is included.\par An algorithm, employing third-order Schr{\"o}der iteration supported by Newton-Raphson iteration, is provided for computing $x$ when $a$, $ P(a, x) $, and $ Q(a, x) $ are given. Three iterations at most are required to obtain 10 significant digit accuracy for $x$.", acknowledgement = ack-nhfb, fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", keywords = "algorithms", review = "ACM CR 8709-0775", subject = "{\bf G.1.2}: Mathematics of Computing, NUMERICAL ANALYSIS, Approximation.", } @Article{DiMarzio:1986:IPA, author = "F. {Di Marzio}", title = "An improved procedure for the accurate evaluation of polygamma functions with integer and half-integer argument", journal = j-COMP-PHYS-COMM, volume = "39", number = "3", pages = "343--345", month = apr, year = "1986", CODEN = "CPHCBZ", DOI = "https://doi.org/10.1016/0010-4655(86)90095-0", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", bibdate = "Mon Feb 13 10:28:13 MST 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/compphyscomm1980.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/0010465586900950", acknowledgement = ack-nhfb, fjournal = "Computer Physics Communications", journal-URL = "http://www.sciencedirect.com/science/journal/00104655", } @Article{Dutka:1986:SRT, author = "Jacques Dutka", title = "On square roots and their representations", journal = j-ARCH-HIST-EXACT-SCI, volume = "36", number = "1", pages = "21--39", month = mar, year = "1986", CODEN = "AHESAN", DOI = "https://doi.org/10.1007/BF00357439", ISSN = "0003-9519 (print), 1432-0657 (electronic)", ISSN-L = "0003-9519", MRclass = "01A05 (11-03 11A63)", MRnumber = "863340 (87m:01003)", MRreviewer = "Donald Cook", bibdate = "Fri Feb 4 21:50:24 MST 2011", bibsource = "http://springerlink.metapress.com/openurl.asp?genre=issue&issn=0003-9519&volume=36&issue=1; https://www.math.utah.edu/pub/tex/bib/archhistexactsci.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "http://www.springerlink.com/openurl.asp?genre=article&issn=0003-9519&volume=36&issue=1&spage=21", acknowledgement = ack-nhfb, fjournal = "Archive for History of Exact Sciences", journal-URL = "http://link.springer.com/journal/407", MRtitle = "On square roots and their representations", } @Article{Evans:1986:RIU, author = "D. J. Evans and G. M. Megson", title = "{Romberg} integration using systolic arrays", journal = j-PARALLEL-COMPUTING, volume = "3", number = "4", pages = "289--304", month = oct, year = "1986", CODEN = "PACOEJ", ISSN = "0167-8191 (print), 1872-7336 (electronic)", ISSN-L = "0167-8191", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, bibno = "11019", catcode = "G.1.1; G.1.2", CRclass = "G.1.1 Interpolation; G.1.1 Extrapolation; G.1.2 Approximation; G.1.2 Elementary function approximation", descriptor = "Mathematics of Computing, NUMERICAL ANALYSIS, Interpolation, Extrapolation; Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation", fjournal = "Parallel Computing", genterm = "theory; design; algorithms", guideno = "1986-10554", journal-URL = "http://www.sciencedirect.com/science/journal/01678191", journalabbrev = "Parallel Comput.", jrldate = "Oct. 1986", subject = "G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS", } @Article{FernandezVelicia:1986:HPA, author = "F. J. {Fern{\'a}ndez Velicia}", title = "High-precision analytic approximations for the {Fermi--Dirac} functions by means of elementary functions", journal = j-PHYS-REV-A-3, volume = "34", number = "5", pages = "4387--4395", year = "1986", CODEN = "PLRAAN", ISSN = "1050-2947 (print), 1094-1622, 1538-4446, 1538-4519", MRclass = "33A70 (82A05)", MRnumber = "MR869021 (88b:33024)", bibdate = "Wed Apr 13 06:46:35 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Physical Review. A. Third Series", journal-URL = "http://pra.aps.org/browse", } @Article{Froman:1986:PIF, author = "Per Olof Fr{\"o}man and Finn Karlsson and Staffan Yngve", title = "Phase-integral formulas for {Bessel} functions and their relation to already existing asymptotic formulas", journal = j-J-MATH-PHYS, volume = "27", number = "11", pages = "2738--2747", month = nov, year = "1986", CODEN = "JMAPAQ", DOI = "https://doi.org/10.1063/1.527296", ISSN = "0022-2488 (print), 1089-7658 (electronic), 1527-2427", ISSN-L = "0022-2488", MRclass = "41A60 (33A40 81C12)", MRnumber = "87j:41073", bibdate = "Mon Oct 31 11:57:50 MDT 2011", bibsource = "http://jmp.aip.org/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jmathphys1985.bib", URL = "http://jmp.aip.org/resource/1/jmapaq/v27/i11/p2738_s1", acknowledgement = ack-nhfb, fjournal = "Journal of Mathematical Physics", journal-URL = "http://jmp.aip.org/", pagecount = "10", } @InProceedings{Gal:1986:CEF, author = "Shmuel Gal", title = "Computing elementary functions: a new approach for achieving high accuracy and good performance", crossref = "Miranker:1986:ASC", pages = "1--16", year = "1986", MRclass = "65D20", MRnumber = "868 283", bibdate = "Mon May 19 13:30:58 1997", bibsource = "ftp://ftp.ira.uka.de/pub/bibliography/Math/elefunt.bib.gz; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, } @InProceedings{Gustavson:1986:FEF, author = "F. G. Gustavson", title = "Fast Elementary Function Algorithms for 370 Machines", crossref = "Miranker:1986:ASC", pages = "17--17", year = "1986", bibdate = "Mon May 19 13:30:58 1997", bibsource = "ftp://ftp.ira.uka.de/pub/bibliography/Misc/MPG/lncs235.bib.gz; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, } @Book{Hochstadt:1986:FMP, author = "Harry Hochstadt", title = "The Functions of Mathematical Physics", publisher = pub-DOVER, address = pub-DOVER:adr, pages = "xi + 322", year = "1986", ISBN = "0-486-65214-9 (paperback), 0-486-16878-6 (e-book)", ISBN-13 = "978-0-486-65214-6 (paperback), 978-0-486-16878-4 (e-book)", LCCN = "QA351 .H68 1986", bibdate = "Tue Dec 5 10:51:16 MST 2023", bibsource = "fsz3950.oclc.org:210/WorldCat; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, tableofcontents = "1: Orthogonal Polynomials \\ 1 Linear Spaces / 1 \\ 2 Orthogonal Polynomials / 6 \\ 3 The Recurrence Formula / 8 \\ 4 The Christoffel--Darboux Formula / 9 \\ 5 The Weierstrass Approximation Theorem / 11 \\ 6 The Zeros of the Orthogonal Polynomials / 14 \\ 7 Approximation Theory / 16 \\ 8 More about the Zeros of the Orthonormal Polynomials / 23 \\ 9 The completeness of the Orthonormal Polynomials in the Space of Square-Integrable Functions / 27 \\ 10 Generalizations and an Application to Conformal Mappings / 32 \\ \\ 2: The Classical Orthogonal Polynomials 1 Rodrigues' Formula and the Classical Orthogonal Polynomials / 39 \\ 2 The Differential Equations Satisfied by the Classical Orthogonal Polynomials / 43 \\ 3 On the Zeros of the Jacobi Polynomials / 45 \\ 4 An Alternative Approach to the Tchebicheff Polynomials / 46 \\ 5 An Application of the Hermite Polynomials to Quantum Mechanics / 49 \\ 6 The Completeness of the Hermite and Laguerre Polynomials / 53 \\ 7 Generating Functions / 57 \\ \\ 3: The Gamma Function 1 Definitions and Basic Properties / 61 \\ 2 Analytic Continuation and Integral Representations / 65 \\ 3 Asymptotic Expansions / 69 \\ 4 Beta Functions / 75 \\ 5 The Logarithmic Derivative of the Gamma Function / 77 \\ 6 Mellin--Barnes Integrals / 78 \\ 7 Mellin Transforms / 80 \\ 8 Applications to Algebraic Equations / 81 \\ \\ 4: Hypergeometric Functions 1 Review of Linear Differential Equations with Regular Singular Points / 88 \\ 2 The Hypergeometric Differential Equation / 90 \\ 3 The Hypergeometric Function / 93 \\ 4 A General Method for Finding Integral Representations / 100 \\ 5 Integral Representations for the Hypergeometric Function / 105 \\ 6 The Twenty-four Solutions of the Hypergeometric / Equation / 106 \\ 7 The Schwarz--Christoffel Transformation / 112 \\ 8 Mappings of Curvilinear Triangles / 119 \\ 9 Group Theoretic Discussion of the Case $ \pi(\alpha_1 + \alpha_2 + \alpha_3) > \pi$ / 130 \\ 10 Nonlinear Transformations of Hypergeometric Functions / 132 \\ \\ 5: The Legendre Functions 1 Laplace's Differential Equation / 138 \\ 2 Maxwell's Theory of Poles / 140 \\ 3 Relationship to the Hypergeometric Functions / 141 \\ 4 Expansion Formulas / 147 \\ 5 The Addition Theorem / 149 \\ 6 Green's Functions / 153 \\ 7 The Complete Solution of Legendre's Differential Equation / 156 \\ 8 Asymptotic Formulas / 161 \\ \\ 6: Spherical Harmonics in $p$ Dimensions 1 Homogeneous Polynomials / 168 \\ 2 Orthogonality of Spherical Harmonics / 171 \\ 3 Legendre Polynomials / 175 \\ 4 Applications to Boundary Value Problems / 183 \\ \\ 7: Confluent Hypergeometric Functions 1 Relationship to the Hypergeometric Functions / 189 \\ 2 Applications of These Functions in Mathematical Physics / 191 \\ 3 Integral Representations / 195 \\ 4 Asymptotic Representations / 198 \\ \\ 8: Bessel Functions 1 Basic Definitions / 200 \\ 2 Integral Representations / 203 \\ 3 Relationship to the Legendre Functions / 205 \\ 4 The Generating Function of the Bessel Function / 207 \\ 5 More Integral Representations / 210 \\ 6 Addition Theorems / 216 \\ 7 The Complete Solution of Bessel's Equation / 223 \\ 8 Asymptotic Expansions for Large Argument / 225 \\ 9 Airy Functions / 230 \\ 10 Asymptotic Expansions for Large Indices and Large Arguments / 235 \\ 11 Some Applications of Bessel Functions in Physical Optics / 241 \\ 12 The Zeros of Bessel Functions / 249 \\ 13 Fourier--Bessel Expansions / 257 \\ 14 Applications in Mathematical Physics / 266 \\ 15 Discontinuous Integrals / 269 \\ \\ 9: Hill's Equation 1 Mathieu's Equation / 281 \\ 2 Hill's Equation / 282 \\ 3 The Discriminant / 287 \\ 4 Expansion Theorems / 299 \\ 5 Inverse Problems / 305 \\ 6 Hill's Equations with Even Coefficients / 309 \\ 7 Mathieu's Equation Revisited / 310 \\ 8 Energy Bands in Crystals / 313 \\ Appendix / 314 \\ \\ Bibliography / 318 \\ \\ Index / 321", } @Article{Hull:1986:VPE, author = "T. E. Hull and A. Abrham", title = "Variable Precision Exponential Function", journal = j-TOMS, volume = "12", number = "2", pages = "79--91", month = jun, year = "1986", CODEN = "ACMSCU", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", MRclass = "65D15 (65D20)", MRnumber = "863 786", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.acm.org/pubs/citations/journals/toms/1986-12-2/p79-hull/; http://www.acm.org/pubs/toc/Abstracts/toms/6498.html", acknowledgement = ack-nhfb, bibno = "91", content = "algorithms; verification; THEORY", CRclass = "G.1.2 Approximation; G.1.2 Elementary function approximation; G.4 Algorithm analysis; G.4 Certification and testing; G.4 Verification", CRnumber = "1986-02428", descriptor = "mathematics of computing, numerical analysis, approximation, elementary function approximation; mathematics of computing, mathematical software, algorithm analysis; mathematics of computing, mathematical software, certification and testing; mathematics of computing, mathematical software, verification", fjournal = "ACM Transactions on Mathematical Software (TOMS)", fortitle = "ACM Trans. Math. Softw.", genterm = "June 1986", guideno = "2", journal-URL = "https://dl.acm.org/loi/toms", keywords = "algorithms; theory; verification", review = "ACM CR 8702-0091", subject = "{\bf G.1.2}: Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation. {\bf G.4}: Mathematics of Computing, MATHEMATICAL SOFTWARE, Algorithm analysis. {\bf G.4}: Mathematics of Computing, MATHEMATICAL SOFTWARE, Certification and testing. {\bf G.4}: Mathematics of Computing, MATHEMATICAL SOFTWARE, Verification.", } @Article{Jacobsen:1986:FRC, author = "Lisa Jacobsen and William B. Jones and Haakon Waadeland", title = "Further results on the computation of incomplete gamma functions", journal = j-LECT-NOTES-MATH, volume = "1199", pages = "67--89", year = "1986", CODEN = "LNMAA2", DOI = "https://doi.org/10.1007/BFb0075936", ISBN = "3-540-16768-4 (print), 3-540-38817-6 (e-book)", ISBN-13 = "978-3-540-16768-6 (print), 978-3-540-38817-3 (e-book)", ISSN = "0075-8434 (print), 1617-9692 (electronic)", ISSN-L = "0075-8434", MRclass = "40A15 (33A10 33A15)", MRnumber = "870245 (88f:40004)", MRreviewer = "Marietta J. Tretter", bibdate = "Thu May 15 18:46:23 MDT 2014", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/lnm1985.bib", URL = "http://link.springer.com/chapter/10.1007/BFb0075936/", acknowledgement = ack-nhfb, book-DOI = "https://doi.org/10.1007/BFb0075930", book-URL = "http://www.springerlink.com/content/978-3-540-38817-3", fjournal = "Lecture Notes in Mathematics", journal-URL = "http://link.springer.com/bookseries/304", } @Misc{Kahan:1986:S, author = "W. Kahan and K. C. Ng", title = "{SQRT}", howpublished = "Web document.", pages = "11", day = "6", month = may, year = "1986", bibdate = "Sat Dec 13 10:38:09 2025", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/k/kahan-william-m.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "https://adampunk.com/documents/softsqrt.pdf", abstract = "Two algorithms are given in this document to implement $ \sqrt {x} $ in software. Both supply $ \sqrt {x} $ correctly rounded. The first algorithm (in Section A) uses Newton iterations and involves four divisions. The second one uses reciproot iterations to avoid division, but requires more multiplications. Both algorithms need the ability to chop results of arithmetic operations instead of round them, and the INEXACT flag to indicate when an arithmetic operation is executed exactly with no roundoff error, all part of the standard. The ability to perform shift, add, subtract and logical AND operations upon 32-bit words is needed too, though not part of the standard.", acknowledgement = ack-nhfb, remark = "See \cite{Hyland:20xx:FIS} for later developments that were influenced by this report.", } @Article{Kushner:1986:ECC, author = "Ed Kushner and Rick Broussard", title = "Efficient computation of the cylindrical {Bessel} functions of complex argument", journal = j-COMP-PHYS-COMM, volume = "42", number = "3", pages = "345--349", month = nov, year = "1986", CODEN = "CPHCBZ", DOI = "https://doi.org/10.1016/0010-4655(86)90004-4", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", bibdate = "Mon Feb 13 10:28:16 MST 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/compphyscomm1980.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/0010465586900044", abstract = "An algorithm that generates the cylindrical Bessel function very accurately for a wide range of complex arguments has been developed by Mason. The Mason algorithm consists of four different methods that apply to different portions of the complex plane. Experience with the Floating Point Systems FPS-364 minisupercomputer indicates several ways by which these methods can be made more efficient. Specific improvements relate to: (1) the method for determination of the point where backward recursion is initiated for the Bessel functions of the first kind; (2) the way that the Bessel functions of the first and second kind are normalized when $ |y| < 5 $ and $ |x| \leq 20 $; and (3) the extent that asymptotic expansions are used when $ |x| > 20 $ and $ |y| < 5 $. The first and third modifications will result in increased efficiency for all architectures. The second modification will be of value for many, but probably not all, architectures.", acknowledgement = ack-nhfb, fjournal = "Computer Physics Communications", journal-URL = "http://www.sciencedirect.com/science/journal/00104655", } @Article{Laforgia:1986:IBF, author = "Andrea Laforgia", title = "Inequalities for {Bessel} functions", journal = j-J-COMPUT-APPL-MATH, volume = "15", number = "1", pages = "75--81", month = may, year = "1986", CODEN = "JCAMDI", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Sat Feb 25 11:59:55 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath1980.bib", URL = "http://www.sciencedirect.com/science/article/pii/0377042786902396", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @Article{Lavoie:1986:SEG, author = "J. L. Lavoie", title = "Some evaluations for the generalized hypergeometric series", journal = j-MATH-COMPUT, volume = "46", number = "173", pages = "215--218", month = jan, year = "1986", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "33A35 (65D20)", MRnumber = "87c:33007", MRreviewer = "S. D. Bajpai", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp1980.bib; JSTOR database", acknowledgement = ack-nhfb, classcodes = "B0200 (Engineering mathematics and mathematical techniques); B0290Z (Other numerical methods); C1100 (Mathematical techniques); C4190 (Other numerical methods)", corpsource = "Laval Univ., Que., Canada", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "evaluation formulae; generalized hypergeometric series; series (mathematics); summation formulae; unit argument; Whipple's theorem", treatment = "T Theoretical or Mathematical", } @Article{Marsaglia:1986:CIG, author = "John C. W. Marsaglia", title = "{C249}. {The} incomplete gamma function and {Ramanujan}'s rational approximation to $ e^x $", journal = j-J-STAT-COMPUT-SIMUL, volume = "24", number = "2", pages = "163--168", year = "1986", CODEN = "JSCSAJ", DOI = "https://doi.org/10.1080/00949658608810899", ISSN = "0094-9655 (print), 1026-7778 (electronic), 1563-5163", ISSN-L = "0094-9655", bibdate = "Tue Apr 22 09:11:07 MDT 2014", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jstatcomputsimul.bib", acknowledgement = ack-nhfb, fjournal = "Journal of Statistical Computation and Simulation", journal-URL = "http://www.tandfonline.com/loi/gscs20", } @Article{Muller:1986:MDC, author = "Jean-Michel Muller", title = "Une m{\'e}thodologie du calcul hardware des fonctions {\'e}l{\'e}mentaires. ({French}) [{A} methodology for the hardware computation of elementary functions]", journal = j-MATH-MODEL-NUM-ANA, volume = "20", number = "4", pages = "667--695", year = "1986", CODEN = "RMMAEV", ISSN = "0764-583X (print), 1290-3841 (electronic)", ISSN-L = "0764-583X", MRclass = "65D20 (41-04)", MRnumber = "88h:65043", MRreviewer = "E. W. Cheney", bibdate = "Sat Jan 11 17:44:01 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Mathematical modelling and numerical analysis = Modelisation math{\'e}matique et analyse num{\'e}rique: $M^2AN$", journal-URL = "http://journals.cambridge.org/action/displayJournal?jid=MZA", language = "Russian", } @Article{Petkovic:1986:SIS, author = "M. S. Petkovi{\'c} and L. V. Stefanovi{\'c}", title = "On some improvements of square root iteration for polynomial complex zeros", journal = j-J-COMPUT-APPL-MATH, volume = "15", number = "1", pages = "13--25", month = may, year = "1986", CODEN = "JCAMDI", DOI = "https://doi.org/10.1016/0377-0427(86)90235-9", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Sat Feb 25 11:59:55 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath1980.bib", URL = "http://www.sciencedirect.com/science/article/pii/0377042786902359", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", keywords = "polynomial root finding", } @Article{Piessens:1986:ATP, author = "Robert Piessens and Shafique Ahmed", title = "Approximation for the turning points of {Bessel} functions", journal = j-J-COMPUT-PHYS, volume = "64", number = "1", pages = "253--257", month = may, year = "1986", CODEN = "JCTPAH", DOI = "https://doi.org/10.1016/0021-9991(86)90029-X", ISSN = "0021-9991 (print), 1090-2716 (electronic)", ISSN-L = "0021-9991", bibdate = "Sun Jan 1 15:59:29 MST 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputphys1980.bib", URL = "http://www.sciencedirect.com/science/article/pii/002199918690029X", acknowledgement = ack-nhfb, fjournal = "Journal of Computational Physics", journal-URL = "http://www.sciencedirect.com/science/journal/00219991/", } @Book{Prudnikov:1986:ISE, author = "Anatolij P. Prudnikov and Jurij A. Bry{\v{c}}kov and Oleg I. Mari{\v{c}}ev", title = "Integrals and series. {Elementary} functions", volume = "1", publisher = "Gordon and Breach Science Publishers", address = "New York, NY, USA", pages = "798", year = "1986", ISBN = "2-88124-089-5", ISBN-13 = "978-2-88124-089-8", LCCN = "QA308.P7813 1986", bibdate = "Thu Nov 2 15:40:35 MDT 2017", bibsource = "fsz3950.oclc.org:210/WorldCat; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, author-dates = "1927--", remark = "Translated from the Russian by N. M. Queen.", seriestableofcontents = "v. 1. Elementary functions \\ v. 2. Special functions \\ v. 3. More special functions \\ v. 4. Direct Laplace transforms \\ v. 5. Inverse Laplace transforms", subject = "Integrals; Series", } @Book{Prudnikov:1986:ISS, author = "Anatolij P. Prudnikov and Jurij A. Bry{\v{c}}kov and Oleg I. Mari{\v{c}}ev", title = "Integrals and series. {Special} functions", volume = "2", publisher = "Gordon and Breach Science Publishers", address = "New York, NY, USA", pages = "750", year = "1986", ISBN = "2-88124-090-9", ISBN-13 = "978-2-88124-090-4", LCCN = "QA308.P7813 1986", MRclass = "26A33, 26A36, 26A39, 26A42, 26B15, 26B20, 26B25", bibdate = "Thu Nov 2 15:43:13 MDT 2017", bibsource = "fsz3950.oclc.org:210/WorldCat; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, remark = "Translated from the Russian by N. M. Queen.", seriestableofcontents = "v. 1. Elementary functions \\ v. 2. Special functions \\ v. 3. More special functions \\ v. 4. Direct Laplace transforms \\ v. 5. Inverse Laplace transforms", subject = "Integrals; Series", } @Article{Reichel:1986:PAU, author = "L. Reichel", title = "On polynomial approximation in the uniform norm by the discrete least squares method", journal = j-BIT, volume = "26", number = "3", pages = "350--368", month = jan, year = "1986", CODEN = "BITTEL, NBITAB", ISSN = "0006-3835 (print), 1572-9125 (electronic)", ISSN-L = "0006-3835", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, bibno = "12404", catcode = "G.1.2; G.1.2; G.1.2", CRclass = "G.1.2 Approximation; G.1.2 Elementary function approximation; G.1.2 Approximation; G.1.2 Least squares approximation; G.1.2 Approximation; G.1.2 Spline and piecewise polynomial approximation", descriptor = "Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation; Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Least squares approximation; Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Spline and piecewise polynomial approximation", fjournal = "BIT (Nordisk tidskrift for informationsbehandling)", genterm = "algorithms", guideno = "1986-03278", journal-URL = "http://link.springer.com/journal/10543", jrldate = "Jan. 1986", subject = "G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS", } @Article{Ronning:1986:CTF, author = "Gerd Ronning", title = "On the curvature of the trigamma function", journal = j-J-COMPUT-APPL-MATH, volume = "15", number = "3", pages = "397--399", month = jul, year = "1986", CODEN = "JCAMDI", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Sat Feb 25 11:59:56 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath1980.bib", URL = "http://www.sciencedirect.com/science/article/pii/0377042786902311", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @Article{S:1986:CEF, author = "A. S. Kuz'menko and K. I. Rogozin", title = "Calculation of elementary functions in a number system with arbitrary basis on the basis of order-differential transformations. ({Russian})", journal = "Prace Nauk. Inst. Cybernet. Tech. Politech. Wroc{\l}aw. Ser. Konfer.", volume = "74", number = "31", pages = "259--262", year = "1986", MRclass = "65G99 (65D20)", MRnumber = "894 691", bibdate = "Sat Jan 11 17:44:01 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, language = "Russian", } @Article{Shore:1986:AID, author = "Haim Shore", title = "An approximation for the inverse distribution function of a combination of random variables, with an application to operating theatres", journal = j-J-STAT-COMPUT-SIMUL, volume = "23", number = "3", pages = "157--181", year = "1986", CODEN = "JSCSAJ", ISSN = "0094-9655 (print), 1563-5163 (electronic)", ISSN-L = "0094-9655", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, affiliation = "Ban-llan University, Israel", bibno = "5358", catcode = "G.3; G.m; G.1.2; G.2.1; J.3; G.3", content = "The author worked on a project to predict the percentage of time that operations are carried out relative to the time that the operating theater is available. The numerator of the percentage is a weighted sum of the times required to carry out different kinds of operations, where the weights are the numbers of operations of each kind to be performed. Since different kinds of operations have different mean times, this sum has a skewed distribution.\par Based on the Central Limit Theorem, the normal distribution is the most widely used approximation to the distribution of a weighted sum of random variables. However, this approximation is not very good if the sum has a skewed distribution.\par In a separate paper [1], the author derived an alternative approximation based on the first four moments of the sum. In the present paper, he applies this approximation to the operating theater problem by estimating the moments of the times of the different kinds of operations. The paper also contains a Monte Carlo comparison of the normal approximation with the proposed approximation for four underlying distributions of the sum.", CRclass = "G.3 Statistical computing; G.1.2 Approximation; G.1.2 Elementary function approximation; G.2.1 Combinatorics; G.2.1 Generating functions; J.3 Health; G.3 Probabilistic algorithms (including Monte Carlo)", CRnumber = "8612-1109", descriptor = "Mathematics of Computing, PROBABILITY AND STATISTICS, Statistical computing; Mathematics of Computing, MISCELLANEOUS; Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation; Mathematics of Computing, DISCRETE MATHEMATICS, Combinatorics, Generating functions; Computer Applications, LIFE AND MEDICAL SCIENCES, Health; Mathematics of Computing, PROBABILITY AND STATISTICS, Probabilistic algorithms (including Monte Carlo)", fjournal = "Journal of Statistical Computation and Simulation", genterm = "algorithms; measurement", journal-URL = "http://www.tandfonline.com/loi/gscs20", journalabbrev = "J. Stat. Comput. Simul.", jrldate = "1986", reviewer = "M. Snyder", subject = "G. Mathematics of Computing; G.3 PROBABILITY AND STATISTICS; G. Mathematics of Computing; G.m MISCELLANEOUS; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; G. Mathematics of Computing; G.2 DISCRETE MATHEMATICS; J. Computer Applications; J.3 LIFE AND MEDICAL SCIENCES; G. Mathematics of Computing; G.3 PROBABILITY AND STATISTICS", } @Article{Skeel:1986:CVS, author = "Robert D. Skeel", title = "Construction of Variable-Stepsize Multistep Formulas", journal = j-MATH-COMPUT, volume = "47", number = "176", pages = "503--510, S45--S52", month = oct, year = "1986", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "65L05", MRnumber = "87j:65080", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, classcodes = "C4110 (Error analysis in numerical methods); C4170 (Differential equations)", corpsource = "Dept. of Comput. Sci., Illnois Univ., Urbana, IL, USA", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "Adams formula; adaptable multistep methods; backward-differentiation; differential equations; error analysis; estimation; first Dahlquist barrier; fixed leading coefficient method; fixed-coefficient methods; fixed-stepsize; formula; formula changing; initial value; interpolatory methods; local error; minimum storage variable-stepsize; multistep formula; Nordsieck stepsize changing technique; problems; step methods; variable; variable coefficient methods; variable-order family of variable-coefficient formulas", treatment = "T Theoretical or Mathematical", } @Article{Skeel:1986:SCV, author = "Robert D. Skeel", title = "Supplement to Construction of Variable-Stepsize Multistep Formulas", journal = j-MATH-COMPUT, volume = "47", number = "176", pages = "S45--S52", month = oct, year = "1986", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Takagi:1986:HAC, author = "Naofumi Takagi and T. Asada and S. Yajima", title = "A Hardware Algorithm for Computing Sine and Cosine Using Redundant Binary Representation", journal = "Transactions IEEE Japan", volume = "J69-D", number = "??", pages = "841--847", month = "????", year = "1986", DOI = "", bibdate = "Wed Nov 12 08:59:45 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "", acknowledgement = ack-nhfb, } @Article{Temme:1986:DIC, author = "N. M. Temme", title = "A double integral containing the modified {Bessel} function: asymptotics and computation", journal = j-MATH-COMPUT, volume = "47", number = "176", pages = "683--691", month = oct, year = "1986", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "33A40 (41A60 65D30)", MRnumber = "87m:33006", MRreviewer = "S. D. Bajpai", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, classcodes = "C4110 (Error analysis in numerical methods); C4130 (Interpolation and function approximation); C4160 (Numerical integration and differentiation)", corpsource = "Centre for Math. and Comput. Sci., Amsterdam, Netherlands", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "distribution function; double integral; error analysis; error function; integral; integration; modified Bessel function; normal; polynomials; probability; series (mathematics); series expansions; two-dimensional", treatment = "T Theoretical or Mathematical", } @Article{Thompson:1986:CBF, author = "I. J. Thompson and A. R. Barnett", title = "{Coulomb} and {Bessel} functions of complex arguments and order", journal = j-J-COMPUT-PHYS, volume = "64", number = "2", pages = "490--509", month = jun, year = "1986", CODEN = "JCTPAH", DOI = "https://doi.org/10.1016/0021-9991(86)90046-X", ISSN = "0021-9991 (print), 1090-2716 (electronic)", ISSN-L = "0021-9991", bibdate = "Sun Jan 1 15:59:30 MST 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputphys1980.bib", URL = "http://www.sciencedirect.com/science/article/pii/002199918690046X", abstract = "The Coulomb wavefunctions, originally constructed for real $ \varrho > 0 $, real $ \eta $ and integer $ \lambda \geq 0 $ are defined for $ \varrho $, $ \eta $, and $ \lambda $ all complex. We examine the complex continuation of a variety of series and continued-fraction expansions for the Coulomb functions and their logarithmic derivatives, and then see how these expansions may be selectively combined to calculate both the regular and irregular functions and their derivatives. The resulting algorithm [46] is a complex generalisation of Steed's method [6, 7] as it appears in the real procedure COULFG [10]. Complex Whittaker, confluent hypergeometric and Bessel functions can also be calculated.", acknowledgement = ack-nhfb, fjournal = "Journal of Computational Physics", journal-URL = "http://www.sciencedirect.com/science/journal/00219991/", } @Book{Varshalovich:1986:QTA, author = "D. A. Varshalovich and A. N. Moskalev and V. K. Khersonskii", title = "Quantum Theory of Angular Momentum: Irreducible Tensors, Spherical Harmonics, Vector Coupling Coefficients, $ 3 n j $ Symbols", publisher = pub-WORLD-SCI, address = pub-WORLD-SCI:adr, pages = "x + 514", year = "1986", ISBN = "9971-5-0107-4", ISBN-13 = "978-9971-5-0107-5", LCCN = "QC793.3.A5 V3713 1986", bibdate = "Tue Aug 5 06:20:04 MDT 2025", bibsource = "fsz3950.oclc.org:210/WorldCat; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, shorttableofcontents = "1: Elements of vector and tensor theory \\ 2: Angular momentum operators \\ 3: Irreducible tensors \\ 4: Wigner D-functions \\ 5: Spherical harmonics \\ 6: Spin functions \\ 7: Tensor spherical harmonics \\ 8: Clebsch--Gordan coefficients and $3jm$ symbols \\ 9: $6j$ symbols and the Racah coefficients \\ 10: $9j$ and $12j$ symbols \\ 11: The graphical method in angular momentum theory \\ 12: Sums involving vector addition and recoupling coefficients \\ 13: Matrix elements of irreducible tensor operators", subject = "Angular momentum (Nuclear physics); Quantum theory; Moment angulaire (Physique nucl{\'e}aire); Th{\'e}orie quantique; Angular momentum (Nuclear physics); Quantum theory.", tableofcontents = "Preface / v \\ Introduction: Basic Concepts / 1 \\ 1: Elements of Vector and Tensor Theory / 3 \\ 1.1. Coordinate Systems. Basis Vectors / 3 \\ 1.2. Vectors. Tensors / 11 \\ 1.3. Differential Operations / 17 \\ 1.4. Rotations of Coordinate System / 21 \\ 2: Angular Momentum Operators / 36 \\ 2.1. Total Angular Momentum Operator / 36 \\ 2.2. Orbital Angular Momentum Operator / 39 \\ 2.3. Spin Angular Momentum Operator / 42 \\ 2.4. Polarization Operators / 44 \\ 2.5. Spin Matrices for $S = 1/2$ / 47 \\ 2.6. Spin Matrices and Polarization Operators for $S = 1$ / 51 \\ 3: Irreducible Tensors / 61 \\ 3.1. Definition and Properties of Irreducible Tensors / 61 \\ 3.2. Relation Between the Irreducible Tensor Algebra and Vector and Tensor Theory / 65 \\ 3.3. Recoupling in Irreducible Tensor Products / 69 \\ 4: Wigner $D$-Functions / 72 \\ 4.1. Definition of $D^J_{M M'}(\alpha, \beta, \gamma)$ / 72 \\ 4.2. Differential Equations for $D^J_{M M'}(\alpha, \beta, \gamma)$ / 74 \\ 4.3. Explicit Forms of the Wigner $D$-Functions / 76 \\ 4.4. Symmetries of $d^J_{M M'}(\beta)$ and $D^J_{M M'}(\alpha, \beta, \gamma)$ / 79 \\ 4.5. Rotation Matrix $U^J_{M M'}$, in Terms of Angles $\omega, \Theta, \Phi$ / 80 \\ 4.6. Sums Involving $D$-Functions / 84 \\ 4.7. Addition of Rotations / 87 \\ 4.8. Recursion Relations for $D^J_{M M'}(\alpha, \beta, \gamma)$ / 90 \\ 4.9. Differential Relations for $D^J_{M M'}(\alpha, \beta, \gamma)$ / 94 \\ 4.10. Orthogonality and Completeness of the $D$-Functions / 94 \\ 4.11. Integrals Involving the $D$-Functions / 96 \\ 4.12. Invariant Summation of Integrals Involving $D^J_{M M'}(\alpha, \beta, \gamma)$ / 97 \\ 4.13. Generating Functions for $d^J_{M M'}(\beta)$ / 98 \\ 4.14. Characters $\chi^J(R)$ of Irreducible Representations of Rotation Group / 99 \\ 4.15. Generalized Characters, $\chi^J_\lambda(R)$ of Irreducible Representations of the Rotation Group / 106 \\ 4.16. $D^J_{M M'}(\alpha, \beta, \gamma)$ for Particular Values of the Arguments / 112 \\ 4.17. Special Cases of $D^J_{M M'}(\alpha, \beta, \gamma)$ for Particular $M$ or $M'$ / 113 \\ 4.18. Asymptotics of $D^J_{M M'}(\alpha, \beta, \gamma)$ / 115 \\ 4.19. Definitions of $D^J_{M M'}(\alpha, \beta, \gamma)$ by Other Authors / 117 \\ 4.20. Special Cases of $d^J_{M M'}(\beta)$ for Particular $J$, $M$ and $M'$ / 117 \\ 4.21. Tables of $d^J_{M M'}(\beta)$ for $\beta = \pi/2$ / 117 \\ 4.22. Special Cases of $U^J_{M M'}(\omega; \Theta, \Phi)$ / 117 \\ 5: Spherical Harmonics / 130 \\ 5.1. Definition / 130 \\ 5.2. Explicit Forms of the Spherical Harmonics and Their Relations to Other Functions / 133 \\ 5.3. Integral Representations of the Spherical Harmonics [4, 22, 27] / 139 \\ 5.4. Symmetry Properties / 140 \\ 5.5. Behaviour of $Y_{l m}(\theta, \phi)$ under Transformations of Coordinate Systems / 141 \\ 5.6. Expansions in Series of the Spherical Harmonics / 143 \\ 5.7. Recursion Relations / 145 \\ 5.8. Differential Relations / 146 \\ 5.9. Some Integrals Involving Spherical Harmonics / 148 \\ 5.10. Sums Involving Spherical Harmonics / 150 \\ 5.11. Generating Functions for $Y_{l m}(\theta, \phi)$ / 151 \\ 5.12. Asymptotic Expressions for $Y_{l m}(\theta, \phi)$ / 152 \\ 5.13. $Y_{l m}(\theta, \phi)$ for Special Values of $l$ and $m$ / 155 \\ 5.14. $Y_{l m}(\theta, \phi)$ and $(\partial / \partial \theta) Y_{l m}(\theta, \phi)$ for Special $\theta$ / 158 \\ 5.15. Zeros of $Y_{l m}(\theta, \phi)$ and $(\partial / \partial \theta) Y_{l m}(\theta, \phi)$ / 158 \\ 5.16. Bipolar and Tripolar Spherical Harmonics / 160 \\ 5.17. Expansions of Functions Which Depend on Two Vectors / 163 \\ 6: Spin Functions / 170 \\ 6.1. Spin Functions of Particles with Arbitrary Spin / 170 \\ 6.2. Spin Functions for $S = 1/2$ / 178 \\ 6.3. Spin Functions for $S = 1$ / 185 \\ 7: Tensor Spherical Harmonics / 196 \\ 7.1. General Properties of Tensor Spherical Harmonics / 196 \\ 7.2. Spinor Spherical Harmonics / 202 \\ 7.3. Vector Spherical Harmonics / 208 \\ 7.4. Other Notations for Tensor Spherical Harmonics / 234 \\ 8: Clebsch--Gordan Coefficients and $3jm$ Symbols / 235 \\ 8.1. Definition / 235 \\ 8.2. Explicit Forms of the Clebsch--Gordan Coefficients and Their Relations to Other Functions / 237 \\ 8.3. Integral Representations / 243 \\ 8.4. Symmetry Properties / 244 \\ 8.5. Explicit Forms of the Clebsch--Gordan Coefficients for Special Values of the Arguments / 248 \\ 8.6. Recursion Relations for the Clebsch--Gordan Coefficients / 252 \\ 8.7. Sums of Products of the Clebsch--Gordan Coefficients / 259 \\ 8.8. Generating Functions / 263 \\ 8.9. Classical Limit and Asymptotic Expressions for the Clebsch--Gordan Coefficients / 264 \\ 8.10. Zeros of the Vector-Addition Coefficients / 268 \\ 8.11. Connection of the Clebsch--Gordan Coefficients and the $3jm$ Symbols with Analogous Functions of Other Authors / 268 \\ 8.12. Algebraic Tables of the Clebsch--Gordan Coefficients / 270 \\ 8.13. Numerical Tables of the Clebsch--Gordan Coefficients / 270 \\ 9: $6j$ Symbols and the Racah Coefficients / 290 \\ 9.1. Definition / 290 \\ 9.2. General Expressions for the $6j$ Symbols. Relations Between the $6j$ Symbols and Other Functions / 293 \\ 9.3. Integral Representations of the $6j$ Symbols / 297 \\ 9.4. Symmetries of the $6j$ Symbols and the Racah Coefficients / 298 \\ 9.5. Explicit Forms of the &j Symbols for Certain Arguments / 299 \\ 9.6. Recursion Relations / 303 \\ 9.7. Generating Function / 305 \\ 9.8. Sums Involving the $6j$ Symbols / 305 \\ 9.9. Asymptotics of the $6j$ Symbols for Large Angular Momenta / 306 \\ 9.10. Relations Between the Wigner $6j$ Symbols and Analogous Functions of Other Authors / 310 \\ 9.11. Tables of Algebraic Expressions for the $6j$ Symbols / 310 \\ 9.12. Numerical Values of the $6j$ Symbols / 310 \\ 10: $9j$ and $12j$ Symbols / 333 \\ 10.1. Definition of the $9j$ Symbols / 333 \\ 10.2. Explicit Forms of the $9j$ Symbols and Their Relations to Other Functions / 336 \\ 10.3. Integral Representations of the $9j$ Symbols / 340 \\ 10.4. Symmetry Properties of the $9j$ Symbols / 342 \\ 10.5. Recursion Relations for the $9j$ Symbols / 345 \\ 10.6. Generating Function of the $9j$ Symbols / 351 \\ 10.7. Asymptotic Expression for a $9j$ Symbol / 351 \\ 10.8. Explicit Forms of the $9j$ Symbols at Some Relations Between Arguments / 352 \\ 10.9. Explicit Forms of the $9j$ Symbols for Special Values of the Arguments / 357 \\ 10.10. Relations Between the Wigner $9j$ Symbols and Analogous Functions of Other Authors / 359 \\ 10.11. Tables of Algebraic Formulas of the $9j$ Symbols / 359 \\ 10.12. Tables of Numerical Values of the $9j$ Symbols / 360 \\ 10.13. $12j$ Symbols / 361 \\ 11: The Graphical Method in Angular Momentum Theory / 412 \\ 11.1. Graphical Representation of Functions / 412 \\ 11.2. Graphical Representation of the Main Operations of the Theory / 419 \\ 11.3. Rules of the Graphical Technique / 424 \\ 11.4. Summary of the Graphical Technique / 446 \\ 12: Sums Involving Vector Addition and Recoupling Coefficients / 452 \\ 12.1. Summation of Products of $3jm$ Symbols / 452 \\ 12.2. Summation of Products of $6j$ and $9j$ Symbols / 462 \\ 13: Matrix Elements of Irreducible Tensor Operators / 475 \\ 13.1. The Wigner--Eckart Theorem and the Evaluation of Matrix Elements / 475 \\ 13.2. Matrix Elements of Basic Tensor Operators / 485 \\ Glossary of Symbols and Notation / 505 \\ References / 509", } @Article{Zaritskaya:1986:ACE, author = "Z. V. Zaritskaya and A. I. Shva{\u\i} and P. {\=E}. Antonyuk", title = "Approximation of certain elementary functions in the metric $ {L} $. ({Russian})", journal = "Vestnik L'vov. Politekhn. Inst.", volume = "202", pages = "38--40", year = "1986", MRclass = "149.41A10 (33A10)", MRnumber = "87j:41030", bibdate = "Sat Jan 11 17:44:01 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, language = "Russian", } @Article{Agarwal:1987:CNS, author = "Ramesh C. Agarwal and James W. Cooley and Fred G. Gustavson and James B. Shearer and Gordon Slishman and Bryant Tuckerman", title = "Clarification: {``New scalar and vector elementary functions for the IBM System/370''} [{IBM J. Res. Develop. {\bf 30} (1986), no. 2, 126--144}]", journal = j-IBM-JRD, volume = "31", number = "2", pages = "274--274", month = mar, year = "1987", CODEN = "IBMJAE", DOI = "https://doi.org/10.1147/rd.312.0274", ISSN = "0018-8646 (print), 2151-8556 (electronic)", ISSN-L = "0018-8646", MRclass = "76W05", MRnumber = "MR894626", bibdate = "Mon Feb 12 08:07:08 2001", bibsource = "http://www.research.ibm.com/journal/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ibmjrd.bib", note = "See \cite{Agarwal:1986:NSV}.", acknowledgement = ack-nhfb, ajournal = "IBM J. Res. Develop.", fjournal = "IBM Journal of Research and Development", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=5288520", } @PhdThesis{Braune:1987:HSF, author = "K. Braune", title = "{Hochgenaue Standardfunktionen f{\"u}r reelle und komplexe Punkte und Intervalle in beliebigen Gleitpunktrastern} \toenglish {High-Accuracy Elementary Functions for Real and Complex Points and Intervals in Arbitrary Floating-Point Systems} \endtoenglish", type = "Dissertation", school = "Universit{\"a}t Karlsruhe", address = "Karlsruhe, Germany", pages = "????", year = "1987", bibdate = "Fri Sep 16 16:30:40 1994", bibsource = "garbo.uwasa.fi:/pc/doc-soft/fpbiblio.txt; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nj, } @Article{Buhring:1987:BUA, author = "Wolfgang B{\"u}hring", title = "The behavior at unit argument of the hypergeometric function {${}_3 F_2$}", journal = j-SIAM-J-MATH-ANA, volume = "18", number = "5", pages = "1227--1234", month = sep, year = "1987", CODEN = "SJMAAH", ISSN = "0036-1410 (print), 1095-7154 (electronic)", ISSN-L = "0036-1410", MRclass = "33A30", MRnumber = "88j:33004", MRreviewer = "K. M. Saksena", bibdate = "Sun Nov 28 19:24:11 MST 2010", bibsource = "http://epubs.siam.org/sam-bin/dbq/toc/SIMA/18/5; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/siamjmathana.bib", acknowledgement = ack-nhfb, fjournal = "SIAM Journal on Mathematical Analysis", journal-URL = "http://epubs.siam.org/sima", } @Article{Carlson:1987:TEI, author = "B. C. Carlson", title = "A Table of Elliptic Integrals of the Second Kind", journal = j-MATH-COMPUT, volume = "49", number = "180", pages = "595--606, S13--S17", month = oct, year = "1987", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "65A05 (33A25 65V05)", MRnumber = "89b:65013", MRreviewer = "F. W. J. Olver", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, classcodes = "C4160 (Numerical integration and differentiation); C7310 (Mathematics)", corpsource = "Dept. of Math., Iowa State Univ., Ames, IA, USA", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "elliptic integrals of the second kind; FORTRAN listings; integration; mathematics computing; standard R-functions", treatment = "P Practical; T Theoretical or Mathematical; X Experimental", } @Article{Crandall:1987:EFE, author = "R. E. Crandall and J. P. Buhler", title = "Elementary function expansions for {Madelung} constants", journal = j-J-PHYS-A, volume = "20", number = "16", pages = "5497--5510", year = "1987", CODEN = "JPHAC5", ISSN = "0305-4470 (print), 1361-6447 (electronic)", ISSN-L = "0305-4470", MRclass = "82A60 (82-08)", MRnumber = "MR924725 (88m:82034)", bibdate = "Wed Apr 13 06:46:35 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Journal of Physics. A. Mathematical and General", journal-URL = "http://iopscience.iop.org/0305-4470", } @Article{DiDonato:1987:AFS, author = "Armido R. {DiDonato} and Alfred H. {Morris Jr.}", title = "{Algorithm 654}: {FORTRAN} Subroutines for Computing the Incomplete Gamma Function Ratios and their Inverse", journal = j-TOMS, volume = "13", number = "3", pages = "318--319", month = sep, year = "1987", CODEN = "ACMSCU", DOI = "https://doi.org/10.1145/29380.214348", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Sun Sep 4 21:43:08 1994", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fortran2.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib", URL = "https://dl.acm.org/doi/pdf/10.1145/29380.214348", acknowledgement = ack-nhfb, fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", keywords = "algorithms", subject = "{\bf G.1.2}: Mathematics of Computing, NUMERICAL ANALYSIS, Approximation. {\bf G.m}: Mathematics of Computing, MISCELLANEOUS.", } @Article{Dunham:1987:PMAa, author = "Charles B. Dunham", title = "Provably monotone approximations", journal = j-SIGNUM, volume = "22", number = "2", pages = "6--11", month = apr, year = "1987", CODEN = "SNEWD6", DOI = "https://doi.org/10.1145/24936.24938", ISSN = "0163-5778 (print), 1558-0237 (electronic)", ISSN-L = "0163-5778", bibdate = "Tue Apr 12 07:50:15 MDT 2005", bibsource = "http://portal.acm.org/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "ACM SIGNUM Newsletter", journal-URL = "https://dl.acm.org/loi/signum", keywords = "theory; verification", subject = "G.1.2 Mathematics of Computing, NUMERICAL ANALYSIS, Approximation", } @Article{Dunham:1987:PMAb, author = "Charles B. Dunham", title = "Provably monotone approximations, {II}", journal = j-SIGNUM, volume = "22", number = "3", pages = "30--31", month = jul, year = "1987", CODEN = "SNEWD6", DOI = "https://doi.org/10.1145/36318.36323", ISSN = "0163-5778 (print), 1558-0237 (electronic)", ISSN-L = "0163-5778", bibdate = "Tue Apr 12 07:50:15 MDT 2005", bibsource = "http://portal.acm.org/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "ACM SIGNUM Newsletter", journal-URL = "https://dl.acm.org/loi/signum", keywords = "theory", subject = "G.1.2 Mathematics of Computing, NUMERICAL ANALYSIS, Approximation", } @Article{Gervais:1987:RAF, author = "R. Gervais and Q. I. Rahman and G. Schmeisser", title = "Representation and approximation of functions via $ (0, 2) $-interpolation", journal = j-J-APPROX-THEORY, volume = "50", number = "2", pages = "89--110", month = jun, year = "1987", CODEN = "JAXTAZ", ISSN = "0021-9045 (print), 1096-0430 (electronic)", ISSN-L = "0021-9045", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, bibno = "30719", catcode = "G.1.2; G.1.1", CRclass = "G.1.2 Approximation; G.1.2 Elementary function approximation; G.1.1 Interpolation; G.1.1 Spline and piecewise polynomial interpolation", descriptor = "Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation; Mathematics of Computing, NUMERICAL ANALYSIS, Interpolation, Spline and piecewise polynomial interpolation", fjournal = "Journal of Approximation Theory", genterm = "theory; verification", guideno = "1987-09238", journal-URL = "http://www.sciencedirect.com/science/journal/00219045", journalabbrev = "J. Approx. Theory", jrldate = "June 1987", subject = "G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS", } @Article{Ifantis:1987:UBF, author = "E. K. Ifantis and P. D. Siafarikas and C. B. Kouris", title = "Upper bounds for the first zeros of {Bessel} functions", journal = j-J-COMPUT-APPL-MATH, volume = "17", number = "3", pages = "355--358", month = mar, year = "1987", CODEN = "JCAMDI", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Sat Feb 25 11:59:57 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath1980.bib", URL = "http://www.sciencedirect.com/science/article/pii/0377042787901117", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @Article{Johnson:1987:AES, author = "Kenneth C. Johnson", title = "{Algorithm 650}: Efficient Square Root Implementation on the 68000", journal = j-TOMS, volume = "13", number = "2", pages = "138--151", month = jun, year = "1987", CODEN = "ACMSCU", DOI = "https://doi.org/10.1145/328512.328520", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", MRclass = "65D15", MRnumber = "898 489", bibdate = "Sun Sep 4 21:36:32 1994", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib", note = "See also \cite{Johnson:1987:CES}.", acknowledgement = ack-nj # " and " # ack-nhfb, fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", } @Article{Johnson:1987:CES, author = "Kenneth C. Johnson", title = "Corrigendum: {``Algorithm 650: efficient square root implementation on the 68000'' [ACM Trans. Math. Software {\bf 13} (1987), no. 2, 138--151]}", journal = j-TOMS, volume = "13", number = "3", pages = "320--320", month = sep, year = "1987", CODEN = "ACMSCU", DOI = "https://doi.org/10.1145/29380.356210", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", MRclass = "320. 65D15", MRnumber = "918 582", bibdate = "Thu Aug 08 15:52.08 2024", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib", note = "See \cite{Johnson:1987:AES}.", acknowledgement = ack-nhfb, fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", } @InProceedings{Kahan:1987:BCC, author = "W. Kahan", title = "Branch Cuts for Complex Elementary Functions or Much Ado About Nothing's Sign Bit", crossref = "Iserles:1987:SAN", volume = "9", pages = "165--211", year = "1987", MRclass = "65E05", MRnumber = "88k:65027", bibdate = "Sat Jan 11 17:44:01 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", series = "Inst. Math. Appl. Conf. Ser. New Ser.", acknowledgement = ack-nhfb, } @Article{Kolbig:1987:BRC, author = "K. S. K{\"o}lbig", title = "Book Review: {{\booktitle{Calculation of Special Functions, the Gamma Function, the Exponential Integrals and Error-Like Functions}} (C. G. van der Laan and N. M. Temme)}", journal = j-SIAM-REVIEW, volume = "29", number = "4", pages = "660--661", month = "????", year = "1987", CODEN = "SIREAD", DOI = "https://doi.org/10.1137/1029138", ISSN = "0036-1445 (print), 1095-7200 (electronic)", ISSN-L = "0036-1445", bibdate = "Sat Mar 29 09:54:19 MDT 2014", bibsource = "http://epubs.siam.org/toc/siread/29/4; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/siamreview.bib", acknowledgement = ack-nhfb, fjournal = "SIAM Review", journal-URL = "http://epubs.siam.org/sirev", onlinedate = "December 1987", } @PhdThesis{Kramer:1987:ISR, author = "W. Kr{\"a}mer", title = "Inverse Standardfunktionen f{\"u}r reelle und komplexe Intervallargumente mit a priori Fehlerabsch{\"a}tzungen f{\"u}r beliebige Datenformate \toenglish {Inverse Elementary Functions for Real and Complex Interval Arguments with A-Priori Error Estimates for Arbitrary Data Formats} \endtoenglish", type = "Dissertation", school = "Universit{\"a}t Karlsruhe", address = "Karlsruhe, Germany", pages = "????", year = "1987", bibdate = "Fri Sep 16 16:30:41 1994", bibsource = "garbo.uwasa.fi:/pc/doc-soft/fpbiblio.txt; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nj, author-dates = "1952--2014 (WK)", } @Article{Lewanowicz:1987:CRR, author = "Stanis{\l}aw Lewanowicz", title = "Corrigendum: {``Recurrence relations for hypergeometric functions of unit argument''} {[Math. Comp. {\bf 45} (1985), no. 172, 521--535, MR 86m:33004]}", journal = j-MATH-COMPUT, volume = "48", number = "178", pages = "853--853", month = apr, year = "1987", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "33A35 (65Q05)", MRnumber = "88a:33013", bibdate = "Wed Jan 15 09:19:34 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp1980.bib", acknowledgement = ack-nhfb, fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @MastersThesis{Liu:1987:BEF, author = "Z. A. Liu", title = "{Berkeley} Elementary Function Test Suite", type = "{M.S.} thesis", school = "Computer Science Division, Department of Electrical Engineering and Computer Science, Univerity of California at Berkeley", address = "Berkeley, CA, USA", month = dec, year = "1987", bibdate = "Mon Sep 12 23:52:34 1994", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nj # "\slash " # ack-nhfb, } @Article{Lo:1987:HGA, author = "Hao-Yung Lo and Jau-Ling Chen", title = "A Hardwired Generalized Algorithm for Generating the Logarithm Base-$k$ by Iteration", journal = j-IEEE-TRANS-COMPUT, volume = "C-36", number = "11", pages = "1363--1367", month = nov, year = "1987", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/TC.1987.5009477", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Sat Jul 9 09:28:57 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput1980.bib", URL = "https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5009477", acknowledgement = ack-nj # "\slash " # ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", } @InProceedings{Mathis:1987:EFP, author = "Robert F. Mathis", title = "Elementary Functions Package for {Ada}", crossref = "ACM:1987:UAA", pages = "95--100", month = dec, year = "1987", bibdate = "Mon May 19 13:30:58 1997", bibsource = "ftp://ftp.ira.uka.de/pub/bibliography/Compiler/ada.bib.gz; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, } @Article{Moricz:1987:ACF, author = "Ferenc Moricz and Xianliang Shi", title = "Approximation to continuous functions by {Cesaro} means of double {Fourier} series and conjugate series", journal = j-J-APPROX-THEORY, volume = "49", number = "4", pages = "346--377", month = apr, year = "1987", CODEN = "JAXTAZ", ISSN = "0021-9045 (print), 1096-0430 (electronic)", ISSN-L = "0021-9045", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, bibno = "30744", catcode = "G.1.2; G.1.2", CRclass = "G.1.2 Approximation; G.1.2 Elementary function approximation; G.1.2 Approximation; G.1.2 Least squares approximation", descriptor = "Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation; Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Least squares approximation", fjournal = "Journal of Approximation Theory", genterm = "theory; verification", guideno = "1987-09224", journal-URL = "http://www.sciencedirect.com/science/journal/00219045", journalabbrev = "J. Approx. Theory", jrldate = "April 1987", subject = "G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS", } @TechReport{Morris:1987:NLM, author = "Alfred H. {Morris, Jr.}", title = "{NSWC} Library of Mathematics Subroutines", type = "Report", number = "NSWC TR 86-251", institution = "Naval Surface Warfare Center", address = "Dahlgren, VA 22448-5000, USA; Silver Spring, MD 20903-5000, USA", pages = "xiii + 424 + 5", month = apr, year = "1987", bibdate = "Sat Jan 10 14:16:12 2026", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fortran2.bib; https://www.math.utah.edu/pub/tex/bib/prng.bib", note = "See also later editions \cite{Morris:1990:NLM,Morris:1993:NLM}.", URL = "http://vandyke.mynetgear.com/NSWC-TR-86-251.pdf; http://vandyke.mynetgear.com/nswclibr.tgz", abstract = "The NSWC library is a library of general-purpose FORTRAN subroutines that provide a basic computational capability in a variety of mathematical activities. Emphasis has been placed on the transportability of the codes. Subroutines are available in the following areas: Elementary Operations, Geometry, Special Functions, Polynomials, Vectors, Matrices, Large Dense Systems of Linear Equations, Banded Matrices, Sparse Matrices, Eigenvalues and Eigenvectors, $ \ell_1 $ Solution of Linear Equations, Least-Squares Solution of Linear Equations, Optimization, Transforms, Approximation of Functions, Curve Fitting, Surface Fitting, Manifold Fitting, Numerical Integration, Integral Equations, Ordinary Differential Equations, Partial Differential Equations, and Random Number Generation.", acknowledgement = ack-nhfb, remark = "The tableofcontents value in this entry is derived from optical character recognition (OCR) of the 9-page listing from the PDF files, with editorial correction of spelling and OCR errors. There are numerous routine names in the compiled library that are not mentioned in the table of contents.", tableofcontents = "Introduction / 1 \\ \\ Elementary Operations \\ \\ Sorting Lists --- ISHELL, SHELL, SHELL2 / 3 \\ Cube Root --- CBRT / 5 \\ Four Quadrant Arctangent --- ARTNQ, DARTNQ / 5 \\ Length of a Two-Dimensional Vector --- CPABS, DCPABS / 5 \\ Square Root of a Double Precision Complex Number --- DCSQRT / 6 \\ Conversion of Polar to Cartesian Coordinates --- POCA / 7 \\ Conversion of Cartesian to Polar Coordinates --- CAPO / 7 \\ Rotation of Axes --- ROTA / 7 \\ Planar Givens Rotations --- SROTG, DROTG / 9 \\ Three Dimensional Rotations --- ROT3 / 11 \\ Rotation of a Point on the Unit Sphere to the North Pole --- CONSTR / 13 \\ Hyperbolic Sine and Cosine Functions --- SNHCSH / 15 \\ Exponentials --- REXP / 17 \\ Logarithms --- ALNREL, RLOG / 19 \\ \\ Geometry \\ \\ The Convex Hull for a Finite Planar Set --- HULL / 21 \\ Areas of Planar Polygons --- PAREA / 23 \\ \\ Special Functions \\ \\ Error Function --- ERF, ERFC, ERFC1, CERF / 25 \\ Normal Distribution Function --- PNDF / 27 \\ Complex Fresnel Integral --- CFRNLI / 29 \\ Real Fresnel Integrals --- FRNL / 31 \\ Exponential Integral Function --- CEXPLL, EXPLI / 33 \\ Sine and Cosine Integral Functions --- SI, CIN / 37 \\ Gamma Function --- CGAMMA, GAMMA, GAMLN / 39 \\ Digamma Function --- CPSI, PSI / 41 \\ Logarithm of the Beta Function --- BETALN / 43 \\ Incomplete Gamma Ratio Functions --- GRATIO / 45 \\ Inverse Incomplete Gamma Ratio Functions --- GAMINV / 47 \\ Incomplete Beta Function --- BRATIO, ISUBX / 49 \\ Bessel Function $J_\nu(z)$ --- CBSSLJ, BSSLJ, BESJ / 51 \\ Bessel Function $Y\_nu(z)$ --- BSSLY / 53 \\ Modified Bessel Function $I_\nu(z)$ --- BSSLI, BESI / 55 \\ Modified Bessel Function $K_\nu(z)$ --- CBSSLK, BSSLK / 57 \\ Complete Complex Elliptic Integrals of the First and Second Kinds --- CK, CKE / 59 \\ Real Elliptic Integrals of the First and Second Kinds --- ELLPI / 63 \\ Real Elliptic Integrals of the Third Kind --- EPL, RJ / 65 \\ Jacobian Elliptic Functions --- ELLPF / 69 \\ Weierstrass Elliptic Function for the Equianharmonic and Lemniscatic Cases --- PEQ, PEQ1, PLEM, PLEM1 / 71 \\ Integral of the Bivariate Density Function over Arbitrary Polygons and Semi-Infinite Angular Regions --- VALR2 / 75 \\ Circular Coverage Function --- CIRCV / 79 \\ Elliptical Coverage Function --- PKILL, PKILL3 / 81 \\ \\ Polynomials \\ \\ Copying Polynomials --- PLCOPY, DPCOPY / 83 \\ Addition of Polynomials --- PADD, DPADD / 85 \\ Subtraction of Polynomials --- PSUBT, DPSUBT / 87 \\ Multiplication of Polynomials --- PMULT, DPMULT / 89 \\ Division of Polynomials --- PDIV, DPDIV / 91 \\ Real Powers of Polynomials --- PLPWR, DPLPWR / 93 \\ Derivatives and Integrals of Polynomials --- MPLNMV / 95 \\ Lagrange Polynomials --- LGRNGN, LGRNGV, LGRNGX / 97 \\ Orthogonal Polynomials on Finite Sets --- ORTHOS, ORTHOV, ORTHOX / 99 \\ \\ Solutions of Nonlinear Equations \\ \\ Zeros of Continuous Functions --- ZEROIN / 101 \\ Solution of Systems of Nonlinear Equations --- HBRD / 103 \\ Solutions of Quadratic, Cubic, and Quartic Equations \\ Double Precision Roots of a Real Polynomial --- RPOLY / 107 \\ Accuracy of the Roots of a Real Polynomial --- RBND / 109 \\ Copying Vectors --- SCOPY, DCOPY, CCOPY / 111 \\ Interchanging Vectors --- SSWAP, DSWAP, CSWAP / 113 \\ Planar Rotation of Vectors --- SROT, DROT, CSROT / 115 \\ Dot Products of Vectors --- SDOT, DDOT, CDOTC, CDOTU / 117 \\ Scaling Vectors --- SSCAL, DSCAL, CSCAL, CSSCAL / 119 \\ Vector Addition --- SAXPY, DAXPY, CAXPY / 121 \\ $L_1$ Norm of a Vector --- SASUM, DASUM, SCASUM / 123 \\ $L_2$ Norm of a Vector --- SNRM2, DNRM2, SCNRM2 / 0 125 \\ $L_\infty$ Norm of a Vector --- ISAMAX, IDAMAX, ICAMAX / 127 \\ \\ Matrices \\ \\ Packing and Unpacking Symmetric Matrices --- MCVFS, --- DMCVES, MCVSF, DMCVSF / 129 \\ Conversion of Real Matrices to and from Double Precision Form --- MCVRD, MCVDR / 131 \\ Storage of Real Matrices in the Complex Matrix Format --- MCVRC / 133 \\ The Real and Imaginary Portions of a Complex Matrix --- CMREAL, CMIMAG / 135 \\ Copying Matrices --- MCOPY, SMCOPY, DMCOPY, CMCOPY / 137 \\ Computation of the Conjugate of a Complex Matrix --- CMCONJ / 139 \\ Transposing Matrices --- TPOSE, DTPOSE, CTPOSE, TIP, DTIP, CTIP / 141 \\ Computing Adjoints of Complex Matrices --- CMADJ, CTRANS / 143 \\ Matrix Addition --- MADD, SMADD, DMADD, CMADD / 145 \\ Matrix Subtraction --- MSUBT, SMSUBT, DMSUBT, CMSUBT / 147 \\ Matrix Multiplication --- MPROD, DMPROD, CMPROD / 149 \\ Product of a Packed Symmetric Matrix and a Vector --- SVPRD, DSVPRD / 151 \\ Transpose Matrix Products --- TMPROD / 153 \\ Symmetric Matrix Products --- SMPROD / 155 \\ Kronecker Product of Matrices --- KPROD, DKPROD, CKPROD / 157 \\ Inverting General Real Matrices and Solving General Systems of Real Linear Equations --- CROUT, KROUT, NPIVOT, MSLV, DMSLV / 159 \\ Solution of Real Equations with Iterative Improvement --- SLVMP / 165 \\ Solution of Almost Block Diagonal Systems of Linear Equations --- ARCECO, ARCESL / 167 \\ Solution of Almost Block Tridiagonal Systems of Linear Equations --- BTSLV / 171 \\ Inverting Symmetric Real Matrices and Solving Symmetric Systems of Real Linear Equations --- SMSLV, DSMSLV / 173 \\ Inverting Positive Definite Symmetric Matrices and Solving Positive Definite Symmetric Systems of Linear Equations --- PCHOL, DPCHOL / 177 \\ Inverting General Complex Matrices and Solving General Systems of Complex Linear Equations --- CMSLV / 179 \\ Solution of Complex Equations with Iterative Improvement --- CSLVMP / 181 \\ Singular Value Decomposition of a Matrix --- SSVDC, DSVDC, CSVDC / 183 \\ Evaluation of the Characteristic Polynomial of a Matrix --- DET, DPDET, CDET / 185 \\ Solution of the Matrix Equation $A X + X B = C$ --- ABSLV, DABSLV / 187 \\ Solution of the Matrix Equation $A^T X + X A = C$ when $C$ is Symmetric --- TASLV, DTASLV / 189 \\ Solution of the Matrix Equation $A X^2 + B X + C = 0$ --- SQUINT / 191 \\ Exponential of a Real Matrix --- MEXP, DMEXP / 193 \\ \\ Large Dense Systems of Linear Equations \\ \\ Solving Systems of 200--400 Linear Equations --- LE, DPLE, CLE / 195 \\ \\ Banded Matrices \\ \\ Band Matrix Storage / 197 \\ Conversion of Banded Matrices to and from the Standard Format --- CVBR, CVBC, CVRB, CVCB, CVRB1, CVCB1 / 199 \\ Conversion of Banded Matrices to and from Sparse Form --- MCVBS, CMCVBS, MCVSB, CMCVSB / 201 \\ Transposing Banded Matrices --- BPOSE, CBPOSE / 203 \\ Addition of Banded Matrices --- BADD, CBADD / 205 \\ Subtraction of Banded Matrices --- BSUBT, CBSUBT / 207 \\ Multiplication of Banded Matrices --- BPROD, CBPROD / 209 \\ Product of a Real Banded Matrix and Vector --- BVPRD, BYPRD1, BTPRD, BTPRD1 / 211 \\ Product of a Complex Banded Matrix and Vector --- CBVPD, CBVPD1, CBTPD, CBTPD1 / 213 \\ Solution of Banded Systems of Real Linear Equations --- BLSV, BLSV1 / 215 \\ Solution of Banded Systems of Complex Linear Equations --- CBSLV, CBSLV1 / 217 \\ \\ Sparse Matrices \\ \\ Storage of Sparse Matrices / 219 \\ Conversion of Sparse Matrices to and from the Standard Format --- CVRS, CVCS, CVSR, CVSC / 221 \\ Computing Conjugates of Sparse Complex Matrices --- CSCONJ / 223 \\ Transposing Sparse Real Matrices --- RPOSE, RPOSE1 / 223 \\ Transposing Sparse Complex Matrices --- CPOSE, CPOSEL / 227 \\ Addition of Sparse Matrices --- RADD, RADDI1, CADD, CADD1 / 229 \\ Subtraction of Sparse Matrices --- RSUB, RSUB1, CSUB, CSUB1 / 231 \\ Multiplication of Sparse Matrices --- RMLT, RMLT1, CMLT, CMLT1 / 233 \\ Product of a Real Sparse Matrix and Vector --- MVPRD, MVPRD1, MTPRD, MTPRD1 / 235 \\ Product of a Complex Sparse Matrix and Vector --- CVPRD, CVPRD1, CTPRD, CTPRD1 / 237 \\ Ordering the Rows of a Sparse Matrix by Increasing Length --- SPORD / 239 \\ Reordering Sparse Matrices into Block Triangular Form --- BLKORD / 241 \\ Solution of Sparse Systems of Real Linear Equations --- SPSLYV, RSLV, TSLV / 243 \\ Solution of Sparse Systems of Complex Linear Equations --- CSPSLV, CSLV, CTSLV / 247 \\ \\ Eigenvalues and Eigenvectors \\ \\ Computation of Eigenvalues of General Real Matrices --- EIG, EIG1 / 231 \\ Computation of Eigenvalues and Eigenvectors of General Real Matrices --- EIGV, EIGV1 /293 \\ Double Precision Computation of Eigenvalues of Real Matrices --- DEIG / 235 \\ Double Precision Computation of Eigenvalues and Eigenvectors of Real Matrices --- DEIGV /237 \\ Computation of Eigenvalues of Symmetric Real Matrices --- SEIG, SEIGI1 / 259 \\ Computation of Eigenvalues and Eigenvectors of Symmetric Real Matrices --- SEIGV, SEIGV / 261 \\ Computation of Eigenvalues of Complex Matrices --- CEIG / 263 \\ Computation of Eigenvalues and Eigenvectors of Complex Matrices --- CEIGV / 265 \\ Double Precision Computation of Eigenvalues of Complex Matrices --- DCEIG / 267 \\ Double Precision Computation of Eigenvalues and Eigenvectors of Complex Matrices --- DCEIGV / 269 \\ \\ $\ell_1$ Solution of Linear Equations \\ \\ $\ell_1$ Solution of Systems of Linear Equations with Equality and Inequality Constraints --- CL1 / 271 \\ \\ Least Squares Solution of Linear Equations \\ \\ Least Squares Solution of Systems of Linear Equations --- LLSQ, HFTI, HFTI2 / 273 \\ Least Squares Solution of Overdetermined Systems of Linear Equations with Iterative Improvement --- LLSQMP / 277 \\ Least Squares Solution of Systems of Linear Equations with Equality and Inequality Constraints --- LSEI / 279 \\ Least Squares Solution of Systems of Linear Equations with Equality and Nonnegativity Constraints --- WNNLS / 285 \\ Least Squares Iterative Improvement Solution of Systems of Linear Equations with Equality Constraints --- L2SLV / 289 \\ Iterative Least Squares Solution of Banded Linear Equations --- BLSQ / 293 \\ Iterative Least Squares Solution of Sparse Linear Equations --- SPLSQ, STLSQ / 295 \\ \\ Optimization \\ \\ Minimization of Functions of a Single Variable --- FMIN / 297 \\ Unconstrained Minimum of the Sum of Squares of Nonlinear Functions --- LMDIFF / 299 \\ Linear Programming --- SMPLX, SSPLX / 301 \\ The Assignment Problem --- ASSGN / 300 \\ \\ Transforms \\ \\ Fast Fourier Transform --- FFT, FFT1 / 307 \\ Multivariate Fast Fourier Transform --- MFFT, MFFT1 / 309 \\ Discrete Cosine and Sine Transforms --- COSQI, COSQB, COSQF, SINQB, SINQF / 311 \\ \\ Approximation of Functions \\ \\ Rational Minimax Approximation of Functions --- CHEBY / 315 \\ $L_p$ Approximation of Functions --- 317 \\ \\ Curve Fitting \\ \\ Linear Interpolation --- TRP / 323 \\ Lagrange Interpolation --- LTRP / 325 \\ Hermite Interpolation --- HTRP / 327 \\ Conversion of Real Polynomials from Newton to Taylor Series Form --- PCOEFF / 329 \\ Least Squares Polynomial Fit --- PFIT / 331 \\ Weighted Least Squares Polynomial Fit --- WPFIT / 333 \\ Cubic Spline Interpolation --- SPLIFT / 335 \\ Weighted Least Squares Cubic Spline Fitting --- SPFIT / 337 \\ Cubic Spline Evaluation --- SCOMP, SCOMP1, SCOMP2 / 339 \\ Cubic Spline Evaluation and Differentiation --- SEVAL, SEVAL1, SEVAL2 / 341 \\ Spline under Tension Interpolation --- CURVI1 / 343 \\ Spline under Tension Evaluation --- CURV2 / 345 \\ Differentiation and Integration of Splines under Tension --- CURVD, CURVI / 347 \\ Two Dimensional Spline under Tension Curve Fitting --- KURV1, KURV2 / 349 \\ Two Dimensional Spline under Tension Closed Curve Fitting --- KURVP1, KURVP2 / 351 \\ Three Dimensional Spline under Tension Curve Fitting --- QURV1, QURV2 / 353 \\ B-Splines / 355 \\ Piecewise Polynomial Interpolation --- BSTRP / 357 \\ Conversion of Piecewise Polynomials from B-Spline to Taylor Series Form --- BSPPP / 359 \\ Piecewise Polynomial Evaluation --- PPVAL / 361 \\ Weighted Least Squares Piecewise Polynomial Fitting --- BSL2 / 363 \\ \\ Surface Fitting over Rectangular Grids \\ \\ B-Splines under Tension / 365 \\ B-Spline under Tension Surface Interpolation --- SURF / 367 \\ B-Spline under Tension Evaluation --- SURF2, NSURF2 / 369 \\ \\ Surface Fitting over Arbitrarily Positioned Data Points \\ \\ Surface Interpolation for Arbitrarily Positioned Data Points --- BVIP, BVIP2 / 371 \\ \\ Manifold Fitting \\ \\ Weighted Least Squares Fitting with Polynomials of n Variables --- MFIT, DMFIT, MEVAL, DMEVAL / 375 \\ \\ Numerical Integration \\ \\ Evaluation of Integrals over Finite Intervals --- QAGS, QSUBA, DQAGS / 379 \\ Evaluation of Integrals over Infinite Intervals --- QAGI, DQAGI / 383 \\ Evaluation of Double Integrals over Triangles --- CUBTRI / 389 \\ \\ Integral Equations \\ \\ Solution of Fredholm Integral Equations of the Second Kind --- IESLV / 39 \\ \\ Ordinary Differential Equations\slash Initial Value Problems \\ \\ The Initial Value Solvers --- Introductory Comments / 395 \\ Adaptive Adams Solution of Nonstiff Differential Equations --- ODE / 397 \\ Adaptive RKF Solution of Nonstiff Differential Equations --- RKF4S / 401 \\ Adaptive RKF Solution of Nonstiff Differential Equations with Global Error Estimation --- GERK / 40S \\ Adaptive Solution of Stiff Differential Equations --- SFODE, SFODE1 / 409 \\ Fourth-Order Runge-Kutta --- RK / 413 \\ Eighth-Order Runge-Kutta --- RK8 / 415 \\ \\ Partial Differential Equations \\ \\ Separable Second-Order Elliptic Equations on Rectangular Domains --- SEPDE / 417 \\ \\ Random Number Generation \\ \\ Uniform Random Number Generator --- URNG / 421 \\ Gaussian Random Number Generator using the Box-M{\"u}ller Transformation --- NRNG / 423", } @Article{Musielak:1987:AEG, author = "J. Musielak", title = "Approximation of elements of a generalized {Orlicz} sequence space by nonlinear, singular kernels", journal = j-J-APPROX-THEORY, volume = "50", number = "4", pages = "366--372", month = aug, year = "1987", CODEN = "JAXTAZ", ISSN = "0021-9045 (print), 1096-0430 (electronic)", ISSN-L = "0021-9045", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, bibno = "30716", catcode = "G.2.1; G.1.5; G.1.2", CRclass = "G.2.1 Combinatorics; G.2.1 Generating functions; G.1.5 Roots of Nonlinear Equations; G.1.5 Convergence; G.1.2 Approximation; G.1.2 Elementary function approximation", descriptor = "Mathematics of Computing, DISCRETE MATHEMATICS, Combinatorics, Generating functions; Mathematics of Computing, NUMERICAL ANALYSIS, Roots of Nonlinear Equations, Convergence; Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation", fjournal = "Journal of Approximation Theory", genterm = "theory; verification", guideno = "1987-09257", journal-URL = "http://www.sciencedirect.com/science/journal/00219045", journalabbrev = "J. Approx. Theory", jrldate = "Aug. 1987", subject = "G. Mathematics of Computing; G.2 DISCRETE MATHEMATICS; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS", } @Article{Pereira:1987:CEC, author = "N. Costa Pereira", title = "Corrigendum: {``Estimates for the Chebyshev function $ \psi (x) - \theta (x) $''} {[Math. Comp. {\bf 44} (1985), no. 169, 211--221, MR 86k:11005]}", journal = j-MATH-COMPUT, volume = "48", number = "177", pages = "447--447", month = jan, year = "1987", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "11A25 (11N45 11Y35 33A70)", MRnumber = "87k:11006", bibdate = "Thu Jun 15 07:27:03 2000", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "See \cite{Pereira:1985:ECF}.", acknowledgement = ack-nhfb, fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Proinov:1987:NIA, author = "Petko D. Proinov", title = "Numerical integration and approximation of differentiable functions, {II}", journal = j-J-APPROX-THEORY, volume = "50", number = "4", pages = "373--393", month = aug, year = "1987", CODEN = "JAXTAZ", ISSN = "0021-9045 (print), 1096-0430 (electronic)", ISSN-L = "0021-9045", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, bibno = "30717", catcode = "G.1.4; G.1.2", CRclass = "G.1.4 Quadrature and Numerical Differentiation; G.1.2 Approximation; G.1.2 Elementary function approximation", descriptor = "Mathematics of Computing, NUMERICAL ANALYSIS, Quadrature and Numerical Differentiation; Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation", fjournal = "Journal of Approximation Theory", genterm = "theory; verification", guideno = "1987-09258", journal-URL = "http://www.sciencedirect.com/science/journal/00219045", journalabbrev = "J. Approx. Theory", jrldate = "Aug. 1987", subject = "G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS", } @Article{Rolfe:1987:FIS, author = "Timothy J. Rolfe", title = "On a Fast Integer Square Root Algorithm", journal = j-SIGNUM, volume = "22", number = "4", pages = "6--11", month = oct, year = "1987", CODEN = "SNEWD6", ISSN = "0163-5778 (print), 1558-0237 (electronic)", ISSN-L = "0163-5778", bibdate = "Tue Apr 12 07:50:16 MDT 2005", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/signum.bib", acknowledgement = ack-nj # " and " # ack-nhfb, fjournal = "ACM SIGNUM Newsletter", journal-URL = "https://dl.acm.org/loi/signum", keywords = "algorithms; performance; theory", subject = "F.2.1 Theory of Computation, ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY, Numerical Algorithms and Problems, Number-theoretic computations", } @Article{Smith:1987:BAM, author = "P. W. Smith and J. J. Swetits", title = "Best approximation by monotone functions", journal = j-J-APPROX-THEORY, volume = "49", number = "4", pages = "398--403", month = apr, year = "1987", CODEN = "JAXTAZ", ISSN = "0021-9045 (print), 1096-0430 (electronic)", ISSN-L = "0021-9045", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, bibno = "30747", catcode = "G.1.2", CRclass = "G.1.2 Approximation; G.1.2 Elementary function approximation", descriptor = "Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation", fjournal = "Journal of Approximation Theory", genterm = "theory; verification", guideno = "1987-09227", journal-URL = "http://www.sciencedirect.com/science/journal/00219045", journalabbrev = "J. Approx. Theory", jrldate = "April 1987", subject = "G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS", } @Book{Spanier:1987:AF, author = "Jerome Spanier and Keith B. Oldham", title = "An Atlas of Functions", publisher = pub-HEMISPHERE, address = pub-HEMISPHERE:adr, pages = "ix + 700", year = "1987", ISBN = "0-89116-573-8, 3-540-17395-1", ISBN-13 = "978-0-89116-573-6, 978-3-540-17395-3", LCCN = "QA331 .S685 1987", bibdate = "Fri Aug 31 16:20:13 MDT 2007", bibsource = "https://www.math.utah.edu/pub/tex/bib/canjstat.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; z3950.loc.gov:7090/Voyager", note = "See also the second edition \cite{Oldham:2009:AF}.", acknowledgement = ack-nhfb, subject = "elementary functions; special functions; cognate functions; complementary incomplete gamma function; complementary modulus; complete beta function; complex argument when; cosecant functions; cotangent functions; digamma function; economized polynomial; error function complement; eta numbers; function and its reciprocal; hypergeometric algorithm; important definite integrals; incomplete elliptic integrals; inverse gudermannian function; inverse hyperbolic functions; negative integer order; other definite integrals; parabolic cylinder function; polygamma functions; purely imaginary argument; reciprocal linear function; reflection formula", tableofcontents = "Preface / ix \\ 0 General Considerations / 1 \\ 1 The Constant Function $c$ / 11 \\ 2 The Factorial Function $n!$ and Its Reciprocal / 19 \\ 3 The Zeta Numbers and Related Functions / 25 \\ 4 The Bernoulli Numbers, $B_n$ / 35 \\ 5 The Euler Numbers, $E_n$ / 39 \\ 6 The Binomial Coefficients $\binom{\nu}{m}$ / 43 \\ 7 The Linear Function $b x + c$ and Its Reciprocal / 53 \\ 8 The Unit-Step $u(x - a)$ and Related Functions / 63 \\ 9 The Integer-Value ${\tt Int}(x)$ and Fractional-Value ${\tt frac}(x)$ Functions / 71 \\ 10 The Dirac Delta Function $\delta(x - a)$ / 79 \\ 11 The Integer Powers $(bx + c)^n$ and $x^n$ / 83 \\ 12 The Square-Root Function $\sqrt{b x + c}$ and Its Reciprocal / 91 \\ 13 The Noninteger Powers $x^\nu$ / 99 \\ 14 The $b \sqrt{a^2 - x^2}$ Function and Its Reciprocal / 107 \\ 15 The $b \sqrt{x^2 + a}$ Function and Its Reciprocal / 115 \\ 16 The Quadratic Function $a x^2 + b x + c$ and Its Reciprocal / 123 \\ 17 The Cubic Function $x^3 + a x^2 + b x + c$ and Higher Polynomials / 131 \\ 18 The Pochhammer Polynomials $(x)_n$ / 149 \\ 19 The Bernoulli Polynomials $B_n(x)$ / 167 \\ 20 The Euler Polynomials $E_n(x)$ / 175 \\ 21 The Legendre Polynomials $P_n(x)$ / 183 \\ 22 The Chebyshev Polynomials $T_n(x)$ and $U_n(x)$ / 193 \\ 23 The Laguerre Polynomials $L_n(x)$ / 209 \\ 24 The Hermite Polynomials $H_n(x)$ / 217 \\ 25 The Logarithmic Function $\ln(x)$ / 225 \\ 26 The Exponential Function $\exp(b x + c)$ / 233 \\ 27 Exponentials of Powers $\exp(-a x^\nu)$ / 253 \\ 28 The Hyperbolic Sine $\sinh(x)$ and Cosine $\cosh(x)$ Functions / 263 \\ 29 The Hyperbolic Secant $\sech(x)$ and Cosecant $\csch(x)$ Functions / 273 \\ 30 The Hyperbolic Tangent $\tanh(x)$ and Cotangent $\coth(x)$ Functions / 279 \\ 31 The Inverse Hyperbolic Functions / 285 \\ 32 The Sine $\sin(x)$ and Cosine $\cos(x)$ Functions / 295 \\ 33 The Secant $\sec(x)$ and Cosecant $\csc(x)$ Functions / 311 \\ 34 The Tangent $\tan(x)$ and Cotangent $\cot(x)$ Functions / 319 \\ 35 The Inverse Trigonometric Functions / 331 \\ 36 Periodic Functions / 343 \\ 37 The Exponential Integral $\Ei(x)$ and Related Functions / 351 \\ 38 Sine and Cosine Integrals / 361 \\ 39 The Fresnel Integrals $S(x)$ and $C(x)$ / 373 \\ 40 The Error Function $\erf(x)$ and Its Complement $\erfc(x)$ / 385 \\ 41 The $\exp(x) \erfc(\sqrt{x})$ and Related Functions / 395 \\ 42 Dawson's Integral / 405 \\ 43 The Gamma Function $\Gamma(x)$ / 411 \\ 44 The Digamma Function $\psi(x)$ / 423 \\ 45 The Incomplete Gamma $\gamma(\nu,x)$ and Related Functions / 435 \\ 46 The Parabolic Cylinder Function $D_\nu(x)$ / 445 \\ 47 The Kummer Function $M(a; c; x)$ / 459 \\ 48 The Tricomi Function $U(a; c; x)$ 471 \\ 49 The Hyperbolic Bessel Functions $I_0(x)$ and $I_1(x)$ / 479 \\ 50 The General Hyperbolic Bessel Function $I_\nu(x)$ / 489 \\ 51 The Basset Function $K_\nu(x)$ / 499 \\ 52 The Bessel Coefficients $J_0(x)$ and $J_1(x)$ / 509 \\ 53 The Bessel Function $J_\nu(x)$ / 521 \\ 54 The Neumann Function $Y_\nu(x)$ / 533 \\ 55 The Kelvin Functions / 543 \\ 56 The Airy Functions $\Ai(x)$ and $\Bi(x)$ / 555 \\ 57 The Struve Function / 563 \\ 58 The Incomplete Beta Function $B(\nu; \mu; x)$ / 573 \\ 59 The Legendre Functions $P_\nu(x)$ and $Q_\nu(x)$ / 581 \\ 60 The Gauss Function $F(a, b; c; x)$ / 599 \\ 61 The Complete Elliptic Integrals $K(p)$ and $E(p)$ / 609 \\ 62 The Incomplete Elliptic Integrals $F(p; \phi)$ and $E(p; \phi)$ / 621 \\ 63 The Jacobian Elliptic Functions / 635 \\ 64 The Hurwitz Function $\zeta(\nu; u)$ / 653 \\ Appendix A Utility Algorithms / 665 \\ Appendix B Some Useful Data / 673 \\ References and Bibliography / 679 \\ Subject Index / 681 \\ Symbol Index / 691", } @Article{Stoyanov:1987:AE, author = "Basil J. Stoyanov and Richard A. Farrell", title = "On the Asymptotic Evaluation of $ \int^{\pi / 2}_0 {J}^2_0 (\gamma \sin x) d x $", journal = j-MATH-COMPUT, volume = "49", number = "179", pages = "275--279", month = jul, year = "1987", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "41A60 (65D30)", MRnumber = "88e:41067", MRreviewer = "Roderick Wong", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, classcodes = "C1120 (Analysis); C4180 (Integral equations)", corpsource = "Appl. Phys. Lab., Johns Hopkins Univ., Laurel, MD, USA", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "analytical expression; asymptotic behavior; asymptotic evaluation; Bessel functions; first kind; integral; integral equations; order Bessel function; positive parameter; zeroth-", treatment = "T Theoretical or Mathematical", } @Article{Takagi:1987:HAC, author = "Naofumi Takagi and T. Asada and S. Yajima", title = "A Hardware Algorithm for Computing Sine and Cosine Using Redundant Binary Representation", journal = j-SYS-COMP-JAPAN, volume = "18", number = "??", pages = "1--9", month = "????", year = "1987", CODEN = "SCJAEP", DOI = "", ISSN = "0882-1666 (print), 1520-684X (electronic)", ISSN-L = "0882-1666", bibdate = "Wed Nov 12 08:59:45 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "", acknowledgement = ack-nhfb, fjournal = "Systems and computers in Japan", } @InProceedings{Tanese:1987:PGA, author = "Reiko Tanese", editor = "John J. Grefenstette", booktitle = "Genetic algorithms and their applications: proceedings of the second International Conference on Genetic Algorithms: July 28--31, 1987 at the Massachusetts Institute of Technology, Cambridge, {MA}", title = "Parallel genetic algorithms for a hypercube", publisher = pub-ERLBAUM, address = pub-ERLBAUM:adr, bookpages = "260", pages = "177--183", year = "1987", ISBN = "0-8058-0158-8, 0-8058-0159-6 (paperback)", ISBN-13 = "978-0-8058-0158-3, 978-0-8058-0159-0 (paperback)", LCCN = "Q334 .I5561 1987", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", price = "US\$39.95", acknowledgement = ack-nhfb, bibno = "42536", catcode = "I.2.6; G.1.0; G.1.2; C.1.2", CRclass = "I.2.6 Learning; I.2.6 Analogies; G.1.0 General; G.1.0 Parallel algorithms; G.1.2 Approximation; G.1.2 Elementary function approximation; C.1.2 Multiple Data Stream Architectures (Multiprocessors); C.1.2 Parallel processors", descriptor = "Computing Methodologies, ARTIFICIAL INTELLIGENCE, Learning, Analogies; Mathematics of Computing, NUMERICAL ANALYSIS, General, Parallel algorithms; Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation; Computer Systems Organization, PROCESSOR ARCHITECTURES, Multiple Data Stream Architectures (Multiprocessors), Parallel processors", genterm = "algorithms; performance; experimentation", guideno = "1988-16888", procdate = "Sponsored by Amer. Assoc. for AI, Naval Res. Lab. and Bolt Beranek & Newman, July 28-31, 1987", procloc = "Cambridge, MA", subject = "I. Computing Methodologies; I.2 ARTIFICIAL INTELLIGENCE; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; C. Computer Systems Organization; C.1 PROCESSOR ARCHITECTURES", } @InCollection{Temme:1987:CIG, author = "N. M. Temme", title = "On the computation of the incomplete gamma functions for large values of the parameters", crossref = "Mason:1987:AAB", pages = "479--489", year = "1987", bibdate = "Fri Oct 18 16:42:44 2024", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, } @Article{Thompson:1987:IEF, author = "Peter Thompson", title = "Implementing an Elementary Function Library", journal = j-SIGNUM, volume = "22", number = "2", pages = "2--5", month = apr, year = "1987", CODEN = "SNEWD6", ISSN = "0163-5778 (print), 1558-0237 (electronic)", ISSN-L = "0163-5778", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); garbo.uwasa.fi:/pc/doc-soft/fpbiblio.txt; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb # " and " # ack-nj, bibno = "24937", catcode = "G.m; D.3.2", CRclass = "D.3.2 Language Classifications; D.3.2 OCCAM", descriptor = "Mathematics of Computing, MISCELLANEOUS; Software, PROGRAMMING LANGUAGES, Language Classifications, OCCAM", fjournal = "ACM SIGNUM Newsletter", genterm = "theory; languages", guideno = "1987-03363", journal-URL = "https://dl.acm.org/loi/signum", journalabbrev = "SIGNUM Newsl.", jrldate = "April 1987", subject = "G. Mathematics of Computing; G.m MISCELLANEOUS; D. Software; D.3 PROGRAMMING LANGUAGES", } @Article{Thompson:1987:MBF, author = "I. J. Thompson and A. R. Barnett", title = "Modified {Bessel} functions {$ I_\nu (z) $} and {$ K_\nu (z) $} of real order and complex argument, to selected accuracy", journal = j-COMP-PHYS-COMM, volume = "47", number = "2--3", pages = "245--257", month = nov # "\slash " # dec, year = "1987", CODEN = "CPHCBZ", DOI = "https://doi.org/10.1016/0010-4655(87)90111-1", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", bibdate = "Mon Feb 13 10:28:21 MST 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/compphyscomm1980.bib; https://www.math.utah.edu/pub/tex/bib/compphyscomm2000.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "See erratum \cite{Thompson:2004:EBB}.", URL = "http://www.sciencedirect.com/science/article/pii/0010465587901111", acknowledgement = ack-nhfb, fjournal = "Computer Physics Communications", journal-URL = "http://www.sciencedirect.com/science/journal/00104655", } @Article{Timan:1987:DFP, author = "A. F. Timan", title = "Distribution of fractional parts and approximation of functions with singularities by {Bernstein} polynomials", journal = j-J-APPROX-THEORY, volume = "50", number = "2", pages = "167--174", month = jun, year = "1987", CODEN = "JAXTAZ", ISSN = "0021-9045 (print), 1096-0430 (electronic)", ISSN-L = "0021-9045", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, bibno = "30725", catcode = "G.1.5; G.1.2", CRclass = "G.1.5 Roots of Nonlinear Equations; G.1.5 Polynomials, methods for; G.1.2 Approximation; G.1.2 Elementary function approximation", descriptor = "Mathematics of Computing, NUMERICAL ANALYSIS, Roots of Nonlinear Equations, Polynomials, methods for; Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation", fjournal = "Journal of Approximation Theory", genterm = "theory; verification", guideno = "1987-09244", journal-URL = "http://www.sciencedirect.com/science/journal/00219045", journalabbrev = "J. Approx. Theory", jrldate = "June 1987", subject = "G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS", } @Article{Visentin:1987:FAE, author = "Kley Visentin and Pablo Mart{\'\i}n", title = "Fractional approximation to elliptic functions", journal = j-J-MATH-PHYS, volume = "28", number = "2", pages = "330--333", month = feb, year = "1987", CODEN = "JMAPAQ", DOI = "https://doi.org/10.1063/1.527661", ISSN = "0022-2488 (print), 1089-7658 (electronic), 1527-2427", ISSN-L = "0022-2488", MRclass = "41A10 (33A65)", MRnumber = "87m:41009", bibdate = "Mon Oct 31 11:57:55 MDT 2011", bibsource = "http://jmp.aip.org/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jmathphys1985.bib", URL = "http://jmp.aip.org/resource/1/jmapaq/v28/i2/p330_s1", acknowledgement = ack-nhfb, fjournal = "Journal of Mathematical Physics", journal-URL = "http://jmp.aip.org/", pagecount = "4", } @Book{Zare:1987:AMU, author = "Richard N. Zare", title = "Angular Momentum: Understanding Spatial Aspects in Chemistry and Physics", publisher = pub-WILEY, address = pub-WILEY:adr, pages = "viii + 349", year = "1987", ISBN = "0-471-85892-7", ISBN-13 = "978-0-471-85892-8", LCCN = "QC793.3.A5 Z37 1987", bibdate = "Tue Aug 5 06:06:20 MDT 2025", bibsource = "fsz3950.oclc.org:210/WorldCat; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://catalog.hathitrust.org/api/volumes/oclc/214728550.html", acknowledgement = ack-nhfb, remark = "Based on lectures given to the Chemistry Department, Cornell University, in the autumn of 1980 as part of their Baker lecture series.", subject = "Angular momentum (Nuclear physics); Moment angulaire (Physique nucl{\'e}aire); Angular momentum (Nuclear physics); Drehimpuls; Quantenmechanik", tableofcontents = "Angular Momentum Operators and Wave Functions \\ Coupling of Two Angular Momentum Vectors \\ Transformation under Rotation \\ The Coupling of More than Two Angular Momentum Vectors \\ Spherical Tensor Operators \\ Energy-level Structure and Wave Functions of a Rigid Rotor \\ Appendix: Computer Programs for 3J, 6J, and 9J Symbols \\ Index", } @Article{Zurawski:1987:DHS, author = "J. H. P. Zurawski and J. B. Gosling", title = "Design of a High-Speed Square Root Multiply and Divide Unit", journal = j-IEEE-TRANS-COMPUT, volume = "C-36", number = "1", pages = "13--23", month = jan, year = "1987", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/TC.1987.5009445", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Sat Jul 9 09:28:49 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput1980.bib", URL = "https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5009445", acknowledgement = ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", } @Article{Zwick:1987:BAC, author = "D. Zwick", title = "Best approximation by convex functions", journal = j-AMER-MATH-MONTHLY, volume = "94", number = "6", pages = "528--534", month = jul, year = "1987", CODEN = "AMMYAE", ISSN = "0002-9890 (print), 1930-0972 (electronic)", ISSN-L = "0002-9890", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, affiliation = "University of Vermont, Vermont, NY", bibno = "43727", catcode = "G.1.2", CRclass = "G.1.2 Approximation; G.1.2 Elementary function approximation", descriptor = "Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation", fjournal = "American Mathematical Monthly", genterm = "theory; verification", guideno = "1988-04405", journal-URL = "https://www.jstor.org/journals/00029890.htm", journalabbrev = "Am. Math. Monthly", jrldate = "June/July 1987", subject = "G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS", } @Book{Aberth:1988:PNA, author = "Oliver Aberth", title = "Precise Numerical Analysis", publisher = pub-WCB, address = pub-WCB:adr, pages = "x + 225", year = "1988", ISBN = "0-697-06760-2", ISBN-13 = "978-0-697-06760-9", LCCN = "QA297 .A28 1988", bibdate = "Mon Oct 24 11:37:20 2011", bibsource = "file://sunrise/u/sy/beebe/tex/bib/all_brec.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", abstract = "Aberth addresses elementary issues of precise floating point computations using variable precision range arithmetic. Numbers are represented as a variable precision number $ \pm $ a range. Rational arithmetic is also considered. Chapters are devoted to \begin{enumerate} \item rootfinding, \item polynomial rootfinding, \item numerical linear algebra, \item differentiation and integration, and \item ordinary differential equations.\end{enumerate} Differentiation is handled by a codelist approach like [Rall81a], and applications to Taylor series are given. Interval techniques for ordinary differential equations are based on using an {\it a priori\/} bound to capture remainder terms. Several methods are illustrated, including Taylor series methods.", acknowledgement = ack-nj, comment = "Text for a one semester, junior level course in numerical analysis. Includes PC disk with software written in PBASIC. Sound introductory level discussion of code lists and error capture techniques.", keywords = "differentiation; differentiation arithmetic; general numerical analysis; integration; interval techniques; linear algebra; ordinary differential equations.; variable precision arithmetic", } @Article{Alonso:1988:SCN, author = "Javier Alonso and Carlos Benitez", title = "Some characteristic and non-characteristic properties of inner product spaces", journal = j-J-APPROX-THEORY, volume = "55", number = "3", pages = "318--325", month = dec, year = "1988", CODEN = "JAXTAZ", ISSN = "0021-9045 (print), 1096-0430 (electronic)", ISSN-L = "0021-9045", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, bibno = "56052", catcode = "G.1.2", CRclass = "G.1.2 Approximation; G.1.2 Elementary function approximation", descriptor = "Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation", fjournal = "Journal of Approximation Theory", genterm = "theory; verification", guideno = "1988-10198", journal-URL = "http://www.sciencedirect.com/science/journal/00219045", journalabbrev = "J. Approx. Theory", jrldate = "Dec. 1988", subject = "G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS", } @Article{Anderson:1988:MRE, author = "Ned Anderson", title = "Minimum relative error approximations for $ 1 / t $", journal = j-NUM-MATH, volume = "54", number = "2", pages = "117--124", month = nov, year = "1988", CODEN = "NUMMA7", DOI = "https://doi.org/10.1007/BF01396969", ISSN = "0029-599X (print), 0945-3245 (electronic)", ISSN-L = "0029-599X", MRclass = "65D15", MRnumber = "90a:65037", MRreviewer = "Mariano Gasca", bibdate = "Mon May 26 11:49:34 MDT 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/nummath.bib", acknowledgement = ack-nhfb, classification = "B0290F (Interpolation and function approximation); C4130 (Interpolation and function approximation)", corpsource = "Corp. Res., Digital Equipment Corp., Hudson, MA, USA", fjournal = "Numerische Mathematik", journal-URL = "http://link.springer.com/journal/211", keywords = "approximation theory; functional equation; geometric convergence rates; iterative methods; minimum relative error approximations; polynomial approximations; polynomials", treatment = "T Theoretical or Mathematical", } @Article{Bailey:1988:CDD, author = "David H. Bailey", title = "The computation of $ \pi $ to $ 29, 360, 000 $ decimal digits using {Borweins}' quartically convergent algorithm", journal = j-MATH-COMPUT, volume = "50", number = "181", pages = "283--296", month = jan, year = "1988", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "11Y60 (11-04 11K16 65-04)", MRnumber = "88m:11114", MRreviewer = "A. J. van der Poorten", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, classcodes = "C1140Z (Other and miscellaneous); C1160 (Combinatorial mathematics); C4130 (Interpolation and function approximation); C5470 (Performance evaluation and testing); C7310 (Mathematics)", corpsource = "NASA Ames Res. Centre, Moffet Field, CA, USA", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "Borwein quartically convergent algorithm; computation of pi; computer testing; Cray 2 computer test; decimal expansion; elliptic integrals; iterative methods; mathematics computing; multiprecision arithmetic; number theory; prime modulus; series (mathematics); statistical analyses; statistical analysis; transform", treatment = "X Experimental", } @Article{Bloom:1988:LCL, author = "Thomas Bloom", title = "The {Lebesgue} constant for {Lagrange} interpolation in the simplex", journal = j-J-APPROX-THEORY, volume = "54", number = "3", pages = "338--353", month = sep, year = "1988", CODEN = "JAXTAZ", ISSN = "0021-9045 (print), 1096-0430 (electronic)", ISSN-L = "0021-9045", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, bibno = "56887", catcode = "G.1.1; G.1.2; G.1.2; G.1.7; G.1.7", CRclass = "G.1.1 Interpolation; G.1.1 Interpolation formulas; G.1.2 Approximation; G.1.2 Minimax approximation and algorithms; G.1.2 Approximation; G.1.2 Elementary function approximation; G.1.7 Ordinary Differential Equations; G.1.7 Convergence and stability; G.1.7 Ordinary Differential Equations; G.1.7 Boundary value problems", descriptor = "Mathematics of Computing, NUMERICAL ANALYSIS, Interpolation, Interpolation formulas; Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Minimax approximation and algorithms; Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation; Mathematics of Computing, NUMERICAL ANALYSIS, Ordinary Differential Equations, Convergence and stability; Mathematics of Computing, NUMERICAL ANALYSIS, Ordinary Differential Equations, Boundary value problems", fjournal = "Journal of Approximation Theory", guideno = "1988-10170", journal-URL = "http://www.sciencedirect.com/science/journal/00219045", journalabbrev = "J. Approx. Theory", jrldate = "Sept. 1988", subject = "G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS", } @Article{Borwein:1988:CFF, author = "J. M. Borwein and P. B. Borwein", title = "On the Complexity of Familiar Functions and Numbers", journal = j-SIAM-REVIEW, volume = "30", number = "4", pages = "589--601", month = dec, year = "1988", CODEN = "SIREAD", DOI = "https://doi.org/10.1137/1030134", ISSN = "0036-1445 (print), 1095-7200 (electronic)", ISSN-L = "0036-1445", MRclass = "68Q25 (03D15 11Y16)", MRnumber = "967961; 89k:68061", MRreviewer = "Klaus W. Wagner", bibdate = "Sat Mar 29 09:54:29 MDT 2014", bibsource = "ACM Computing Archive CD-ROM database (1991); Compendex database; http://epubs.siam.org/toc/siread/30/4; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/siamreview.bib", abstract = "This paper examines low-complexity approximations to familiar functions and numbers. The intent is to suggest that it is possible to base a taxonomy of such functions and numbers on their computational complexity. A central theme is that traditional methods of approximation are often very far from optimal, while good or optimal methods are often very far from obvious. For most functions, provably optimal methods are not known; however the gap between what is known and what is possible is often small. A considerable number of open problems are posed and a number of related examples are presented.", acknowledgement = ack-nhfb, affiliationaddress = "Halifax, NS, Can", bibno = "58008", catcode = "G.1.2; F.2.1; F.1.3", classification = "921", CRclass = "G.1.2 Approximation; G.1.2 Elementary function approximation; F.2.1 Numerical Algorithms and Problems; F.1.3 Complexity Classes", descriptor = "Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation; Theory of Computation, ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY, Numerical Algorithms and Problems; Theory of Computation, COMPUTATION BY ABSTRACT DEVICES, Complexity Classes", fjournal = "SIAM Review", genterm = "algorithms; theory; performance", guideno = "1988-13907", journal-URL = "http://epubs.siam.org/sirev", journalabbrev = "SIAM Rev.", journalabr = "SIAM Rev", jrldate = "Dec. 1988", keywords = "Algebraic Approximation; Approximation Theory; Computation of Digits; Familiar Functions; Low Complexity Approximation; Mathematical Techniques; Rational Approximation; Reduced Complexity Approximation", onlinedate = "December 1988", subject = "G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; F. Theory of Computation; F.2 ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY; F. Theory of Computation; F.1 COMPUTATION BY ABSTRACT DEVICES", } @Article{Borwein:1988:PAE, author = "Peter B. Borwein", title = "{Pad{\'e}} approximants for the $q$-elementary functions", journal = j-CONST-APPROX, volume = "4", number = "4", pages = "391--402", year = "1988", ISSN = "0176-4276 (print), 1432-0940 (electronic)", ISSN-L = "0176-4276", MRclass = "41A21 (33A10 41A20)", MRnumber = "89f:41022", MRreviewer = "Annie A. M. Cuyt", bibdate = "Sat Jan 11 17:44:01 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Constructive Approximation", journal-URL = "http://link.springer.com/journal/365", } @Article{Brezinski:1988:BRS, author = "C. Brezinski", title = "Book Review: {{\booktitle{Special functions of mathematical physics}}: A. F. Nikiforov and V. B. Uvarov, Birkh{\"a}user, Basel, 1988, xviii + 427 pages, Sfr 168.00, ISBN 3-7643-3183-6}", journal = j-MATH-COMPUT-SIMUL, volume = "30", number = "4", pages = "376--376", month = sep, year = "1988", CODEN = "MCSIDR", DOI = "https://doi.org/10.1016/S0378-4754(98)90019-2", ISSN = "0378-4754 (print), 1872-7166 (electronic)", ISSN-L = "0378-4754", bibdate = "Mon Aug 18 16:03:56 MDT 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcomputsimul1980.bib", URL = "https://www.sciencedirect.com/science/article/pii/S0378475498900192", acknowledgement = ack-nhfb, ajournal = "Math. Comput. Simul.", fjournal = "Mathematics and Computers in Simulation", journal-URL = "https://www.sciencedirect.com/science/journal/03784754", } @InCollection{Brezinski:1988:NAC, author = "Claude Brezinski", booktitle = "{Nonlinear numerical methods and rational approximation (Wilrijk, 1987)}", title = "A new approach to convergence acceleration methods", volume = "43", publisher = "Reidel", address = "Dordrecht", pages = "373--405", year = "1988", MRclass = "65Bxx (40A25)", MRnumber = "1005369 (90m:65010)", MRreviewer = "John P. Coleman", bibdate = "Thu Dec 1 17:36:45 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", series = "Math. Appl.", acknowledgement = ack-nhfb, keywords = "convergence acceleration", } @Article{Carlson:1988:TEI, author = "B. C. Carlson", title = "A Table of Elliptic Integrals of the Third Kind", journal = j-MATH-COMPUT, volume = "51", number = "183", pages = "267--280, S1--S5", month = jul, year = "1988", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "33A25 (65A05)", MRnumber = "89k:33003", MRreviewer = "F. W. J. Olver", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, classcodes = "B0290R (Integral equations); B0220 (Analysis); B0290D (Functional analysis); B0290M (Numerical integration and differentiation); C4180 (Integral equations); C1120 (Analysis); C4120 (Functional analysis); C4160 (Numerical integration and differentiation)", corpsource = "Iowa State Univ., Ames, IA, USA", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "Cauchy principal values; elliptic integrals; FORTRAN listings; function evaluation; integral equations; integration; points; real singular; recurrence relations; standard R-functions", treatment = "P Practical; T Theoretical or Mathematical", } @Article{Cartwright:1988:JTC, author = "Donald I. Cartwright and Krzysztof Kucharski", title = "{Jackson's Theorem} for compact connect {Lie} groups", journal = j-J-APPROX-THEORY, volume = "55", number = "3", pages = "352--359", month = dec, year = "1988", CODEN = "JAXTAZ", ISSN = "0021-9045 (print), 1096-0430 (electronic)", ISSN-L = "0021-9045", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, bibno = "56056", catcode = "G.2.m; G.1.2", CRclass = "G.2.m Miscellaneous; G.1.2 Approximation; G.1.2 Elementary function approximation", descriptor = "Mathematics of Computing, DISCRETE MATHEMATICS, Miscellaneous; Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation", fjournal = "Journal of Approximation Theory", genterm = "verification; theory", guideno = "1988-10202", journal-URL = "http://www.sciencedirect.com/science/journal/00219045", journalabbrev = "J. Approx. Theory", jrldate = "Dec. 1988", subject = "G. Mathematics of Computing; G.2 DISCRETE MATHEMATICS; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS", } @Article{Chellali:1988:ACN, author = "Mustapha Chellali", title = "Acc{\'e}l{\'e}ration de calcul de nombres de {Bernoulli}. ({French}) [{Bernoulli} number calculation acceleration]", journal = j-J-NUMBER-THEORY, volume = "28", number = "3", pages = "347--362", month = mar, year = "1988", CODEN = "JNUTA9", DOI = "https://doi.org/10.1016/0022-314X(88)90047-9", ISSN = "0022-314X (print), 1096-1658 (electronic)", ISSN-L = "0022-314X", bibdate = "Wed Jul 15 08:46:57 MDT 2020", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jnumbertheory1980.bib", URL = "http://www.sciencedirect.com/science/article/pii/0022314X88900479", acknowledgement = ack-nhfb, ajournal = "J. Number Theory", fjournal = "Journal of Number Theory", journal-URL = "http://www.sciencedirect.com/science/journal/0022314X", language = "French", } @Article{Chiccoli:1988:EGE, author = "C. Chiccoli and S. Lorenzutta and G. Maino", title = "On the evaluation of generalized exponential integrals {$ E_\nu (x) $}", journal = j-J-COMPUT-PHYS, volume = "78", number = "2", pages = "278--287", month = oct, year = "1988", CODEN = "JCTPAH", DOI = "https://doi.org/10.1016/0021-9991(88)90050-2", ISSN = "0021-9991 (print), 1090-2716 (electronic)", ISSN-L = "0021-9991", bibdate = "Sun Jan 1 15:59:42 MST 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputphys1980.bib", URL = "http://www.sciencedirect.com/science/article/pii/0021999188900502", acknowledgement = ack-nhfb, fjournal = "Journal of Computational Physics", journal-URL = "http://www.sciencedirect.com/science/journal/00219991/", } @Article{Cover:1988:DII, author = "Thomas M. Cover and Joy A. Thomas", title = "Determinant inequalities via information theory", journal = j-SIAM-J-MAT-ANA-APPL, volume = "9", number = "3", pages = "384--392", month = jul, year = "1988", CODEN = "SJMAEL", ISSN = "0895-4798 (print), 1095-7162 (electronic)", ISSN-L = "0895-4798", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, bibno = "58040", catcode = "H.1.1; G.1.3; F.2.1; G.1.2", CRclass = "H.1.1 Systems and Information Theory; H.1.1 Information theory; G.1.3 Numerical Linear Algebra; G.1.3 Determinants; F.2.1 Numerical Algorithms and Problems; F.2.1 Computations on matrices; G.1.2 Approximation; G.1.2 Elementary function approximation", descriptor = "Information Systems, MODELS AND PRINCIPLES, Systems and Information Theory, Information theory; Mathematics of Computing, NUMERICAL ANALYSIS, Numerical Linear Algebra, Determinants; Theory of Computation, ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY, Numerical Algorithms and Problems, Computations on matrices; Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation", fjournal = "SIAM Journal on Matrix Analysis and Applications", genterm = "algorithms; theory; performance", guideno = "1988-13777", journal-URL = "http://epubs.siam.org/simax", journalabbrev = "SIAM J. Matrix Anal. Appl.", jrldate = "July 1988", subject = "H. Information Systems; H.1 MODELS AND PRINCIPLES; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; F. Theory of Computation; F.2 ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS", } @InProceedings{Davenport:1988:ICF, author = "J. H. Davenport", editor = "N. M. Stephens and M. P. Thorne", booktitle = "Computers in mathematical research: based on the proceedings of a conference organized by the Institute of Mathematics and its Applications on computers in mathematical research, held at University College, Cardiff in September 1986", title = "Integration in closed form", volume = "14", publisher = pub-CLARENDON, address = pub-CLARENDON:adr, bookpages = "235", year = "1988", ISBN = "0-19-853620-8", ISBN-13 = "978-0-19-853620-8", LCCN = "QA11.A1 C618 1986", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", price = "US\$57.50", series = "Institute of Mathematics and its applications conference series, new series", acknowledgement = ack-nhfb, affiliation = "Univ. of Bath", bibno = "52474", catcode = "J.2; F.2.1; G.1.2; G.1.2; I.2.9; F.2.1", CRclass = "J.2 Mathematics and statistics; F.2.1 Numerical Algorithms and Problems; F.2.1 Number-theoretic computations; G.1.2 Approximation; G.1.2 Rational approximation; G.1.2 Approximation; G.1.2 Elementary function approximation; I.2.9 Robotics; F.2.1 Numerical Algorithms and Problems; F.2.1 Computations in finite fields", descriptor = "Computer Applications, PHYSICAL SCIENCES AND ENGINEERING, Mathematics and statistics; Theory of Computation, ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY, Numerical Algorithms and Problems, Number-theoretic computations; Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Rational approximation; Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation; Computing Methodologies, ARTIFICIAL INTELLIGENCE, Robotics; Theory of Computation, ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY, Numerical Algorithms and Problems, Computations in finite fields", genterm = "algorithms; theory", guideno = "1988-16247", page = "119--134", procdate = "Sept. 1986", procloc = "Cardiff, Wales", subject = "J. Computer Applications; J.2 PHYSICAL SCIENCES AND ENGINEERING; F. Theory of Computation; F.2 ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; I. Computing Methodologies; I.2 ARTIFICIAL INTELLIGENCE; F. Theory of Computation; F.2 ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY", waffil = "Univ. College, Cardiff, Wales; Univ. College, Cardiff, Wales", } @TechReport{DiDonato:1988:SDC, author = "Armido I. DiDonato and Alfred H. {Morris, Jr.}", title = "Significant Digit Computation of the Incomplete Beta Function Ratios", type = "Technical Report", number = "NSWC TR 88-365", institution = "Naval Surface Warfare Center (K33)", address = "Dahlgren, VA 22448-5000, USA", month = nov, year = "1988", bibdate = "Sat Nov 15 10:30:20 2014", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.dtic.mil/dtic/tr/fulltext/u2/a210118.pdf", abstract = "An algorithm is given for evaluating the incomplete beta function ratio $ I_x(a, b) $ and its complement $ 1 - I_x(a, b) $. Two new procedures are used with classical results. A listing of a transportable Fortran subroutine using this algorithm is given. The subroutine is accurate to 14 significant digits when the precision is not restricted by inherent error.", acknowledgement = ack-nhfb, keywords = "bratio; continued fraction; expm1; gamma; incomplete gamma function; ln; log1p; r1mach; spmpar", } @Article{Dunham:1988:PMA, author = "Charles B. Dunham", title = "Provably monotone approximations, {III}", journal = j-SIGNUM, volume = "23", number = "1", pages = "10--10", month = jan, year = "1988", CODEN = "SNEWD6", DOI = "https://doi.org/10.1145/43931.43934", ISSN = "0163-5778 (print), 1558-0237 (electronic)", ISSN-L = "0163-5778", bibdate = "Tue Apr 12 07:50:16 MDT 2005", bibsource = "http://portal.acm.org/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "ACM SIGNUM Newsletter", journal-URL = "https://dl.acm.org/loi/signum", keywords = "theory", subject = "G.1.2 Mathematics of Computing, NUMERICAL ANALYSIS, Approximation", } @TechReport{Duprat:1988:EPE, author = "J. Duprat and J. M. Muller", title = "Evaluation of Polynomials and elementary Functions by integrated Circuits", number = "RR698-I", institution = "IMAG", address = "Grenoble, France", month = jan, year = "1988", bibdate = "Mon Oct 24 11:37:20 2011", bibsource = "ftp://ftp.ira.uka.de/pub/bibliography/Theory/eureca.bib.gz; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, } @Article{Duprat:1988:HPE, author = "Jean Duprat and Jean-Michel Muller", title = "Hardwired polynomial evaluation", journal = j-J-PAR-DIST-COMP, volume = "5", number = "3", pages = "291--309", month = jun, year = "1988", CODEN = "JPDCER", ISSN = "0743-7315 (print), 1096-0848 (electronic)", ISSN-L = "0743-7315", bibdate = "Sat Apr 12 19:06:31 MDT 1997", bibsource = "Compendex database; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, affiliationaddress = "CNRS, Grenoble, Fr", classification = "721; 722; 723; 921; C4130 (Interpolation and function approximation); C5230 (Digital arithmetic methods)", corpsource = "Inst. Nat. Polytech. de Grenoble, France", fjournal = "Journal of Parallel and Distributed Computing", journal-URL = "http://www.sciencedirect.com/science/journal/07437315", journalabr = "J Parallel Distrib Comput", keywords = "computer architecture; computers, digital --- Circuits; digital arithmetic; elementary functions; hardwired polynomial evaluation; mathematical functions; mathematical techniques; Polynomials; polynomials; special-purpose circuits; VLSI implementation", treatment = "P Practical", } @Article{Feng:1988:AIN, author = "Y. Y. Feng and J. Kozak", title = "An approach to the interpolation of nonuniformly spaced data", journal = j-J-COMPUT-APPL-MATH, volume = "23", number = "2", pages = "169--178", month = aug, year = "1988", CODEN = "JCAMDI", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, bibno = "56475", catcode = "G.1.1; G.1.2; G.1.7; G.1.1; G.1.4", CRclass = "G.1.1 Interpolation; G.1.1 Spline and piecewise polynomial interpolation; G.1.2 Approximation; G.1.2 Elementary function approximation; G.1.7 Ordinary Differential Equations; G.1.7 Boundary value problems; G.1.1 Interpolation; G.1.1 Smoothing; G.1.4 Quadrature and Numerical Differentiation; G.1.4 Error analysis", descriptor = "Mathematics of Computing, NUMERICAL ANALYSIS, Interpolation, Spline and piecewise polynomial interpolation; Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation; Mathematics of Computing, NUMERICAL ANALYSIS, Ordinary Differential Equations, Boundary value problems; Mathematics of Computing, NUMERICAL ANALYSIS, Interpolation, Smoothing; Mathematics of Computing, NUMERICAL ANALYSIS, Quadrature and Numerical Differentiation, Error analysis", fjournal = "Journal of Computational and Applied Mathematics", genterm = "algorithms; theory", guideno = "1988-10490", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", journalabbrev = "J. Comput. Appl. Math.", jrldate = "August 1988", subject = "G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS", } @Article{Gawronski:1988:ZLT, author = "Wolfgang Gawronski and Ulrich Stadtm{\"u}ller", title = "On the zeros of {Lerch}'s transcendental function of real parameters", journal = j-J-APPROX-THEORY, volume = "53", number = "3", pages = "354--364", month = jun, year = "1988", CODEN = "JAXTAZ", ISSN = "0021-9045 (print), 1096-0430 (electronic)", ISSN-L = "0021-9045", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, affiliation = "Univ. of Trier, Trier, FRG; Univ. of Ulm, Ulm, FRG", bibno = "49727", catcode = "G.1.2", CRclass = "G.1.2 Approximation; G.1.2 Elementary function approximation", descriptor = "Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation", fjournal = "Journal of Approximation Theory", genterm = "verification; theory", guideno = "1988-10146", journal-URL = "http://www.sciencedirect.com/science/journal/00219045", journalabbrev = "J. Approx. Theory", jrldate = "June 1988", subject = "G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS", } @Article{Gillman:1988:ARG, author = "E. Gillman and H. R. Fiebig", title = "Accurate recursive generation of spherical {Bessel} and {Neumann} functions for a large range of indices", journal = j-COMPUT-PHYS, volume = "2", number = "1", pages = "62--??", month = jan, year = "1988", CODEN = "CPHYE2", DOI = "https://doi.org/10.1063/1.168296", ISSN = "0894-1866 (print), 1558-4208 (electronic)", ISSN-L = "0894-1866", bibdate = "Wed Apr 10 08:45:10 MDT 2019", bibsource = "https://www.math.utah.edu/pub/tex/bib/computphys.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "https://aip.scitation.org/doi/10.1063/1.168296", acknowledgement = ack-nhfb, ajournal = "Comput. Phys", fjournal = "Computers in Physics", journal-URL = "https://aip.scitation.org/journal/cip", } @Article{Guerrero:1988:HOT, author = "Antonio L. Guerrero and Pablo Martin", title = "Higher order two-point quasi-fractional approximations to the {Bessel} functions {$ J_0 (x) $} and {$ J_1 (x) $}", journal = j-J-COMPUT-PHYS, volume = "77", number = "1", pages = "276--281", month = jul, year = "1988", CODEN = "JCTPAH", DOI = "https://doi.org/10.1016/0021-9991(88)90168-4", ISSN = "0021-9991 (print), 1090-2716 (electronic)", ISSN-L = "0021-9991", bibdate = "Sun Jan 1 15:59:41 MST 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputphys1980.bib", URL = "http://www.sciencedirect.com/science/article/pii/0021999188901684", acknowledgement = ack-nhfb, fjournal = "Journal of Computational Physics", journal-URL = "http://www.sciencedirect.com/science/journal/00219991/", remark = "This work produces only 3D approximations.", } @Article{Hautot:1988:CAC, author = "A. Hautot", title = "Convergence acceleration of continued fractions of {Poincar{\'e}} type", journal = j-APPL-NUM-MATH, volume = "4", number = "2--4", pages = "309--322", month = jun, year = "1988", CODEN = "ANMAEL", ISSN = "0168-9274 (print), 1873-5460 (electronic)", ISSN-L = "0168-9274", MRclass = "65B05 (40A15)", MRnumber = "90b:65005", MRreviewer = "Gh. Adam", bibdate = "Sat Feb 8 10:09:54 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/applnummath.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Applied Numerical Mathematics: Transactions of IMACS", journal-URL = "http://www.sciencedirect.com/science/journal/01689274/", keywords = "convergence acceleration", } @Article{Heilmann:1988:SSM, author = "Margareta Heilmann", title = "{$ L_p $}-saturation of some modified {Bernstein} operators", journal = j-J-APPROX-THEORY, volume = "54", number = "3", pages = "260--273", month = sep, year = "1988", CODEN = "JAXTAZ", ISSN = "0021-9045 (print), 1096-0430 (electronic)", ISSN-L = "0021-9045", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, bibno = "56880", catcode = "G.1.2; G.1.9; G.1.7; G.1.7", CRclass = "G.1.2 Approximation; G.1.2 Elementary function approximation; G.1.9 Integral Equations; G.1.9 Integro-differential equations; G.1.7 Ordinary Differential Equations; G.1.7 Boundary value problems; G.1.7 Ordinary Differential Equations; G.1.7 Convergence and stability", descriptor = "Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation; Mathematics of Computing, NUMERICAL ANALYSIS, Integral Equations, Integro-differential equations; Mathematics of Computing, NUMERICAL ANALYSIS, Ordinary Differential Equations, Boundary value problems; Mathematics of Computing, NUMERICAL ANALYSIS, Ordinary Differential Equations, Convergence and stability", fjournal = "Journal of Approximation Theory", genterm = "algorithms; theory", guideno = "1988-10163", journal-URL = "http://www.sciencedirect.com/science/journal/00219045", journalabbrev = "J. Approx. Theory", jrldate = "Sept. 1988", subject = "G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS", } @Article{Horwitz:1988:TPF, author = "Alan L. Horwitz and Lee A. Rubel", title = "Totally positive functions and totally bounded functions on $ [ - 1, 1] $", journal = j-J-APPROX-THEORY, volume = "52", number = "2", pages = "204--216", month = feb, year = "1988", CODEN = "JAXTAZ", ISSN = "0021-9045 (print), 1096-0430 (electronic)", ISSN-L = "0021-9045", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, affiliation = "Pennsylvania State Univ., University Park; Univ. of Illinois, Urbana", bibno = "49653", catcode = "G.1.2", CRclass = "G.1.2 Approximation; G.1.2 Elementary function approximation", descriptor = "Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation", fjournal = "Journal of Approximation Theory", genterm = "verification; theory", guideno = "1988-10106", journal-URL = "http://www.sciencedirect.com/science/journal/00219045", journalabbrev = "J. Approx. Theory", jrldate = "February 1988", subject = "G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS", } @Article{Ifantis:1988:BFP, author = "E. K. Ifantis and P. D. Siafarikas", title = "Bounds for the first positive zero of a mixed {Bessel} function", journal = j-J-COMPUT-APPL-MATH, volume = "21", number = "2", pages = "245--249", month = feb, year = "1988", CODEN = "JCAMDI", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Sat Feb 25 12:20:38 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath1980.bib", URL = "http://www.sciencedirect.com/science/article/pii/0377042788902737", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @Article{Jacobsen:1988:CAL, author = "Lisa Jacobsen and Haakon Waadeland", title = "Convergence acceleration of limit periodic continued fractions under asymptotic side conditions", journal = j-NUM-MATH, volume = "53", number = "3", pages = "285--298", month = jul, year = "1988", CODEN = "NUMMA7", ISSN = "0029-599X (print), 0945-3245 (electronic)", ISSN-L = "0029-599X", MRclass = "65B99 (30B70 40A15)", MRnumber = "89h:65010", MRreviewer = "Claude Brezinski", bibdate = "Mon May 26 11:49:34 MDT 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, classification = "B0290 (Numerical analysis); C4100 (Numerical analysis)", corpsource = "Dept. of Math. and Stat., Trondheim Univ., Dragvoll, Norway", fjournal = "Numerische Mathematik", journal-URL = "http://link.springer.com/journal/211", keywords = "asymptotic side conditions; convergence acceleration; convergence of numerical methods; hypergeometric functions; limit periodic continued fractions; regular C-fraction expansions", treatment = "T Theoretical or Mathematical", } @InProceedings{Johnsson:1988:DPP, author = "S. L. Johnsson", editor = "J. R. Rice", booktitle = "Mathematical aspects of scientific software", title = "Data parallel programming and basic linear algebra subroutines", volume = "14", publisher = pub-SV, address = pub-SV:adr, bookpages = "vi + 208", pages = "183--196", year = "1988", ISBN = "0-387-96706-0", ISBN-13 = "978-0-387-96706-6", LCCN = "QA76.76.D47 M366 1988", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", series = "The IMS volumes in mathematics and its applications", acknowledgement = ack-nhfb, bibno = "42725", catcode = "G.1.2; D.1.m; G.4", CRclass = "G.1.2 Approximation; G.1.2 Elementary function approximation; D.1.m Miscellaneous", descriptor = "Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation; Software, PROGRAMMING TECHNIQUES, Miscellaneous; Mathematics of Computing, MATHEMATICAL SOFTWARE", genterm = "theory; algorithms; design; languages", guideno = "1988-02324", subject = "G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; D. Software; D.1 PROGRAMMING TECHNIQUES; G. Mathematics of Computing; G.4 MATHEMATICAL SOFTWARE", waffil = "Purdue Univ., West Lafayette, IN", } @Article{Kirby:1988:ELA, author = "James C. Kirby", title = "An Efficient Logarithm Algorithm for Calculators", journal = j-COLLEGE-MATH-J, volume = "19", number = "3", pages = "257--260", month = may, year = "1988", CODEN = "????", DOI = "https://doi.org/10.1080/07468342.1988.11973125", ISSN = "0746-8342 (print), 1931-1346 (electronic)", ISSN-L = "0746-8342", bibdate = "Thu Feb 14 09:50:43 MST 2019", bibsource = "https://www.math.utah.edu/pub/tex/bib/collegemathj.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.tandfonline.com/doi/abs/10.1080/07468342.1988.11973125", acknowledgement = ack-nhfb, fjournal = "College Mathematics Journal", journal-URL = "https://maa.tandfonline.com/loi/ucmj20; https://www.jstor.org/journal/collmathj", onlinedate = "30 Jan 2018", } @Article{Kowalski:1988:ASP, author = "Marek Kowalski and Waldemar Sielski", title = "Approximation of smooth periodic functions in several variables", journal = j-J-COMPLEXITY, volume = "4", number = "4", pages = "356--372", month = dec, year = "1988", CODEN = "JOCOEH", ISSN = "0885-064X (print), 1090-2708 (electronic)", ISSN-L = "0885-064X", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, bibno = "56811", catcode = "G.1.2", CRclass = "G.1.2 Approximation; G.1.2 Elementary function approximation", descriptor = "Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation", fjournal = "Journal of complexity", genterm = "algorithms; verification; theory", guideno = "1988-10411", journal-URL = "http://www.sciencedirect.com/science/journal/0885064X", journalabbrev = "J. Complexity", jrldate = "Dec. 1988", subject = "G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS", } @Article{Kramer:1988:ISF, author = "W. Kr{\"a}mer", title = "Inverse standard functions for real and complex point and interval arguments with dynamic accuracy", journal = j-COMPUTING-SUPPLEMENTUM, pages = "185--212", year = "1988", CODEN = "COSPDM", ISSN = "0344-8029", bibdate = "Thu Dec 14 17:19:38 MST 1995", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "Computer Arithmetic and Scientific Computation.", abstract = "Algorithms to compute inverse standard functions to arbitrary accuracy with safe error bounds are given. Not only approximation errors but also all possible rounding errors are considered. The desired accuracy of the function as well as the base of the number system used are parameters of the error formula. For implementation it is only assumed that the four elementary arithmetic operations are performed with a certain number of correct digits of the function value. The interval routines are constructed out of the point routines considering the monotonicity behaviour of the functions. The ambiguity of the complex functions is briefly discussed.", acknowledgement = ack-nhfb, affiliation = "Karlsruhe Univ., West Germany", author-dates = "1952--2014 (WK)", classification = "B0290B (Error analysis in numerical methods); B0290F (Interpolation and function approximation); C4110 (Error analysis in numerical methods); C4130 (Interpolation and function approximation)", confdate = "30 Sept.-2 Oct. 1987", conflocation = "Karlsruhe, West Germany", confsponsor = "Karlsruhe Univ.; GAMM Committee", fjournal = "Computing. Supplementum", issue = "no.6 p. 185-212", keywords = "Approximation errors; Dynamic accuracy; Error formula; Interval arguments; Inverse standard functions; Monotonicity behaviour; Rounding errors", pubcountry = "Austria", thesaurus = "Error analysis; Function approximation", } @Article{Laforgia:1988:MRI, author = "Andrea Laforgia and Silvana Sismondi", title = "Monotonicity results and inequalities for the gamma and error functions", journal = j-J-COMPUT-APPL-MATH, volume = "23", number = "1", pages = "25--33", month = jul, year = "1988", CODEN = "JCAMDI", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Sat Feb 25 12:20:39 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath1980.bib", URL = "http://www.sciencedirect.com/science/article/pii/0377042788903287", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @Article{Lembarki:1988:CAL, author = "Alami Lembarki", title = "Convergence acceleration of limit $k$-periodic continued fractions", journal = j-APPL-NUM-MATH, volume = "4", number = "2--4", pages = "337--349", month = jun, year = "1988", CODEN = "ANMAEL", ISSN = "0168-9274 (print), 1873-5460 (electronic)", ISSN-L = "0168-9274", MRclass = "65B05 (40A15)", MRnumber = "89j:65012", MRreviewer = "Thomas A. Atchison", bibdate = "Sat Feb 8 10:09:54 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/applnummath.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Applied Numerical Mathematics: Transactions of IMACS", journal-URL = "http://www.sciencedirect.com/science/journal/01689274/", keywords = "convergence acceleration", } @InCollection{Levrie:1988:CAM, author = "Paul Levrie and Robert Piessens", booktitle = "{Nonlinear numerical methods and rational approximation (Wilrijk, 1987)}", title = "Convergence acceleration for {Miller}'s algorithm", volume = "43", publisher = "Reidel", address = "Dordrecht, The Netherlands", pages = "349--370", year = "1988", MRclass = "65B10 (26C15 39A10)", MRnumber = "1005368 (90m:65012)", MRreviewer = "Pierre Hillion", bibdate = "Thu Dec 1 17:36:45 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", series = "Math. Appl.", acknowledgement = ack-nhfb, keywords = "convergence acceleration", } @Article{Lou:1988:ETR, author = "Lou van den Dries", title = "On the elementary theory of restricted elementary functions", journal = j-J-SYMBOLIC-LOGIC, volume = "53", number = "3", pages = "796--808", year = "1988", CODEN = "JSYLA6", ISSN = "0022-4812 (print), 1943-5886 (electronic)", ISSN-L = "0022-4812", MRclass = "03C65 (03C40 03C68 12L12)", MRnumber = "89i:03074", MRreviewer = "M. Yasuhara", bibdate = "Sat Jan 11 17:44:01 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Journal of Symbolic Logic", journal-URL = "http://projecteuclid.org/euclid.jsl; http://www.jstor.org/journal/jsymboliclogic", } @PhdThesis{Marsaglia:1988:CES, author = "John Christopher Winston Marsaglia", title = "Computer Evaluation of the special functions of probability and statistics", type = "{Ph.D.} Dissertation", school = "Department of Computer Science, Washington State University", address = "Pullman, WA, USA", pages = "vii + 79", month = aug, year = "1988", bibdate = "Wed Jun 22 07:17:49 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, keywords = "$\exp(x)$; $\Phi(x)$; Bernoulli numbers; chi-square distribution; continued fraction; continuous Poisson distribution; Erlang distribution; exponential distribution; Gamma function; incomplete Gamma function; normal distribution; normal probability distribution; Poisson distribution; Stirling's approximation", } @Article{Milone:1988:EDF, author = "L. A. Milone and A. A. E. Milone", title = "Evaluation of {Dawson}'s function", journal = j-ASTROPHYS-SPACE-SCI, volume = "147", number = "1", pages = "189--191", year = "1988", CODEN = "APSSBE", DOI = "https://doi.org/10.1007/bf00656618", ISSN = "0004-640X (print), 1572-946X (electronic)", ISSN-L = "0004-640X", bibdate = "Sat Feb 17 11:54:06 2024", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, ajournal = "Astrophys. Space Sci.", fjournal = "Astrophysics and Space Science", journal-URL = "http://link.springer.com/journal/10509", } @Article{Mimachi:1988:PRI, author = "Katsuhisa Mimachi", title = "A proof of {Ramanujan}'s identity by use of loop integrals", journal = j-SIAM-J-MATH-ANA, volume = "19", number = "6", pages = "1490--1493", month = nov, year = "1988", CODEN = "SJMAAH", ISSN = "0036-1410 (print), 1095-7154 (electronic)", ISSN-L = "0036-1410", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, bibno = "58904", catcode = "G.1.9; G.1.2; F.2.2", CRclass = "G.1.9 Integral Equations; G.1.9 Integro-differential equations; G.1.2 Approximation; G.1.2 Elementary function approximation; F.2.2 Nonnumerical Algorithms and Problems; F.2.2 Geometrical problems and computations", descriptor = "Mathematics of Computing, NUMERICAL ANALYSIS, Integral Equations, Integro-differential equations; Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation; Theory of Computation, ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY, Nonnumerical Algorithms and Problems, Geometrical problems and computations", fjournal = "SIAM Journal on Mathematical Analysis", genterm = "algorithms; theory", guideno = "1988-13744", journal-URL = "http://epubs.siam.org/sima", journalabbrev = "SIAM J. Math. Anal.", jrldate = "Nov. 1988", subject = "G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; F. Theory of Computation; F.2 ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY", } @Article{Muroi:1988:ECR, author = "Kazuo Muroi", title = "Extraction of Cube Roots in {Babylonian} Mathematics", journal = j-CENTAURUS, volume = "31", number = "3", pages = "181--188", month = oct, year = "1988", CODEN = "CENTA4", DOI = "https://doi.org/10.1111/j.1600-0498.1988.tb00736.x", ISSN = "0008-8994 (print), 1600-0498 (electronic)", ISSN-L = "0008-8994", bibdate = "Sat Jul 27 18:43:36 MDT 2013", bibsource = "https://www.math.utah.edu/pub/tex/bib/centaurus.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Centaurus: An International Journal of the History of Science and its Cultural Aspects", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1111/(ISSN)1600-0498/", onlinedate = "26 Jul 2007", } @Book{Nikiforov:1988:SFM, author = "Arnol'd F. Nikiforov and Vasilij B. Uvarov", title = "Special functions of mathematical physics: a unified introduction with applications", publisher = pub-BIRKHAUSER, address = pub-BIRKHAUSER:adr, pages = "xviii + 427", year = "1988", ISBN = "3-7643-3183-6, 0-8176-3183-6", ISBN-13 = "978-3-7643-3183-2, 978-0-8176-3183-3", LCCN = "QC20.7.F87 N692e", bibdate = "Sat Oct 30 18:34:41 MDT 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; z3950.gbv.de:20011/gvk", note = "Translated from the Russian by Ralph P. Boas.", URL = "http://www.gbv.de/dms/hbz/toc/ht002696178", acknowledgement = ack-nhfb, } @Article{Olver:1988:EBL, author = "F. W. J. Olver", title = "Error Bounds for Linear Recurrence Relations", journal = j-MATH-COMPUT, volume = "50", number = "182", pages = "481--499", month = apr, year = "1988", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "65Q05 (39A10 65G05)", MRnumber = "89e:65146", MRreviewer = "B. Choczewski", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, classcodes = "B0290B (Error analysis in numerical methods); C4110 (Error analysis in numerical methods)", corpsource = "Maryland Univ., College Park, MD, USA", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "a posteriori methods; analysis; Bessel function; bounds; computational complexity; difference equations; error; homogeneous second order; Legendre function; linear recurrence relations; monotonic systems; numerical examples; O(r) arithmetic operations; oscillatory systems; realistic error; relations; rounded interval arithmetic", treatment = "T Theoretical or Mathematical", } @Article{Polyak:1988:SOM, author = "R. A. Polyak", title = "Smooth optimization methods for minimax problems", journal = j-SIAM-J-CONTROL-OPTIM, volume = "26", number = "6", pages = "1274--1286", month = nov, year = "1988", CODEN = "SJCODC", ISSN = "0363-0129 (print), 1095-7138 (electronic)", ISSN-L = "0363-0129", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, bibno = "57989", catcode = "G.1.6; G.1.2; F.2.1; G.1.2", CRclass = "G.1.6 Optimization; G.1.6 Nonlinear programming; G.1.2 Approximation; G.1.2 Minimax approximation and algorithms; F.2.1 Numerical Algorithms and Problems; F.2.1 Computation of transforms; G.1.2 Approximation; G.1.2 Elementary function approximation", descriptor = "Mathematics of Computing, NUMERICAL ANALYSIS, Optimization, Nonlinear programming; Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Minimax approximation and algorithms; Theory of Computation, ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY, Numerical Algorithms and Problems, Computation of transforms; Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation", fjournal = "SIAM Journal on Control and Optimization", genterm = "algorithms; theory; performance", guideno = "1988-13621", journal-URL = "http://epubs.siam.org/sicon", journalabbrev = "SIAM J. Control Optim.", jrldate = "Nov. 1988", subject = "G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; F. Theory of Computation; F.2 ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS", } @InCollection{Powell:1988:RBF, author = "J. D. Powell", editor = "D. F. (David Francis) Griffiths and G. A. Watson", booktitle = "Numerical analysis 1987", title = "Radial basis function approximations to polynomials", volume = "170", publisher = "Longman, Inc.", address = "New York, NY, USA", bookpages = "300", pages = "223--241", year = "1988", ISBN = "0-582-02157-X", ISBN-13 = "978-0-582-02157-0", LCCN = "QA297.N828 1988", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", price = "US\$54.95", series = "Pitman research notes in mathematics series", acknowledgement = ack-nhfb, bibno = "54996", catcode = "G.1.2; G.1.2; G.1.9; G.1.1; G.3; G.1.7", CRclass = "G.1.2 Approximation; G.1.2 Elementary function approximation; G.1.2 Approximation; G.1.2 Spline and piecewise polynomial approximation; G.1.9 Integral Equations; G.1.9 Integro-differential equations; G.1.1 Interpolation; G.1.1 Spline and piecewise polynomial interpolation; G.3 Statistical computing; G.1.7 Ordinary Differential Equations; G.1.7 Convergence and stability", descriptor = "Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation; Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Spline and piecewise polynomial approximation; Mathematics of Computing, NUMERICAL ANALYSIS, Integral Equations, Integro-differential equations; Mathematics of Computing, NUMERICAL ANALYSIS, Interpolation, Spline and piecewise polynomial interpolation; Mathematics of Computing, PROBABILITY AND STATISTICS, Statistical computing; Mathematics of Computing, NUMERICAL ANALYSIS, Ordinary Differential Equations, Convergence and stability", genterm = "algorithms; performance", guideno = "1988-01246", subject = "G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; G.3 PROBABILITY AND STATISTICS", waffil = "Univ. of Dundee; Univ. of Dundee", } @Article{Proinov:1988:ISF, author = "Petko D. Proinov", title = "Integration of smooth functions and $ \phi $-discrepancy", journal = j-J-APPROX-THEORY, volume = "52", number = "3", pages = "284--292", month = mar, year = "1988", CODEN = "JAXTAZ", ISSN = "0021-9045 (print), 1096-0430 (electronic)", ISSN-L = "0021-9045", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, affiliation = "Univ. of Plovdiv, Plovdiv, Bulgaria", bibno = "49659", catcode = "G.1.2", CRclass = "G.1.2 Approximation; G.1.2 Elementary function approximation", descriptor = "Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation", fjournal = "Journal of Approximation Theory", genterm = "verification; theory", guideno = "1988-10111", journal-URL = "http://www.sciencedirect.com/science/journal/00219045", journalabbrev = "J. Approx. Theory", jrldate = "March 1988", subject = "G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS", } @Article{Puoskari:1988:MCB, author = "M. Puoskari", title = "A method for computing {Bessel} function integrals", journal = j-J-COMPUT-PHYS, volume = "75", number = "2", pages = "334--344", month = apr, year = "1988", CODEN = "JCTPAH", DOI = "https://doi.org/10.1016/0021-9991(88)90116-7", ISSN = "0021-9991 (print), 1090-2716 (electronic)", ISSN-L = "0021-9991", bibdate = "Sun Jan 1 15:59:40 MST 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputphys1980.bib", URL = "http://www.sciencedirect.com/science/article/pii/0021999188901167", acknowledgement = ack-nhfb, fjournal = "Journal of Computational Physics", journal-URL = "http://www.sciencedirect.com/science/journal/00219991/", } @Article{Richardson:1988:NMT, author = "Daniel Richardson", title = "Nonstandard models of the theory of elementary functions of a real variable", journal = j-Z-MATH-LOGIK-GRUNDL-MATH, volume = "34", number = "4", pages = "355--372", year = "1988", CODEN = "ZMLGAQ", ISBN = "0044-3050", ISBN-13 = "0044-3050", ISSN = "0044-3050", MRclass = "03B30 (03C62 03H15 26E35)", MRnumber = "90a:03009", MRreviewer = "Reuven H. Gurevi{\v{c}}", bibdate = "Sat Jan 11 17:44:01 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "{Zeitschrift f{\"u}r mathematische Logik und Grundlagen der Mathematik}", } @Article{Ruscheweyh:1988:EST, author = "Stephan Ruscheweyh", title = "Extension of {Szeg{\H{o}}}'s theorem on the sections of univalent functions", journal = j-SIAM-J-MATH-ANA, volume = "19", number = "6", pages = "1442--1449", month = nov, year = "1988", CODEN = "SJMAAH", ISSN = "0036-1410 (print), 1095-7154 (electronic)", ISSN-L = "0036-1410", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, bibno = "58899", catcode = "G.1.2; G.1.9; F.2.2; F.2.2", CRclass = "G.1.2 Approximation; G.1.2 Elementary function approximation; G.1.9 Integral Equations; G.1.9 Integro-differential equations; F.2.2 Nonnumerical Algorithms and Problems; F.2.2 Computations on discrete structures; F.2.2 Nonnumerical Algorithms and Problems; F.2.2 Geometrical problems and computations", descriptor = "Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation; Mathematics of Computing, NUMERICAL ANALYSIS, Integral Equations, Integro-differential equations; Theory of Computation, ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY, Nonnumerical Algorithms and Problems, Computations on discrete structures; Theory of Computation, ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY, Nonnumerical Algorithms and Problems, Geometrical problems and computations", fjournal = "SIAM Journal on Mathematical Analysis", genterm = "algorithms; theory", guideno = "1988-13739", journal-URL = "http://epubs.siam.org/sima", journalabbrev = "SIAM J. Math. Anal.", jrldate = "Nov. 1988", subject = "G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; F. Theory of Computation; F.2 ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY; F. Theory of Computation; F.2 ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY", } @Article{Schappacher:1988:EIG, author = "Norbert Schappacher", title = "Elliptic integrals and the gamma function", journal = j-LECT-NOTES-MATH, volume = "1301", pages = "117--127", year = "1988", CODEN = "LNMAA2", DOI = "https://doi.org/10.1007/BFb0082098", ISBN = "3-540-18915-7 (print), 3-540-38842-7 (e-book)", ISBN-13 = "978-3-540-18915-2 (print), 978-3-540-38842-5 (e-book)", ISSN = "0075-8434 (print), 1617-9692 (electronic)", ISSN-L = "0075-8434", bibdate = "Fri May 9 19:07:24 MDT 2014", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/lnm1985.bib", URL = "http://link.springer.com/chapter/10.1007/BFb0082098/", acknowledgement = ack-nhfb, book-DOI = "https://doi.org/10.1007/BFb0082094", book-URL = "http://www.springerlink.com/content/978-3-540-38842-5", fjournal = "Lecture Notes in Mathematics", journal-URL = "http://link.springer.com/bookseries/304", } @InProceedings{Schwarz:1988:CLI, author = "Jerry Schwarz", title = "A {C++} Library for Infinite Precision Floating Point", crossref = "USENIX:1988:UPC", bookpages = "362", pages = "271--281", year = "1988", bibdate = "Tue Dec 12 09:20:21 MST 1995", bibsource = "ftp://ftp.uu.net/library/bibliography; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", abstract = "The Real library supports infinite precision floating point computation in C++. Arbitrary precision rational arithmetic and transcendental functions are supported.", acknowledgement = ack-nhfb, affiliation = "AT\&T Bell Laboratories, Murray Hill", classification = "C5230 (Digital arithmetic methods); C6130 (Data handling techniques)", confdate = "17--21 Oct. 1988", conflocation = "Denver, CO, USA", keywords = "C++ library; Infinite precision floating point; Rational arithmetic; Real library; Transcendental functions", pubcountry = "USA", thesaurus = "C language; Digital arithmetic; Object-oriented programming; Subroutines", } @InProceedings{Sobczyk:1988:SMA, author = "Kazimierz Sobczyk", editor = "W. Schiehlen and W. Wedig", booktitle = "Analysis and estimation of stochastic mechanical systems", title = "Stochastic modelling and analysis of fatigue", volume = "303", publisher = pub-SV, address = pub-SV:adr, bookpages = "350", pages = "269--313", year = "1988", ISBN = "0-387-82058-2", ISBN-13 = "978-0-387-82058-3", LCCN = "TA350.3.I5 no.303; TJ173 .A531 1988", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", series = "Courses and lectures", acknowledgement = ack-nhfb, bibno = "58064", catcode = "J.2; I.6.3; G.1.8; G.3; G.1.2; G.1.9", CRclass = "J.2 Engineering; I.6.3 Applications; G.1.8 Partial Differential Equations; G.1.8 Difference methods; G.3 Probabilistic algorithms (including Monte Carlo); G.1.2 Approximation; G.1.2 Elementary function approximation; G.1.9 Integral Equations; G.1.9 Integro-differential equations", descriptor = "Computer Applications, PHYSICAL SCIENCES AND ENGINEERING, Engineering; Computing Methodologies, SIMULATION AND MODELING, Applications; Mathematics of Computing, NUMERICAL ANALYSIS, Partial Differential Equations, Difference methods; Mathematics of Computing, PROBABILITY AND STATISTICS, Probabilistic algorithms (including Monte Carlo); Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation; Mathematics of Computing, NUMERICAL ANALYSIS, Integral Equations, Integro-differential equations", genterm = "algorithms; theory; design; measurement; reliability", guideno = "1988-17476", procdate = "1987", procloc = "International Centre for Mechanical Sciences in Udine", subject = "J. Computer Applications; J.2 PHYSICAL SCIENCES AND ENGINEERING; I. Computing Methodologies; I.6 SIMULATION AND MODELING; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; G. Mathematics of Computing; G.3 PROBABILITY AND STATISTICS; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS", waffil = "Univ. of Stuttgart; Univ. of Karlsruhe", } @Article{Stephens:1988:SCR, author = "A. J. Stephens and H. C. Williams", title = "Some computational results on a problem concerning powerful numbers", journal = j-MATH-COMPUT, volume = "50", number = "182", pages = "619--632", month = apr, year = "1988", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "11R11 (11A51 11R27 11Y16 11Y40)", MRnumber = "89d:11091", MRreviewer = "H. J. Godwin", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, classcodes = "B0210 (Algebra); B0250 (Combinatorial mathematics); C1110 (Algebra); C1160 (Combinatorial mathematics)", corpsource = "Dept. of Comput. Sci., Manitoba Univ., Winnipeg, Man., Canada", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "computational complexity; continued; fractions; free integer; fundamental unit; number theory; positive square-; real quadratic number fields; regulator algorithm Amdahl 5850; step algorithm; time complexity $O(D^{1/4+\epsilon})$", treatment = "T Theoretical or Mathematical; X Experimental", } @InProceedings{Stillinger:1988:CPS, author = "Frank H. Stillinger", editor = "Donald G. Truhlar", booktitle = "Mathematical frontiers in computational chemical physics", title = "Collective phenomena in statistical mechanics and the geometry of potential energy hypersurfaces", volume = "15", publisher = pub-SV, address = pub-SV:adr, bookpages = "xii + 349", pages = "157--173", year = "1988", ISBN = "0-387-96782-6", ISBN-13 = "978-0-387-96782-0", LCCN = "QD455.3.M3 M38 1988", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "Proceedings of the Workshop on Atomic and Molecular Structure and Dynamics, held June 15--July 24, 1987, at the Institute for Mathematics and Its Applications, University of Minnesota.", price = "US\$36.80", series = "The IMA volumes in mathematics and its applications", acknowledgement = ack-nhfb, bibno = "58051", catcode = "J.2; J.2; J.2; F.2.2; G.3; G.1.2", CRclass = "J.2 Physics; J.2 Chemistry; J.2 Engineering; F.2.2 Nonnumerical Algorithms and Problems; F.2.2 Geometrical problems and computations; G.3 Statistical computing; G.1.2 Approximation; G.1.2 Elementary function approximation", descriptor = "Computer Applications, PHYSICAL SCIENCES AND ENGINEERING, Physics; Computer Applications, PHYSICAL SCIENCES AND ENGINEERING, Chemistry; Computer Applications, PHYSICAL SCIENCES AND ENGINEERING, Engineering; Theory of Computation, ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY, Nonnumerical Algorithms and Problems, Geometrical problems and computations; Mathematics of Computing, PROBABILITY AND STATISTICS, Statistical computing; Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation", genterm = "algorithms; theory", guideno = "1988-17776", procdate = "1982", procloc = "Univ. of Minnesota, Minneapolis", subject = "J. Computer Applications; J.2 PHYSICAL SCIENCES AND ENGINEERING; J. Computer Applications; J.2 PHYSICAL SCIENCES AND ENGINEERING; J. Computer Applications; J.2 PHYSICAL SCIENCES AND ENGINEERING; F. Theory of Computation; F.2 ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY; G. Mathematics of Computing; G.3 PROBABILITY AND STATISTICS; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS", waffil = "Univ. of Minnesota, Minneapolis", } @Article{Sun:1988:SAF, author = "Xiehua Sun", title = "On the simultaneous approximation of functions and their derivatives by the {Szasz--Mirakyan} operator", journal = j-J-APPROX-THEORY, volume = "55", number = "3", pages = "279--288", month = dec, year = "1988", CODEN = "JAXTAZ", ISSN = "0021-9045 (print), 1096-0430 (electronic)", ISSN-L = "0021-9045", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, bibno = "56048", catcode = "G.1.2", CRclass = "G.1.2 Approximation; G.1.2 Elementary function approximation", descriptor = "Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation", fjournal = "Journal of Approximation Theory", genterm = "verification; theory", guideno = "1988-10194", journal-URL = "http://www.sciencedirect.com/science/journal/00219045", journalabbrev = "J. Approx. Theory", jrldate = "Dec. 1988", subject = "G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS", } @TechReport{Tang:1988:PIG, author = "Ping Tak Peter Tang", title = "Portable Implementation of a Generic Exponential Function", type = "Technical report", number = "ANL-88-3", institution = inst-ANL, address = inst-ANL:adr, year = "1988", bibdate = "Fri Dec 28 11:27:51 2007", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, } @Book{vanRijckevorsek:1988:CCA, editor = "Jan L. A. van Rijckevorsek and Jan de Leeus", title = "Component and correspondence analysis: dimension reduction by functional approximation", publisher = pub-WILEY, address = pub-WILEY:adr, pages = "xiii + 146", year = "1988", ISBN = "0-471-91847-4", ISBN-13 = "978-0-471-91847-9", LCCN = "QA278.5 .C6571 1988", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", price = "US\$40", series = "Wiley series in probability and mathematical statistics", acknowledgement = ack-nhfb, bibno = "59092", catcode = "G.1.2; G.3", CRclass = "G.1.2 Approximation; G.1.2 Elementary function approximation; G.3 Statistical computing", descriptor = "Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation; Mathematics of Computing, PROBABILITY AND STATISTICS, Statistical computing", genterm = "theory; algorithms", guideno = "1988-03042", subject = "G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; G. Mathematics of Computing; G.3 PROBABILITY AND STATISTICS", } @Book{Vilenkin:1988:SFT, author = "N. Ja. (Naum Jakovlevich) Vilenkin", title = "Special functions and the theory of group representations", volume = "22", publisher = pub-AMS, address = pub-AMS:adr, pages = "x + 613", year = "1988", ISBN = "0-8218-1572-5", ISBN-13 = "978-0-8218-1572-4", LCCN = "QA3 .A5 v.22 1988", bibdate = "Sat Oct 30 17:01:56 MDT 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; prodorbis.library.yale.edu:7090/voyager", note = "Reprint of 1968 edition.", series = "Translations of mathematical monographs", acknowledgement = ack-nhfb, subject = "Representations of groups; Functions, Special", } @Article{Wong:1988:AE, author = "R. Wong", title = "Asymptotic Expansion of $ \int^{\pi / 2}_0 {J}^2_\nu (\lambda \cos \theta) d \theta $", journal = j-MATH-COMPUT, volume = "50", number = "181", pages = "229--234", month = jan, year = "1988", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "41A60 (33A40)", MRnumber = "89g:41022", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, classcodes = "A0260 (Numerical approximation and analysis); A0270 (Computational techniques); C4130 (Interpolation and function approximation); C4160 (Numerical integration and differentiation)", corpsource = "Dept. of Appl. Math., Manitoba Univ., Winnipeg, Man., Canada", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "asymptotic expansion; Bessel function; Bessel functions; crystallography; diffraction theory; function approximation; integral; integration", treatment = "T Theoretical or Mathematical", } @InProceedings{Ahmed:1989:EEF, author = "H. M. Ahmed", title = "Efficient Elementary Function Generation with Multipliers", crossref = "Ercegovac:1989:PSC", pages = "52--59", year = "1989", bibdate = "Sat Nov 27 14:19:10 MST 2004", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb # " and " # ack-nj, } @Article{Armbruster:1989:KSD, author = "Dieter Armbruster and John Guckenheimer and Philip Holmes", title = "{Kuramoto--Sivashinsky} dynamics on the center-unstable manifold", journal = j-SIAM-J-APPL-MATH, volume = "49", number = "3", pages = "676--691", month = jun, year = "1989", CODEN = "SMJMAP", ISSN = "0036-1399 (print), 1095-712X (electronic)", ISSN-L = "0036-1399", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, affiliation = "Cornell Univ., Ithaca, NY; Cornell Univ., Ithaca, NY; Cornell Univ., Ithaca, NY", bibno = "64938", catcode = "G.1.7; G.1.7; G.1.0; J.2; G.1.2; G.1.8", CRclass = "G.1.7 Ordinary Differential Equations; G.1.7 Convergence and stability; G.1.7 Ordinary Differential Equations; G.1.7 Boundary value problems; G.1.0 General; G.1.0 Numerical algorithms; J.2 Physics; G.1.2 Approximation; G.1.2 Elementary function approximation; G.1.8 Partial Differential Equations", descriptor = "Mathematics of Computing, NUMERICAL ANALYSIS, Ordinary Differential Equations, Convergence and stability; Mathematics of Computing, NUMERICAL ANALYSIS, Ordinary Differential Equations, Boundary value problems; Mathematics of Computing, NUMERICAL ANALYSIS, General, Numerical algorithms; Computer Applications, PHYSICAL SCIENCES AND ENGINEERING, Physics; Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation; Mathematics of Computing, NUMERICAL ANALYSIS, Partial Differential Equations", fjournal = "SIAM Journal on Applied Mathematics", genterm = "algorithms; theory; experimentation", guideno = "1989-09707", journal-URL = "http://epubs.siam.org/siap", journalabbrev = "SIAM J. Appl. Math.", jrldate = "June 1989", subject = "G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; J. Computer Applications; J.2 PHYSICAL SCIENCES AND ENGINEERING; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS", } @Article{Avellaneda:1989:OBE, author = "Marco Avellaneda and Graeme W. Milton", title = "Optimal bounds on the effective bulk modulus of polycrystals", journal = j-SIAM-J-APPL-MATH, volume = "49", number = "3", pages = "824--837", month = jun, year = "1989", CODEN = "SMJMAP", ISSN = "0036-1399 (print), 1095-712X (electronic)", ISSN-L = "0036-1399", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, bibno = "64947", catcode = "G.1.8; J.2; G.1.2; G.1.6", CRclass = "G.1.8 Partial Differential Equations; G.1.8 Difference methods; J.2 Physics; G.1.2 Approximation; G.1.2 Elementary function approximation; G.1.6 Optimization; G.1.6 Constrained optimization", descriptor = "Mathematics of Computing, NUMERICAL ANALYSIS, Partial Differential Equations, Difference methods; Computer Applications, PHYSICAL SCIENCES AND ENGINEERING, Physics; Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation; Mathematics of Computing, NUMERICAL ANALYSIS, Optimization, Constrained optimization", fjournal = "SIAM Journal on Applied Mathematics", genterm = "algorithms; theory; experimentation", guideno = "1989-09716", journal-URL = "http://epubs.siam.org/siap", journalabbrev = "SIAM J. Appl. Math.", jrldate = "June 1989", subject = "G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; J. Computer Applications; J.2 PHYSICAL SCIENCES AND ENGINEERING; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS", } @Article{Bauer:1989:BKR, author = "Friedrich L. Bauer", title = "{Eine Bemerkung zu Koechers Reihen f{\"u}r die Eulersche Konstante}. ({German}) [{A} remark on {Koecher}'s series for the {Euler}'s constant]", journal = "Bayer. Akad. Wiss. Math.-Natur. Kl. Sitzungsber.", pages = "27--33 (1990)", year = "1989", ISSN = "0340-7586", MRclass = "11Y60 (65D20) 26A06 11Y60 65D20", MRnumber = "1086008", MRreviewer = "F. Beukers", bibdate = "Thu Aug 20 18:22:34 2020", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/b/bauer-friedrich-ludwig.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", ZMID = "00005959", ZMnumber = "0777.26004", acknowledgement = ack-nhfb, author-dates = "Friedrich (``Fritz'') Ludwig Bauer (10 June 1924--26 March 2015)", fjournal = "Bayerische Akademie der Wissenschaften. Mathematisch-Naturwissenschaftliche Klasse. Sitzungsberichte", keywords = "asymptotic expansion; Euler's constant; series representation", language = "German", } @Article{Belaga:1989:TMM, author = "E. G. Belaga", title = "Through the mincing machine with a {Boolean} layer cake: nonstandard computations over {Boolean} circuits in the lower-bounds-to-circuit-size complexity proving", journal = j-ACTA-INFO, volume = "26", number = "4", pages = "381--407", month = feb, year = "1989", CODEN = "AINFA2", ISSN = "0001-5903 (print), 1432-0525 (electronic)", ISSN-L = "0001-5903", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, affiliation = "Univ. Louis Pasteur, Strasbourg Cedex, France and Univ. su Pisa, Pisa, Italy", bibno = "69310", catcode = "F.4.1; B.6.1; F.1.1; F.1.3; F.2.2; F.1.2; G.1.2; F.2.2", CRclass = "F.4.1 Mathematical Logic; F.4.1 Computational logic; B.6.1 Design Styles; B.6.1 Combinational logic; F.1.1 Models of Computation; F.1.1 Unbounded-action devices; F.1.3 Complexity Classes; F.2.2 Nonnumerical Algorithms and Problems; F.2.2 Complexity of proof procedures; F.1.2 Modes of Computation; F.1.2 Alternation and nondeterminism; G.1.2 Approximation; G.1.2 Elementary function approximation; F.2.2 Nonnumerical Algorithms and Problems; F.2.2 Computations on discrete structures", descriptor = "Theory of Computation, MATHEMATICAL LOGIC AND FORMAL LANGUAGES, Mathematical Logic, Computational logic; Hardware, LOGIC DESIGN, Design Styles, Combinational logic; Theory of Computation, COMPUTATION BY ABSTRACT DEVICES, Models of Computation, Unbounded-action devices; Theory of Computation, COMPUTATION BY ABSTRACT DEVICES, Complexity Classes; Theory of Computation, ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY, Nonnumerical Algorithms and Problems, Complexity of proof procedures; Theory of Computation, COMPUTATION BY ABSTRACT DEVICES, Modes of Computation, Alternation and nondeterminism; Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation; Theory of Computation, ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY, Nonnumerical Algorithms and Problems, Computations on discrete structures", fjournal = "Acta Informatica", genterm = "algorithms; theory", guideno = "1989-03239", journal-URL = "http://www.springerlink.com/content/0001-5903", journalabbrev = "Acta Inf.", jrldate = "Feb. 1989", subject = "F. Theory of Computation; F.4 MATHEMATICAL LOGIC AND FORMAL LANGUAGES; B. Hardware; B.6 LOGIC DESIGN; F. Theory of Computation; F.1 COMPUTATION BY ABSTRACT DEVICES; F. Theory of Computation; F.1 COMPUTATION BY ABSTRACT DEVICES; F. Theory of Computation; F.2 ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY; F. Theory of Computation; F.1 COMPUTATION BY ABSTRACT DEVICES; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; F. Theory of Computation; F.2 ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY", } @Article{Birge:1989:SUB, author = "John R. Birge and Roger J. Wets", title = "Sublinear upper bounds for stochastic programs with recourse", journal = j-MATH-PROG, volume = "43", number = "2", pages = "131--149", month = feb, year = "1989", CODEN = "MHPGA4", ISSN = "0025-5610", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, affiliation = "Univ. of Michigan, Ann Arbor; Univ. of California, Davis", bibno = "65226", catcode = "G.1.6; G.1.7; G.3; G.1.6; G.1.2", CRclass = "G.1.6 Optimization; G.1.6 Linear programming; G.1.7 Ordinary Differential Equations; G.1.7 Convergence and stability; G.3 Probabilistic algorithms (including Monte Carlo); G.1.6 Optimization; G.1.6 Gradient methods; G.1.2 Approximation; G.1.2 Elementary function approximation", descriptor = "Mathematics of Computing, NUMERICAL ANALYSIS, Optimization, Linear programming; Mathematics of Computing, NUMERICAL ANALYSIS, Ordinary Differential Equations, Convergence and stability; Mathematics of Computing, PROBABILITY AND STATISTICS, Probabilistic algorithms (including Monte Carlo); Mathematics of Computing, NUMERICAL ANALYSIS, Optimization, Gradient methods; Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation", fjournal = "Mathematical Programming", genterm = "algorithms; theory; performance", guideno = "1989-09042", journal-URL = "http://link.springer.com/journal/10107", journalabbrev = "Math. Program.", jrldate = "February 1989", subject = "G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; G. Mathematics of Computing; G.3 PROBABILITY AND STATISTICS; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS", } @Article{Borwein:1989:MI, author = "J. M. Borwein and P. B. Borwein", title = "On the Mean Iteration $ (a, b) \leftarrow \big (\frac {a + 3b}{4}, \frac {\sqrt {ab} + b}{2} \big) $", journal = j-MATH-COMPUT, volume = "53", number = "187", pages = "311--326", month = jul, year = "1989", CODEN = "MCMPAF", DOI = "https://doi.org/10.2307/2008364", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "30D05 (33A25)", MRnumber = "968148, 90a:30075", MRreviewer = "Carl C. Cowen", bibdate = "Wed Aug 10 11:09:47 2016", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib; https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp1980.bib; JSTOR database", URL = "http://docserver.carma.newcastle.edu.au/1586/", acknowledgement = ack-nhfb, classcodes = "C4130 (Interpolation and function approximation)", corpsource = "Dept. of Math. Stat. and Comput. Sci., Dalhousie Univ., Halifax, NS, Canada", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "computation; convergence of numerical methods; converging process; iterative methods; iterative process; limit; mean iteration; nontrivial identifications; quadratically; symbolic; uniformizing parameters", treatment = "T Theoretical or Mathematical", } @Article{Borwein:1989:RME, author = "J. M. Borwein and P. B. Borwein and D. H. Bailey", title = "{Ramanujan}, modular equations, and approximations to $ \pi $ or how to compute one billion digits of $ \pi $", journal = j-AMER-MATH-MONTHLY, volume = "96", number = "3", pages = "201--219", month = mar, year = "1989", CODEN = "AMMYAE", ISSN = "0002-9890 (print), 1930-0972 (electronic)", ISSN-L = "0002-9890", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, affiliation = "Dalhousie Univ., Halifax; Dalhousie Univ., Halifax", bibno = "65243", catcode = "I.1.2; G.1.2; G.1.8; G.1.4; I.1.3; F.2.1; F.2.1", CRclass = "I.1.2 Algorithms; I.1.2 Algebraic algorithms; G.1.2 Approximation; G.1.2 Elementary function approximation; G.1.8 Partial Differential Equations; G.1.8 Elliptic equations; G.1.4 Quadrature and Numerical Differentiation; G.1.4 Multiple quadrature; I.1.3 Languages and Systems; F.2.1 Numerical Algorithms and Problems; F.2.1 Computation of transforms; F.2.1 Numerical Algorithms and Problems; F.2.1 Number-theoretic computations", descriptor = "Computing Methodologies, ALGEBRAIC MANIPULATION, Algorithms, Algebraic algorithms; Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation; Mathematics of Computing, NUMERICAL ANALYSIS, Partial Differential Equations, Elliptic equations; Mathematics of Computing, NUMERICAL ANALYSIS, Quadrature and Numerical Differentiation, Multiple quadrature; Computing Methodologies, ALGEBRAIC MANIPULATION, Languages and Systems; Theory of Computation, ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY, Numerical Algorithms and Problems, Computation of transforms; Theory of Computation, ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY, Numerical Algorithms and Problems, Number-theoretic computations", fjournal = "American Mathematical Monthly", genterm = "algorithms; theory", guideno = "1989-03459", journal-URL = "https://www.jstor.org/journals/00029890.htm", journalabbrev = "Am. Math. Monthly", jrldate = "March 1989", subject = "F. Theory of Computation; F.2 ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; I. Computing Methodologies; I.1 ALGEBRAIC MANIPULATION", } @Article{Bos:1989:CPR, author = "L. Bos", title = "A characteristic of points in {$ R^2 $} having {Lebesgue} function of polynomial growth", journal = j-J-APPROX-THEORY, volume = "56", number = "3", pages = "316--329", month = mar, year = "1989", CODEN = "JAXTAZ", ISSN = "0021-9045 (print), 1096-0430 (electronic)", ISSN-L = "0021-9045", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, bibno = "72254", catcode = "F.2.1; G.1.1; G.1.2; F.2.2; G.1.2; G.1.3", CRclass = "F.2.1 Numerical Algorithms and Problems; F.2.1 Computations on matrices; G.1.1 Interpolation; G.1.1 Interpolation formulas; G.1.2 Approximation; G.1.2 Elementary function approximation; F.2.2 Nonnumerical Algorithms and Problems; F.2.2 Geometrical problems and computations; G.1.2 Approximation; G.1.2 Chebyshev approximation and theory; G.1.3 Numerical Linear Algebra; G.1.3 Sparse and very large systems", descriptor = "Theory of Computation, ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY, Numerical Algorithms and Problems, Computations on matrices; Mathematics of Computing, NUMERICAL ANALYSIS, Interpolation, Interpolation formulas; Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation; Theory of Computation, ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY, Nonnumerical Algorithms and Problems, Geometrical problems and computations; Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Chebyshev approximation and theory; Mathematics of Computing, NUMERICAL ANALYSIS, Numerical Linear Algebra, Sparse and very large systems", fjournal = "Journal of Approximation Theory", genterm = "algorithms; theory", guideno = "1989-07812", journal-URL = "http://www.sciencedirect.com/science/journal/00219045", journalabbrev = "J. Approx. Theory", jrldate = "Mar. 1989", subject = "F. Theory of Computation; F.2 ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; F. Theory of Computation; F.2 ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS", } @Book{Boyd:1989:CFS, author = "John Philip Boyd", title = "{Chebyshev} and {Fourier} spectral methods", volume = "49", publisher = pub-SV, address = pub-SV:adr, pages = "xvi + 798", year = "1989", ISBN = "0-387-51487-2, 3-540-51487-2", ISBN-13 = "978-0-387-51487-1, 978-3-540-51487-9", LCCN = "QA404.5 .B69 1989", bibdate = "Sat Feb 17 14:00:30 MST 2024", bibsource = "fsz3950.oclc.org:210/WorldCat; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", series = "Lecture notes in engineering", acknowledgement = ack-nhfb, subject = "Chebyshev polynomials; Fourier analysis; Spectral theory (Mathematics); Polyn{\'y}omes de Tchebychev; Analyse de Fourier; Spectre (Math{\'y}ematiques); Chebyshev polynomials; Fourier analysis; Spectral theory (Mathematics)", } @Article{Bronstein:1989:AIE, author = "Manuel Bronstein", title = "An algorithm for the integration of elementary functions", journal = j-LECT-NOTES-COMP-SCI, volume = "378", pages = "491--497", year = "1989", CODEN = "LNCSD9", ISSN = "0302-9743 (print), 1611-3349 (electronic)", ISSN-L = "0302-9743", MRclass = "65D30", MRnumber = "91a:65050", bibdate = "Sat Jan 11 17:44:01 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "EUROCAL '87 (Leipzig, 1987).", acknowledgement = ack-nhfb, fjournal = "Lecture Notes in Computer Science", journal-URL = "http://link.springer.com/bookseries/558", } @InProceedings{Bronstein:1989:SRE, author = "M. Bronstein", title = "Simplification of real elementary functions", crossref = "ACM:1989:PAI", pages = "207--211", year = "1989", bibdate = "Tue Sep 17 06:46:18 MDT 1996", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/issac.bib", abstract = "The author describes an algorithm, based on Risch's real structure theorem, that determines explicitly all the algebraic relations among a given set of real elementary functions. He provides examples from its implementation in the Scratchpad computer algebra system that illustrate the advantages over the use of complex logarithms and exponentials.", acknowledgement = ack-nhfb, affiliation = "IBM Res. Div., T. J. Watson Res. Center, Yorktown Heights, NY, USA", classification = "C1110 (Algebra); C7310 (Mathematics)", keywords = "Computer algebra system; Real elementary functions; Real structure theorem; Scratchpad", thesaurus = "Functions; Mathematics computing; Symbol manipulation", } @Article{Cao:1989:ABS, author = "J.-D. Cao and H. H. Gonska", title = "Approximation by {Boolean} sums of positive linear operators. {II}. {Gopengauz}-type estimates", journal = j-J-APPROX-THEORY, volume = "57", number = "1", pages = "77--89", month = apr, year = "1989", CODEN = "JAXTAZ", ISSN = "0021-9045 (print), 1096-0430 (electronic)", ISSN-L = "0021-9045", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, bibno = "72266", catcode = "G.1.2; G.1.2; G.3; G.2.1", CRclass = "G.1.2 Approximation; G.1.2 Nonlinear approximation; G.1.2 Approximation; G.1.2 Elementary function approximation; G.3 Statistical computing; G.2.1 Combinatorics; G.2.1 Generating functions", descriptor = "Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Nonlinear approximation; Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation; Mathematics of Computing, PROBABILITY AND STATISTICS, Statistical computing; Mathematics of Computing, DISCRETE MATHEMATICS, Combinatorics, Generating functions", fjournal = "Journal of Approximation Theory", genterm = "algorithms; theory; measurement", guideno = "1989-07823", journal-URL = "http://www.sciencedirect.com/science/journal/00219045", journalabbrev = "J. Approx. Theory", jrldate = "April 1989", subject = "G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; G. Mathematics of Computing; G.3 PROBABILITY AND STATISTICS; G. Mathematics of Computing; G.2 DISCRETE MATHEMATICS", } @Article{Carlson:1989:TEI, author = "B. C. Carlson", title = "A Table of Elliptic Integrals: Cubic Cases", journal = j-MATH-COMPUT, volume = "53", number = "187", pages = "327--333", month = jul, year = "1989", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "65A05 (33A25 65D20)", MRnumber = "89m:65009", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, classcodes = "C4180 (Integral equations); C1120 (Analysis)", corpsource = "Dept. of Math., Iowa State Univ., Ames, IA, USA", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "cubic polynomial; elliptic; elliptic integral; first kind; Fortran codes; functions; integral equations; integrals; integration interval; R-; rational integrands; real zeros; second kind; square root; table; third kind", treatment = "T Theoretical or Mathematical", } @Article{Chen:1989:EMB, author = "X. R. Chen and P. R. Krishnaiah and W. W. Liang", title = "Estimation of multivariate binary density using orthogonal functions", journal = j-J-MULTIVAR-ANAL, volume = "31", number = "2", pages = "178--186", month = nov, year = "1989", CODEN = "JMVAAI", ISSN = "0047-259x (print), 1095-7243 (electronic)", ISSN-L = "0047-259X", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, affiliation = "Univ. of Pittsburgh, Pittsburgh, PA; Univ. of Pittsburgh, Pittsburgh, PA; Univ. of Pittsburgh, Pittsburgh, PA", bibno = "69313", catcode = "D.3.3; G.3; G.1.3; G.1.2", CRclass = "D.3.3 Language Constructs; D.3.3 Procedures, functions, and subroutines; G.3 Statistical computing; G.1.3 Numerical Linear Algebra; G.1.3 Linear systems (direct and iterative methods); G.1.2 Approximation; G.1.2 Elementary function approximation", descriptor = "Software, PROGRAMMING LANGUAGES, Language Constructs, Procedures, functions, and subroutines; Mathematics of Computing, PROBABILITY AND STATISTICS, Statistical computing; Mathematics of Computing, NUMERICAL ANALYSIS, Numerical Linear Algebra, Linear systems (direct and iterative methods); Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation", fjournal = "Journal of Multivariate Analysis", genterm = "algorithms; theory; verification", guideno = "1989-08483", journalabbrev = "J. Multivariate Anal.", jrldate = "Nov. 1989", subject = "D. Software; D.3 PROGRAMMING LANGUAGES; G. Mathematics of Computing; G.3 PROBABILITY AND STATISTICS; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS", } @Article{Chen:1989:FCR, author = "S.-G. Chen and P. Y. Hsieh", title = "Fast computation of the $ {N} $ th root", journal = j-COMPUT-MATH-APPL, volume = "17", number = "10", pages = "1423--1427", month = "????", year = "1989", CODEN = "CMAPDK", DOI = "https://doi.org/10.1016/0898-1221(89)90024-2", ISSN = "0898-1221 (print), 1873-7668 (electronic)", ISSN-L = "0898-1221", bibdate = "Thu Dec 29 08:01:37 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/0898122189900242", acknowledgement = ack-nhfb, fjournal = "Computers and Mathematics with Applications", journal-URL = "http://www.sciencedirect.com/science/journal/08981221", remark = "From the abstract: ``A new class of iterative methods for computing a differentiable function is proposed, which is based on Pad{\'e} approximation to Taylor's series of the function. It leads to a faster algorithm than Newton's method for $ x^{1 / N} $ and a different interpretation of Newton's method.''", } @Article{Chen:1989:FCTa, author = "S.-G. Chen and P. Y. Hsieh", title = "Fast computation of the {$N$}-th root", journal = j-COMPUT-MATH-APPL, volume = "17", number = "10", pages = "1423--1427", month = "????", year = "1989", CODEN = "CMAPDK", DOI = "https://doi.org/10.1016/0898-1221(89)90024-2", ISSN = "0898-1221 (print), 1873-7668 (electronic)", ISSN-L = "0898-1221", bibdate = "Wed Mar 1 19:01:11 MST 2017", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; https://www.math.utah.edu/pub/tex/bib/computmathappl1980.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "http://www.sciencedirect.com/science/article/pii/0898122189900242", acknowledgement = ack-nj # " and " # ack-nhfb, fjournal = "Computers and Mathematics with Applications", journal-URL = "http://www.sciencedirect.com/science/journal/08981221", remark = "From the abstract: ``A new class of iterative methods for computing a differentiable function is proposed, which is based on Pad{\'e} approximation to Taylor's series of the function. It leads to a faster algorithm than Newton's method for $ x^{1 / N} $ and a different interpretation of Newton's method.''", } @InProceedings{Considine:1989:CTF, author = "V. Considine", booktitle = "{International Conference on Acoustics, Speech, and Signal Processing}", title = "{CORDIC} trigonometric function generator for {DSP}", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "2381--2384 (vol. 4)", year = "1989", DOI = "https://doi.org/10.1109/ICASSP.1989.266946", bibdate = "Mon Oct 27 10:32:44 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, keywords = "Computer architecture; Digital arithmetic; Digital signal processing; Digital systems; Equations; Polynomials; Signal generators; Signal processing algorithms; Signal sampling; Table lookup", } @Article{Corliss:1989:IIV, author = "George Corliss and Gary Krenz", editor = "L. Gatteschi", title = "Indefinite Integration with Validation", journal = j-TOMS, volume = "15", number = "4", pages = "375--393", month = dec, year = "1989", CODEN = "ACMSCU", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", MRclass = "65D30 (65-04)", MRnumber = "1 062 497", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.acm.org/pubs/citations/journals/toms/1989-15-4/p375-corliss/; http://www.acm.org/pubs/toc/Abstracts/toms/76915.html", acknowledgement = ack-nhfb, bibno = "393", content = "ALGORITHMS; THEORY", CRclass = "G.1.4 Quadrature and Numerical Differentiation; G.1.2 Approximation; G.1.2 Elementary function approximation; G.1.2 Approximation; G.1.2 Chebyshev approximation and theory", CRnumber = "1989-03199", descriptor = "mathematics of computing, numerical analysis, quadrature and numerical differentiation; mathematics of computing, numerical analysis, approximation, elementary function approximation; mathematics of computing, numerical analysis, approximation, Chebyshev approximation and theory", fjournal = "ACM Transactions on Mathematical Software (TOMS)", fortitle = "ACM Trans. Math. Softw.", guideno = "4", journal-URL = "https://dl.acm.org/loi/toms", keywords = "algorithms; theory", review = "ACM CR 9007-0598", subject = "{\bf G.1.4}: Mathematics of Computing, NUMERICAL ANALYSIS, Quadrature and Numerical Differentiation. {\bf G.1.2}: Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation. {\bf G.1.2}: Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Chebyshev approximation and theory.", waffil = "G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS", } @Article{Critchfield:1989:CEF, author = "Charles L. Critchfield", title = "Computation of elliptic functions", journal = j-J-MATH-PHYS, volume = "30", number = "2", pages = "295--297", month = feb, year = "1989", CODEN = "JMAPAQ", DOI = "https://doi.org/10.1063/1.528444", ISSN = "0022-2488 (print), 1089-7658 (electronic), 1527-2427", ISSN-L = "0022-2488", MRclass = "33A25 (65D20)", MRnumber = "89k:33004", MRreviewer = "H. Hochstadt", bibdate = "Mon Oct 31 11:58:32 MDT 2011", bibsource = "http://jmp.aip.org/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jmathphys1985.bib", URL = "http://jmp.aip.org/resource/1/jmapaq/v30/i2/p295_s1", acknowledgement = ack-nhfb, fjournal = "Journal of Mathematical Physics", journal-URL = "http://jmp.aip.org/", pagecount = "3", } @Misc{Darley:1989:FPI, author = "H. M. Darley and others", title = "Floating Point\slash Integer Processor with Divide and Square Root Functions", year = "1989", bibdate = "Thu Apr 2 08:38:35 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", note = "U.S. Patent No. 4,878,190.", acknowledgement = ack-sfo # " and " # ack-nhfb, } @Article{Dehling:1989:FLI, author = "Herold Dehling", title = "The functional law of the iterated logarithm for {von Mises} functionals and multiple {Wiener} integrals", journal = j-J-MULTIVAR-ANAL, volume = "28", number = "2", pages = "177--189", month = feb, year = "1989", CODEN = "JMVAAI", ISSN = "0047-259x (print), 1095-7243 (electronic)", ISSN-L = "0047-259X", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, affiliation = "Univ. of G{\"o}ttingen, G{\"o}ttingen, FRG", bibno = "64336", catcode = "G.3; G.1.9; G.1.2", CRclass = "G.3 Statistical computing; G.1.9 Integral Equations; G.1.9 Integro-differential equations; G.1.2 Approximation; G.1.2 Elementary function approximation", descriptor = "Mathematics of Computing, PROBABILITY AND STATISTICS, Statistical computing; Mathematics of Computing, NUMERICAL ANALYSIS, Integral Equations, Integro-differential equations; Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation", fjournal = "Journal of Multivariate Analysis", genterm = "algorithms; theory; measurement", guideno = "1989-08462", journalabbrev = "J. Multivariate Anal.", jrldate = "February 1989", subject = "G. Mathematics of Computing; G.3 PROBABILITY AND STATISTICS; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS", } @Article{Demirbas:1989:MSE, author = "K. Demirbas", title = "Multidimensional state estimation using stacks for dynamic systems with interference", journal = j-AUTOMATICA, volume = "25", number = "4", pages = "617--621", month = jul, year = "1989", CODEN = "ATCAA9", ISSN = "0005-1098 (print), 1873-2836 (electronic)", ISSN-L = "0005-1098", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, bibno = "72661", catcode = "G.1.2; G.1.2; G.1.3; G.1.5; H.1.1", CRclass = "G.1.2 Approximation; G.1.2 Elementary function approximation; G.1.2 Approximation; G.1.2 Linear approximation; G.1.3 Numerical Linear Algebra; G.1.3 Linear systems (direct and iterative methods); G.1.5 Roots of Nonlinear Equations; H.1.1 Systems and Information Theory; H.1.1 Information theory", descriptor = "Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation; Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Linear approximation; Mathematics of Computing, NUMERICAL ANALYSIS, Numerical Linear Algebra, Linear systems (direct and iterative methods); Mathematics of Computing, NUMERICAL ANALYSIS, Roots of Nonlinear Equations; Information Systems, MODELS AND PRINCIPLES, Systems and Information Theory, Information theory", fjournal = "Automatica: the journal of IFAC, the International Federation of Automatic Control", genterm = "algorithms; theory", guideno = "1989-03952", jrldate = "July 1989", subject = "G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; H. Information Systems; H.1 MODELS AND PRINCIPLES", } @TechReport{Dritz:1989:RPS, author = "K. W. Dritz", title = "Rationale for the Proposed Standard for a Generic Package of Elementary Functions for {Ada}", type = "Report", number = "ANL-89/2 Rev. 1", institution = "Argonne National Laboratory, Mathematics and Computer Science Division", address = "Argonne, IL, USA", pages = "????", month = oct, year = "1989", bibdate = "Thu Sep 01 12:08:24 1994", bibsource = "garbo.uwasa.fi:/pc/doc-soft/fpbiblio.txt; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nj, } @InProceedings{Duprat:1989:SRA, author = "J. Duprat and Y. Herreros and J.-M. Muller", title = "Some results about on-line computation of functions", crossref = "Ercegovac:1989:PSC", pages = "112--118", year = "1989", bibdate = "Sat Nov 27 14:19:10 MST 2004", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, } @Article{Dyn:1989:PEB, author = "N. Dyn and A. Ron", title = "Periodic exponential box splines on a three direction mesh", journal = j-J-APPROX-THEORY, volume = "56", number = "3", pages = "287--296", month = mar, year = "1989", CODEN = "JAXTAZ", ISSN = "0021-9045 (print), 1096-0430 (electronic)", ISSN-L = "0021-9045", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, bibno = "72251", catcode = "G.1.2; F.2.2; G.1.2; G.1.2; G.1.2; G.2.1; G.3", CRclass = "G.1.2 Approximation; G.1.2 Spline and piecewise polynomial approximation; F.2.2 Nonnumerical Algorithms and Problems; F.2.2 Geometrical problems and computations; G.1.2 Approximation; G.1.2 Chebyshev approximation and theory; G.1.2 Approximation; G.1.2 Elementary function approximation; G.1.2 Approximation; G.1.2 Linear approximation; G.2.1 Combinatorics; G.2.1 Recurrences and difference equations; G.3 Statistical computing", descriptor = "Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Spline and piecewise polynomial approximation; Theory of Computation, ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY, Nonnumerical Algorithms and Problems, Geometrical problems and computations; Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Chebyshev approximation and theory; Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation; Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Linear approximation; Mathematics of Computing, DISCRETE MATHEMATICS, Combinatorics, Recurrences and difference equations; Mathematics of Computing, PROBABILITY AND STATISTICS, Statistical computing", fjournal = "Journal of Approximation Theory", genterm = "algorithms; theory; design", guideno = "1989-07809", journal-URL = "http://www.sciencedirect.com/science/journal/00219045", journalabbrev = "J. Approx. Theory", jrldate = "Mar. 1989", subject = "F. Theory of Computation; F.2 ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; G.2 DISCRETE MATHEMATICS; G.3 PROBABILITY AND STATISTICS", } @Article{Eberlein:1989:SAC, author = "E. Eberlein", title = "Strong approximation of continuous time stochastic processes", journal = j-J-MULTIVAR-ANAL, volume = "31", number = "2", pages = "220--235", month = nov, year = "1989", CODEN = "JMVAAI", ISSN = "0047-259x (print), 1095-7243 (electronic)", ISSN-L = "0047-259X", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, affiliation = "Univ. Freiburg, Freiburg, W. Germany", bibno = "69316", catcode = "D.4.8; F.1.2; G.1.2", CRclass = "D.4.8 Performance; D.4.8 Stochastic analysis; F.1.2 Modes of Computation; F.1.2 Probabilistic computation; G.1.2 Approximation; G.1.2 Elementary function approximation", descriptor = "Software, OPERATING SYSTEMS, Performance, Stochastic analysis; Theory of Computation, COMPUTATION BY ABSTRACT DEVICES, Modes of Computation, Probabilistic computation; Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation", fjournal = "Journal of Multivariate Analysis", genterm = "algorithms; performance; theory", guideno = "1989-08486", journalabbrev = "J. Multivariate Anal.", jrldate = "Nov. 1989", subject = "D. Software; D.4 OPERATING SYSTEMS; F. Theory of Computation; F.1 COMPUTATION BY ABSTRACT DEVICES; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS", } @Article{Egger:1989:PAC, author = "A. Egger and R. Huotari", title = "The {Polya} algorithm on convex sets", journal = j-J-APPROX-THEORY, volume = "56", number = "2", pages = "212--216", month = feb, year = "1989", CODEN = "JAXTAZ", ISSN = "0021-9045 (print), 1096-0430 (electronic)", ISSN-L = "0021-9045", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, bibno = "72242", catcode = "F.2.2; F.2.2; G.1.2; G.1.2; G.1.5", CRclass = "F.2.2 Nonnumerical Algorithms and Problems; F.2.2 Geometrical problems and computations; F.2.2 Nonnumerical Algorithms and Problems; F.2.2 Computations on discrete structures; G.1.2 Approximation; G.1.2 Elementary function approximation; G.1.2 Approximation; G.1.2 Minimax approximation and algorithms; G.1.5 Roots of Nonlinear Equations; G.1.5 Convergence", descriptor = "Theory of Computation, ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY, Nonnumerical Algorithms and Problems, Geometrical problems and computations; Theory of Computation, ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY, Nonnumerical Algorithms and Problems, Computations on discrete structures; Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation; Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Minimax approximation and algorithms; Mathematics of Computing, NUMERICAL ANALYSIS, Roots of Nonlinear Equations, Convergence", fjournal = "Journal of Approximation Theory", genterm = "algorithms; theory", guideno = "1989-07801", journal-URL = "http://www.sciencedirect.com/science/journal/00219045", journalabbrev = "J. Approx. Theory", jrldate = "Feb. 1989", subject = "F. Theory of Computation; F.2 ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS", } @Article{Ehrenmark:1989:ONF, author = "Ulf T. Ehrenmark", title = "Overconvergence of the near-field expansion for linearized waves normally incident on a sloping beach", journal = j-SIAM-J-APPL-MATH, volume = "49", number = "3", pages = "799--815", month = jun, year = "1989", CODEN = "SMJMAP", ISSN = "0036-1399 (print), 1095-712X (electronic)", ISSN-L = "0036-1399", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, bibno = "64945", catcode = "G.1.7; G.1.2; F.2.1; G.1.2; F.2.2", CRclass = "G.1.7 Ordinary Differential Equations; G.1.7 Convergence and stability; G.1.2 Approximation; G.1.2 Minimax approximation and algorithms; F.2.1 Numerical Algorithms and Problems; F.2.1 Computation of transforms; G.1.2 Approximation; G.1.2 Elementary function approximation; F.2.2 Nonnumerical Algorithms and Problems; F.2.2 Geometrical problems and computations", descriptor = "Mathematics of Computing, NUMERICAL ANALYSIS, Ordinary Differential Equations, Convergence and stability; Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Minimax approximation and algorithms; Theory of Computation, ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY, Numerical Algorithms and Problems, Computation of transforms; Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation; Theory of Computation, ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY, Nonnumerical Algorithms and Problems, Geometrical problems and computations", fjournal = "SIAM Journal on Applied Mathematics", genterm = "algorithms; theory; experimentation", guideno = "1989-09714", journal-URL = "http://epubs.siam.org/siap", journalabbrev = "SIAM J. Appl. Math.", jrldate = "June 1989", subject = "G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; F. Theory of Computation; F.2 ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; F. Theory of Computation; F.2 ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY", } @Article{Einmahl:1989:ERK, author = "U. Einmahl", title = "Extensions of results of {Komlos}, {Major}, and {Tusnady} to the multivariate case", journal = j-J-MULTIVAR-ANAL, volume = "28", number = "1", pages = "20--68", month = jan, year = "1989", CODEN = "JMVAAI", ISSN = "0047-259x (print), 1095-7243 (electronic)", ISSN-L = "0047-259X", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, affiliation = "Univ. zu Koln, West Germany", bibno = "66700", catcode = "G.3; G.1.2; F.2.1", CRclass = "G.3 Statistical computing; G.1.2 Approximation; G.1.2 Elementary function approximation; F.2.1 Numerical Algorithms and Problems; F.2.1 Computation of transforms", descriptor = "Mathematics of Computing, PROBABILITY AND STATISTICS, Statistical computing; Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation; Theory of Computation, ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY, Numerical Algorithms and Problems, Computation of transforms", fjournal = "Journal of Multivariate Analysis", genterm = "algorithms; theory", guideno = "1989-08456", journalabbrev = "J. Multivariate Anal.", jrldate = "Jan. 1989", subject = "G. Mathematics of Computing; G.3 PROBABILITY AND STATISTICS; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; F. Theory of Computation; F.2 ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY", } @Article{Epperson:1989:UIM, author = "J. F. Epperson", title = "On the use of iteration methods for approximating the natural logarithm", journal = j-AMER-MATH-MONTHLY, volume = "96", number = "9", pages = "831--835", month = nov, year = "1989", CODEN = "AMMYAE", ISSN = "0002-9890 (print), 1930-0972 (electronic)", ISSN-L = "0002-9890", MRclass = "26A06 (26A09)", MRnumber = "91a:26002", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, bibno = "71703", catcode = "G.1.2; G.1.2; K.3.2", CRclass = "G.1.2 Approximation; G.1.2 Elementary function approximation; G.1.2 Approximation; G.1.2 Spline and piecewise polynomial approximation; K.3.2 Computer and Information Science Education", descriptor = "Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation; Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Spline and piecewise polynomial approximation; Computing Milieux, COMPUTERS AND EDUCATION, Computer and Information Science Education", fjournal = "American Mathematical Monthly", genterm = "algorithms; theory", guideno = "1989-03518", journal-URL = "https://www.jstor.org/journals/00029890.htm", journalabbrev = "Am. Math. Monthly", jrldate = "Nov. 1989", subject = "G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; K. Computing Milieux; K.3 COMPUTERS AND EDUCATION", } @InProceedings{Ercegovac:1989:FRD, author = "M. D. Ercegovac and T. Lang", title = "On-the-fly rounding for division and square root", crossref = "Ercegovac:1989:PSC", pages = "169--173", year = "1989", bibdate = "Sat Jul 16 15:28:13 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "http://www.acsel-lab.com/arithmetic/arith9/papers/ARITH9_Ercegovac_rounding.pdf", acknowledgement = ack-nhfb, keywords = "ARITH-9", summary = "In division and square root implementation based on digit-recurrence algorithms, the result is obtained in digit-serial form, from most significant digit to least significant. To reduce the complexity of the result-digit selection and to allow the \ldots{}", } @InProceedings{Ercegovac:1989:IMC, author = "M. D. Ercegovac and T. Lang", booktitle = "{IEEE} International Symposium on Circuits and Systems, 8--11 May 1989", title = "Implementation of module combining multiplication, division, and square root", volume = "1", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "150--153", year = "1989", CODEN = "????", ISSN = "????", bibdate = "Sat Jul 16 15:28:13 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, summary = "The implementation of a module that performs radix-$2$ multiplication, division, and square root is presented. The module is compact because most of the components are shared by all three operations, the complexity being similar to that of a radix-$2$ \ldots{}", } @InProceedings{Ercegovac:1989:RSR, author = "Milo{\v{s}} D. Ercegovac and Tomas Lang", title = "Radix-4 square root without initial {PLA}", crossref = "Ercegovac:1989:PSC", pages = "162--168", year = "1989", bibdate = "Sat Jul 16 15:28:13 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "http://www.acsel-lab.com/arithmetic/arith9/papers/ARITH9_Ercegovac_radix4.pdf", acknowledgement = ack-nhfb, keywords = "ARITH-9", summary = "A systematic derivation of a radix-$4$ square root algorithm using redundance in the partial residuals and the result is presented. Unlike other similar schemes, the algorithm does not use a table-lookup or programmable logic array (PLA) for the \ldots{}", } @InProceedings{Fandrianto:1989:AHS, author = "Jan Fandrianto", title = "Algorithms for high-speed shared radix 8 division and radix 8 square root", crossref = "Ercegovac:1989:PSC", pages = "68--75", year = "1989", bibdate = "Sat Jul 16 15:28:13 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "http://www.acsel-lab.com/arithmetic/arith9/papers/ARITH9_Fandrianto.pdf", acknowledgement = ack-sfo # " and " # ack-nhfb, keywords = "ARITH-9", summary = "An algorithm for performing radix-$8$ division and square root in a shared hardware is described. To achieve short iteration cycle time, it utilizes an optimized `next quotient/root prediction PLA' generally used in a radix-$4$ SRT division with minimal \ldots{}", } @Article{Ge:1989:OCL, author = "Renpu Ge", title = "Optimal choice of linear interval extension", journal = j-APPL-MATH-COMP, volume = "30", number = "2", pages = "165--189", month = mar, year = "1989", CODEN = "AMHCBQ", ISSN = "0096-3003 (print), 1873-5649 (electronic)", ISSN-L = "0096-3003", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, bibno = "64710", catcode = "G.1.6; G.1.2; G.1.2; G.2.1; F.2.2; G.1.2; G.1.1; G.1.0", CRclass = "G.1.6 Optimization; G.1.6 Linear programming; G.1.2 Approximation; G.1.2 Linear approximation; G.1.2 Approximation; G.1.2 Elementary function approximation; G.2.1 Combinatorics; G.2.1 Combinatorial algorithms; F.2.2 Nonnumerical Algorithms and Problems; F.2.2 Computations on discrete structures; G.1.2 Approximation; G.1.2 Minimax approximation and algorithms; G.1.1 Interpolation; G.1.1 Difference formulas; G.1.0 General; G.1.0 Numerical algorithms", descriptor = "Mathematics of Computing, NUMERICAL ANALYSIS, Optimization, Linear programming; Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Linear approximation; Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation; Mathematics of Computing, DISCRETE MATHEMATICS, Combinatorics, Combinatorial algorithms; Theory of Computation, ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY, Nonnumerical Algorithms and Problems, Computations on discrete structures; Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Minimax approximation and algorithms; Mathematics of Computing, NUMERICAL ANALYSIS, Interpolation, Difference formulas; Mathematics of Computing, NUMERICAL ANALYSIS, General, Numerical algorithms", fjournal = "Applied Mathematics and Computation", genterm = "algorithms; theory; measurement", guideno = "1989-03670", journal-URL = "http://www.sciencedirect.com/science/journal/00963003", journalabbrev = "Appl. Math. Comput.", jrldate = "March 1989", subject = "F. Theory of Computation; F.2 ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; G.2 DISCRETE MATHEMATICS", } @Article{Gersch:1989:SPT, author = "W. Gersch and G. Kitagawa", title = "Smoothness priors transfer function estimation", journal = j-AUTOMATICA, volume = "25", number = "4", pages = "603--608", month = jul, year = "1989", CODEN = "ATCAA9", ISSN = "0005-1098 (print), 1873-2836 (electronic)", ISSN-L = "0005-1098", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, bibno = "72658", catcode = "G.1.1; G.1.2; G.1.6", CRclass = "G.1.1 Interpolation; G.1.1 Smoothing; G.1.2 Approximation; G.1.2 Elementary function approximation; G.1.6 Optimization; G.1.6 Gradient methods", descriptor = "Mathematics of Computing, NUMERICAL ANALYSIS, Interpolation, Smoothing; Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation; Mathematics of Computing, NUMERICAL ANALYSIS, Optimization, Gradient methods", fjournal = "Automatica: the journal of IFAC, the International Federation of Automatic Control", genterm = "algorithms; theory", guideno = "1989-03949", jrldate = "July 1989", subject = "G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS", } @InCollection{Glover:1989:THN, author = "Keith Glover", editor = "Jan C. Willems", booktitle = "From data to model", title = "A tutorial on {Hankel}-norm approximation", publisher = pub-SV, address = pub-SV:adr, bookpages = "246", pages = "26--48", year = "1989", ISBN = "0-387-51571-2", ISBN-13 = "978-0-387-51571-7", LCCN = "QA279 .F76 1989", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, bibno = "70545", catcode = "G.1.2; F.2.1; F.2.1", CRclass = "G.1.2 Approximation; G.1.2 Elementary function approximation; F.2.1 Numerical Algorithms and Problems; F.2.1 Computations on matrices; F.2.1 Numerical Algorithms and Problems; F.2.1 Computation of transforms", descriptor = "Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation; Theory of Computation, ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY, Numerical Algorithms and Problems, Computations on matrices; Theory of Computation, ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY, Numerical Algorithms and Problems, Computation of transforms", genterm = "algorithms; theory", guideno = "1989-01692", subject = "G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; F. Theory of Computation; F.2 ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY; F. Theory of Computation; F.2 ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY", waffil = "Univ. of Groningen, Groningen, The Netherlands", } @InProceedings{Glynn:1989:OSS, author = "P. W. Glynn", editor = "Edward A. MacNair and Kenneth J. Musselman and Philip Heidelberger", booktitle = "1989 Winter Simulation Conference proceedings: December 4--6, 1989, the Capital Hilton Hotel, Washington, {DC}", title = "Optimization of stochastic systems via simulation", publisher = pub-ACM, address = pub-ACM:adr, bookpages = "xx + 1139", pages = "90--105", year = "1989", ISBN = "0-911801-58-8", ISBN-13 = "978-0-911801-58-3", LCCN = "QA76.9.C65 W56 1989", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "IEEE order no. 89CH2778-9.", URL = "https://ieeexplore.ieee.org/servlet/opac?punumber=5823", acknowledgement = ack-nhfb, bibno = "76750", catcode = "I.6.3; G.1.6; G.3; G.1.2", CRclass = "I.6.3 Applications; G.1.6 Optimization; G.1.2 Approximation; G.1.2 Elementary function approximation", descriptor = "Computing Methodologies, SIMULATION AND MODELING, Applications; Mathematics of Computing, NUMERICAL ANALYSIS, Optimization; Mathematics of Computing, PROBABILITY AND STATISTICS; Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation", genterm = "algorithms; design; performance", guideno = "1989-12012", procdate = "December 4-6, 1989", procloc = "Washington, D. C.", subject = "I. Computing Methodologies; I.6 SIMULATION AND MODELING; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; G. Mathematics of Computing; G.3 PROBABILITY AND STATISTICS; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS", } @Article{Gomes:1989:GGL, author = "M. I. Gomes", title = "Generalized {Gumbel} and likelihood ratio test statistics in the multivariate {GEV} model", journal = j-COMPUT-STAT-DATA-ANAL, volume = "7", number = "3", pages = "259--267", month = feb, year = "1989", CODEN = "CSDADW", ISSN = "0167-9473 (print), 1872-7352 (electronic)", ISSN-L = "0167-9473", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, bibno = "70043", catcode = "G.3; G.1.7; I.5.1; G.1.6; G.1.2", CRclass = "G.3 Statistical computing; G.1.7 Ordinary Differential Equations; G.1.7 Convergence and stability; I.5.1 Models; I.5.1 Statistical; G.1.6 Optimization; G.1.6 Nonlinear programming; G.1.2 Approximation; G.1.2 Elementary function approximation", descriptor = "Mathematics of Computing, PROBABILITY AND STATISTICS, Statistical computing; Mathematics of Computing, NUMERICAL ANALYSIS, Ordinary Differential Equations, Convergence and stability; Computing Methodologies, PATTERN RECOGNITION, Models, Statistical; Mathematics of Computing, NUMERICAL ANALYSIS, Optimization, Nonlinear programming; Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation", fjournal = "Computational Statistics \& Data Analysis", genterm = "algorithms; measurement; reliability; theory", guideno = "1989-04403", journal-URL = "http://www.sciencedirect.com/science/journal/01679473", journalabbrev = "Comput. Stat. Data Anal.", jrldate = "Feb. 1989", subject = "G. Mathematics of Computing; G.3 PROBABILITY AND STATISTICS; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; I. Computing Methodologies; I.5 PATTERN RECOGNITION; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS", } @InProceedings{Gonzaga:1989:ASL, author = "Cl{\'o}vis C. Gonzaga", title = "An algorithm for solving linear programming programs in {$ O(n^3 L) $} operations", crossref = "Megiddo:1989:PMP", pages = "1--28", year = "1989", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, bibno = "74172", catcode = "G.1.6; G.1.2; F.2.1; I.1.2; I.1.2", CRclass = "G.1.6 Optimization; G.1.6 Linear programming; G.1.2 Approximation; G.1.2 Elementary function approximation; F.2.1 Numerical Algorithms and Problems; F.2.1 Computations on matrices; I.1.2 Algorithms; I.1.2 Algebraic algorithms; I.1.2 Algorithms; I.1.2 Analysis of algorithms", descriptor = "Mathematics of Computing, NUMERICAL ANALYSIS, Optimization, Linear programming; Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation; Theory of Computation, ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY, Numerical Algorithms and Problems, Computations on matrices; Computing Methodologies, ALGEBRAIC MANIPULATION, Algorithms, Algebraic algorithms; Computing Methodologies, ALGEBRAIC MANIPULATION, Algorithms, Analysis of algorithms", genterm = "algorithms; theory", guideno = "1989-12474", procdate = "March 1-4, 1987", procloc = "Pacific Grove, CA", subject = "G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; F. Theory of Computation; F.2 ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY; I. Computing Methodologies; I.1 ALGEBRAIC MANIPULATION; I. Computing Methodologies; I.1 ALGEBRAIC MANIPULATION", } @TechReport{Gragg:1989:FSE, author = "W. Gragg and B. Neta", title = "{Fortran} Subroutines for the Evaluation of the Confluent Hypergeometric Functions", number = "NPS-MA-89-014", institution = inst-MATH-NPS, address = inst-MATH-NPS:adr, year = "1989", bibdate = "Fri Nov 11 14:50:24 MST 1994", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/n/neta-beny.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "Software available URL http://math.nps.navy.mil", } @Article{Guo:1989:RCS, author = "S.-S. Guo and M. K. Khan", title = "On the rate of convergence of some operators on functions of bounded variation", journal = j-J-APPROX-THEORY, volume = "58", number = "1", pages = "90--101", month = jul, year = "1989", CODEN = "JAXTAZ", ISSN = "0021-9045 (print), 1096-0430 (electronic)", ISSN-L = "0021-9045", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, bibno = "69424", catcode = "F.2.1; G.1.2", CRclass = "F.2.1 Numerical Algorithms and Problems; F.2.1 Computations on polynomials; G.1.2 Approximation; G.1.2 Elementary function approximation", descriptor = "Theory of Computation, ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY, Numerical Algorithms and Problems, Computations on polynomials; Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation", fjournal = "Journal of Approximation Theory", genterm = "algorithms; theory", guideno = "1989-07854", journal-URL = "http://www.sciencedirect.com/science/journal/00219045", journalabbrev = "J. Approx. Theory", jrldate = "July 1989", subject = "F. Theory of Computation; F.2 ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS", } @Article{Hornik:1989:MFN, author = "K. Hornik and M. Stinchcombe and H. White", title = "Multilayer feedforward networks are universal approximators", journal = j-NEURAL-NETWORKS, volume = "2", number = "5", pages = "359--366", year = "1989", CODEN = "NNETEB", ISSN = "0893-6080 (print), 1879-2782 (electronic)", ISSN-L = "0893-6080", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, affiliation = "Technisce Univ. Wien, Vienna, Austria; Univ. of California, San Diego; Univ. of California, San Diego", bibno = "70408", catcode = "F.2.1; G.1.2; F.1.1; I.2.4", CRclass = "F.2.1 Numerical Algorithms and Problems; G.1.2 Approximation; G.1.2 Elementary function approximation; F.1.1 Models of Computation; F.1.1 Unbounded-action devices; I.2.4 Knowledge Representation Formalisms and Methods", descriptor = "Theory of Computation, ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY, Numerical Algorithms and Problems; Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation; Theory of Computation, COMPUTATION BY ABSTRACT DEVICES, Models of Computation, Unbounded-action devices; Computing Methodologies, ARTIFICIAL INTELLIGENCE, Knowledge Representation Formalisms and Methods", fjournal = "Neural Networks", genterm = "design; performance", guideno = "1989-09273", journalabbrev = "Neural Networks", jrldate = "1989", subject = "F. Theory of Computation; F.2 ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; F. Theory of Computation; F.1 COMPUTATION BY ABSTRACT DEVICES; I. Computing Methodologies; I.2 ARTIFICIAL INTELLIGENCE", } @Article{Jamieson:1989:RCI, author = "M. J. Jamieson", title = "Rapidly converging iterative formulae for finding square roots and their computational efficiencies", journal = j-COMP-J, volume = "32", number = "1", pages = "93--94", month = feb, year = "1989", CODEN = "CMPJA6", DOI = "https://doi.org/10.1093/comjnl/32.1.93", ISSN = "0010-4620 (print), 1460-2067 (electronic)", ISSN-L = "0010-4620", MRclass = "65H05", MRnumber = "89k:65063", bibdate = "Tue Mar 25 13:51:56 MST 1997", bibsource = "Compendex database; http://www3.oup.co.uk/computer_journal/hdb/Volume_32/Issue_01/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "This work generalizes the Pythagorean sums in \cite{Dubrulle:1983:CNM,Moler:1983:RSR}.", URL = "http://www3.oup.co.uk/computer_journal/hdb/Volume_32/Issue_01/tiff/93.tif; http://www3.oup.co.uk/computer_journal/hdb/Volume_32/Issue_01/tiff/94.tif", abstract = "A derivation is given of rapidly converging iterative formulae for finding square roots which include, as special cases, some recently published examples. Their computational efficiencies are investigated for sequential and parallel implementation. It is concluded that the most efficient method is equivalent to sequential application of the Newton Raphson formula; a simple modification is suggested which brings the advantage of root bracketing at little extra computational cost.", acknowledgement = ack-nhfb, affiliation = "Dept. of Comput. Sci., Glasgow Univ., UK", affiliationaddress = "Glasgow, Scotl", classcodes = "B0290F (Interpolation and function approximation); C4130 (Interpolation and function approximation)", classification = "723; 921; B0290F (Interpolation and function approximation); C4130 (Interpolation and function approximation)", corpsource = "Dept. of Comput. Sci., Glasgow Univ., UK", fjournal = "The Computer Journal", journal-URL = "http://comjnl.oxfordjournals.org/", keywords = "computational; Computational efficiencies; Computational Efficiency; Computer Metatheory; Convergence; convergence of numerical methods; Converging iterative formulae; converging iterative formulae; efficiencies; formula; function approximation; Iterative Methods; iterative methods; Newton Raphson; Newton Raphson formula, Mathematical Techniques; Parallel implementation; parallel implementation; Square Roots; Square roots; square roots", thesaurus = "Convergence of numerical methods; Function approximation; Iterative methods", treatment = "P Practical", } @Article{Jeffries:1989:GFA, author = "John S. Jeffries and Donald R. Smith", title = "A {Green} function approach for a singularly perturbed vector boundary-value problem", journal = j-ADV-APPL-MATH, volume = "10", number = "1", pages = "1--50", month = mar, year = "1989", CODEN = "????", ISSN = "0196-8858 (print), 1090-2074 (electronic)", ISSN-L = "0196-8858", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, affiliation = "Univ. of California at San Diego, La Jolla; Univ. of California at San Diego, La Jolla", bibno = "64833", catcode = "G.1.7; G.1.7; F.2.1; G.1.2; G.1.3; G.1.2; G.1.3", CRclass = "G.1.7 Ordinary Differential Equations; G.1.7 Boundary value problems; G.1.7 Ordinary Differential Equations; G.1.7 Convergence and stability; F.2.1 Numerical Algorithms and Problems; F.2.1 Computation of transforms; G.1.2 Approximation; G.1.2 Nonlinear approximation; G.1.3 Numerical Linear Algebra; G.1.3 Eigenvalues; G.1.2 Approximation; G.1.2 Elementary function approximation; G.1.3 Numerical Linear Algebra; G.1.3 Linear systems (direct and iterative methods)", descriptor = "Mathematics of Computing, NUMERICAL ANALYSIS, Ordinary Differential Equations, Boundary value problems; Mathematics of Computing, NUMERICAL ANALYSIS, Ordinary Differential Equations, Convergence and stability; Theory of Computation, ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY, Numerical Algorithms and Problems, Computation of transforms; Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Nonlinear approximation; Mathematics of Computing, NUMERICAL ANALYSIS, Numerical Linear Algebra, Eigenvalues; Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation; Mathematics of Computing, NUMERICAL ANALYSIS, Numerical Linear Algebra, Linear systems (direct and iterative methods)", fjournal = "Advances in Applied Mathematics", genterm = "algorithms; theory", guideno = "1989-03271", journal-URL = "http://www.sciencedirect.com/science/journal/01968858", journalabbrev = "Adv. Appl. Math.", jrldate = "March 1989", subject = "F. Theory of Computation; F.2 ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS", } @Article{Johnson:1989:IMA, author = "K. R. Johnson", title = "An Iterative Method for Approximating Square Roots", journal = j-MATH-MAG, volume = "62", number = "4", pages = "253--259", month = oct, year = "1989", CODEN = "MAMGA8", ISSN = "0025-570X", bibdate = "Thu Sep 1 10:15:42 1994", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nj, fjournal = "Mathematics Magazine", journal-URL = "http://www.maa.org/pubs/mathmag.html", } @Article{Kaishev:1989:SSC, author = "A. I. Kaishev", title = "A sharpened scheme for constructing an a posteriori interval extension of an elementary function. ({Russian})", journal = "Voprosy Kibernet. (Moscow)", volume = "149", pages = "14--18", year = "1989", ISBN = "0134-6388", ISBN-13 = "0134-6388", MRclass = "65G10", MRnumber = "91i:65090", MRreviewer = "I. N. Molchanov", bibdate = "Sat Jan 11 17:44:01 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, language = "Russian", } @InProceedings{Kak:1989:BAS, author = "S. C. Kak and A. O. Barbir", booktitle = "Proceedings of the Twenty-First Southeastern Symposium on System Theory, 26--28 March 1989", title = "The {Brahmagupta} algorithm for square rooting", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "456--459", year = "1989", CODEN = "????", ISSN = "????", bibdate = "Sat Jul 16 15:28:13 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, summary = "An algorithm for square root evaluation is introduced. Novel features of the algorithm include suitability for parallel processing and multi-initial guesses of the root. An extension of the algorithm to the nth rooting is provided. A VLSI \ldots{}", } @Article{Kogan:1989:GBF, author = "B. J. Kogan", title = "General background of functional memory algorithms", journal = j-TRANS-SOC-COMP-SIM, volume = "5", number = "4", pages = "285--317", month = oct, year = "1989", CODEN = "TSCSEV", ISSN = "0740-6797", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, affiliation = "Univ. of California, Los Angeles", bibno = "69149", catcode = "C.3; G.1.2; G.1.2; E.4", CRclass = "C.3 Signal processing systems; G.1.2 Approximation; G.1.2 Elementary function approximation; G.1.2 Approximation; G.1.2 Linear approximation; E.4 Data compaction and compression", descriptor = "Computer Systems Organization, SPECIAL-PURPOSE AND APPLICATION-BASED SYSTEMS, Signal processing systems; Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation; Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Linear approximation; Data, CODING AND INFORMATION THEORY, Data compaction and compression", fjournal = "Transactions of the Society for Computer Simulation", genterm = "algorithms; design", guideno = "1989-10680", journalabbrev = "Trans. Soc. Comput. Simul.", jrldate = "Oct. 1989", subject = "C. Computer Systems Organization; C.3 SPECIAL-PURPOSE AND APPLICATION-BASED SYSTEMS; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; E. Data; E.4 CODING AND INFORMATION THEORY", } @InProceedings{Kojima:1989:PDI, author = "M. Kojima and S. Mizuno and A. Yoshise", title = "A primal-dual interior point algorithm for linear programming", crossref = "Megiddo:1989:PMP", bookpages = "x + 158", pages = "29--47", year = "1989", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, bibno = "74173", catcode = "G.1.6; F.2.1; G.1.2; F.2.2; F.2.1; I.1.2; I.1.2", CRclass = "G.1.6 Optimization; G.1.6 Linear programming; F.2.1 Numerical Algorithms and Problems; G.1.2 Approximation; G.1.2 Elementary function approximation; F.2.2 Nonnumerical Algorithms and Problems; F.2.2 Geometrical problems and computations; F.2.1 Numerical Algorithms and Problems; F.2.1 Computations on matrices; I.1.2 Algorithms; I.1.2 Algebraic algorithms; I.1.2 Algorithms; I.1.2 Analysis of algorithms", descriptor = "Mathematics of Computing, NUMERICAL ANALYSIS, Optimization, Linear programming; Theory of Computation, ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY, Numerical Algorithms and Problems; Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation; Theory of Computation, ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY, Nonnumerical Algorithms and Problems, Geometrical problems and computations; Theory of Computation, ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY, Numerical Algorithms and Problems, Computations on matrices; Computing Methodologies, ALGEBRAIC MANIPULATION, Algorithms, Algebraic algorithms; Computing Methodologies, ALGEBRAIC MANIPULATION, Algorithms, Analysis of algorithms", genterm = "algorithms; theory", guideno = "1989-12475", procdate = "March 1-4, 1987", procloc = "Pacific Grove, CA", subject = "G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; F. Theory of Computation; F.2 ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; F. Theory of Computation; F.2 ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY; F. Theory of Computation; F.2 ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY; I. Computing Methodologies; I.1 ALGEBRAIC MANIPULATION; I. Computing Methodologies; I.1 ALGEBRAIC MANIPULATION", } @Article{Kraaikamp:1989:SEP, author = "Cor Kraaikamp", title = "Statistic and ergodic properties of {Minkowski}'s diagonal continued fraction", journal = j-THEOR-COMP-SCI, volume = "65", number = "2", pages = "197--212", day = "28", month = jun, year = "1989", CODEN = "TCSCDI", ISSN = "0304-3975 (print), 1879-2294 (electronic)", ISSN-L = "0304-3975", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, affiliation = "Amsterdam Univ., Amsterdam, The Netherlands and Univ. de Provence, Marseille, France", bibno = "70095", catcode = "G.1.2", CRclass = "G.1.2 Approximation; G.1.2 Elementary function approximation", descriptor = "Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation", fjournal = "Theoretical Computer Science", genterm = "algorithms; theory", guideno = "1989-10594", journal-URL = "http://www.sciencedirect.com/science/journal/03043975", journalabbrev = "Theor. Comput. Sci.", jrldate = "28 June 1989", subject = "G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS", } @Article{Kreider:1989:WSA, author = "K. L. Kreider", title = "A wave splitting approach to time dependent inverse scattering for the stratified cylinder", journal = j-SIAM-J-APPL-MATH, volume = "49", number = "3", pages = "932--943", month = jun, year = "1989", CODEN = "SMJMAP", ISSN = "0036-1399 (print), 1095-712X (electronic)", ISSN-L = "0036-1399", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, bibno = "64953", catcode = "G.1.7; G.1.2; J.2; J.2; F.2.2; G.1.3; I.1.1", CRclass = "G.1.7 Ordinary Differential Equations; G.1.7 Initial value problems; G.1.2 Approximation; G.1.2 Elementary function approximation; J.2 Physics; J.2 Electronics; F.2.2 Nonnumerical Algorithms and Problems; F.2.2 Geometrical problems and computations; G.1.3 Numerical Linear Algebra; G.1.3 Linear systems (direct and iterative methods); I.1.1 Expressions and Their Representation; I.1.1 Simplification of expressions", descriptor = "Mathematics of Computing, NUMERICAL ANALYSIS, Ordinary Differential Equations, Initial value problems; Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation; Computer Applications, PHYSICAL SCIENCES AND ENGINEERING, Physics; Computer Applications, PHYSICAL SCIENCES AND ENGINEERING, Electronics; Theory of Computation, ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY, Nonnumerical Algorithms and Problems, Geometrical problems and computations; Mathematics of Computing, NUMERICAL ANALYSIS, Numerical Linear Algebra, Linear systems (direct and iterative methods); Computing Methodologies, ALGEBRAIC MANIPULATION, Expressions and Their Representation, Simplification of expressions", fjournal = "SIAM Journal on Applied Mathematics", genterm = "algorithms; theory; experimentation; measurement", guideno = "1989-09722", journal-URL = "http://epubs.siam.org/siap", journalabbrev = "SIAM J. Appl. Math.", jrldate = "June 1989", subject = "G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; J. Computer Applications; J.2 PHYSICAL SCIENCES AND ENGINEERING; J. Computer Applications; J.2 PHYSICAL SCIENCES AND ENGINEERING; F. Theory of Computation; F.2 ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; I. Computing Methodologies; I.1 ALGEBRAIC MANIPULATION", } @Article{Lin:1989:ANT, author = "Jinn Tyan Lin", title = "Approximating the normal tail probability and its inverse for use on a pocket calculator", journal = j-APPL-STAT, volume = "38", number = "1", pages = "69--70", year = "1989", CODEN = "APSTAG", ISSN = "0035-9254 (print), 1467-9876 (electronic)", ISSN-L = "0035-9254", MRclass = "62E15", MRnumber = "983 303", bibdate = "Sat Apr 21 10:25:25 MDT 2001", bibsource = "https://www.math.utah.edu/pub/tex/bib/as1980.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "Applied Statistics", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1111/(ISSN)1467-9876/issues", } @PhdThesis{Littlestone:1989:MBL, author = "N. Littlestone", title = "Mistake bounds and logarithmic linear-threshold learning algorithms", type = "{Ph.D} Thesis", school = "University of California at Santa Cruz", address = "Santa Cruz, CA, USA", pages = "????", year = "1989", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, bibno = "76493", catcode = "I.2.6; J.4; H.1.2; G.1.2", CRclass = "I.2.6 Learning; J.4 Psychology; H.1.2 User/Machine Systems; H.1.2 Human information processing; G.1.2 Approximation; G.1.2 Elementary function approximation", descriptor = "Computing Methodologies, ARTIFICIAL INTELLIGENCE, Learning; Computer Applications, SOCIAL AND BEHAVIORAL SCIENCES, Psychology; Information Systems, MODELS AND PRINCIPLES, User/Machine Systems, Human information processing; Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation", genterm = "algorithms; human factors; performance", guideno = "1989-12934", source = "UMI Order No: GAX89-26506", subject = "I. Computing Methodologies; I.2 ARTIFICIAL INTELLIGENCE; J. Computer Applications; J.4 SOCIAL AND BEHAVIORAL SCIENCES; H. Information Systems; H.1 MODELS AND PRINCIPLES; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS", } @Article{Lo:1989:RBP, author = "Shaw-Hwa Lo and Jane-Ling Wang", title = "Representations for the bivariate product limit estimators and the bootstrap versions", journal = j-J-MULTIVAR-ANAL, volume = "28", number = "2", pages = "211--226", month = feb, year = "1989", CODEN = "JMVAAI", ISSN = "0047-259x (print), 1095-7243 (electronic)", ISSN-L = "0047-259X", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, affiliation = "Univ. of California, Davis; Univ. of California, Davis", bibno = "64339", catcode = "G.3; G.1.2; G.1.4; G.1.7", CRclass = "G.3 Statistical computing; G.1.2 Approximation; G.1.2 Elementary function approximation; G.1.4 Quadrature and Numerical Differentiation; G.1.4 Gaussian quadrature; G.1.7 Ordinary Differential Equations; G.1.7 Convergence and stability", descriptor = "Mathematics of Computing, PROBABILITY AND STATISTICS, Statistical computing; Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation; Mathematics of Computing, NUMERICAL ANALYSIS, Quadrature and Numerical Differentiation, Gaussian quadrature; Mathematics of Computing, NUMERICAL ANALYSIS, Ordinary Differential Equations, Convergence and stability", fjournal = "Journal of Multivariate Analysis", genterm = "algorithms; theory; measurement", guideno = "1989-08465", journalabbrev = "J. Multivariate Anal.", jrldate = "February 1989", subject = "G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; G.3 PROBABILITY AND STATISTICS", } @Article{Lorentz:1989:NA, author = "G. G. Lorentz", title = "Notes on approximation", journal = j-J-APPROX-THEORY, volume = "56", number = "3", pages = "360--365", month = mar, year = "1989", CODEN = "JAXTAZ", ISSN = "0021-9045 (print), 1096-0430 (electronic)", ISSN-L = "0021-9045", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, bibno = "72258", catcode = "G.1.1; G.1.2; G.1.1", CRclass = "G.1.1 Interpolation; G.1.1 Smoothing; G.1.2 Approximation; G.1.2 Elementary function approximation; G.1.1 Interpolation; G.1.1 Spline and piecewise polynomial interpolation", descriptor = "Mathematics of Computing, NUMERICAL ANALYSIS, Interpolation, Smoothing; Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation; Mathematics of Computing, NUMERICAL ANALYSIS, Interpolation, Spline and piecewise polynomial interpolation", fjournal = "Journal of Approximation Theory", genterm = "algorithms; theory", guideno = "1989-07816", journal-URL = "http://www.sciencedirect.com/science/journal/00219045", journalabbrev = "J. Approx. Theory", jrldate = "Mar. 1989", subject = "G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS", } @InCollection{Lovelace:1989:SAE, author = "Augusta Ada Lovelace", title = "Sketch of the {Analytical Engine} (1843)", crossref = "Campbell-Kelly:1989:WCB-3", pages = "89--170", year = "1989", bibdate = "Tue Jan 22 17:54:41 2013", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/l/lovelace-ada-augusta.bib; https://www.math.utah.edu/pub/tex/bib/adabooks.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, keywords = "Bernoulli numbers", } @InProceedings{Lu:1989:VMI, author = "P. Y. Lu and K. Dawallu", title = "A {VLSI} Module for {IEEE} Floating-Point Multiplication\slash Division\slash Square Root", crossref = "IEEE:1989:PII", bookpages = "xvii + 587", pages = "366--368", year = "1989", bibdate = "Wed Nov 06 12:08:38 2002", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", abstract = "The major objective of this VLSI module design is to determine how to modify a fast floating-point multiplier so that it can perform division and square root in accordance with IEEE standards. This has been achieved by applying the Newton-Ralphson iteration only on the mantissa and adjusting the iterated result by a rounding algorithm. Using 1.0- mu m CMOS standard cell technology, the total area of this module is approximately 7.0 mm*6.5 mm, which is just 25\% larger than the floating-point multiplier. The module can compute multiplication, division, and square root in 3, 31, and 43 cycles, respectively. The cycle time, under nominal conditions, is expected to be 20 ns. (2 Refs.)", acknowledgement = ack-nhfb # " and " # ack-nj, affiliation = "LSI Logic Corp., Menlo Park, CA, USA", classification = "B1265B (Logic circuits); B2570D (CMOS integrated circuits); C4130 (Interpolation and function approximation); C5230 (Digital arithmetic methods)", keywords = "1 Micron; 20 Ns; 7 To 6.5 mm; CMOS standard cell technology; Cycle time; Fast floating-point multiplier; Floating point division; Floating point square root; IEEE standards; Iterated result; Mantissa; Multiplier modification; Newton-Ralphson iteration; Rounding algorithm; VLSI module design", numericalindex = "Time 2.0E-08 s; Size 1.0E-06 m; Size 6.5E-03 to 7.0E-03 m", thesaurus = "Cellular arrays; CMOS integrated circuits; Digital arithmetic; Dividing circuits; Iterative methods; Modules; Multiplying circuits; VLSI", } @Article{Macleod:1989:SAA, author = "Allan J. Macleod", title = "Statistical Algorithms: {Algorithm AS 245}: a Robust and Reliable Algorithm for the Logarithm of the Gamma Function", journal = j-APPL-STAT, volume = "38", number = "2", pages = "397--402", month = jun, year = "1989", CODEN = "APSTAG", ISSN = "0035-9254 (print), 1467-9876 (electronic)", ISSN-L = "0035-9254", bibdate = "Sat Apr 21 10:25:27 MDT 2001", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", URL = "http://lib.stat.cmu.edu/apstat/245", acknowledgement = ack-nhfb, fjournal = "Applied Statistics", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1111/(ISSN)1467-9876/issues", } @InProceedings{Mansour:1989:CAS, author = "Y. Mansour and B. Schieber and P. Tiwari", title = "The complexity of approximating the square root", crossref = "IEEE:1989:ASF", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "325--330", year = "1989", CODEN = "????", ISSN = "????", bibdate = "Sat Jul 16 15:28:13 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, summary = "The authors prove upper and lower bounds for approximately computing the square root using a given set of operations. The bounds are extended to hold for approximating the kth root, for any fixed k. Several tools from approximation \ldots{}", } @Article{Martin:1989:TPQ, author = "Pablo Martin and Antonio Luis Guerrero", title = "Two-point quasi-fractional approximations to the {Bessel} function {$ J_\nu (x) $} of fractional order", journal = j-J-COMPUT-PHYS, volume = "85", number = "2", pages = "487--492", month = dec, year = "1989", CODEN = "JCTPAH", DOI = "https://doi.org/10.1016/0021-9991(89)90161-7", ISSN = "0021-9991 (print), 1090-2716 (electronic)", ISSN-L = "0021-9991", bibdate = "Sun Jan 1 15:59:48 MST 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputphys1980.bib", URL = "http://www.sciencedirect.com/science/article/pii/0021999189901617", acknowledgement = ack-nhfb, fjournal = "Journal of Computational Physics", journal-URL = "http://www.sciencedirect.com/science/journal/00219991/", remark = "This work produces only 3D approximations.", } @InProceedings{Megiddo:1989:POS, author = "N. Megiddo", title = "Pathways to the optimal set in linear programming", crossref = "Megiddo:1989:PMP", pages = "131--158", year = "1989", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, bibno = "74179", catcode = "G.1.6; G.1.2; G.2.2; G.1.2; G.1.5; F.2.1; I.1.2; G.4; G.4", CRclass = "G.1.6 Optimization; G.1.6 Linear programming; G.1.2 Approximation; G.1.2 Elementary function approximation; G.2.2 Graph Theory; G.2.2 Path and circuit problems; G.1.2 Approximation; G.1.2 Spline and piecewise polynomial approximation; G.1.5 Roots of Nonlinear Equations; G.1.5 Iterative methods; F.2.1 Numerical Algorithms and Problems; F.2.1 Computations on matrices; I.1.2 Algorithms; I.1.2 Analysis of algorithms; G.4 Algorithm analysis; G.4 Efficiency", descriptor = "Mathematics of Computing, NUMERICAL ANALYSIS, Optimization, Linear programming; Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation; Mathematics of Computing, DISCRETE MATHEMATICS, Graph Theory, Path and circuit problems; Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Spline and piecewise polynomial approximation; Mathematics of Computing, NUMERICAL ANALYSIS, Roots of Nonlinear Equations, Iterative methods; Theory of Computation, ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY, Numerical Algorithms and Problems, Computations on matrices; Computing Methodologies, ALGEBRAIC MANIPULATION, Algorithms, Analysis of algorithms; Mathematics of Computing, MATHEMATICAL SOFTWARE, Algorithm analysis; Mathematics of Computing, MATHEMATICAL SOFTWARE, Efficiency", genterm = "algorithms; performance; theory", guideno = "1989-12481", procdate = "March 1-4, 1987", procloc = "Pacific Grove, CA", subject = "G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; G. Mathematics of Computing; G.2 DISCRETE MATHEMATICS; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; F. Theory of Computation; F.2 ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY; I. Computing Methodologies; I.1 ALGEBRAIC MANIPULATION; G. Mathematics of Computing; G.4 MATHEMATICAL SOFTWARE; G. Mathematics of Computing; G.4 MATHEMATICAL SOFTWARE", } @PhdThesis{Miler:1989:EEM, author = "T. H. Miler", title = "Error evaluation of microcomputer intrinsic functions", type = "{Ph.D} Thesis", school = "University of Idaho", address = "Moscow, ID", pages = "????", year = "1989", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, bibno = "76168", catcode = "G.4; G.1.2; J.2", CRclass = "G.4 Reliability and robustness; G.1.2 Approximation; G.1.2 Elementary function approximation; J.2 Mathematics and statistics", descriptor = "Mathematics of Computing, MATHEMATICAL SOFTWARE, Reliability and robustness; Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation; Computer Applications, PHYSICAL SCIENCES AND ENGINEERING, Mathematics and statistics", genterm = "algorithms; reliability", guideno = "1989-12941", source = "UMI order no: GAX89-22813", subject = "G. Mathematics of Computing; G.4 MATHEMATICAL SOFTWARE; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; J. Computer Applications; J.2 PHYSICAL SCIENCES AND ENGINEERING", } @InProceedings{Montuschi:1989:EIH, author = "Paolo Montuschi and Luigi Cinimera", title = "On the efficient implementation of higher radix square root algorithms", crossref = "Ercegovac:1989:PSC", pages = "154--161", year = "1989", bibdate = "Sat Jul 16 15:28:13 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "http://www.acsel-lab.com/arithmetic/arith9/papers/ARITH9_Montuschi.pdf", acknowledgement = ack-nhfb, keywords = "ARITH-9", summary = "Square root nonrestoring algorithms operating with a radix higher than two (but power of 2) are discussed. Formulas are derived delimiting the feasibility space of the class of algorithms considered as a function of the different parameters. This \ldots{}", } @Book{Moshier:1989:MPM, author = "Stephen L. B. Moshier", title = "Methods and Programs for Mathematical Functions", publisher = pub-ELLIS-HORWOOD, address = pub-ELLIS-HORWOOD:adr, pages = "vii + 415", year = "1989", ISBN = "0-7458-0289-3", ISBN-13 = "978-0-7458-0289-3", LCCN = "QA331 .M84 1989", MRclass = "*65D20, 26-04, 33-04, 65-02, 65C99", bibdate = "Thu Sep 01 10:33:40 1994", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/master.bib", price = "US\pounds 48.00", URL = "http://www.moshier.net/; http://www.netlib.org/cephes", ZMnumber = "0701.65011", acknowledgement = ack-nj, shorttableofcontents = "Preface / vii \\ 1: Floating Point Arithmetic / 1 \\ 2: Approximation Methods / 75 \\ 3: Software Notes / 129 \\ 4: Elementary Functions / 143 \\ 5: Probability Distributions and Related Functions / 201 \\ 6: Bessel Functions / 263 \\ 7: Other Special Functions / 333 \\ Bibliography / 411 \\ Index / 413", tableofcontents = "Preface / vii \\ 1: Floating Point Arithmetic / 1 \\ 1.1 Numeric Data Structures / 1 \\ 1.2 Rounding / 5 \\ 1.3 Addition and Subtraction / 6 \\ 1.4 Multiplication / 7 \\ 1.4.1 Long Multiplication in Binary Radix / 8 \\ 1.4.2 Multiplication in Word Integer Radix / 8 \\ 1.4.3 Fast Multiplication / 9 \\ 1.5 Division / 10 \\ 1.5.1 Long Division / 10 \\ 1.5.2 Division by Taylor Series / 11 \\ 1.5.3 Newton--Raphson Division / 11 \\ 1.6 C Language / 12 \\ 1.7 An Extended Double Arithmetic: ieee.c / 13 \\ 1.8 Binary - Decimal Conversion / 46 \\ 1.8.1 etoasc.c / 47 \\ 1.8.2 asctoe.c / 54 \\ 1.9 Analysis of Error / 58 \\ 1.9.1 Roundoff and Cancellation / 58 \\ 1.9.2 Error Propagation / 60 \\ 1.9.3 Error as a Random Variable / 61 \\ 1.9.4 Order of Summation / 62 \\ 1.10 Complex Arithmetic / 62 \\ 1.10.1 cmplx.c / 64 \\ 1.10.2 Absolute Value: cabs.c / 67 \\ 1.11 Rational Arithmetic / 69 \\ 1.11.1 euclid.c / 70 \\ 2: Approximation Methods / 75 \\ 2.1 Power Series / 75 \\ 2.2 Chebyshev Expansions / 76 \\ 2.2.1 chbevl.c / 79 \\ 2.3 Pad{\'e} Approximations / 80 \\ 2.4 Least Maximum Approximations / 82 \\ 2.4.1 Best Polynomial Approximations / 82 \\ 2.4.2 Best Rational Approximations / 85 \\ 2.4.3 Special Rational Forms / 87 \\ 2.5 A Program to Find Best Approximations: remes.c / 88 \\ 2.6 Forms of Approximation / 111 \\ 2.7 Asymptotic Expansions / 113 \\ 2.8 Continued Fractions / 114 \\ 2.8.1 Continued Fractions from Recurrences / 115 \\ 2.8.2 Recurrences from Differential Equations / 116 \\ 2.8.3 Computing Continued Fractions / 117 \\ 2.9 Polynomials / 117 \\ 2.9.1 polevl.c / 118 \\ 2.10 Newton--Raphson Iterations / 119 \\ 2.10.1 Division / 120 \\ 2.10.2 Exponent Separation / 121 \\ 2.10.3 Square Root / 122 \\ 2.10.4 sqrt.c / 123 \\ 2.10.5 Longhand Square Root / 124 \\ 2.10.6 esqrt.c / 124 \\ 2.10.7 Cube Root / 126 \\ 2.10.8 cbrt.c / 127 \\ 3: Software Notes / 129 \\ 3.1 Design Strategy / 129 \\ 3.2 Testing / 131 \\ 3.3 System Utilities / 132 \\ 3.3.1 mconf.h / 132 \\ 3.3.2 mtherr.c / 134 \\ 3.3.3 const.c / 136 \\ 3.4 Arithmetic Utilities / 137 \\ 3.4.1 efloor.c / 138 \\ 3.4.2 efrexp.c / 140 \\ 3.4.3 eldexp.c / 140 \\ 4: Elementary Functions / 143 \\ 4.1 $e^x$ / 143 \\ 4.1.1 exp.c / 145 \\ 4.2 $\ln x$ / 147 \\ 4.2.1 log.c / 149 \\ 4.3 Argument Transformation for Circular Functions / 152 \\ 4.4 Sine and cosine / 153 \\ 4.4.1 sin.c / 154 \\ 4.4.2 cos.c / 156 \\ 4.5 Tangent and Cotangent / 157 \\ 4.5.1 tan.c / 158 \\ 4.6 Complex Circular Functions / 161 \\ 4.7 $\sin^{-1} x $ / 162 \\ 4.7.1 asin.c / 163 \\ 4.8 $\cos^{-1} x $ / 165 \\ 4.8.1 acos.c / 165 \\ 4.9 $\tan^{-1} x$ / 166 \\ 4.9.1 atan.c / 168 \\ 4.9.2 atan2.c / 169 \\ 4.10 Complex Inverse Circular Functions / 170 \\ 4.11 $\sinh x$ / 170 \\ 4.11.1 sinh.c / 171 \\ 4.12 $\cosh x$ / 172 \\ 4.12.1 cosh.c / 173 \\ 4.13 $\tanh x$ / 173 \\ 4.13.1 tanh.c / 174 \\ 4.14 $\sinh^{-1} x $ / 175 \\ 4.14.1 asinh.c / 176 \\ 4.15 $\cosh^{-1} x $ / 177 \\ 4.15.1 acosh.c / 178 \\ 4.16 $\tanh^{-1} x$ / 179 \\ 4.16.1 atanh.c / 180 \\ 4.17 Power Function / 181 \\ 4.17.1 Real Exponent / 182 \\ 4.17.2 pow.c / 182 \\ 4.17.3 Integer Exponent / 189 \\ 4.17.4 powi.c / 190 \\ 4.18 Testing / 192 \\ 4.19 Single Precision Polynomial Approximations / 193 \\ 4.19.1 $\cos x$ / 193 \\ 4.19.2 $\cosh^{-1} x $ / 193 \\ 4.19.3 $\exp x$ / 196 \\ 4.19.4 $\ln x$ / 196 \\ 4.19.5 $\sin x$ / 197 \\ 4.19.6 $\sin^{-1} x $ / 197 \\ 4.19.7 Square Root / 197 \\ 4.19.8 $\tan x$ / 198 \\ 4.19.9 $\tan^{-1} x$ / 198 \\ 4.19.10 $\tanh x$ / 199 \\ 4.19.11 $tanh^{-1} x$ / 199 \\ 5: Probability Distributions and Related Functions / 201 \\ 5.1 $n!$ / 202 \\ 5.1.1 fac.c / 204 \\ 5.2 $\Gamma(x)$ / 206 \\ 5.2.1 gamma.c / 210 \\ 5.2.2 lgam.c / 214 \\ 5.3 Incomplete Gamma Integral / 217 \\ 5.3.1 igamc.c / 218 \\ 5.3.2 igam.c / 220 \\ 5.3.3 Functional Inverse of Incomplete Gamma Integral / 221 \\ 5.3.4 igami.c / 221 \\ 5.4 Gamma Distribution / 222 \\ 5.4.1 gdtr c / 222 \\ 5.4.2 gdtrc.c / 223 \\ 5.5 $\chi^2$ Distribution / 223 \\ 5.5.1 chdtrc.c / 224 \\ 5.5.2 chdtr.c / 224 \\ 5.5.3 chdtrl.c / 224 \\ 5.6 Poisson Distribution / 225 \\ 5.6.1 pdtrc.c / 225 \\ 5.6.2 pdtr.c / 226 \\ 5.6.3 pdtri.c / 226 \\ 5.7 Beta Function / 227 \\ 5.7.1 beta.c / 227 \\ 5.8 Incomplete Beta Integral / 229 \\ 5.8.1 ibet.c / 231 \\ 5.8.2 Functional Inverse of Incomplete Beta Integral / 238 \\ 5.9 Beta Distribution / 241 \\ 5.9.1 btdtr.c / 241 \\ 5.10 Binomial Distribution / 241 \\ 5.10.1 bdtrc.c / 242 \\ 5.10.2 bdtr.c / 243 \\ 5.10.3 bdtri.c / 244 \\ 5.11 Negative Binomial Distribution / 244 \\ 5.11.1 nbdtr.c / 245 \\ 5.11.2 nbdtrc.c / 245 \\ 5.12 F Distribution / 246 \\ 5.12.1 fdtrc.c / 247 \\ 5.12.2 fdtr.c / 247 \\ 5.12.3 fdtrci.c / 248 \\ 5.13 Student's $t$ distribution / 249 \\ 5.13.1 stdtr.c / 250 \\ 5.14 Gaussian Distribution / 252 \\ 5.14.1 ndtr.c / 254 \\ 5.14.2 erfc.c / 256 \\ 5.14.3 erf.c / 257 \\ 5.14.4 Functional Inverse of Gaussian Distribution / 258 \\ 5.14.5 ndtri.c / 259 \\ 6: Bessel Functions / 263 \\ 6.1 $J_0(x)$ / 263 \\ 6.1.1 jO.c / 265 \\ 6.2 $Y_0(x)$ / 268 \\ 6.2.1 yO.c / 269 \\ 6.3 Modulus and Phase / 270 \\ 6.4 $J_1(x)$ / 271 \\ 6.4.1 jl.c / 272 \\ 6.5 $Y_1(x)$ / 275 \\ 6.5.1 yl.c / 275 \\ 6.6 $J_n(x)$ / 276 \\ 6.1 $I_0(x)$ / 277 \\ 6.7.1 i0.c / 278 \\ 6.8 $I_1(x)$ / 281 \\ 6.8.1 i1.c / 283 \\ 6.9 $I_\nu(x)$ / 285 \\ 6.9.1 iv.c / 286 \\ 6.10 $K_0(x)$ / 287 \\ 6.10.1 kO.c / 287 \\ 6.11 $K_1(x)$ / 291 \\ 6.11.1 kl.c / 291 \\ 6.12 $K_n(x)$ / 294 \\ 6.12.1 kn.c / 295 \\ 6.13 $J_\nu(x)$ / 299 \\ 6.13.1 jv.c / 301 \\ 6.14 Airy Functions / 315 \\ 6.14.1 airy.c / 322 \\ 6.15 $Y_n(x)$ / 328 \\ 6.15.1 yn.c / 329 \\ 6.16 Testing / 330 \\ 7: Other Special Functions / 333 \\ 7.1 Hypergeometric Functions / 333 \\ 7.1.1 $_2F_1$ / 334 \\ 7.1.2 hyp2fi.c / 335 \\ 7.1.3 $_1F_1$ / 341 \\ 7.1.4 hyplfi.c / 342 \\ 7.1.5 $_2F_0$ / 346 \\ 7.1.6 hyp2ffi.c / 346 \\ 7.2 Struve Functions / 348 \\ 7.2.1 hypl1f2.c / 348 \\ 7.2.2 hyp3f0.c / 349 \\ 7.2.3 yv.c / 351 \\ 7.2.4 struve.c / 351 \\ 7.3 $\psi(x)$ / 352 \\ 7.3.1 psi.c / 354 \\ 7.4 Exponential Integral / 355 \\ 7.4.1 en.c / 356 \\ 7.5 Sine and Cosine Integrals / 360 \\ 7.5.1 sici.c / 362 \\ 7.5.2 Hyperbolic Sine and Cosine Integrals / 367 \\ 7.5.3 shichi.c / 370 \\ 7.6 Dilogarithm / 374 \\ 7.6.1 spence.c / 375 \\ 7.7 Dawson's Integral / 377 \\ 7.7.1 dawsn.c / 378 \\ 7.8 Fresnel Integrals / 381 \\ 7.8.1 fresnl.c / 383 \\ 7.9 Elliptic Functions / 387 \\ 7.9.1 $K(m)$ / 387 \\ 7.9.2 ellpk.c / 388 \\ 7.9.3 $F(\phi|m)$ / 389 \\ 7.9.4 ellik.c / 390 \\ 7.9.5 $E(m)$ / 392 \\ 7.9.6 ellpe.c / 392 \\ 7.9.7 $E(\phi|m)$ / 393 \\ 7.9.8 ellie.c / 394 \\ 7.9.9 Jacobian Elliptic Functions / 396 \\ 7.9.10 ellpj.c / 398 \\ 7.10 Zeta Functions / 400 \\ 7.10.1 hurwiz.c / 400 \\ 7.10.2 Riemann Zeta Function / 402 \\ 7.10.3 zetac.c / 405 \\ Bibliography / 411 \\ Index / 413", } @Article{Norton:1989:PCA, author = "Robert M. Norton", title = "Pocket-Calculator Approximation for Areas under the Standard Normal Curve", journal = j-AMER-STAT, volume = "43", number = "1", pages = "24--26", month = feb, year = "1989", CODEN = "ASTAAJ", ISSN = "0003-1305 (print), 1537-2731 (electronic)", ISSN-L = "0003-1305", bibdate = "Fri Jan 27 12:40:30 MST 2012", bibsource = "http://www.jstor.org/journals/00031305.html; http://www.jstor.org/stable/i326443; https://www.math.utah.edu/pub/tex/bib/amstat1980.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.jstor.org/stable/2685163", acknowledgement = ack-nhfb, fjournal = "The American Statistician", journal-URL = "http://www.tandfonline.com/loi/utas20", } @Article{Rhee:1989:MII, author = "W. T. Rhee and M. Talagrand", title = "Martingale inequalities, interpolation and {NP}-complete problems", journal = j-MATH-OP-RES, volume = "14", number = "1", pages = "91--96", month = feb, year = "1989", CODEN = "MOREDQ", ISSN = "0364-765x (print), 1526-5471 (electronic)", ISSN-L = "0364-765X", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, affiliation = "Univ., Paris, Paris, France; Ohio State Univ., Columbus", bibno = "67063", catcode = "G.2.2; H.1.1; G.1.1; G.1.2; G.3; I.6.4; F.1.3", CRclass = "G.2.2 Graph Theory; G.2.2 Path and circuit problems; H.1.1 Systems and Information Theory; H.1.1 General systems theory; G.1.1 Interpolation; G.1.1 Interpolation formulas; G.1.2 Approximation; G.1.2 Elementary function approximation; G.3 Probabilistic algorithms (including Monte Carlo); I.6.4 Model Validation and Analysis; F.1.3 Complexity Classes; F.1.3 Reducibility and completeness", descriptor = "Mathematics of Computing, DISCRETE MATHEMATICS, Graph Theory, Path and circuit problems; Information Systems, MODELS AND PRINCIPLES, Systems and Information Theory, General systems theory; Mathematics of Computing, NUMERICAL ANALYSIS, Interpolation, Interpolation formulas; Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation; Mathematics of Computing, PROBABILITY AND STATISTICS, Probabilistic algorithms (including Monte Carlo); Computing Methodologies, SIMULATION AND MODELING, Model Validation and Analysis; Theory of Computation, COMPUTATION BY ABSTRACT DEVICES, Complexity Classes, Reducibility and completeness", fjournal = "Mathematics of Operations Research", genterm = "algorithms; theory; measurement", guideno = "1989-09079", journal-URL = "http://pubsonline.informs.org/loi/moor", journalabbrev = "Math. Oper. Res.", jrldate = "Feb. 1989", subject = "F. Theory of Computation; F.1 COMPUTATION BY ABSTRACT DEVICES; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; G.2 DISCRETE MATHEMATICS; G.3 PROBABILITY AND STATISTICS; H. Information Systems; H.1 MODELS AND PRINCIPLES; I. Computing Methodologies; I.6 SIMULATION AND MODELING", } @Article{Ruymgaart:1989:SPB, author = "F. H. Ruymgaart", title = "Some properties of bivariate empirical hazard processes under random censoring", journal = j-J-MULTIVAR-ANAL, volume = "28", number = "2", pages = "271--281", month = feb, year = "1989", CODEN = "JMVAAI", ISSN = "0047-259x (print), 1095-7243 (electronic)", ISSN-L = "0047-259X", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, bibno = "64343", catcode = "G.3; G.1.2", CRclass = "G.3 Statistical computing; G.1.2 Approximation; G.1.2 Elementary function approximation", descriptor = "Mathematics of Computing, PROBABILITY AND STATISTICS, Statistical computing; Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation", fjournal = "Journal of Multivariate Analysis", genterm = "algorithms; theory; measurement", guideno = "1989-08469", journalabbrev = "J. Multivariate Anal.", jrldate = "February 1989", subject = "G. Mathematics of Computing; G.3 PROBABILITY AND STATISTICS; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS", } @Article{Rybicki:1989:DIS, author = "George B. Rybicki", title = "{Dawson}'s Integral and the Sampling Theorem", journal = j-COMPUT-PHYS, volume = "3", number = "2", pages = "85--87", month = mar, year = "1989", CODEN = "CPHYE2", DOI = "https://doi.org/10.1063/1.4822832", ISSN = "0894-1866 (print), 1558-4208 (electronic)", ISSN-L = "0894-1866", bibdate = "Wed Apr 10 08:45:17 MDT 2019", bibsource = "https://www.math.utah.edu/pub/tex/bib/computphys.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "https://aip.scitation.org/doi/10.1063/1.4822832", acknowledgement = ack-nhfb, ajournal = "Comput. Phys", fjournal = "Computers in Physics", journal-URL = "https://aip.scitation.org/journal/cip", } @InCollection{Saigo:1989:FID, author = "Megumi Saigo", title = "Fractional integrals and derivatives associated with elementary functions and {Bessel} functions", crossref = "Srivastava:1989:UFF", pages = "283--306", year = "1989", MRclass = "26A33 (33C10)", MRnumber = "93h:26011", bibdate = "Sat Jan 11 17:44:01 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", series = "Ellis Horwood Ser. Math. Appl.", acknowledgement = ack-nhfb, } @Article{Sala:1989:TJA, author = "Kenneth L. Sala", title = "Transformations of the {Jacobian} amplitude function and its calculation via the arithmetic-geometric mean", journal = j-SIAM-J-MATH-ANA, volume = "20", number = "6", pages = "1514--1528", month = nov, year = "1989", CODEN = "SJMAAH", ISSN = "0036-1410 (print), 1095-7154 (electronic)", ISSN-L = "0036-1410", MRclass = "33A25 (42A16 70D99)", MRnumber = "90j:33003", MRreviewer = "J. M. H. Peters", bibdate = "Sun Nov 28 19:24:55 MST 2010", bibsource = "http://epubs.siam.org/sam-bin/dbq/toc/SIMA/20/6; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "SIAM Journal on Mathematical Analysis", journal-URL = "http://epubs.siam.org/sima", } @Article{Smith:1989:EMP, author = "David M. Smith", title = "Efficient multiple-precision evaluation of elementary functions", journal = j-MATH-COMPUT, volume = "52", number = "185", pages = "131--134", month = jan, year = "1989", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "65D15 (26-04)", MRnumber = "90c:65034", MRreviewer = "Menachem Dishon", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, classcodes = "C4120 (Functional analysis)", corpsource = "Dept. of Math., Loyola Univ., Los Angeles, CA, USA", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "arithmetic; base b; elementary functions; function evaluation; multiple-precision evaluation", treatment = "T Theoretical or Mathematical", } @InProceedings{Stearns:1989:SFD, author = "C. C. Stearns", title = "Subtractive floating-point division and square root for {VLSI DSP}", crossref = "IEE:1989:EEC", pages = "405--409", year = "1989", bibdate = "Tue Dec 12 09:17:24 MST 1995", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", abstract = "This paper describes recent architectural developments in VLSI design for real-time digital signal processing. In particular, floating point division and floating point square root architectures applicable to both adaptive filtering, standard deviation computations, and general purpose processing are discussed. Emphasis here is on the internal architectures of the arithmetic units not on their applications. The research presented in this paper has been proven feasible and reliable from extensive gate-level simulation and fabrication in silicon.", acknowledgement = ack-nhfb, classification = "B1265F (Microprocessors and microcomputers); B1270F (Digital filters); B2570D (CMOS integrated circuits); C5230 (Digital arithmetic methods); C5240 (Digital filters); C5260 (Digital signal processing)", keywords = "Adaptive filtering; Arithmetic units; CMOS technology; Floating point division; Floating point square root architectures; Gate-level simulation; General purpose processing; Real-time digital signal processing; Semiconductor; Standard deviation computations; VLSI DSP", thesaurus = "Adaptive filters; CMOS integrated circuits; Digital arithmetic; Digital signal processing chips; VLSI", } @TechReport{Tang:1989:TCA, author = "Ping Tak Peter Tang", title = "Testing Computer Arithmetic by Elementary Number Theory", institution = "Mathematics and Computer Science Division, Argonne National Laboratory", address = "Argonne, IL, USA", pages = "????", month = aug, year = "1989", bibdate = "Fri Jun 11 12:38:06 1999", bibsource = "garbo.uwasa.fi:/pc/doc-soft/fpbiblio.txt; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nj, } @Article{Tang:1989:TDI, author = "Ping Tak Peter Tang", title = "Table-Driven Implementation of the Exponential Function in {IEEE} Floating-Point Arithmetic", journal = j-TOMS, volume = "15", number = "2", pages = "144--157", month = jun, year = "1989", CODEN = "ACMSCU", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Sun Sep 04 22:47:40 1994", bibsource = "garbo.uwasa.fi:/pc/doc-soft/fpbiblio.txt; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://doi.acm.org/10.1145/63522.214389; http://www.acm.org/pubs/citations/journals/toms/1989-15-2/p144-tang/", abstract = "Algorithms and implementation details for the exponential function in both single- and double-precision of IEEE 754 arithmetic are presented here. With a table of moderate size, the implementations need only working-precision arithmetic and are provably accurate to within 0.54 ulp as long as the final result does not underflow. When the final result suffers gradual underflow, the error is still no worse than 0.77 ulp.", acknowledgement = ack-nj, fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", keywords = "algorithms", subject = "{\bf G.1.0}: Mathematics of Computing, NUMERICAL ANALYSIS, General, Computer arithmetic. {\bf G.1.0}: Mathematics of Computing, NUMERICAL ANALYSIS, General, Error analysis. {\bf G.1.0}: Mathematics of Computing, NUMERICAL ANALYSIS, General, Numerical algorithms. {\bf G.4}: Mathematics of Computing, MATHEMATICAL SOFTWARE, Algorithm analysis.", } @TechReport{Thomas:1989:SNL, author = "Marlin A. Thomas and Gary W. Gemmill and John R. Crigler", title = "{STATLIB}: {NSWC} Library of Statistical Programs and Subroutines", type = "Technical Report", number = "NSWC TR 89-97", institution = "Naval Surface Warfare Center", address = "Dahlgren, VA 22448-5000, USA and Silver Spring, MD 20903-5000, USA", pages = "viii + 280", month = aug, year = "1989", bibdate = "Sat Nov 15 10:39:12 2014", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/prng.bib", URL = "http://www.dtic.mil/dtic/tr/fulltext/u2/a221538.pdf", abstract = "This document provides a description of each program and subroutine in STATLIB, THE Naval Surface Warfare Center library of statistical programs and subroutines for its general purpose computers. The Library contains thirty-four programs and twenty-four subroutines for statistical analysis, probability evaluation, and random number generation. It was written to enable Center Scientists and Engineers to efficiently perform a wide variety of analyses and to generate pseudo random numbers from many different probability distributions.", acknowledgement = ack-nhfb, onlinedate = "255", tableofcontents = "Introduction / 1 \\ Overview of STATLIB / 1 \\ Origin of STATLIB / 1 \\ Establishment of STATLIB / 1 \\ Commercial Statistical Packages at NSWC / 2 \\ Using STATLIB / 5 \\ Library Organization / 5 \\ How to Call It / 5 \\ Information Needed to Run It / 5 \\ Examples / 9 \\ Program / 9 \\ Subroutine / 10 \\ Descriptions and Input Guides / 21 \\ Programs / 23 \\ Regression Analysis / 25 \\ GEMREG General Multiple Regression / 29 \\ DAMRCA Dahlgren Multiple Regression Comprehensive Analysis / 35 \\ WEPORU Uncorrelated Weighted Polynomial Regression / 41 \\ WEPORC Correlated Weighted Polynomial Regression / 45 \\ MROP Multiple Regression Using Orthogonal Polynomials / 49 \\ CANON Canonical Analysis of Second Order Response Functions / 57 \\ DURBWAT Durbin--Watson Test for Independence of Residuals / 61 \\ NEARNEB Near Neighbor Estimation of Experimental Error / 63 \\ Goodness of Fit Analysis / 67 \\ UNORGOF Univariate Normal Goodness of Fit / 69 \\ BNORGOF Bivariate Normal Goodness of Fit / 75 \\ EXPGOF Exponential Goodness of Fit / 81 \\ WBLGOF Weibull Goodness ot / 83 \\ PERGOF Pearson System Goodness of Fit / 87 \\ UNKSGOF Univariate Normal Kolmogorov--Smirnov Test of Fit / 93 \\ RANDOM Test of Fit for Uniform Random Number Generators / 99 \\ Power Evaluation / 103 \\ DISCRETE POWER EVALUATION / 107 \\ BINIPOW Power of the Test on a Binomial Proportion / 109 \\ BIN2POW Power of the Test on the Difference of Two Binomial Proportions / 113 \\ POIIPOW Power of the Test on the Poisson Parameter / 121 \\ Continuous Power Evaluation / 125 \\ NORIPOW Power of the One-Sample Normal Test on the Mean / 127 \\ NOR2PWE Power of the Two-Simple Normal Test on Means with ample Sizes / 131 \\ NOR2PWU Power of the Two-Sample Normal Test on Means with Unequal Sample Sizes / 135 \\ T1POW Power of the One-Sample $t$ Test on the Mean / 141 \\ T2POW Power of the Two-Sample (Pooled) $t$ Test on Means / 147 \\ CHIVPOW Power of the Chi-square Test on the Variance / 153 \\ FVARPOW Power of the $F$ Test for the Equality of Variances / 157 \\ FEMPOW Power of the Test for One-Way Fixed Effects Analysis of Variance / 163 \\ REMPOW Power of the Test for One-Way Random Effects Analysis of Variance / 167 \\ Probability Evaluation / 171 \\ BINVARP Binomial Probability Distribution with Unequal Single Trial Probabilities / 173 \\ NEGBIN Negative Binomial Probability Distribution / 177 \\ Confidence Limit Evaluation / 179 \\ BINCL Confidence Limits for the Binomial Parameter p / 181 \\ CEPCL Confidence Limits for the CEP (Circular Probable Error) / 185 \\ SEPCL Confidence Limits for the SEP (Spherical Probable Error) / 191 \\ Miscellaneous Statistical Analysis / 197 \\ LD50EST Estimation of LD50 (Lethal Dose 50th Percentile) / 199 \\ FFAC2K Analysis of the 2**k Fractional Factorial Experiment / 203 \\ Subroutines / 211 \\ Random Number Generation / 213 \\ Discrete Random Number Generators, / 217 \\ RANARB Arbitrary (User Specified) Discrete Distribution / 219 \\ RANBER Bernoulli Distribution / 221 \\ RANBIN Binomial Distribution / 223 \\ RANGEO Geometric Distribution / 225 \\ RANHYP Hypergeometric Distribution / 227 \\ RANNBI Negative Binomial Distribution / 229 \\ RANPOI Poisson Distribution / 231 \\ RANUWO Discrete Uniform Distribution (Without Replacement) / 233 \\ RANUWR Discrete Uniform Distribution (With Replacement) / 235 \\ Continuous Random Number Generators / 237 \\ RANBET Beta Distribution / 239 \\ RANCSQ Chi-square Distribution / 241 \\ RANEXP Exponential Distribution / 243 \\ RANFDI $F$ Distribution / 245 \\ RANGAM Gamma Distribution / 247 \\ RANLGS Logistic Distribution / 249 \\ RANLOG Lognormal Distribution / 251 \\ RANNOR Normal Distribution / 255 \\ RANNVE Multivariate Normal Distribution / 257 \\ RANPDI Pearson Distributions / 261 \\ RANTDI Student's $t$ Distribution / 265 \\ RANUNI Continuous Uniform Distribution (On a Line) / 267 \\ RANCIR Continuous Uniform Distribution (Within a Circle) / 269 \\ RANWEI Three-parameter Weibull Distribution / 271 \\ RANMK1 1st Order Markov Process / 273 \\ Glossary / 275 \\ Distribution / 277", } @Article{Ubhaya:1989:LAN, author = "V. A. Ubhaya", title = "{$ L_p $} approximation from nonconvex subsets of special classes of functions", journal = j-J-APPROX-THEORY, volume = "57", number = "2", pages = "223--238", month = may, year = "1989", CODEN = "JAXTAZ", ISSN = "0021-9045 (print), 1096-0430 (electronic)", ISSN-L = "0021-9045", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, bibno = "72277", catcode = "G.1.2", CRclass = "G.1.2 Approximation; G.1.2 Elementary function approximation", descriptor = "Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation", fjournal = "Journal of Approximation Theory", genterm = "verification; theory", guideno = "1989-07833", journal-URL = "http://www.sciencedirect.com/science/journal/00219045", journalabbrev = "J. Approx. Theory", jrldate = "May 1989", subject = "G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS", } @InProceedings{Vaidya:1989:LWB, author = "P. M. Vaidya", title = "A locally well-behaved potential function and a simple {Newton}-type method for finding the center of a polytype", crossref = "Megiddo:1989:PMP", pages = "79--90", year = "1989", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, bibno = "74176", catcode = "G.1.6; G.1.6; F.2.2; G.1.2; G.1.5; G.1.5; I.1.2", CRclass = "G.1.6 Optimization; G.1.6 Linear programming; G.1.6 Optimization; G.1.6 Gradient methods; F.2.2 Nonnumerical Algorithms and Problems; F.2.2 Geometrical problems and computations; G.1.2 Approximation; G.1.2 Elementary function approximation; G.1.5 Roots of Nonlinear Equations; G.1.5 Convergence; G.1.5 Roots of Nonlinear Equations; G.1.5 Iterative methods; I.1.2 Algorithms; I.1.2 Nonalgebraic algorithms", descriptor = "Mathematics of Computing, NUMERICAL ANALYSIS, Optimization, Linear programming; Mathematics of Computing, NUMERICAL ANALYSIS, Optimization, Gradient methods; Theory of Computation, ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY, Nonnumerical Algorithms and Problems, Geometrical problems and computations; Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation; Mathematics of Computing, NUMERICAL ANALYSIS, Roots of Nonlinear Equations, Convergence; Mathematics of Computing, NUMERICAL ANALYSIS, Roots of Nonlinear Equations, Iterative methods; Computing Methodologies, ALGEBRAIC MANIPULATION, Algorithms, Nonalgebraic algorithms", genterm = "algorithms; theory", guideno = "1989-12478", procdate = "March 1-4, 1987", procloc = "Pacific Grove, CA", subject = "F. Theory of Computation; F.2 ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; I. Computing Methodologies; I.1 ALGEBRAIC MANIPULATION", xxpages = "131--158", } @Article{VanHalen:1989:AAA, author = "P. {Van Halen}", title = "Accurate analytical approximations for error function and its integral", journal = j-ELECT-LETTERS, volume = "25", number = "9", pages = "561--563", day = "27", month = apr, year = "1989", CODEN = "ELLEAK", DOI = "https://doi.org/10.1049/el:19890383", ISSN = "0013-5194 (print), 1350-911X (electronic)", ISSN-L = "0013-5194", bibdate = "Sat Dec 16 18:15:17 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "https://ieeexplore.ieee.org/document/19780/", acknowledgement = ack-nhfb, fjournal = "Electronics Letters", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=2220; https://digital-library.theiet.org/journal/el", } @PhdThesis{Vavasis:1989:CFP, author = "S. A. Vavasis", title = "Complexity of fixed point computations", type = "{Ph.D} Thesis", school = "Stanford University", address = "Stanford, CA, USA", pages = "????", year = "1989", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, bibno = "76220", catcode = "J.4; G.1.2; F.1.3; G.1.2; G.2.2", CRclass = "J.4 Economics; G.1.2 Approximation; G.1.2 Nonlinear approximation; F.1.3 Complexity Classes; G.1.2 Approximation; G.1.2 Elementary function approximation; G.2.2 Graph Theory; G.2.2 Network problems", descriptor = "Computer Applications, SOCIAL AND BEHAVIORAL SCIENCES, Economics; Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Nonlinear approximation; Theory of Computation, COMPUTATION BY ABSTRACT DEVICES, Complexity Classes; Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation; Mathematics of Computing, DISCRETE MATHEMATICS, Graph Theory, Network problems", genterm = "algorithms; theory", guideno = "1989-12859", source = "UMI order no: GAX89-19486", subject = "J. Computer Applications; J.4 SOCIAL AND BEHAVIORAL SCIENCES; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; F. Theory of Computation; F.1 COMPUTATION BY ABSTRACT DEVICES; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; G. Mathematics of Computing; G.2 DISCRETE MATHEMATICS", } @InProceedings{Vial:1989:APP, author = "J.-P. Vial", title = "Approximate projections in a projective method for the linear feasibility problem", crossref = "Megiddo:1989:PMP", bookpages = "x + 158", pages = "65--78", year = "1989", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, bibno = "74175", catcode = "G.1.6; F.2.2; I.1.2; G.1.6; G.1.2; F.2.1; G.1.2", CRclass = "G.1.6 Optimization; G.1.6 Linear programming; F.2.2 Nonnumerical Algorithms and Problems; F.2.2 Geometrical problems and computations; I.1.2 Algorithms; I.1.2 Nonalgebraic algorithms; G.1.6 Optimization; G.1.6 Constrained optimization; G.1.2 Approximation; G.1.2 Elementary function approximation; F.2.1 Numerical Algorithms and Problems; F.2.1 Computations on matrices; G.1.2 Approximation; G.1.2 Minimax approximation and algorithms", descriptor = "Mathematics of Computing, NUMERICAL ANALYSIS, Optimization, Linear programming; Theory of Computation, ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY, Nonnumerical Algorithms and Problems, Geometrical problems and computations; Computing Methodologies, ALGEBRAIC MANIPULATION, Algorithms, Nonalgebraic algorithms; Mathematics of Computing, NUMERICAL ANALYSIS, Optimization, Constrained optimization; Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation; Theory of Computation, ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY, Numerical Algorithms and Problems, Computations on matrices; Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Minimax approximation and algorithms", genterm = "algorithms; experimentation; measurement; performance; theory", guideno = "1989-12477", procdate = "March 1-4, 1987", procloc = "Pacific Grove, CA", subject = "G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; F. Theory of Computation; F.2 ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY; I. Computing Methodologies; I.1 ALGEBRAIC MANIPULATION; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; F. Theory of Computation; F.2 ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS", } @Article{vonRosen:1989:MLE, author = "D. von Rosen", title = "Maximum likelihood estimators in multivariate linear normal models", journal = j-J-MULTIVAR-ANAL, volume = "31", number = "2", pages = "187--200", month = nov, year = "1989", CODEN = "JMVAAI", ISSN = "0047-259x (print), 1095-7243 (electronic)", ISSN-L = "0047-259X", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, affiliation = "Univ. of Stockholm, Stockholm, Sweden", bibno = "69314", catcode = "G.1.6; G.1.2; G.1.2", CRclass = "G.1.6 Optimization; G.1.6 Linear programming; G.1.2 Approximation; G.1.2 Linear approximation; G.1.2 Approximation; G.1.2 Elementary function approximation", descriptor = "Mathematics of Computing, NUMERICAL ANALYSIS, Optimization, Linear programming; Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Linear approximation; Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation", fjournal = "Journal of Multivariate Analysis", genterm = "algorithms; theory", guideno = "1989-08484", journalabbrev = "J. Multivariate Anal.", jrldate = "Nov. 1989", subject = "G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS", } @Book{Wang:1989:SF, author = "Z. X. Wang and D. R. Guo", title = "Special Functions", publisher = pub-WORLD-SCI, address = pub-WORLD-SCI:adr, pages = "xvi + 695", year = "1989", ISBN = "1-283-63565-8, 661394811X, 981-277-936-1 (e-book), 9971-5-0659-9 (hardcover), 9971-5-0667-X (paperback)", ISBN-13 = "978-1-283-63565-3, 661394811X, 978-981-277-936-6 (e-book), 978-9971-5-0659-9 (hardcover), 978-9971-5-0667-4 (paperback)", LCCN = "QA331 .W296 1989", bibdate = "Mon Sep 3 16:10:24 MDT 2007", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; z3950.loc.gov:7090/Voyager", note = "Translation to English by D.R. Guo and X. J. Xia.", acknowledgement = ack-nhfb, remark = "Reprinted in 2010. Library catalogs record two rather different pages values. The one chosen here matches the tableofcontents value.", shorttableofcontents = "1: The expansion of functions in infinite series and infinite products \\ 2: Linear ordinary differential equations of the second order \\ 3: The gamma function \\ 4: Hypergeometric function \\ 5: Legendre functions \\ 6: Confluent hypergeometric functions \\ 7: Bessel functions \\ 8: Weierstrass elliptic functions \\ 9: Theta functions \\ 10: Jacobian elliptic functions \\ 11: Lame functions \\ 12: Mathieu functions", subject = "Functions", tableofcontents = "1: THE EXPANSION OF FUNCTIONS IN INFINITE SERIES AND INFINITE PRODUCTS \\ 1.1. Bernoulli Polynomials and Bernoulli Numbers / 1 \\ 1.2. Euler Polynomials and Euler Numbers / 5 \\ 1.3. Euler--Maclaurin Formula / 8 \\ 1.4. Lagrange's Expansion Formula / 14 \\ 1.5. Expansion of Meromorphic Functions in Rational Fractions / 17 \\ 1.6. Infinite Product / 21 \\ 1.7. The Expansion of a Function in Infinite Product Weierstrass Theorem / 25 \\ 1.8. Asymptotic Expansion / 29 \\ 1.9. The Asymptotic Expansion of the Laplace Integral, Watson's Lemma / 34 \\ 1.10. Expansion in Terms of Functions of an Orthonormal Set / 36 \\ Exercise 1 / 41 \\ 2: LINEAR ORDINARY DIFFERENTIAL EQUATIONS OF THE SECOND ORDER \\ 2.1. Singular Points of Linear Ordinary Differential Equations of the Second Order / 47 \\ 2.2. Solution of the Equation in the Vicinity of an Ordinary Point / 48 \\ 2.3. Solutions of the Equation in the Vicinity of a Singular Point / 51 \\ 2.4. Regular Solution. Regular Singularities / 55 \\ 2.5. Frobenius Method / 61 \\ 2.6. Point at Infinity / 63 \\ 2.7. Equations of Fuchsian Type / 64 \\ 2.8. Equations of Fuchsian Type Having Five Regular Singular Points / 66 \\ 2.9. Equations of Fuchsian Type Having Three Regular Singularities / 68 \\ 2.10. Irregular Singularities. Formal Regular Solution / 71 \\ 2.11. Irregular Singularities. Normal Solutions and Subnormal Solutions / 73 \\ 2.12. Method of Solution by Integrals. Basic Principle / 78 \\ 2.13. Equations of Laplacian Type and Laplace Transform / 81 \\ 2.14. Euler Transform / 85 \\ Exercise 2 / 88 \\ 3: THE GAMMA FUNCTION \\ 3.1. Definition of the Gamma Function / 93 \\ 3.2. Recurrence Relation / 94 \\ 3.3. The Infinite-Product Expression of Euler / 95 \\ 3.4. Weierstrass' Infinite Product / 97 \\ 3.5. Relation between the $\Gamma$-Function and the Trigonometric Function / 98 \\ 3.6. Multiplication Formula / 99 \\ 3.7. Contour Integral / 101 \\ 3.8. Euler Integral of the First Kind. $B$-Function / 103 \\ 3.9. Double-Contour Integral / 105 \\ 3.10. Dirichlet Integral / 106 \\ 3.11. Logarithmic Derivative of the $\Gamma$-Function / 107 \\ 3.12. Asymptotic Expansions / 111 \\ 3.13. Another Derivation of the Asymptotic Expansion / 112 \\ 3.14. Riemann $\zeta$ Function / 114 \\ 3.15. The Functional Equation of the $\zeta$-Function / 115 \\ 3.16. The Value of $\zeta(s,a)$ when $s$ is an Integer / 117 \\ 3.17. Hermite Formula / 118 \\ 3.18. Relation to the $\Gamma$-Function / 120 \\ 3.19. Euler Product of the $\zeta$-Function / 123 \\ 3.20. Riemann Integral of the $\zeta$-Function / 124 \\ 3.21. Another Derivation of the Asymptotic Expansion of the $\Gamma$-Function / 125 \\ 3.22. Evaluation of the $\zeta$-Function / 128 \\ Exercise 3 / 128 \\ 4: HYPERGEOMETRIC FUNCTION \\ 4.1. Hypergeometric Series and Hypergeometric Function / 135 \\ 4.2. Recurrence Relations / 137 \\ 4.3. Other Solutions of the Hypergeometric Equation Expressed in Terms of Hypergeometric Functions / 139 \\ 4.4. The Second Solution of the Hypergeometric Equation when the Difference of the Exponents is an Integer / 144 \\ 4.5. Integral Representation of the Hypergeometric Function / 150 \\ 4.6. Barnes' Integral Representation of the Hypergeometric Function / 153 \\ 4.7. The Value of $F(\alpha, \beta, \gamma, 1)$ / 156 \\ 4.8. Connections between the Fundamental Solutions at the Singular Points $0, 1, \infty$. Analytic Continuation / 159 \\ 4.9. When $\gamma - \alpha - \beta$, $\alpha - \beta$ are Integers / 162 \\ 4.10. Jacobi Polynomials / 169 \\ 4.11. Chebyshev Polynomials / 173 \\ 4.12. Quadratic Transformations / 177 \\ 4.13. Kummer's Formula and Summation Formula Derived from It / 184 \\ 4.14. Asymptotic Expansions for Large Parameters / 186 \\ 4.15. Generalized Hypergeometric Series / 189 \\ 4.16. Hypergeometric Series with Two Variables / 191 \\ 4.17. The Transformation Formulae of $F_1$ and $F_2$ / 195 \\ 4.18. Reducible Cases / 197 \\ Exercise 4 / 202 \\ 5: LEGENDRE FUNCTIONS \\ 5.1. Legendre Equation / 210 \\ 5.2. Legendre Polynomials / 212 \\ 5.3. The Generating Function of $P_n(i)$. Rodrigues Formula / 215 \\ 5.4. Integral Representations of $P_n(x)$ / 216 \\ 5.5. Recurrence Relations of $P_n(x)$ / 218 \\ 5.6. Legendre Polynomials as a Complete Set of Orthonormal Functions / 219 \\ 5.7. Zeros of $P_n(x)$ / 223 \\ 5.8. Legendre Functions of the Second Kind, $Q_n(x)$ / 224 \\ 5.9. Recurrence Relations of $Q_n(x)$ / 230 \\ 5.10. Expansion of the Function $(x - t)^{-1}$ in Terms of Legendre Functions. Neumann Expansion / 231 \\ 5.11. Associate Legendre Functions $P_l^m(x)$ / 233 \\ 5.12. Orthogonality Relations of $P_l^m(x)$ / 235 \\ 5.13. Recurrence Relations for $P_l^m(x)$ and $Q_l^m(x)$ / 239 \\ 5.14. Addition Formula / 241 \\ 5.15. Spherical Surface Harmonics $Y_{lm}(\theta, \phi)$ / 244 \\ 5.16. The General Associate Legendre Functions $P_\nu^\mu(z)$ / 247 \\ 5.17. $Q_\nu^\mu(z)$ / 251 \\ 5.18. Definition of $P_\nu^\mu(x)$ on the Cut: $-\infty < x < 1$ / 255 \\ 5.19. Definition of $Q_\nu^\mu(x)$ on the Cut: $-\infty < x < 1$ / 258 \\ 5.20. Other Integral Expressions for $P_\nu(z)$ and $P_\nu^m(z)$ / 262 \\ 5.21. Addition Formulae / 267 \\ 5.22. Asymptotic Expansions of $P_\nu^\mu(\cos \theta)$ and $Q_\nu^\mu(\cos \theta)$ when $\nu \to \infty$ / 270 \\ 5.23. Ultra-Spherical Polynomials $C_n^\lambda(x)$ / 274 \\ Exercise 5 / 277 \\ 6: CONFLUENT HYPERGEOMETRIC FUNCTIONS \\ 6.1. Confluent Hypergeometric Functions / 296 \\ 6.2. Relations among the Consecutive Functions / 299 \\ 6.3. Whittaker Equation and Whittaker Functions $M_{k,m}(z)$ / 300 \\ 6.4. Integral Representations / 302 \\ 6.5. Whittaker Functions $W_{k,m}(z)$ / 305 \\ 6.6. Asymptotic Expansion of $W_{k,m}(z)$ when $z \to \infty$ / 307 \\ 6.7. Barnes' Integral Representation of $W_{k,m}(z)$ / 310 \\ 6.8. Relations between $W_{\pm k,m}(\pm z)$ and $M_{\pm k,m}(\pm z)$. Asymptotic Expansion of $F(\alpha, \gamma, z)$. Stokes Phenomenon / 313 \\ 6.9. The Case when $\gamma$ (or $2m$) is an Integer / 316 \\ 6.10. The Asymptotic Expansions of $F(\alpha, \gamma, z)$ for Large $|\alpha|$, $|\gamma|$ / 318 \\ 6.11. Differential Equations Reducible to the Confluent Hypergeometric Equation / 318 \\ 6.12. Weber Equation. Parabolic Cylinder Functions $D_n(z)$ / 320 \\ 6.13. Hermite Functions and Hermite Polynomials / 325 \\ 6.14. Laguerre Polynomials / 327 \\ 6.15. Other Special Functions Expressible by Whittaker Functions / 332 \\ Exercise 6 / 335 \\ 7: BESSEL FUNCTIONS \\ 7.1. Bessel Equation. Its Relation to the Confluent Hypergeometric Equation / 345 \\ 7.2. Bessel Functions of the First Kind: $J_{\pm \nu}(z)$, $2\nu \neq$ integer / 347 \\ 7.3. Bessel Functions of Order Half an Odd Integer: $J_{n + 1/2}(z)$ $(n = 0, \pm 1, \pm 2, \ldots{})$ / 350 \\ 7.4. Integral Representations of $J_\nu(z)$ / 351 \\ 7.5. Bessel Functions of Integral Order $J_n(z)$ $(n = 0, 1, 2, \ldots{})$ / 359 \\ 7.6. Bessel Functions of the Second Kind $Y_\nu(z)$ / 365 \\ 7.7. Bessel Functions of the Third Kind (Hankel Functions) $H_\nu^{(1)}(z)$, $H_\nu^{(2)}(z)$ / 368 \\ 7.8. Modified (or Imaginary Argument) Bessel Functions $I_\nu(z)$ and $K_\nu(z)$. Thomson Functions $\ber_\nu(z)$ and $\bei_\nu(z)$; etc. / 374 \\ 7.9. Spherical Bessel Functions $j_l(z)$, $n_l(z)$, $h_l^{(1)}(z), $h_l^{(2)}(z) / 376 \\ 7.10. Asymptotic Expansions for the Case $|z| \to \infty$ / 378 \\ 7.11. The Method of Steepest Descent / 381 \\ 7.12. Asymptotic Expansions of Bessel Functions of Order $\nu$ for Large $|\nu|$ and $|z|$ / 384 \\ 7.13. Addition Formulae / 395 \\ 7.14. Integrals Containing Bessel Functions. (1) Finite Integrals / 399 \\ 7.15. Integrals Containing Bessel Functions. (2) Infinite Integrals / 401 \\ 7.16. Neumann Expansion / 412 \\ 7.17. Kapteyn Expansion / 415 \\ 7.18. The Zeros of Bessel Functions / 420 \\ 7.19. Fourier--Bessel Expansion / 424 \\ Exercise 7 / 425 \\ 8: WEIERSTRASS ELLIPTIC FUNCTIONS \\ 8.1. Elliptic Integrals and Elliptic Functions / 456 \\ 8.2. The Periods of Elliptic Integrals / 460 \\ 8.3. The General Properties of Doubly-Periodic Functions and Elliptic Functions / 462 \\ 8.4. The Function $\wp(z)$ / 466 \\ 8.5. Algebraic Relation between $\wp(z)$ and $\wp'(z)$ / 468 \\ 8.6. The Function $\zeta(z)$ / 471 \\ 8.7. The Function $\sigma(z)$ / 473 \\ 8.8. Homogeneity of the Weierstrass Elliptic Function / 476 \\ 8.9. Representations of a General Elliptic Function / 476 \\ 8.10. Addition Formulae / 481 \\ 8.11. Expressing the Coordinates of a Cubic Curve by Elliptic Functions / 485 \\ 8.12. The Problem of a Quartic Polynomial / 486 \\ 8.13. Curves of Genus (Deficiency) 1 / 489 \\ Exercise 8 / 493 \\ 9: THETA FUNCTIONS \\ 9.1. The Theta Function $\theta(\nu)$ / 498 \\ 9.2. The Functions $\vartheta_k(\nu)$ / 500 \\ 9.3. Elliptic Functions Represented by Theta Functions / 502 \\ 9.4. Relations among the Squares of $\vartheta_k(\nu)$ / 503 \\ 9.5. Addition Formulae / 504 \\ 9.6. Differential Equations Satisfied by Theta Functions / 506 \\ 9.7. The Values of Some Constants / 508 \\ 9.8. Legendre's Elliptic Integral of the First Kind / 510 \\ 9.9. Jacobi's Imaginary Transformation / 514 \\ 9.10. Transformation of Landen-Type / 516 \\ 9.11. Representation of Theta Functions by Infinite Product / 517 \\ 9.12. Fourier Expansion of the Logarithmic Derivatives of Theta Functions / 521 \\ 9.13. The Functions $\Theta(u)$ and $H(u)$ / 522 \\ Exercise 9 / 523 \\ 10: JACOBIAN ELLIPTIC FUNCTIONS \\ 10.1. Jacobian Elliptic Functions $\sn u$, $\cn u$, $\dn u$ / 530 \\ 10.2. Geometric Representations of Jacobian Elliptic Functions / 532 \\ 10.3. Complete Elliptic Integrals / 535 \\ 10.4. Addition Formulae / 537 \\ 10.5. The Periodicity of Jacobian Elliptic Functions / 539 \\ 10.6. The Zeros and Poles of Jacobian Elliptic Functions / 540 \\ 10.7. Transformations of Elliptic Functions / 542 \\ 10.8. Reductions of Elliptic Integrals / 545 \\ 10.9. Elliptic Integral of the Second Kind / 552 \\ 10.10. Elliptic Integral of the Third Kind / 553 \\ 10.11. Properties of the Function $E(u)$ / 555 \\ 10.12. Differential Equations Satisfied by $K$ and $E$ with Respect to $k$ and Expansions of $K$ and $E$ with Respect to $k$ / 558 \\ 10.13. Relations between Jacobian Elliptic Functions and Theta Functions / 561 \\ 10.14. Expressing Jacobian Elliptic Functions in Infinite Products and Fourier Series / 566 \\ Exercise 10 / 569 \\ 11: LAM{\'E} FUNCTIONS \\ 11.1. Ellipsoidal Coordinates / 575 \\ 11.2. Representing the Coordinates with Elliptic Functions / 578 \\ 11.3. Lam{\'e} Equation / 580 \\ 11.4. Four Types of Lam{\'e} Functions / 583 \\ 11.5. Ellipsoidal Harmonics / 589 \\ 11.6. Niven's Representation / 591 \\ 11.7. On the Zeros of Lam{\'e} Polynomials / 595 \\ 11.8. Lam{\'e} Functions of the Second Kind / 597 \\ 11.9. Generalized Lam{\'e} Functions / 599 \\ 11.10. Integral Equations of Lam{\'e} Functions / 602 \\ 11.11. The Integral Representation of Ellipsoidal Harmonics / 604 \\ Exercise 11 / 607 \\ 12: MATHIEU FUNCTIONS \\ 12.1. Mathieu Equation / 610 \\ 12.2. General Properties of the Solution. Fundamental Solutions / 612 \\ 12.3. Floquet Solution / 614 \\ 12.4. Periodic Solutions of Mathieu Equation / 615 \\ 12.5. Fourier Expansion of the Floquet Solution / 617 \\ 12.6. Formulae for Computing Eigenvalues $\lambda(q)$ / 620 \\ 12.7. Mathieu Functions $\ce_m(z)$, $m = 0, 1, 2, \ldots{}$ and $\se_m(z)$, $m = 1, 2 \ldots{}$ / 624 \\ 12.8. Expansion of $\lambda_\nu(q)$ in Powers of $q$ / 627 \\ 12.9. Fourier Expansions of $\ce_m(z)$ and $\se_m(z)$ for Small $q$ / 630 \\ 12.10. Infinite Determinant / 632 \\ 12.11. Hill Equation / 633 \\ 12.12. Stable and Unstable Solutions of Mathieu Equation. Stable Region and Unstable Region / 637 \\ 12.13. Approximate Solutions of Mathieu Equation for $\lambda \gg q > 0$ / 640 \\ 12.14. Integral Equations for Mathieu Functions / 643 \\ Exercise 12 / 645 \\ Appendices \\ Appendix I. Roots of a Cubic Equation / 654 \\ Appendix II. Roots of the Quartic Equation / 656 \\ Appendix III. Orthogonal Curvilinear Coordinate Systems / 658 \\ Bibliography / 677 \\ Glossary / 679 \\ Index / 683", xxpages = "xiii + 422", } @Article{Wasilkowski:1989:AIV, author = "G. W. Wasilkowski", title = "On adaptive information with varying cardinality for linear problems with elliptically contoured measures", journal = j-J-COMPLEXITY, volume = "5", number = "3", pages = "363--368", month = sep, year = "1989", CODEN = "JOCOEH", ISSN = "0885-064X (print), 1090-2708 (electronic)", ISSN-L = "0885-064X", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, affiliation = "Univ. of Kentucky, Lexington, KY", bibno = "68756", catcode = "G.1.2; G.1.2; F.1.3", CRclass = "G.1.2 Approximation; G.1.2 Elementary function approximation; G.1.2 Approximation; G.1.2 Nonlinear approximation; F.1.3 Complexity Classes", descriptor = "Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation; Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Nonlinear approximation; Theory of Computation, COMPUTATION BY ABSTRACT DEVICES, Complexity Classes", fjournal = "Journal of complexity", genterm = "algorithms; theory", guideno = "1989-08045", journal-URL = "http://www.sciencedirect.com/science/journal/0885064X", journalabbrev = "J. Complexity", jrldate = "Sept. 1989", subject = "G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; F. Theory of Computation; F.1 COMPUTATION BY ABSTRACT DEVICES", } @Article{Weniger:1989:NST, author = "Ernst Joachim Weniger", title = "Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series", journal = j-COMPUT-PHYS-REP, volume = "10", number = "5--6", pages = "189--371", month = dec, year = "1989", CODEN = "CPHREF", DOI = "https://doi.org/10.1016/0167-7977(89)90011-7", ISSN = "0167-7977 (print), 1878-1004 (electronic)", ISSN-L = "0167-7977", bibdate = "Thu Dec 01 10:13:37 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "Available as math.NA/0306302.", acknowledgement = ack-nhfb, fjournal = "Computer Physics Reports", keywords = "convergence acceleration", } @Article{Yortsos:1989:LSI, author = "Y. C. Yortsos and F. J. Hickernell", title = "Linear stability of immiscible displacement in porous media", journal = j-SIAM-J-APPL-MATH, volume = "49", number = "3", pages = "730--748", month = jun, year = "1989", CODEN = "SMJMAP", ISSN = "0036-1399 (print), 1095-712X (electronic)", ISSN-L = "0036-1399", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, bibno = "64942", catcode = "G.1.8; G.1.0; J.2; G.1.2; G.1.4; G.1.8", CRclass = "G.1.8 Partial Differential Equations; G.1.8 Parabolic equations; G.1.0 General; G.1.0 Stability (and instability); J.2 Earth and atmospheric sciences; G.1.2 Approximation; G.1.2 Elementary function approximation; G.1.4 Quadrature and Numerical Differentiation; G.1.4 Finite difference methods; G.1.8 Partial Differential Equations; G.1.8 Difference methods", descriptor = "Mathematics of Computing, NUMERICAL ANALYSIS, Partial Differential Equations, Parabolic equations; Mathematics of Computing, NUMERICAL ANALYSIS, General, Stability (and instability); Computer Applications, PHYSICAL SCIENCES AND ENGINEERING, Earth and atmospheric sciences; Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation; Mathematics of Computing, NUMERICAL ANALYSIS, Quadrature and Numerical Differentiation, Finite difference methods; Mathematics of Computing, NUMERICAL ANALYSIS, Partial Differential Equations, Difference methods", fjournal = "SIAM Journal on Applied Mathematics", genterm = "algorithms; theory; experimentation", guideno = "1989-09711", journal-URL = "http://epubs.siam.org/siap", journalabbrev = "SIAM J. Appl. Math.", jrldate = "June 1989", subject = "G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; J. Computer Applications; J.2 PHYSICAL SCIENCES AND ENGINEERING", } @Article{Zalik:1989:NIE, author = "R. A. Zalik", title = "A new inequality for entire functions", journal = j-J-APPROX-THEORY, volume = "58", number = "3", pages = "281--283", month = sep, year = "1989", CODEN = "JAXTAZ", ISSN = "0021-9045 (print), 1096-0430 (electronic)", ISSN-L = "0021-9045", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, bibno = "72294", catcode = "G.1.2", CRclass = "G.1.2 Approximation; G.1.2 Elementary function approximation", descriptor = "Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation", fjournal = "Journal of Approximation Theory", genterm = "verification; theory", guideno = "1989-07871", journal-URL = "http://www.sciencedirect.com/science/journal/00219045", journalabbrev = "J. Approx. Theory", jrldate = "Sept. 1989", subject = "G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS", } @Book{Ziemer:1989:WDF, author = "William P. Ziemer", title = "Weakly differentiable functions: {Sobolev} spaces and functions of bounded variation", volume = "120", publisher = pub-SV, address = pub-SV:adr, pages = "xvi + 308", year = "1989", ISBN = "0-387-97017-7", ISBN-13 = "978-0-387-97017-2", LCCN = "QA323 .Z53 1989", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", series = "Graduate texts in mathematics", acknowledgement = ack-nhfb, affiliation = "Indiana Univ., Bloomington", bibno = "69369", catcode = "G.1.2; G.1.8; G.1.5", CRclass = "G.1.2 Approximation; G.1.2 Elementary function approximation; G.1.8 Partial Differential Equations; G.1.5 Roots of Nonlinear Equations", descriptor = "Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation; Mathematics of Computing, NUMERICAL ANALYSIS, Partial Differential Equations; Mathematics of Computing, NUMERICAL ANALYSIS, Roots of Nonlinear Equations", genterm = "algorithms; theory", guideno = "1989-01732", subject = "G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS", } @Article{Amos:1990:RPP, author = "Donald E. Amos", title = "Remark on ``{Algorithm 644}: a Portable Package for {Bessel} Functions of a Complex Argument and Nonnegative Order''", journal = j-TOMS, volume = "16", number = "4", pages = "404--404", month = dec, year = "1990", CODEN = "ACMSCU", DOI = "https://doi.org/10.1145/98267.98299", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Tue Mar 09 10:26:24 1999", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "See \cite{Amos:1986:APP,Amos:1995:RAP,Kodama:2007:RA}.", URL = "http://www.acm.org/pubs/citations/journals/toms/1990-16-4/p404-amos/", acknowledgement = ack-nhfb, fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", keywords = "algorithms; theory", subject = "{\bf F.2.2}: Theory of Computation, ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY, Nonnumerical Algorithms and Problems.", } @Article{Anderson:1990:FIC, author = "G. D. Anderson and M. K. Vamanamurthy and M. Vuorinen", title = "Functional Inequalities for Complete Elliptic Integrals and Their Ratios", journal = j-SIAM-J-MATH-ANA, volume = "21", number = "2", pages = "536--549", month = mar, year = "1990", CODEN = "SJMAAH", ISSN = "0036-1410 (print), 1095-7154 (electronic)", ISSN-L = "0036-1410", MRclass = "33E05 (30C62 33C75)", MRnumber = "91d:33039", MRreviewer = "K. C. Gupta", bibdate = "Sat Dec 5 18:14:13 MST 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "SIAM Journal on Mathematical Analysis", journal-URL = "http://epubs.siam.org/sima", } @Article{Anonymous:1990:BRPb, author = "Anonymous", title = "Book Review: {{\booktitle{Properties of estimators for the gamma function}}: K. O. Bowman and L. R. Shenton, Marcel Dekker, New York, 1988. 268 pp., ISBN 0-8247-7556-2}", journal = j-MATH-COMPUT-SIMUL, volume = "32", number = "4", pages = "420--420", month = oct, year = "1990", CODEN = "MCSIDR", DOI = "https://doi.org/10.1016/0378-4754(90)90150-H", ISSN = "0378-4754 (print), 1872-7166 (electronic)", ISSN-L = "0378-4754", bibdate = "Mon Aug 18 12:50:00 MDT 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcomputsimul1990.bib", URL = "https://www.sciencedirect.com/science/article/pii/037847549090150H", acknowledgement = ack-nhfb, ajournal = "Math. Comput. Simul.", fjournal = "Mathematics and Computers in Simulation", journal-URL = "https://www.sciencedirect.com/science/journal/03784754", } @Article{Bellalij:1990:SPC, author = "M. Bellalij", title = "A simultaneous process for convergence acceleration and error control", journal = j-J-COMPUT-APPL-MATH, volume = "33", number = "2", pages = "217--231", day = "21", month = dec, year = "1990", CODEN = "JCAMDI", DOI = "https://doi.org/10.1016/0377-0427(90)90370-F", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", MRclass = "65B99 (65G10)", MRnumber = "1090897 (92a:65029)", MRreviewer = "Thomas A. Atchison", bibdate = "Thu Dec 1 17:36:45 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", keywords = "convergence acceleration", } @Article{Bronstein:1990:IEF, author = "Manuel Bronstein", title = "Integration of elementary functions", journal = j-J-SYMBOLIC-COMP, volume = "9", number = "2", pages = "117--173", month = feb, year = "1990", CODEN = "JSYCEH", ISSN = "0747-7171 (print), 1095-855X (electronic)", ISSN-L = "0747-7171", MRclass = "12H05 (68Q40)", MRnumber = "91h:12017", MRreviewer = "Alexandru Buium", bibdate = "Sat May 10 15:54:09 MDT 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; Theory/cathode.bib", acknowledgement = ack-nhfb, classcodes = "C1120 (Analysis); C4160 (Numerical integration and differentiation)", corpsource = "Dept. of Math. Sci, IBM Res. Div., Thomas J. Watson Res. Center, Yorktown Heights, NY, USA", fjournal = "Journal of Symbolic Computation", journal-URL = "http://www.sciencedirect.com/science/journal/07477171", keywords = "algebraic extension; algebraic function; decision procedure; elementary function field; elementary functions; exponential; finite terms; indefinite; integration; integration Risch ODEs oderef; logarithm; proof; Trager", treatment = "T Theoretical or Mathematical", } @Article{Carre:1990:PEF, author = "C. Carre", title = "Plethysm of elementary functions", journal = "Bayreuth. Math. Schr.", volume = "31", pages = "1--18", year = "1990", ISSN = "0172-1062", MRclass = "20C30 (05E05 22E45)", MRnumber = "91f:20013", MRreviewer = "John R. Stembridge", bibdate = "Sat Jan 11 17:44:01 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, } @Article{Ciminiera:1990:HRS, author = "L. Ciminiera and P. Montuschi", title = "Higher radix square rooting", journal = j-IEEE-TRANS-COMPUT, volume = "39", number = "10", pages = "1220--1231", month = oct, year = "1990", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/12.59853", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Thu Jul 7 14:20:04 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput1990.bib", URL = "https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=59853", acknowledgement = ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", summary = "A general discussion on nonrestoring square root algorithms is presented, showing bounds and constraints delimiting the space of feasible algorithms, for all the choices of radix, digit set and representation of the partial remainder. Two classes of \ldots{}", } @Article{Cody:1990:PEP, author = "W. J. Cody", title = "Performance Evaluation of Programs for the Error and Complementary Error Functions", journal = j-TOMS, volume = "16", number = "1", pages = "29--37", month = mar, year = "1990", CODEN = "ACMSCU", DOI = "https://doi.org/10.1145/77626.77628", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", MRclass = "65-04 (65G05)", MRnumber = "1 073 407", bibdate = "Tue Oct 09 09:29:47 2007", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/c/cody-william-j.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib", URL = "http://www.acm.org/pubs/citations/journals/toms/1990-16-1/p29-cody/; http://www.acm.org/pubs/toc/Abstracts/0098-3500/77628.html", abstract = "This paper presents methods for performance evaluation of computer programs for the functions $ \textrm {erf}(x) $, $ \textrm {erfc}(x) $, and $ \e^{x^2} \textrm {erfc}(x) $. Accuracy estimates are based on comparisons using power series expansions and an expansion in the repeated integrals of $ \textrm {erfc}(x) $. Some suggestions for checking robustness are also given. Details of a specific implementation of a test program are included.", acknowledgement = ack-nhfb, affiliation = "Argonne Nat. Lab., IL, USA", classification = "B0290B (Error analysis in numerical methods); B0290F (Interpolation and function approximation); C4110 (Error analysis in numerical methods); C4130 (Interpolation and function approximation)", fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", keywords = "Complementary error functions; Computer programs; FORTRAN; Power series expansions; Repeated integrals; Robustness; Test program", subject = "{\bf G.4}: Mathematics of Computing, MATHEMATICAL SOFTWARE, Certification and testing. {\bf G.4}: Mathematics of Computing, MATHEMATICAL SOFTWARE, Reliability and robustness. {\bf G.1.0}: Mathematics of Computing, NUMERICAL ANALYSIS, General, Numerical algorithms.", thesaurus = "Error analysis; Function approximation; Performance evaluation", } @TechReport{DiDonato:1990:SDC, author = "Armido R. DiDonato", title = "Significant Digit Computation of the Elliptical Coverage Function", type = "Technical Report", number = "NAVSWC TR 90-513", institution = "Naval Surface Warfare Center", address = "Dahlgren, VA 22448-5000, USA and Silver Spring, MD 20903-5000, USA", pages = "v + 13 + A-7 + B-9 + 5", month = sep, year = "1990", bibdate = "Sat Nov 15 10:55:35 2014", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.dtic.mil/dtic/tr/fulltext/u2/a230523.pdf", acknowledgement = ack-nhfb, } @Article{Dunham:1990:FPF, author = "C. B. Dunham", title = "Feasibility of ``perfect'' function evaluation", journal = j-SIGNUM, volume = "25", number = "4", pages = "25--26", month = oct, year = "1990", CODEN = "SNEWD6", DOI = "https://doi.org/10.1145/122272.122276", ISSN = "0163-5778 (print), 1558-0237 (electronic)", ISSN-L = "0163-5778", bibdate = "Tue Apr 12 07:50:19 MDT 2005", bibsource = "garbo.uwasa.fi:/pc/doc-soft/fpbiblio.txt; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb # " and " # ack-nj, fjournal = "ACM SIGNUM Newsletter", journal-URL = "https://dl.acm.org/loi/signum", } @Article{Dvorak:1990:NCI, author = "Steven L. Dvorak and Edward F. Kuester", title = "Numerical computation of the incomplete {Lipschitz--Hankel} integral {$ {\rm Je}_0 (a, z) $}", journal = j-J-COMPUT-PHYS, volume = "87", number = "2", pages = "301--327", month = apr, year = "1990", CODEN = "JCTPAH", DOI = "https://doi.org/10.1016/0021-9991(90)90255-Y", ISSN = "0021-9991 (print), 1090-2716 (electronic)", ISSN-L = "0021-9991", bibdate = "Mon Jan 2 07:55:40 MST 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputphys1990.bib", URL = "http://www.sciencedirect.com/science/article/pii/002199919090255Y", acknowledgement = ack-nhfb, fjournal = "Journal of Computational Physics", journal-URL = "http://www.sciencedirect.com/science/journal/00219991/", } @Article{Ercegovac:1990:RSR, author = "M. D. Ercegovac and T. Lang", title = "Radix-$4$ square root without initial {PLA}", journal = j-IEEE-TRANS-COMPUT, volume = "39", number = "8", pages = "1016--1024", month = aug, year = "1990", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/12.57040", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Thu Jul 7 14:20:03 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput1990.bib", URL = "https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=57040", acknowledgement = ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", } @Article{Germain-Bonne:1990:CAN, author = "B. Germain-Bonne", title = "Convergence acceleration of number-machine sequences", journal = j-J-COMPUT-APPL-MATH, volume = "32", number = "1--2", pages = "83--88", day = "26", month = nov, year = "1990", CODEN = "JCAMDI", DOI = "https://doi.org/10.1016/0377-0427(90)90419-Z", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", MRclass = "65B05", MRnumber = "1091778 (91m:65007)", MRreviewer = "W. Govaerts", bibdate = "Thu Dec 1 17:36:45 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "Extrapolation and rational approximation (Luminy, 1989)", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", keywords = "convergence acceleration", } @Article{Hashemian:1990:SRA, author = "R. Hashemian", title = "Square Rooting Algorithms for Integer and Floating-Point Numbers", journal = j-IEEE-TRANS-COMPUT, volume = "C-39", number = "8", pages = "1025--1029", month = aug, year = "1990", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/12.57041", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Sat Jul 16 15:28:13 MDT 2005", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", abstract = "An algorithm for evaluating the square root of integers and real numbers is developed. The procedure consists of two parts: one to obtain a close estimate of the square root and the other to modify the initial value, iteratively, until a precise root is evaluated. The major effort in this development has been concentrated on two objectives: high speed and no division operation other than division by 2. The first objective is achieved through a simple two-step procedure for getting the first estimate, and then modifying it by employing a fast converging iteration technique. The second objective is also fulfilled through applying bit-shift operation rather than division operation. The algorithm is simulated for both integer and real numbers, and the results are compared to two methods being widely used. The results (tabulated) show considerable improvement in speed compared to these other two methods.", acknowledgement = ack-nhfb # " and " # ack-nj, affiliation = "Dept. of Electr. Eng., Northern Illinois Univ., Dekalb, IL, USA", ajournal = "IEEE Trans. Comput.", classification = "C1160 (Combinatorial mathematics); C4130 (Interpolation and function approximation); C5230 (Digital arithmetic methods)", fjournal = "IEEE Transactions on Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", keywords = "Bit-shift operation; Close estimate; Division by 2; Fast converging iteration; Floating-point numbers; Initial value modification; Integer numbers; Precise root; Real numbers; Square rooting algorithms", summary = "An algorithm for evaluating the square root of integers and real numbers is developed. The procedure consists of two parts: one to obtain a close estimate of the square root and the other to modify the initial value, iteratively, until a precise \ldots{}", thesaurus = "Digital arithmetic; Iterative methods; Number theory", } @MastersThesis{Holton:1990:IJE, author = "P. G. W. Holton", title = "An Introduction to the {Jacobian} Elliptic Functions with some Applications", type = "Thesis (M.Sc.)", school = "University of Newcastle upon Tyne", address = "Newcastle upon Tyne, UK", pages = "117", year = "1990", LCCN = "????", bibdate = "Wed Mar 15 06:50:49 MDT 2017", bibsource = "fsz3950.oclc.org:210/WorldCat; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, } @Article{Ifantis:1990:DIP, author = "E. K. Ifantis and P. D. Siafarikas", title = "Differential inequalities for the positive zeros of {Bessel} functions", journal = j-J-COMPUT-APPL-MATH, volume = "30", number = "2", pages = "139--143", day = "28", month = may, year = "1990", CODEN = "JCAMDI", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Sat Feb 25 12:20:45 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath1990.bib", URL = "http://www.sciencedirect.com/science/article/pii/037704279090022R", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @InProceedings{Kiernan:1990:FAE, author = "J. M. Kiernan and T. B. Blachowiak", title = "Fast, Accurate Elementary Functions For the {Cray Y-MP} Computer", crossref = "Cray:1990:PCU", pages = "243--252", year = "1990", bibdate = "Thu Sep 1 10:15:30 1994", bibsource = "garbo.uwasa.fi:/pc/doc-soft/fpbiblio.txt; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nj, } @Article{Kolbig:1990:BRC, author = "K. S. K{\"o}lbig", title = "Book Review: {{\booktitle{Computation of Functions on Electronic Computers --- Handbook}} (in Russian). Naukova Dumka, Kiev, 194, 599pp, by B. A. Popov, G. S. Tesler}", journal = j-MATH-COMPUT, volume = "55", number = "191", pages = "395--397", month = jul, year = "1990", CODEN = "MCMPAF", DOI = "https://doi.org/10.2307/2008818", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Jan 24 08:35:37 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp1990.bib", URL = "http://www.jstor.org/stable/2008818", acknowledgement = ack-nhfb, fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Koren:1990:EEF, author = "Israel Koren and Ofra Zinaty", title = "Evaluating Elementary Functions in a Numerical Coprocessor Based on Rational Approximations", journal = j-IEEE-TRANS-COMPUT, volume = "39", number = "8", pages = "1030--1037", month = aug, year = "1990", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/12.57042", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Thu Sep 1 10:15:30 1994", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; garbo.uwasa.fi:/pc/doc-soft/fpbiblio.txt; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput1990.bib", URL = "https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=57042", acknowledgement = ack-nj # "\slash " # ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", remark = "Contains coefficients of rational polynomial fits for sine (81.35 bits), cosine (75.84 bits), tangent (77.93 bits), log (80.26 bits), exp2 (71.72 bits), and asin (68.3 bits).", } @Article{Lentz:1990:CFC, author = "William J. Lentz", title = "Continued fraction calculation of spherical {Bessel} functions", journal = j-COMPUT-PHYS, volume = "4", number = "4", pages = "403--407", month = jul, year = "1990", CODEN = "CPHYE2", DOI = "https://doi.org/10.1063/1.168382", ISSN = "0894-1866 (print), 1558-4208 (electronic)", ISSN-L = "0894-1866", MRnumber = "33C10 30B70 65-04", bibdate = "Wed Mar 22 14:36:46 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/computphys.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "https://aip.scitation.org/doi/10.1063/1.168382", ZMnumber = "0703.33001", abstract = "An efficient new method of calculating spherical Bessel functions of complex argument based on continued fractions is developed. The method does not depend on recurrence relations, and it allows accurate calculations on computers with differing word lengths. The method may be easily extended to other types of Bessel functions and to complex orders.", acknowledgement = ack-nhfb, ajournal = "Comput. Phys.", fjournal = "Computers in Physics", journal-URL = "https://aip.scitation.org/journal/cip", } @Article{Levrie:1990:CAN, author = "Paul Levrie and Adhemar Bultheel", title = "Convergence Acceleration for the Numerical Solution of Second-Order Linear Recurrence Relations", journal = j-SIAM-J-NUMER-ANAL, volume = "27", number = "1", pages = "166--177", month = feb, year = "1990", CODEN = "SJNAAM", ISSN = "0036-1429 (print), 1095-7170 (electronic)", ISSN-L = "0036-1429", MRclass = "65Q05 (40A15 65B99)", MRnumber = "91a:65244", bibdate = "Fri Oct 16 06:57:22 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "SIAM Journal on Numerical Analysis", journal-URL = "http://epubs.siam.org/sinum", keywords = "convergence acceleration", } @Article{Lin:1990:MSL, author = "Jinn Tyan Lin", title = "Miscellanea: a Simpler Logistic Approximation to the Normal Tail Probability and its Inverse", journal = j-APPL-STAT, volume = "39", number = "2", pages = "255--257", year = "1990", CODEN = "APSTAG", ISSN = "0035-9254 (print), 1467-9876 (electronic)", ISSN-L = "0035-9254", MRclass = "62E15", MRnumber = "1 060 209", bibdate = "Sat Apr 21 10:25:45 MDT 2001", bibsource = "https://www.math.utah.edu/pub/tex/bib/as1990.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "Applied Statistics", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1111/(ISSN)1467-9876/issues", } @Article{Magnus:1990:CFL, author = "Arne Magnus and John McCabe", title = "On a continued fraction for $ \log^2_e(1 + x) $", journal = j-J-COMPUT-APPL-MATH, volume = "30", number = "1", pages = "81--86", day = "10", month = apr, year = "1990", CODEN = "JCAMDI", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Sat Feb 25 12:20:45 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath1990.bib", URL = "http://www.sciencedirect.com/science/article/pii/037704279090007M", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @Article{Markstein:1990:CEF, author = "P. W. Markstein", title = "Computation of elementary functions on the {IBM RISC System\slash 6000} processor", journal = j-IBM-JRD, volume = "34", number = "1", pages = "111--119", month = jan, year = "1990", CODEN = "IBMJAE", ISSN = "0018-8646 (print), 2151-8556 (electronic)", ISSN-L = "0018-8646", MRclass = "65-04 (65D20)", MRnumber = "1 057 659", bibdate = "Sat Jan 11 17:44:01 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", abstract = "The additional speed and precision of the IBM RISC System\slash 6000 floating-point unit have motivated reexamination of algorithms to perform division, square root, and the elementary functions. New results are obtained which avoid the necessity of doing special testing to get the last bit rounded correctly in accordance with all of the IEEE rounding modes in the case of division and square root. For the elementary function library, a technique is described for always getting the last bit rounded correctly in the selected IEEE rounding mode.", acknowledgement = ack-nhfb, affiliation = "IBM Res. Div., Austin, TX, USA", classification = "C5230 (Digital arithmetic methods)", fjournal = "IBM Journal of Research and Development", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=5288520", keywords = "Division; Elementary functions; Floating-point unit; IBM RISC System\slash 6000 processor; IEEE rounding modes; IEEE rounding modes, IBM RISC System/6000 processor; Square root", thesaurus = "Digital arithmetic; IBM computers; Reduced instruction set computing", } @Article{Matos:1990:CAM, author = "Ana C. Matos", title = "A convergence acceleration method based on a good estimation of the absolute value of the error", journal = j-IMA-J-NUMER-ANAL, volume = "10", number = "2", pages = "243--251", year = "1990", CODEN = "IJNADH", ISSN = "0272-4979 (print), 1464-3642 (electronic)", ISSN-L = "0272-4979", MRclass = "65B99", MRnumber = "91m:65010", MRreviewer = "K. B{\"o}hmer", bibdate = "Sat Dec 23 17:06:35 MST 2000", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; MathSciNet database", acknowledgement = ack-nhfb, fjournal = "IMA Journal of Numerical Analysis", journal-URL = "http://imajna.oxfordjournals.org/content/by/year", keywords = "convergence acceleration", } @InProceedings{Matula:1990:HPD, author = "D. Matula", title = "Highly parallel divide and square root algorithms for a new generation floating point processor", crossref = "Ullrich:1990:CCA", pages = "??--??", year = "1990", bibdate = "Thu Apr 2 08:38:35 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-sfo # " and " # ack-nhfb, } @Article{McConnell:1990:LEP, author = "C. R. McConnell", title = "Letter to the {Editor}: Pocket computer approximation for areas under the standard normal curve", journal = j-AMER-STAT, volume = "44", number = "1", pages = "63--63", month = feb, year = "1990", CODEN = "ASTAAJ", ISSN = "0003-1305 (print), 1537-2731 (electronic)", ISSN-L = "0003-1305", bibdate = "Sat Dec 16 17:19:37 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.jstor.org/stable/2684963", acknowledgement = ack-nhfb, fjournal = "The American Statistician", journal-URL = "http://www.tandfonline.com/loi/utas20", } @InProceedings{Meyer:1990:APN, author = "R. Meyer and R. Mehling", editor = "????", booktitle = "Proceedings of the {1990 International Conference on Acoustics, Speech, and Signal Processing, Albuquerque, New Mexico, 1990}", title = "Architecture and Performance of a New Arithmetic Unit for the Computation of Elementary Functions", publisher = "????", address = "????", pages = "1783--1786", year = "1990", DOI = "", ISBN = "", ISBN-13 = "", LCCN = "", bibdate = "Wed Nov 12 10:00:53 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "", acknowledgement = ack-nhfb, } @Article{Montuschi:1990:SSR, author = "P. Montuschi and P. M. Mezzalama", title = "Survey of square rooting algorithms", journal = j-IEE-PROC-COMPUT-DIGIT-TECH, volume = "137", number = "1", pages = "31--40", month = jan, year = "1990", CODEN = "ICDTEA", ISSN = "1350-2387 (print), 1359-7027 (electronic)", ISSN-L = "1350-2387", bibdate = "Sat Jul 16 15:28:13 MDT 2005", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, fjournal = "IEE Proceedings. Computers and Digital Techniques", summary = "The paper reviews the algorithms for the computation of square roots for binary machines. After an initial classification, the algorithms are analysed in detail by considering their specific peculiarities and properties. Finally, some comments are \ldots{}", } @InCollection{Mora:1990:EFI, author = "Gerardo Mora and Edwin Castro and Ioan Muntean", booktitle = "Mathematics in Costa Rica, Vol. 1 (Spanish) (San Jos{\'e}, 1990)", title = "Elementary functions. {I}. ({Spanish})", publisher = "Univ. Costa Rica", address = "San Jos{\'e}, Costa Rica", pages = "76--86", year = "1990", MRclass = "26A09", MRnumber = "1 111 714", bibdate = "Sat Jan 11 17:44:01 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, language = "Spanish", } @TechReport{Morris:1990:NLM, author = "Alfred H. {Morris, Jr.}", title = "{NSWC} Library of Mathematics Subroutines", type = "Report", number = "NSWC TR 90-21", institution = "Naval Surface Warfare Center", address = "Dahlgren, VA 22448-5000, USA; Silver Spring, MD 20903-5000, USA", pages = "xii + 492 + 9", month = jan, year = "1990", bibdate = "Tue Jun 13 08:47:19 2023", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fortran2.bib", note = "See also later edition \cite{Morris:1993:NLM}.", URL = "https://apps.dtic.mil/sti/citations/ADA476840; https://apps.dtic.mil/sti/pdfs/ADA476840.pdf; https://people.math.sc.edu/Burkardt/f_src/nswc/nswc.f90; https://people.math.sc.edu/Burkardt/f_src/nswc/nswc.html", abstract = "The NSWC library is a library of general-purpose Fortran subroutines that provide a basic computational capability in a variety of mathematical activities and emphasis has been placed on the transportability of the codes. Subroutines are available in the following areas Elementary Operations, Geometry, Special Functions, Polynomials, Vectors, Matrices, Large Dense Systems of Linear Equations, Banded Matrices, Sparse Matrices, Eigenvalues and Eigenvectors, Solution of Linear Equations, Least-Squares Solution of Linear Equations, Optimization, Transforms, Approximation of Functions, Curve Fitting, Surface Fitting, Manifold Fitting, Numerical Integration, Integral Equations, Ordinary--Differential Equations, Partial Differential Equations, and Random Number Generation.", acknowledgement = ack-nhfb, remark = "The single Fortran 90 file of 99,021 nonblank lines compiles into a library of 905 distinct functions and subroutines: the source code appears to contain 139 functions and 762 subroutines (901 total entry points). The tableofcontents values in this entry is derived from optical character recognition (OCR) of the 8-page listing from the PDF files, with editorial correction of spelling and OCR errors, and matching of uppercase software names against the entry points of the compiled library. There are about five names in the PDF table of contents that disagree with subroutine names in the source code: they have been edited here to reflect the correct code names. There are numerous routine names in the compiled library that are not mentioned in the table of contents.", tableofcontents = "Introduction / 1 \\ \\ Elementary Operations \\ \\ Machine Constants --- SPMPAR, DPMPAR, IPMPAR / 3 \\ Sorting Lists --- ISHELL, SHELL, AORD, RISORT, SHELL2, DSORT, DAORD, DISORT, DDSORT / 5 \\ Cube Root --- CBRT, DCBRT / 7 \\ Four Quadrant Arctangent --- ARTNQ, DARTNQ / 7 \\ Length of a Two-Dimensional Vector --- CPABS, DCPABS / 7 \\ Reciprocal of a Complex Number --- CREC, DCREC / 9 \\ Square Root of a Double Precision Complex Number --- DCSQRT / 9 \\ Conversion of Polar to Cartesian Coordinates --- POCA / 11 \\ Conversion of Cartesian to Polar Coordinates --- CAPO / 11 \\ Rotation of Axes --- ROTA / 11 \\ Planar Givens Rotations --- SROTG, DROTG / 13 \\ Three Dimensional Rotations --- ROT3 / 15 \\ Rotation of a Point on the Unit Sphere to the North Pole --- CONSTR / 17 \\ Hyperbolic Sine and Cosine Functions --- SNHCSH / 19 \\ Exponentials --- REXP, DREXP / 21 \\ Logarithms --- ALNREL, RLOG, RLOG1, DLNREL, DRLOG, DRLOG1 / 23 \\ \\ Geometry \\ \\ Determining if a Point is Inside or Outside a Polygon --- LOCPT / 25 \\ The Convex Hull for a Finite Planar Set --- HULL / 27 \\ Areas of Planar Polygons --- PAREA / 29 \\ Hamiltonian Circuits --- HC / 31 \\ \\ Special Functions \\ \\ Error Function --- CERF, CERFC, ERF, ERFC, ERFC1, DCERF, DCERFC, DERF, DERFC, DERFC1 / 35 \\ Inverse Error Function --- ERFINV / 41 \\ Normal Probability Distribution Function --- PNDF / 43 \\ Inverse Normal Probability Distribution Function --- PNINV / 45 \\ Dawson's Integral --- DAWSON / 47 \\ Complex Fresnel Integral --- CFRNLI / 49 \\ Real Fresnel Integrals --- FRESNEL / 51 \\ Exponential Integral Function --- CEXPLI, EXPLI, DEI, DEI1 / 53 \\ Sine and Cosine Integral Functions --- SI, CIN / 57 \\ Dilogarithm Function --- CLI, ALI / 59 \\ Gamma Function --- CGAMMA, GAMMA, GAMLN, DCGAMA, DGAMMA, DGAMLN / 61 \\ Diganma Function --- CPSI, PSI, DCPSI, DPSI / 65 \\ Logarithm of the Beta Function --- BETALN, DBETLN / 67 \\ Incomplete Gamma Ratio Functions --- GRATIO, RCOMP / 69 \\ Inverse Incomplete Gamma Ratio Function --- GAMINV / 71 \\ Incomplete Beta Function --- BRATIO, ISUBX, BRCOMP / 73 \\ Bessel Function $J_\nu(z)$ --- CBSSLJ, BSSLJ, BESJ / 75 \\ Bessel Function $Y_\nu(z)$ --- BSSLY / 77 \\ Modified Bessel Function $I_\nu(z)$ --- BSSLI, BESI / 79 \\ Modified Bessel Function $K_\nu(z)$ --- CBSSLK, BSSLK / 81 \\ Airy functions --- CAI, CBI, AI, AIE, BI, BIE / 83 \\ Complete Complex Elliptic Integrals of the First and Second Kinds --- CK, CKE / 87 \\ Real Elliptic Integrals of the First and Second Kinds --- ELLPI, RFVAL, RDVAL, DELLPI, DRFVAL, DRDVAL / 91 \\ Real Elliptic Integrals of the Third Kind --- EPI, RJVAL, DEPI, DRJVAL / 95 \\ Jacobian Elliptic Functions --- ELLPF, ELPFC1 / 99 \\ Weierstrass Elliptic Function for the Equianharmonic and Lemniscatic Cases --- PEQ, PEQ1, PLEM, PLEM1 / 103 \\ Integral of the Bivariate Density Function over Arbitrary Polygons and Semi-infinite Angular Regions --- VALR2 / 107 \\ Circular Coverage Function --- CIRCV / 109 \\ Elliptical Coverage Function --- PKILL, PKILL3 / 111 \\ \\ Polynomials \\ \\ Copying Polynomials --- PLCOPY, DPCOPY / 113 \\ Addition of Polynomials --- PADD, DPADD / 115 \\ Subtraction of Polynomials --- PSUBT, DPSUBT / 117 \\ Multiplication of Polynomials --- PMULT, DPMULT / 119 \\ Division of Polynomials --- PDIV, DPDIV / 121 \\ Real Powers of Polynomials --- PLPWR, DPLPWR / 123 \\ Inverses of Power Series --- PINV, DPINV / 125 \\ Derivatives and Integrals of Polynomials --- MPLNMV / 127 \\ Evaluation of Chebyshev Expansions --- CSEVL, DCSEVL / 129 \\ Lagrange Polynomials --- LGRNGN, LGRNGV, LGRNGX / 131 \\ Orthogonal Polynomials on Finite Sets --- ORTHOS, ORTHOV, ORTHOX / 133 \\ \\ Solutions of Nonlinear Equations \\ \\ Zeros of Continuous Functions --- ZEROIN / 135 \\ Solution, of Systems of Nonlinear Equations --- HBRD / 137 \\ Solutions of Quadratic, Cubic, and Quartic Equations --- QDCRT, CBCRT, QTCRT, DQDCRT, DCBCRT, DQTCRT / 139 \\ Double Precision Roots of Polynomials --- DRPOLY, DCPOLY / 141 \\ Accuracy of the Roots of a Real Polynomial --- RBND / 143 \\ \\ Vectors \\ \\ Copying Vectors --- SCOPY, DCOPY, CCOPY / 145 \\ Interchanging Vectors --- SSWAP, DSWAP, CSWAP / 147 \\ Planar Rotation of Vectors --- SROT, DROT, CSROT / 149 \\ Dot Products of Vectors --- SDOT, DDOT, CDOTC, CDOTU / 151 \\ Scaling Vectors --- SSCAL, DSCAL, CSCAL, CSSCAL / 153 \\ Vector Addition --- SAXPY, DAXPY, CAXPY / 155 \\ $L_1$ Norm of a Vector-- SASUM, DASUM, SCASUM / 157 \\ $L_2$ Norm of a Vector --- SNRM2, DNRM2, SCNRM2 / 159 \\ $L_\infty$ Norm of a Vector --- ISAMAX, IDAMAX, ICAMAX / 161 \\ \\ Matrices \\ \\ Packing and Unpacking Symmetric Matrices --- MCVFS, DMCVFS, MCVSF, DMCVSF / 163 \\ Conversion of Real Matrices to and from Double Precision Form --- MCVRD, MCVDR / 165 \\ Storage of Real Matrices in the Complex Matrix Format --- MCVRC / 167 \\ The Real and Imaginary Parts of a Complex Matrix --- CMREAL, CMIMAG / 169 \\ Copying Matrices --- MCOPY, SMCOPY, DMCOPY, CMCOPY / 171 \\ Computation of the Conjugate of a Complex Matrix --- CMCONJ / 173 \\ Transposing Matrices --- TPOSE, DTPOSE, CTPOSE, TIP, DTIP, CTIP / 175 \\ Computing Adjoints of Complex Matrices --- CMADJ, CTRANS / 177 \\ Matrix Addition --- MADD, SMADD, DMADD, CMADD / 179 \\ Matrix Subtraction --- MSUBT, SMSUBT, DMSUBT, CMSUBT / 181 \\ Matrix Multiplication-- MTMS, DMTMS, CMTMS, MPROD, DMPROD, CMPROD / 183 \\ Product of a Packed Symmetric Matrix and a Vector --- SVPRD, DSVPRD / 185 \\ Transpose Matrix Products --- TMPROD / 187 \\ Symmetric Matrix Products --- SMPROD / 189 \\ Kronecker Product of Matrices --- KPROD, DKPROD, CKPROD / 191 \\ Inverting General Real Matrices and Solving General Systems of Real Linear Equations --- CROUT, KROUT, NPIVOT, MSLV, DSMSLV / 193 \\ Solutions of Real Equations with Iterative Improvement --- SLVMP / 197 \\ Solution of Almost Block Diagonal Systems of Linear Equations --- ARCECO, ARCESL / 199 \\ Solution of Almost Block Tridiagonal Systems of Linear Equations --- BTSLV / 201 \\ Inverting Symmetric Real Matrices and Solving Symmetric Systems of Real Linear Equations --- SMSLV, DSMSLV / 203 \\ Inverting Positive Definite Symmetric Matrices and Solving Positive Definite Symmetric Systems of Linear Equations --- PCHOL, DPCHOL / 207 \\ Solution of Toeplitz Systems of Linear Equations --- TOPLX, DTOPLX / 209 \\ Inverting General Complex Matrices and Solving General Systems of Complex Linear Equations --- CMSLV, CMSLV1, DCMSLV / 211 \\ Solution of Complex Equations with Iterative Improvement --- CSLVMP / 215 \\ Singular Value Decomposition of a Matrix --- SSVDC, DSVDC, CSVDC / 217 \\ Evaluation of the Characteristic Polynomial of a Matrix --- DET, DPDET, CDET / 219 \\ Solution of the Matrix Equation $A X + X B = C$ --- ABSLV, DABSLV / 221 \\ Solution of the Matrix Equation $A^t X + X B = C$ when $C$ is Symmetric --- TASLV, DTASLV / 223 \\ Solution of the Matrix Equation $A X^2 + X B + C = 0$ --- SQUINT / 225 \\ Exponential of a Real Matrix --- MEXP, DMEXP / 227 \\ \\ Large Dense Systems of Linear Equations \\ \\ Solving systems of 200--400 Linear Equations --- LE, DPLE, CLE / 229 \\ \\ Banded Matrices \\ \\ Band Matrix Storage / 231 \\ Conversion of Banded Matrices to and from the Standard Format --- CVBR, CVBC, CVRB, CVCB, CVRB1, CVCB1 / 233 \\ Conversion of Banded Matrices to and from Sparse Form --- MCVBS, CMCVBS, MCVSB, CMCVSB / 235 \\ Transposing Banded Matrices --- BPOSE, CBPOSE / 237 \\ Addition of Banded Matrices --- BADD, CBADD / 239 \\ Subtraction of Banded Matrices --- BSUBT, CBSUBT / 241 \\ Multiplication of Banded Matrices --- BPROD, CBPROD / 243 \\ Product of a Real Banded Matrix and Vector --- BVPRD, BVPRD1, BTPRD, BTPRD1 / 245 \\ Product of a Complex Banded Matrix and Vector --- CBVPD, CBVPD1, CBTPD, CBTPD1 / 247 \\ Solution of Banded Systems of Real Linear Equations --- BSLV, BSLV1 / 249 \\ Solution of Banded Systems of Complex Linear Equations --- CBSLV, CBSLV1 / 251 \\ \\ Sparse Matrices \\ \\ Storage of Sparse Matrices / 253 \\ Conversion of Sparse Matrices to and from the Standard Format --- CVRS, CVDS, CVCS, CVSR, CVSD, CVSC / 255 \\ Conversion of Sparse Real Matrices to and from Double Precision Form --- SCVRD, SCVDR / 257 \\ The Real and Imaginary Parts of a Sparse Complex Matrix --- CSREAL, CSIMAG / 259 \\ \\ Computing $A + i D$ for Sparse Real Matrices $A$ and $B$ --- SCVRC / 261 \\ Copying Sparse Matrices --- RSCOPY, DSCOPY, CSCOPY / 263 \\ Computing Conjugates of Sparse Complex Matrices --- SCONJ / 265 \\ Transposing Sparse Real Matrices --- RPOSE, RPOSE1 / 267 \\ Transposing Sparse Double Precision Matrices --- DPOSE, DPOSE1 / 269 \\ Transposing Sparse Complex Matrices --- CPOSE, CPOSE1 / 271 \\ Addition of Sparse Matrices --- SADD, DSADD, CSADD / 273 \\ Subtraction of Sparse Matrices --- SSUBT, DSSUBT, CSSUBT / 275 \\ Multiplication of Sparse Matrices --- SPROD, DSPROD, CSPROD / 277 \\ Product of a Real Sparse Matrix and Vector --- MVPRD, MVPRD1, MTPRD, MTPRD1 / 279 \\ Product of a Double Precision Sparse Matrix and Vector --- DVPRD, DVPRD1, DTPRD, DTPRD1 / 281 \\ Product of a Complex Sparse Matrix and Vector --- CVPRD, CVPRD1, CTPRD, CTPRD1 / 283 \\ Ordering the Rows of a Sparse Matrix by Increasing Length --- SPORD / 285 \\ Reordering Sparse Matrices into Block Triangular Form --- BLKORD / 287 \\ Solution of Sparse Systems of Real Linear Equations --- SPSLV, RSLV, TSLV / 289 \\ Double Precision Solution of Sparse Systems of Real Linear Equations --- DSPSLV, DSLV, DTSLV / 293 \\ Solution of Sparse Systems of Complex Linear Equations --- CSPSLV, CSLV, CTSLV / 297 \\ \\ Eigenvalues and Eigenvectors \\ \\ Computation of Eigenvalues of General Real Matrices --- EIG, EIG1 / 301 \\ Computation of Eigenvalues and Eigenvectors of General Real Matrices --- EIGV, EIGV1 / 303 \\ Double Precision Computation of Eigenvalues of Real Matrices --- DEIG / 305 \\ Double Precision Computation of Eigenvalues and Eigenvectors of Real Matrices --- DEIGV / 307 \\ Computation of Eigenvalues of Symmetric Real Matrices --- SEIG, SEIG1 / 309 \\ Computation of Eigenvalues and Eigenvectors of Symmetric Real Matrices --- SEIGV, SEIGV1 / 311 \\ Computation of Eigenvalues of Complex Matrices --- CEIG / 313 \\ Computation of Eigenvalues and Eigenvectors of Complex Matrices --- CEIGV / 315 \\ Double Precision Computation of Eigenvalues of Complex Matrices --- DCEIG / 317 \\ Double Precision Computation of Eigenvalues and Eigenvectors of Complex Matrices --- DCEIGV / 319 \\ \\ $\ell_1$ Solution of Linear Equations \\ \\ $\ell_1$ Solution of Systems of Linear Equations with Equality and Inequality Constraints --- CL1 / 321 \\ \\ Least Squares Solution of Linear Equations \\ \\ Least Squares Solution of Systems of Linear Equations --- LLSQ, HFTI, HFTI2 / 323 \\ Least Squares Solution of Overdetermined Systems of Linear Equations with Iterative Improvement --- LLSQMP / 327 \\ Double Precision Least Squares Solution of Systems of Linear Equations --- DLLSQ, DHFTI, DHFTI2 / 329 \\ Least Squares Solution of Systems of Linear Equations with Equality and Inequality Constraints --- LSEI / 333 \\ Least Squares Solution of Systems of Linear Equations with Equality and Nonnegativity Constraints --- WNNLS / 337 \\ Least Squares Iterative Improvement Solution of Systems of Linear Equations with Equality Constraints --- L2SLV / 341 \\ Iterative Least Squares Solution of Banded Linear Equations --- BLSQ / 345 \\ Iterative Least Squares Solution of Sparse Linear Equations --- SPLSQ, STLSQ / 347 \\ \\ Optimization \\ \\ Minimization of Functions of a Single Variable --- FMIN / 349 \\ Minimization of Functions of n Variables --- OPTF / 351 \\ Unconstrained Minimum of the Sum of Squares of Nonlinear Functions --- LMDIFF / 353 \\ Linear Programming --- SMPLX, SSPLX / 355 \\ The Assignment Problem --- ASSGN / 359 \\ $0$--$1$ Knapsack Problem --- MKP / 361 \\ \\ Transforms \\ \\ Inversion of the Laplace Transform --- LAINV / 363 \\ Fast Fourier Transform --- FFT, FFT1 / 367 \\ Multivariate Fast Fourier Transform --- MFFT, MFFT1 / 369 \\ Discrete Cosine and Sine Transforms --- COSQI, COSQB, COSQF, SINQB, SINQF / 371 \\ \\ Approximation of Functions \\ \\ Rational Minimax Approximation of Functions --- CHEBY / 375 \\ $L_p$ Approximation of Functions --- ADAPT / 377 \\ Calculation of the Taylor Series of a Complex Analytic Function --- CPSC, DCPSC / 381 \\ \\ Curve Fitting \\ \\ Linear Interpolation --- TRP / 385 \\ Lagrange Interpolation --- LTRP / 387 \\ Hermite Interpolation --- HTRP / 389 \\ Conversion of Real Polynomials from Newton to Taylor Series Form --- PCOEFF / 391 \\ Least Squares Polynomial Fit --- PFIT / 393 \\ Weighted Least Squares Polynomial Fit --- WPFIT / 395 \\ Cubic Spline Interpolation --- CBSPL, SPLIFT / 397 \\ Weighted Least Squares Cubic Spline Fitting --- SPFIT / 399 \\ Cubic Spline Evaluation --- SCOMP, SCOMP1, SCOMP2 / 401 \\ Cubic Spline Evaluation and Differentiation --- SEVAL, SEVAL1, SEVAL2 / 403 \\ Integrals of Cubic Splines --- CSINT, CSINT1, CSINT2 / 405 \\ N-Dimensional Cubic Spline Closed Curve Fitting --- CSLOOP, LOPCMP, LOPDF / 407 \\ Spline under Tension Interpolation --- CURV1 / 409 \\ Spline under Tension Evaluation --- CURV2 / 411 \\ Differentiation and Integration of Splines under Tension --- CURVD, CURVI / 413 \\ Two Dimensional Spline under Tension Curve Fitting --- KURV1, KURV2 / 415 \\ Two Dimensional Spline under Tension Closed Curve Fitting --- KURVP1, KURVP2 / 417 \\ Three Dimensional Spline under Tension Curve Fitting --- QURV1, QURV2 / 419 \\ B-Splines / 421 \\ Piecewise Polynomial Interpolation --- BSTRP / 423 \\ Conversion of Piecewise Polynomials from B-Spline to Taylor Series Form --- BSPP / 425 \\ Piecewise Polynomial Evaluation --- PPVAL / 427 \\ Weighted Least Squares Piecewise Polynomial Fitting --- BSL2 / 429 \\ \\ Surface Fitting over Rectangular Grids \\ \\ Bi-Splines under Tension / 431 \\ Bi-Spline under Tension Surface Interpolation --- SURF / 433 \\ Bi-Spline under Tension Evaluation --- SURF2, NSURF2 / 435 \\ \\ Surface Fitting over Arbitrarily Positioned Data Points \\ \\ Surface Interpolation for Arbitrarily Positioned Data Points --- BVIP, BVIP2 / 437 \\ \\ Manifold Fitting \\ \\ Weighted Least Squares Fitting with Polynomials of $n$ Variables --- MFIT, DMFIT, MEVAL, DMEVAL / 441 \\ \\ Numerical Integration \\ \\ Evaluation of Integrals over Finite Intervals --- QAGS, QSUBA, DQAGS / 445 \\ \\ Evaluation of Integrals over Infinite Intervals --- QAGI, DQAGI / 449 \\ Evaluation of Double Integrals over Triangles --- CUBTRI / 453 \\ \\ Integral Equations \\ \\ Solution of Fredholm Integral Equations of the Second Kind --- IESLV / 455 \\ \\ Ordinary Differential Equations/Initial Value Problems \\ \\ The Initial Value Solvers --- Introductory Comments / 459 \\ Adaptive Adams Solution of Nonstiff Differential Equations --- ODE / 461 \\ Adaptive RKF Solution of Nonstiff Differential Equations --- RKF45 / 465 \\ Adaptive RKF Solution of Nonstiff Differential Equations with Global Error Estimation --- GERK / 469 \\ Adaptive Solution of Stiff Differential Equations --- SFODE, SFODE1 / 473 \\ Fourth-Order Runge-Kutta --- RK / 477 \\ Eighth-Order Runge-Kutta --- RK8 / 479 \\ \\ Partial Differential Equations \\ \\ Separable Second-Order Elliptic Equations on Rectangular Domains --- SEPDE / 481 \\ \\ Random Number Generation \\ \\ Uniform Random Number Generator --- URNG / 485 \\ Gaussian Random Number Generator using the Box--M{\"u}ller Transformation --- NRNG / 487 \\ \\ Appendix. Installation of the NSWC Library / 489 \\ \\ Index / 491 \\ \\ Distribution", } @MastersThesis{Muller:1990:HCA, author = "Volker M{\"u}ller", title = "{Hochgenaue CORDIC-Algorithmen f{\"u}r reelle Standardfunktionen mittels dynamischer Defektberechnung}. ({German}) [{High}-accuracy {CORDIC} Algorithms for Real Elementary Functions by Means of Dynamic Error Computation]", type = "{Diplomarbeit}", school = "Institut f{\"u}r angewandte Mathematik, Universit{\"a}t Karlsruhe", address = "Karlsruhe, Germany", pages = "????", month = dec, year = "1990", bibdate = "Fri Sep 16 16:30:40 1994", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; garbo.uwasa.fi:/pc/doc-soft/fpbiblio.txt; https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nj, language = "German", } @Article{Osada:1990:CAM, author = "Naoki Osada", title = "A Convergence Acceleration Method for Some Logarithmically Convergent Sequences", journal = j-SIAM-J-NUMER-ANAL, volume = "27", number = "1", pages = "178--189", month = feb, year = "1990", CODEN = "SJNAAM", DOI = "https://doi.org/10.1137/0727012", ISSN = "0036-1429 (print), 1095-7170 (electronic)", ISSN-L = "0036-1429", MRclass = "65B05", MRnumber = "1034928 (91b:65002)", bibdate = "Thu Dec 1 17:36:45 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "SIAM Journal on Numerical Analysis", journal-URL = "http://epubs.siam.org/sinum", keywords = "convergence acceleration", } @Article{Palmore:1990:CAC, author = "J. Palmore and C. Herring", title = "Computer arithmetic, chaos and fractals", journal = j-PHYSICA-D, volume = "42", number = "1--3", pages = "99--110", month = jun, year = "1990", CODEN = "PDNPDT", DOI = "https://doi.org/10.1016/0167-2789(90)90069-2", ISSN = "0167-2789 (print), 1872-8022 (electronic)", ISSN-L = "0167-2789", bibdate = "Tue Dec 12 09:17:24 MST 1995", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "Ninth Annual International Conference of the Center for Nonlinear Studies on Self-Organizing, Collective and Cooperative Phenomena in Natural and Artificial Networks", abstract = "The authors explore aspects of computer arithmetic from the viewpoint of dynamical systems. They demonstrate the effects of finite precision arithmetic in three uniformly hyperbolic chaotic dynamical systems: Bernoulli shifts, cat maps, and pseudorandom number generators. They show that elementary floating-point operations in binary computer arithmetic possess an inherently fractal structure. Each of these dynamical systems allows us to compare the exact results in integer arithmetic with those obtained by using floating-point arithmetic.", acknowledgement = ack-nhfb, affiliation = "Dept. of Math., Illinois Univ., Urbana, IL, USA", classification = "C1160 (Combinatorial mathematics); C5230 (Digital arithmetic methods)", confdate = "22-26 May 1989", conflocation = "Los Alamos, NM, USA", fjournal = "Physica. D, Nonlinear phenomena", journal-URL = "http://www.sciencedirect.com/science/journal/01672789", keywords = "Bernoulli shifts; Binary computer arithmetic; Cat maps; Chaos; Computer arithmetic; Dynamical systems; Elementary floating-point operations; Finite precision arithmetic; Floating-point arithmetic; Fractal structure; Integer arithmetic; Pseudorandom number generators; Self-similar structure; Uniformly hyperbolic chaotic dynamical systems", pubcountry = "Netherlands", thesaurus = "Chaos; Digital arithmetic; Fractals; Random number generation; Roundoff errors", } @Article{Poppe:1990:AEC, author = "G. P. M. Poppe and C. M. J. Wijers", title = "{Algorithm 680}: Evaluation of the Complex Error Function", journal = j-TOMS, volume = "16", number = "1", pages = "47--47", month = mar, year = "1990", CODEN = "ACMSCU", DOI = "https://doi.org/10.1145/77626.77630", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", MRclass = "47. 65G05 (65-04)", MRnumber = "91h:65068b", bibdate = "Sun Sep 04 23:03:20 1994", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib", note = "See remark \cite{Zaghloul:2019:RO}.", URL = "http://www.acm.org/pubs/citations/journals/toms/1990-16-1/p47-poppe/", acknowledgement = ack-nhfb, fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", keywords = "algorithms", subject = "{\bf G.1.2}: Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Rational approximation.", } @Article{Poppe:1990:MEC, author = "G. P. M. Poppe and C. M. J. Wijers", title = "More Efficient Computation of the Complex Error Function", journal = j-TOMS, volume = "16", number = "1", pages = "38--46", month = mar, year = "1990", CODEN = "ACMSCU", DOI = "https://doi.org/10.1145/77626.77629", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", MRclass = "65G05 (65D20)", MRnumber = "91h:65068a", bibdate = "Sun Sep 04 23:03:20 1994", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib", URL = "http://www.acm.org/pubs/citations/journals/toms/1990-16-1/p38-poppe/", acknowledgement = ack-nhfb, fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", keywords = "algorithms", subject = "{\bf G.4}: Mathematics of Computing, MATHEMATICAL SOFTWARE, Algorithm analysis. {\bf G.1.2}: Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Rational approximation.", } @Article{Press:1990:EI, author = "William H. Press and Saul A. Teukolsky", title = "Elliptic Integrals", journal = j-COMPUT-PHYS, volume = "4", number = "1", pages = "92--??", month = jan, year = "1990", CODEN = "CPHYE2", DOI = "https://doi.org/10.1063/1.4822893", ISSN = "0894-1866 (print), 1558-4208 (electronic)", ISSN-L = "0894-1866", bibdate = "Wed Apr 10 08:45:21 MDT 2019", bibsource = "https://www.math.utah.edu/pub/tex/bib/computphys.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "https://aip.scitation.org/doi/10.1063/1.4822893", acknowledgement = ack-nhfb, ajournal = "Comput. Phys", fjournal = "Computers in Physics", journal-URL = "https://aip.scitation.org/journal/cip", } @Article{Reemtsen:1990:MFR, author = "Rembert Reemtsen", title = "Modifications of the First {Remez} Algorithm", journal = j-SIAM-J-NUMER-ANAL, volume = "27", number = "2", pages = "507--518", month = apr, year = "1990", CODEN = "SJNAAM", DOI = "https://doi.org/10.1137/0727031", ISSN = "0036-1429 (print), 1095-7170 (electronic)", ISSN-L = "0036-1429", MRclass = "65D15", MRnumber = "91a:65039", bibdate = "Fri Oct 16 06:57:22 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/siamjnumeranal.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "SIAM Journal on Numerical Analysis", journal-URL = "http://epubs.siam.org/sinum", } @Article{Revfeim:1990:LEM, author = "K. J. A. Revfeim", title = "Letter to the {Editor}: More approximations for the cumulative and inverse normal distribution", journal = j-AMER-STAT, volume = "44", number = "1", pages = "63--63", month = feb, year = "1990", CODEN = "ASTAAJ", ISSN = "0003-1305 (print), 1537-2731 (electronic)", ISSN-L = "0003-1305", bibdate = "Fri Jan 27 14:51:19 MST 2012", bibsource = "http://www.jstor.org/journals/00031305.html; http://www.jstor.org/stable/i326447; https://www.math.utah.edu/pub/tex/bib/amstat1990.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.jstor.org/stable/2684963", acknowledgement = ack-nhfb, fjournal = "The American Statistician", journal-URL = "http://www.tandfonline.com/loi/utas20", } @Article{Sedogbo:1990:CAS, author = "Guy Antoine Sedogbo", title = "Convergence acceleration of some logarithmic sequences", journal = j-J-COMPUT-APPL-MATH, volume = "32", number = "1--2", pages = "253--260", day = "26", month = nov, year = "1990", CODEN = "JCAMDI", DOI = "https://doi.org/10.1016/0377-0427(90)90435-3", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", MRclass = "65B10 (40A25 65B99)", MRnumber = "1091794 (91m:65009)", bibdate = "Thu Dec 1 17:36:45 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "Extrapolation and rational approximation (Luminy, 1989)", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", keywords = "convergence acceleration", } @Book{Swartzlander:1990:CA, author = "Earl E. Swartzlander", title = "Computer arithmetic", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "various", year = "1990", ISBN = "0-8186-8931-5 (v. 1), 0-8186-5931-9 (v. 1 microfiche)", ISBN-13 = "978-0-8186-8931-4 (v. 1), 978-0-8186-5931-7 (v. 1 microfiche)", LCCN = "QA76.6.C633 1990", bibdate = "Sat Feb 24 15:01:45 MST 1996", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "Two volumes.", series = "IEEE Computer Society Press tutorial", acknowledgement = ack-nhfb, annote = "Vol. 1 is a reprint. Originally published: Stroudsburg, Pa.: Dowden, Hutchinson and Ross, c1980. Originally published in series: Benchmark papers in electrical engineering and computer science; 21. Vol 2 is a sequel to the earlier collection. Vol. 1: 2nd ed.", keywords = "Computer arithmetic.; Electronic digital computers --- Programming.; Floating-point arithmetic.", tableofcontents = "Arithmetic operations in a binary computer / R. F. Shaw \\ High-speed arithmetic in binary computers / O. L. MacSorley \\ Fast carry logic for digital computers / B. Gilchrist, J. H. Pomerene, and S. Y. Wong \\ A logic for high-speed addition / A. Weinberger and J. L. Smith \\ Conditional-sum addition logic / J. Sklansky \\ An evaluation of several two-summand binary adders / J. Sklansky \\ Adder with distributed control / A. Svoboda \\ Multiple addition by residue threshold functions and their representation by array logic / I. T. Ho and T. C. Chen \\ Counting responders in an associative memory / C. C. Foster and F. D. Stockton \\ Parallel counters / E. E. Swartzlander, Jr. \\ A signed binary multiplication technique / A. D. Booth \\ Multiplying made easy for digital assemblies / C. Ghest. A binary multiplication scheme based on squaring / T. C. Chen \\ A suggestion for a fast multiplier / C. S. Wallace \\ Some schemes for parallel multipliers / L. Dadda \\ On parallel digital multipliers / L. Dadda \\ A compact high-speed parallel multiplication scheme / W. J. Stenzel, W. J. Kubitz, and G. H. Garcia \\ A two's complement parallel array multiplication algorithm / C. R. Baugh and B. A. Wooley \\ Comments on ``A two's complement parallel array multiplication algorithm'' / P. E. Blankenship \\ The quasi-serial multiplier / E. E. Swartzlander, Jr. \\ The two's complement quasi-serial multiplier / T. G. McDaneld and R. K. Guha \\ A new class of digital division methods / J. E Robertson \\ An algorithm for rapid binary division / J. B. Wilson and R. S. Ledley. Digit-by-digit transcendental-function computation / R. J. Linhardt and H. S. Miller \\ A unified algorithm for elementary functions / J. S. Walther \\ Some properties of iterative square-rooting methods using high-speed multiplication /C. V. Ramamoorthy, J. R. Goodman, and K. H. Kim \\ Radix-16 evaluation of certain elementary functions / M. D. Ercegovac \\ On the distribution of numbers / R. W. Hamming \\ An analysis of floating-point addition / D. W. Sweeney \\ The IBM\ldots{}Model 91: floating-point execution unit / S. F. Anderson \ldots{} [et al.] \\ Design of large high-speed floating-point-arithmetic units / J. B. Gosling \\ Analysis of rounding methods in floating-point arithmetic / D. J. Kuck,D. S. Parker, Jr., and A. H. Sameh. \\ cos x, tan-p1s x, and cot-p1s x / W. H. Specker.", } @Article{Tang:1990:AET, author = "Ping Tak Peter Tang", title = "Accurate and Efficient Testing of the Exponential and Logarithm Functions", journal = j-TOMS, volume = "16", number = "3", pages = "185--200", month = sep, year = "1990", CODEN = "ACMSCU", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", MRclass = "65-04 (65G99)", MRnumber = "1 070 797", bibdate = "Sun Sep 04 23:14:59 1994", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://doi.acm.org/10.1145/79505.79506; http://www.acm.org/pubs/citations/journals/toms/1990-16-3/p185-tang/", abstract = "Table-driven techniques can be used to test highly accurate implementation of EXP LOG. The largest error observed in EXP and LOG accurately to within 1/500 unit in the last place are reported in our tests. Methods to verify the tests' reliability are discussed. Results of applying the tests to our own as well as to a number of other implementations of EXP and LOG are presented.", acknowledgement = ack-nhfb, fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", keywords = "algorithms; languages; verification", subject = "{\bf G.1.0}: Mathematics of Computing, NUMERICAL ANALYSIS, General, Numerical algorithms. {\bf G.1.0}: Mathematics of Computing, NUMERICAL ANALYSIS, General, Error analysis. {\bf G.4}: Mathematics of Computing, MATHEMATICAL SOFTWARE, Certification and testing. {\bf G.4}: Mathematics of Computing, MATHEMATICAL SOFTWARE, Portability.", } @InProceedings{Tang:1990:FAL, author = "Ping Tak Peter Tang", title = "A fast algorithm for linear complex {Chebyshev} approximation", crossref = "Mason:1990:AAI", pages = "265--274", year = "1990", bibdate = "Wed Nov 29 14:12:06 2023", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, } @TechReport{Tang:1990:PGE, author = "Ping Tak Peter Tang", title = "A Portable Generic Elementary Function Package in {Ada} and an Accurate Test Suite", type = "Technical report", number = "ANL-90/35", institution = inst-ANL, address = inst-ANL:adr, pages = "iii + 35", month = nov, year = "1990", bibdate = "Fri Dec 28 11:36:25 2007", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.osti.gov/bridge/servlets/purl/6310184-4n5sOR/6310184.PDF", abstract = "A comprehensive set of elementary functions has been implemented portably in Ada. The high accuracy of the implementation has been confirmed by rigorous analysis. Moreover, we present new test methods that are efficient and offer a high resolution of 0.005 unit in the last place, Tbese test methods have been implemented portably here and confirm the accuracy of our implemented functions. Reports on the accuracy of other function libraries obtained by our test programs are also presented.", acknowledgement = ack-nhfb, } @TechReport{Tang:1990:SSI, author = "Ping Tak Peter Tang", title = "Some Software Implementations of the Functions Sine and Cosine", type = "Technical report", number = "ANL-90/3", institution = inst-ANL, address = inst-ANL:adr, month = apr, year = "1990", bibdate = "Fri Dec 28 11:21:38 2007", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www-fp.mcs.anl.gov/division/publications/abstracts/abstracts90.htm", abstract = "This document presents several software implementations of the elementary functions sin and cos designed to fit a large class of machines. Implementation details are provided. Also provided is a detailed error analysis that bounds the errors of these implementations, over the full range of input arguments, from 0.721 to 0.912 units in the last place. Tests performed on these codes give results that are consistent with the error bounds.", acknowledgement = ack-nhfb, xxnote = "Where is this? I can find no electronic version online, other than the abstract at the given URL.", } @Article{Tang:1990:TDI, author = "Ping Tak Peter Tang", title = "Table-Driven Implementation of the Logarithm Function in {IEEE} Floating-Point Arithmetic", journal = j-TOMS, volume = "16", number = "4", pages = "378--400", month = dec, year = "1990", CODEN = "ACMSCU", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Sun Sep 04 23:26:09 1994", bibsource = "garbo.uwasa.fi:/pc/doc-soft/fpbiblio.txt; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://doi.acm.org/10.1145/98267.98294; http://www.acm.org/pubs/citations/journals/toms/1990-16-4/p378-tang/", abstract = "Algorithms and implementation details for the logarithm functions in both single and double precision of IEEE 754 arithmetic are presented here. With a table of moderate size, the implementation need only working- precision arithmetic and are provably accurate to within 0.57 ulp.", acknowledgement = ack-nj, fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", keywords = "algorithms; design; performance; reliability; standardization; theory; verification", subject = "{\bf G.1.0}: Mathematics of Computing, NUMERICAL ANALYSIS, General, Computer arithmetic. {\bf G.1.0}: Mathematics of Computing, NUMERICAL ANALYSIS, General, Error analysis. {\bf G.1.0}: Mathematics of Computing, NUMERICAL ANALYSIS, General, Numerical algorithms. {\bf G.4}: Mathematics of Computing, MATHEMATICAL SOFTWARE, Algorithm analysis.", } @Article{Todd:1990:WMP, author = "John Todd", title = "The {Weierstrass} mean. {I}. The periods of $ \wp (z \vert e_1, e_2, e_3) $", journal = j-NUM-MATH, volume = "57", number = "8", pages = "737--746", month = aug, year = "1990", CODEN = "NUMMA7", ISSN = "0029-599X (print), 0945-3245 (electronic)", ISSN-L = "0029-599X", MRclass = "65D20 (33E05)", MRnumber = "91m:65057", MRreviewer = "Syvert P. N{\o}rsett", bibdate = "Mon May 26 11:49:34 MDT 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/nummath.bib", acknowledgement = ack-nhfb, classification = "B0290 (Numerical analysis); C4100 (Numerical analysis)", corpsource = "California Inst. of Technol., Pasadena, CA, USA", fjournal = "Numerische Mathematik", journal-URL = "http://link.springer.com/journal/211", keywords = "convergence; elliptic objects; limits; numerical methods; Weierstrass mean", treatment = "T Theoretical or Mathematical", } @InProceedings{Watson:1990:NMC, author = "G. Alistair Watson", title = "Numerical methods for {Chebyshev} approximation of complex-valued functions", crossref = "Mason:1990:AAI", pages = "246--264", year = "1990", bibdate = "Wed Nov 29 14:09:32 2023", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, } @Article{Wells:1990:LE, author = "Martin T. Wells and Ram C. Tiwari and I. Arizono and H. Ohta and James W. Mergerson and Gabriella M. Belli and Christopher Cox and Murray A. Jorgensen and Jacques Benichou and Mitchell H. Gail and Warren F. Kuhfeld and Brian Dawkins and Walter B. Studdiford and Colin Goodall and W. D. Kaigh and Stephen W. Looney and Robert Kinnison and James A. Gibbons and Joel R. Levin and Ronald C. Serlin and K. J. A. Revfeim and Charles R. McConnell and Robert M. Norton and R. W. Farebrother and I. J. Good and Stanley Lebergott and Vedula N. Murty", title = "Letters to the Editor", journal = j-AMER-STAT, volume = "44", number = "1", pages = "56--65", month = feb, year = "1990", CODEN = "ASTAAJ", ISSN = "0003-1305 (print), 1537-2731 (electronic)", ISSN-L = "0003-1305", bibdate = "Fri Jan 27 14:51:19 MST 2012", bibsource = "http://www.jstor.org/journals/00031305.html; http://www.jstor.org/stable/i326447; https://www.math.utah.edu/pub/tex/bib/amstat1990.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.jstor.org/stable/2684963", acknowledgement = ack-nhfb, fjournal = "The American Statistician", journal-URL = "http://www.tandfonline.com/loi/utas20", } @Article{Weniger:1990:RAM, author = "Ernst Joachim Weniger and Ji{\v{r}}i C{\'\i}{\v{z}}ek", title = "Rational approximations for the modified {Bessel} function of the second kind", journal = j-COMP-PHYS-COMM, volume = "59", number = "3", pages = "471--493", month = jul, year = "1990", CODEN = "CPHCBZ", DOI = "https://doi.org/10.1016/0010-4655(90)90089-J", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", bibdate = "Mon Feb 13 21:29:12 MST 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/compphyscomm1990.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/001046559090089J", abstract = "Various different rational approximations for the modified Bessel function $ K_\nu (z) $ are compared with respect to their ability of computing $ K_\nu (z) $ efficiently and reliably in the troublesome region of moderately large arguments $z$. The starting point for the construction of the rational approximations is the asymptotic series $_2 F_0$ for $ K_\nu (z) $, which diverges for all finite arguments $z$ but is Borel summable and Stieltjes summable. The numerical tests showed that Pad{\'e} approximants for $ K_\nu (z) $ are significantly less efficient than the other rational approximations which were considered. The best results were produced by some recently derived sequence transformations (E. J. Weniger, Comput. Phys. Rep. {\bf 10} (1989) 189), which are closely related to Levin's sequence transformations (D. Levin, Int. J. Comput. Math. B {\bf 3} (1973) 371).", acknowledgement = ack-nhfb, fjournal = "Computer Physics Communications", journal-URL = "http://www.sciencedirect.com/science/journal/00104655", } @Article{Bartoloni:1991:MFU, author = "A. Bartoloni and C. Battista and S. Cabasino and N. Cabibbo and F. Del Prete and F. Marzano and P. S. Paolucci and R. Sarno and G. Salina and G. M. Todesco and M. Torelli and R. Tripiccione and W. Tross and E. Zanetti", title = "{MAD}, a floating-point unit for massively-parallel processors", journal = "Particle World", volume = "2", number = "3", pages = "65--73", month = "????", year = "1991", CODEN = "PARWEG", ISSN = "1043-6790", bibdate = "Tue Dec 12 09:26:54 MST 1995", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", abstract = "The authors describe in detail the architecture and implementation of the MAD chip. It is a floating point unit, used as the elementary processing element of the APE100 array processor. The design has been accurately tailored to the requirements of a SIMD floating point intensive machine.", acknowledgement = ack-nhfb, affiliation = "Roma Univ., Italy", classification = "B1265F (Microprocessors and microcomputers); C5130 (Microprocessor chips); C5220P (Parallel architecture); C5230 (Digital arithmetic methods); C7320 (Physics and Chemistry)", keywords = "APE100 array processor; Architecture; Elementary processing element; Floating-point unit; Massively-parallel processors; SIMD floating point intensive machine", pubcountry = "UK", thesaurus = "Digital arithmetic; Microprocessor chips; Parallel architectures; Physics computing", } @TechReport{Beebe:1991:ASR, author = "Nelson H. F. Beebe", title = "Accurate Square Root Computation", institution = inst-CSC, address = inst-CSC:adr, pages = "23", day = "4", month = feb, year = "1991", bibdate = "Sat Feb 8 10:28:55 2020", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/b/beebe-nelson-h-f.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", note = "Supplemental class notes prepared for Mathematics 118.", } @Article{Boersma:1991:UAB, author = "J. Boersma", title = "Uniform asymptotics of a {Bessel}-function series occurring in a transmission-line problem", journal = j-J-COMPUT-APPL-MATH, volume = "37", number = "1--3", pages = "143--159", day = "18", month = nov, year = "1991", CODEN = "JCAMDI", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Sat Feb 25 13:02:22 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath1990.bib", URL = "http://www.sciencedirect.com/science/article/pii/037704279190113X", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @InProceedings{Bohlender:1991:SEF, author = "G. Bohlender and W. Walter and P. Kornerup and D. W. Matula", title = "Semantics for exact floating point operations", crossref = "Kornerup:1991:PIS", bookpages = "xiii + 282", pages = "22--26", year = "1991", bibdate = "Wed Dec 13 13:13:34 MST 1995", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "IEEE Catalog number 91CH3015-5.", abstract = "Semantics are given for the four elementary arithmetic operations and the square root, to characterize what are termed exact floating point operations. The operands of the arithmetic operations and the argument of the square root are all floating point numbers in one format. In every case, the result is a pair of floating point numbers in the same format with no accuracy lost in the computation. These semantics make it possible to realize the following principle: it shall be a user option to discard any information in the result of a floating point arithmetic operation. The reliability and portability previously associated with only mathematical software implementations in integer arithmetic can thus be attained exploiting the generally higher efficiency of floating point hardware.", acknowledgement = ack-nhfb, affiliation = "Inst. fur Angewandte Math., Karlsruhe Univ., Germany", classification = "C1160 (Combinatorial mathematics); C5230 (Digital arithmetic methods)", confdate = "26-28 June 1991", conflocation = "Grenoble, France", confsponsor = "IEEE; CNRS; IMAG", keywords = "Argument; Elementary arithmetic operations; Exact floating point operations; Floating point arithmetic; Floating point hardware; Floating point numbers; Integer arithmetic; Mathematical software; Operands; Portability; Reliability; Semantics; Square root", pubcountry = "USA", thesaurus = "Digital arithmetic; Number theory", } @Book{Brezinski:1991:EMT, author = "Claude Brezinski and Michela {Redivo Zaglia}", title = "Extrapolation Methods: Theory and Practice", volume = "2", publisher = pub-NORTH-HOLLAND, address = pub-NORTH-HOLLAND:adr, pages = "ix + 464", year = "1991", ISBN = "0-444-88814-4", ISBN-13 = "978-0-444-88814-3", LCCN = "QA281 .B74 1991", bibdate = "Mon May 24 09:18:52 MDT 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; melvyl.cdlib.org:210/CDL90", series = "Studies in computational mathematics", acknowledgement = ack-nhfb, subject = "extrapolation; data processing", tableofcontents = "Preface / v \\ 1 INTRODUCTION TO THE THEORY / 1 \\ 1.1 First steps / 1 \\ 1.2 What is an extrapolation method? / 5 \\ 1.3 What is an extrapolation algorithm? / 8 \\ 1.4 Quasi-linear sequence transformations / 11 \\ 1.5 Sequence transformations as ratios of determinants / 18 \\ 1.6 Triangular recursive schemes / 21 \\ 1.7 Normal forms of the algorithms / 26 \\ 1.8 Progressive forms of the algorithms / 28 \\ 1.9 Particular rules of the algorithms / 34 \\ 1.10 Accelerability and non-accelerability / 39 \\ 1.11 Optimality / 42 \\ 1.12 Asymptotic behaviour of sequences / 47 \\ \\ 2 SCALAR EXTRAPOLATION ALGORITHMS / 55 \\ 2.1 The $E$-algorithm / 55 \\ 2.2 Richardson extrapolation process T2 \\ 2.3 The $\epsilon$-algorithm / 78 \\ 2.4 The $G$-transformation / 95 \\ 2.5 Rational extrapolation / 101 \\ 2.6 Generalizations of the $\epsilon$-algorithm / 108 \\ 2.7 Levin's transforms / 113 \\ 2.8 Overholt's process / 119 \\ 2.9 $\Theta$-type algorithms / 121 \\ 2.10 The iterated $\Delta^2$ process / 128 \\ 2.11 Miscellaneous algorithms / 131 \\ \\ 3 SPECIAL DEVICES / 145 \\ 3.1 Error estimates and acceleration / 145 \\ 3.2 Convergence tests and acceleration 151 \\ 3.3 Construction of asymptotic expansions / 159 \\ 3.4 Construction of extrapolation processes / 165 \\ 3.5 Extraction procedures / 171 \\ 3.6 Automatic selection / 178 \\ 3.7 Composite sequence transformations / 185 \\ 3.8 Error control / 193 \\ 3.9 Contractive sequence transformations / 201 \\ 3.10 Least squares extrapolation / 210 \\ \\ 4 VECTOR EXTRAPOLATION ALGORITHMS / 213 \\ 4.1 The vector $\epsilon$-algorithm / 216 \\ 4.2 The topological $\epsilon$-algorithm / 220 \\ 4.3 The vector $E$-algorithm / 228 \\ 4.4 The recursive projection algorithm / 233 \\ 4.5 The H-algorithm / 238 \\ 4.6 The Ford--Sidi algorithms / 244 \\ 4.7 Miscellaneous algorithms / 247 \\ \\ 5 CONTINUOUS PREDICTION ALGORITHMS / 253 \\ 5.1 The Taylor expansion / 254 \\ 5.2 Confluent Overholt's process / 255 \\ 5.3 Confluent $\epsilon$-algorithms / 256 \\ 5.4 Confluent $\rho$-algorithm / 262 \\ 5.5 Confluent $G$-transform / 265 \\ 5.6 Confluent $E$-algorithm / 266 \\ 5.7 $\Theta$-type confluent algorithms / 267 \\ \\ 6 APPLICATIONS / 269 \\ 6.1 Sequences and series / 270 \\ 6.1.1 Simple sequences / 270 \\ 6.1.2 Double sequences / 278 \\ 6.1.3 Chebyshev and Fourier series / 282 \\ 6.1.4 Continued fractions / 284 \\ 6.1.5 Vector sequences / 208 \\ 6.2 Systems of equations / 302 \\ 6.2.1 Linear systems / 303 \\ 6.2.2 Projection methods / 2307 \\ 6.2.3 Regularization and penalty techniques / 309 \\ 6.2.4 Nonlinear equations / 315 \\ 6.2.5 Continuation methods / 330 \\ 6.3 Eigenelements / 332 \\ 6.3.1 Eigenvalues and eigenvectors / 333 \\ 6.3.2 Derivatives of eigensystems / 336 \\ 6.4 Integral and differential equations / 338 \\ 6.4.1 Implicit Runge--Kutta methods / 339 \\ 6.4.2 Boundary value problems / 340 \\ 6.4.3 Nonlinear methods / 346 \\ 6.4.4 Laplace transform inversion / 348 \\ 6.4.5 Partial differential equations / 352 \\ 6.5 Interpolation and approximation / 354 \\ 6.6 Statistics / 357 \\ 6.6.1 The jackknife / 358 \\ 6.6.2 ARMA models / 359 \\ 6.6.3 Monte--Carlo methods / 361 \\ 6.7 Integration and differentiation / 365 \\ 6.7.1 Acceleration of quadrature formulae / 366 \\ 6.7.2 Nonlinear quadrature formulae / 372 \\ 6.7.3 Cauchy's principal values / 373 \\ 6.7.4 Infinite integrals / 378 \\ 6.7.5 Multiple integrals / 387 \\ 6.7.6 Numerical differentiation / 389 \\ 6.8 Prediction / 5389 \\ \\ 7 SOFTWARE / 397 \\ 7.1 Programming the algorithms / 397 \\ 7.2 Computer arithmetic / 400 \\ 73 Programs / 403 \\ \\ Bibliography / 413 \\ \\ Index / 455", } @Article{Bunch:1991:DFA, author = "K. J. Bunch and W. N. Cain and R. W. Grow", title = "A data fitting approach to series convergence acceleration", journal = j-APPL-MATH-COMP, volume = "42", number = "2 (part II)", pages = "189--195", month = "????", year = "1991", CODEN = "AMHCBQ", DOI = "https://doi.org/10.1016/0096-3003(91)90050-W", ISSN = "0096-3003 (print), 1873-5649 (electronic)", ISSN-L = "0096-3003", MRclass = "65B10 (65D10)", MRnumber = "1094414 (91k:65015)", bibdate = "Thu Dec 1 17:36:45 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Applied Mathematics and Computation", journal-URL = "http://www.sciencedirect.com/science/journal/00963003", keywords = "convergence acceleration", } @Article{Carlson:1991:TEI, author = "B. C. Carlson", title = "A table of elliptic integrals: {One} quadratic factor", journal = j-MATH-COMPUT, volume = "56", number = "193", pages = "267--280", month = jan, year = "1991", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "33E05 (65A05)", MRnumber = "92b:33056", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, classcodes = "B0290R (Integral equations); B0290M (Numerical integration and differentiation); C4180 (Integral equations); C4160 (Numerical integration and differentiation)", corpsource = "Dept. of Math., Iowa State Univ., Ames, IA, USA", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "conjugate complex zeros; elliptic integrals; Fortran programs; integral equations; integration; polynomial; R-functions; square root", treatment = "P Practical", } @TechReport{Cody:1991:CPT, author = "W. J. Cody", title = "{CELEFUNT}: a Portable Test Package for Complex Elementary Functions", type = "Technical Report", number = "ANL-91/1", institution = inst-ANL, address = inst-ANL:adr, pages = "iii + 21", month = jan, year = "1991", bibdate = "Fri Sep 23 23:39:07 1994", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/c/cody-william-j.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, } @Article{Corless:1991:NEA, author = "R. M. Corless and D. J. Jeffrey and H. Rasmussen", title = "Numerical evaluation of {Airy} functions with complex arguments", journal = j-J-COMPUT-PHYS, volume = "93", number = "1", pages = "252--253", month = mar, year = "1991", CODEN = "JCTPAH", DOI = "https://doi.org/10.1016/0021-9991(91)90089-4", ISSN = "0021-9991 (print), 1090-2716 (electronic)", ISSN-L = "0021-9991", bibdate = "Mon Jan 2 07:55:47 MST 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputphys1990.bib", URL = "http://www.sciencedirect.com/science/article/pii/0021999191900894", acknowledgement = ack-nhfb, fjournal = "Journal of Computational Physics", journal-URL = "http://www.sciencedirect.com/science/journal/00219991/", } @Article{Crenshaw:1991:SRS, author = "J. W. Crenshaw", title = "Square roots are simple?", journal = j-EMBED-SYS-PROG, volume = "4", number = "11", pages = "30--52", month = nov, year = "1991", CODEN = "EYPRE4", ISSN = "1040-3272", bibdate = "Wed Sep 14 19:14:52 1994", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nj, fjournal = "Embedded Systems Programming", } @Article{DeDoelder:1991:SSC, author = "P. J. {De Doelder}", title = "On some series containing $ \psi (x) - \psi (y >) $ and $ (\psi (x) - \psi (y >))^2 $ for certain values of $x$ and $y$", journal = j-J-COMPUT-APPL-MATH, volume = "37", number = "1--3", pages = "125--141", day = "18", month = nov, year = "1991", CODEN = "JCAMDI", DOI = "https://doi.org/10.1016/0377-0427(91)90112-W", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Sat Feb 25 13:02:22 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath1990.bib", URL = "http://www.sciencedirect.com/science/article/pii/037704279190112W", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @Article{Dritz:1991:IPS, author = "Kenneth W. Dritz", title = "Introduction to the proposed standard for the elementary functions in {Ada}", journal = j-SIGADA-LETTERS, volume = "11", number = "7", pages = "3--8", month = "Fall", year = "1991", CODEN = "AALEE5", ISSN = "1094-3641 (print), 1557-9476 (electronic)", ISSN-L = "1094-3641", bibdate = "Thu Mar 20 07:41:09 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, classcodes = "C6140D (High level languages); C7310 (Mathematics)", corpsource = "Dept. of Math. and Comput. Sci., Argonne Nat. Lab., IL, USA", fjournal = "ACM SIGAda Ada Letters", journal-URL = "http://portal.acm.org/citation.cfm?id=J32", keywords = "Ada; committees; elementary functions; generic package; ISO standard; mathematics computing; secondary; standards", treatment = "P Practical", } @Article{Dritz:1991:PSGa, author = "K. W. Dritz", title = "Proposed standard for a generic package of elementary functions for {Ada}", journal = j-SIGADA-LETTERS, volume = "11", number = "7", pages = "9--46", month = "Fall", year = "1991", CODEN = "AALEE5", ISSN = "1094-3641 (print), 1557-9476 (electronic)", ISSN-L = "1094-3641", bibdate = "Thu Mar 20 07:41:09 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, classcodes = "C6140D (High level languages); C6110B (Software engineering techniques); C7310 (Mathematics)", corpsource = "Div. of Math. and Comput. Sci., Argonne Nat. Lab., IL, USA", fjournal = "ACM SIGAda Ada Letters", journal-URL = "http://portal.acm.org/citation.cfm?id=J32", keywords = "ACM SIGAda Numerics Working; Ada; Ada-Europe Numerics Working Group; basic; elementary functions; ELEMENTARY-FUNCTIONS-; EXCEPTIONS; generic package; GENERIC-ELEMENTARY-FUNCTIONS; Group; international standard; joint proposal; mathematical routines; mathematics computing; NRG; Rapporteur Group; reusable applications; SC22; software reusability; specification; standards; WG9 Numerics", treatment = "P Practical", } @Article{Dritz:1991:PSGb, author = "K. W. Dritz", title = "Proposed standard for a generic package of primitive functions for {Ada}", journal = j-SIGADA-LETTERS, volume = "11", number = "7", pages = "66--82", month = "Fall", year = "1991", CODEN = "AALEE5", ISSN = "1094-3641 (print), 1557-9476 (electronic)", ISSN-L = "1094-3641", bibdate = "Thu Mar 20 07:41:09 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, classcodes = "C6140D (High level languages); C7310 (Mathematics)", corpsource = "Div. of Math. and Comput. Sci., Argonne Nat. Lab., IL, USA", fjournal = "ACM SIGAda Ada Letters", journal-URL = "http://portal.acm.org/citation.cfm?id=J32", keywords = "Ada; compliable Ada; elementary functions; generic package; GENERIC-; mathematical; mathematics computing; primitive functions; primitive operations; PRIMITIVE-FUNCTIONS; software; specification; standards", treatment = "P Practical", } @Article{Dritz:1991:RPS, author = "K. W. Dritz", title = "Rationale for the proposed standard for a generic package of elementary functions for {Ada}", journal = j-SIGADA-LETTERS, volume = "11", number = "7", pages = "47--65", month = "Fall", year = "1991", CODEN = "AALEE5", ISSN = "1094-3641 (print), 1557-9476 (electronic)", ISSN-L = "1094-3641", bibdate = "Thu Mar 20 07:41:09 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, classcodes = "C6140D (High level languages); C7310 (Mathematics); C6110 (Systems analysis and programming)", corpsource = "Div. of Math. and Comput. Sci., Argonne Nat. Lab., IL, USA", fjournal = "ACM SIGAda Ada Letters", journal-URL = "http://portal.acm.org/citation.cfm?id=J32", keywords = "ACM SIGAda Numerics Working Group; Ada; Ada-Europe Numerics; collateral; elementary functions standard; mathematics computing; numerical software; portability; programming; robustness; standards; Working Group", treatment = "P Practical", } @Article{Duprat:1991:WND, author = "J. Duprat and J.-M. Muller", title = "Writing numbers differently for faster calculation", journal = j-TECHNIQUE-SCI-INFORMATIQUES, volume = "10", number = "3", pages = "211--224", month = "????", year = "1991", CODEN = "TTSIDJ", ISSN = "0752-4072, 0264-7419", ISSN-L = "0752-4072", bibdate = "Tue Dec 12 09:20:21 MST 1995", bibsource = "https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", abstract = "Instead of Avizienis' or the carry save methods a borrow save (BS) notation is proposed. Examples are given of BS addition, subtraction, shifting and multiplication with the necessary elementary cells being proposed and circuits for testing zero and sign being described. Floating point arithmetic is explained, involving pseudo normalisation and applications are covered including the Cordic algorithm.", acknowledgement = ack-nhfb, affiliation = "Ecole Normale Superieure de Lyon, France", classification = "C5230 (Digital arithmetic methods)", fjournal = "Technique et science informatiques : TSI", keywords = "Addition; Borrow save; Carry save methods; Cordic algorithm; Floating point arithmetic; Multiplication; Pseudo normalisation; Shifting; Subtraction; Zero", language = "French", pubcountry = "France", thesaurus = "Digital arithmetic", } @InProceedings{Ferguson:1991:AMA, author = "Warren E. {Ferguson, Jr.} and Tom Brightman", title = "Accurate and Monotone Approximations of Some Transcendental Functions", crossref = "Kornerup:1991:PIS", pages = "237--244", year = "1991", DOI = "https://doi.org/10.1109/ARITH.1991.145566", bibdate = "Sat Nov 27 12:40:58 MST 2004", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "https://ieeexplore.ieee.org/document/145566", acknowledgement = ack-nj # " and " # ack-nhfb, keywords = "CORDIC algorithms", remark = "From page. 237: ``For example, an approximation of the sine function on $ [ - \pi / 4, \pi / 4] $ that is accurate to 66 bits of precision will necessarily be monotonic when rounded to 64 bits of precision. We have used this technique to establish the monotonicity of a Cyrix FasMath coprocessor's polynomial based approximations of transcendental functions [2]. Since this technique does not depend on how the approximation is determined, then it also can be applied to approximations derived by other means, e.g.,, CORDIC based approximations.''", } @Article{Gal:1991:AEM, author = "Shmuel Gal and Boris Bachelis", title = "An Accurate Elementary Mathematical Library for the {IEEE} Floating Point Standard", journal = j-TOMS, volume = "17", number = "1", pages = "26--45", month = mar, year = "1991", CODEN = "ACMSCU", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", MRclass = "65D20 (65-04 65D15)", MRnumber = "92a:65069", bibdate = "Sun Sep 04 23:33:02 1994", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.acm.org/pubs/toc/Abstracts/toms/103151.html", abstract = "The algorithms used by the IBM Israel Scientific Center for the elementary mathematical library using the IEEE standard for binary floating point arithmetic are described. The algorithms are based on the ``accurate tables method.'' This methodology achieves high performance and produces very accurate results. It overcomes one of the main problems encountered in elementary mathematical functions computations: achieving last bit accuracy. The results obtained are correctly rounded for almost all argument values. \par Our main idea in the accurate tables method is to use ``nonstandard tables,'' which are different from the natural tables of equally spaced points in which the rounding error prevents obtaining last bit accuracy. In order to achieve a small error we use the following idea: Perturb the original, equally spaced, points in such a way that the table value (or tables values in case we need several tables) will be very close to numbers which can be exactly represented by the computer (much closer than the usual double precision representation). Thus we were able to control the error introduced by the computer representation of real numbers and extended the accuracy without actually using extended precision arithmetic.", acknowledgement = ack-nhfb, fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", keywords = "algorithms; theory", subject = "{\bf G.1.0}: Mathematics of Computing, NUMERICAL ANALYSIS, General, Computer arithmetic. {\bf G.1.2}: Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation.", } @Article{Gray:1991:GMA, author = "H. L. Gray and Suojin Wang", title = "A General Method for Approximating Tail Probabilities", journal = j-J-AM-STAT-ASSOC, volume = "86", number = "413", pages = "159--166", month = mar, year = "1991", CODEN = "JSTNAL", ISSN = "0162-1459 (print), 1537-274X (electronic)", ISSN-L = "0162-1459", bibdate = "Wed Jan 25 08:06:12 MST 2012", bibsource = "http://www.jstor.org/journals/01621459.html; http://www.jstor.org/stable/i314297; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jamstatassoc1990.bib", URL = "http://www.jstor.org/stable/2289726", acknowledgement = ack-nhfb, fjournal = "Journal of the American Statistical Association", journal-URL = "http://www.tandfonline.com/loi/uasa20", } @InProceedings{Gustafson:1991:CAA, author = "Sven-{\AA}ke Gustafson and Frank Stenger", title = "Convergence acceleration applied to {Sinc} approximation with application to approximation of $ |x|^\alpha $", crossref = "Bowers:1991:CCI", pages = "161--171", year = "1991", MRclass = "41A30 (93B40)", MRnumber = "MR1140021", bibdate = "Thu May 10 16:31:10 2007", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/s/stenger-frank.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", ZMnumber = "0746.41034", abstract = "The author studies mainly the role of Chebyshev acceleration of Sinc approximation. Then he considers various methods of approximating $ \vert x \vert^\alpha $ and applies Chebyshev acceleration to the various type of approximants for the case of $ \alpha = 1 $.", acknowledgement = ack-nhfb, classmath = "*41A65 (Abstract approximation theory)", keywords = "Chebyshev acceleration; convergence acceleration", reviewer = "Zhang Ganglu (Dongying)", } @Article{Hamza:1991:MBD, author = "K. M. Hamza and M. A. H. Abdul-Karim", title = "Microprocessor Based Direct Square Root Extractor", journal = "Modelling", volume = "34", number = "1", pages = "45--48", month = "????", year = "1991", bibdate = "Thu Sep 1 10:15:42 1994", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nj, } @Article{Ifantis:1991:PZS, author = "E. K. Ifantis and C. G. Kokologiannaki and C. B. Kouris", title = "On the positive zeros of the second derivative of {Bessel} functions", journal = j-J-COMPUT-APPL-MATH, volume = "34", number = "1", pages = "21--31", day = "10", month = feb, year = "1991", CODEN = "JCAMDI", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Sat Feb 25 12:20:48 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath1990.bib", URL = "http://www.sciencedirect.com/science/article/pii/0377042791901449", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @Article{Ikebe:1991:CZO, author = "Yasuhiko Ikebe and Yasushi Kikuchi and Issei Fujishiro", title = "Computing zeros and orders of {Bessel} functions", journal = j-J-COMPUT-APPL-MATH, volume = "38", number = "1--3", pages = "169--184", day = "23", month = dec, year = "1991", CODEN = "JCAMDI", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Sat Feb 25 13:02:23 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath1990.bib", URL = "http://www.sciencedirect.com/science/article/pii/037704279190169K", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @Article{Iserles:1991:CDC, author = "A. Iserles", title = "Complex dynamics of convergence acceleration", journal = j-IMA-J-NUMER-ANAL, volume = "11", number = "2", pages = "205--240", year = "1991", CODEN = "IJNADH", ISSN = "0272-4979 (print), 1464-3642 (electronic)", ISSN-L = "0272-4979", MRclass = "65B05 (65E05)", MRnumber = "92h:65006", bibdate = "Sat Dec 23 17:06:35 MST 2000", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; MathSciNet database", acknowledgement = ack-nhfb, fjournal = "IMA Journal of Numerical Analysis", journal-URL = "http://imajna.oxfordjournals.org/content/by/year", keywords = "convergence acceleration", } @Article{Laforgia:1991:BMB, author = "Andrea Laforgia", title = "Bounds for modified {Bessel} functions", journal = j-J-COMPUT-APPL-MATH, volume = "34", number = "3", pages = "263--267", day = "26", month = apr, year = "1991", CODEN = "JCAMDI", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Sat Feb 25 12:20:49 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath1990.bib", URL = "http://www.sciencedirect.com/science/article/pii/037704279190087Z", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @Article{Levrie:1991:CAF, author = "Paul Levrie", title = "Convergence acceleration for $n$-fractions", journal = j-APPL-NUM-MATH, volume = "7", number = "6", pages = "481--492", month = jun, year = "1991", CODEN = "ANMAEL", ISSN = "0168-9274 (print), 1873-5460 (electronic)", ISSN-L = "0168-9274", MRclass = "40A15 (65B05)", MRnumber = "92k:40001", MRreviewer = "Claude Brezinski", bibdate = "Sat Feb 8 10:09:54 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/applnummath.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Applied Numerical Mathematics: Transactions of IMACS", journal-URL = "http://www.sciencedirect.com/science/journal/01689274/", keywords = "convergence acceleration", } @Article{Levrie:1991:CFC, author = "Paul Levrie", title = "$ {G} $-continued fractions and convergence acceleration in the solution of third-order linear recurrence relations of {Poincar{\'e}} type", journal = j-APPL-NUM-MATH, volume = "8", number = "3", pages = "225--242", month = oct, year = "1991", CODEN = "ANMAEL", DOI = "https://doi.org/10.1090/surv/037", ISSN = "0168-9274 (print), 1873-5460 (electronic)", ISSN-L = "0168-9274", MRclass = "65B99 (65Q05)", MRnumber = "92m:65012", MRreviewer = "J. Albrycht", bibdate = "Sat Feb 8 10:09:54 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/applnummath.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Applied Numerical Mathematics: Transactions of IMACS", journal-URL = "http://www.sciencedirect.com/science/journal/01689274/", keywords = "convergence acceleration", } @InProceedings{Lyons:1991:FMF, author = "Ken Lyons", title = "A fast method for finding an integer square root", crossref = "Koopman:1991:PST", pages = "27--30", year = "1991", bibdate = "Tue May 4 05:57:50 MDT 1999", bibsource = "http://www.acm.org/pubs/toc/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "http://www.acm.org:80/pubs/citations/proceedings/plan/259965/p27-lyons/", acknowledgement = ack-nhfb, } @InProceedings{Markstein:1991:WFF, author = "V. Markstein and P. Markstein and T. Nguyen and S. Poole", title = "Wide Format Floating-Point Math Libraries", crossref = "IEEE:1991:PSA", pages = "130--138", year = "1991", bibdate = "Wed Dec 13 18:34:51 1995", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", abstract = "The authors present the performance and accuracy evaluations of eleven transcendental functions found in 64- and 128-bit floating-point formats in math libraries on the Cray Y-MP, the IBM 3090E/VF, the Convex C-240, the Hewlett--Packard 9000/720, and the IBM System/6000. Both architecture and algorithms are shown to impact the results.", acknowledgement = ack-nhfb # " and " # ack-nj, affiliation = "ISQUARE, Inc., Austin, TX, USA", classification = "C5230 (Digital arithmetic methods); C5470 (Performance evaluation and testing); C7310 (Mathematics)", confdate = "18-22 Nov. 1991", conflocation = "Albuquerque, NM, USA", confsponsor = "IEEE; ACM", keywords = "128 Bit; 64 Bit; Accuracy evaluations; Convex C-240; Cray Y-MP; Floating-point formats; Hewlett--Packard 9000/720; IBM 3090E/VF; IBM System/6000; Math libraries; Performance; Transcendental functions; Wide format floating point math libraries", numericalindex = "Word length 6.4E+01 bit; Word length 1.28E+02 bit", pubcountry = "USA", thesaurus = "Digital arithmetic; Mathematics computing; Parallel processing; Performance evaluation", } @Article{Maximon:1991:EIP, author = "Leonard C. Maximon", title = "On the evaluation of the integral over the product of two spherical {Bessel} functions", journal = j-J-MATH-PHYS, volume = "32", number = "3", pages = "642--648", month = mar, year = "1991", CODEN = "JMAPAQ", DOI = "https://doi.org/10.1063/1.529405", ISSN = "0022-2488 (print), 1089-7658 (electronic), 1527-2427", ISSN-L = "0022-2488", MRclass = "33C55", MRnumber = "92f:33018", MRreviewer = "S. K. Chatterjea", bibdate = "Tue Nov 1 08:57:23 MDT 2011", bibsource = "http://jmp.aip.org/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jmathphys1990.bib", URL = "http://jmp.aip.org/resource/1/jmapaq/v32/i3/p642_s1", acknowledgement = ack-nhfb, fjournal = "Journal of Mathematical Physics", journal-URL = "http://jmp.aip.org/", pagecount = "7", } @Article{McQuillan:1991:HPV, author = "S. E. McQuillan and J. V. McCanny and R. F. Woods", title = "High performance {VLSI} architecture for division and square root", journal = j-ELECT-LETTERS, volume = "27", number = "1", pages = "19--21", day = "3", month = jan, year = "1991", CODEN = "ELLEAK", DOI = "https://doi.org/10.1049/el:19910013", ISSN = "0013-5194 (print), 1350-911X (electronic)", ISSN-L = "0013-5194", bibdate = "Sat Jul 16 15:28:13 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, fjournal = "Electronics Letters", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=2220; https://digital-library.theiet.org/journal/el", summary = "A novel high performance bit parallel architecture to perform square root and division is proposed. Relevant VLSI design issues have been addressed. By employing redundant arithmetic and a semisystolic schedule, the throughput has been made independent of the size of the array.", } @InProceedings{McQuillan:1991:VAM, author = "S. E. McQuillan and J. V. McCanny", booktitle = "1991 International Conference on Acoustics, Speech, and Signal Processing: {ICASSP-91, 14--17} April 1991", title = "A {VLSI} architecture for multiplication, division and square root", volume = "2", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "1205--1208", year = "1991", CODEN = "????", ISSN = "????", bibdate = "Sat Jul 16 15:28:14 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, summary = "A high-performance VLSI architecture to perform combined multiply-accumulate, divide, and square root operations is proposed. The circuit is highly regular, requires only minimal control, and can be reconfigured for every cycle. The execution time \ldots{}", } @Article{Midy:1991:CSE, author = "P. Midy and Y. Yakovlev", title = "Computing some elementary functions of a complex variable", journal = j-MATH-COMPUT-SIMUL, volume = "33", number = "1", pages = "33--49", year = "1991", CODEN = "MCSIDR", ISSN = "0378-4754 (print), 1872-7166 (electronic)", ISSN-L = "0378-4754", MRclass = "65Y10 (65D20)", MRnumber = "MR1122989", bibdate = "Sat Jan 11 17:44:01 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Mathematics and Computers in Simulation", journal-URL = "http://www.sciencedirect.com/science/journal/03784754", } @Article{Montuschi:1991:OAE, author = "P. Montuschi and M. Mezzalama", title = "Optimal Absolute Error Starting Values for {Newton--Raphson} Calculation of Square Root", journal = j-COMPUTING, volume = "46", number = "1", pages = "67--86", month = mar, year = "1991", CODEN = "CMPTA2", ISSN = "0010-485X (print), 1436-5057 (electronic)", ISSN-L = "0010-485X", MRclass = "65H05 (65G99)", MRnumber = "92a:65161", bibdate = "Tue Oct 12 16:33:42 MDT 1999", bibsource = "Compendex database; http://springerlink.metapress.com/openurl.asp?genre=journal&issn=0010-485X; https://www.math.utah.edu/pub/tex/bib/computing.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; MathSciNet database; OCLC Contents1st database", acknowledgement = ack-nhfb, affiliation = "Politecnico di Torino", affiliationaddress = "Torino, Italy", classification = "723; 921", fjournal = "Computing", journal-URL = "http://link.springer.com/journal/607", journalabr = "Comput Vienna New York", keywords = "Absolute Error; Computer Programming --- Algorithms; Mathematical Techniques; Newton--Raphson Method; Optimization; Square Roots", } @InProceedings{Montuschi:1991:SRD, author = "Paolo Montuschi and Luigi Ciminiera", title = "Simple radix 2 division and square root with skipping of some addition steps", crossref = "Kornerup:1991:PIS", pages = "202--209", year = "1991", bibdate = "Sat Jul 16 15:28:14 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "http://www.acsel-lab.com/arithmetic/arith10/papers/ARITH10_Montuschi.pdf", acknowledgement = ack-nhfb, keywords = "ARITH-10", summary = "The authors present a novel algorithm for shared radix 2 division and square root whose main characteristic is the ability to avoid any addition when the digit 0 has been selected. The solution presented uses a redundant representation of the \ldots{}", } @Article{OGrady:1991:HOA, author = "E. Pearse O'Grady and Baek-Kyu Young", title = "A hardware-oriented algorithm for floating-point function generation", journal = j-IEEE-TRANS-COMPUT, volume = "40", number = "2", pages = "237--241", month = feb, year = "1991", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/12.73596", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Sat Jul 16 08:40:52 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", abstract = "An algorithm is presented for performing accurate, high-speed, floating-point function generation for univariate functions defined at arbitrary breakpoints. Rapid identification of the breakdown interval, which includes the input argument, is the key operation in the algorithm. A hardware implementation which makes extensive use of read/write memories illustrates the algorithm.", acknowledgement = ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", } @Article{Okabe:1991:LDC, author = "Y. Okabe and N. Takagi and S. Yaima", key = "OTY91", title = "Log-Depth Circuits for Elementary Functions Using Residue Number System", journal = j-ELECTRON-COMMUN-JPN, volume = "74", number = "8", pages = "31--37", year = "1991", CODEN = "ECOJAL", ISSN = "0424-8368", bibdate = "Mon May 19 15:16:09 1997", bibsource = "ftp://ftp.ira.uka.de/pub/bibliography/Theory/arith.bib.gz; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "Translated from Denshi Joho Tsushin Gakkai Ronbunshi, vol.\ 21-DI, no.\ 9, September 1990, pp.\ 723-728", acknowledgement = ack-nhfb, fjournal = "Electronics and communications in Japan", } @Article{Olver:1991:UEIb, author = "F. W. J. Olver", title = "Uniform, Exponentially Improved, Asymptotic Expansions for the Confluent Hypergeometric Function and Other Integral Transforms", journal = j-SIAM-J-MATH-ANA, volume = "22", number = "5", pages = "1475--1489", month = sep, year = "1991", CODEN = "SJMAAH", ISSN = "0036-1410 (print), 1095-7154 (electronic)", ISSN-L = "0036-1410", MRclass = "41A60 (33C15)", MRnumber = "92g:41035", MRreviewer = "Hans-J{\"u}rgen Glaeske", bibdate = "Sun Nov 28 19:25:21 MST 2010", bibsource = "http://epubs.siam.org/sam-bin/dbq/toc/SIMA/22/5; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/siamjmathana.bib", acknowledgement = ack-nhfb, fjournal = "SIAM Journal on Mathematical Analysis", journal-URL = "http://epubs.siam.org/sima", } @Article{Press:1991:BFF, author = "William H. Press and Saul A. Teukolsky", title = "{Bessel} Functions of Fractional Order", journal = j-COMPUT-PHYS, volume = "5", number = "2", pages = "244--??", month = mar, year = "1991", CODEN = "CPHYE2", DOI = "https://doi.org/10.1063/1.4822982", ISSN = "0894-1866 (print), 1558-4208 (electronic)", ISSN-L = "0894-1866", bibdate = "Wed Apr 10 08:45:28 MDT 2019", bibsource = "https://www.math.utah.edu/pub/tex/bib/computphys.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "https://aip.scitation.org/doi/10.1063/1.4822982", acknowledgement = ack-nhfb, ajournal = "Comput. Phys", fjournal = "Computers in Physics", journal-URL = "https://aip.scitation.org/journal/cip", } @Article{Press:1991:MBF, author = "William H. Press and Saul A. Teukolsky", title = "Modified {Bessel} Functions of Fractional Order", journal = j-COMPUT-PHYS, volume = "5", number = "3", pages = "330--??", month = may, year = "1991", CODEN = "CPHYE2", DOI = "https://doi.org/10.1063/1.4822991", ISSN = "0894-1866 (print), 1558-4208 (electronic)", ISSN-L = "0894-1866", bibdate = "Wed Apr 10 08:45:29 MDT 2019", bibsource = "https://www.math.utah.edu/pub/tex/bib/computphys.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "https://aip.scitation.org/doi/10.1063/1.4822991", acknowledgement = ack-nhfb, ajournal = "Comput. Phys", fjournal = "Computers in Physics", journal-URL = "https://aip.scitation.org/journal/cip", } @Book{Saan:1991:VFP, author = "T. Saan", title = "{{\cyr Vychislenie {\`e}lementarnykh funktsi{\u\i}s pomoshch'yu drobno-ratsional'nykh priblizheni{\u\i}}}. ({Russian}) [Calculation of elementary functions by means of rational approximations]", publisher = "{\`E}ston. Nauchno-Proizvod. Ob\cdprime ed. Vychisl. Tekhn. Inform., Tartu", pages = "139", year = "1991", MRclass = "65-04 (65D15)", MRnumber = "94f:65008", MRreviewer = "W. Govaerts", bibdate = "Sat Jan 11 17:44:01 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, language = "Russian", } @Article{Smith:1991:AFP, author = "David M. Smith", title = "Algorithm 693: {A FORTRAN} Package for Floating-Point Multiple-Precision Arithmetic", journal = j-TOMS, volume = "17", number = "2", pages = "273--283", month = jun, year = "1991", CODEN = "ACMSCU", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Sun Sep 04 23:44:20 1994", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.acm.org/pubs/toc/Abstracts/toms/108585.html", abstract = "FM is a collection of FORTRAN-77 routines which performs floating-point multiple-precision arithmetic and elementary functions. Results are almost always correctly rounded, and due to improved algorithms used for elementary functions, reasonable efficiency is obtained.", acknowledgement = ack-nhfb, affiliation = "Loyola Marymount Univ., Los Angeles, CA, USA", classification = "C4130 (Interpolation and function approximation); C5230 (Digital arithmetic methods); C7310 (Mathematics)", fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", keywords = "Accuracy; Algorithms; Elementary functions; Floating-point multiple-precision arithmetic; FM; FORTRAN-77 routines; Mathematical library; Portable software; Rounding off", subject = "{\bf G.1.0}: Mathematics of Computing, NUMERICAL ANALYSIS, General, Numerical algorithms. {\bf D.3.2}: Software, PROGRAMMING LANGUAGES, Language Classifications, FORTRAN 77.", thesaurus = "Digital arithmetic; Function approximation; Mathematics computing; Software packages; Subroutines", } @Article{Squire:1991:ANS, author = "Jon S. Squire", title = "{Ada} numerics standardization and testing", journal = j-SIGADA-LETTERS, volume = "11", number = "7", address = "New York, NY, USA", pages = "1--286", year = "1991", CODEN = "AALEE5", ISSN = "1094-3641 (print), 1557-9476 (electronic)", ISSN-L = "1094-3641", bibdate = "Sat Feb 24 15:01:45 MST 1996", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, annote = "``A special edition from SIGAda \ldots{} presented by SIGAda Numerics Working Group and Ada-Europe Numerics Working Group and ISO- IEC/JTC1/SC22/WG9 Numerics Rapporteur Group.''--Cover. Includes bibliographies. Introduction to the proposed standard for the elementary functions in Ada / Kenneth W. Dritz --- Proposed standard for a generic package of elementary functions for Ada / edited by Kenneth W. Dritz --- Rationale for the proposed standard for a generic package of elementary functions for Ada; Proposed standard for a generic package of primitive functions for Ada; Rationale for the proposed standard for a generic package of primitive functions for Ada / Kenneth W. Dritz --- Proposed standard for packages of real and complex type declarations and basic operations for Ada (including vector and matrix types) / edited by Graham S. Hodgson --- Rationale for the proposed standard for packages of real and complex type declarations and basic operations for Ada (including vector and matrix types) / Graham S. Hodgson. Proposed standard for a generic package of complex elementary functions / edited by Jon S. Squire --- Rationale for the proposed standard for a generic package of complex elementary functions / Jon S. Squire --- A portable generic elementary function package in Ada and an accurate test suite / Ping Tak Peter Tang --- Towards validation of generic elementary functions and other standard Ada numerics packages / Jon S. Squire --- Floating point attributes in Ada / Dik T. Winter --- An Ada math library for real-time avionics / Donald A. Celarier and Donald W. Sando --- Predifined floating point type names, uniformity rapporteur group UI-48 / edited by Jon S. Squire.", fjournal = "ACM SIGAda Ada Letters", journal-URL = "http://portal.acm.org/citation.cfm?id=J32", keywords = "Ada (Computer program language)", } @Article{Squire:1991:PSG, author = "J. S. Squire", title = "Proposed standard for a generic package of complex elementary functions ({Ada})", journal = j-SIGADA-LETTERS, volume = "11", number = "7", pages = "140--165", month = "Fall", year = "1991", CODEN = "AALEE5", ISSN = "1094-3641 (print), 1557-9476 (electronic)", ISSN-L = "1094-3641", bibdate = "Thu Mar 20 07:41:09 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, classcodes = "C6140D (High level languages); C7310 (Mathematics); C6110B (Software engineering techniques)", fjournal = "ACM SIGAda Ada Letters", journal-URL = "http://portal.acm.org/citation.cfm?id=J32", keywords = "ACM SIGAda; Ada; Ada-Europe Numerics Working Group; applications; complex elementary functions; complex mathematical routines; COMPLEX-ELEMENTARY-FUNCTIONS; generic package; GENERIC-; international standard; joint proposal; mathematics computing; Numerics Working Group; portable; reusable; software reusability; standards; WG9 Numerics Rapporteur Group", treatment = "P Practical", } @Article{Squire:1991:RPS, author = "Jon S. Squire", title = "Rationale for the proposed standard for a generic package of complex elementary functions ({Ada})", journal = j-SIGADA-LETTERS, volume = "11", number = "7", pages = "166--179", month = "Fall", year = "1991", CODEN = "AALEE5", ISSN = "1094-3641 (print), 1557-9476 (electronic)", ISSN-L = "1094-3641", bibdate = "Thu Mar 20 07:41:09 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/sigada.bib", acknowledgement = ack-nhfb, classcodes = "C6140D (High level languages); C7310 (Mathematics); C6110B (Software engineering techniques)", fjournal = "ACM SIGAda Ada Letters", journal-URL = "http://portal.acm.org/citation.cfm?id=J32", keywords = "ACM SIGAda Numerics Working Group; Ada; Ada-; basic complex mathematical; complex; COMPLEX-ELEMENTARY-FUNCTIONS; elementary functions; error bounds; Europe Numerics Working Group; generic package; GENERIC-; mathematics computing; proposed standard; reusable applications; routines; software reusability; specification; standards", treatment = "P Practical", } @Article{Squire:1991:TVG, author = "J. S. Squire", title = "Towards validation of generic elementary functions and other standard {Ada} numerics packages", journal = j-SIGADA-LETTERS, volume = "11", number = "7", pages = "217--243", month = "Fall", year = "1991", CODEN = "AALEE5", ISSN = "1094-3641 (print), 1557-9476 (electronic)", ISSN-L = "1094-3641", bibdate = "Thu Mar 20 07:41:09 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, classcodes = "C6150G (Diagnostic, testing, debugging and evaluating systems); C7310 (Mathematics); C6140D (High level languages)", fjournal = "ACM SIGAda Ada Letters", journal-URL = "http://portal.acm.org/citation.cfm?id=J32", keywords = "Ada; Ada listings; computing; conformance testing; conformance tests; generic elementary functions; implementors guide; mathematics; program testing; proposed ISO; prototype tests; standard Ada numerics packages; standards; test suite", treatment = "P Practical", } @Article{Takagi:1991:RCM, author = "N. Takagi and T. Asada and S. Yajima", title = "Redundant {CORDIC} Methods with a Constant Scale Factor for Sine and Cosine Computation", journal = j-IEEE-TRANS-COMPUT, volume = "40", number = "9", pages = "989--995", month = sep, year = "1991", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/12.83660", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Thu Jul 7 12:52:24 MDT 2011", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput1990.bib", URL = "https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=83660", acknowledgement = ack-nj # "\slash " # ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", } @Article{Tang:1991:PGE, author = "Ping Tak Peter Tang", title = "A portable generic elementary function package in {Ada} and an accurate test suite", journal = j-SIGADA-LETTERS, volume = "11", number = "7", pages = "181--216", month = "Fall", year = "1991", CODEN = "AALEE5", ISSN = "1094-3641 (print), 1557-9476 (electronic)", ISSN-L = "1094-3641", bibdate = "Thu Mar 20 07:41:09 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, classcodes = "C7310 (Mathematics); C6140D (High level languages); C6110B (Software engineering techniques); C6150G (Diagnostic, testing, debugging and evaluating systems)", fjournal = "ACM SIGAda Ada Letters", journal-URL = "http://portal.acm.org/citation.cfm?id=J32", keywords = "accurate test; Ada; function libraries; mathematics computing; portability; portable generic elementary function package; program testing; resolution; rigorous analysis; software; suite; test programs", treatment = "P Practical", } @InProceedings{Tang:1991:TLA, author = "Ping Tak Peter Tang", title = "Table-Lookup Algorithms for Elementary Functions and Their Error Analysis", crossref = "Kornerup:1991:PIS", pages = "232--236", year = "1991", bibdate = "Sat Nov 27 12:40:58 MST 2004", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nj # " and " # ack-nhfb, } @InProceedings{Wong:1991:FHA, author = "W. F. Wong and E. Goto", title = "Fast Hardware-based Algorithms for Elementary Function Computations", crossref = "Anonymous:1991:PIS", pages = "56--65", year = "1991", bibdate = "Sat Jan 11 10:14:06 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, searchkey = "ti:elementary function", } @Article{Zeilberger:1991:MPP, author = "Doron Zeilberger", title = "A {Maple} program for proving hypergeometric identities", journal = j-SIGSAM, volume = "25", number = "3", pages = "4--13", month = jul, year = "1991", CODEN = "SIGSBZ", ISSN = "0163-5824 (print), 1557-9492 (electronic)", ISSN-L = "0163-5824", bibdate = "Fri Feb 8 18:27:01 MST 2002", bibsource = "http://portal.acm.org/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", abstract = "Gives the listing of a MAPLE program for implementing an algorithm for proving any terminating definite hypergeometric identity, and more generally, for finding the linear recurrence satisfied by any definite hypergeometric sum R(n):= Sigma /sub k/F(n,k), where F(n,k) has the form x/sup k/( Pi /sub i=1//sup m/( alpha /sub i/n+ beta /sub i/k+c/sub i/)!/ Pi /sub i'=1//sup m'/( alpha '/sub i'/n+ beta '/sub i'/k+c'/sub i'/)!). The algorithm for definite hypergeometric summation relies on Gosper's (1978) ingenious decision procedure for indefinite summation, but not in the obvious way!.", acknowledgement = ack-nhfb, affiliation = "Dept. of Math. and Comput. Sci., Drexel Univ., Philadelphia, PA, USA", classcodes = "C7310 (Mathematics)", classification = "C7310 (Mathematics)", corpsource = "Dept. of Math. and Comput. Sci., Drexel Univ., Philadelphia, PA, USA", fjournal = "SIGSAM Bulletin", issue = "97", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J1000", keywords = "definite; Definite hypergeometric summation; hypergeometric identity; hypergeometric summation; linear recurrence; Linear recurrence; manipulation; MAPLE program; mathematics computing; proofs; Proofs; public domain software; shareware; Shareware; symbol; terminating definite; Terminating definite hypergeometric identity; theorem proving", thesaurus = "Mathematics computing; Public domain software; Symbol manipulation; Theorem proving", treatment = "P Practical", } @Article{Ziv:1991:FEE, author = "Abraham Ziv", title = "Fast Evaluation of Elementary Mathematical Functions with Correctly Rounded Last Bit", journal = j-TOMS, volume = "17", number = "3", pages = "410--423", month = sep, year = "1991", CODEN = "ACMSCU", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Thu Sep 1 10:15:31 1994", bibsource = "garbo.uwasa.fi:/pc/doc-soft/fpbiblio.txt; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.acm.org/pubs/toc/Abstracts/toms/116813.html", acknowledgement = ack-nj, fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", keywords = "algorithms; standardization; theory", subject = "{\bf G.1.0}: Mathematics of Computing, NUMERICAL ANALYSIS, General, Numerical algorithms. {\bf G.1.2}: Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation. {\bf G.4}: Mathematics of Computing, MATHEMATICAL SOFTWARE, Efficiency.", } @Book{Achieser:1992:TA, author = "N. I. Achieser", title = "Theory of Approximation", publisher = pub-DOVER, address = pub-DOVER:adr, pages = "x + 307", year = "1992", ISBN = "0-486-67129-1 (paperback)", ISBN-13 = "978-0-486-67129-1 (paperback)", LCCN = "QA221 .A533 1992", bibdate = "Fri Oct 20 08:06:59 MDT 2023", bibsource = "fsz3950.oclc.org:210/WorldCat; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", series = "Dover books on advanced mathematics", acknowledgement = ack-nhfb, remark = "Translation of Russian original, Lek{\"e}t{\`\i}sii po teorii approksima{\"e}t{\`\i}sii. Reprint of English translation \cite{Achieser:1956:TA}.", subject = "Mathematical analysis", tableofcontents = "Approximation Problems in Linear Normalized Spaces \\ Formulation of the Principal Problem in the Theory of Approximation / 1 \\ The Concept of Metric Space / 1 \\ The Concept of Linear Normalized Space / 2 \\ Examples of Linear Normalized Spaces / 3 \\ The Inequalities of Holder and Minkowski / 4 \\ Additional Examples of Linear Normalized Spaces / 7 \\ Hilbert Space / 8 \\ The Fundamental Theorem of Approximation Theory in Linear Normalized Spaces / 10 \\ Strictly Normalized Spaces / 11 \\ An Example of Approximation in the Space $L^p$ / 12 \\ Geometric Interpretation / 13 \\ Separable and Complete Spaces / 14 \\ Approximation Theorems in Hilbert Space / 15 \\ An Example of Approximation in Hilbert Space / 19 \\ More About the Approximation Problem in Hilbert Space / 21 \\ Orthonormalized Vector Systems in Hilbert Space / 22 \\ Orthogonalization of Vector Systems / 23 \\ Infinite Orthonormalized Systems / 25 \\ An Example of a Non-Separable System / 29 \\ Weierstrass' First Theorem / 29 \\ Weierstrass' Second Theorem / 32 \\ The Separability of the Space C / 33 \\ The Separability of the Space $L^p$ / 34 \\ Generalization of Weierstrass' Theorem to the Space $L^p$ / 37 \\ The Completeness of the Space $L^p$ / 38 \\ Examples of Complete Orthonormalized Systems in L[superscript 2] / 40 \\ Muntz's Theorem / 43 \\ The Concept of the Linear Functional / 46 \\ F. Riesz's Theorem / 47 \\ A Criterion for the Closure of a Set of Vectors in Linear Normalized Spaces / 49 \\ P. L. Tchebysheff's Domain of Ideas \\ Statement of the Problem / 51 \\ A Generalization of the Theorem of de la Vallee-Poussin / 52 \\ The Existence Theorem / 53 \\ Tchebysheff's Theorem / 55 \\ A Special Case of Tchebysheff's Theorem / 57 \\ The Tchebysheff Polynomials of Least Deviation from Zero / 57 \\ A Further Example of P. Tchebysheff's Theorem / 58 \\ An Example for the Application of the General Theorem of de la Vallee-Poussin / 60 \\ An Example for the Application of P. L. Tchebysheff's General Theorem / 62 \\ The Passage to Periodic Functions / 64 \\ An Example of Approximating with the Aid of Periodic Functions / 66 \\ The Weierstrass Function / 66 \\ Haar's Problem / 67 \\ Proof of the Necessity of Haar's Condition / 68 \\ Proof of the Sufficiency of Haar's Condition / 69 \\ An Example Related to Haar's Problem / 72 \\ P. L. Tchebysheff's Systems of Functions / 73 \\ Generalization of P. L. Tchebysheff's Theorem / 74 \\ On a Question Pertaining to the Approximation of a Continuous Function in the Space $L$ / 76 \\ A. A. Markoff's Theorem / 82 \\ Special Cases of the Theorem of A. A. Markoff / 85 \\ Elements of Harmonic Analysis \\ The Simplest Properties of Fourier Series / 89 \\ Fourier Series for Functions of Bounded Variation / 93 \\ The Parseval Equation for Fourier Series / 97 \\ Examples of Fourier Series / 98 \\ Trigonometric Integrals / 101 \\ The Riemann--Lebesgue Theorem / 103 \\ Plancherel's Theory / 104 \\ Watson's Theorem / 106 \\ Plancherel's Theorem / 108 \\ Fejer's Theorem / 110 \\ Integral-Operators of the Fejer Type / 113 \\ The Theorem of Young and Hardy / 116 \\ Examples of Kernels of the Fejer Type / 118 \\ The Fourier Transformation of Integrable Functions / 120 \\ The Faltung of two Functions / 122 \\ V. A. Stekloff's Functions / 123 \\ Multimonotonic Functions / 125 \\ Conjugate Functions / 126 \\ Certain Extremal Properties of Integral Transcendental Functions of the Exponential Type \\ Integral Functions of the Exponential Type / 130 \\ The Borel Transformation / 132 \\ The Theorem of Wiener and Paley / 134 \\ Integral Functions of the Exponential Type which are Bounded along the Real Axis / 137 \\ S. N. Bernstein's Inequality / 140 \\ B. M. Levitan's Polynomials / 146 \\ The Theorem of Fejer and Riesz. A Generalization of This Theorem / 152 \\ A Criterion for the Representation of Continuous Functions as Fourier--Stieltjes Integrals / 154 \\ Questions Regarding the Best Harmonic Approximation of Functions Preliminary Remarks / 160 \\ The Modulus of Continuity / 161 \\ The Generalization to the Space $L^p$ ($p \geq 1$) / 162 \\ An Example of Harmonic Approximation / 165 \\ Some Estimates for Fourier Coefficients / 169 \\ More about V. A. Stekloff's Functions / 173 \\ Two Lemmas / 175 \\ The Direct Problem of Harmonic Approximation / 176 \\ A Criterion due to B. Sz.-Nagy / 183 \\ The Best Approximation of Differentiable Functions / 187 \\ Direct Observations Concerning Periodic Functions / 195 \\ Jackson's Second Theorem / 199 \\ The Generalized Fejer Method / 201 \\ Berstein's Theorem / 206 \\ Priwaloff's Theorem / 210 \\ Generalizations of Bernstein's Theorems to the Space $L^p$ ($p \geq 1$) / 211 \\ The Best Harmonic Approximation of Analytic Functions / 214 \\ A Different Formulation of the Result of the Preceding Section / 218 \\ The Converse of Bernstein's Theorem / 221 \\ Wiener's Theorem on Approximation \\ Wiener's Problem / 224 \\ The Necessity of Wiener's Condition / 224 \\ Some Definitions and Notation / 225 \\ Several Lemmas / 227 \\ The Wiener--Levy Theorem / 230 \\ Proof of the Sufficiency of Wiener's Condition / 233 \\ Wiener's General Tauber Theorem / 234 \\ Weakly Decreasing Functions / 235 \\ Remarks on the Terminology / 237 \\ Ikehara's Theorem / 238 \\ Carleman's Tauber Theorem / 241 \\ Various Addenda and Problems \\ Elementary Extremal Problems and Certain Closure Criteria / 243 \\ Szego's Theorem and Some of Its Applications / 256 \\ Further Examples of Closed Sequences of Functions / 267 \\ The Caratheodory--Fejer Problem and Similar Problems / 270 \\ Solotareff's Problems and Related Problems / 280 \\ The Best Harmonic Approximation of the Simplest Analytic Functions / 289 \\ Notes / 296 \\ Index / 306", } @Article{Anderson:1992:FIH, author = "G. D. Anderson and M. K. Vamanamurthy and M. Vuorinen", title = "Functional Inequalities for Hypergeometric Functions and Complete Elliptic Integrals", journal = j-SIAM-J-MATH-ANA, volume = "23", number = "2", pages = "512--524", month = mar, year = "1992", CODEN = "SJMAAH", ISSN = "0036-1410 (print), 1095-7154 (electronic)", ISSN-L = "0036-1410", MRclass = "33C05 (33C75)", MRnumber = "93b:33001", MRreviewer = "J. M. H. Peters", bibdate = "Sat Dec 5 18:14:13 MST 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "SIAM Journal on Mathematical Analysis", journal-URL = "http://epubs.siam.org/sima", } @Book{Baker:1992:CMF, author = "Louis Baker", title = "{C} Mathematical Function Handbook", publisher = pub-MCGRAW-HILL, address = pub-MCGRAW-HILL:adr, pages = "xviii + 757", year = "1992", ISBN = "0-07-911158-0", ISBN-13 = "978-0-07-911158-6", LCCN = "QA351.B17 1991; QA351 .B17 1992", bibdate = "Fri Aug 31 18:54:02 MDT 2007", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; melvyl.cdlib.org:210/CDL90", series = "McGraw-Hill programming tools for scientists and engineers", acknowledgement = ack-nhfb, remark = "System requirements for computer disk: PC; C or C++ compiler.", subject = "Functions, Special; Computer programs; C (Computer program language)", tableofcontents = "Preface / xv \\ 1. Special Functions and Numerical Analysis / 1 \\ \\ Correspondence with Abramowitz and Stegun / 1 \\ Numerical Analysis / 1 \\ The IEEE-754 Standard / 1 \\ Practical Considerations / 3 \\ Reference / 6 \\ \\ 2. Special Functions in C and C++ / 8 \\ C and C++ / 8 \\ Portability, ANSI C, and C++ / 8 \\ Infinite Loops / 9 \\ Header Files COMPLEX.H, CMLIB.H, PROTOM.H / 10 \\ Error Handling / 10 \\ Pitfalls with Special Functions / 11 \\ Normalization Conventions / 12 \\ Tips and Pitfalls in C / 12 \\ Calling C from C++ / 14 \\ References / 15 \\ Cmlib.h / 16 \\ COMPLEX.H Header File / 17 \\ Prototypes for \booktitle{C Mathematical Function Handbook} / 19 \\ \\ 3. Elementary Analytical Methods / 33 \\ Powers and Roots / 33 \\ Complex Numbers / 34 \\ Roots of Polynomials / 35 \\ Quadratics / 35 \\ Cubics / 35 \\ Quartics (biquadratics) / 36 \\ Implementation Considerations / 37 \\ Ouintics / 38 \\ References / 38 \\ Complex Variable Auxiliary Routines / 39 \\ Powers and Roots / 45 \\ Polynomial Root Finders / 48 \\ Test Driver for Programs of Chapters 3-4 / 60 \\ \\ 4. Elementary Transcendental Functions / 72 \\ Elementary Functions / 72 \\ Complex Elementary Functions / 73 \\ Gudermannian / 74 \\ References / 74 \\ \\ 5. Exponential Integral and Relatives / 75 \\ Exponential and Related Integrals / 75 \\ Methods / 76 \\ References / 76 \\ Shi(x) and Chi(x) / 76 \\ Exponential Integral / 78 \\ Test Driver Results: Exponential Integral and Relatives / 95 \\ \\ 6. Gamma Function and Related Integrals / 100 \\ Gamma Function and Relatives / 100 \\ The Pochhammer Symbol / 100 \\ Methods / 101 \\ Asymptotic Series / 101 \\ Reference / 101 \\ Gamma Function and Relatives / 102 \\ Digamma and First 2 Polygamma Functions / 111 \\ \\ 7. Error Function and Relatives / 115 \\ Error Function and Relatives / 115 \\ Methods / 117 \\ Recurrence Relations / 117 \\ C Code / 118 \\ References / 118 \\ Figure: Dawson's Integral / 118 \\ Plasma Dispersion Function / 120 \\ Iterated Error Function / 127 \\ Boehmer (generalized Fresnel) Integral / 129 \\ Complementary Error Function for Complex Arguments / 133 \\ Test Driver Chapter 7 / 134 \\ \\ 8. Legendre Functions / 142 \\ Legendre Functions / 142 \\ Derivatives / 143 \\ Applications / 143 \\ Methods / 144 \\ References / 144 \\ Legendre and Associated Legendre Functions / 146 \\ Legendre Functions for $|x| > 1$ / 155 \\ Toroidal $P|x| > 1$ / 163 \\ Mehler (Conical Legendre) Functions / 164 \\ Test Driver for Legendre Functions of Chapter 8 / 166 \\ \\ 9. Bessel Functions / 177 \\ Struve Functions / 178 \\ Anger and Weber Functions / 179 \\ Relationship to Confluent Hypergeometric Function / 179 \\ Derivatives / 179 \\ Other Related Functions / 179 \\ Zeros of Bessel Functions / 180 \\ Applications / 180 \\ Methods / 180 \\ References / 181 \\ Bessel Functions for Complex Arguments / 182 \\ Bessel Functions: Rational Approximations / 194 \\ Bessel Function Tables as a Function of (Integral) n / 201 \\ Zeros of Bessel Functions / 205 \\ Test Driver for Bessel Functions / 211 \\ Test Driver Bessel Zero / 220 \\ Output of Bessel Function Test Driver / 221 \\ Output of Test Driver for Zerobess() / 237 \\ \\ 10. Bessel Functions of Fractional Order / 239 \\ Introduction / 239 \\ Spherical Bessel Functions / 239 \\ Airy Functions / 239 \\ Applications / 240 \\ Asymptotics / 240 \\ Caustics / 240 \\ Schroedinger's Equation, Turning Points and the WKB Method / 242 \\ Methods / 243 \\ The $|A|$ Function / 243 \\ References / 244 \\ Spherical Bessel Functions and Allied Routines / 245 \\ Airy, Bessel Functions and Integrals Thereof / 250 \\ \\ 11. Integrals of Bessel Functions / 265 \\ Introduction / 265 \\ Applications / 265 \\ Methods / 265 \\ Bickley Functions / 265 \\ Adaptive Quadrature / 266 \\ Repeated Integrals of Jn / 266 \\ Other Integrals / 266 \\ References / 266 \\ Figures of Integrals of Bessel Functions / 267 \\ Integrals of Bessel Functions / 277 \\ Adaptive Integration Routine / 242 \\ \\ 12. Struve and Anger--Weber Functions / 284 \\ Introduction / 284 \\ Struve Functions / 284 \\ Anger--Weber Functions / 284 \\ Methods / 285 \\ References / 285 \\ Figures / 286 \\ Struve Functions General Order / 289 \\ Struve Functions Lowest Order / 291 \\ Integrals of Struve HO, HO/t, LO / 295 \\ Integral of Anger--Weber Function / 297 \\ \\ 13. Confluent Hypergeometric Functions and Relatives / 301 \\ Introduction / 301 \\ Airy Functions / 303 \\ Applications / 303 \\ Methods / 304 \\ References / 304 \\ Confluent Hypergeometric Function Complex Arguments / 305 \\ Test Drive Confluent Hypergeometric Function / 318 \\ Confluent Hypergeometric Function U / 322 \\ Test Driver U.c / 331 \\ Test Output: Confluent-Hypergeometric Function / 332 \\ Test Output: U / 334 \\ \\ 14. Coulomb Wave Functions / 335 \\ Introduction / 335 \\ Methods / 336 \\ References / 337 \\ Coulomb Wave Functions / 339 \\ Test Driver Coulomb Wave Functions / 343 \\ Test Output: Coulomb wave Functions / 344 \\ \\ 15. The Hypergeometric Function / 345 \\ Introduction / 345 \\ Applications / 346 \\ Methods / 346 \\ References / 346 \\ Hypergeometric Function Complex Arguments / 347 \\ Legendre Function $P$ for Complex Parameters / 364 \\ Legendre $Q$ for Complex Arguments / 366 \\ Test Driver $_2F_1$ / 369 \\ Output: Gauss Hypergeometric Functions / 370 \\ Real Hypergeometric Function and Relatives / 371 \\ \\ 16. The Elliptic Functions / 373 \\ Introduction / 373 \\ Applications / 376 \\ Methods / 376 \\ References / 377 \\ Figure / 378 \\ Basic Elliptic Functions Real Arguments / 380 \\ Elliptic Integral of Third Kind / 391 \\ Jacobian Elliptic Function Complex Argument / 397 \\ Complex Elliptic Theta Functions / 406 \\ Output: Test Driver for Elliptic Functions / 411 \\ \\ 17. The Elliptic Integrals / 416 \\ Introduction / 415 \\ Caveat / 416 \\ References / 417 \\ Figures / 417 \\ \\ 18. The Weierstrass Elliptic Function and Relatives / 425 \\ Introduction / 425 \\ Methods / 427 \\ References / 427 \\ Inverse of Weierstrass Elliptic P . / 428 \\ Brent Root Finder / 430 \\ \\ 19. The Parabolic Cylinder Functions / 432 \\ Introduction / 432 \\ Methods / 432 \\ Figures / 432 \\ Parabolic Cylinder Functions / 435 \\ Test Driver Parabolic Cylinder Function / 442 \\ \\ 20. The Mathieu Functions / 444 \\ Introduction / 444 \\ Method / 443 \\ References / 445 \\ Figures / 445 \\ Mathieu Functions / 449 \\ Mathieu Functions Test Driver / 465 \\ Test Output: Mathieu Functions / 467 \\ \\ 21. The Spheroidal Wave Functions / 475 \\ Introduction / 475 \\ Spheroidal Wave Functions / 475 \\ Generalized Spheroidal Wave Functions / 477 \\ Caveat / 477 \\ Method / 477 \\ References / 478 \\ Figures / 478 \\ Spheroidal Wave Functions / 486 \\ Legptable Pn,m for n = m to ntop / 500 \\ Test Driver Spheroidal Wave Functions / 501 \\ Test Output: Spheroidal Wave Functions / 504 \\ \\ 22. Orthogonal Polynomials / 509 \\ Introduction / 509 \\ Contents / 509 \\ Applications / 509 \\ Method / 509 \\ Orthogonal Polynomials / 510 \\ Test Driver Orthogonal Polynomials / 513 \\ \\ 23. Bernoulli and Euler Numbers and Polynomials, Riemann Zeta Function / 516 \\ Introduction / 516 \\ Riemann Zeta Function / 516 \\ Bernoulli Polynomials and Numbers / 517 \\ Euler Polynomials and Numbers / 517 \\ Methods / 517 \\ References / 517 \\ Riemann Zeta (real arguments) / 519 \\ Test Driver Riemann Zeta (Real) / 527 \\ Test Output: Riemann Zeta, Bernoulli, Euler Numbers / 324 \\ \\ 24. Combinatorics. Stirling Numbers / 529 \\ Introduction / 529 \\ Methods / 529 \\ References / 529 \\ Stirling Numbers First and Second Kind / 530 \\ Fibonacci Numbers / 531 \\ Binomial Coefficients / 532 \\ Test Driver for Stirling Numbers / 533 \\ Test Output: Stirling, Fibonacci, Binomial Coefficients / 534 \\ \\ 25. Numerical Analysis / 535 \\ Discussion / 535 \\ Reference / 536 \\ \\ 26. Statistical Functions, Probability Distributions, and Random Variables / 537 \\ Introduction / 537 \\ Methods / 537 \\ References / 537 \\ Random Numbers / 538 \\ Random Distributions / 544 \\ Test Driver for Random Number / 551 \\ Test Output: Random Numbers and Distributions / 560 \\ Calculator Version of Stat.c Routines / 565 \\ \\ 27. Miscellaneous Functions / 592 \\ Introduction / 592 \\ Debye Functions / 592 \\ Method / 593 \\ Sievert / 593 \\ Method / 593 \\ Abramowitz and Kruse--Ramsey / 593 \\ Method / 593 \\ Ritchie / 593 \\ Method / 593 \\ Dilogarithm and Polylogarithms / 593 \\ Method / 594 \\ Claussen / 594 \\ Method / 594 \\ Lobachevsky / 594 \\ Method / 594 \\ Clebsch--Gordon and Relatives / 594 \\ Method / 595 \\ v and Relatives / 595 \\ Method / 596 \\ References / 596 \\ Figures / 595 \\ Sievert Integral Chapter 27 / 605 \\ Dilogarithm / 606 \\ Polylogarithm Function / 607 \\ ``Abramowitz'' Functions f1, f2, f3 from Chapter 27 / 608 \\ Nu and Mu Integrals of Erdelyi Vol III p. 217 / 612 \\ Lobachevsky Function / 614 \\ Test Driver Miscellaneous Functions / 616 \\ Test Driver Ritchie's Integral / 619 \\ Test Output: Miscellaneous Functions / 620 \\ Test Output: Ritchie's Integral / 624 \\ Clebsch--Gordon, Wigner, Related Coefficients / 625 \\ Test Driver Wigner / 631 \\ Wigner Sample Output / 635 \\ \\ 28. Scales of Notation / 636 \\ \\ 29. C++ Programs / 637 \\ Hurwitz Zeta and Lerch Phi Transcendent / 638 \\ Methods / 638 \\ Meijer $G$ and Generalized Hypergeometric Functions / 638 \\ Methods / 639 \\ Elliptic Functions / 639 \\ Applications / 639 \\ Methods / 639 \\ Fock Functions / 640 \\ Methods / 541 \\ References / 641 \\ // Complex.hpp / 643 \\ Elliptic Functions / 646 \\ Arithmetic--Geometric Mean (AGM) Supplemental / 656 \\ Test Driver for: Elliptic Functions / 662 \\ // Cvector.hpp / 665 \\ // cmatrix.hpp / 666 \\ Test Output: Elliptic Functions / 667 \\ Complex C++ Utilities / 669 \\ Fock Functions / 674 \\ Test Output: Fock Functions / 682 \\ ``Abramowitz'' Functions f1, f2, f3 from Chapter 27 / 697 \\ Test Output: Abramowitz Functions, Complex Arguments / 701 \\ Riemann and Hurwitz / 712 \\ Riemann and Hurwitz: Pad{\'e} / 716 \\ Lerch Phi Transcendent / 720 \\ Generalized Hypergeometric Function and Meijer $G$ / 723 \\ Test Output: Meijer $G$ / 730 \\ Generalized Hypergeometric Function and Meijer $G$ / 731 \\ Test Output: Meijer $G$ with Pad{\'e} / 738 \\ MacRobert E Function / 739 \\ \\ 30. Xref / 740 \\ Other Listings / 741 \\ Adaptive Quadrature (from C loops) / 741 \\ Test Driver Chaps. 16--18 / 743 \\ Index to C Functions / 747 \\ Index / 753", } @Article{Baker:1992:LCE, author = "H. G. Baker", title = "Less Complex Elementary Functions", journal = j-SIGPLAN, volume = "27", number = "11", pages = "15--16", month = nov, year = "1992", CODEN = "SINODQ", ISSN = "0362-1340 (print), 1523-2867 (print), 1558-1160 (electronic)", ISSN-L = "0362-1340", bibdate = "Thu Sep 08 08:11:27 1994", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nj # " and " # ack-nhfb, fjournal = "ACM SIGPLAN Notices", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J706", } @Article{Bohman:1992:FRP, author = "Jan Bohman and Carl-Erik Fr{\"o}berg", title = "The {$ \Gamma $}-function revisited: power series expansions and real-imaginary zero lines", journal = j-MATH-COMPUT, volume = "58", number = "197", pages = "315--322", month = jan, year = "1992", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "33B15 (11Y70 65D20)", MRnumber = "92e:33001", MRreviewer = "A. de Castro Brzezicki", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Borwein:1992:FEG, author = "J. M. Borwein and I. J. Zucker", title = "Fast evaluation of the gamma function for small rational fractions using complete elliptic integrals of the first kind", journal = j-IMA-J-NUMER-ANAL, volume = "12", number = "4", pages = "519--526", year = "1992", CODEN = "IJNADH", ISSN = "0272-4979 (print), 1464-3642 (electronic)", ISSN-L = "0272-4979", MRclass = "65D20", MRnumber = "93g:65028", bibdate = "Sat Dec 23 17:06:35 MST 2000", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; MathSciNet database", acknowledgement = ack-nhfb, fjournal = "IMA Journal of Numerical Analysis", journal-URL = "http://imajna.oxfordjournals.org/content/by/year", } @Article{Buhring:1992:GHF, author = "Wolfgang B{\"u}hring", title = "Generalized hypergeometric functions at unit argument", journal = j-PROC-AM-MATH-SOC, volume = "114", number = "1", pages = "145--153", month = "????", year = "1992", CODEN = "PAMYAR", ISSN = "0002-9939 (print), 1088-6826 (electronic)", ISSN-L = "0002-9939", MRclass = "33C20", MRnumber = "MR1068116 (92c:33004)", bibdate = "Thu Dec 01 09:52:06 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", ZMnumber = "Zbl 0754.33003", acknowledgement = ack-nhfb, fjournal = "Proceedings of the American Mathematical Society", journal-URL = "http://www.ams.org/journals/proc", remark = "The paper treats $_{p + 1F}_p$ (or equivalently, $_p F_{p - 1}$ ).", } @Article{Carlson:1992:TEI, author = "B. C. Carlson", title = "A Table of Elliptic Integrals: Two Quadratic Factors", journal = j-MATH-COMPUT, volume = "59", number = "199", pages = "165--180", month = jul, year = "1992", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "65D20 (33C75 33E05)", MRnumber = "92k:65027", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Corless:1992:NEAa, author = "R. M. Corless and D. J. Jeffrey and H. Rasmussen", title = "Numerical evaluation of {Airy} functions with complex arguments", journal = j-J-COMPUT-PHYS, volume = "98", number = "2", pages = "347--347", month = feb, year = "1992", CODEN = "JCTPAH", DOI = "https://doi.org/10.1016/0021-9991(92)90150-W", ISSN = "0021-9991 (print), 1090-2716 (electronic)", ISSN-L = "0021-9991", bibdate = "Mon Jan 2 07:55:53 MST 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputphys1990.bib", URL = "http://www.sciencedirect.com/science/article/pii/002199919290150W", acknowledgement = ack-nhfb, fjournal = "Journal of Computational Physics", journal-URL = "http://www.sciencedirect.com/science/journal/00219991/", } @Article{Corless:1992:NEAb, author = "R. M. Corless and D. J. Jeffrey and H. Rasmussen", title = "Numerical evaluation of {Airy} functions with complex arguments", journal = j-J-COMPUT-PHYS, volume = "99", number = "1", pages = "106--114", month = mar, year = "1992", CODEN = "JCTPAH", DOI = "https://doi.org/10.1016/0021-9991(92)90279-8", ISSN = "0021-9991 (print), 1090-2716 (electronic)", ISSN-L = "0021-9991", MRclass = "65D20 (33E30)", MRnumber = "92k:65028", bibdate = "Mon Jan 2 07:55:54 MST 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputphys1990.bib", URL = "http://www.sciencedirect.com/science/article/pii/0021999192902798", acknowledgement = ack-nhfb, fjournal = "Journal of Computational Physics", journal-URL = "http://www.sciencedirect.com/science/journal/00219991/", keywords = "Maple", } @Article{Croft:1992:ACA, author = "A. Croft", title = "An application of convergence acceleration techniques to a class of two-point boundary value problems on a semi-infinite domain", journal = j-NUMER-ALGORITHMS, volume = "2", number = "3--4", pages = "307--320", month = sep, year = "1992", CODEN = "NUALEG", ISSN = "1017-1398 (print), 1572-9265 (electronic)", ISSN-L = "1017-1398", MRclass = "65L10 (65B99)", MRnumber = "93g:65097", bibdate = "Fri Nov 6 18:06:29 MST 1998", bibsource = "http://www.math.psu.edu/dna/contents/na.html; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, classification = "C4130 (Interpolation and function approximation); C4170 (Differential equations)", corpsource = "Dept. of Math. Sci., Leicester Polytech., UK", fjournal = "Numerical Algorithms", journal-URL = "http://link.springer.com/journal/11075", keywords = "boundary condition; boundary-value problems; convergence acceleration; convergence acceleration algorithms; convergence of numerical methods; extrapolate; extrapolation; semi-infinite domain; two-point boundary value problems; unbounded domains", pubcountry = "Switzerland", treatment = "T Theoretical or Mathematical", } @Article{Dattoli:1992:GFM, author = "G. Dattoli and C. Chiccoli and S. Lorenzutta and G. Maino and M. Richetta and A. Torre", title = "Generating functions of multivariable generalized {Bessel} functions and {Jacobi}-elliptic functions", journal = j-J-MATH-PHYS, volume = "33", number = "1", pages = "25--36", month = jan, year = "1992", CODEN = "JMAPAQ", DOI = "https://doi.org/10.1063/1.529959", ISSN = "0022-2488 (print), 1089-7658 (electronic), 1527-2427", ISSN-L = "0022-2488", MRclass = "33E05 (33C10 34B30 42A16 42A85)", MRnumber = "92m:33037", MRreviewer = "J. M. H. Peters", bibdate = "Tue Nov 1 08:57:37 MDT 2011", bibsource = "http://jmp.aip.org/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jmathphys1990.bib", URL = "http://jmp.aip.org/resource/1/jmapaq/v33/i1/p25_s1", acknowledgement = ack-nhfb, fjournal = "Journal of Mathematical Physics", journal-URL = "http://jmp.aip.org/", pagecount = "12", } @Article{DiDonato:1992:ASD, author = "Armido R. {DiDonato} and Alfred H. {Morris, Jr.}", title = "{Algorithm 708}: Significant Digit Computation of the Incomplete Beta Function Ratios", journal = j-TOMS, volume = "18", number = "3", pages = "360--373", month = sep, year = "1992", CODEN = "ACMSCU", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Sat Nov 19 13:14:47 1994", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "See also \cite{Brown:1994:CAS}.", URL = "http://doi.acm.org/10.1145/131766.131776; http://www.acm.org/pubs/citations/journals/toms/1992-18-3/p360-didonato/", abstract = "An algorithm is given for evaluating the incomplete beta function ratio $ I_x(a, b) $ and its complement $ 1 - I^x(a, b) $. A new continued fraction and a new asymptotic series are used with classical results. A transportable Fortran subroutine based on this algorithm is currently in use. It is accurate to 14 significant digits when precision is not restricted by inherent error.", acknowledgement = ack-nhfb, fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", keywords = "algorithms", subject = "{\bf G.1.2}: Mathematics of Computing, NUMERICAL ANALYSIS, Approximation.", } @InProceedings{Dubois:1992:CFQ, author = "D. Dubois and H. Prade", title = "Calculation with fuzzy quantities", crossref = "EC2:1992:DJN", bookpages = "384", pages = "24--27", year = "1992", bibdate = "Thu Dec 14 17:22:18 MST 1995", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", abstract = "In some instances where numerical data are either provisional, incomplete, or variable within a limited range, the classical calculation of confidence intervals can be extended in a fuzzy-set approach, distinguishing between more or less plausible values. The simultaneous use of relatively wide intervals containing all possible values, and generally much narrower intervals covering only the most likely ones, can give sufficiently informative results. Some precautions advisable in arithmetic operations on imprecisely known quantities are outlined. Examples of application include provisional budgeting, resource estimation, evaluation of candidates, and extension of PERT to projects involving precedence among elementary tasks with uncertain durations and/or starting times. Computer-aided engineering design can also benefit from fuzzy specifications for values eventually to be optimised.", acknowledgement = ack-nhfb, affiliation = "IRIT, Paul Sabatier Univ., Toulouse, France", classification = "C1160 (Combinatorial mathematics); C4210 (Formal logic); C7310 (Mathematics)", confdate = "2-3 Nov. 1992", conflocation = "Nimes, France", keywords = "Arithmetic operations; Candidates; Confidence intervals; Engineering design; Fuzzy quantities; Fuzzy specifications; Fuzzy-set approach; Imprecisely known quantities; Numerical data; PERT; Provisional budgeting; Resource estimation", language = "French", pubcountry = "France", thesaurus = "Fuzzy logic; Fuzzy set theory; Statistical analysis", } @Article{Feinsilver:1992:BFR, author = "P. Feinsilver and R. Schott", title = "On {Bessel} functions and rate of convergence of zeros of {Lommel} polynomials", journal = j-MATH-COMPUT, volume = "59", number = "199", pages = "153--156", month = jul, year = "1992", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "33C10 (33C45)", MRnumber = "93a:33007", MRreviewer = "Boro D{\"o}ring", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, affiliation = "Southern Illinois Univ., Carbondale, IL, USA", classcodes = "B0220 (Analysis); B0290P (Differential equations); B0290F (Interpolation and function approximation); B0210 (Algebra); C1120 (Analysis); C4170 (Differential equations); C4130 (Interpolation and function approximation); C1110 (Algebra)", classification = "B0210 (Algebra); B0220 (Analysis); B0290F (Interpolation and function approximation); B0290P (Differential equations); C1110 (Algebra); C1120 (Analysis); C4130 (Interpolation and function approximation); C4170 (Differential equations)", corpsource = "Southern Illinois Univ., Carbondale, IL, USA", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "average case analysis; Average case analysis; Bessel function; Bessel functions; convergence of numerical methods; convergence rate; Convergence rate; data structures; differential equations; dynamic; Dynamic data structures; Lommel; Lommel polynomials; Maple program; polynomials; rate of convergence; Rate of convergence; zeros; Zeros", thesaurus = "Bessel functions; Convergence of numerical methods; Differential equations; Polynomials", treatment = "T Theoretical or Mathematical", } @Article{Fillebrown:1992:FCB, author = "Sandra Fillebrown", title = "Faster computation of {Bernoulli} numbers", journal = j-J-ALG, volume = "13", number = "3", pages = "431--445", month = sep, year = "1992", CODEN = "JOALDV", DOI = "https://doi.org/10.1016/0196-6774(92)90048-H", ISSN = "0196-6774 (print), 1090-2678 (electronic)", ISSN-L = "0196-6774", bibdate = "Tue Dec 11 09:15:18 MST 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jalg.bib", URL = "http://www.sciencedirect.com/science/article/pii/019667749290048H", acknowledgement = ack-nhfb, fjournal = "Journal of Algorithms", journal-URL = "http://www.sciencedirect.com/science/journal/01966774", remark = "The author gives algorithms for computing Bernoulli numbers of high order that require at most $ \lfloor 2 n \lg n \rfloor $ bits. One algorithm requires $ O(n^2 \log n) $ multiplications of numbers of $ O(n \log n) $ bits, and the other need $ O(n) $ multiplications of numbers of $ O(n \log n) $ bits.", } @Article{Giordano:1992:FMC, author = "Carla Giordano and Lucia G. Rodon{\`o}", title = "Further monotonicity and convexity properties of the zeros of cylinder functions", journal = j-J-COMPUT-APPL-MATH, volume = "42", number = "2", pages = "245--251", day = "12", month = oct, year = "1992", CODEN = "JCAMDI", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Sat Feb 25 12:20:54 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath1990.bib", URL = "http://www.sciencedirect.com/science/article/pii/037704279290078C", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @Article{Ifantis:1992:DIP, author = "E. K. Ifantis and P. D. Siafarikas", title = "A differential inequality for the positive zeros of {Bessel} functions", journal = j-J-COMPUT-APPL-MATH, volume = "44", number = "1", pages = "115--120", day = "9", month = dec, year = "1992", CODEN = "JCAMDI", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Sat Feb 25 12:20:56 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath1990.bib", URL = "http://www.sciencedirect.com/science/article/pii/0377042792900553", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @Article{Jiang:1992:CCM, author = "Thomas J. Jiang and Joseph B. Kadane and James M. Dickey", title = "Computation of {Carlson}'s Multiple Hypergeometric Function {$R$} for {Bayesian} Applications", journal = j-J-COMPUT-GRAPH-STAT, volume = "1", number = "3", pages = "231--251", month = sep, year = "1992", CODEN = "????", DOI = "https://doi.org/10.1080/10618600.1992.10474583", ISSN = "1061-8600 (print), 1537-2715 (electronic)", ISSN-L = "1061-8600", MRclass = "33C90 (62F15 65D20)", MRnumber = "95e:33021", MRreviewer = "P. N. Rathie", bibdate = "Thu Aug 13 10:27:39 MDT 2020", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputgraphstat.bib", URL = "http://www.tandfonline.com/doi/abs/10.1080/10618600.1992.10474583", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Graphical Statistics", journal-URL = "http://www.amstat.org/publications/jcgs/; http://www.tandfonline.com/loi/ucgs20", onlinedate = "21 Feb 2012", xxtitle = "Computation of {Carlson}'s multiple hypergeometric function {$ {\cal R} $} for {Bayesian} applications", } @Book{Johnson:1992:UDD, author = "Norman Lloyd Johnson and Samuel Kotz and Adrienne W. Kemp", title = "Univariate Discrete Distributions", publisher = pub-WILEY, address = pub-WILEY:adr, edition = "Second", pages = "xx + 565", year = "1992", ISBN = "0-471-54897-9 (hardcover)", ISBN-13 = "978-0-471-54897-3 (hardcover)", LCCN = "QA273.6 .J64 1992", bibdate = "Sat Feb 7 17:19:01 MST 2015", bibsource = "https://www.math.utah.edu/pub/tex/bib/computstatdataanal1990.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; z3950.loc.gov:7090/Voyager", series = "Wiley series in probability and mathematical statistics. Applied probability and statistics", URL = "http://www.loc.gov/catdir/description/wiley031/92011685.html; http://www.loc.gov/catdir/enhancements/fy0607/92011685-b.html; http://www.loc.gov/catdir/toc/onix01/92011685.html", acknowledgement = ack-nhfb, remark = "Revised edition of \booktitle{Discrete distributions}, Norman L. Johnson, Samuel Kotz. 1969.", shorttableofcontents = "Preface \\ Preliminary Information \\ Families of Discrete Distributions \\ Binomial Distributions \\ Poisson Distributions \\ Neggative Binomial Distributions \\ Hypergeometric Distributions \\ Logarithmic and Lagrangian Distributions \\ Mixture Distributions \\ Stopped-Sum Distributions \\ Matching, Occupancy, Runs, and q-Series Distributions \\ Parametric Regression Models and Miscellanea \\ Bibliography \\ Abbreviations \\ Index", subject = "Distribution (Probability theory)", tableofcontents = "Preface / xv \\ List of Tables / xix \\ 1. Preliminary Information / 1 \\ A. Mathematical Preliminaries \\ B. Probability and Statistical Preliminaries \\ C. Computer Generation of Univariate Discrete Random Variables \\ 2. Families of Discrete Distributions / 69 \\ 1. Lattice Distributions \\ 2. Power Series Distributions \\ 3. Difference Equation Systems \\ 4. Kemp Families \\ 5. Distributions Based on Lagrangian Expansions \\ 6. Factorial Series Distributions \\ 3. Binomial Distribution / 105 \\ 1. Definition \\ 2. Historical Remarks and Genesis \\ 3. Moments \\ 4. Properties \\ 5. Order Statistics \\ 6. Approximations, Bounds, and Transformations \\ 7. Computation and Tables \\ 8. Estimation \\ 9. Characterizations \\ 10. Applications \\ 11. Truncated Binomial Distributions \\ 12. Other Related Distributions \\ 4. Poisson Distribution / 151 \\ 1. Definition \\ 2. Historical Remarks and Genesis \\ 3. Moments \\ 4. Properties \\ 5. Approximations, Bounds, and Transformations \\ 6. Computation and Tables \\ 7. Estimation \\ 8. Characterizations \\ 9. Applications \\ 10. Truncated and Misrecorded Poisson Distributions \\ 11. Poisson-Stopped-Sum Distributions \\ 12. Other Related Distributions \\ 5. Negative Binomial Distribution / 199 \\ 1. Definition \\ 2. Geometric Distribution \\ 3. Historical Remarks and Genesis \\ 4. Moments \\ 5. Properties \\ 6. Approximations and Transformations \\ 7. Computation and Tables \\ 8. Estimation \\ 9. Characterizations \\ 10. Applications \\ 11. Truncated Negative Binomial Distributions \\ 12. Other Related Distributions \\ 6. Hypergeometric Distributions / 237 \\ 1. Definition \\ 2. Historical Remarks and Genesis \\ 3. Moments \\ 4. Properties \\ 5. Approximations and Bounds \\ 6. Tables and Computation \\ 7. Estimation \\ 8. Characterizations \\ 9. Applications \\ 10. Special Cases \\ 11. Extended Hypergeometric Distributions \\ 12. Other Related Distributions \\ 7. Logarithmic Distribution / 285 \\ 1. Definition \\ 2. Historical Remarks and Genesis \\ 3. Moments \\ 4. Properties \\ 5. Approximations and Bounds \\ 6. Computation and Tables \\ 7. Estimation \\ 8. Characterizations \\ 9. Applications \\ 10. Truncated and Modified Logarithmic Distributions \\ 11. Other Related Distributions \\ 8. Mixture Distributions / 305 \\ 1. Introduction \\ 2. Finite Mixtures of Discrete Distributions \\ 3. Continuous and Countable Mixtures of Discrete Distributions \\ 9. Generalized (Stopped-Sum) Distributions / 343 \\ 1. Introduction \\ 2. Damage Processes \\ 3. Poisson-Stopped-Sum Distributions: Generalized Poisson Distributions \\ 4. Hermite Distribution \\ 5. Poisson-Binomial Distribution \\ 6. Neyman Type A Distribution \\ 7. Polya--Aeppli Distribution \\ 8. Poisson--Pascal Distribution: Poisson-Negative Binomial Distribution, Generalized Polya--Aeppli Distribution \\ 9. Generalizations of the Neyman Type A Distribution \\ 10. Thomas Distribution \\ 11. Lagrangian Poisson Distribution: Shifted Borel--Tanner Distribution \\ 12. Other Families of Stopped-Sum Distributions \\ 10. Matching, Occupancy, and Runs Distributions / 405 \\ 1. Introduction \\ 2. Probabilities of Combined Events \\ 3. Matching Distributions \\ 4. Occupancy Distributions \\ 5. Runs Distributions \\ 6. Distributions of Order k \\ 11. Miscellaneous Discrete Distributions / 433 \\ 1. Absorption Distribution \\ 2. Dandekar's Modified Binomial and Poisson Distributions \\ 3. Digammma and Trigamma Distributions \\ 4. Discrete Ad{\`e}s Distribution \\ 5. Discrete Student's $t$-Distribution \\ 6. Geeta Distribution \\ 7. Gegenbauer Distribution: Negative Binomial* Pseudo-Negative Binomial Convolution \\ 8. Gram-Charlier Type B Distributions \\ 9. ``Interrupted'' Distributions \\ 10. Lost-Games Distributions \\ 11. Naor's Distribution \\ 12. Partial-Sums Distributions \\ 13. Queueing Theory Distributions \\ 14. Record-Value Distributions \\ 15. Sichel Distribution: Poisson-Inverse Gaussian Distribution \\ 16. Skellam's Gene Frequency Distribution \\ 17. Steyn's Two-Parameter Power Series Distributions \\ 18. Univariate Multinomial-Type Distributions \\ 19. Urn Models with Stochastic Replacements \\ 20. Zipf and Zeta Distributions \\ Bibliography / 473 \\ Abbreviations / 549 \\ Index / 551", } @Article{Kearfott:1992:IPF, author = "Baker Kearfott and Milind Dawande and Kaisheng Du and Chen-Yi Hu", title = "{INTLIB}: a Portable {FORTRAN} 77 Elementary Function Library", journal = j-INTERVAL-COMP, volume = "3", number = "5", pages = "96--105", year = "1992", ISSN = "0135-4868", MRclass = "65G10", MRnumber = "1 253 132", bibdate = "Sat Jan 11 17:44:01 MST 1997", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/k/kearfott-r-baker.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "Interval '92 (Moscow, 1992).", acknowledgement = ack-nhfb, fjournal = "Interval Computations = Interval'nye vychisleniia", } @Article{Kzaz:1992:CAS, author = "M. Kzaz", title = "Convergence acceleration of some {Gaussian} quadrature formulas for analytic functions", journal = j-APPL-NUM-MATH, volume = "10", number = "6", pages = "481--496", month = nov, year = "1992", CODEN = "ANMAEL", ISSN = "0168-9274 (print), 1873-5460 (electronic)", ISSN-L = "0168-9274", MRclass = "65B10 (65D32)", MRnumber = "93j:65004", MRreviewer = "J. Kofro{\v{n}}", bibdate = "Sat Feb 8 10:09:54 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/applnummath.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Applied Numerical Mathematics: Transactions of IMACS", journal-URL = "http://www.sciencedirect.com/science/journal/01689274/", keywords = "convergence acceleration", } @Article{Lang:1992:HRS, author = "T. Lang and P. Montuschi", title = "Higher radix square root with prescaling", journal = j-IEEE-TRANS-COMPUT, volume = "41", number = "8", pages = "996--1009", month = aug, year = "1992", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/12.156542", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Sat Jul 16 15:28:13 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", summary = "A scheme for performing higher radix square root based on prescaling of the radicand is presented to reduce the complexity of the result-digit selection. The scheme requires several steps, namely multiplication for prescaling the radicand, square \ldots{}", } @Article{Lee:1992:LCF, author = "Chu-In Charles Lee", title = "On {Laplace} continued fraction for the normal integral", journal = j-ANN-INST-STAT-MATH-TOKYO, volume = "44", number = "1", pages = "107--120", month = mar, year = "1992", CODEN = "AISXAD", DOI = "https://doi.org/10.1007/BF00048673", ISSN = "0020-3157 (print), 1572-9052 (electronic)", ISSN-L = "0020-3157", bibdate = "Sat Jan 31 16:59:48 MST 2015", bibsource = "https://www.math.utah.edu/pub/tex/bib/anninststatmath.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://link.springer.com/article/10.1007/BF00048673", acknowledgement = ack-nhfb, fjournal = "Annals of the Institute of Statistical Mathematics", journal-URL = "http://link.springer.com/journal/10463", } @InProceedings{Liu:1992:QBS, author = "K. J. R. Liu and E. Frantzeskakis", booktitle = "Workshop on {VLSI} Signal Processing, V, 1992", title = "Qrd-based Square Root Free and Division Free Algorithms and Architectures", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "459--468", year = "1992", CODEN = "????", ISSN = "????", bibdate = "Sat Jul 16 15:28:14 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, summary = "Not \ldots{}", } @Misc{Lynch:1992:HSD, author = "T. Lynch and S. McIntyre and K. Tseng and S. Shaw and T. Hurson", title = "High speed divider with square root capability", year = "1992", bibdate = "Thu Apr 2 08:38:35 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", note = "U.S. Patent No. 5,128,891.", acknowledgement = ack-sfo # " and " # ack-nhfb, } @Article{Martin:1992:TPQa, author = "Pablo Martin and Ricardo P{\'e}rez and Antonio L. Guerrero", title = "Two-point quasi-fractional approximations to the {Airy} function {$ {\rm Ai}(x) $}", journal = j-J-COMPUT-PHYS, volume = "98", number = "2", pages = "349--349", month = feb, year = "1992", CODEN = "JCTPAH", DOI = "https://doi.org/10.1016/0021-9991(92)90165-U", ISSN = "0021-9991 (print), 1090-2716 (electronic)", ISSN-L = "0021-9991", bibdate = "Mon Jan 2 07:55:53 MST 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputphys1990.bib", URL = "http://www.sciencedirect.com/science/article/pii/002199919290165U", acknowledgement = ack-nhfb, fjournal = "Journal of Computational Physics", journal-URL = "http://www.sciencedirect.com/science/journal/00219991/", } @Article{Martin:1992:TPQb, author = "Pablo Mart{\'\i}n and Ricardo P{\'e}rez and Antonio L. Guerrero", title = "Two-point quasi-fractional approximations to the {Airy} function {$ {\rm Ai}(x) $}", journal = j-J-COMPUT-PHYS, volume = "99", number = "2", pages = "337--340", month = apr, year = "1992", CODEN = "JCTPAH", DOI = "https://doi.org/10.1016/0021-9991(92)90212-H", ISSN = "0021-9991 (print), 1090-2716 (electronic)", ISSN-L = "0021-9991", bibdate = "Mon Jan 2 07:55:55 MST 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputphys1990.bib", URL = "http://www.sciencedirect.com/science/article/pii/002199919290212H", acknowledgement = ack-nhfb, fjournal = "Journal of Computational Physics", journal-URL = "http://www.sciencedirect.com/science/journal/00219991/", } @Article{Matos:1992:CAP, author = "Ana C. Matos", title = "Convergence and acceleration properties for the vector $ \epsilon $-algorithm", journal = j-NUMER-ALGORITHMS, volume = "3", number = "1--4", pages = "313--319", month = dec, year = "1992", CODEN = "NUALEG", ISSN = "1017-1398 (print), 1572-9265 (electronic)", ISSN-L = "1017-1398", MRclass = "65B99", MRnumber = "93h:65006", bibdate = "Fri Nov 6 18:06:29 MST 1998", bibsource = "http://www.math.psu.edu/dna/contents/na.html; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "Extrapolation and rational approximation (Puerto de la Cruz, 1992).", acknowledgement = ack-nhfb, classification = "B0290F (Interpolation and function approximation); C4130 (Interpolation and function approximation)", conflocation = "Puerto de la Cruz, Spain; 13-17 Jan. 1992", conftitle = "International Mathematical Congress on Extrapolation and Rational Approximation", corpsource = "Fac. de Ciencias, Porto Univ., Portugal", fjournal = "Numerical Algorithms", journal-URL = "http://link.springer.com/journal/11075", keywords = "acceleration; convergence acceleration; convergence of numerical methods; convergence speed; exactness; extrapolation; extrapolation algorithm; speed of convergence; vector $\epsilon$-algorithm; vector sequences", pubcountry = "Switzerland", treatment = "T Theoretical or Mathematical", } @Article{McQuillan:1992:VMH, author = "S. E. McQuillan and J. V. McCanny", title = "{VLSI} module for high-performance multiply, square root and divide", journal = j-IEE-PROC-COMPUT-DIGIT-TECH, volume = "139", number = "6", pages = "505--510", month = nov, year = "1992", CODEN = "ICDTEA", ISSN = "1350-2387 (print), 1359-7027 (electronic)", ISSN-L = "1350-2387", bibdate = "Sat Jul 16 15:28:13 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "IEE Proceedings. Computers and Digital Techniques", summary = "A high-performance VLSI architecture to perform multiply-accumulate, division and square root operations is proposed. The circuit is highly regular, requires only minimal control and ean be pipelined right down to the bit level. The system can also \ldots{}", } @Article{Mikami:1992:NDO, author = "N. Mikami and M. Kobayashi and Y. Yokoyama", title = "A New {DSP}-Oriented Algorithm for Calculation of the Square Root Using a Nonlinear Digital Filter", journal = j-IEEE-TRANS-SIG-PROC, volume = "40", number = "7", pages = "1663--1669", month = jul, year = "1992", CODEN = "ITPRED", DOI = "https://doi.org/10.1109/78.143438", ISSN = "1053-587X (print), 1941-0476 (electronic)", ISSN-L = "1053-587X", bibdate = "Sat Jul 16 15:28:13 MDT 2005", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nj # " and " # ack-nhfb, fjournal = "IEEE Transactions on Signal Processing", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=78", summary = "A high-speed algorithm for calculating the square root is proposed. This algorithm, which can be regarded as calculation of the step response of a kind of nonlinear IIR filter, requires no divisions. Therefore, it is suitable for a VLSI digital \ldots{}", } @Article{Mitchell:1992:VFA, author = "H. B. Mitchell", title = "Very fast accurate square-root algorithm for use with gradient edge operators", journal = j-ELECT-LETTERS, volume = "28", number = "10", pages = "922--923", day = "7", month = may, year = "1992", CODEN = "ELLEAK", DOI = "https://doi.org/10.1049/el:19920584", ISSN = "0013-5194 (print), 1350-911X (electronic)", ISSN-L = "0013-5194", bibdate = "Sat Jul 16 15:28:13 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Electronics Letters", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=2220; https://digital-library.theiet.org/journal/el", summary = "Commonly used gradient edge operators such as the Sobel, Prewitt and Roberts operators all required a square root operation; this is, however, computationally intensive and, consequently, simple but very inaccurate approximations are often used \ldots{}", } @Article{Paris:1992:EIA, author = "R. B. Paris and A. D. Wood", title = "Exponentially-improved asymptotics for the gamma function", journal = j-J-COMPUT-APPL-MATH, volume = "41", number = "1--2", pages = "135--143", day = "20", month = aug, year = "1992", CODEN = "JCAMDI", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Sat Feb 25 12:20:53 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath1990.bib", URL = "http://www.sciencedirect.com/science/article/pii/037704279290243Q", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @Article{Paszkowski:1992:CAC, author = "Stefan Paszkowski", title = "Convergence acceleration of continued fractions of {Poincar{\'e}}'s type $1$", journal = j-NUMER-ALGORITHMS, volume = "2", number = "2", pages = "155--170", month = "????", year = "1992", CODEN = "NUALEG", ISSN = "1017-1398 (print), 1572-9265 (electronic)", ISSN-L = "1017-1398", MRclass = "65B05 (40A15)", MRnumber = "93c:65006", MRreviewer = "A. Bultheel", bibdate = "Fri Nov 6 18:06:29 MST 1998", bibsource = "http://www.math.psu.edu/dna/contents/na.html; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, classification = "B0290F (Interpolation and function approximation); C4130 (Interpolation and function approximation)", corpsource = "Instytut Niskich Temp. i Badan Strukturalnych PAN, Wroclaw, Poland", fjournal = "Numerical Algorithms", journal-URL = "http://link.springer.com/journal/11075", keywords = "asymptotic behaviour; continued fractions; convergence acceleration; convergence of numerical methods; function approximation", pubcountry = "Switzerland", treatment = "T Theoretical or Mathematical", } @Book{Prudnikov:1992:IS, author = "Anatolij P. Prudnikov and Jurij A. Bry{\v{c}}kov and Oleg I. Mari{\v{c}}ev", title = "Integrals and series. {More} special functions", volume = "3", publisher = "Gordon and Breach Science Publishers", address = "New York, NY, USA", pages = "xx + 618", year = "1992", ISBN = "2-88124-097-6", ISBN-13 = "978-2-88124-097-3", LCCN = "QA308 P68 1986", bibdate = "Thu Nov 2 15:54:36 MDT 2017", bibsource = "fsz3950.oclc.org:210/WorldCat; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, remark = "Translated from the Russian by N. M. Queen.", seriestableofcontents = "v. 1. Elementary functions \\ v. 2. Special functions \\ v. 3. More special functions \\ v. 4. Direct Laplace transforms \\ v. 5. Inverse Laplace transforms", subject = "Mathematics", } @Book{Rockett:1992:CF, author = "Andrew Mansfield Rockett and Peter Sz{\"u}sz", title = "Continued Fractions", publisher = pub-WORLD-SCI, address = pub-WORLD-SCI:adr, pages = "ix + 188", year = "1992", ISBN = "981-02-1047-7", ISBN-13 = "978-981-02-1047-2", LCCN = "QA295.R6 1992; QA295 .R6 1992", bibdate = "Wed Apr 15 16:49:47 MDT 2009", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; melvyl.cdlib.org:210/CDL90", acknowledgement = ack-nhfb, remark = "From the first line of the Preface: ``The theory of continued fractions does not receive the attention it deserves.''.", subject = "continued fractions; processes, infinite", tableofcontents = "I. Introduction \\ 1. What is a continued fraction? \\ 2. Regular continued fractions \\ 3. The transformation $ T(x) = \{ 1 / x \} $ \\ 4. The quantity $ D_k = B_k t - A_k $ \\ 5. Convergents to a number and its reciprocal \\ 6. The ratio $ \xi_k = B_{k - 1} / B_k $ \\ 7. The golden ratio and the Fibonacci sequence \\ 8. The continued fraction for $ e $ \\ \\ II. The Law of Best Approximation \\ 1. Best approximation \\ 2. The first proof \\ 3. A theorem of Lagrange \\ 4. Ostrowski's algorithm and a second proof. \\ 5. The approximation $ B ||b t|| < K $ \\ 6. The $t$-expansion of a real number \\ \\ III. Periodic Continued Fractions \\ 1. The classical theorems \\ 2. Period lengths \\ 3. Second order linear recurrences \\ \\ IV. Applications \\ 1. Gear ratio problems \\ 2. Pell's equation \\ 3. Fermat's theorem on the sum of two squares \\ 4. Hall's theorem \\ 5. A theorem of Hurwitz \\ 6. The Lagrange and Markov spectra \\ 7. Asymmetric approximation and S{\`e}gre's theorem \\ 8. Approximation by non-convergents \\ 9. Inhomogeneous approximation \\ 10. Szekeres' empty parallelogram theorem \\ \\ V. Metrical Theory \\ 1. Numbers with bounded partial quotients \\ 2. The Borel--Cantelli lemma \\ 3. Random variables and expectations \\ 4. Chebyshev's inequality and large number laws \\ 5. The Gauss--Kuzmin theorem \\ 6. The distribution of $ \xi_n $ \\ 7. Partial quotients are weakly dependent \\ 8. Khintchine's theorem \\ 9. The Khintchine--L{\'e}vy theorem for $ \sqrt[n]{B_n} $ \\ \\ VI. Applications to Metrical Diophantine Approximation \\ Notes \\ Bibliography \\ Symbols \\ Index", } @Article{Salwin:1992:UPE, author = "Arthur E. Salwin", title = "Using the Proposed Elementary Functions Standard to Build a Strongly Typed Trig Package", journal = j-SIGADA-LETTERS, volume = "12", number = "5", pages = "59--63", month = sep # "\slash " # oct, year = "1992", CODEN = "AALEE5", ISSN = "1094-3641 (print), 1557-9476 (electronic)", ISSN-L = "1094-3641", bibdate = "Sat Aug 9 09:05:46 MDT 2003", bibsource = "http://portal.acm.org/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/sigada.bib", acknowledgement = ack-nhfb, classcodes = "C6140D (High level languages)", corpsource = "Mitre Corp., McLean, VA, USA", fjournal = "ACM SIGAda Ada Letters", journal-URL = "http://portal.acm.org/citation.cfm?id=J32", keywords = "Ada; compiler; elementary functions standard; standards; strong typing; strongly typed trig package; trigonometric functions", treatment = "P Practical", } @Article{Saunders:1992:EFS, author = "L. R. Saunders", title = "An Exact Formula for the Symmetrical Incomplete Beta Function Where the Parameter Is an Integer or Half-Integer", journal = j-AUST-J-STAT, volume = "34", number = "2", pages = "261--264", month = jun, year = "1992", CODEN = "AUJSA3", DOI = "https://doi.org/10.1111/j.1467-842X.1992.tb01358.x", ISSN = "0004-9581", ISSN-L = "0004-9581", bibdate = "Fri Jul 15 14:28:59 MDT 2016", bibsource = "https://www.math.utah.edu/pub/tex/bib/anzjs.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Australian Journal of Statistics", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1111/(ISSN)1467-842X/issues", } @MastersThesis{Schulte:1992:AHD, author = "Michael Joseph Schulte and Function generation", title = "Algorithms and hardware designs for parallel elementary function generation", type = "Thesis ({M.S.} in Engin.)", school = "University of Texas at Austin", address = "Austin, TX, USA", pages = "ix + 73", year = "1992", bibdate = "Sat Jan 11 10:14:06 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, keywords = "Computer input-output equipment -- Design and construction.; Computer science -- Mathematics.; Numerical analysis.; Parallel processing (Electronic computers)", searchkey = "ti:elementary n1 function", } @Article{Tang:1992:TDI, author = "Ping Tak Peter Tang", title = "Table-Driven Implementation of the {{\tt Expm1}} Function in {IEEE} Floating-Point Arithmetic", journal = j-TOMS, volume = "18", number = "2", pages = "211--222", month = jun, year = "1992", CODEN = "ACMSCU", DOI = "https://doi.org/10.1145/146847.146928", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", MRclass = "65D15", MRnumber = "1 167 891", bibdate = "Sat Feb 24 15:01:45 MST 1996", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "See independent analysis and accuracy confirmation of this algorithm in \cite{Kramer:1998:PWC}.", URL = "http://www.acm.org/pubs/citations/journals/toms/1992-18-2/p211-tang/", abstract = "Algorithms and implementation details for the function $ e^x - 1 $ in both single and double precision of IEEE 754 arithmetic are presented here. With a table of moderate size, the implementations need only working-precision arithmetic and are provably accurate to within 0.58 ulp.", acknowledgement = ack-nhfb, fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", keywords = "algorithms", subject = "{\bf G.1.0}: Mathematics of Computing, NUMERICAL ANALYSIS, General, Computer arithmetic. {\bf G.1.0}: Mathematics of Computing, NUMERICAL ANALYSIS, General, Error analysis. {\bf G.1.0}: Mathematics of Computing, NUMERICAL ANALYSIS, General, Numerical algorithms. {\bf G.4}: Mathematics of Computing, MATHEMATICAL SOFTWARE, Algorithm analysis.", } @Article{Temme:1992:AII, author = "N. M. Temme", title = "Asymptotic Inversion of Incomplete Gamma Functions", journal = j-MATH-COMPUT, volume = "58", number = "198", pages = "755--764", month = apr, year = "1992", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "33B20", MRnumber = "93a:33003", MRreviewer = "F. W. J. Olver", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Wong:1992:DSR, author = "W. F. Wong and E. Goto", title = "Division and square-rooting using a split multiplier", journal = j-ELECT-LETTERS, volume = "28", number = "18", pages = "1758--1759", day = "27", month = aug, year = "1992", CODEN = "ELLEAK", DOI = "https://doi.org/10.1049/el:19921119", ISSN = "0013-5194 (print), 1350-911X (electronic)", ISSN-L = "0013-5194", bibdate = "Sat Jul 16 15:28:13 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Electronics Letters", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=2220; https://digital-library.theiet.org/journal/el", summary = "A modification is proposed to the traditional design of a fast floating point multiplication circuit such that instead of just performing A$\times$B where A and B are m bits long, it is also capable of \ldots{}", } @TechReport{Wood:1992:CP, author = "David C. Wood", title = "The Computation of Polylogarithms", type = "Report", institution = "University of Kent", address = "Canterbury, Kent CT2 7NZ, UK", year = "1992", bibdate = "Fri Jun 30 10:12:54 2023", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "https://www.cs.kent.ac.uk/pubs/1992/110/content.pdf", abstract = "The polylogarithm function, $ \Li_p(z) $, is defined, and a number of algorithms are derived for its computation, valid in different ranges of its real parameter $p$ and complex argument $z$.", acknowledgement = ack-nhfb, keywords = "polylogarithm", remark = "Undated, but its URL suggests the year. The PDF file was created 20-Mar-2014. The latest reference is to a 1992 journal article.", } @InProceedings{Woods:1992:HPD, author = "R. F. Woods and S. E. McQuillan and J. Dowling and J. V. McCanny", booktitle = "Proceedings of Fifth Annual {IEEE} International {ASIC} Conference and Exhibit, 1992", title = "High performance {DSP} {ASIC} for multiply, divide and square root", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "209--213", year = "1992", CODEN = "????", ISSN = "????", bibdate = "Sat Jul 16 15:28:14 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, summary = "The design of a high-speed ASIC that combines the operations of multiplication, division and square root is described. The chip is based on a systolic array architecture that uses a redundant number system and allows multiplication, division, and \ldots{}", } @Article{Yeyios:1992:TSA, author = "A. K. Yeyios", title = "On two sequences of algorithms for approximating square roots", journal = j-J-COMPUT-APPL-MATH, volume = "40", number = "1", pages = "63--72", month = jun, year = "1992", CODEN = "JCAMDI", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Thu Sep 1 10:15:56 1994", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nj, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @Article{Alzer:1993:SGF, author = "Horst Alzer", title = "Some gamma function inequalities", journal = j-MATH-COMPUT, volume = "60", number = "201", pages = "337--346", month = jan, year = "1993", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "33B15 (26D20)", MRnumber = "93f:33001", MRreviewer = "Aurelio Cannizzo", bibdate = "Sat Jan 11 13:29:06 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Periodical{Anonymous:1993:ITS, author = "Anonymous", title = "Integral transforms and special functions", publisher = "Gordon and Breach Science Publishers", address = "Yverdon, Switzerland", year = "1993", ISSN = "1065-2469, 1476-8291", ISSN-L = "1065-2469", bibdate = "Mon Oct 24 11:37:20 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "Appears with variable frequency from 1993--2001, and six times yearly from 2002--date.", acknowledgement = ack-nhfb, } @Article{Arenstorf:1993:SMZ, author = "R. F. Arenstorf and L. L. Brewer", title = "A study of the motion of zeros of the {Epstein} zeta function associated to $ m^2 + y^2 n^2 $ as $y$ varies from $1$ to $ \sqrt {6}$", journal = j-COMPUT-MATH-APPL, volume = "26", number = "5", pages = "57--69", month = sep, year = "1993", CODEN = "CMAPDK", DOI = "https://doi.org/10.1016/0898-1221(93)90074-6", ISSN = "0898-1221 (print), 1873-7668 (electronic)", ISSN-L = "0898-1221", bibdate = "Wed Mar 1 19:11:16 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/computmathappl1990.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/0898122193900746", acknowledgement = ack-nhfb, fjournal = "Computers and Mathematics with Applications", journal-URL = "http://www.sciencedirect.com/science/journal/08981221", } @Article{Bailey:1993:AMT, author = "D. H. Bailey", title = "Algorithm 719: Multiprecision Translation and Execution of {FORTRAN} Programs", journal = j-TOMS, volume = "19", number = "3", pages = "288--319", month = sep, year = "1993", CODEN = "ACMSCU", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Wed Dec 13 18:37:31 1995", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", abstract = "The author describes two Fortran utilities for multiprecision computation. The first is a package of Fortran subroutines that perform a variety of arithmetic operations and transcendental functions on floating point numbers of arbitrarily high precision. This package is in some cases over 200 times faster than that of certain other packages that have been developed for this purpose. The second utility is a translator program, which facilitates the conversion of ordinary Fortran programs to use this package. By means of source directives (special comments) in the original Fortran program, the user declares the precision level and specifies which variables in each subprogram are to be treated as multiprecision. The translator program reads this source program and outputs a program with the appropriate multiprecision subroutine calls. This translator supports multiprecision integer, real, and complex datatypes. The required array space for multiprecision data types is automatically allocated. In the evaluation of computational expressions, all of the usual conventions for operator precedence and mixed mode operations are upheld. Furthermore, most of the Fortran-77 intrinsics, such as ABS, MOD, NINT, COS, EXP are supported and produce true multiprecision values.", acknowledgement = ack-nhfb # " and " # ack-nj, affiliation = "NASA Ames Res. Center, Moffett Field, CA, USA", classification = "C5230 (Digital arithmetic methods); C6120 (File organisation); C6140D (High level languages); C6150C (Compilers, interpreters and other processors); C7310 (Mathematics)", fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", keywords = "Algorithm 719; Arithmetic operations; Array space; Complex data types; Computational expressions; Floating point numbers; Fortran programs; Fortran subroutines; Fortran utilities; Fortran-77 intrinsics; Mixed mode operations; Multiprecision computation; Multiprecision data types; Multiprecision subroutine calls; Multiprecision translation; Operator precedence; Source directives; Transcendental functions; Translator program", pubcountry = "USA", thesaurus = "Data structures; Digital arithmetic; FORTRAN; Mathematics computing; Program interpreters; Subroutines", } @Article{Barrera:1993:IBS, author = "Tony Barrera and Pelle Olsson", title = "An Integer Based Square Root Algorithm", journal = j-BIT, volume = "33", number = "2", pages = "253--261", month = jun, year = "1993", CODEN = "BITTEL, NBITAB", DOI = "https://doi.org/10.1007/BF01989748", ISSN = "0006-3835 (print), 1572-9125 (electronic)", ISSN-L = "0006-3835", MRclass = "68M07", MRnumber = "1 326 017", bibdate = "Wed Jan 4 18:52:23 MST 2006", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; garbo.uwasa.fi:/pc/doc-soft/fpbiblio.txt; http://springerlink.metapress.com/openurl.asp?genre=issue&issn=0006-3835&volume=33&issue=2; https://www.math.utah.edu/pub/tex/bib/bit.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "http://www.mai.liu.se/BIT/contents/bit33.html; http://www.springerlink.com/openurl.asp?genre=article&issn=0006-3835&volume=33&issue=2&spage=253", abstract = "The authors propose a fast integer based method for computing square roots of floating point numbers. This implies high accuracy and robustness, since no precision will be lost during the computation. Only integer addition and shifts are necessary to obtain the square root. Comparisons made with the modified Newton method indicate that the suggested method is twice as fast for computing floating point square roots. (5 Refs.)", acknowledgement = ack-nhfb # " and " # ack-nj, affiliation = "AB Consonant, Uppsala, Sweden", classification = "C5230 (Digital arithmetic methods)", fjournal = "BIT (Nordisk tidskrift for informationsbehandling)", journal-URL = "http://link.springer.com/journal/10543", keywords = "Floating point numbers; floating-point arithmetic; Integer based square root algorithm; Modified Newton method; Robustness", pubcountry = "Denmark", thesaurus = "Digital arithmetic", xxpages = "254--261??", } @InCollection{Bohlender:1993:PAF, author = "G. Bohlender and D. Cordes and A. Knofel and U. Kulisch and R. Lohner and W. V. Walter", title = "Proposal for accurate floating-point vector arithmetic", crossref = "Adams:1993:ACA", bookpages = "x + 612", pages = "87--102", year = "1993", bibdate = "Tue Dec 12 09:27:13 MST 1995", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", abstract = "Many computers provide accurate and reliable scalar arithmetic for floating point numbers. An accurate definition of the four elementary floating-point operations +, -, *, / is given in the IEEE standards for floating-point arithmetic and was well established long before. An increasing number of computers (especially PC's and workstations) feature IEEE arithmetic. In many numerical algorithms, however, compound operations such as the summation of a sequence of numbers or the dot product of two vectors are highly common. A simulation of these compound operations by means of elementary floating-point operations leads to accumulation of rounding errors and may suffer from catastrophic cancellation of leading digits. Existing standards for floating-point arithmetic do not improve this situation. The goal of the proposal is to define vector operations in a manner consistent with the elementary scalar arithmetic operations. The rounding modes and accuracy requirements as well as the data formats of the operands and results of the vector operations described in the proposal are chosen to be fully consistent with the existing scalar floating-point arithmetic.", acknowledgement = ack-nhfb, affiliation = "Inst. fur Angewandte Math., Karlsruhe Univ., Germany", classification = "C5230 (Digital arithmetic methods); C6130 (Data handling techniques); C7310 (Mathematics)", keywords = "Accuracy requirements; Catastrophic cancellation; Compound operations; Data formats; Dot product; Elementary floating-point operations; Elementary scalar arithmetic operations; Floating point numbers; IEEE arithmetic; IEEE standards; Leading digits; Numerical algorithms; Operands; Rounding errors; Rounding modes; Scalar floating-point arithmetic; Sequence; Standards; Summation; Vector operations", pubcountry = "USA", thesaurus = "Digital arithmetic; Mathematics computing; Roundoff errors; Standards", } @Article{Cody:1993:ACP, author = "W. J. Cody", title = "{Algorithm 714}: {CELEFUNT}: a Portable Test Package for Complex Elementary Functions", journal = j-TOMS, volume = "19", number = "1", pages = "1--21", month = mar, year = "1993", CODEN = "ACMSCU", DOI = "https://doi.org/10.1145/151271.151272", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Tue Sep 20 18:24:35 1994", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/c/cody-william-j.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib", URL = "http://www.acm.org/pubs/citations/journals/toms/1993-19-1/p1-cody/; http://www.acm.org/pubs/toc/Abstracts/toms/151272.html", abstract = "This paper discusses CELEFUNT, a package of Fortran programs for testing complex elementary functions.", abstract-2 = "The author discusses CELEFUNT, a package of Fortran programs for testing complex elementary functions. CELEFUNT is a collection of test programs for the complex floating-point elementary functions required by the 1978 ANSI Fortran Standard (CABS), CSQRT, CLOG, CEXP, CSIN/CCOS, and the complex power function.", acknowledgement = ack-nhfb, affiliation = "Div. of Math. and Comput. Sci., Argonne Nat. Lab., IL, USA", classification = "C4100 (Numerical analysis); C5230 (Digital arithmetic methods); C7310 (Mathematics)", fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", keywords = "algorithms; CABS; CELEFUNT; CEXP; CLOG; Complex elementary functions; Complex power function; CSIN/CCOS; CSQRT; Floating-point elementary functions; Fortran programs; measurement; performance; Portable test package", subject = "{\bf G.4}: Mathematics of Computing, MATHEMATICAL SOFTWARE, Certification and testing. {\bf G.1.0}: Mathematics of Computing, NUMERICAL ANALYSIS, General, Numerical algorithms.", thesaurus = "Conformance testing; Digital arithmetic; FORTRAN; Mathematics computing; Numerical analysis; Program testing; Software packages", } @Article{Cody:1993:ASP, author = "W. J. {Cody, Jr.}", title = "Algorithm 715: {SPECFUN}: a Portable {FORTRAN} Package of Special Function Routines and Test Drivers", journal = j-TOMS, volume = "19", number = "1", pages = "22--32", month = mar, year = "1993", CODEN = "ACMSCU", DOI = "https://doi.org/10.1145/151271.151273", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Tue Sep 20 18:24:38 1994", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/c/cody-william-j.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib", URL = "http://www.acm.org/pubs/toc/Abstracts/0098-3500/151273.html", abstract = "SPECFUN is a package containing transportable FORTRAN special function programs for real arguments and accompanying test drivers. Components include Bessel functions, exponential integrals, error functions and related functions, and gamma functions and related functions.", acknowledgement = ack-nhfb, affiliation = "Div. of Math. and Comput. Sci., Argonne Nat. Lab., IL, USA", classification = "C4100 (Numerical analysis); C7310 (Mathematics)", fjournal = "ACM Transactions on Mathematical Software", journal-URL = "http://portal.acm.org/toc.cfm?idx=J782", keywords = "algorithms; Bessel functions; Error functions; Exponential integrals; Gamma functions; Portable FORTRAN package; Real arguments; SPECFUN; Special function routines; Test drivers", pubcountry = "USA", subject = "{\bf G.4}: Mathematics of Computing, MATHEMATICAL SOFTWARE, Certification and testing. {\bf G.1.0}: Mathematics of Computing, NUMERICAL ANALYSIS, General, Numerical algorithms.", thesaurus = "FORTRAN; Mathematics computing; Numerical analysis; Software packages; Software portability", } @Article{duToit:1993:BFI, author = "C. F. du Toit", title = "{Bessel} functions {$ J_n(z) $} and {$ Y_n(z) $} of integer order and complex argument", journal = j-COMP-PHYS-COMM, volume = "78", number = "1--2", pages = "181--189", month = dec, year = "1993", CODEN = "CPHCBZ", DOI = "https://doi.org/10.1016/0010-4655(93)90153-4", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", bibdate = "Mon Feb 13 21:29:41 MST 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/compphyscomm1990.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/0010465593901534", abstract = "This paper describes computer subroutines which were developed to compute Bessel functions of the first and second kind ($ J_n(z) $ and $ Y_n(z) $, respectively) for a complex argument $z$ and a range of integer orders. A novel way of determining the starting point of backward recurrence is used, and the algorithm for $ Y_n(z) $ improves on previous algorithms in terms of accuracy and restrictions on the range of orders.", acknowledgement = ack-nhfb, fjournal = "Computer Physics Communications", journal-URL = "http://www.sciencedirect.com/science/journal/00104655", } @Book{Feinsilver:1993:ASO, author = "Philip J. (Philip Joel) Feinsilver and Ren{\'e} Schott", title = "Algebraic Structures and Operator Calculus", volume = "241, 292, 347", publisher = pub-KLUWER, address = pub-KLUWER:adr, pages = "?????", year = "1993, 1994, 1996", ISBN = "0-7923-2116-2 (v. 1), 0-7923-2921-X (v. 2), 0-7923-3834-0 (v. 3)", ISBN-13 = "978-0-7923-2116-3 (v. 1), 978-0-7923-2921-3 (v. 2), 978-0-7923-3834-5 (v. 3)", LCCN = "QA432 .F45 1993", bibdate = "Sat Oct 30 17:31:34 MDT 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; prodorbis.library.yale.edu:7090/voyager", note = "Volume 1: Representations and probability theory. Volume 2: Special functions and computer science. Volume 3: Representations of Lie groups", series = "Mathematics and its applications", acknowledgement = ack-nhfb, subject = "Calculus, Operational; Probabilities; Representations of groups", } @Article{Fowkes:1993:HEA, author = "Raymond E. Fowkes", title = "Hardware Efficient Algorithms for Trigonometric Functions", journal = j-IEEE-TRANS-COMPUT, volume = "42", number = "2", pages = "235--239", month = feb, year = "1993", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/12.204796", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Thu Jul 7 07:58:47 MDT 2011", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput1990.bib", URL = "https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=204796", acknowledgement = ack-nj # "\slash " # ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", } @Article{Frappier:1993:QFI, author = "Cl{\'e}ment Frappier and Patrick Olivier", title = "A quadrature formula involving zeros of {Bessel} functions", journal = j-MATH-COMPUT, volume = "60", number = "201", pages = "303--316", month = jan, year = "1993", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "41A55 (65D32)", MRnumber = "93d:41025", MRreviewer = "Hans Strauss", bibdate = "Tue Mar 25 15:38:13 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, classcodes = "B0290F (Interpolation and function approximation); B0290M (Numerical integration and differentiation); C4130 (Interpolation and function approximation); C4160 (Numerical integration and differentiation)", corpsource = "Dept. de Math. Appliqu{\'e}es, Montreal, Que., Canada", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "Bessel functions; integration; interpolation; poles and; polynomials; quadrature formula; sampling theorem; zeros", treatment = "T Theoretical or Mathematical", } @InProceedings{Han:1993:CAS, author = "Weimin Han and Florian A. Potra", title = "Convergence acceleration for some rootfinding methods", crossref = "Albrecht:1993:VNT", volume = "9", pages = "67--78", year = "1993", CODEN = "COSPDM", ISSN = "0344-8029", bibdate = "Sun Oct 17 11:55:48 MDT 1999", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", series = j-COMPUTING-SUPPLEMENTUM, acknowledgement = ack-nhfb, keywords = "convergence acceleration", } @Article{Higginbotham:1993:ISR, author = "T. F. Higginbotham", title = "The integer square root of {$N$} via a binary search", journal = j-SIGCSE, volume = "25", number = "4", pages = "41--45", month = dec, year = "1993", CODEN = "SIGSD3", DOI = "https://doi.org/10.1145/164205.164229", ISSN = "0097-8418 (print), 2331-3927 (electronic)", ISSN-L = "0097-8418", bibdate = "Sat Nov 17 18:57:24 MST 2012", bibsource = "http://portal.acm.org/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/sigcse1990.bib", abstract = "An algorithm is presented which may be used to find the integer square root of N. The method is intended for use on a binary computer, where only addition, subtraction, multiplication, or division by 2 is required. The problem arose when the author was working on factoring large numbers, where the machine, the Honeywell DPS 8, had double precision integer addition and subtraction, and the simulation of multiplication was easy. The actual factoring of the large number was to be Fermat's Method, requiring only addition and subtraction, but the integer square root is required in order to test for termination. The algorithm is implemented in FORTRAN for ease of reading. Students enjoy the unconventional approach to solving this problem. It isn't long before some of them think of other unusual solutions.", acknowledgement = ack-nhfb, fjournal = "SIGCSE Bulletin (ACM Special Interest Group on Computer Science Education)", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J688", } @Article{Hu:1993:BRS, author = "Chen-Yi Hu and R. Baker Kearfott and Abdulhamid Awad", title = "On Bounding the Range of Some Elementary Functions in {FORTRAN} 77", journal = j-INTERVAL-COMP, volume = "1993", number = "3", pages = "29--39", year = "1993", ISSN = "0135-4868", MRclass = "65G10", MRnumber = "1 305 844", bibdate = "Wed Dec 4 11:13:33 1996", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/k/kearfott-r-baker.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "Proceedings of the International Conference on Numerical Analysis with Automatic Result Verification (Lafayette, LA, 1993)", acknowledgement = ack-nhfb, fjournal = "Interval Computations = Interval'nye vychisleniia", } @InProceedings{Hu:1993:NES, author = "Xiaobo Hu and Steven C. Bass", title = "A neglected error source in the {CORDIC} algorithm", crossref = "Swartzlander:1993:SCA", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "766--769 (vol. 1)", year = "1993", DOI = "https://doi.org/10.1109/ISCAS.1993.393834", bibdate = "Wed Nov 12 10:45:34 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, keywords = "Circuits; Computer errors; Computer science; Cost function; Digital arithmetic; Error analysis; Hardware; Iterative algorithms; Laboratories; Signal processing algorithms", } @TechReport{Karp:1993:HPD, author = "A. H. Karp and P. Markstein", title = "High precision division and square root", number = "HPL-93-42", institution = "Hewlett--Packard Lab.", address = "Palo Alto, CA, USA", pages = "20", month = jun, year = "1993", bibdate = "Tue Dec 12 09:27:13 MST 1995", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", abstract = "The authors present division and square root algorithms for calculations with more bits than are handled by the floating point hardware. These algorithms avoid the need to multiply two high precision numbers, speeding up the last iteration by as much as a factor of ten.", acknowledgement = ack-nhfb, classification = "C5230 (Digital arithmetic methods)", keywords = "Division; Floating point hardware; Square root algorithms", thesaurus = "Digital arithmetic", } @InCollection{Kramer:1993:MPC, author = "Walter Kr{\"a}mer", booktitle = "Mathematics in Science and Engineering: Scientific Computing with Automatic Result Verification", title = "Multiple-Precision Computations with Result Verification", volume = "189", publisher = "Elsevier BV", address = "Amsterdam, The Netherlands", pages = "325--356", year = "1993", DOI = "https://doi.org/10.1016/s0076-5392(08)62851-9", bibdate = "Tue Mar 14 19:20:47 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, keywords = "arithmetic-geometric mean iteration; computation of $ e^\pi $; computation of a large number of digits of $\pi$; computation of elliptic integrals; computation of guaranteed bounds for the natural logarithm; interval arithmetic; PASCAL-XSC", } @Article{Laforgia:1993:AMR, author = "Andrea Laforgia and Maria Luisa Mathis", title = "Additional monotonicity results for the zeros of {Bessel} functions", journal = j-J-COMPUT-APPL-MATH, volume = "47", number = "1", pages = "135--139", day = "28", month = jun, year = "1993", CODEN = "JCAMDI", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Sat Feb 25 12:20:58 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath1990.bib", URL = "http://www.sciencedirect.com/science/article/pii/037704279390095S", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @Article{Lee:1993:DAE, author = "Joong-Eon Lee and Oh-Young Kwon and Tack-Don Han", title = "Design of an area efficient unit for floating-point division and square root", journal = j-J-KOREA-INFO-SCI-SOCIETY, volume = "20", number = "7", pages = "1060--1071", month = jul, year = "1993", CODEN = "HJKHDC", ISSN = "0258-9125", bibdate = "Tue Dec 12 09:27:13 MST 1995", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", abstract = "The authors propose an algorithm for a high performance floating point division and square root unit that uses a parallel multiplier. The basic algorithm used in the design is the continued-product normalization method. In this method, an arbitrary number is constantly multiplied to the divisor and dividend and dividend/divisor ends up with quotient/1 and the desired result attained. However this method requires computation of x*(2-x) and x*(3-x)/2 and this is quite an overhead. Therefore they propose a new algorithm to compute (2-x) and (3-x)/2 by using the modified Booth algorithm. When applied to the continued-product normalization method, this algorithm can maximize the inherent parallelism of the continued-product normalization method, and reduce computation time by effectively applying pipelining, and also achieve area efficient design by eliminating one register and one carry propagate adder needed for computing (2-x) and (3-x)/2. When the designed unit is used with the seed generator which has the accuracy of 2/sup -7/, division can be executed in eight cycles and the square root operation in 13 cycles.", acknowledgement = ack-nhfb, classification = "B1265B (Logic circuits); C4240P (Parallel programming and algorithm theory); C5120 (Logic and switching circuits); C5230 (Digital arithmetic methods)", fjournal = "Journal of the Korea Information Science Society = Chongbo Kwahakhoe nonmunji", keywords = "Area efficient unit; Continued-product normalization method; Floating-point division; Modified Booth algorithm; Parallel multiplier; Pipelining; Seed generator; Square root", language = "Korean", pubcountry = "South Korea", thesaurus = "Adders; Digital arithmetic; Parallel algorithms", } @Article{Li:1993:CAF, author = "Y. Li and X. Dong and S. Pan", title = "Computation of Auxiliary Functions in {STO} Molecular Integrals up to Arbitrary Accuracy. {I}. {Evaluation} of Incomplete Gamma Function {E$_n$ (X)} by Forward Recursion", journal = j-IJQC, volume = "45", number = "1", pages = "3--??", year = "1993", CODEN = "IJQCB2", ISSN = "0020-7608 (print), 1097-461X (electronic)", ISSN-L = "0020-7608", bibdate = "Wed Jan 3 14:24:13 MST 1996", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/ijqc.bib", acknowledgement = ack-nhfb, fjournal = "International Journal of Quantum Chemistry", journal-URL = "http://www.interscience.wiley.com/jpages/0020-7608/", } @TechReport{Litvinov:1993:ACR, author = "Grigori L. Litvinov", title = "Approximate construction of rational approximations and the effect of error autocorrection", type = "Technical report", number = "8", institution = "Institute of Mathematics, University of Oslo", address = "Oslo, Norway", month = may, year = "1993", bibdate = "Tue Mar 24 20:51:52 2009", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "See \cite{Litvinov:1994:ACR}.", acknowledgement = ack-nhfb, } @Article{Liu:1993:DSC, author = "Hui Min Liu", title = "Determination of several classes of elementary functions by functional inequalities. ({Chinese})", journal = "Hunan Jiaoyu Xueyuan Xuebao (Ziran Kexue)", volume = "11", number = "2", pages = "30--35, 12", year = "1993", ISSN = "1001-6074", MRclass = "26A09 (39B72)", MRnumber = "94g:26003", MRreviewer = "Ling Yau Chan", bibdate = "Sat Jan 11 17:44:01 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, } @InProceedings{Louie:1993:DRS, author = "M. E. Louie and M. D. Ercegovac", booktitle = "Proceedings of the {IEEE} Workshop on {FPGAs} for Custom Computing Machines, 5--7 April 1993", title = "A digit-recurrence square root implementation for field programmable gate arrays", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "178--183", year = "1993", CODEN = "????", ISSN = "????", bibdate = "Sat Jul 16 15:28:14 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, summary = "Creating efficient arithmetic processors requires a pairing of high speed arithmetic algorithms with optimal mapping strategies for a given technology. The authors propose bit reduction as key to an efficient pairing process for lookup table based \ldots{}", } @InProceedings{Lozier:1993:ABF, author = "Daniel W. Lozier and F. W. J. Olver", title = "{Airy} and {Bessel} Functions by Parallel Integration of {ODEs}", crossref = "Sincovec:1993:PSS", pages = "530--538", year = "1993", bibdate = "Fri Jul 09 06:36:27 2004", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, } @Book{Mathai:1993:HGS, author = "A. M. Mathai", title = "A handbook of generalized special functions for statistical and physical sciences", publisher = pub-CLARENDON, address = pub-CLARENDON:adr, pages = "xi + 235", year = "1993", ISBN = "0-19-853595-3", ISBN-13 = "978-0-19-853595-9", LCCN = "QA351 .M35 1993", bibdate = "Sat Oct 30 18:57:40 MDT 2010", bibsource = "http://cat.cisti-icist.nrc-cnrc.gc.ca/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; z3950.loc.gov:7090/Voyager", series = "Oxford science publications", URL = "http://www.loc.gov/catdir/enhancements/fy0635/92036065-d.html; http://www.loc.gov/catdir/enhancements/fy0635/92036065-t.html", acknowledgement = ack-nhfb, subject = "Functions, Special; Handbooks, manuals, etc", tableofcontents = "I Mathematical preliminaries \\ 1.1 The gamma function 1 \\ 1.2 Bernoulli polynomials 6 \\ 1.3 Asymptotic expansions of gamma functions 9 \\ 1.4 The psi function 11 \\ 1.5 The generalized zeta functions 12 \\ 1.6 The beta function 15 \\ 1.7 Calculation of residues for gamma functions 16 \\ 1.8 The Mellin transform 23 \\ 1.9 Density functions 24 \\ 1.10 Methods of deriving distributions 49 \\ 2 The G-function \\ 2.1 The G-function 60 \\ 2.2 Some basic properties of the G-function 69 \\ 2.3 The Mellin transform of a G-function 78 \\ 2.4 Properties connected with the derivatives of a G-function 94 \\ 2.5 Series representations for a G-function 96 \\ 2.6 G-functions as multiple integrals or as solutions of integral equations 106 \\ 2.7 Differential equation for a G-function 111 \\ 2.8 Asymptotic expansions for a G-function 112 \\ 3 Elementary special functions and the G-function \\ 3.1 Gamma and related functions: notations and definitions 117 \\ 3.2 Hypergeometric functions: notations and special cases 118 \\ 3.3 Confluent hypergeometric function and related functions 119 \\ 3.4 Exponential integral and related functions 121 \\ 3.5 Bessel functions and associated functions 121 \\ 3.6 Other special functions 122 \\ 3.7 Orthogonal polynomials 124 \\ 3.8 Elementary special functions expressed in terms of G-functions 127 \\ 3.9 G-functions expressed in terms of elementary special functions 129 \\ 3.10 Some integrals involving G-functions 132 \\ 3.11 The H-function 140 \\ 3.12 Computational aspects of G- and H-functions 144 \\ 3.13 Orders of the special functions for small and large values of the argument 145 \\ 4 Generalizations to matrix variables \\ 4.1 Scalar functions of a symmetric positive definite matrix 152 \\ 4.2 Scalar functions of matrix arguments 158 \\ 4.3 Laplace transform 160 \\ 4.4 Hypergeometric functions of matrix arguments 171 \\ 4.5 Generalized matrix transform or M-transform 177 \\ 4.6 Zonal polynomial 194 \\ 4.7 Matrix variate Dirichlet distribution 197 \\ 4.8 Hypergeometric functions of many scalar variables 205 \\ 4.9 Hypergeometric functions of many matrix arguments 215 \\ 4.10 G- and H-functions of two variables 217 \\ Bibliography 227 \\ Glossary of symbols 231 \\ Author index 233 \\ Subject index 234", } @Article{Mazenc:1993:CFU, author = "Christophe Mazenc and Xavier Merrheim and Jean-Michel Muller", title = "Computing functions $ \cos^{-1} $ and $ \sin^{-1} $ using {CORDIC}", journal = j-IEEE-TRANS-COMPUT, volume = "42", number = "1", pages = "118--122", month = jan, year = "1993", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/12.192222", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Thu Jul 7 07:58:47 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput1990.bib", URL = "https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=192222", acknowledgement = ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", keywords = "Arithmetic; Array signal processing; Concurrent computing; Instruments; Logic arrays; Signal processing algorithms; Silicon compounds; Systolic arrays; Throughput; Very large scale integration", remark = "From the abstract: ``a slight modification of the algorithm enables the computation of the functions $ \acos (x) $, $ \asin (x) $, $ \sqrt {1 - t^2} $, $ \acosh (x) $, $ \asinh (x) $, and $ \sqrt {1 + t^2} $.''", } @InProceedings{McQuillan:1993:NAV, author = "S. E. McQuillan and J. V. McCanny and R. Hamill", title = "New algorithms and {VLSI} architectures for {SRT} division and square root", crossref = "Swartzlander:1993:SCA", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "80--86", year = "1993", CODEN = "????", ISSN = "????", bibdate = "Sat Jul 16 15:28:14 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.acsel-lab.com/arithmetic/arith11/papers/ARITH11_McQuillan.pdf", acknowledgement = ack-sfo # " and " # ack-nhfb, keywords = "ARITH-11", summary = "Radix two algorithms for SRT division and square-rooting are developed. For these schemes, the result digits and the residuals are computed concurrently and the computations in adjacent rows are overlapped. Consequently, their performance should \ldots{}", } @Article{Montuschi:1993:RIT, author = "P. Montuschi and L. Ciminiera", title = "Reducing iteration time when result digit is zero for radix $2$ {SRT} division and square root with redundant remainders", journal = j-IEEE-TRANS-COMPUT, volume = "42", number = "2", pages = "239--246", month = feb, year = "1993", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/12.204797", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Sat Jul 16 15:28:13 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "See remark \cite{Montuschi:1995:RRI}.", acknowledgement = ack-sfo # " and " # ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", summary = "A new architecture is presented for shared radix 2 division and square root whose main characteristic is the ability to avoid any addition/subtraction, when the digit 0 has been selected. The solution presented uses a redundant representation of the \ldots{}", } @TechReport{Morris:1993:NLM, author = "Alfred H. {Morris, Jr.}", title = "{NSWC} Library of Mathematics Subroutines", type = "Report", number = "NSWCDD/TR-92/425", institution = "Naval Surface Warfare Center", address = "Dahlgren, VA 22448-5000, USA; Silver Spring, MD 20903-5000, USA", pages = "xvi + 608", month = jan, year = "1993", bibdate = "Tue Jun 13 08:47:19 2023", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fortran2.bib", note = "See also earlier editions \cite{Morris:1984:NLM,Morris:1987:NLM,Morris:1990:NLM}.", URL = "https://apps.dtic.mil/sti/tr/pdf/ADA261511.pdf; https://ntrl.ntis.gov/NTRL/dashboard/searchResults/titleDetail/ADA261511.xhtml", abstract = "The NSWC library 1s a library of general purpose Fortran subroutines that provide a basic computational capability for a variety of mathematical activities. Emphasis has been placed on the transportability of the codes. Subroutines are available in the following areas: elementary operations, geometry, special functions, polynomials, vectors, matrices, large dense systems of linear equations, banded matrices, sparse matrices, eigenvalues and eigenvectors, $\ell_1$ solution of linear equations, least-squares solution of linear equations, optimization, transforms, approximation of functions, curve fitting, surface fitting, manifold fitting, numerical integration, integral equations, ordinary differential equations, partial differential equations, and random number generation.", acknowledgement = ack-nhfb, pdfpages = "463", remark = "[10-Jan-2026] Despite several Web searches, a machine-readable freely downloadable copy of this report's associated software has not yet been located, but the report itself was found today in the DTIC archives. The entries for the earlier editions [Morris:1987:NLM,Morris:1990:NLM] have links to a PDF file and the Fortran 90 source code. Another entry [Miller:2004:AMF] has links to some of the source code, without indication of software version dates. There is a large page gap in the PDF file: 624 numbered pages, but only 463 are present.", tableofcontents = "Introduction / 1 Elementary Operations Machine Constants --- SPMPAR, DPMPAR, IPMPAR / 3 Argument Bounds for the Exponential Function --- EPSLN, EXPARG, DEPSLN, DXPARG / 5 Sorting Lists --- ISHELL, SHELL, AORD, RISORT, SHELL2, DSORT, DAORD, DISORT, DDSORT, QSORTI, QSORTR, QSORTD, JORDER, RORDER, DORDER / 7 Cube Root --- CBRT, DCBRT / 11 Four Quadrant Arctangent --- ARTNQ, DARTNQ / 11 Length of a Two Dimensional Vector --- CPABS, DCPABS / 11 Reciprocal of a Complex Number --- CREC, DCREC / 13 Division of Complex Numbers --- 13 Square Root of a Double Precision Complex Number --- DCSQRT / 13 Conversion of Polar to Cartesian Coordinates --- POCA / 15 Conversion of Cartesian to Polar Coordinates --- CAPO / 15 Rotation of Axes --- ROTA / 15 Planar Givens Rotations --- SROTG, DROTG / 17 Three Dimensional Rotations --- ROT3 / 19 Rotation of a Point on the Unit Sphere to the North Pole --- CONSTR / 21 Computation of the Angle Between Two Vectors --- ANG / 23 Trigonometric Functions --- SIN1, COS1, DSIN1, DCOSI / 25 Hyperbolic Sine and Cosine Functions --- SNHCSH / 27 Exponentials --- REXP, DREXP / 29 Logarithms --- ALNREL, RLOC, RLOG1, DLNREL, DRLOG, DRLOG1 / 31 Geometry Determining if a Point is Inside or Outside a Polygon --- LOCPT / 33 Intersection of a Straight Line and Polygonal Path --- PFIND / 35 The Convex Hull for a Infinite Planar Set --- HULL / 37 Areas of Planar Polygons --- PAREA / 39 Hamiltonian Circuits --- HC / 41 Special Functions Error Function --- CERF, CERFC, ERF, ERFC, ERFCI1, DCERF, DCERFC, DCERFC1 / 45 Inverse Error Function --- ERFI, DERFI / 51 Difference of Error Functions --- AERF, DAERF / 53 Normal Probability Distribution Function --- PNDEF / 55 Inverse Normal Probability Distribution Function --- PNI, DPNI/ 57 Dawson's Integral --- DAW, DPDAW / 59 Complex Fresnel Integral --- CFRNLI / 61 Real Fresnel Integrals --- FRNL / 63 Exponential Integral Function --- CEXPLI, EXPLI, DEI, DEI1 / 65 Sine and Cosine Integral Functions --- SI, CIN / 69 Exponential Integral Function --- CEXEXI / 71 Dilogarithm Function --- CLI, ALI / 72 Gamma Function --- CGAMMA GAMMA, GAMLN, DCGAMA, DGAMMA, DGAMLN / 15 Digamma Function --- CPSI, PSI, DCPSI, DPSI / 19 Derivatives of the Digamma Function --- PSIDF / 81 Incomplete Gamma Ratio Functions --- GRATIO, RCOMP, DGRAT, DRCOMP / 83 Inverse Incomplete Gamma Ratio Function --- GAMINV, DGINV / 85 Logarithm of the Beta Function --- BETALN, DBETLN / 87 Incomplete Beta Function --- BRATIO, ISUBX, BRCOMP / 89 Bessel Function $J_\nu(z)$ --- CBSSLJ, BSSLJ, BESI / 91 Bessel Function $Y_\nu(z)$ --- BSSLY / 93 Modified Bessel Function $I_\nu(z)$ --- CBSSLI, BSSLI, BESI / 95 Modified Bessel Function $K_\nu(z)$ --- CBESK, CBSSLK, BSSLK / 97 Airy functions --- CAI, CBI, AI, AIE, BI, BIE / 99 Complete Complex Elliptic Integrals of the First and Second Kinds --- CK, CKE / 103 Real Elliptic Integrals of the First and Second Kinds --- ELLPI, RFVAL, RDVAL, DELLPI, DRFVAL, DRDVAL / 107 Real Elliptic Integrals of the Third Kind --- EPI, RJVAL, DEPI, DRJVAL / 111 Jacobian Elliptic Functions --- ELLPF, ELPFC1 / 115 Weierstrass Elliptic Function for the Equianharmonic and Lemniscatic Cases --- PEQ, PEQ1, PLEM, PLEM1 / 119 Integral of the Bivariate Density Function over Arbitrary Polygons and Semi-infinite Angular Regions --- VALR2 / 123 A Circular Coverage Function --- CIRCV / 125 Elliptical Coverage Function --- PKILL / 127 Polynomials Copying Polynomials --- PLCOPY, DPCOPY / 129 Addition of Polynomials --- PADD, DPADD / 131 Subtraction of Polynomials --- PSUBT, DPSUBT / 133 Multiplication of Polynomials --- PMULT, DPMULT / 135 Division of Polynomials --- PDIV, DPDIV / 137 Real Powers of Polynomials --- PLPWR, DPLPWR / 139 Inverses of Power Series --- PINV, DPINV / 141 Derivatives and Integrals of Polynomials --- MPLNMV / 143 Evaluation of Chebyshev Expansions --- CSEVL, DCSEVL / 145 Lagrange Polynomials --- LGRNGN, LGRNGV, LGRNGX / 147 Orthogonal Polynomials on Finite Sets --- ORTHOS, QRTHOV, ORTHOX / 149 Solutions of Nonlinear Equations Zeros of Continuous Functions --- ZEROIN, DZERO / 151 Solution of Systems of Nonlinear Equations --- HBRD / 153 Solutions of Quadratic, Cubic, and Quartic Equations --- QDCRT, CBCRT, QTCRT, DQDCRT, DCBCRT, DQTCRT / 155 Double Precision Roots of Polynomials --- DRPOLY, DCPOLY / 157 Accuracy of the Roots of Polynomials --- RBND, CBND / 159 Vectors Copying Vectors --- SCOPY, DCOPY, CCOPY / 181 Interchanging Vectors --- SSWAP, DSWAP, CSWAP / 163 Planar Rotation of Vectors --- SROT, DROT, CSROT / 165 Modified Givens Rotations --- SROTMG, DROTMG, SKROTM, DROTM / 167 Dot Products of Vectors --- SDOT, DDOT, CDOTC, CDOTU / 171 Scaling Vectors --- SSCAL, DSCAL, CSCAL, CSSCAL / 173 Vector Addition --- SAXPY, DAXPY, CAXPY / 175 Norm of a Vector --- SASUM, DASUM, SCASUM / 177 Norm of a Vector --- SNRM2, DNRM2, SCNRM2 / 179 Norm of a Vector --- ISAMAX, IDAMAX, ICAMAX / 181 Matrices Packing and Unpacking Symmetric Matrices --- MCVFS, DMCVFS, MCVSF, DMCVSF / 183 Conversion of Real Matrices to and from Double Precision Form --- MCVRD, MCVDR / 185 Storage of Real Matrices in the Complex Matrix Format --- MCVRC / 187 The Real and Imaginary Parts of a Complex Matrix --- CMREAL, CMIMAG / 189 Copying Matrices --- MCOPY SMCOPY, DMCOPY, CMCOPY / 181 Computation of the Conjugate of a Complex Matrix --- CMCONJ / 193 Transposing Matrices --- TPOSE, DTPOSE, CTPOSE, TIP, DTIP, CTIP / 195 Computing Adjoints of Complex Matrices --- - CMADJ, CTRANS / 197 Matrix Addition --- MADD, SMADD, DMADD, CMADD / 199 Matrix Subtraction --- MSUBT, SMSUBT, DMSUBT, CMSUBT / 201 SAT Matrix Multiplication --- MTMS, DMTMS, CMTMS MPROD, DMPROD, CMPROD / 203 Product of a Packed Symmetric Matrix and a Vector --- SVPRD, ESVPRD / 205 Transpose Matrix Products --- TMPROD / 207 Symmetric Matrix Products --- SMPROD / 209 Kronecker Product of Matrices --- KPROD, DKPROD, KPROI (???) / 211 Rank of a Real Matrix --- RNK, DRNK / 213 Inverting General Real Matrices and Solving General Systems of Real Linear Equations --- CROUT, KROUT, NPIVOT, MSLV, DMSLV, MSLV1, DMSLV1 / 215 Solution of Real Equations with Iterative Improvement --- SLVMP / 221 Solution of Almost Block Diagonal Systems of Linear Equations --- ARCECO, ARCESL / 223 Solution of Almost Block Tridiagonal Systems of Linear Equations --- BTSLV / 225 Inverting Symmetric Real Matrices and Solving Symmetric Systems of Real Linear Equations --- SMSLV, DSMSLV / 227 Inverting Positive Definite Symmetric Matrices and Solving Positive Definite Symmetric Systems of Linear Equations --- PCHOL, DPCHOL / 231 Solution of Toeplitz Systems of Linear Equations --- TOPLX, DTOPLX / 233 Inverting General Complex Matrices and Solving General Systems of Complex Linear Equations --- CMSLV, CMSLV1, DCMSLV 235 Solution of Complex Equations with Iterative Improvement --- CSLVMP / 239 Singular Value Decomposition of a Matrix --- SSVDC, DSVDC, CSVDC / 241 Evaluation of the Characteristic Polynomial of a Matrix --- DET, DPDET, CDET / 243 Solution of the Matrix Equation $A X + X B = C$ --- ABSLV, DABSLV / 245 Solution of the Matrix Equation $A^T X + X A = C$ when $C$ is Symmetric --- TASLV, DTASLV / 247 Solution of the Matrix Equation $A X^2 + B X + C = 0$ --- SQUINT / 249 Exponential of a Real Matrix --- MEXP, DMEXP / 251 Large Dense Systems of Linear Equations Solving systems of 200--400 Linear Equations --- LE, DPLE, CLE / 253 Banded Matrices Band Matrix Storage / 255 Conversion of Banded Matrices to and from the Standard Format --- CVBR, CVBD, CVBC, CVRB, CVDB, CVCB, CVRB1, CVDB1, CVCB1 / 257 Conversion of Banded Matrices to and from Sparse Form --- MCVBS, DMCVBS, CMCVBS MCVSB, DMCVSB, CMCVSB / 259 Conversion of Banded Real Matrices to and from Double Precision Form --- BCVRD, BCVDR / 261 The Real and Imaginary Parts of a Banded Complex Matrix --- BREAL, BIMAG / 263 Computing $A + B^%$ for Banded Real Matrices $A$ and $B$ --- BCVRC / 265 Transposing Banded Matrices --- BPOSE, DBPOSE, CBPOSE / 267 A Addition of Banded Matrices --- BADD, DBADD, CBADD / 269 RE Subtraction of Banded Matrices --- BSUBT, DBSUBT, CBSURT / 271 ER Multiplication of Banded Matrices --- BPROD, DBPROD, CBPROL / 273 Product of a Real Banded Matrix and Vector --- BVPRD, BVPRDI, BTPRD, BTPRD1 / 275 Product of a Double Precision Banded Matrix and Vector --- DBVPD, DBVPD1, DBTPD, DBTPD1 / 277 Product of a Complex Banded Matrix and Vector --- CBVPD1, CBTPD, CBTPD1 / 279 $L_1$ Norm of a Real Banded --- B1NRM, DB1NRM / 281 $L_\infty$ Norm of a Real Banded --- BNRM, DBNRM / 283 Solution of Banded Systems of Real Linear Equations --- BSLV, BSLV1 / 285 Computation of the Condition Number of a Real Banded Matrix --- DB1CND / 287 Double Precision Solution of Banded Systems of Real Linear Equations --- DBSLV, DBSLVI1 / 289 Computation of the Condition Number of a Double Precision Banded Matrix --- DBICND / 201 Solution of Banded Systems of Complex Linear Equations --- CBSLV, CBSLV1 / 208 Sparse Matrices Storage of Sparse Matrices / 290 Conversion of Sparse Matrices to and from the Standard Format --- CVRS, CVDS, CVCS, CVSR, CVSD, CVSC / 207 Conversion of Sparse Real Matrices to and from Double Precision Form --- SCVRD, SCVDR / 299 The Real and Imaginary Parts of a Sparse Complex Matrix --- CSREAL, CSIMAG / 301 Computing $A + B^T$ for Sparse Real Matrices $A$ and $B$ --- SCVRC / 303 Copying Sparse Matrices --- RSCOPY, DSCOPY, CSCOPY / 305 Computing Conjugates of Sparse Complex Matrices --- SCONJ / 307 Transposing Sparse Real Matrices --- RPOSE, RPOSEL / 309 Transposing Sparse Double Precision Matrices --- DPOSE, DPOSE1 / 311 Transposing Sparse Complex Matrices --- CPOSE, CPOSEL / 313 Addition of Sparse Matrices --- SADD, DSADD, CSADD / 315 Subtraction of Sparse Matrices --- SSUBT, DSSUBT, CSSUBT / 317 Multiplication of Sparse Matrices --- SPROD, DSPROD, CSPROD / 319 Product of a Real Sparse Matrix and Vector --- MVPRD, MVPRDI, MTPRD, MTPRD1 / 321 Product of a Double Precision Sparse Matrix and Vector --- DVPRD, DVPRD1, DTPRD, DTPRD1 / 323 Product of a Complex Sparse Matrix and Vector --- CVPRD, CVPRD1, CTPRD, CTPRD1 / 325 $L_1$ Norm of a Sparse Real Matrix --- SINRM, DSINRM / 327 $L_\infty$ Norm of a Sparse Real Matrix --- SNRM, DSNRM / 329 Ordering the Rows of a Sparse Matrix by Increasing Length --- SPORD / 331 Reordering Sparse Matrices into Block Triangular Form --- BLKORD / 333 Solution of Sparse Systems of Real Linear Equations --- SPSLV, RSLV, TSLV / 335 Computation of the Condition Number of a Real Sparse Matrix --- S1CND / 339 Double Precision Solution of Sparse Systems of Real Linear Equations --- DSPSLV, DSLV, DTSLV / 341 Computation of the Condition Number of a Double Precision Sparse Matrix --- DSICND / 345 Solution of Sparse Systems of Complex Linear Equations --- CSPSLV, CSLV, CTSLV / 347 Eigenvalues and Eigenvectors Computation of Eigenvalues of General Real Matrices --- EIG, EIG1 / 351 Computation of Eigenvalues and Eigenvectors of General Real Matrices --- EIGV, EIGV1 / 353 Double Precision Computation of Eigenvalues of Real Matrices --- DEIG / 355 Double Precision Computation of Eigenvalues and Eigenvectors of Real Matrices --- DEIGV / 357 Computation of Eigenvalues of Symmetric Real Matrices --- SEIG, SEIG1 / 359 Computation of Eigenvalues and Eigenvectors of Symmetric Real Matrices --- SEIGV, SEIGV1 / 361 Double Precision Computation of Eigenvalues of Symmetric Real Matrices --- DSEIG / 363 Double Precision Computation of Eigenvalues and Eigenvectors of Symmetric Real Matrices --- DSEIGV / 365 Computation of Eigenvalues of Complex Matrices --- CEIG / 367 Computation of Eigenvalues and Eigenvectors of Complex Matrices --- CEIGV / 369 Double Precision Computation of Eigenvalues of Complex Matrices --- DCEIG / 371 Double Precision Computation of Eigenvalues and Eigenvectors of Complex Matrices --- DCEIGV / 373 $\ell_1$ Solution of Linear Equations Solution of Systems of Linear Equations with Equality and Inequality Constraints --- CL1 / 375 Least Squares Solution of Linear Equations Least Squares Solution of Systems of Linear Equations --- LLSQ, LSQR, HFTI, HFTI2 / 377 Least Squares Solution of Overdetermined Systems of Linear Equations with Iterative Improvement --- LLSQMP / 383 Double Precision Least Squares Solution of Systems of Linear Equations --- DLLSQ, DLSQR, DHFTI, DHFTI2 / 385 Least Squares Solution of Systems of Linear Equations with Equality and Inequality Constraints --- LSEI / 391 Least Squares Solution of Systems of Linear Equations with Equality and Nonnegativity Constraints --- WNNLS / 395 Least Squares Iterative Improvement Solution of Systems of Linear Equations with Equality Constraints --- L2SLV / 399 Iterative Least Squares Solution of Banded Linear Equations --- BLSQ / 403 Iterative Least Squares Solution of Sparse Linear Equations --- SPLSQ, STLSQ / 405 Optimization Minimization of Functions of a Single Variable --- FMIN / 407 Minimization of Functions of $n$ Variables --- OPTF / 409 Unconstrained Minimum of the Sum of Squares of Nonlinear Functions --- LMDIFF / 411 Linear Programming --- SMPLX, SSPLX / 413 The Assignment Problem --- ASSGN / 417 A $0$--$1$ Knapsack Problem --- MKP / 419 Transforms Inversion of the Laplace Transform --- LAINV / 421 Fast Fourier Transform --- FFT, FFT1 / 425 Multivariate Fast Fourier Transform --- MFFT, MFFT1 / 427 Discrete Cosine and Sine Transforms --- COSQI, COSQB, COSQF, SINQB, SINQF / 429 Approximation of Functions Rational Minimax Approximation of Functions --- CHEBY / 433 Approximation of Functions --- ADAPT / 435 Calculation of the Taylor Series of a Complex Analytic Function --- CPSC, DCPSC / 439 Curve Fitting Linear Interpolation --- TRP / 443 Lagrange Interpolation --- LTRP / 445 Hermite Interpolation --- HTRP / 447 Conversion of Real Polynomials from Newton to Taylor Series Form --- PCOEFF / 449 Least Squares Polynomial Fit --- PFIT / 451 Weighted Least Squares Polynomial Fit --- WPFIT / 453 Cubic Spline Interpolation --- CBSPL, SPLIFT / 455 Weighted Least Squares Cubic Spline Fitting --- SPFITT / 457 Least Squares Cubic Spline Fitting with Equality and Inequality Constraints --- CSPFIT / 459 Cubic Spline Evaluation --- SCOMP, SCOMP1, SCOMP2 / 461 Cubic Spline Evaluation and Differentiation--- SEVAL, SEVAL1, SEVAL2 / 463 Integrals of Cubic Splines --- CSINT, CSINT1, CSINT2 / 465 Periodic Cubic Spline Interpolation --- PDSPL / 467 Least Squares Periodic Cubic Spline Fitting --- PDFIT / 469 Periodic Cubic Spline Evaluation and Differentiation --- PSCMP, PSEVL / 471 $N$-Dimensional Cubic Spline Closed Curve Fitting --- CSLOOP, LOPCMP, LOPDF / 473 Spline under Tension Interpolation --- CURV1 / 475 Spline under Tension Evaluation --- CURV2 / 477 Differentiation and Integration of Splines under Tension --- CURVD, CURVI / 479 Two Dimensional Spline under Tension Curve Fitting --- KURV1, KURV2 / 481 Two Dimensional Spline under Tension Closed Curve Fitting --- KURVP1, KURVP2 / 483 Three Dimensional Spline under Tension Curve Fitting --- QURV1, QURV2 / 485 B-Splines / 487 Finding the Interval that Contains a Point --- INTRVL / 489 Evaluation and Differentiation of Piecewise Polynomials from their B-Spline Representations --- BVAL / 491 Evaluation of the Indefinite Integral of a Piecewise Polynomial from its B-Spline Representation --- BVAL1 / 493 Conversion of Piecewise Polynomials from B-Spline to Taylor Series Form --- BSPP / 495 Evaluation of Piecewise Polynomials from their Taylor Series Representation --- PPVAL / 497 Piecewise Polynomial Interpolation --- BSTRE / 499 Weighted Least Squares Piecewise Polynomial Fitting --- BSLSQ / 501 Least Squares Piecewise Polynomial Fitting with Equality and Inequality Constraints --- BFIT / 503 Surface Fitting over Rectangular Grids Bicubic Splines and Bisplines under Tension / 505 Weighted Least Squares Bicubic Spline Fitting --- SPFET2 / 507 Evaluation and Differentiation of Bicubic Splines --- CSURF, CSURF1, CSRF, CSRF2 / 509 Bispline under Tension Surface Interpolation --- SURF / 513 Bispline under Tension Evaluation --- SURF2, NSURF2 / 515 Bivariate B-Spline Piecewise Polynomial Interpolation --- BSTRP2 / 517 Bivariate B-Spline Piecewise Polynomial Least Squares Fitting --- RSLSQ2 / 519 Evaluation and Differentiation of Bivariate Piecewise Polynomials from their B-Spline Representation --- BVAL2 / 521 Surface Fitting over Arbitrarily Positioned Data Points Surface Interpolation for Arbitrarily Positioned Data Points --- TRMESH, GRADG, GRADL, SFVAL, SFVAL2 / 523 Manifold Fitting Weighted Least Squares Fitting with Polynomials of $n$ Variables --- MFIT, DMFIT, MEVAL, DMEVAL / 527 Numerical Integration Evaluation of Integrals over Finite Intervals --- QAGS, QXGS, QSUBA, DQAGS, DQXGS / 531 Evaluation of Integrals over Infinite Intervals --- QAGI, DQAGI / 539 Evaluation of Double Integrals over Triangles --- CUBTRI / 543 Integral Equations Solution of Fredholm Integral Equations of the Second Kind --- IESLV / 545 Ordinary Differential Equations / Initial Value Problems The Initial Value Solvers --- Introductory Comments / 549 Adaptive Adams Solution of Nonstiff Differential Equations --- ODE / 551 Adaptive Block RKF Solution of Nonstiff Differential Equations --- BRKF45 / 555 Adaptive RKF Solution of Nonstiff Differential Equations --- RKF45 / 559 Adaptive RKF Solution of Nonstiff Differential Equations with Global Error Estimation --- GERK / 563 Adaptive Solution of Stiff Differential Equations --- SFODE, SFODE1 / 567 Fourth-Order Runge-Kutta --- RK / 571 Eighth-Order Runge-Kutta --- RK8 / 573 Partial Differential Equations Separable Second-Order Elliptic Equations on Rectangular Domains --- SEPDE / 575 Discrete Random Number Generation Uniform Random Selection of Values from a Finite Set of Integers --- URGET / 579 Continuous Random Number Generation Uniform Random Number Generator --- URNG, DURNG 581 Generating Points Uniformly in a Square --- URNG2, DURNG2 / 583 Generating Points Uniformly in a Circle --- RCIR, DRCIR / 585 Normal Random Number Generator --- RNOR, DRNOR, NRNG, DNRNG / 587 Multivariate Normal Random Vector Generator --- NRVG, DNRVG, NRVG1, DNRVG1 / 589 Exponential Random Number Generator --- RANEXP, DRNEXP / 593 Gamma Random Number Generator and the Chi-Square Distribution --- RGAM, DRGAM / 595 Beta Random Number Generator --- RBETA, DRBETA / 597 F-Distribution Random Number Generator --- FRAN, DFRAN / 599 Student $t$-Distribution Random Number Generator --- TRAN, DTRAN / 601 First Order Markov Random Number Generator --- RMK1, DRMK1 / 603 Appendix. Installation of the NSWC Library and Conversion of Codes from Single to Double Precision Form / 605 Index / 607", } @Article{Muller:1993:NAC, author = "J{\"u}rgen M{\"u}ller", title = "On numerical analytic continuation and convergence acceleration by summability methods", journal = "Analysis", volume = "13", number = "3", pages = "279--291", year = "1993", ISSN = "0174-4747", MRclass = "40G10 (40A30 41A25 65B10)", MRnumber = "1245757 (94j:40013)", MRreviewer = "S. Sridhar", bibdate = "Thu Dec 1 17:36:45 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Analysis. International Mathematical Journal of Analysis and its Applications", keywords = "convergence acceleration", } @Article{Perger:1993:NEG, author = "Warren F. Perger and Atul Bhalla and Mark Nardin", title = "A numerical evaluator for the generalized hypergeometric series", journal = j-COMP-PHYS-COMM, volume = "77", number = "2", pages = "249--254", month = oct, year = "1993", CODEN = "CPHCBZ", DOI = "https://doi.org/10.1016/0010-4655(93)90008-Z", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", bibdate = "Thu Dec 01 09:22:29 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/compphyscomm1990.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Computer Physics Communications", journal-URL = "http://www.sciencedirect.com/science/journal/00104655", remark = "Uses extended precision complex arithmetic.", } @Article{Petkovic:1993:SII, author = "Miodrag S. Petkovi{\'c} and Carsten Carstensen", title = "Some improved inclusion methods for polynomial roots with {Weierstrass}' corrections", journal = j-COMPUT-MATH-APPL, volume = "25", number = "3", pages = "59--67", month = feb, year = "1993", CODEN = "CMAPDK", ISSN = "0898-1221 (print), 1873-7668 (electronic)", ISSN-L = "0898-1221", bibdate = "Wed Mar 1 19:11:11 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/computmathappl1990.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/089812219390143J", acknowledgement = ack-nhfb, fjournal = "Computers and Mathematics with Applications", journal-URL = "http://www.sciencedirect.com/science/journal/08981221", } @Book{Povolotskii:1993:OFR, author = "A. I. Povolotski{\u\i} and G. A. Sviridyuk", title = "{{\cyr Odnomerny{\u\i} matematicheski{\u\i} analiz {\`e}lementarnykh funktsi{\u\i}}}. ({Russian}) [One-dimensional mathematical analysis of elementary functions] {{\cyr Nepreryvnye funktsii. Differentsiruemye funktsii. Integriruemye funktsii}}. [Continuous functions. Differentiable functions. Integrable functions]", publisher = "Chelyabinsk. Gos. Univ.", address = "Chelyabinsk, USSR", pages = "92", year = "1993", ISBN = "5-230-17764-0", ISBN-13 = "978-5-230-17764-7", MRclass = "26-01 (00A05)", MRnumber = "94e:26001", MRreviewer = "J{\'o}zef Kalinowski", bibdate = "Sat Jan 11 17:44:01 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, language = "Russian", } @Article{Ratis:1993:CCH, author = "Yu. L. Ratis and P. Fern{\'a}ndez de C{\'o}rdoba", title = "A code to calculate (high order) {Bessel} functions based on the continued fractions method", journal = j-COMP-PHYS-COMM, volume = "76", number = "3", pages = "381--388", month = aug, year = "1993", CODEN = "CPHCBZ", DOI = "https://doi.org/10.1016/0010-4655(93)90062-H", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", bibdate = "Mon Feb 13 21:29:39 MST 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/compphyscomm1990.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/001046559390062H", abstract = "We have developed a fast code to calculate Bessel functions of integer and fractional order based on the continued fractions method. This algorithm is specially useful in the case of Bessel functions of high order because it does not require any recalculation using normalization relations.", acknowledgement = ack-nhfb, fjournal = "Computer Physics Communications", journal-URL = "http://www.sciencedirect.com/science/journal/00104655", } @Book{Saurer:1993:BSF, author = "Josef Saurer", title = "Bases of special functions and their domains of convergence", volume = "73", publisher = "Akademie Verlag GmbH", address = "Berlin, Germany", pages = "158", year = "1993", ISBN = "3-05-501613-0", ISBN-13 = "978-3-05-501613-4", ISSN = "0138-3019", LCCN = "QA351 .S28 1993", bibdate = "Sat Oct 30 18:53:24 MDT 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; z3950.loc.gov:7090/Voyager", series = "Mathematical research", acknowledgement = ack-nhfb, remark = "Revised version of the author's thesis--Universit{\"a}t Essen, 1992.", subject = "Functions, Special; Analytic functions; Eigenfunction expansions; Convergence; Mathematical physics", tableofcontents = "Introduction 9 \\ 1 Foundations of the theory 15 \\ 1.1 Holomorphic operator functions in Frechet spaces 15 \\ 1.2 Floquet eigenvalue problems (with a regular singular point) 22 \\ 1.3 Relationships between differential operators corresponding to Floquet eigenvalue problems for systems of differential equations and scalar differential equations 26 \\ 1.4 Biholomorphic images of functions 30 \\ 1.5 An expansion theorem 37 \\ 2 First order differential systems with a regular singular point 43 \\ 2.1 Fundamental properties 44 \\ 2.2 Construction of fundamental systems depending holomorphically on parameters 46 \\ 2.3 A family of second order differential equations 55 \\ 2.4 Differential equations and special functions of mathematical physics 57 \\ 2.4.1 Bessel equation, Bessel function 57 \\ 2.4.2 Whittaker equation, Whittaker function 58 \\ 2.4.3 Hypergeometric equation, hypergeometric function 60 \\ 2.4.4 Generalised spherical function 61 \\ 3 Floquet eigenvalue problems for first order differential systems with a regular singular point 63 \\ 3.1 Construction of biorthogonal canonical systems of eigen- and associated vectors of the operator functions $T$ and $T^*$ 67 \\ 3.2 A general expansion theorem 73 \\ 3.3 Floquet eigenvalue problems and expansion theorems for a family of second order differential equations 75 \\ 4 Domains of convergence of the eigenfunction expansions 83 \\ 4.1 The Bessel and Whittaker case 85 \\ 4.2 The hypergeometric case 95 \\ 4.3 Typical domains of convergence 105 \\ 5 Examples of expansions in series of special functions 109 \\ 5.1 Expansions in series of Bessel functions 109 \\ 5.2 Expansions in series of Whittaker functions 115 \\ 5.3 Expansions in series of hypergeometric functions 119 \\ 6 First order differential systems for products of vector-valued functions 127 \\ 6.1 Products of vectors and sums of matrices of different dimensions 128 \\ 6.2 Construction of the first order differential system 132 \\ 7 Floquet eigenvalue problems and expansions in series, of m - fold products of special functions 135 \\ 7.1 Construction of biorthogonal canonical systems of eigen- and associated vectors of the operator functions $T$ and $T^*$ 142 \\ 7.2 Application 145 \\ References 151 \\ Notation index 154 \\ Index 157", } @InProceedings{Schulte:1993:ERC, author = "M. Schulte and E. Swartzlander", title = "Exact rounding of certain elementary functions", crossref = "Swartzlander:1993:SCA", bookpages = "xii + 284", pages = "138--145", year = "1993", bibdate = "Thu Dec 14 11:25:18 MST 1995", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://mesa.ece.wisc.edu/publications/cp_1993-01.pdf", abstract = "An algorithm is described which produces exactly rounded results for the functions of reciprocal, square root, 2/sup x/, and log 2/sup x/. Hardware designs based on this algorithm are presented for floating point numbers with 16- and 24-b significands. These designs use a polynomial approximation in which coefficients are originally selected based on the Chebyshev series approximation and are then adjusted to ensure exactly rounded results for all inputs. To reduce the number of terms in the approximation, the input interval is divided into subintervals of equal size and different coefficients are used for each subinterval. For floating point numbers with 16-b significands, the exactly rounded value of the function can be computed in 51 ns on a 20-mm/sup 2/ chip. For floating point numbers with 24-b significands, the functions can be computed in 80 ns on a 98-mm/sup 2/ chip.", acknowledgement = ack-nhfb, affiliation = "Dept. of Electr. and Comput. Eng., Texas Univ., Austin, TX, USA", classification = "C4120 (Functional analysis); C5230 (Digital arithmetic methods)", confdate = "29 June--2 July 1993", conflocation = "Windsor, Ont., Canada", confsponsor = "IEEE Comput. Soc.; IEEE Tech. Committee on VLSI; Natural Sci. and Eng. Res.; Council of Canada", keywords = "Elementary functions; Exact rounding; Floating point numbers; Polynomial approximation; Reciprocal; Rounded results; Square root", thesaurus = "Floating point arithmetic; Function evaluation", } @Article{Schulte:1993:PHD, author = "Michael J. Schulte and Earl E. {Swartzlander, Jr.}", title = "Parallel hardware designs for correctly rounded elementary functions", journal = j-INTERVAL-COMP, volume = "4", pages = "65--88", year = "1993", ISSN = "0135-4868", MRclass = "65G10 (65C20)", MRnumber = "1 305 859", bibdate = "Sat Jan 11 17:44:01 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "Proceedings of the International Conference on Numerical Analysis with Automatic Result Verification (Lafayette, LA, 1993)", acknowledgement = ack-nhfb, fjournal = "Interval Computations = Interval'nye vychisleniia", } @TechReport{Schwarz:1993:HRAa, author = "E. Schwarz", title = "High-radix algorithms for high-order arithmetic operations", type = "Technical Report", number = "CSL-TR-93-559", institution = "Computer Systems Laboratory, Stanford University", address = "Stanford, CA, USA", month = jan, year = "1993", bibdate = "Thu Apr 2 08:38:35 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-sfo # " and " # ack-nhfb, } @PhdThesis{Schwarz:1993:HRAb, author = "Eric Mark Schwarz", title = "High-radix algorithms for high-order arithmetic operations", type = "Thesis ({Ph.D.})", school = "Department of Electrical Engineering, Stanford University", address = "Stanford, CA, USA", pages = "224", month = apr, year = "1993", bibdate = "Mon Jan 07 22:38:06 2008", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", abstract = "Many common algorithms for high-order arithmetic operations require an initial approximation. The Newton--Raphson algorithm starts with an approximation and then quadratically converges on the solution. The initial approximation determines the number of iterations of the algorithm and is typically implemented as a look-up table in the form of a ROM or PLA. A novel method is suggested which describes high-order arithmetic operations with a partial product array. This method applies to the operations of division, reciprocal, square root, natural logarithm, exponential, and trigonometric functions. The partial product array of Boolean elements which describes the operation can be summed on an existing floating-point multiplier. The hardware needed is only the logic gates to create the Boolean elements in the array and a multiplexor, and the latency is that of the multiplier. Thus, by reusing a floating-point multiplier, a high-precision approximation to a high-order arithmetic operation can be implemented with a low marginal cost.\par This dissertation describes the implementation and shows a method for deriving partial product arrays to approximate arithmetic operations. Then the proposed method is applied and evaluated for several operations. The proposed method yields a minimum approximation of twelve bits correct for the reciprocal operation and sixteen bits for the square root operation. The proposed method is shown to be as small as 0.05\% the size (in gates) of an equivalent precision look-up table and has up to four times the accuracy (in bits) as an equivalent latency polynomial approximation. Also, three new iterative algorithms to increase the precision of the approximations and a theoretical analysis of the partial product array representation are detailed. Thus, high-radix algorithms of many arithmetic operations are possible at low cost.", acknowledgement = ack-nhfb, keywords = "division; elementary functions; exponential; logarithm; PPA (partial product array); reciprocal square root; square root", remark = "AAT 9317816. ProQuest document ID 746798521.", } @InProceedings{Schwarz:1993:HSA, author = "E. M. Schwarz and M. J. Flynn", title = "Hardware starting approximation for the square root operation", crossref = "Swartzlander:1993:SCA", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "103--111", year = "1993", CODEN = "????", ISSN = "????", bibdate = "Sat Jul 16 15:28:14 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-sfo # " and " # ack-nhfb, summary = "A method for obtaining high-precision approximations of high-order arithmetic operations is presented. These approximations provide an accurate starting approximation for high-precision iterative algorithms, which translates into few iterations and \ldots{}", } @TechReport{Schwarz:1993:UFM, author = "Eric Mark Schwarz and M. J. (Michael J.) Flynn", title = "Using a floating-point multiplier's internals for high-radix division and square root", type = "Technical report", number = "CSL-TR-93-554", institution = "Computer Systems Laboratory, Stanford University", address = "Stanford, CA, USA", pages = "iv + 45", year = "1993", bibdate = "Sat Feb 24 15:01:45 MST 1996", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, keywords = "Computer arithmetic.", remark = "``January 1993.'' Abstract: ``A method for obtaining high-precision approximations of high-order arithmetic operations at low-cost is presented in this study. Specifically, high-precision approximations of the reciprocal (12 bits worst case) and square root (16 bits) operations are obtained using the internal hardware of a floating-point multiplier without the use of look-up tables. The additional combinatorial logic necessary is very small due to the reuse of existing hardware. These low-cost high-precision approximations are used by iterative algorithms to perform the operations of division and square root. The method presented also applies to several other high-order arithmetic operations. Thus, high-radix algorithms for high-order arithmetic operations such as division and square root are possible at low-cost.''", } @Article{Sellers:1993:CDC, author = "H. Sellers", title = "The {C$^2$-DIIS} Convergence Acceleration Algorithm", journal = j-IJQC, volume = "45", number = "1", pages = "31--??", year = "1993", CODEN = "IJQCB2", ISSN = "0020-7608 (print), 1097-461X (electronic)", ISSN-L = "0020-7608", bibdate = "Wed Jan 3 14:24:13 MST 1996", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "International Journal of Quantum Chemistry", journal-URL = "http://www.interscience.wiley.com/jpages/0020-7608/", keywords = "convergence acceleration", } @Article{Shishkov:1993:RDB, author = "Dimit{\cdprime}r Shishkov", title = "Reduction of domains of basic elementary functions to arbitrary small intervals", journal = "Annuaire Univ. Sofia Fac. Math. Inform.", volume = "87", number = "1--2", pages = "3--32 (1999)", year = "1993", ISSN = "0205-0808", MRclass = "65D20", MRnumber = "MR1745336", bibdate = "Wed Apr 13 06:46:35 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Godishnik na Sofi\u\i skiya Universitet ``Sv. Kliment Okhridski''. Fakultet po Matematika i Informatika. Annuaire de l'Universit{\'e} de Sofia ``St. Kliment Ohridski''. Facult{\'e} de Math{\'e}matiques et Informatique", } @Article{Snyder:1993:AFI, author = "W. Van Snyder", title = "{Algorithm 723}: {Fresnel} Integrals", journal = j-TOMS, volume = "19", number = "4", pages = "452--456", month = dec, year = "1993", CODEN = "ACMSCU", DOI = "https://doi.org/10.1145/168173.168193", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Thu Apr 29 15:24:56 1999", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib", note = "See remarks \cite{Snyder:1996:RAF,Snyder:2021:CRA}.", URL = "http://www.acm.org/pubs/citations/journals/toms/1993-19-4/p452-van_snyder/", abstract = "An implementation of approximations for Fresnel integrals and associated functions is described. The approximations were originally developed by W. J. Cody, but a Fortran implementation using them has not previously been published.", acknowledgement = ack-nhfb, fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", keywords = "algorithms; special functions", subject = "{\bf G.1.2}: Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Rational approximation. {\bf G.4}: Mathematics of Computing, MATHEMATICAL SOFTWARE, Certification and testing.", } @Article{Thompson:1993:CCQ, author = "William J. Thompson", title = "Cutting Corners: Quick Square Roots and Trig Functions", journal = j-COMPUT-PHYS, volume = "7", number = "1", pages = "18--??", month = jan, year = "1993", CODEN = "CPHYE2", DOI = "https://doi.org/10.1063/1.4823136", ISSN = "0894-1866 (print), 1558-4208 (electronic)", ISSN-L = "0894-1866", bibdate = "Wed Apr 10 08:45:39 MDT 2019", bibsource = "https://www.math.utah.edu/pub/tex/bib/computphys.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "https://aip.scitation.org/doi/10.1063/1.4823136", acknowledgement = ack-nhfb, ajournal = "Comput. Phys", fjournal = "Computers in Physics", journal-URL = "https://aip.scitation.org/journal/cip", } @Article{Vedder:1993:IAN, author = "John D. Vedder", title = "An invertible approximation to the normal distribution function", journal = j-COMPUT-STAT-DATA-ANAL, volume = "16", number = "1", pages = "119--123", month = jun, year = "1993", CODEN = "CSDADW", ISSN = "0167-9473 (print), 1872-7352 (electronic)", ISSN-L = "0167-9473", bibdate = "Fri Feb 6 11:39:39 MST 2015", bibsource = "https://www.math.utah.edu/pub/tex/bib/computstatdataanal1990.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/016794739390248R", acknowledgement = ack-nhfb, fjournal = "Computational Statistics \& Data Analysis", journal-URL = "http://www.sciencedirect.com/science/journal/01679473", } @Article{Zaitsev:1993:IMM, author = "A. V. Zaitsev", title = "Implementation of {Miller}'s method for evaluation of {Bessel} functions of first kind", journal = j-J-SOV-MATH, volume = "63", number = "5", pages = "558--560", month = feb, year = "1993", CODEN = "JSOMAR", DOI = "https://doi.org/10.1007/bf01142530", ISSN = "0090-4104 (print), 2376-5798 (electronic)", ISSN-L = "0090-4104", bibdate = "Wed Mar 1 09:29:38 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Journal of Soviet Mathematics", journal-URL = "http://link.springer.com/journal/10958", } @Article{Anonymous:1994:C, author = "Anonymous", title = "Corrigenda", journal = j-TOMS, volume = "20", number = "4", pages = "553--553", month = dec, year = "1994", CODEN = "ACMSCU", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Tue Mar 14 16:17:03 1995", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "See \cite{Hull:1994:ICE}", acknowledgement = ack-rfb # "\slash " # ack-nhfb, fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", } @Article{Bajard:1994:BNH, author = "Jean-Claude Bajard and Sylvanus Kla and Jean-Michel Muller", title = "{BKM}: a New Hardware Algorithm for Complex Elementary Functions", journal = j-IEEE-TRANS-COMPUT, volume = "43", number = "8", pages = "955--963", month = aug, year = "1994", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/12.295857", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", MRclass = "68M07", MRnumber = "1 294 301", bibdate = "Sat Jan 11 17:44:01 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput1990.bib; OCLC Proceedings database", URL = "https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=295857", acknowledgement = ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", remark = "Selected revised and extended papers from ARITH'11 \cite{Swartzlander:1993:SCA}.", } @Article{Bender:1994:DAS, author = "Carl M. Bender and Stefan Boettcher", title = "Determination of $ f(\infty) $ from the asymptotic series for $ f(x) $ about $ x = 0 $", journal = j-J-MATH-PHYS, volume = "35", number = "4", pages = "1914--1921", month = apr, year = "1994", CODEN = "JMAPAQ", DOI = "https://doi.org/10.1063/1.530577", ISSN = "0022-2488 (print), 1089-7658 (electronic), 1527-2427", ISSN-L = "0022-2488", MRclass = "41A60 (41A21 65D15 81Q15)", MRnumber = "95d:41063", bibdate = "Tue Nov 1 08:58:10 MDT 2011", bibsource = "http://jmp.aip.org/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jmathphys1990.bib", URL = "http://jmp.aip.org/resource/1/jmapaq/v35/i4/p1914_s1", acknowledgement = ack-nhfb, fjournal = "Journal of Mathematical Physics", journal-URL = "http://jmp.aip.org/", pagecount = "8", } @Article{Brown:1994:CAS, author = "Barry W. Brown and Lawrence Levy", title = "Certification of {Algorithm 708}: Significant Digit Computation of the Incomplete Beta", journal = j-TOMS, volume = "20", number = "3", pages = "393--397", month = sep, year = "1994", CODEN = "ACMSCU", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Sat Nov 19 12:53:17 1994", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "See \cite{DiDonato:1992:ASD}.", URL = "http://doi.acm.org/10.1145/192115.192155; http://www.acm.org/pubs/citations/journals/toms/1994-20-3/p393-brown/", acknowledgement = ack-rfb # "\slash " # ack-nhfb, fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", keywords = "algorithms; continued fractions; F-distribution", subject = "G.1.2 [Numerical Analysis]: Approximation", } @Article{Bruno:1994:AAF, author = "Oscar P. Bruno and Fernando Reitich", title = "Approximation of analytic functions: a method of enhanced convergence", journal = j-MATH-COMPUT, volume = "63", number = "207", pages = "195--213", month = jul, year = "1994", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "30B10 (41A21 41A25)", MRnumber = "94m:30003", MRreviewer = "A. Edrei", bibdate = "Tue Mar 25 15:38:13 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp1990.bib", acknowledgement = ack-nhfb, affiliation = "Sch. of Math., Georgia Inst. of Technol., Atlanta, GA, USA", ajournal = "Math. Comput.", classcodes = "B0290F (Interpolation and function approximation); C4130 (Interpolation and function approximation)", corpsource = "Sch. of Math., Georgia Inst. of Technol., Atlanta, GA, USA", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "analytic functions; approximation; conformal; conformal maps; convergence of numerical methods; convergence rates; enhanced convergence; Euler transform; function approximation; Pad{\'e}; perturbation methods; perturbation techniques; power; series; series (mathematics); series expansions; Stieltjes-type functions; Taylor expansion; transformations; truncated enhanced", treatment = "T Theoretical or Mathematical", } @Article{Carlson:1994:AAS, author = "B. C. Carlson and J. L. Gustafson", title = "Asymptotic Approximations for Symmetric Elliptic Integrals", journal = j-SIAM-J-MATH-ANA, volume = "25", number = "2", pages = "288--303", month = mar, year = "1994", CODEN = "SJMAAH", ISSN = "0036-1410 (print), 1095-7154 (electronic)", ISSN-L = "0036-1410", MRclass = "33E05 (41A60)", MRnumber = "95b:33056", MRreviewer = "Bruce C. Berndt", bibdate = "Sat Dec 5 18:14:13 MST 1998", bibsource = "http://epubs.siam.org/sam-bin/dbq/toc/SIMA/25/2; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://epubs.siam.org/sam-bin/dbq/article/22847", acknowledgement = ack-nhfb, fjournal = "SIAM Journal on Mathematical Analysis", journal-URL = "http://epubs.siam.org/sima", } @Article{Chaudhry:1994:GIG, author = "M. Aslam Chaudhry and S. M. Zubair", title = "Generalized incomplete gamma functions with applications", journal = j-J-COMPUT-APPL-MATH, volume = "55", number = "1", pages = "99--123", day = "31", month = oct, year = "1994", CODEN = "JCAMDI", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Sat Feb 25 12:24:33 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath1990.bib", URL = "http://www.sciencedirect.com/science/article/pii/0377042794901872", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @Article{Chen:1994:ABB, author = "Yang Chen and Mourad E. H. Ismail and K. A. Muttalib", title = "Asymptotics of basic {Bessel} functions and $q$-{Laguerre} polynomials", journal = j-J-COMPUT-APPL-MATH, volume = "54", number = "3", pages = "263--272", day = "20", month = oct, year = "1994", CODEN = "JCAMDI", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Sat Feb 25 12:24:33 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath1990.bib", URL = "http://www.sciencedirect.com/science/article/pii/037704279200128V", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @InProceedings{Chen:1994:EDU, author = "Sau-Gee Chen and Chieh-Chih Li", booktitle = "{Proceedings of TENCON '94. IEEE Region 10's Ninth Annual International Conference. Theme: `Frontiers of Computer Technology'}", title = "Efficient designs of unified $2$'s complement division and square root algorithm and architecture", volume = "2", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "943--947", year = "1994", CODEN = "????", ISSN = "????", bibdate = "Sat Jul 16 15:28:14 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, summary = "Efficient unified 2's complement division and square root algorithm, and their architectures are proposed in this work. The designs are high speed, small area and high compatibility. The architectures provide bit level pipelined operation, as well \ldots{}", } @Article{Cortadella:1994:HRD, author = "J. Cortadella and T. Lang", title = "High-Radix Division and Square-Root with Speculation", journal = j-IEEE-TRANS-COMPUT, volume = "43", number = "8", pages = "919--931", month = aug, year = "1994", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/12.295854", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Sat Jul 16 15:28:13 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; OCLC Proceedings database", acknowledgement = ack-sfo # " and " # ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", remark = "Selected revised and extended papers from ARITH'11 \cite{Swartzlander:1993:SCA}.", summary = "The speed of high-radix digit-recurrence dividers and square-root units is mainly determined by the complexity of the result-digit selection. We present a scheme in which a simpler function speculates the result digit, and, when this speculation is \ldots{}", } @Article{Damnjanovic:1994:EFL, author = "Zlatan Damnjanovic", title = "Elementary functions and loop programs", journal = j-NOTRE-DAME-J-FORM-LOG, volume = "35", number = "4", pages = "496--522", year = "1994", CODEN = "NDJFAM", ISSN = "0029-4527 (print), 1939-0726 (electronic)", ISSN-L = "0029-4527", MRclass = "03D20 (68Q15)", MRnumber = "96i:03036", MRreviewer = "John P. Helm", bibdate = "Sat Jan 11 17:44:01 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Notre Dame journal of formal logic", journal-URL = "http://projecteuclid.org/all/euclid.ndjfl", } @Article{Das:1994:ATE, author = "Mrinal Kanti Das", title = "Analysis of two elementary functions", journal = j-INT-J-MATH-EDU-SCI-TECH, volume = "25", number = "1", pages = "17--24", year = "1994", CODEN = "IJMEBM", ISSN = "0020-739X (print), 1464-5211 (electronic)", ISSN-L = "0020-739X", MRclass = "26-01 (33B10)", MRnumber = "1 257 731", bibdate = "Sat Jan 11 17:44:01 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "International Journal of Mathematical Education in Science and Technology", journal-URL = "http://www.tandfonline.com/loi/tmes20", } @TechReport{Dunham:1994:PMAa, author = "Charles B. Dunham", title = "Provably Monotone Approximations, {IV}", type = "Technical report", number = "TR-417", institution = "Department of Computer Science, University of Western Ontario", address = "London, Ontario, Canada", day = "8", month = mar, year = "1994", bibdate = "Tue Apr 12 11:26:47 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.csd.uwo.ca/tech-reports/", acknowledgement = ack-nhfb, } @TechReport{Dunham:1994:PMAb, author = "Charles B. Dunham", title = "Provably Monotone Approximations, {IV}, Revised", type = "Technical report", number = "TR-422", institution = "Department of Computer Science, University of Western Ontario", address = "London, Ontario, Canada", day = "4", month = apr, year = "1994", bibdate = "Tue Apr 12 11:26:47 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.csd.uwo.ca/tech-reports/", acknowledgement = ack-nhfb, } @Article{Dunkl:1994:AHI, author = "Charles F. Dunkl and Donald E. Ramirez", title = "{Algorithm 736}: Hyperelliptic Integrals and the Surface Measure of Ellipsoids", journal = j-TOMS, volume = "20", number = "4", pages = "427--435", month = dec, year = "1994", CODEN = "ACMSCU", DOI = "https://doi.org/10.1145/198429.198431", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", MRclass = "65D30", MRnumber = "1 368 025", bibdate = "Tue Mar 14 16:16:51 1995", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.acm.org/pubs/citations/journals/toms/1994-20-4/p427-dunkl/", acknowledgement = ack-rfb # "\slash " # ack-nhfb, fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", keywords = "elliptic integral; expected radius; Lauricella's hypergeometric function; optimal designs; surface measure", subject = "G.1.4 [Numerical Analysis]: Quadrature and Numerical Differentiation -- multiple quadrature; G.3 [Mathematics of Computing]: Probability and Statistics", } @Article{Dunkl:1994:CHI, author = "Charles F. Dunkl and Donald E. Ramirez", title = "Computing Hyperelliptic Integrals for Surface Measure of Ellipsoids", journal = j-TOMS, volume = "20", number = "4", pages = "413--426", month = dec, year = "1994", CODEN = "ACMSCU", DOI = "https://doi.org/10.1145/198429.198430", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", MRclass = "65D30", MRnumber = "1 368 024", bibdate = "Tue Mar 14 16:16:49 1995", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.acm.org/pubs/citations/journals/toms/1994-20-4/p413-dunkl/", acknowledgement = ack-rfb # "\slash " # ack-nhfb, fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", keywords = "elliptic integral; expected radius; Lauricella's hypergeometric function; optimal designs; surface measure", subject = "G.1.4 [Numerical Analysis]: Quadrature and Numerical Differentiation -- multiple quadrature; G.3 [Mathematics of Computing]: Probability and Statistics", } @Book{Ercegovac:1994:DSR, author = "Milo{\v{s}} D. (Dragutin) Ercegovac and Tomas Lang", title = "Division and Square Root: Digit-recurrence Algorithms and Implementations", publisher = pub-KLUWER, address = pub-KLUWER:adr, pages = "x + 230", year = "1994", ISBN = "0-7923-9438-0", ISBN-13 = "978-0-7923-9438-9", LCCN = "QA76.9.C62 E73 1994", bibdate = "Fri Mar 27 09:46:24 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, tableofcontents = "Preface / 1 \\ 1. General Comments / 5 \\ 2. Division by Digit Recurrence / 5 \\ 3. Theory of Digit-Recurrence Division / 19 \\ 4. Division With Scaling and Prediction /65 \\ 5. Higher Radix Division / 91 \\ 6. On-The-Fly Conversion and Round / 121 \\ 7. Square Root by Digit Recurrence / 135 \\ 8. Implementations of Square Root / 153 \\ A: Restoring and Non-Restoring Division / 182 \\ B: Evaluation of Some Implementations / 182 \\ Bibliography / 207 \\ Index / 227", } @Article{Everitt:1994:GBF, author = "W. N. Everitt and C. Markett", title = "On a generalization of {Bessel} functions satisfying higher-order differential equations", journal = j-J-COMPUT-APPL-MATH, volume = "54", number = "3", pages = "325--349", day = "20", month = oct, year = "1994", CODEN = "JCAMDI", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Sat Feb 25 12:24:33 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath1990.bib", URL = "http://www.sciencedirect.com/science/article/pii/0377042794902550", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @Article{Fukushima:1994:NCI, author = "Toshio Fukushima and Hideharu Ishizaki", title = "Numerical computation of incomplete elliptic integrals of a general form", journal = j-CELEST-MECH-DYN-ASTR, volume = "59", number = "3", pages = "237--251", month = jul, year = "1994", CODEN = "CLMCAV", DOI = "https://doi.org/10.1007/BF00692874", ISSN = "0923-2958 (print), 1572-9478 (electronic)", ISSN-L = "0923-2958", MRclass = "33E05 65R20 65D32 (33E30 70-08 70E15)", MRnumber = "1285916 (95c:65041)", bibdate = "Wed Oct 20 21:26:45 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.springerlink.com/content/0923-2958/", ZMnumber = "Zbl 0818.33013", abstract = "We present an algorithm to compute the incomplete elliptic integral of a general form. The algorithm efficiently evaluates some linear combinations of incomplete elliptic integrals of all kinds to a high precision. Some numerical examples are given as illustrations. This enables us to numerically calculate the values and the partial derivatives of incomplete elliptic integrals of all kinds, which are essential when dealing with many problems in celestial mechanics, including the analytic solution of the torque-free rotational motion of a rigid body around its barycenter.", acknowledgement = ack-nhfb, fjournal = "Celestial Mechanics \& Dynamical Astronomy. An International Journal of Space Dynamics", keywords = "Incomplete elliptic integrals; numerical computation", } @Article{Hahn:1994:UDF, author = "H. Hahn and D. Timmermann and B. J. Hosticka and B. Rix", title = "A unified and division-free {CORDIC} argument reduction method with unlimited convergence domain including inverse hyperbolic functions", journal = j-IEEE-TRANS-COMPUT, volume = "43", number = "11", pages = "1339--1344", month = nov, year = "1994", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/12.324568", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Thu Jul 7 07:13:58 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput1990.bib", URL = "https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=324568", acknowledgement = ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", } @InProceedings{Homeier:1994:NCA, author = "Herbert H. H. Homeier", editor = "Ralf Gruber and Marco Tomassini", booktitle = "Proceedings of the 6th {Joint EPS-APS International Conference} on {Physics Computing, Physics Computing} '94", title = "Nonlinear convergence acceleration for orthogonal series", publisher = "European Physical Society, Boite Postale 69, CH-1213 Petit-Lancy, Geneva, Switzerland", address = "Lugano", pages = "47--50", year = "1994", ISBN = "2-88270-011-3", ISBN-13 = "978-2-88270-011-7", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/h/homeier-herbert-h-h.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.chemie.uni-regensburg.de/pub/preprint/preprint.html#TCNA942", keywords = "convergence acceleration", tech = "Technical Report TC-NA-94-2, Institut f{\"u}r {Physikalische} und {Theoretische Chemie, Universit{\"a}t Regensburg, D-93040 Regensburg}, 1994", } @Article{Hull:1994:ICE, author = "T. E. Hull and Thomas F. Fairgrieve and Ping Tak Peter Tang", title = "Implementing Complex Elementary Functions Using Exception Handling", journal = j-TOMS, volume = "20", number = "2", pages = "215--244", month = jun, year = "1994", CODEN = "ACMSCU", DOI = "https://doi.org/10.1145/178365.178404", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Tue Mar 21 15:10:29 1995", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib", note = "See correction \cite{Anonymous:1994:C}, and improved analysis, tightened bounds, and exhibition of worst cases for complex square roots \cite{Jeannerod:2017:REC}.", URL = "http://www.acm.org/pubs/citations/journals/toms/1994-20-2/p215-hull/", abstract = "Algorithms are developed for reliable and accurate evaluations of the complex elementary functions required in Fortran 77 and Fortran 90, namely cabs, csqrt, cexp, clog, csin, and ccos. The algorithms are presented in a pseudocode that has a convenient exception-handling facility. A tight error bound is derived for each algorithm. Corresponding Fortran programs for an IEEE environment have also been developed to illustrate the practicality of the algorithms, and these programs have been tested very carefully to help confirm the correctness of the algorithms and their error bounds. The results are of these tests are included in the paper, but the Fortran programs are not; the programs are available from Fairgrieve, (tff@cs.toronto.edu).", acknowledgement = ack-nhfb, fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", keywords = "algorithms; complex elementary functions; design; implementation", subject = "G.1.0 [Numerical Analysis]: General--error analysis; numerical algorithms; G.1.2 [Numerical Analysis]: Approximation--elementary function approximation; G.4 [Mathematics of Computing]: Mathematical Software--algorithm analysis; reliability and robustness; verification", } @Article{Iserles:1994:CAD, author = "A. Iserles", title = "Convergence acceleration as a dynamical system", journal = j-APPL-NUM-MATH, volume = "15", number = "2", pages = "101--121", day = "13", month = sep, year = "1994", CODEN = "ANMAEL", ISSN = "0168-9274 (print), 1873-5460 (electronic)", ISSN-L = "0168-9274", MRclass = "58F23 (30D05 58F08 65D99 65H99)", MRnumber = "95i:58155", MRreviewer = "Peter M. Makienko", bibdate = "Wed Jul 28 14:35:48 MDT 1999", bibsource = "http://www.elsevier.com/cgi-bin/cas/tree/store/apnum/cas_free/browse/browse.cgi?year=1994&volume=15&issue=2; https://www.math.utah.edu/pub/tex/bib/applnummath.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "Innovative methods in numerical analysis (Bressanone, 1992).", URL = "http://www.elsevier.com/cgi-bin/cas/tree/store/apnum/cas_sub/browse/browse.cgi?year=1994&volume=15&issue=2&aid=496", acknowledgement = ack-nhfb, fjournal = "Applied Numerical Mathematics: Transactions of IMACS", journal-URL = "http://www.sciencedirect.com/science/journal/01689274/", keywords = "convergence acceleration", } @Article{Jablonski:1994:NES, author = "Aleksander Jablonski", title = "Numerical Evaluation of Spherical {Bessel} Functions of the First Kind", journal = j-J-COMPUT-PHYS, volume = "111", number = "2", pages = "256--259", month = apr, year = "1994", CODEN = "JCTPAH", DOI = "https://doi.org/10.1006/jcph.1994.1060", ISSN = "0021-9991 (print), 1090-2716 (electronic)", ISSN-L = "0021-9991", bibdate = "Mon Jan 2 07:54:54 MST 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputphys1990.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0021999184710606", abstract = "Calculations of cross sections for elastic scattering of electrons require frequent evaluations of the spherical Bessel functions, $ j_l(x) $ and $ n_l(x) $, in a wide range of the argument $x$ and the order $l$. It turns out that the usual algorithms providing the values of the spherical Bessel function of the first kind, $ j_l(x) $, have a rather limited range of stability. It is shown that there is no algorithm implementing a single method which can be used in calculations associated with the theory of elastic scattering of electrons. An attempt is made to select different areas of stability from different algorithms in order to create a relatively fast and universal algorithm.", acknowledgement = ack-nhfb, fjournal = "Journal of Computational Physics", journal-URL = "http://www.sciencedirect.com/science/journal/00219991/", } @InProceedings{Jain:1994:SRR, author = "V. K. Jain and Lei Lin", booktitle = "{IEEE} International Conference on Acoustics, Speech, and Signal Processing: {ICASSP-94, 19--22} April 1994", title = "Square-root, reciprocal, sine\slash cosine, arctangent cell for signal and image processing", volume = "2", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "II/521--II/524", year = "1994", CODEN = "????", ISSN = "????", bibdate = "Sat Jul 16 15:28:14 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, summary = "This paper discusses an efficient interpolation method for nonlinear function generation. Based on this, a 24 bit VLSI cell, capable of computing the (1) square root, (2) reciprocal, (3) sine/cosine, and (4) arctangent functions, is presented for \ldots{}", } @Misc{Karp:1994:FPA, author = "Alan H. Karp and Peter Markstein and Dennis Brzezinski", title = "Floating point arithmetic unit using modified {Newton--Raphson} technique for division and square root", howpublished = "US Patent 5,341,321", day = "23", month = aug, year = "1994", bibdate = "Thu Oct 17 10:20:52 2019", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", note = "Patent filed 5 May 1993, granted to Hewlett-Packard Company on 23 August 1994. Patent expired 5-May-2013. See criticism in \cite{Zimmermann:2005:XXX}.", URL = "http://patft.uspto.gov/netahtml/PTO/search-bool.html; https://patents.google.com/patent/US5341321A", abstract = "A floating point processing system which uses a multiplier unit and an adder unit to perform floating point division and square root operations using both a conventional and a modified form of the Newton--Raphson method. The modified form of the Newton--Raphson method is used in place of the final iteration of the conventional Newton--Raphson so as to compute high precision approximated results with a substantial improvement in speed. The invention computes approximated results faster and simplifies hardware requirements because no multiplications of numbers of the precision of the result are required.", acknowledgement = ack-nhfb, } @Article{Kearfott:1994:AIP, author = "R. B. Kearfott and M. Dawande and K. Du and C. Hu", title = "Algorithm 737: {INTLIB}: a Portable {Fortran}-77 Elementary Function Library", journal = j-TOMS, volume = "20", number = "4", pages = "447--459", month = dec, year = "1994", CODEN = "ACMSCU", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Sat May 20 15:54:18 1995", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", accepted = "December 1993", acknowledgement = ack-rfb # "\slash " # ack-nhfb, fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", keywords = "BLAS; Fortran 77; Fortran 90; interval arithmetic; operator overloading; standard functions", subject = "D.2.2 [Software Engineering]: Tools and Techniques -- software libraries; D.2.7 [Software Engineering]: Distribution and Maintenance -- documentation; portability; G.1.0 [Numerical Analysis]: General -- computer arithmetic; G.1.2 [Numerical Analysis]: Approximation -- elementary function approximation", } @Article{Khajah:1994:UHP, author = "H. G. Khajah and E. L. Ortiz", title = "Ultra-high precision computations", journal = j-COMPUT-MATH-APPL, volume = "27", number = "7", pages = "41--57", month = apr, year = "1994", CODEN = "CMAPDK", DOI = "https://doi.org/10.1016/0898-1221(94)90148-1", ISSN = "0898-1221 (print), 1873-7668 (electronic)", ISSN-L = "0898-1221", bibdate = "Mon Jun 13 22:03:39 2011", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/l/lanczos-cornelius.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/0898122194901481", abstract = "We describe a machine independent Fortran subroutine which performs the four basic arithmetic operations with a degree of accuracy prescribed by the user. Tables of Chebyshev expansions of orders 48 and 50 for some basic mathematical functions are obtained as a result of applying this subroutine in conjunction with the recursive formulation of the Tau Method. A recently devised technique for the sharp determination of upper and lower error bounds for Tau Method approximations enables us to find the degree $n$ required to achieve a prescribed accuracy $ \epsilon $ over a given interval $ [a, b] $. A number of practical illustrations are given.", acknowledgement = ack-nhfb, affiliation = "Dept. of Math., Imperial Coll. of Sci., Technol. and Med., London, UK", classification = "C6140D (High level languages); C7310 (Mathematics)", fjournal = "Computers and Mathematics with Applications", journal-URL = "http://www.sciencedirect.com/science/journal/08981221", keywords = "$\cos(\pi x)$; $\erf(x) / x$; $\exp(-x^2)$; $\exp(x)$; $\sin(\pi x)$; $x \exp(x^2) erfc(x)$; $z \exp(z) \Ei(-z)$; Arithmetic operations; Chebyshev expansions; Lower error bounds; Machine independent Fortran subroutine; Mathematical functions; Tau method; Upper error bounds", pubcountry = "UK", thesaurus = "FORTRAN; Mathematics computing", } @Article{Lewanowicz:1994:SAS, author = "Stanis{\l}aw Lewanowicz", title = "A simple approach to the summation of certain slowly convergent series", journal = j-MATH-COMPUT, volume = "63", number = "208", pages = "741--745", month = oct, year = "1994", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "65B10", MRnumber = "95a:65010", MRreviewer = "Thomas A. Atchison", bibdate = "Sat Jan 11 13:29:06 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp1990.bib", acknowledgement = ack-nhfb, fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Litvinov:1994:ACR, author = "Grigori L. Litvinov", title = "Approximate construction of rational approximations and the effect of error autocorrection", journal = j-RUSS-J-MATH-PHYS, volume = "1", number = "3", pages = "313--352", month = "????", year = "1994", CODEN = "RJMPEL", ISSN = "1061-9208 (print), 1555-6638 (electronic)", ISSN-L = "1061-9208", bibdate = "Tue Mar 24 20:54:11 2009", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://arxiv.org/abs/math/0101042", abstract = "Several construction methods for rational approximations to functions of one real variable are described in the present paper; the computational results that characterize the comparative accuracy of these methods are presented; an effect of error autocorrection is considered. This effect occurs in efficient methods of rational approximation (e.g., Pad{\'e} approximations, linear and nonlinear Pad{\'e} Chebyshev approximations) where very significant errors in the coefficients do not affect the accuracy of the approximation. The matter of import is that the errors in the numerator and the denominator of a fractional rational approximant compensate each other. This effect is related to the fact that the errors in the coefficients of a rational approximant are not distributed in an arbitrary way but form the coefficients of a new approximant to the approximated function. Understanding of the error autocorrection mechanism allows to decrease this error by varying the approximation procedure depending on the form of the approximant. Some applications are described in the paper. In particular, a method of implementation of basic calculations on decimal computers that uses the technique of rational approximations is described in the Appendix.\par To a considerable extent the paper is a survey and the exposition is as elementary as possible.", acknowledgement = ack-nhfb, fjournal = "Russian Journal of Mathematical Physics", } @TechReport{Lozier:1994:NESa, author = "D. W. Lozier and F. W. J. Olver", title = "Numerical evaluation of special functions", type = "Report", number = "NISTIR 5383", institution = "Computing and Applied Mathematics Laboratory, U. S. Department of Commerce", address = "Washington, DC, USA", pages = "47", month = mar, year = "1994", MRclass = "65D20 (33-00)", bibdate = "Thu Nov 16 07:52:34 2023", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/o/olver-frank-w-j.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "https://math.nist.gov/~DLozier/publications/nistir5383.pdf", abstract = "Higher transcendental functions continue to play varied and important roles in investigations by engineers, mathematicians, scientists and statisticians. The purpose of this paper is to assist in locating useful approximations and software for the numerical generation of these functions, and to offer some suggestions for future developments in this field.", acknowledgement = ack-nhfb, author-dates = "Frank William John Olver (15 December 1924--23 April 2013)", tableofcontents = "1. Introduction / 3 \\ 2. Mathematical Developments / 5 \\ 3. Packages, Libraries and Systems / 6 \\ 3.1. Software Packages / 6 \\ 3.2. Intermediate Libraries / 8 \\ 3.3. Comprehensive Libraries / 8 \\ 3.4. Interactive Systems / 12 \\ 4. Functions of One Variable / 15 \\ 4.1. Airy Functions / 15 \\ 4.2. Error Functions, Dawson's Integral, Fresnel Integrals, Goodwin--Staton Integral / 15 \\ 4.3. Exponential Integrals, Logarithmic Integral, Sine and Cosine Integrals / 16 \\ 4.4. Gamma, Psi, and Polygamma Functions / 16 \\ 4.5. Landau Density and Distribution Functions / 16 \\ 4.6. Polylogarithms, Clausen Integral / 16 \\ 4.7. Zeta Function / 17 \\ 4.8. Additional Functions of One Variable / 17 \\ 5. Functions of Two or More Variables / 17 \\ 5.1. Bessel Functions / 17 \\ 5.2. Coulomb Wave Functions / 18 \\ 5.3. Elliptic Integrals and Functions / 18 \\ 5.4. Fermi--Dirac, Bose--Einstein, and Debye Integrals / 19 \\ 5.5. Hypergeometric and Concuent Hypergeometric Functions / 19 \\ 5.6. Incomplete Bessel Functions, Incomplete Beta Function / 19 \\ 5.7. Incomplete Gamma Functions, Generalized Exponential Integrals / 20 \\ 5.8. Legendre Functions and Associated Legendre Functions / 20 \\ 5.9. Mathieu, Lam{\'e}, and Spheroidal Wave Functions / 20 \\ 5.10. Orthogonal Polynomials / 21 \\ 5.11. Polylogarithms (Generalized) / 21 \\ 5.12. Struve and Anger--Weber Functions / 21 \\ 5.13. Weber Parabolic Cylinder Functions / 21 \\ 5.14. Zeta Function (Generalized) / 21 \\ 5.15. Additional Functions of Two or More Variables / 21 \\ 6. Testing and Library Construction / 22 \\ 7. Future Trends / 22 \\ Acknowledgments / 23 \\ A Note on the Reference Acronyms / 23 \\ References / 23--47", } @InProceedings{Lozier:1994:NESb, author = "D. W. Lozier and F. W. J. Olver", title = "Numerical evaluation of special functions", crossref = "Gautschi:1994:MCH", volume = "48", pages = "79--125", year = "1994", DOI = "https://doi.org/10.1090/psapm/048/1314844", MRclass = "65D20 (30-04 33-04 41-04)", MRnumber = "95m:65036 (1314844)", MRreviewer = "John P. Coleman", bibdate = "Fri Jul 9 05:44:10 2004", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/o/olver-frank-w-j.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp1990.bib", series = "Proc. Sympos. Appl. Math.", URL = "http://math.nist.gov/mcsd/Reports/2001/nesf/", abstract = "Higher transcendental functions continue to play varied and important roles in investigations by engineers, mathematicians, scientists, and statisticians. The purpose of this paper is to assist in locating useful approximations and software for the numerical generation of these functions, and to offer some suggestions for future developments in the field.", acknowledgement = ack-nhfb, author-dates = "Frank William John Olver (15 December 1924--23 April 2013)", remark = "The references list contains about 400 entries which should ultimately be incorporated in this BibTeX bibliography collection.", } @TechReport{Lozier:1994:SNS, author = "Daniel W. Lozier", title = "Software Needs in Special Functions", type = "Technical Report", number = "NISTIR 5490", institution = pub-NIST, address = pub-NIST:adr, pages = "16", month = aug, year = "1994", bibdate = "Fri Jul 09 05:47:26 2004", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "Published in \cite{Lozier:1996:SNS}.", URL = "http://math.nist.gov/acmd/Staff/DLozier/publications/nistir5490.ps", acknowledgement = ack-nhfb, } @Article{MacLeod:1994:CIA, author = "Allan J. MacLeod", title = "Computation of inhomogeneous {Airy} functions", journal = j-J-COMPUT-APPL-MATH, volume = "53", number = "1", pages = "109--116", day = "29", month = jul, year = "1994", CODEN = "JCAMDI", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Sat Feb 25 12:24:31 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath1990.bib", URL = "http://www.sciencedirect.com/science/article/pii/0377042794901961", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @Article{MacLeod:1994:TBT, author = "Allan J. MacLeod", title = "Table-based tests for {Bessel} function software", journal = j-ADV-COMPUT-MATH, volume = "2", number = "2", pages = "251--260", month = mar, year = "1994", CODEN = "ACMHEX", DOI = "https://doi.org/10.1007/BF02521111", ISSN = "1019-7168 (print), 1572-9044 (electronic)", ISSN-L = "1019-7168", MRclass = "65-04 (33-04 33C10 65D20)", MRnumber = "1269384", bibdate = "Sat Feb 3 18:21:41 MST 2018", bibsource = "https://www.math.utah.edu/pub/tex/bib/advcomputmath.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://link.springer.com/article/10.1007/BF02521111", acknowledgement = ack-nhfb, fjournal = "Advances in Computational Mathematics", journal-URL = "http://link.springer.com/journal/10444", } @InProceedings{Magnus:1994:ASA, author = "Alphonse P. Magnus", title = "Asymptotics and super asymptotics of best rational approximation error norms for the exponential function (the `$ 1 / 9 $' problem) by the {Carath{\'e}odory--Fej{\'e}r} method", crossref = "Cuyt:1994:NNM", pages = "173--185", year = "1994", bibdate = "Mon Nov 24 21:30:41 2008", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", ZMnumber = "809.41015", acknowledgement = ack-nhfb, } @Article{Marsaglia:1994:REI, author = "George Marsaglia and Arif Zaman and John C. W. Marsaglia", title = "Rapid evaluation of the inverse of the normal distribution function", journal = j-STAT-PROB-LETT, volume = "19", number = "4", pages = "259--266", day = "15", month = mar, year = "1994", CODEN = "SPLTDC", DOI = "https://doi.org/10.1016/0167-7152(94)90174-0", ISSN = "0167-7152 (print), 1879-2103 (electronic)", ISSN-L = "0167-7152", MRclass = "65U05", MRnumber = "1 278 658", bibdate = "Thu Dec 22 07:42:24 2011", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/prng.bib; MathSciNet database", ZMnumber = "0798.65132", abstract = "This is an interesting article with direct application in generating normal random variable by computer programs. The suggested applications are related to Monte Carlo simulation based on massively parallel systems or supercomputers. The idea is to replace larger programs with complicated computations and with difficulties in accuracy controlling by simpler arithmetic programs that use tabled constants. These seem to be the normal evolution since memory becomes cheaper and cheaper.\par The authors compute the inverse of the cPhi function $$ c P h i(x) = (2 / \pi)^{1 / 2} \int^\infty_x \exp ( - t^2 / 2) d t = u, $$ using a uniform random variable as input and the truncated Taylor series development of it. In order to increase the speed the coefficients of the truncated Taylor series $$ x(u_0 + h) = x(u_0) + x'(u_0) \cdot h + {1 \over 2} x''(u_0) \cdot h^2 + {1 \over 6} x'''(u_0) \cdot h^3, $$ are predetermined for 1024 points. And here comes another bright idea: the 1024 points are chosen based on the representation of the uniform random variable in modern computers as floating point variable of the form: $ u = 2^{-k} ((1 / 2) + (j / 64)) + 2^{-k} \cdot (m / 2^{24}) $ with $ 0 \le k & l t; 32 $, $ 0 \le j & l t; 32 $ and $ 0 \le m & l t; 2^{18} $ and considering 32 bit representation.\par With this assumptions and the truncation to the third power of $h$ of the Taylor series, the authors show that the error does not exceed the limit of single precision accuracy. Furthermore the calculations are speeded up based on reducing multiplications. A number of FORTRAN programs are also presented in order to evaluate the complementary normal distribution function cPhi (several versions) with great accuracy, create the constant tables, and generate the normal distribution variable. These simple programs give the user the possibility to completely control the accuracy.", acknowledgement = ack-nhfb, fjournal = "Statistics \& Probability Letters", journal-URL = "http://www.sciencedirect.com/science/journal/01677152", keywords = "cPhi function; FORTRAN programs; massive parallel systems; Monte Carlo simulation; normal distribution function; normal random variable; supercomputers; truncated Taylor series", ZMclass = "*65C99 Numerical simulation 65C05 Monte Carlo methods 60-04 Machine computation, programs (probability theory) 60E05 General theory of probability distributions 62E17 Approximations to statistical distributions (nonasymptotic)", ZMreviewer = "A. Pasculescu (Bucuresti)", } @Article{Merrheim:1994:CEF, author = "X. Merrheim", title = "The computation of elementary functions in radix $ 2^p $", journal = j-COMPUTING, volume = "53", number = "3--4", pages = "219--232", year = "1994", CODEN = "CMPTA2", ISSN = "0010-485X (print), 1436-5057 (electronic)", ISSN-L = "0010-485X", MRclass = "68M07", MRnumber = "95j:68028", bibdate = "Sat Jan 11 17:44:01 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "International Symposium on Scientific Computing, Computer Arithmetic and Validated Numerics (Vienna, 1993).", acknowledgement = ack-nhfb, fjournal = "Computing", journal-URL = "http://link.springer.com/journal/607", } @Article{Narayanaswami:1994:AE, author = "Chandrasekhar Narayanaswami and William Luken", title = "Approximating $ x^n $ efficiently", journal = j-INFO-PROC-LETT, volume = "50", number = "4", pages = "205--210", day = "25", month = may, year = "1994", CODEN = "IFPLAT", ISSN = "0020-0190 (print), 1872-6119 (electronic)", ISSN-L = "0020-0190", MRclass = "65D20 (41-04 65B99)", MRnumber = "95b:65031", bibdate = "Wed Nov 11 12:16:26 MST 1998", bibsource = "Compendex database; http://www.elsevier.com:80/inca/publications/store/5/0/5/6/1/2/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, affiliation = "IBM Advanced Workstations and Systems Div", affiliationaddress = "Austin, TX, USA", classification = "721.1; 723.2; 723.5; 741.2; 921.1; 921.6; B0290F (Interpolation and function approximation); C4130 (Interpolation and function approximation); C6130B (Graphics techniques)", corpsource = "IBM Adv. Workstations and Syst. Div., Austin, TX, USA", fjournal = "Information Processing Letters", journal-URL = "http://www.sciencedirect.com/science/journal/00200190", journalabr = "Inf Process Lett", keywords = "$x^n$ approximation; approximation theory; Approximation theory; Color computer graphics; Computational complexity; Computational methods; computer graphics; elementary functions; floating-point arithmetic; Function evaluation; graphics modeling; Image quality; Light intensity computation; look-up tables; performance requirements; Polynomial evaluation; Polynomials; polynomials; power function; scientific applications; Semiconducting silicon; Table lookup", treatment = "T Theoretical or Mathematical", } @Article{Nishioka:1994:EFB, author = "Keiji Nishioka", title = "Elementary functions based on elliptic curves", journal = j-TOKYO-J-MATH, volume = "17", number = "2", pages = "439--446", year = "1994", ISSN = "0387-3870", MRclass = "12H05", MRnumber = "96b:12011", MRreviewer = "Alexandru Buium", bibdate = "Sat Jan 11 17:44:01 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Tokyo journal of mathematics", } @Article{Ohta:1994:INP, author = "Shigemi Ohta and Eiichi Goto and Weng Fai Wong and Nobuaki Yoshida", title = "Improvement and new proposal on fast evaluation of elementary functions. ({Japanese})", journal = j-TRANS-INFO-PROCESSING-SOC-JAPAN, volume = "35", number = "5", pages = "926--933", month = may, year = "1994", CODEN = "JSGRD5", ISSN = "0387-5806", ISSN-L = "0387-5806", MRclass = "65D20", MRnumber = "95f:65045", bibdate = "Sat Jan 11 17:44:01 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", abstract = "Wong, Gore, and Yoshida (ibid., vol. 34, no. 7, pp. 1570-1579, 1993) introduced fast methods for numerical evaluation of elementary functions based on table lookup. They are called ATA (add/table-lookup/add) and ATA-M (add/table-lookup/add and multiply) methods for single- and double-precision calculations respectively. In this paper, an improvement to these methods that shrinks the size of the table by a factor of about 3/16 is presented. Another method called the `split parallel multiplication method', which is characterized by simpler table lookup than ATA-M and by split and parallel use of double-precision floating point circuitry, is also introduced, These new methods fit on to integrated circuits of a size comparable with commercially available floating-point accelerators. Methods for accelerating double-precision division, generating uniform pseudo-random numbers in double-precision, and accelerating the multiplication of single-precision complex numbers using the same circuitry are proposed.", acknowledgement = ack-nhfb, affiliation = "RIKEN, Inst. of Phys. and Chem. Res., Saitama, Japan", classification = "C4120 (Functional analysis); C5230 (Digital arithmetic methods); C6130 (Data handling techniques)", fjournal = "Transactions of the Information Processing Society of Japan", keywords = "Add/table-lookup/add method; Add/table-lookup/add/multiply method; ATA method; ATA-M method; Double-precision calculations; Double-precision division; Double-precision floating point circuitry; Elementary functions evaluation; Floating-point accelerators; Integrated circuits; Numerical evaluation; Single-precision calculations; Single-precision complex number multiplication; Split parallel multiplication method; Table size; Uniform pseudo-random number generation", language = "Japanese", pubcountry = "Japan", thesaurus = "Digital arithmetic; Function evaluation; Random number generation; Table lookup", } @InProceedings{Olver:1994:GEI, author = "F. W. J. Olver", title = "The generalized exponential integral", crossref = "Zahar:1994:ACF", pages = "497--510", year = "1994", MRclass = "33B20 (34E05 41A60)", MRnumber = "1333639", MRreviewer = "Richard B. Paris", bibdate = "Sat Feb 18 15:02:52 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", series = "International Series of Numerical Mathematics", acknowledgement = ack-nhfb, } @Article{Osada:1994:CAM, author = "Naoki Osada", title = "Convergence acceleration methods", journal = "S{\=u}rikaisekikenky{\=u}sho K{\=o}ky{\=u}roku", volume = "880", number = "??", pages = "28--43", month = "????", year = "1994", MRclass = "65B05", MRnumber = "1366233", bibdate = "Thu Dec 1 17:36:45 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "The state of the art of scientific computing and its prospects (Japanese) (Kyoto, 1993)", acknowledgement = ack-nhfb, fjournal = "S{\=u}rikaisekikenky{\=u}sho K{\=o}ky{\=u}roku", keywords = "convergence acceleration", } @InProceedings{Rappoport:1994:TMC, author = "Juri M. Rappoport", title = "The {Tau-Method} and the Computation of the {Bessel} Functions of the Complex Order", crossref = "Brown:1994:PCL", pages = "353--355", year = "1994", bibdate = "Sat Jun 11 17:22:09 2011", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/l/lanczos-cornelius.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; MathSciNet database", acknowledgement = ack-nhfb, } @Article{Schulte:1994:HDE, author = "M. J. Schulte and E. E. {Swartzlander, Jr.}", title = "Hardware Design for Exactly Rounded Elementary Functions", journal = j-IEEE-TRANS-COMPUT, volume = "43", number = "8", pages = "964--973", month = aug, year = "1994", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/12.295858", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Tue Dec 12 09:29:07 MST 1995", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", abstract = "This paper presents hardware designs that produce exactly rounded results for the functions of reciprocal, square-root, 2/sup x/, and log/sub 2/(x). These designs use polynomial approximation in which the terms in the approximation are generated in parallel, and then summed by using a multi-operand adder. To reduce the number of terms in the approximation, the input interval is partitioned into subintervals of equal size, and different coefficients are used for each subinterval. The coefficients used in the approximation are initially determined based on the Chebyshev series approximation. They are then adjusted to obtain exactly rounded results for all inputs. Hardware designs are presented, and delay and area comparisons are made based on the degree of the approximating polynomial and the accuracy of the final result. For single-precision floating point numbers, a design that produces exactly rounded results for all four functions has an estimated delay of 80 ns and a total chip area of 98 mm/sup 2/ in a 1.0-micron CMOS technology. Allowing the results to have a maximum error of one unit in the last place reduces the computational delay by 5\% to 30\% and the area requirements by 33\% to 77\%.", acknowledgement = ack-nhfb # " and " # ack-nj, affiliation = "Dept. of Electr. and Comput. Eng., Texas Univ., Austin, TX, USA", ajournal = "IEEE Trans. Comput.", classification = "B0290F (Interpolation and function approximation); B1265B (Logic circuits); B2570D (CMOS integrated circuits); C4130 (Interpolation and function approximation); C5120 (Logic and switching circuits); C5230 (Digital arithmetic methods)", fjournal = "IEEE Transactions on Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", keywords = "1 Micron; 1.0-Micron CMOS technology; Argument reduction; Chebyshev series approximation; Chip area; Computational delay; Computer arithmetic; Exact rounding; Exactly rounded elementary functions; Hardware designs; Multi-operand adder; Parallel multiplier; Polynomial approximation; Reciprocal; Single-precision floating point numbers; Special-purpose hardware; Square-root", numericalindex = "Size 1.0E-06 m", pubcountry = "USA", thesaurus = "Approximation theory; Chebyshev approximation; CMOS integrated circuits; Digital arithmetic; Multiplying circuits; Polynomials; Summing circuits", } @InProceedings{Skaf:1994:LHI, author = "Ali Skaf and Jean-Michel Muller and Alain Guyot", editor = "Anonymous", booktitle = "{ESSCIRC '94: Twentieth European Solid-State Circuits Conference. Ulm, Germany. September 20--22, 1994}", title = "On-Line Hardware Implementation for Complex Exponential and Logarithm", publisher = "{\'E}ditions Fronti{\`e}res", address = "B. P. 33. 91192 Gif-sur-Yvette Cedex, France", pages = "252--255", year = "1994", ISBN = "2-86332-160-9", ISBN-13 = "978-2-86332-160-7", bibdate = "Fri Sep 29 10:32:36 2023", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, } @Article{Sorenson:1994:TFG, author = "J. Sorenson", title = "Two Fast {GCD} Algorithms", journal = j-J-ALG, volume = "16", number = "1", pages = "110--144", month = jan, year = "1994", CODEN = "JOALDV", DOI = "https://doi.org/10.1006/jagm.1994.1006", ISSN = "0196-6774 (print), 1090-2678 (electronic)", ISSN-L = "0196-6774", bibdate = "Tue Dec 11 09:15:38 MST 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jalg.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0196677484710066", acknowledgement = ack-nhfb, fjournal = "Journal of Algorithms", journal-URL = "http://www.sciencedirect.com/science/journal/01966774", } @Article{Spouge:1994:CGD, author = "John L. Spouge", title = "Computation of the Gamma, Digamma, and Trigamma Functions", journal = j-SIAM-J-NUMER-ANAL, volume = "31", number = "3", pages = "931--944", month = jun, year = "1994", CODEN = "SJNAAM", ISSN = "0036-1429 (print), 1095-7170 (electronic)", ISSN-L = "0036-1429", MRclass = "33B15 (30E10 33-04 40-04 65D20)", MRnumber = "95g:33002", MRreviewer = "E. Kaucher", bibdate = "Mon Jan 20 15:27:00 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "SIAM Journal on Numerical Analysis", journal-URL = "http://epubs.siam.org/sinum", } @InProceedings{Temme:1994:CAI, author = "N. M. Temme", title = "Computational aspects of incomplete gamma functions with large complex parameters", crossref = "Zahar:1994:ACF", pages = "551--562", year = "1994", bibdate = "Sat Feb 18 15:02:52 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", series = "International Series of Numerical Mathematics", acknowledgement = ack-nhfb, } @Article{Temme:1994:SAI, author = "N. M. Temme", title = "A Set of Algorithms for the Incomplete Gamma Functions", journal = j-PROBAB-ENGRG-INFORM-SCI, volume = "8", number = "2", pages = "291--307", month = apr, year = "1994", CODEN = "????", DOI = "https://doi.org/10.1017/S0269964800003417", ISSN = "0269-9648 (print), 1469-8951 (electronic)", ISSN-L = "0269-9648", bibdate = "Thu Aug 24 08:18:58 MDT 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/probab-engrg-inform-sci.bib", URL = "https://www.cambridge.org/core/product/F5677268895A0805A0BDF31E4B20A106", acknowledgement = ack-nhfb, ajournal = "Probab. Engrg. Inform. Sci.", fjournal = "Probability in the Engineering and Informational Sciences", journal-URL = "http://www.journals.cambridge.org/jid_PES", onlinedate = "01 July 2009", } @Article{Timmermann:1994:CFV, author = "D. Timmermann and B. Rix and H. Hahn and B. J. Hosticka", title = "A {CMOS} floating-point vector-arithmetic unit", journal = j-IEEE-J-SOLID-STATE-CIRCUITS, volume = "29", number = "5", pages = "634--639", month = may, year = "1994", CODEN = "IJSCBC", DOI = "https://doi.org/10.1109/4.284719", ISSN = "0018-9200 (print), 1558-173X (electronic)", ISSN-L = "0018-9200", bibdate = "Tue Dec 12 09:29:07 MST 1995", bibsource = "https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", abstract = "This work describes a floating-point arithmetic unit based on the CORDIC algorithm. The unit computes a full set of high level arithmetic and elementary functions: multiplication, division, (co)sine, hyperbolic (co)sine, square root, natural logarithm, inverse (hyperbolic) tangent, vector norm, and phase. The chip has been integrated in 1.6 mu m double-metal n-well CMOS technology and achieves a normalized peak performance of 220 MFLOPS.", acknowledgement = ack-nhfb, affiliation = "Fraunhofer Inst. of Microelectron. Circuits and Syst., Duisburg, Germany", classification = "B1265B (Logic circuits); B2570D (CMOS integrated circuits); C5120 (Logic and switching circuits); C5220P (Parallel architecture); C5230 (Digital arithmetic methods)", fjournal = "IEEE Journal of Solid-State Circuits", keywords = "1.6 Micron; 220 MFLOPS; CORDIC algorithm; Cosine; Division; Double-metal n-well CMOS technology; Floating-point vector-arithmetic unit; Hyperbolic sine; Inverse tangent; Multiplication; Natural logarithm; Phase; Sine; Square root; Vector norm", numericalindex = "Size 1.6E-06 m; Computer speed 2.2E+08 FLOPS", pubcountry = "USA", thesaurus = "CMOS integrated circuits; Digital arithmetic; Integrated logic circuits; Parallel architectures; Pipeline processing; Vector processor systems", } @Article{Turner:1994:SRM, author = "Stephen M. Turner", title = "Square roots mod $p$", journal = j-AMER-MATH-MONTHLY, volume = "101", number = "5", pages = "443--449", month = may, year = "1994", CODEN = "AMMYAE", ISSN = "0002-9890 (print), 1930-0972 (electronic)", ISSN-L = "0002-9890", MRclass = "11A07", MRnumber = "95c:11004", MRreviewer = "David Lee Hilliker", bibdate = "Wed Dec 3 17:17:33 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, fjournal = "American Mathematical Monthly", journal-URL = "https://www.jstor.org/journals/00029890.htm", } @Book{Watanabe:1994:MSP, author = "T. (Tsutomu) Watanabe and Makoto Natori and Tsutomu Oguni", title = "Mathematical Software for the {P.C.} and Work Stations: a Collection of {Fortran 77} Programs", publisher = pub-NORTH-HOLLAND, address = pub-NORTH-HOLLAND:adr, pages = "xiv + 387", month = jun, year = "1994", ISBN = "0-444-82000-0", ISBN-13 = "978-0-444-82000-6", LCCN = "QA 76.73 F25 F6813 1994", bibdate = "Sun Sep 28 10:42:07 MDT 1997", bibsource = "http://www.amazon.com/exec/obidos/ISBN=0444820000/wholesaleproductA/; http://www.cbooks.com/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "Translation of: FORTRAN 77 ni yoru suchi keisan sofutowea.", price = "US\$178.50", URL = "http://www.cbooks.com/sqlnut/SP/search/gtsumt?source=&isbn=0444820000", acknowledgement = ack-nhfb, alttitle = "{Fortran 77} ni yoru suchi keisan sofutowea. English.", keywords = "Fortran 77 (computer program language); Numerical analysis --- Use of --- Computers; {Fortran 77} (Computer program language)", } @InProceedings{Wong:1994:FEE, author = "W. F. Wong and E. Goto", title = "Fast evaluation of the elementary functions in double precision", crossref = "Mudge:1994:PTS", bookpages = "xi + 621", pages = "349--358", year = "1994", bibdate = "Tue Dec 12 09:29:07 MST 1995", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", abstract = "One of the most spectacular development in computer technology is the growth in memory density and speed. It is with this development in mind that we intend to tackle the old problem of computing the elementary functions. Since the dawn of computing, the fast and accurate computation of the elementary functions has been a constant concern of numerical computing. It now seems possible to use tables of sizes in the range of megabits to aid in such computation. To this end, in this paper, we propose a method called ATA-M (Add-Table Lookup-Add with Multiplication) for evaluating polynomials with the aid of tables. When applied to the elementary functions, we obtained a set of algorithms which computes the reciprocal, square root, exponential, sine, cosine, logarithm, are tangent and the hyperbolic functions in about 3 to 4 double precision floating point multiplication time and utilizing about 2 Mbyte of tables.", acknowledgement = ack-nhfb, affiliation = "Dept. of Inf. Syst. and Comput. Sci., Nat. Univ. of Singapore, Singapore", classification = "C4130 (Interpolation and function approximation); C5230 (Digital arithmetic methods)", confdate = "4-7 Jan. 1994", conflocation = "Wailea, HI, USA", confsponsor = "IEEE; ACM; Univ. Hawaii; Univ. Hawaii Coll. Bus. Admin", keywords = "Add-Table Lookup-Add with Multiplication; ATA-M; Double precision; Elementary functions; Floating point multiplication time; Hyperbolic functions; Memory density; Memory speed; Numerical computing; Polynomials", pubcountry = "USA", thesaurus = "Digital arithmetic; Polynomials; Table lookup", } @Article{Wong:1994:FHB, author = "W. F. Wong and E. Goto", title = "Fast Hardware-Based Algorithms for Elementary Function Computations Using Rectangular Multipliers", journal = j-IEEE-TRANS-COMPUT, volume = "43", number = "3", pages = "278--294", month = mar, year = "1994", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/12.272429", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Thu Jul 7 07:13:54 MDT 2011", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput1990.bib", URL = "https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=272429", abstract = "As the name suggests, elementary functions play a vital role in scientific computations. Yet due to their inherent nature, they are a considerable computing task by themselves. Not surprisingly, since the dawn of computing, the goal of speeding up elementary function computation has been pursued. This paper describes new hardware based algorithms for the computation of the common elementary functions, namely division, logarithm, reciprocal square root, arc tangent, sine and cosine. These algorithms exploit microscopic parallelism using specialized hardware with heavy use of truncation based on detailed accuracy analysis. The contribution of this work lies in the fact that these algorithms are very fast and yet are accurate. If we let the time to perform an IEEE Standard 754 double precision floating point multiplication be $ \tau_\times $, our algorithms to achieve roughly $ 3.68 \tau_\times $, $ 4.56 \tau_\times $, $ 5.25 \tau_\times $, $ 3.69 \tau_\times $, $ 7.06 \tau_\times $, and $ 6.5 \tau_\times $, for division, logarithm, square root, exponential, are tangent and complex exponential (sine and cosine) respectively. The trade-off is the need for tables and some specialized hardware. The total amount of tables required, however, is less than 128 Kbytes. We discuss the hardware, algorithmic and accuracy aspects of these algorithms.", acknowledgement = ack-nj # " and " # ack-nhfb, affiliation = "Dept. of Inf. Syst. and Comput. Sci., Nat. Univ. of Singapore, Singapore", ajournal = "IEEE Trans. Comput.", classification = "C4110 (Error analysis in numerical methods); C5230 (Digital arithmetic methods)", fjournal = "IEEE Transactions on Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", keywords = "Arc tangent; Common elementary functions; Cosine; Elementary function computations; Floating point multiplication; Hardware-based algorithms; Microscopic parallelism; Reciprocal square root; Rectangular multipliers; Scientific computations; Sine", pubcountry = "USA", thesaurus = "Digital arithmetic; Error analysis", } @Article{Xu:1994:VPC, author = "Guo Liang Xu and Jia Kai Li", title = "Variable precision computation of elementary functions. ({Chinese})", journal = j-J-NUMER-METHODS-COMPUT-APPL, volume = "15", number = "3", pages = "161--171", year = "1994", ISSN = "1000-3266", MRclass = "65D20 (65Y20)", MRnumber = "MR1357336 (96i:65013)", bibdate = "Sat Jan 11 17:44:01 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Journal on Numerical Methods and Computer Applications. Shuzhi Jisuan yu Jisuanji Yingyong", } @Article{Abad:1995:CRC, author = "Julio Abad and Javier Sesma", title = "Computation of the Regular Confluent Hypergeometric Function", journal = j-MATHEMATICA-J, volume = "5", number = "4", pages = "??--??", month = "Fall", year = "1995", CODEN = "????", ISSN = "1047-5974 (print), 1097-1610 (electronic)", ISSN-L = "1047-5974", bibdate = "Sat Nov 6 13:34:06 MDT 2010", bibsource = "http://www.mathematica-journal.com/issue/v5i4/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.mathematica-journal.com/issue/v5i4/article/abad/index.html", acknowledgement = ack-nhfb, fjournal = "Mathematica Journal", journal-URL = "http://www.mathematica-journal.com/", } @Article{Amos:1995:RAP, author = "D. E. Amos", title = "A Remark on {Algorithm 644}: a Portable Package for {Bessel} Functions of a Complex Argument and Nonnegative Order", journal = j-TOMS, volume = "21", number = "4", pages = "388--393", month = dec, year = "1995", CODEN = "ACMSCU", DOI = "https://doi.org/10.1145/212066.212078", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Tue Mar 09 10:24:54 1999", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib", note = "See \cite{Amos:1986:APP,Amos:1990:RPP,Kodama:2007:RA}.", URL = "http://www.acm.org/pubs/citations/journals/toms/1995-21-4/p388-amos/", acknowledgement = ack-rfb, fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", keywords = "complex Airy Functions; complex Bessel functions; derivatives of Airy functions; H, I, J, K, and Y Bessel functions; log gamma function", subject = "G.1.0 [Numerical Analysis]: General -- numerical algorithms; G.1.m [Numerical Analysis]: Miscellaneous; G.m [Mathematics of Computing]: Miscellaneous", } @Article{Bagby:1995:CNP, author = "Richard J. Bagby", title = "Calculating normal probabilities", journal = j-AMER-MATH-MONTHLY, volume = "102", number = "1", pages = "46--48", month = jan, year = "1995", CODEN = "AMMYAE", ISSN = "0002-9890 (print), 1930-0972 (electronic)", ISSN-L = "0002-9890", MRclass = "65D20", MRnumber = "96f:65021", bibdate = "Wed Dec 3 17:17:33 MST 1997", bibsource = "http://www.jstor.org/journals/00029890.htm; https://www.math.utah.edu/pub/tex/bib/amermathmonthly1990.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "American Mathematical Monthly", journal-URL = "https://www.jstor.org/journals/00029890.htm", } @Article{Baratchart:1995:RIE, author = "L. Baratchart and E. B. Saff and F. Wielonsky", title = "Rational interpolation of the exponential function", journal = j-CAN-J-MATH, volume = "47", number = "??", pages = "1121--1147", month = "????", year = "1995", CODEN = "CJMAAB", DOI = "https://doi.org/10.4153/CJM-1995-058-6", ISSN = "0008-414X (print), 1496-4279 (electronic)", ISSN-L = "0008-414X", bibdate = "Sat Sep 10 15:39:05 MDT 2011", bibsource = "http://cms.math.ca/cjm/v47/; https://www.math.utah.edu/pub/tex/bib/canjmath1990.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Canadian Journal of Mathematics = Journal canadien de math{\'e}matiques", journal-URL = "http://cms.math.ca/cjm/", } @TechReport{Borwein:1995:EARa, author = "Peter Borwein", title = "An Efficient Algorithm for the {Riemann} Zeta Function", type = "Report", institution = "Department of Mathematics \& Statistics, Simon Fraser University", address = "Burnaby, BC V5A 1S6, Canada", pages = "9", day = "20", month = jan, year = "1995", bibdate = "Thu Sep 01 18:09:22 2022", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://docserver.carma.newcastle.edu.au/107", abstract = "A very simple class of algorithms for the computation of the Riemann-zeta function to arbitrary precision in arbitrary domains is proposed. These algorithms out perform the standard methods based on Euler--Maclaurin summation, are easier to implement and are easier to analyse.", acknowledgement = ack-nhfb, author-dates = "10 May 1953--23 August 2020", ORCID-numbers = "Borwein, Jonathan/0000-0002-1263-0646", } @InProceedings{Borwein:1995:EARb, author = "P. Borwein", editor = "????", booktitle = "{CMS} Conference Proceedings", title = "An efficient algorithm for the {Riemann} zeta function", volume = "27", publisher = "Canadian Mathematical Society", address = "616 Cooper Street, Ottawa, ON, K1R 5J2, Canada", pages = "29--34", month = jan, year = "1995", bibdate = "Wed Jun 28 08:27:14 2023", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "https://web.archive.org/web/20140602151514/http://www.cecm.sfu.ca/personal/pborwein/PAPERS/P155.pdf", acknowledgement = ack-nhfb, xxURL = "http://www.cecm.sfu.ca/personal/pborwein/PAPERS/P155.pdf", } @Article{Carlson:1995:NCR, author = "B. C. Carlson", title = "Numerical computation of real or complex elliptic integrals", journal = j-NUMER-ALGORITHMS, volume = "10", number = "1--2", pages = "13--26", month = jul, year = "1995", CODEN = "NUALEG", DOI = "https://doi.org/10.1007/BF02198293", ISSN = "1017-1398 (print), 1572-9265 (electronic)", ISSN-L = "1017-1398", MRclass = "33Exx (33-04 65D20)", MRnumber = "1 345 407", bibdate = "Fri Nov 6 18:06:29 MST 1998", bibsource = "http://www.math.psu.edu/dna/contents/na.html; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "Special functions (Torino, 1993)", abstract = "Algorithms for numerical computation of symmetric elliptic integrals of all three kinds are improved in several ways and extended to complex values of the variables (with some restrictions in the case of the integral of the third kind). Numerical check values, consistency checks, and relations to Legendre's integrals and Bulirsch's integrals are included.", acknowledgement = ack-nhfb, classification = "B0290R (Integral equations); C4180 (Integral equations)", conflocation = "Torino, Italy; 14-15 Oct. 1993", conftitle = "International Joint Symposium on Special Functions and Artificial Intelligence", corpsource = "Ames Lab., Iowa State Univ., Ames, IA, USA", fjournal = "Numerical Algorithms", journal-URL = "http://link.springer.com/journal/11075", keywords = "Bulirsch's integrals; complex elliptic integrals; complex values; consistency checks; elliptic equations; integral equations; Legendre's integrals; numerical analysis; numerical check values; numerical computation algorithms; real elliptic integrals", pubcountry = "Switzerland", treatment = "T Theoretical or Mathematical", } @Article{Chaudhry:1995:DGI, author = "M. Aslam Chaudhry and S. M. Zubair", title = "On the decomposition of generalized incomplete gamma functions with applications to {Fourier} transforms", journal = j-J-COMPUT-APPL-MATH, volume = "59", number = "3", pages = "253--284", day = "30", month = may, year = "1995", CODEN = "JCAMDI", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Sat Feb 25 12:24:37 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath1990.bib", URL = "http://www.sciencedirect.com/science/article/pii/037704279400026W", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @InProceedings{Chen:1995:UCA, author = "San-Gee Chen and Chieh-Chih Li", booktitle = "{IEEE} Signal Processing Society Workshop on {VLSI} Signal Processing, {VIII, 1995}", title = "A unified cellular array for multiplication, division and square root", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "533--541", year = "1995", CODEN = "????", ISSN = "????", bibdate = "Sat Jul 16 15:28:14 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, summary = "A unified fast, small-area processor capable of executing multiplication, division and square-root operations, all starting from MSD is proposed. Unlike the existing designs which require both addition and subtraction operations, and complicated \ldots{}", } @Article{Das:1995:IFC, author = "D. Das and K. Mukhopadhyaya and B. P. Sinha", title = "Implementation of four common functions on an {LNS} co-processor", journal = j-IEEE-TRANS-COMPUT, volume = "44", number = "1", pages = "155--161", month = jan, year = "1995", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/12.367997", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Sat Jul 16 16:14:38 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", summary = "We propose a scheme for evaluating four commonly used functions namely, (1) inverse trigonometric functions, (2) trigonometric functions, (3) the exponential function, and (4) the logarithmic function with the help of a logarithmic number system (\ldots{}).", } @Article{Daumas:1995:MRR, author = "Marc Daumas and Christophe Mazenc and Xavier Merrheim and Jean-Michel Muller", title = "Modular range reduction: a new algorithm for fast and accurate computation of the elementary functions", journal = j-J-UCS, volume = "1", number = "3", pages = "162--175 (electronic)", year = "1995", CODEN = "????", ISSN = "0948-6968", ISSN-L = "0948-6968", MRclass = "68M07 (68Q20)", MRnumber = "1 390 003", bibdate = "Sat Jan 11 17:44:01 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "J.UCS: Journal of Universal Computer Science", journal-URL = "http://www.jucs.org/jucs", } @Article{Doman:1995:SAP, author = "B. G. S. Doman and C. J. Pursglove and W. M. Coen", title = "A Set of {Ada} Packages for High Precision Calculations", journal = j-TOMS, volume = "21", number = "4", pages = "416--431", month = dec, year = "1995", CODEN = "ACMSCU", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Tue Nov 14 09:57:55 1995", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-rfb # "\slash " # ack-nhfb, fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", keywords = "accuracy; Ada; arithmetic elementary-function evaluation; floating-point; multiple-precision portable software", subject = "G.1.0 [Numerical Analysis]: General -- computer arithmetic; G.1.2 [Numerical Analysis]: Approximation -- elementary function approximation; G.4 [Mathematics of Computing]: Mathematical Software -- algorithm analysis; efficiency; portability", } @Article{Driver:1995:NQH, author = "Kathy Driver", title = "Nondiagonal quadratic {Hermite--Pad{\'e}} approximation to the exponential function", journal = j-J-COMPUT-APPL-MATH, volume = "65", number = "1--3", pages = "125--134", day = "29", month = dec, year = "1995", CODEN = "JCAMDI", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Sat Feb 25 13:02:25 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath1990.bib", URL = "http://www.sciencedirect.com/science/article/pii/0377042795001069", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @Article{Goano:1995:ACC, author = "Michele Goano", title = "{Algorithm 745}: Computation of the Complete and Incomplete {Fermi--Dirac} Integral", journal = j-TOMS, volume = "21", number = "3", pages = "221--232", month = sep, year = "1995", CODEN = "ACMSCU", DOI = "https://doi.org/10.1145/210089.210090", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Tue Mar 09 10:19:43 1999", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "See remark \cite{Goano:1997:RA7}", URL = "http://www.acm.org/pubs/citations/journals/toms/1995-21-3/p221-goano/", acknowledgement = ack-rfb # "\slash " # ack-nhfb, fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", keywords = "asymptotic expansions; confluent hypergeometric functions; convergence acceleration; e[k] transforms; epsilon algorithm; Euler transformation; Fermi--Dirac integral; incomplete Fermi--Dirac integral; incomplete gamma function; Levin's u transform; Riemann's zeta function", subject = "G.1.2 [Mathematics of Computing]: Approximation; G.4 [Mathematics of Computing]: Mathematical Software; J.2 [Computer Applications]: Physical Sciences and Engineering", } @Article{Hobson:1995:EMR, author = "R. F. Hobson and M. W. Fraser", title = "An efficient maximum-redundancy radix-$8$ {SRT} division and square-root method", journal = j-IEEE-J-SOLID-STATE-CIRCUITS, volume = "30", number = "1", pages = "29--38", month = jan, year = "1995", CODEN = "IJSCBC", DOI = "https://doi.org/10.1109/4.350197", ISSN = "0018-9200 (print), 1558-173X (electronic)", ISSN-L = "0018-9200", bibdate = "Sat Jul 16 15:28:13 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", abstract = "A new approach to integrating hardware multiplication, division, and square-root is presented. We use a fully integrated control path which simultaneously reduces part of the redundant partial-remainder and performs a truncated multiplication of the next quotient or square-root digit by the divisor or square-root value. A separate (parallel) full precision iterative multiplier is used for partial remainder production. Strategic details of a radix-8 implementation are discussed. It is shown that a maximally redundant digit set is a viable choice for high performance in this case.", acknowledgement = ack-nhfb, affiliation = "Sch. of Comput. Sci., Simon Fraser Univ., Burnaby, BC, Canada", classification = "B1265B (Logic circuits); B2570D (CMOS integrated circuits); C5230 (Digital arithmetic methods)", fjournal = "IEEE Journal of Solid-State Circuits", keywords = "1.2 Mum; CMOS adder cell; CMOS divider; Division; IEEE floating point algorithm; Integrated control path; Maximally redundant digit set; Maximum-redundancy radix-8 SRT algorithm; Multiplication; Parallel iterative multiplier; Partial remainder production; Redundant partial-remainder; Square-root method; Table lookup", numericalindex = "Size 1.2E-06 m", pubcountry = "USA", summary = "A new approach to integrating hardware multiplication, division, and square-root is presented. We use a fully integrated control path which simultaneously reduces part of the redundant partial-remainder and performs a truncated multiplication of the \ldots{}", thesaurus = "Adders; CMOS digital integrated circuits; Digital arithmetic; Dividing circuits; Floating point arithmetic; Multiplying circuits", } @InProceedings{Ito:1995:EIA, author = "M. Ito and N. Takagi and S. Yajima", title = "Efficient Initial Approximation and Fast Converging Methods for Division and Square Root", crossref = "Knowles:1995:PSC", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "2--9", month = jul, year = "1995", CODEN = "????", ISSN = "????", bibdate = "Sat Jul 16 15:28:14 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; OCLC Proceedings database", acknowledgement = ack-sfo # " and " # ack-nhfb, summary = "Efficient initial approximations and fast converging algorithms are important to achieve the desired precision faster at lower hardware cost in multiplicative division and square root. In this paper, a new initial approximation method for division, \ldots{}", } @Book{Jeffrey:1995:HMF, author = "Alan Jeffrey", title = "Handbook of Mathematical Formulas and Integrals", publisher = pub-ACADEMIC, address = pub-ACADEMIC:adr, pages = "xxiv + 410", year = "1995", ISBN = "0-08-052301-3, 0-12-382580-6 (e-book), 1-4832-9514-1 (e-book)", ISBN-13 = "978-0-08-052301-9, 978-0-12-382580-3 (e-book), 978-1-4832-9514-5 (e-book)", LCCN = "QA47 .J38 1995", bibdate = "Wed Jun 12 14:33:38 MDT 2024", bibsource = "fsz3950.oclc.org:210/WorldCat; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/book/9780123825803", abstract = "If there is a formula to solve a given problem in mathematics, you will find it in Alan Jeffrey's \booktitle{Handbook of Mathematical Formulas and Integrals}. Thanks to its unique thumb-tab indexing feature, answers are easy to find based upon the type of problem they solve. The Handbook covers important formulas, functions, relations, and methods from algebra, trigonometric and exponential functions, combinatorics, probability, matrix theory, calculus and vector calculus, both ordinary and partial differential equations, Fourier series, orthogonal polynomials, and Laplace transforms. Based on Gradshteyn and Ryzhik's Table of Integrals, Series, and Products, Fifth Edition (edited by Jeffrey), but far more accessible and written with particular attention to the needs of students and practicing scientists and engineers, this book is an essential resource. Affordable and authoritative, it is the first place to look for help and a rewarding place to browse.", acknowledgement = ack-nhfb, subject = "Mathematics; Tables; Formulae; Math{\'e}matiques; Formules; formulas (algorithms); Mathematics.", tableofcontents = "0: Quick Reference List of Frequently Used Data \\ 0.1: Useful Identities / 1 \\ 0.2: Complex Relationships / 2 \\ 0.3: Constants / 2 \\ 0.4: Derivatives of Elementary Functions / 3 \\ 0.5: Rules of Differentiation and Integration / 3 \\ 0.6: Standard Integrals / 4 \\ 0.7: Standard Series / 11 \\ 0.8: Geometry / 13 \\ 1: Numerical, Algebraic, and Analytical Results for Series and Calculus \\ 1.1: Algebraic Results Involving Real and Complex Numbers / 25 \\ 1.2: Finite Sums / 29 \\ 1.3: Bernoulli and Euler Numbers and Polynomials 37 \\ 1.4: Determinants / 47 \\ 1.5: Matrices / 55 \\ 1.6: Permutations and Combinations / 62 \\ 1.7: Partial Fraction Decomposition / 63 \\ 1.8: Convergence of Series / 66 \\ 1.9: Infinite Products / 71 \\ 1.10: Functional Series / 73 \\ 1.11: Power Series / 74 \\ 1.12: Taylor Series / 79 \\ 1.13: Fourier Series / 81 \\ 1.14: Asymptotic Expansions / 85 \\ 1.15: Basic Results from the Calculus / 86 \\ 2: Functions and Identities \\ 2.1: Complex Numbers and Trigonometric and Hyperbolic Functions / 101 \\ 2.2: Logarithms and Exponentials / 112 \\ 2.3: The Exponential Function / 114 \\ 2.4: Trigonometric Identities / 115 \\ 2.5: Hyperbolic Identities / 121 \\ 2.6: The Logarithm / 126 \\ 2.7: Inverse Trigonometric and Hyperbolic Functions / 128 \\ 2.8: Series Representations of Trigonometric and Hyperbolic Functions / 133 \\ 2.9: Useful Limiting Values and Inequalities Involving Elementary Functions / 136 \\ 3: Derivatives of Elementary Functions \\ 3.1: Derivatives of Algebraic, Logarithmic, and Exponential Functions / 139 \\ 3.2: Derivatives of Trigonometric Functions / 140 \\ 3.3: Derivatives of Inverse Trigonometric Functions / 140 \\ 3.4: Derivatives of Hyperbolic Functions / 141 \\ 3.5: Derivatives of Inverse Hyperbolic Functions 142 \\ 4: Indefinite Integrals of Algebraic Functions \\ 4.1: Algebraic and Transcendental Functions / 145 \\ 4.2: Indefinite Integrals of Rational Functions 146 \\ 4.3: Nonrational Algebraic Functions / 158 \\ 5: Indefinite Integrals of Exponential Functions \\ 5.1: Basic Results / 167 \\ 6: Indefinite Integrals of Logarithmic Functions \\ 6.1: Combinations of Logarithms and Polynomials 173 \\ 7: Indefinite Integrals of Hyperbolic Functions \\ 7.1: Basic Results / 179 \\ 7.2: Integrands Involving Powers of sinh(bx) or cosh(bx) / 180 \\ 7.3: Integrands Involving (a [plus or minus] bx)[superscript m] sinh(cx) or (a + bx)[superscript m] cosh(cx) / 181 \\ 7.4: Integrands Involving x[superscript m] sinh[superscript n] x or x[superscript m] cosh[superscript n] x / 183 \\ 7.5: Integrands Involving x[superscript m] sinh[superscript -n] x or x[superscript m] cosh[superscript -n] x / 183 \\ 7.6: Integrands Involving (1 [plus or minus] cosh x)[superscript -m] / 185 \\ 7.7: Integrands Involving sinh(ax)cosh[superscript -n] x or cosh(ax)sinh[superscript -n] x / 185 \\ 7.8: Integrands Involving sinh(ax + b) and cosh(cx + d) / 186 \\ 7.9: Integrands Involving tanh kx and coth kx 188 \\ 7.10: Integrands Involving (a + bx)[superscript m] sinh kx or (a + bx)[superscript m] cosh kx / 189 \\ 8: Indefinite Integrals Involving Inverse Hyperbolic Functions \\ 8.1: Basic Results / 191 \\ 8.2: Integrands Involving x[superscript -n] arcsinh(x/a) or x[superscript -n] arccosh(x/a) 193 \\ 8.3: Integrands Involving x[superscript n] arctanh(x/a) or x[superscript n] arccoth(x/a) 194 \\ 8.4: Integrands Involving x[superscript -n] arctanh(x/a) or x[superscript -n] arccoth(x/a) / 195 \\ 9: Indefinite Integrals of Trigonometric Functions \\ 9.1: Basic Results / 197 \\ 9.2: Integrands Involving Powers of x and Powers of sin x or cos x / 197 \\ 9.3: Integrands Involving tan x and/or cot x 205 \\ 9.4: Integrands Involving sin x and cos x / 207 \\ 9.5: Integrands Involving Sines and Cosines with Linear Arguments and Powers of x / 211 \\ 10: Indefinite Integrals of Inverse Trigonometric Functions \\ 10.1: Integrands Involving Powers of x and Powers of Inverse Trigonometric Functions / 215 \\ 11: The Gamma, Beta, Pi, and Psi Functions \\ 11.1: The Euler Integral and Limit and Infinite Product Representations for [Gamma] (x) / 221 \\ 12: Elliptic Integrals and Functions \\ 12.1: Elliptic Integrals / 229 \\ 12.2: Jacobian Elliptic Functions / 235 \\ 12.3: Derivatives and Integrals / 237 \\ 12.4: Inverse Jacobian Elliptic Functions / 237 \\ 13: Probability Integrals and the Error Function \\ 13.1: Normal Distribution / 239 \\ 13.2: The Error Function / 242 \\ 14: Fresnel Integrals, Sine and Cosine Integrals \\ 14.1: Definitions, Series Representations, and Values at Infinity / 245 \\ 14.2: Definitions, Series Representations, and Values at Infinity / 247 \\ 15: Definite Integrals \\ 15.1: Integrands Involving Powers of x / 249 \\ 15.2: Integrands Involving Trigonometric Functions 251 \\ 15.3: Integrands Involving the Exponential Function / 254 \\ 15.4: Integrands Involving the Hyperbolic Function 256 \\ 15.5: Integrands Involving the Logarithmic Function / 256 \\ 16: Different Forms of Fourier Series \\ 16.1: Fourier Series for f(x) on -[pi] [less than or equal] x [less than or equal] [pi] / 257 \\ 16.2: Fourier Series for f(x) on -L [less than or equal] x [less than or equal] L / 258 \\ 16.3: Fourier Series for f(x) on a [less than or equal] x [less than or equal] b / 258 \\ 16.4: Half-Range Fourier Cosine Series for f(x) on 0 [less than or equal] x [less than or equal] [pi] 259 \\ 16.5: Half-Range Fourier Cosine Series for f(x) on 0 [less than or equal] x [less than or equal] L 259 \\ 16.6: Half-Range Fourier Sine Series for f(x) on 0 [less than or equal] x [less than or equal] [pi] 260 \\ 16.7: Half-Range Fourier Sine Series for f(x) on 0 [less than or equal] x [less than or equal] L 260 \\ 16.8: Complex (Exponential) Fourier Series for f(x) on -[pi] [less than or equal] x [less than or equal] [pi] / 260 \\ 16.9: Complex (Exponential) Fourier Series for f(x) on -L [less than or equal] x [less than or equal] L 261 \\ 16.10: Representative Examples of Fourier Series 261 \\ 16.11: Fourier Series and Discontinuous Functions 265 \\ 17: Bessel Functions \\ 17.1: Bessel's Differential Equation / 269 \\ 17.2: Series Expansions for J[subscript v](x) and Y[subscript v](x) / 270 \\ 17.3: Bessel Functions of Fractional Order / 272 \\ 17.4: Asymptotic Representations for Bessel Functions / 273 \\ 17.5: Zeros of Bessel Functions / 273 \\ 17.6: Bessel's Modified Equation / 274 \\ 17.7: Series Expansions for I[subscript v](x) and K[subscript v](x) / 276 \\ 17.8: Modified Bessel Functions of Fractional Order / 277 \\ 17.9: Asymptotic Representations of Modified Bessel Functions / 278 \\ 17.10: Relationships between Bessel Functions 278 \\ 17.11: Integral Representations of J[subscript n](x), I[subscript n](x), and K[subscript n](x) / 281 \\ 17.12: Indefinite Integrals of Bessel Functions 281 \\ 17.13: Definite Integrals Involving Bessel Functions / 282 \\ 17.14: Spherical Bessel Functions / 283 \\ 18: Orthogonal Polynomials \\ 18.2: Legendre Polynomials P[subscript n](x) 286 \\ 18.3: Chebyshev Polynomials T[subscript n](x) and U[subscript n](x) / 290 \\ 18.4: Laguerre Polynomials L[subscript n](x) 294 \\ 18.5: Hermite Polynomials H[subscript n](x) / 296 \\ 19: Laplace Transformation \\ 20: Fourier Transforms \\ 21: Numerical Integration \\ 21.1: Classical Methods / 315 \\ 22: Solutions of Standard Ordinary Differential Equations \\ 22.2: Separation of Variables / 323 \\ 22.3: Linear First-Order Equations / 323 \\ 22.4: Bernoulli's Equation / 324 \\ 22.5: Exact Equations / 325 \\ 22.6: Homogeneous Equations / 325 \\ 22.7: Linear Differential Equations / 326 \\ 22.8: Constant Coefficient Linear Differential Equations \\ Homogeneous Case / 327 \\ 22.9: Linear Homogeneous Second-Order Equation 330 \\ 22.10: Constant Coefficient Linear Differential Equations \\ Inhomogeneous Case / 331 \\ 22.11: Linear Inhomogeneous Second-Order Equation 333 \\ 22.12: Determination of Particular Integrals by the Method of Undetermined Coefficients / 334 \\ 22.13: The Cauchy-Euler Equation / 336 \\ 22.14: Legendre's Equation / 337 \\ 22.15: Bessel's Equations / 337 \\ 22.16: Power Series and Frobenius Methods / 339 \\ 22.17: The Hypergeometric Equation / 344 \\ 22.18: Numerical Methods / 345 \\ 23: Vector Analysis \\ 23.1: Scalars and Vectors / 353 \\ 23.2: Scalar Products / 358 \\ 23.3: Vector Products / 359 \\ 23.4: Triple Products / 360 \\ 23.5: Products of Four Vectors / 361 \\ 23.6: Derivatives of Vector Functions of a Scalar t / 361 \\ 23.7: Derivatives of Vector Functions of Several Scalar Variables / 362 \\ 23.8: Integrals of Vector Functions of a Scalar Variable t / 363 \\ 23.9: Line Integrals / 364 \\ 23.10: Vector Integral Theorems / 366 \\ 23.11: A Vector Rate of Change Theorem / 368 \\ 23.12: Useful Vector Identities and Results / 368 \\ 24: Systems of Orthogonal Coordinates \\ 24.1: Curvilinear Coordinates / 369 \\ 24.2: Vector Operators in Orthogonal Coordinates 371 \\ 24.3: Systems of Orthogonal Coordinates / 371 \\ 25: Partial Differential Equations and Special Functions \\ 25.1: Fundamental Ideas / 381 \\ 25.2: Method of Separation of Variables / 385 \\ 25.3: The Sturm--Liouville Problem and Special Functions / 387 \\ 25.4: A First-Order System and the Wave Equation 390 \\ 25.5: Conservation Equations (Laws) / 391 \\ 25.6: The Method of Characteristics / 392 \\ 25.7: Discontinuous Solutions (Shocks) / 396 \\ 25.8: Similarity Solutions / 398 \\ 25.9: Burgers's Equation, the KdV Equation, and the KdVB Equation / 400 \\ 26: The z-Transform \\ 26.1: The z-Transform and Transform Pairs / 403 \\ 27: Numerical Approximation \\ 27.2: Economization of Series / 411 \\ 27.3: Pade Approximation / 413 \\ 27.4: Finite Difference Approximations to Ordinary and Partial Derivatives / 415", } @Article{Krattenthaler:1995:HHM, author = "C. Krattenthaler", title = "{HYP} and {HYPQ}: {Mathematica} packages for the manipulation of binomial sums and hypergeometric series, respectively $q$-binomial sums and basic hypergeometric series", journal = j-J-SYMBOLIC-COMP, volume = "20", number = "5--6", pages = "737--744", month = nov # "--" # dec, year = "1995", CODEN = "JSYCEH", ISSN = "0747-7171 (print), 1095-855X (electronic)", ISSN-L = "0747-7171", MRclass = "05Axx (11Bxx 33-04 33Cxx 33Dxx)", MRnumber = "1 395 424", bibdate = "Sat May 10 15:54:09 MDT 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "Symbolic computation in combinatorics $ \Delta_1 $ (Ithaca, NY, 1993).", acknowledgement = ack-nhfb, classcodes = "C7310 (Mathematics computing); C1100 (Mathematical techniques)", corpsource = "Inst. fur Math., Wien Univ., Austria", fjournal = "Journal of Symbolic Computation", journal-URL = "http://www.sciencedirect.com/science/journal/07477171", keywords = "basic hypergeometric; binomial sums; HYP; hypergeometric series; HYPQ; Mathematica packages; mathematics computing; packages; q-binomial sums; series; series (mathematics); software; symbol manipulation", treatment = "T Theoretical or Mathematical", } @InProceedings{Kwan:1995:CII, author = "H. Kwan and R. L. {Nelson, Jr.} and E. E. {Swartzlander, Jr.}", title = "Cascaded Implementation of an Iterative Inverse-Square-Root Algorithm, with Overflow Lookahead", crossref = "Knowles:1995:PSC", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "115--122", year = "1995", CODEN = "????", ISSN = "????", bibdate = "Sat Jul 16 15:28:14 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; OCLC Proceedings database", acknowledgement = ack-nhfb, summary = "We present an unconventional method of computing the inverse of the square root. It implements the equivalent of two iterations of a well-known multiplicative method to obtain 24-bit mantissa accuracy. We implement each ``iteration'' as a \ldots{}", } @InProceedings{Lang:1995:VHR, author = "T. Lang and P. Montuschi", title = "Very-High Radix Combined Division and Square Root with Prescaling and Selection by Rounding", crossref = "Knowles:1995:PSC", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "124--131", year = "1995", CODEN = "????", ISSN = "????", bibdate = "Sat Jul 16 15:28:14 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; OCLC Proceedings database", acknowledgement = ack-nhfb, summary = "An algorithm for square root with prescaling is developed and combined with a similar scheme for division. An implementation is described, evaluated and compared with other combined div/sqrt \ldots{}", } @Book{Lau:1995:NLC, author = "H. T. (Hang Tong) Lau", title = "A Numerical Library in {C} for Scientists and Engineers", publisher = pub-CRC, address = pub-CRC:adr, pages = "xvii + 795", year = "1995", ISBN = "0-8493-7376-X", ISBN-13 = "978-0-8493-7376-3", LCCN = "QA76.73.C15 L38 1995", bibdate = "Fri Sep 26 14:29:10 MDT 2008", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; z3950.loc.gov:7090/Voyager", URL = "http://www.loc.gov/catdir/enhancements/fy0744/94037928-d.html", acknowledgement = ack-nhfb, shorttableofcontents = "Introduction \\ 1 Elementary Procedures \\ 2 Algebraic Evaluations \\ 3 Linear Algebra \\ 4 Analytic Evaluations \\ 5 Analytic Problems \\ 6 Special Functions \\ 7 Interpolation and Approximation", subject = "C (Computer program language)", tableofcontents = "1. Elementary Procedures \\ 1.1. Real vector and matrix Initialization \\ 1.2. Real vector and matrix Duplication \\ 1.3. Real vector and matrix Multiplication \\ 1.4. Real vector vector products \\ 1.5. Real matrix vector products \\ 1.6. Real matrix matrix products \\ 1.7. Real vector and matrix Elimination \\ 1.8. Real vector and matrix Interchanging \\ 1.9. Real vector and matrix Rotation \\ 1.10. Real vector and matrix Norms \\ 1.11. Real vector and matrix Scaling \\ 1.12. Complex vector and matrix Multiplication \\ 1.13. Complex vector and matrix Scalar products \\ 1.14. Complex vector and matrix Elimination \\ 1.15. Complex vector and matrix Rotation \\ 1.16. Complex vector and matrix Norms \\ 1.17. Complex vector and matrix Scaling \\ 1.18. Complex monadic operations \\ 1.19. Complex dyadic operations \\ 1.20. Long integer arithmetic \\ 2. Algebraic Evaluations \\ 2.1. Evaluation of polynomials in Grunert form \\ 2.2. Evaluation of general orthogonal polynomials \\ 2.3. Evaluation of Chebyshev polynomials \\ 2.4. Evaluation of Fourier series \\ 2.5. Evaluation of continued fractions \\ 2.6. Transformation of polynomial representation \\ 2.7. Operations on orthogonal polynomials \\ 3. Linear Algebra \\ 3.1. Full real general matrices \\ 3.2. Real Symmetric positive definite matrices \\ 3.3. General real symmetric matrices \\ 3.4. Real full rank overdetermined systems \\ 3.5. Other real matrix problems \\ 3.6. Real sparse non-symmetric band matrices \\ 3.7. Real sparse non-symmetric tridiagonal matrices \\ 3.8. Sparse symmetric positive definite band matrices \\ 3.9. Symmetric positive definite tridiagonal matrices \\ 3.10. Sparse real matrices \\ Iterative methods \\ 3.11. Similarity transformation \\ 3.12. Other transformations \\ 3.13. The (ordinary) eigenvalue problem \\ 3.14. The generalized eigenvalue problem \\ 3.15. Singular values \\ 3.16. Zeros of polynomials \\ 4. Analytic Evaluations \\ 4.1. Evaluation of an infinite series \\ 4.2. Quadrature \\ 4.3. Numerical differentiation \\ 5. Analytic Problems \\ 5.1. Non-linear equations \\ 5.2. Unconstrained optimization \\ 5.3. Overdetermined nonlinear systems \\ 5.4. Differential equations \\ Initial value problems \\ 5.5. Two point boundary value problems \\ 5.6. Two-dimensional boundary value problems \\ 5.7. Parameter estimation in differential equations \\ 6. Special Functions \\ 6.1. Elementary functions \\ 6.2. Exponential integral \\ 6.3. Gamma function \\ 6.4. Error function \\ 6.5. Bessel functions of integer order \\ 6.6. Bessel functions of real order \\ 7. Interpolation and Approximation \\ 7.1. Real data in one dimension \\ Appendix B: Prototype Declarations \\ Appendix C: Procedure Descriptions \\ Appendix D: Memory Management Utilities", } @InProceedings{Leeser:1995:VSR, author = "M. Leeser and J. O'Leary", booktitle = "Proceedings of the {IEEE} International Conference on Computer Design: {VLSI} in Computers and Processors, {ICCD '95}", title = "Verification of a subtractive radix-$2$ square root algorithm and implementation", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "526--531", year = "1995", CODEN = "????", ISSN = "????", bibdate = "Sat Jul 16 15:28:14 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, summary = "Many modern microprocessors implement floating point square root hardware using subtractive algorithms. Such processors include the HP PA7200, the MIPS R4400, and the Intel Pentium. The Intel Pentium division bug highlights the importance of \ldots{}", } @Article{Lether:1995:MAZ, author = "F. G. Lether and P. R. Wenston", title = "Minimax approximations to the zeros of {$ P_n(x) $} and {Gauss--Legendre} quadrature", journal = j-J-COMPUT-APPL-MATH, volume = "59", number = "2", pages = "245--252", day = "19", month = may, year = "1995", CODEN = "JCAMDI", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Sat Feb 25 12:24:37 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath1990.bib", URL = "http://www.sciencedirect.com/science/article/pii/0377042794000305", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @Article{Lewanowicz:1995:AMC, author = "Stanis{\l}aw Lewanowicz and Stefan Paszkowski", title = "An analytic method for convergence acceleration of certain hypergeometric series", journal = j-MATH-COMPUT, volume = "64", number = "210", pages = "691--713", month = apr, year = "1995", CODEN = "MCMPAF", DOI = "https://doi.org/10.2307/2153446", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "33C45 (65B10 65D20)", MRnumber = "1277769 (95h:33006)", MRreviewer = "Anton Hut'a", bibdate = "Thu Dec 1 17:36:45 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp1990.bib", acknowledgement = ack-nhfb, affiliation = "Inst. of Comput. Sci., Wroclaw Univ., Poland", classcodes = "B0290 (Numerical analysis); C4100 (Numerical analysis)", corpsource = "Inst. of Comput. Sci., Wroclaw Univ., Poland", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "analytic method; convergence acceleration; convergence of numerical methods; fast converging expansions; hypergeometric; iterated transformation; mathematical constants; series; series (mathematics)", treatment = "T Theoretical or Mathematical", } @Article{Liu:1995:SRV, author = "S.-I. Liu", title = "Square-rooting and vector summation circuits using current conveyors", journal = "IEE Proceedings on Circuits, Devices and Systems [see also IEE Proceedings G - Circuits, Devices and Systems]", volume = "142", number = "4", pages = "223--226", month = aug, year = "1995", CODEN = "????", ISSN = "????", bibdate = "Sat Jul 16 15:28:13 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, summary = "New analogue squaring, square-rooting and vector summation circuits using current conveyors (CCIIs) are presented. They consist of MOS transistors biased in the triode region, a buffered unity-gain inverting amplifier, resistors and CCIIs. A general \ldots{}", } @Article{Louie:1995:VPS, author = "Marianne E. Louie and Milo{\v{s}} D. Ercegovac", title = "A Variable-Precision Square Root Implementation for Field Programmable Gate Arrays", journal = j-J-SUPERCOMPUTING, volume = "9", number = "3", pages = "315--336", month = sep, year = "1995", CODEN = "JOSUED", DOI = "https://doi.org/10.1007/BF01212874", ISSN = "0920-8542 (print), 1573-0484 (electronic)", ISSN-L = "0920-8542", bibdate = "Wed Jul 6 11:13:09 MDT 2005", bibsource = "http://springerlink.metapress.com/openurl.asp?genre=issue&issn=0920-8542&volume=9&issue=3; http://www.wkap.nl/issuetoc.htm/0920-8542+9+3+1995; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/jsuper.bib", URL = "http://www.springerlink.com/openurl.asp?genre=article&issn=0920-8542&volume=9&issue=3&spage=315; http://www.wkap.nl/oasis.htm/95692", acknowledgement = ack-nhfb, affiliation = "Dept. of Comput. Sci., California Univ., Los Angeles, CA, USA", classification = "C5120 (Logic and switching circuits); C5230 (Digital arithmetic methods)", corpsource = "Dept. of Comput. Sci., California Univ., Los Angeles, CA, USA", fjournal = "The Journal of Supercomputing", journal-URL = "http://link.springer.com/journal/11227", keywords = "digital arithmetic; field programmable gate arrays; square root; square root implementation; variable-precision; Xilinx XC4010", treatment = "P Practical", } @Article{Lucas:1995:EII, author = "S. K. Lucas and H. A. Stone", title = "Evaluating infinite integrals involving {Bessel} functions of arbitrary order", journal = j-J-COMPUT-APPL-MATH, volume = "64", number = "3", pages = "217--231", year = "1995", CODEN = "JCAMDI", DOI = "https://doi.org/10.1016/0377-0427(95)00142-5", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Thu Jul 8 13:22:49 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/B6TYH-4002HHC-J/2/54f1e67d9bea3e951acc3c39556ab452", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", keywords = "$\varepsilon$-algorithm; Bessel functions; Bessel zeros; infinite integration; mW transform; quadrature", remark = "This paper examines several methods for accurately integrating oscillatory functions, such as products of $ f(x) $ with a trigonometric function or a Bessel function. It also discusses finding zeros of Bessel functions, and sequence acceleration techniques.", } @Article{Luther:1995:HAT, author = "Wolfram Luther", title = "Highly accurate tables for elementary functions", journal = j-BIT-NUM-MATH, volume = "35", number = "3", pages = "352--360", month = sep, year = "1995", CODEN = "BITTEL, NBITAB", DOI = "https://doi.org/10.1007/BF01732609", ISSN = "0006-3835 (print), 1572-9125 (electronic)", ISSN-L = "0006-3835", MRclass = "65D20 (68U05)", MRnumber = "97h:65024", bibdate = "Wed Jan 4 18:52:24 MST 2006", bibsource = "http://springerlink.metapress.com/openurl.asp?genre=issue&issn=0006-3835&volume=35&issue=3; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.mai.liu.se/BIT/contents/bit35.html; http://www.springerlink.com/openurl.asp?genre=article&issn=0006-3835&volume=35&issue=3&spage=352", acknowledgement = ack-nhfb, journal-URL = "http://link.springer.com/journal/10543", keywords = "elementary functions", } @InProceedings{Lynch:1995:KTF, author = "T. Lynch and A. Ahmed and M. Schulte and T. Callaway", title = "The {K5} Transcendental Functions", crossref = "Knowles:1995:PSC", pages = "163--171", year = "1995", bibdate = "Mon May 20 06:05:24 MDT 2002", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; OCLC Proceedings database", URL = "http://mesa.ece.wisc.edu/publications/cp_1995-04.pdf", acknowledgement = ack-nhfb, } @Article{Maroni:1995:IRB, author = "P. Maroni", title = "An integral representation for the {Bessel} form", journal = j-J-COMPUT-APPL-MATH, volume = "57", number = "1--2", pages = "251--260", day = "20", month = feb, year = "1995", CODEN = "JCAMDI", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Sat Feb 25 12:24:35 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath1990.bib", URL = "http://www.sciencedirect.com/science/article/pii/0377042793E0249L", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @InProceedings{Matsubara:1995:NBS, author = "G. Matsubara and N. Ide and H. Tago and S. Suzuki and N. Goto", title = "30-ns 55-b Shared Radix $2$ Division and Square Root Using a Self-Timed Circuit", crossref = "Knowles:1995:PSC", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "98--105", year = "1995", CODEN = "????", ISSN = "????", bibdate = "Sat Jul 16 15:28:14 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; OCLC Proceedings database", acknowledgement = ack-nhfb, summary = "A shared radix 2 division and square root implementation using a self-timed circuit is presented. The same execution time for division and square root is achieved by using an on-the-fly digit decoding and a root multiple generation technique. Most \ldots{}", } @Article{Miller:1995:RCF, author = "A. R. Miller and I. S. Moskowitz", title = "Reduction of a class of {Fox--Wright} psi functions for certain rational parameters", journal = j-COMPUT-MATH-APPL, volume = "30", number = "11", pages = "73--82", month = dec, year = "1995", CODEN = "CMAPDK", ISSN = "0898-1221 (print), 1873-7668 (electronic)", ISSN-L = "0898-1221", bibdate = "Wed Mar 1 21:48:19 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/computmathappl1990.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/089812219500165U", acknowledgement = ack-nhfb, fjournal = "Computers and Mathematics with Applications", journal-URL = "http://www.sciencedirect.com/science/journal/08981221", remark = "From the abstract: ``The Fox--Wright Psi function is a special case of Fox's $H$-function and a generalization of the generalized hypergeometric function. In the present paper, we show that the Psi function reduces to a single generalized hypergeometric function when certain of its parameters are integers and to a finite sum of generalized hypergeometric functions when these parameters are rational numbers.''", } @Article{Montuschi:1995:RRI, author = "P. Montuschi and L. Ciminiera", title = "A remark on {``Reducing iteration time when result digit is zero for radix-$2$ SRT division and square root with redundant remainders''}", journal = j-IEEE-TRANS-COMPUT, volume = "44", number = "1", pages = "144--146", month = jan, year = "1995", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/12.368000", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Sat Jul 16 15:28:13 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "See \cite{Montuschi:1993:RIT}.", acknowledgement = ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", summary = "In a previous paper by P. Montuschi and L. Ciminiera (ibid., vol. 42, no.2 p239-246, Feb 1993), an architecture for shared radix 2 division and square root has been presented whose main characteristic is the ability to avoid any addition/subtraction, \ldots{}", } @Article{Muldoon:1995:EZB, author = "Martin E. Muldoon", title = "Electrostatics and zeros of {Bessel} functions", journal = j-J-COMPUT-APPL-MATH, volume = "65", number = "1--3", pages = "299--308", day = "29", month = dec, year = "1995", CODEN = "JCAMDI", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Sat Feb 25 13:02:25 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath1990.bib", URL = "http://www.sciencedirect.com/science/article/pii/0377042795001182", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @Article{OLeary:1995:NRI, author = "J. O'Leary and M. Leeser and J. Hickey and M. Aagaard", title = "Non-Restoring Integer Square Root: a Case Study in Design by Principled Optimization", journal = j-LECT-NOTES-COMP-SCI, volume = "901", pages = "52--??", year = "1995", CODEN = "LNCSD9", ISSN = "0302-9743 (print), 1611-3349 (electronic)", ISSN-L = "0302-9743", bibdate = "Sat May 11 13:45:32 MDT 1996", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, fjournal = "Lecture Notes in Computer Science", journal-URL = "http://link.springer.com/bookseries/558", } @Article{Paszkowski:1995:QHS, author = "S. Paszkowski", title = "Quasipower and hypergeometric series---construction and evaluation", journal = j-NUMER-ALGORITHMS, volume = "10", number = "3--4", pages = "337--361", month = oct, year = "1995", CODEN = "NUALEG", ISSN = "1017-1398 (print), 1572-9265 (electronic)", ISSN-L = "1017-1398", MRclass = "41A58 (33-04 65B99)", MRnumber = "96k:41042", MRreviewer = "Walter Schempp", bibdate = "Fri Nov 6 18:06:29 MST 1998", bibsource = "http://www.math.psu.edu/dna/contents/na.html; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/numeralgorithms.bib", acknowledgement = ack-nhfb, classification = "B0290F (Interpolation and function approximation); B0290Z (Other numerical methods); C4130 (Interpolation and function approximation); C4190 (Other numerical methods)", corpsource = "Inst. for Low Temp. and Structure Res., Polish Acad. of Sci., Wroclaw, Poland", fjournal = "Numerical Algorithms", journal-URL = "http://link.springer.com/journal/11075", keywords = "approximation theory; Euler's dilogarithm; Hermite-Pade approximation; hypergeometric series; Levin's transforms; Pade approximants; power series; quasipower; recurrence relations; series (mathematics); transforms", pubcountry = "Switzerland", treatment = "P Practical; T Theoretical or Mathematical", } @InProceedings{Prabhu:1995:MRD, author = "J. A. Prabhu and G. B. Zyner", title = "{167 MHz} Radix-$8$ Divide and Square Root Using Overlapped Radix-$2$ Stages", crossref = "Knowles:1995:PSC", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "155--162", year = "1995", CODEN = "????", ISSN = "????", bibdate = "Sat Jul 16 15:28:14 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; OCLC Proceedings database", acknowledgement = ack-nhfb, summary = "UltraSPARC's IEEE-754 compliant floating point divide and square root implementation is presented. Three overlapping stages of SRT radix-$2$ quotient selection logic enable an effective radix-$8$ calculation at 167 MHz while only a single radix-$2$ \ldots{}", } @InProceedings{Schwarz:1995:RQC, author = "E. M. Schwarz", title = "Rounding for quadratically converging algorithms for division and square root", crossref = "Singh:1995:CRT", volume = "1", pages = "600--603", month = oct, year = "1995", CODEN = "????", ISSN = "????", bibdate = "Sat Jul 16 15:28:14 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-sfo # " and " # ack-nhfb, summary = "Exactly rounded results are necessary for many architectures such as IEEE 754 standard. For division and square root, rounding is easy to perform if a remainder is available. But for quadratically converging algorithms, the remainder is not \ldots{}", } @Article{Sidhu:1995:EIF, author = "Satinder S. Sidhu", title = "Elliptic Integrals and Functions", journal = j-COMPUT-PHYS, volume = "9", number = "3", pages = "268--276", month = may # "\slash " # jun, year = "1995", CODEN = "CPHYE2", ISSN = "0894-1866 (print), 1558-4208 (electronic)", ISSN-L = "0894-1866", bibdate = "Thu Feb 02 18:05:53 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Computers in Physics", } @Article{Smith:1995:CFA, author = "Roger Alan Smith", title = "A Continued-Fraction Analysis of Trigonometric Argument Reduction", journal = j-IEEE-TRANS-COMPUT, volume = "44", number = "11", pages = "1348--1351", month = nov, year = "1995", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/12.475133", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Fri Dec 08 10:21:28 2006", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", abstract = "The calculation of a trigonometric function of a large argument x is effectively carried out by finding the integer $N$ and $ 0 \leq \alpha < 1 $ such that $ x = (N + \alpha) \pi / 4 $. This reduction modulo $ \pi / 4 $ makes it possible to calculate a trigonometric function of a reduced argument, either $ \alpha \pi / 4 $ or $ (1 - \alpha) \pi / 4 $, which lies in the interval $ (0, \pi / 4) $. Payne and Hanek [1] described an efficient algorithm for computing $ \alpha $ to a predetermined level of accuracy. They noted that if $x$ differs only slightly from an integral multiple $ \pi / 2 $, the reduction must be carried out quite accurately to avoid loss of significance in the reduced argument. We present a simple method using continued fractions for determining, for all numbers $x$ for which the greatest number of insignificant leading bits occur. Applications are made IEEE single-precision and double-precision formats and two extended-precision formats.", acknowledgement = ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", keywords = "argument reduction; computer arithmetic; continued fractions; nonlinear optimization; Payne/Hanek radian reduction; range reduction; trigonometric functions", } @InProceedings{Soderquist:1995:APC, author = "Peter Soderquist and Miriam Leeser", title = "An Area\slash Performance Comparison of Subtractive and Multiplicative Divide\slash Square Root Implementations", crossref = "Knowles:1995:PSC", pages = "132--139", month = jul, year = "1995", bibdate = "Mon May 20 06:05:24 MDT 2002", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; OCLC Proceedings database", URL = "http://www.acsel-lab.com/arithmetic/arith12/papers/ARITH12_Soderquist.pdf", acknowledgement = ack-sfo # " and " # ack-nhfb, keywords = "ARITH-12", } @Book{Varchenko:1995:MHF, author = "A. N. (Aleksandr Nikolaevich) Varchenko", title = "Multidimensional Hypergeometric Functions and Representation Theory of {Lie} Algebras and Quantum Groups", volume = "21", publisher = pub-WORLD-SCI, address = pub-WORLD-SCI:adr, pages = "ix + 371", year = "1995", ISBN = "981-02-1880-X", ISBN-13 = "978-981-02-1880-5", LCCN = "QA353.H9 V37 1995", bibdate = "Sat Oct 30 21:12:24 MDT 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; z3950.loc.gov:7090/Voyager", series = "Advanced series in mathematical physics", acknowledgement = ack-nhfb, subject = "Hypergeometric functions; Kac-Moody algebras; Representations of Lie algebras; Representations of quantum groups", tableofcontents = "1. Introduction \\ 2. Construction of complexes calculating homology of the complement of a configuration \\ 3. Construction of homology complexes for discriminantal configuration \\ 4. Algebraic interpretation of chain complexes of a discriminantal configuration \\ 5. Quasiisomorphism of two-sided Hochschild complexes to suitable one-sided Hochschild complexes \\ 6. Bundle properties of a discriminantal configuration \\ 7. R-matrix for the two-sided Hochschild complexes \\ 8. Monodromy \\ 9. R-matrix operator as the canonical element, quantum doubles \\ 10. Hypergeometric integrals \\ 11. Kac--Moody Lie algebras without Serre's relations and their doubles \\ 12. Hypergeometric integrals of a discriminantal configuration \\ 13. Resonances at infinity \\ 14. Degenerations of discriminantal configurations \\ 15. Remarks on homology groups of a configuration with coefficients in local systems more general than complex one-dimensional", } @Book{Vilenkin:1995:RLG, author = "N. Ja. (Naum Jakovlevich) Vilenkin and A. U. (Anatolii Ulsianovich) Klimyk", title = "Representation of {Lie} Groups and Special Functions: Recent Advances", volume = "316", publisher = pub-KLUWER, address = pub-KLUWER:adr, pages = "xvi + 497", year = "1995", ISBN = "0-7923-3210-5", ISBN-13 = "978-0-7923-3210-7", LCCN = "QA176 .V55 1995", bibdate = "Sat Oct 30 16:43:03 MDT 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; z3950.loc.gov:7090/Voyager", series = "Mathematics and its applications", URL = "http://www.loc.gov/catdir/enhancements/fy0823/95108075-d.html; http://www.loc.gov/catdir/enhancements/fy0823/95108075-t.html", acknowledgement = ack-nhfb, remark = "Translated to English from Russian by V. A. Groza and A. A. Groza.", subject = "Representations of Lie groups; Functions, Special; Integral transforms", tableofcontents = "Preface / xiii \\ 1. $h$-Harmonic Polynomials, $h$-Hankel Transform, and Coxeter Groups / 1 \\ 2. Symmetric Polynomials and Symmetric Functions / 67 \\ 3. Hypergeometric Functions Related to Jack Polynomials / 185 \\ 4. Clebsch--Gordan Coefficients and Racah Coefficients of Finite Dimensional Representations / 265 \\ 5. Clebsch--Gordan Coefficients of the Group $U(n)$ and Related Generalizations of Hypergeometric Functions / 317 \\ 6. Gel'fand Hypergeometric Functions / 393 \\ Bibliography / 463 \\ Supplementary Bibliography / 484 \\ Bibliography Notes / 488 \\ Subject Index / 494", } @Article{Vrahatis:1995:RPP, author = "M. N. Vrahatis and O. Ragos and T. Skiniotis and F. A. Zafiropoulos and T. N. Grapsa", title = "{RFSFNS}: a portable package for the numerical determination of the number and the calculation of roots of {Bessel} functions", journal = j-COMP-PHYS-COMM, volume = "92", number = "2--3", pages = "252--266", month = dec, year = "1995", CODEN = "CPHCBZ", DOI = "https://doi.org/10.1016/0010-4655(95)00115-9", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", bibdate = "Mon Feb 13 21:30:01 MST 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/compphyscomm1990.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "See erratum \cite{Vrahatis:1999:ESP}.", URL = "http://www.sciencedirect.com/science/article/pii/0010465595001159", acknowledgement = ack-nhfb, fjournal = "Computer Physics Communications", journal-URL = "http://www.sciencedirect.com/science/journal/00104655", } @Book{Watson:1995:TTB, author = "G. N. (George Neville) Watson", title = "A Treatise on the Theory of {Bessel} Functions", publisher = pub-CAMBRIDGE, address = pub-CAMBRIDGE:adr, edition = "Second", pages = "vi + 804", year = "1995", ISBN = "0-521-48391-3 (paperback), 0-521-06743-X (hardcover)", ISBN-13 = "978-0-521-48391-9 (paperback), 978-0-521-06743-0 (hardcover)", LCCN = "QA408 .W2 1995", bibdate = "Sat Apr 19 09:15:26 MDT 2008", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcw.bib; z3950.loc.gov:7090/Voyager", series = "Cambridge mathematical library", URL = "http://www.loc.gov/catdir/description/cam028/96139881.html; http://www.loc.gov/catdir/toc/cam029/96139881.html", acknowledgement = ack-nhfb, remark = "First published 1922. Second edition 1944. Reprinted 1966.", subject = "Bessel functions", tableofcontents = "1. Bessel functions before 1826 \\ 2. The Bessel coefficients \\ 3. Bessel functions \\ 4. Differential equations \\ 5. Miscellaneous properties of Bessel functions \\ 6. Integral representations of Bessel functions \\ 7. Asymptotic expansions of Bessel functions \\ 8. Bessel functions of large order \\ 9. Polynomials associated with Bessel functions \\ 10. Functions associated with Bessel functions \\ 11. Addition theorems \\ 12. Definite integrals \\ 13. Infinitive integrals \\ 14. Multiple integrals \\ 15. The zeros of Bessel functions \\ 16. Neumann series and Lommel's functions of two variables \\ 17. Kapteyn series \\ 18. Series of Fourier-Bessel and Dini \\ 19. Schl{\"o}mlich series \\ 20. The tabulation of Bessel functions \\ Tables of Bessel functions \\ Bibliography \\ Indices", xxauthor = "G. N. Watson", } @Article{Wong:1995:EHS, author = "W. F. Wong and Yoshio Oyanagi and Eiichi Goto", title = "Evaluation of the {Hitachi S-3800} Supercomputer Using Six Benchmarks", journal = j-IJSAHPC, volume = "9", number = "1", pages = "58--70", month = "Spring", year = "1995", CODEN = "IJSAE9", ISSN = "0890-2720", bibdate = "Tue Feb 18 09:07:32 MST 1997", bibsource = "Compendex database; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", abstract = "The S-3000 series is the third generation of Hitachi's supercomputers. It is claimed to be currently the fastest single processor supercomputer. In this paper, we introduce the S-3000 and, using six benchmarks we designed, evaluate a member of this series of supercomputer, the top of the range S-3800, against its predecessor, the Hitachi HITAC S-820. Our purpose is to determine in what areas the S-3800 is an improvement over its predecessor. The suite of benchmarks include kernels for random number generation, elementary function computation, FFT, dense matrix operations, SOR, and list vector (scatter\slash gather) operations. The use of small-to medium-sized kernels, as opposed to large application benchmarks, help to better understand the behavior of the machine. Our findings support the claim that the S-3000 series is at least twice as fast as the previous generation of Hitachi supercomputers.", acknowledgement = ack-nhfb, affiliation = "Dept. of Inf. Syst. and Comput. Sci., Nat. Univ. of Singapore", affiliationaddress = "Singapore", classification = "722.4; 921.3; 921.6; 922.2", fjournal = "International Journal of Supercomputer Applications and High Performance Computing", journal-URL = "http://hpc.sagepub.com/content/by/year", journalabr = "Int J Supercomput Appl High Perform Comput", keywords = "Benchmarks; Computer selection and evaluation; Computer testing; Fast Fourier transforms; Fastest single processor; Hitachi supercomputer; Matrix algebra; Medium sized kernels; Performance; Random number generation; Supercomputers; Vectors", } @Article{Wong:1995:FEE, author = "W. F. Wong and E. Goto", title = "Fast evaluation of the elementary functions in single precision", journal = j-IEEE-TRANS-COMPUT, volume = "44", number = "3", pages = "453--457", month = mar, year = "1995", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/12.372037", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Thu Dec 14 11:25:18 MST 1995", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", abstract = "In this paper we introduce a new method for the fast evaluation of the elementary functions in single precision based on the evaluation of truncated Taylor series using a difference method. We assume the availability of large and fast (at least for read purposes) memory. We call this method the ATA (Add-Table lookup-Add) method. As the name implies, the hardware required for the method are adders (both two/ and multi/operand adders) and fast tables. For IEEE single precision numbers our initial estimates indicate that we can calculate the basic elementary functions, namely reciprocal, square root, logarithm, exponential, trigonometric and inverse trigonometric functions, within the latency of two to four floating point multiplies.", acknowledgement = ack-nhfb, affiliation = "Dept. of Inf. Syst. and Comput. Sci., Nat. Univ. of Singapore, Singapore", ajournal = "IEEE Trans. Comput.", classification = "C4110 (Error analysis in numerical methods); C5230 (Digital arithmetic methods)", fjournal = "IEEE Transactions on Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", keywords = "Adders; Difference method; Elementary functions; Fast evaluation; Floating point multiplies; Inverse trigonometric functions; Logarithm functions; Reciprocal; Single precision; Square root; Truncated Taylor series", pubcountry = "USA", thesaurus = "Error analysis; Floating point arithmetic", } @Article{Ypma:1995:HDN, author = "Tjalling J. Ypma", title = "Historical Development of the {Newton--Raphson} Method", journal = j-SIAM-REVIEW, volume = "37", number = "4", pages = "531--551", month = dec, year = "1995", CODEN = "SIREAD", DOI = "https://doi.org/10.1137/1037125", ISSN = "0036-1445 (print), 1095-7200 (electronic)", ISSN-L = "0036-1445", MRclass = "01A05 (65-03)", MRnumber = "97b:01003", MRreviewer = "M. Z. Nashed", bibdate = "Sat Mar 29 09:55:35 MDT 2014", bibsource = "Compendex database; http://epubs.siam.org/toc/siread/37/4; http://www.siam.org/journals/sirev/sirev374.htm; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/siamreview.bib", URL = "http://epubs.siam.org/23425.htm; http://link.aip.org/link/?SIR/37/531/1", abstract = "This expository paper traces the development of the Newton--Raphson method for solving nonlinear algebraic equations through the extant notes, letters, and publications of Isaac Newton, Joseph Raphson, and Thomas Simpson. It is shown how Newton's formulation differed from the iterative process of Raphson, and that Simpson was the first to give a general formulation, in terms of fluxional calculus, applicable to nonpolynomial equations. Simpson's extension of the method to systems of equations is exhibited.", acknowledgement = ack-nhfb, affiliation = "Western Washington Univ", affiliationaddress = "Bellingham, WA, USA", classification = "921.1; 921.2; 921.6", fjournal = "SIAM Review", journal-URL = "http://epubs.siam.org/sirev", journalabr = "SIAM Rev", keywords = "Algebra; Algorithms; Approximation theory; Differentiation (calculus); Finite difference method; Fluxional calculus; Isaac Newton; Iterative methods; Joseph Raphson; Linearization; Newton--Raphson method; Nonlinear algebraic equations; Nonlinear equations; Nonpolynomial equation; Polynomials; Secant method; Thomas Simpson", onlinedate = "December 1995", } @Article{Zhang:1995:TMAa, author = "J. Zhang and J. A. Belward", title = "Tau-method approximations for the {Bessel} function {$ Y_0 (z) $}", journal = j-COMPUT-MATH-APPL, volume = "30", number = "7", pages = "5--14", month = oct, year = "1995", CODEN = "CMAPDK", DOI = "https://doi.org/10.1016/0898-1221(95)00120-N", ISSN = "0898-1221 (print), 1873-7668 (electronic)", ISSN-L = "0898-1221", bibdate = "Sun Jun 12 08:33:42 2011", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/l/lanczos-cornelius.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/089812219500120N", acknowledgement = ack-nhfb, fjournal = "Computers and Mathematics with Applications", journal-URL = "http://www.sciencedirect.com/science/journal/08981221", } @Article{Zhang:1995:TMAb, author = "J. Zhang", title = "Tau-method approximations for the {Bessel} function {$ Y_1 (z) $}", journal = j-COMPUT-MATH-APPL, volume = "30", number = "7", pages = "15--19", month = oct, year = "1995", CODEN = "CMAPDK", DOI = "https://doi.org/10.1016/0898-1221(95)00121-E", ISSN = "0898-1221 (print), 1873-7668 (electronic)", ISSN-L = "0898-1221", bibdate = "Sun Jun 12 09:26:01 2011", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/l/lanczos-cornelius.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/089812219500121E", acknowledgement = ack-nhfb, fjournal = "Computers and Mathematics with Applications", journal-URL = "http://www.sciencedirect.com/science/journal/08981221", remark = "From p. 18 (conclusions section), ``We may use this method to approximate the Bessel functions of other integer orders. \ldots{} It is therefore advisable to use the recurrence relations of the Bessel functions to compute function values for $ n > 1 $ \ldots''. [This is a similar limitation as with Chebyshev and minimax polynomial approximations: they are valid only for a single order the Bessel function.]", } @InProceedings{Ahrendt:1996:FHC, author = "Timm Ahrendt", title = "Fast High-Precision Computations of Complex Square Roots", crossref = "LakshmanYN:1996:IPI", pages = "142--149", year = "1996", bibdate = "Thu Mar 12 08:43:16 MST 1998", bibsource = "http://www.acm.org/pubs/toc/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "http://www.acm.org:80/pubs/citations/proceedings/issac/236869/p142-ahrendt/", acknowledgement = ack-nhfb, keywords = "algebraic computation; algorithms; ISSAC; measurement; SIGNUM; SIGSAM; symbolic computation", subject = "{\bf I.1.2} Computing Methodologies, SYMBOLIC AND ALGEBRAIC MANIPULATION, Algorithms, Algebraic algorithms. {\bf G.1.0} Mathematics of Computing, NUMERICAL ANALYSIS, General, Numerical algorithms. {\bf F.1.1} Theory of Computation, COMPUTATION BY ABSTRACT DEVICES, Models of Computation, Bounded-action devices. {\bf G.1.5} Mathematics of Computing, NUMERICAL ANALYSIS, Roots of Nonlinear Equations, Iterative methods. {\bf G.1.2} Mathematics of Computing, NUMERICAL ANALYSIS, Approximation.", xxtitle = "Fast high-precision computation of complex square roots", } @Book{Baker:1996:PA, author = "George A. (George Allen) {Baker, Jr.} and P. R. Graves-Morris", title = "{Pad{\'e}} approximants", volume = "59", publisher = pub-CAMBRIDGE, address = pub-CAMBRIDGE:adr, edition = "Second", pages = "xiv + 746", year = "1996", DOI = "https://doi.org/10.1017/CBO9780511530074", ISBN = "1-107-08857-7 (e-book), 0-511-53007-2 (e-book), 0-511-95902-8 (e-book), 0-521-13509-5 (paperback), 0-521-45007-1 (hardcover)", ISBN-13 = "978-1-107-08857-3 (e-book), 978-0-511-53007-4 (e-book), 978-0-511-95902-8 (e-book), 978-0-521-13509-2 (paperback), 978-0-521-45007-2 (hardcover)", LCCN = "QA221 .B35 1996", bibdate = "Sat May 3 06:18:56 MDT 2025", bibsource = "fsz3950.oclc.org:210/WorldCat; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", series = "Encyclopedia of mathematics and its applications", abstract = "The Pad{\'e} approximant of a given power series is a rational function of numerator degree $L$ and denominator degree $M$ whose power series agrees with the given one up to degree $ L + M$ inclusively. A collection of Pad{\'e} approximants formed by using a suitable set of values of $L$ and $M$ often provides a means of obtaining information about the function outside its circle of convergence, and of more rapidly evaluating the function within its circle of convergence. Applications of these ideas in physics, chemistry, electrical engineering, and other areas have led to a large number of generalizations of Pad{\'e} approximants that are tailor-made for specific applications. Applications to statistical mechanics and critical phenomena are extensively covered, and there are newly extended sections devoted to circuit design, matrix Pad{\'e} approximation, computational methods, and integral and algebraic approximants. The book is written with a smooth progression from elementary ideas to some of the frontiers of research in approximation theory. Its main purpose is to make the various techniques described accessible to scientists, engineers, and other researchers who may wish to use them, while also presenting the rigorous mathematical theory.", acknowledgement = ack-nhfb, author-dates = "1932--", remark = "The first edition of this book was reviewed in 1982 as 'the most extensive treatment of Pad{\'e} approximants actually available'. This second edition has been thoroughly updated, with a substantial chapter on multiseries approximants. Applications to statistical mechanics and critical phenomena are extensively covered, and there are extended sections devoted to circuit design, matrix Pad{\`e} approximation, and computational methods. This succinct and straightforward treatment will appeal to scientists, engineers, and mathematicians alike.", subject = "Pad{\'e} approximant; Approximation theory; Functions; Approximants de Pad{\'e}; Th{\'e}orie de l'approximation; Fonctions (Math{\'e}matiques); functions (mathematics); Approximation theory; Functions; Pad{\'e} approximant", tableofcontents = "1. Introduction and definitions \\ 2. Elementary developments \\ 3. Pad{\`e} approximants and numerical methods \\ 4. Connection with continued fractions \\ 5. Stieltjes series and Polya series \\ 6. Convergence theory \\ 7. Extensions of Pad{\`e} approximations \\ 8. Multiseries approximants \\ 9. Connection with integral equations and quantum mechanics \\ 10. Connection with numerical analysis \\ 11. Connection with quantum field theory \\ Appendix: A FORTRAN function", } @Article{Balla:1996:SCB, author = "K. Balla and V. H. Linh", title = "The simultaneous computation of {Bessel} functions of first and second kind", journal = j-COMPUT-MATH-APPL, volume = "31", number = "4--5", pages = "87--97", month = feb # "\slash " # mar, year = "1996", CODEN = "CMAPDK", ISSN = "0898-1221 (print), 1873-7668 (electronic)", ISSN-L = "0898-1221", bibdate = "Wed Mar 1 21:48:26 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/computmathappl1990.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/0898122195002200", acknowledgement = ack-nhfb, fjournal = "Computers and Mathematics with Applications", journal-URL = "http://www.sciencedirect.com/science/journal/08981221", } @InCollection{Barnett:1996:CSB, author = "A. R. Barnett", title = "The Calculation of Spherical {Bessel} and {Coulomb} Functions", crossref = "Bartschat:1996:CAP", chapter = "9", pages = "181--202", year = "1996", DOI = "https://doi.org/10.1007/978-3-642-61010-3_9", bibdate = "Fri Apr 25 14:36:10 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, keywords = "COUL90; Lentz continued fraction algorithm; RICBES; SPBESJY; Steen continued fraction algorithm", } @Article{Cappuccino:1996:DDH, author = "G. Cappuccino and P. Corsonello and G. Cocorullo", title = "Design and demonstration of high throughput square rooting circuit", journal = j-ELECT-LETTERS, volume = "32", number = "5", pages = "434--436", month = feb, year = "1996", CODEN = "ELLEAK", DOI = "https://doi.org/10.1049/el:19960316", ISSN = "0013-5194 (print), 1350-911X (electronic)", ISSN-L = "0013-5194", bibdate = "Sat Jul 16 15:28:13 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Electronics Letters", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=2220; https://digital-library.theiet.org/journal/el", } @Article{Chaudhry:1996:ACF, author = "M. A. Chaudhry and N. M. Temme and E. J. M. Veling", title = "Asymptotics and closed form of a generalized incomplete gamma function", journal = j-J-COMPUT-APPL-MATH, volume = "67", number = "2", pages = "371--379", day = "29", month = mar, year = "1996", CODEN = "JCAMDI", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Sat Feb 25 12:27:48 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath1990.bib", URL = "http://www.sciencedirect.com/science/article/pii/0377042795000186", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @Article{Dawid:1996:DCA, author = "Herbert Dawid and Heinrich Meyr", title = "The Differential {CORDIC} Algorithm: Constant Scale Factor Redundant Implementation Without Correcting Iterations", journal = j-IEEE-TRANS-COMPUT, volume = "45", number = "3", pages = "307--318", month = mar, year = "1996", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/12.485569", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Wed Jul 6 19:47:09 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput1990.bib", URL = "https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=485569", ZMnumber = "1057.68740", acknowledgement = ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", keywords = "68W05,68W35,68M07; Computational complexity; Computer architecture; Delay; Digital arithmetic; Iterative algorithms; Iterative methods; Logic; Parallel architectures; Signal processing algorithms; Student members", ZBmath = "1966848", } @InProceedings{Guyot:1996:STD, author = "A. Guyot and M. Renaudin and B. {El Hassan} and V. Levering", booktitle = "Proceedings of the Ninth International Conference on {VLSI} Design, 1996", title = "Self timed division and square-root extraction", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "376--381", year = "1996", CODEN = "????", ISSN = "????", bibdate = "Sat Jul 16 15:28:14 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, summary = "This paper describes a self-timed integrated circuit for division and square-root extraction. First it concentrates on the development and the proof of a new mathematical algorithm. Then the design methodology and the architecture of a self-timed \ldots{}", } @Article{Heinrich:1996:AAF, author = "Peter Heinrich", title = "Algorithm Alley: a Fast Integer Square Root", journal = j-DDJ, volume = "21", number = "4", pages = "113--114, 130", month = apr, year = "1996", CODEN = "DDJOEB", ISSN = "1044-789X", bibdate = "Mon Sep 2 09:09:39 MDT 1996", bibsource = "http://www.ddj.com/index/author/index.htm; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, fjournal = "Dr. Dobb's Journal of Software Tools", } @TechReport{Hickey:1996:FSP, author = "Timothy J. Hickey and Qun Ju", title = "Fast, Sound, and Precise Narrowing of the Exponential Function", type = "Technical report", institution = "Computer Science Department, Brandeis University", address = "Waltham, MA, USA 02254", month = mar, year = "1996", bibdate = "Sat Nov 05 15:42:23 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.cs.brandeis.edu/~tim/Papers/eiianuia.ps.gz", acknowledgement = ack-nhfb, } @Article{Homeier:1996:CAI, author = "H. H. H. Homeier", title = "On Convergence Acceleration for the Iterative Solution of the Inverse {Dyson} Equation", journal = j-J-MOL-STRUCT-THEOCHEM, volume = "368", pages = "81--91", year = "1996", CODEN = "THEODJ", ISSN = "0166-1280 (print), 1872-7999 (electronic)", ISSN-L = "0166-1280", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/h/homeier-herbert-h-h.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "Proceedings of the 2nd {Electronic Computational Chemistry Conference}.", URL = "http://www.chemie.uni-regensburg.de/pub/preprint/preprint.html#TCQM954", fjournal = "Journal of molecular structure. Theochem", journal-URL = "http://www.sciencedirect.com/science/journal/01661280", keywords = "convergence acceleration", tech = "Technical Report TC-QM-95-4, Institut f{\"u}r {Physikalische} und {Theoretische Chemie, Universit{\"a}t Regensburg, D-93040 Regensburg}, 1995", } @TechReport{Homeier:1996:KMP, author = "H. H. H. Homeier", title = "{Zur Konvergenzverbesserung der M{\o}ller--Plesset St{\"o}rungsreihe} ({English}: {On} Convergence Acceleration of the {M{\o}ller--Plesset} Perturbation Series)", number = "Homeier:1996:KMP", institution = inst-IPTC, address = inst-IPTC:adr, year = "1996", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/h/homeier-herbert-h-h.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "Poster CP 14.77, {Fr{\"u}hjahrstagung des Arbeitskreises Festk{\"o}rperphysik bei der DPG, Regensburg 1996}. Abstract: Verhandl. DPG (VI) 31, 2165-2166 (1996).", URL = "http://www.chemie.uni-regensburg.de/pub/preprint/preprint.html#TCQM962", keywords = "convergence acceleration", } @Article{Ito:1996:SRI, author = "Masayuki Ito and Naofumi Takagi and Shuzo Yajima", title = "Square rooting by iterative multiply-additions", journal = j-INFO-PROC-LETT, volume = "60", number = "5", pages = "267--269", day = "8", month = dec, year = "1996", CODEN = "IFPLAT", ISSN = "0020-0190 (print), 1872-6119 (electronic)", ISSN-L = "0020-0190", MRclass = "68M07", MRnumber = "97i:68014", bibdate = "Wed Nov 11 12:16:26 MST 1998", bibsource = "http://www.elsevier.com:80/inca/publications/store/5/0/5/6/1/2/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, classification = "C4130 (Interpolation and function approximation); C5230 (Digital arithmetic methods)", corpsource = "Dept. of Inf. Sci., Kyoto Univ., Japan", fjournal = "Information Processing Letters", journal-URL = "http://www.sciencedirect.com/science/journal/00200190", keywords = "computer arithmetic; convergence of numerical methods; digital arithmetic; iterative methods; iterative multiply-additions; linear converging ratio; multiplicative methods; Newton--Raphson method; read-only storage; ROM sizes; square root algorithm", treatment = "T Theoretical or Mathematical", } @Article{Jeffrey:1996:UBL, author = "D. J. Jeffrey and D. E. G. Hare and Robert M. Corless", title = "Unwinding the branches of the {Lambert $W$} function", journal = j-MATH-SCI, volume = "21", number = "1", pages = "1--7", month = jun, year = "1996", ISSN = "0312-3685 (print), 1475-6080 (electronic)", ISSN-L = "0312-3685", bibdate = "Sat Oct 06 09:02:19 2018", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.appliedprobability.org/data/files/TMS%20articles/21_1_1.pdf", acknowledgement = ack-nhfb, fjournal = "The Mathematical Scientist", journal-URL = "http://www.appliedprobability.org/content.aspx?Group=tms&Page=allissues", } @Unpublished{Kahan:1996:TCR, author = "W. Kahan", title = "A Test for Correctly Rounded {SQRT}", pages = "4", year = "1996", bibdate = "Mon Apr 25 05:47:38 2005", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/k/kahan-william-m.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "Lecture notes.", URL = "http://www.cs.berkeley.edu/~wkahan/SQRTest.ps", acknowledgement = ack-nhfb, keywords = "floating-point arithmetic; rounding errors", } @Article{Kalantari:1996:HOI, author = "B. Kalantari and I. Kalantari", title = "High order iterative methods for approximating square roots", journal = j-BIT-NUM-MATH, volume = "36", number = "2", pages = "395--399", month = jun, year = "1996", CODEN = "BITTEL, NBITAB", DOI = "https://doi.org/10.1007/BF01731991", ISSN = "0006-3835 (print), 1572-9125 (electronic)", ISSN-L = "0006-3835", MRclass = "65D15 (65H99)", MRnumber = "97k:65039", bibdate = "Wed Jan 4 18:52:24 MST 2006", bibsource = "http://springerlink.metapress.com/openurl.asp?genre=issue&issn=0006-3835&volume=36&issue=2; https://www.math.utah.edu/pub/tex/bib/bit.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "http://www.mai.liu.se/BIT/contents/bit36.html; http://www.springerlink.com/openurl.asp?genre=article&issn=0006-3835&volume=36&issue=2&spage=395", acknowledgement = ack-nhfb, journal-URL = "http://link.springer.com/journal/10543", } @Article{Kantabutra:1996:HCE, author = "Vitit Kantabutra", title = "On hardware for computing exponential and trigonometric functions", journal = j-IEEE-TRANS-COMPUT, volume = "45", number = "3", pages = "328--339", month = mar, year = "1996", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/12.485571", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Wed Jul 6 19:47:09 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput1990.bib", URL = "https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=485571", abstract = "This paper presents new, fast hardware for computing the exponential function, sine, and cosine. The main new idea is to use low-precision arithmetic components to approximate high precision computations, and then to correct very quickly the approximation error periodically so that the effect is to get high precision computation at near low-precision speed. The algorithm used in the paper is a nontrivial modification of the well-known CORDIC algorithm, and might be applicable to the computation of other functions than the ones presented.", acknowledgement = ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", } @Article{Kolbig:1996:PF, author = "K. S. K{\"o}lbig", title = "The polygamma function $ \psi^{(k)}(x) $ for $ x = 1 / 4 $ and $ x = 3 / 4 $", journal = j-J-COMPUT-APPL-MATH, volume = "75", number = "1", pages = "43--46", day = "12", month = nov, year = "1996", CODEN = "JCAMDI", DOI = "https://doi.org/10.1016/S0377-0427(96)00055-6", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Sat Feb 25 12:35:58 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath1990.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0377042796000556", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @Article{Lang:1996:BPU, author = "T. Lang and R. Wong", title = "``{Best} possible'' upper bounds for the first two positive zeros of the {Bessel} function {$ J_v(x) $}: the infinite case", journal = j-J-COMPUT-APPL-MATH, volume = "71", number = "2", pages = "311--329", day = "27", month = jul, year = "1996", CODEN = "JCAMDI", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Sat Feb 25 12:35:56 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath1990.bib", URL = "http://www.sciencedirect.com/science/article/pii/0377042795002200", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @Article{Lehoucq:1996:CEU, author = "R. B. Lehoucq", title = "The Computation of Elementary Unitary Matrices", journal = j-TOMS, volume = "22", number = "4", pages = "393--400", month = dec, year = "1996", CODEN = "ACMSCU", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Mon Oct 24 11:37:20 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", abstract = "The construction of elementary unitary matrices that transform a complex vector to a multiple of $ e_1 $, the first column of the identity matrix, is studied. We present four variants and their software implementation, including a discussion on the {LAPACK} subroutine {CLARFG}. Comparisons are also given.", accepted = "June 1996", acknowledgement = ack-rfb # "\slash " # ack-nhfb, fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", keywords = "algorithms", subject = "{\bf F.2}: Theory of Computation, ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY, Numerical Algorithms and Problems, Computations on matrices. {\bf G.1.3}: Mathematics of Computing, NUMERICAL ANALYSIS, Numerical Linear Algebra. {\bf G.4}: Mathematics of Computing, MATHEMATICAL SOFTWARE, Algorithm analysis.", } @Article{Lether:1996:RAF, author = "Frank G. Lether", title = "Rational approximation formulas for computing the positive zeros of {$ J_0 (x) $}", journal = j-J-COMPUT-APPL-MATH, volume = "67", number = "1", pages = "167--172", day = "20", month = feb, year = "1996", CODEN = "JCAMDI", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Sat Feb 25 12:27:48 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath1990.bib", URL = "http://www.sciencedirect.com/science/article/pii/0377042795002197", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @InProceedings{Li:1996:NNR, author = "Yamin Li and Wanming Chu", booktitle = "Proceedings of the {IEEE} International Conference on Computer Design: {VLSI} in Computers and Processors: {ICCD '96}", title = "A new non-restoring square root algorithm and its {VLSI} implementations", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "538--544", year = "1996", CODEN = "????", ISSN = "????", bibdate = "Sat Jul 16 15:28:14 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, summary = "We present a new non-restoring square root algorithm that is very efficient to implement. The new algorithm presented here has the following features unlike other square root algorithms. First, the focus of the ``non-restoring'' is on the {\&} \ldots{}", } @Article{Lorch:1996:BPU, author = "Lee Lorch and Riccardo Uberti", title = "``{Best} possible'' upper bounds for the first positive zeros of {Bessel} functions --- the finite part", journal = j-J-COMPUT-APPL-MATH, volume = "75", number = "2", pages = "249--258", day = "28", month = nov, year = "1996", CODEN = "JCAMDI", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Sat Feb 25 12:35:58 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath1990.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0377042796000659", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @TechReport{Lozier:1996:PST, author = "Daniel W. Lozier", title = "A Proposed Software Test Service for Special Functions", type = "Technical Report", number = "NISTIR 5916", institution = pub-NIST, address = pub-NIST:adr, pages = "11", month = oct, year = "1996", bibdate = "Fri Jul 09 06:02:16 2004", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "Published in \cite{Lozier:1997:PST}.", URL = "http://math.nist.gov/acmd/Staff/DLozier/publications/nistir5916.ps", acknowledgement = ack-nhfb, } @Article{Lozier:1996:SNS, author = "Daniel W. Lozier", title = "Software Needs in Special Functions", journal = j-J-COMPUT-APPL-MATH, volume = "66", number = "??", pages = "345--358", month = "????", year = "1996", CODEN = "JCAMDI", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Fri Jul 09 05:51:55 2004", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", remark = "See preprint \cite{Lozier:1994:SNS}.", } @Article{Macleod:1996:AMS, author = "Allan J. Macleod", title = "{Algorithm 757}: {MISCFUN}, a software package to compute uncommon special functions", journal = j-TOMS, volume = "22", number = "3", pages = "288--301", month = sep, year = "1996", CODEN = "ACMSCU", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Sat Aug 31 16:07:02 MDT 1996", bibsource = "http://www.acm.org/pubs/contents/journals/toms/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib", URL = "http://doi.acm.org/10.1145/232826.232846; http://www.acm.org/pubs/citations/journals/toms/1996-22-3/p288-macleod/", abstract = "MISCFUN (MISCellaneous FUNctions) is a Fortran package for the evaluation of several special functions, which are not used often enough to have been included in the standard libraries or packages. The package uses Chebyshev expansions as the underlying method of approximation, with the Chebyshev coefficients given to 20D. A wide variety of functions are included, and the package is designed so that other functions can be added in a standard manner.", acknowledgement = ack-nhfb, fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", keywords = "algorithms", subject = "{\bf D.3.2}: Software, PROGRAMMING LANGUAGES, Language Classifications, FORTRAN. {\bf G.1.2}: Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Chebyshev approximation and theory. {\bf G.4}: Mathematics of Computing, MATHEMATICAL SOFTWARE, Certification and testing.", } @Article{Oleksy:1996:CAM, author = "Cz. Oleksy", title = "A convergence acceleration method of {Fourier} series", journal = j-COMP-PHYS-COMM, volume = "96", number = "1", pages = "17--26", year = "1996", CODEN = "CPHCBZ", DOI = "https://doi.org/10.1016/0010-4655(96)00044-6", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", MRclass = "65B05", MRnumber = "1396682 (97c:65012)", bibdate = "Thu Dec 1 17:36:45 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/compphyscomm1990.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Computer Physics Communications", journal-URL = "http://www.sciencedirect.com/science/journal/00104655", keywords = "convergence acceleration", } @Article{Panteliou:1996:DII, author = "S. D. Panteliou and A. D. Dimarogonas and I. N. Katz", title = "Direct and inverse interpolation for {Jacobian} elliptic functions, zeta function of {Jacobi} and complete elliptic integrals of the second kind", journal = j-COMPUT-MATH-APPL, volume = "32", number = "8", pages = "51--57", month = oct, year = "1996", CODEN = "CMAPDK", ISSN = "0898-1221 (print), 1873-7668 (electronic)", ISSN-L = "0898-1221", bibdate = "Wed Mar 1 21:48:33 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/computmathappl1990.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/0898122196001666", acknowledgement = ack-nhfb, fjournal = "Computers and Mathematics with Applications", journal-URL = "http://www.sciencedirect.com/science/journal/08981221", } @Book{Patel:1996:HND, author = "Jagdish K. Patel and Campbell B. Read", title = "Handbook of the Normal Distribution", volume = "150", publisher = pub-DEKKER, address = pub-DEKKER:adr, edition = "Second revised and expanded", pages = "ix + 431", year = "1996", ISBN = "0-8247-9342-0 (hardcover)", ISBN-13 = "978-0-8247-9342-5 (hardcover)", LCCN = "QA273.6 .P373 1996", bibdate = "Sat Dec 16 17:22:16 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; z3950.loc.gov:7090/Voyager", series = "Statistics, textbooks and monographs", URL = "http://www.loc.gov/catdir/enhancements/fy0647/95049404-d.html", acknowledgement = ack-nhfb, subject = "Gaussian distribution", tableofcontents = "Genesis: an historical background \\ Basic properties \\ Expansions and algorithms \\ Characterizations \\ Sampling distributions \\ Limit theorems and expansions \\ Normal approximations to distributions \\ Order statistics from normal samples \\ The bivariate normal distribution \\ Bivariate normal sampling distributions \\ Point estimation \\ Statistical intervals", } @Article{Plofker:1996:ESM, author = "Kim Plofker", title = "An Example of the Secant Method of Iterative Approximation in a {Fifteenth-Century Sanskrit} Text", journal = j-HIST-MATH, volume = "23", number = "3", pages = "246--256", month = aug, year = "1996", CODEN = "HIMADS", ISSN = "0315-0860 (print), 1090-249X (electronic)", ISSN-L = "0315-0860", bibdate = "Wed Jun 26 06:19:07 MDT 2013", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/histmath.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0315086096900269", acknowledgement = ack-nhfb, fjournal = "Historia Mathematica", journal-URL = "http://www.sciencedirect.com/science/journal/03150860", } @Article{Qiu:1996:SEC, author = "S.-L. Qiu and M. K. Vamanamurthy", title = "Sharp Estimates for Complete Elliptic Integrals", journal = j-SIAM-J-MATH-ANA, volume = "27", number = "3", pages = "823--834", month = may, year = "1996", CODEN = "SJMAAH", ISSN = "0036-1410 (print), 1095-7154 (electronic)", ISSN-L = "0036-1410", MRclass = "33E05", MRnumber = "97f:33033", MRreviewer = "G. D. Anderson", bibdate = "Sat Dec 5 18:14:13 MST 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "SIAM Journal on Mathematical Analysis", journal-URL = "http://epubs.siam.org/sima", } @InProceedings{Rao:1996:RTS, author = "V. M. Rao and B. Nowrouzian", booktitle = "Canadian Conference on Electrical and Computer Engineering. 26--29 May 1996", title = "Rounding techniques for signed binary arithmetic", volume = "1", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "294--297", year = "1996", CODEN = "????", ISSN = "????", bibdate = "Sat Jul 16 11:25:04 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", abstract = "This paper is concerned with the derivation of the relationship that exists between the number truncation in two's complement (TC) arithmetic and the corresponding truncation in signed-binary (SB) arithmetic. The resulting relationship is subsequently exploited and applied to the development of a pair of novel techniques for SB rounding. These techniques are then translated into algorithm suitable for two-level logic implementation. Finally, the resulting algorithms are applied to the design and implementation of a high-speed SB-kernel based TC multiply-accumulate arithmetic architecture.", acknowledgement = ack-nhfb, } @Article{Schatzman:1996:ADF, author = "James C. Schatzman", title = "Accuracy of the discrete {Fourier} transform and the fast {Fourier} transform", journal = j-SIAM-J-SCI-COMP, volume = "17", number = "5", pages = "1150--1166", month = sep, year = "1996", CODEN = "SJOCE3", DOI = "https://doi.org/10.1137/S1064827593247023", ISSN = "1064-8275 (print), 1095-7197 (electronic)", ISSN-L = "1064-8275", MRclass = "65T20 (42A65 42C10 68Q25)", MRnumber = "97e:65155", bibdate = "Fri Dec 4 14:47:53 MST 1998", bibsource = "http://epubs.siam.org/toc/sjoce3/17/5; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/siamjscicomput.bib", URL = "http://epubs.siam.org/sam-bin/dbq/article/24702", acknowledgement = ack-nhfb, ajournal = "SIAM J. Sci. Comput.", fjournal = "SIAM Journal on Scientific Computing", journal-URL = "http://epubs.siam.org/sisc", remark = "This article analyzes errors in computing one-dimensional Fourier transforms, the fast way (FFT) and the slow way. The author identifies two main causes of accuracy loss in the computed transforms: (1) inaccurate sine and cosine functions, and (2) failure to use accurate summation methods, such as Kahan's compensated summation.", } @Article{Schwarz:1996:HSA, author = "Eric M. Schwarz and Michael J. Flynn", title = "Hardware Starting Approximation Method and Its Application to the Square Root Operation", journal = j-IEEE-TRANS-COMPUT, volume = "45", number = "12", pages = "1356--1369", month = dec, year = "1996", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/12.545966", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Sat Jul 16 15:28:13 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", summary = "Quadratically converging algorithms for high-order arithmetic operations typically are accelerated by a starting approximation. The higher the precision of the starting approximation, the less number of iterations required for convergence. \ldots{}", } @Article{Sinclair:1996:ORS, author = "R. Sinclair", title = "Optimization of reciprocals and square roots on the {i860} microprocessor", journal = j-INT-J-HIGH-SPEED-COMPUTING, volume = "8", number = "1", pages = "57--64", year = "1996", CODEN = "IHSCEZ", ISSN = "0129-0533", bibdate = "Mon Feb 25 11:19:22 MST 2002", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; OCLC Article1st database", acknowledgement = ack-nhfb, fjournal = "International Journal of High Speed Computing (IJHSC)", journal-URL = "http://www.worldscientific.com/worldscinet/ijhsc", } @Article{Snyder:1996:RAF, author = "W. Van Snyder", title = "Remark on {Algorithm 723}: {Fresnel} Integrals", journal = j-TOMS, volume = "22", number = "4", pages = "498--500", month = dec, year = "1996", CODEN = "ACMSCU", DOI = "https://doi.org/10.1145/235815.235825", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib", note = "See \cite{Snyder:1993:AFI}.", abstract = "{\it Algorithm 723: Fresnel Integrals} has been improved to provide more precise results for $ x \gg 0 $.", acknowledgement = ack-rfb, fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", keywords = "algorithms, performance", subject = "{\bf D.3.2}: Software, PROGRAMMING LANGUAGES, Language Classifications, FORTRAN. {\bf G.1.2}: Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Rational approximation. {\bf G.4}: Mathematics of Computing, MATHEMATICAL SOFTWARE, Certification and testing.", } @Book{Temme:1996:SFI, author = "N. M. Temme", title = "Special Functions: an Introduction to the Classical Functions of Mathematical Physics", publisher = pub-WILEY, address = pub-WILEY:adr, pages = "xii + 374", year = "1996", DOI = "https://doi.org/10.1002/9781118032572", ISBN = "0-471-11313-1", ISBN-13 = "978-0-471-11313-3", LCCN = "QC20.7.F87 T46 1996", bibdate = "Mon Nov 24 21:41:54 MST 2008", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; z3950.bibsys.no:2100/BIBSYS; z3950.loc.gov:7090/Voyager", acknowledgement = ack-nhfb, shorttableofcontents = "Bernoulli, Euler and Stirling numbers \\ Useful methods and techniques \\ The gamma function \\ Differential equations \\ Hypergeometric functions \\ Orthogonal polynomials \\ Confluent hypergeometric functions \\ Legendre functions \\ Bessel functions \\ Separating the wave equation \\ Special statistical distribution functions \\ Elliptic integrals and elliptic functions \\ Numerical aspects of special functions", subject = "functions, special; mathematical physics; boundary value problems", tableofcontents = "1 Bernoulli, Euler and Stirling Numbers / 1 \\ 1.1. Bernoulli Numbers and Polynomials / 2 \\ 1.1.1. Definitions and Properties / 3 \\ 1.1.2. A Simple Difference Equation / 6 \\ 1.1.3. Euler's Summation Formula / 9 \\ 1.2. Euler Numbers and Polynomials / 14 \\ 1.2.1. Definitions and Properties / 15 \\ 1.2.2. Boole's Summation Formula / 17 \\ 1.3. Stirling Numbers / 18 \\ 1.4. Remarks and Comments for Further Reading / 21 \\ 1.5. Exercises and Further Examples / 22 \\ 2 Useful Methods and Techniques / 29 \\ 2.1. Some Theorems from Analysis / 29 \\ 2.2. Asymptotic Expansions of Integrals / 31 \\ 2.2.1. Watson's Lemma / 32 \\ 2.2.2. The Saddle Point Method / 34 \\ 2.2.3. Other Asymptotic Methods / 38 \\ 2.3. Exercises and Further Examples / 39 \\ 3 The Gamma Function / 41 \\ 3.1. Introduction / 41 \\ 3.1.1. The Fundamental Recursion Property / 42 \\ 3.1.2. Another Look at the Gamma Function / 42 \\ 3.2. Important Properties / 43 \\ 3.2.1. Prym's Decomposition / 43 \\ 3.2.2. The Cauchy--Saalsch{\"u}tz Representation / 44 \\ 3.2.3. The Beta Integral / 45 \\ 3.2.4. The Multiplication Formula / 46 \\ 3.2.5. The Reflection Formula / 46 \\ 3.2.6. The Reciprocal Gamma Function / 48 \\ 3.2.7. A Complex Contour for the Beta Integral / 49 \\ 3.3. Infinite Products / 50 \\ 3.3.1. Gauss' Multiplication Formula / 52 \\ 3.4. Logarithmic Derivative of the Gamma Function / 53 \\ 3.5. Riemann's Zeta Function / 57 \\ 3.6. Asymptotic Expansions / 61 \\ 3.6.1. Estimations of the Remainder / 64 \\ 3.6.2. Ratio of Two Gamma Functions / 66 \\ 3.6.3. Application of the Saddle Point Method / 69 \\ 3.7. Remarks and Comments for Further Reading / 71 \\ 3.8. Exercises and Further Examples / 72 \\ 4 Differential Equations / 79 \\ 4.1. Separating the Wave Equation / 79 \\ 4.1.1. Separating the Variables / 81 \\ 4.2. Differential Equations in the Complex Plane / 83 \\ 4.2.1. Singular Points / 83 \\ 4.2.2. Transformation of the Point at Infinity / 84 \\ 4.2.3. The Solution Near a Regular Point / 85 \\ 4.2.4. Power Series Expansions Around a Regular Point / 90 \\ 4.2.5. Power Series Expansions Around a Regular Singular Point / 92 \\ 4.3. Sturm's Comparison Theorem / 97 \\ 4.4. Integrals as Solutions of Differential Equations / 98 \\ 4.5. The Liouville Transformation / 103 \\ 4.6. Remarks and Comments for Further Reading / 104 \\ 4.7. Exercises and Further Examples / 104 \\ 5 Hypergeometric Functions / 107 \\ 5.1. Definitions and Simple Relations / 107 \\ 5.2. Analytic Continuation / 109 \\ 5.2.1. Three Functional Relations / 110 \\ 5.2.2. A Contour Integral Representation / 111 \\ 5.3. The Hypergeometric Differential Equation / 112 \\ 5.4. The Singular Points of the Differential Equation / 114 \\ 5.5. The Riemann--Papperitz Equation / 116 \\ 5.6. Barnes' Contour Integral for $F(a, b; c; z)$ / 119 \\ 5.7. Recurrence Relations / 121 \\ 5.8. Quadratic Transformations / 122 \\ 5.9. Generalized Hypergeometric Functions / 124 \\ 5.9.1. A First Introduction to g-functions / 125 \\ 5.10. Remarks and Comments for Further Reading / 127 \\ 5.11. Exercises and Further Examples / 128 \\ 6 Orthogonal Polynomials / 133 \\ 6.1. General Orthogonal Polynomials / 133 \\ 6.1.1. Zeros of Orthogonal Polynomials / 137 \\ 6.1.2. Gauss Quadrature / 138 \\ 6.2. Classical Orthogonal Polynomials / 141 \\ 6.3. Definitions by the Rodrigues Formula / 142 \\ 6.4. Recurrence Relations / 146 \\ 6.5. Differential Equations / 149 \\ 6.6. Explicit Representations / 151 \\ 6.7. Generating Functions / 154 \\ 6.8. Legendre Polynomials / 156 \\ 6.8.1. The Norm of the Legendre Polynomials / 156 \\ 6.8.2. Integral Expressions for the Legendre Polynomials / 156 \\ 6.8.3. Some Bounds on Legendre Polynomials / 157 \\ 6.8.4. An Asymptotic Expansion as n is Large / 158 \\ 6.9. Expansions in Terms of Orthogonal Polynomials / 160 \\ 6.9.1. An Optimal Result in Connection with Legendre Polynomials / 160 \\ 6.9.2. Numerical Aspects of Chebyshev Polynomials / 162 \\ 6.10. Remarks and Comments for Further Reading / 164 \\ 6.11. Exercises and Further Examples / 164 \\ 7 Confluent Hypergeometric Functions / 171 \\ 7.1. The $M$-function / 172 \\ 7.2. The $U$-function / 175 \\ 7.3. Special Cases and Further Relations / 177 \\ 7.3.1. Whittaker Functions / 178 \\ 7.3.2. Coulomb Wave Functions / 178 \\ 7 3.3. Parabolic Cylinder Functions / 179 \\ 7 3.4. Error Functions / 180 \\ 7.3.5. Exponential Integrals / 180 \\ 7.3.6. Fresnel Integrals / 182 \\ 7.3.7. Incomplete Gamma Functions / 185 \\ 7.3.8. Bessel Functions / 186 \\ 7.3.9. Orthogonal Polynomials / 186 \\ 7.4. Remarks and Comments for Further Reading / 186 \\ 7.5. Exercises and Further Examples / 187 \\ 8 Legendre Functions / 193 \\ 8.1. The Legendre Differential Equation / 194 \\ 8.2. Ordinary Legendre Functions / 194 \\ 8.3. Other Solutions of the Differential Equation / 196 \\ 8.4. A Few More Series Expansions / 198 \\ 8.5. The function $Q_n(z)$ / 200 \\ 8.6. Integral Representations / 202 \\ 8.7. Associated Legendre Functions / 209 \\ 8.8. Remarks and Comments for Further Reading / 213 \\ 8.9. Exercises and Further Examples / 214 \\ 9 Bessel Functions / 219 \\ 9.1. Introduction / 219 \\ 9.2. Integral Representations / 220 \\ 9.3. Cylinder Functions / 223 \\ 9.4. Further Properties / 227 \\ 9.5. Modified Bessel Functions / 232 \\ 9.6. Integral Representations for the $I$- and $K$-Functions / 234 \\ 9.7. Asymptotic Expansions / 238 \\ 9.8. Zeros of Bessel Functions / 241 \\ 9.9. Orthogonality Relations, Fourier--Bessel Series / 244 \\ 9.10. Remarks and Comments for Further Reading / 247 \\ 9.11. Exercises and Further Examples / 247 \\ 10 Separating the Wave Equation / 257 \\ 10.1. General Transformations / 258 \\ 10.2. Special Coordinate Systems / 259 \\ 10.2.1. Cylindrical Coordinates / 259 \\ 10.2.2. Spherical Coordinates / 261 \\ 10.2.3. Elliptic Cylinder Coordinates / 263 \\ 10.2.4. Parabolic Cylinder Coordinates / 264 \\ 10.2.5. Oblate Spheroidal Coordinates / 266 \\ 10.3. Boundary Value Problems / 268 \\ 10.3.1. Heat Conduction in a Cylinder / 268 \\ 10.3.2. Diffraction of a Plane Wave Due to a Sphere / 270 \\ 10.4. Remarks and Comments for Further Reading / 271 \\ 10.5. Exercises and Further Examples / 272 \\ 11 Special Statistical Distribution Functions / 275 \\ 11.1. Error Functions / 275 \\ 11.1.1. The Error Function and Asymptotic Expansions / 276 \\ 11.2. Incomplete Gamma Functions / 277 \\ 11.2.1. Series Expansions / 279 \\ 11.2.2. Continued Fraction for $\Gamma(a, z)$ / 280 \\ 11.2.3. Contour Integral for the Incomplete Gamma Functions / 282 \\ 11.2.4. Uniform Asymptotic Expansions / 283 \\ 11.2.5. Numerical Aspects / 286 \\ 11.3. Incomplete Beta Functions / 288 \\ 11.3.1. Recurrence Relations / 289 \\ 11.3.2. Contour Integral for the Incomplete Beta Function / 290 \\ 11.3.3. Asymptotic Expansions / 291 \\ 11.3.4. Numerical Aspects / 297 \\ 11.4. Non-Central Chi-Squared Distribution / 298 \\ 11.4.1. A Few More Integral Representations / 300 \\ 11.4.2. Asymptotic Expansion; $m$ Fixed, $j$ Large / 302 \\ 11.4.3. Asymptotic Expansion; $j$ Large, $m$ Arbitrary / 303 \\ 11.4.4. Numerical Aspects / 305 \\ 11.5. An Incomplete Bessel Function / 308 \\ 11.6. Remarks and Comments for Further Reading / 309 \\ 11.7. Exercises and Further Examples / 310 \\ 12 Elliptic Integrals and Elliptic Functions / 319 \\ 12.1. Complete Integrals of the First and Second Kind / 315 \\ 12.1.1. The Simple Pendulum / 316 \\ 12.1.2. Arithmetic Geometric Mean / 318 \\ 12.2. Incomplete Elliptic Integrals / 321 \\ 12.3. Elliptic Functions and Theta Functions / 322 \\ 12.3.1. Elliptic Functions / 323 \\ 12.3.2. Theta Functions / 324 \\ 12.4. Numerical Aspects / 328 \\ 12.5. Remarks and Comments for Further Reading / 329 \\ 12.6. Exercises and Further Examples / 330 \\ 13 Numerical Aspects of Special Functions / 333 \\ 13.1. A Simple Recurrence Relation / 334 \\ 13.2. Introduction to the General Theory / 335 \\ 13.3. Examples / 338 \\ 13.4. Miller's Algorithm / 343 \\ 13.5. How to Compute a Continued Fraction / 347 \\ Bibliography / 349 \\ Notations and Symbols / 361 \\ Index / 365", } @Article{Temme:1996:UAI, author = "N. M. Temme", title = "Uniform asymptotics for the incomplete gamma functions starting from negative values of the parameters", journal = j-METHODS-APPL-ANAL, volume = "3", number = "3", pages = "335--344", year = "1996", DOI = "https://doi.org/10.4310/MAA.1996.v3.n3.a3", ISSN = "1073-2772 (print), 1945-0001 (electronic)", ISSN-L = "1073-2772", MRclass = "33B20 (41A60)", MRnumber = "1421474", MRreviewer = "Richard B. Paris", bibdate = "Sat Feb 18 15:19:00 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Methods and Applications of Analysis", journal-URL = "http://www.intlpress.com/MAA/", } @Article{Waissi:1996:SAS, author = "Gary R. Waissi and Donald F. Rossin", title = "A sigmoid approximation of the standard normal integral", journal = j-APPL-MATH-COMP, volume = "77", number = "1", pages = "91--95", month = jun, year = "1996", CODEN = "AMHCBQ", DOI = "https://doi.org/10.1016/0096-3003(95)00190-5", ISSN = "0096-3003 (print), 1873-5649 (electronic)", ISSN-L = "0096-3003", bibdate = "Tue Nov 20 21:02:39 MST 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/applmathcomput1995.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/0096300395001905", acknowledgement = ack-nhfb, fjournal = "Applied Mathematics and Computation", journal-URL = "http://www.sciencedirect.com/science/journal/00963003/", } @Book{Walz:1996:AE, author = "Guido Walz", title = "Asymptotics and Extrapolation", volume = "88", publisher = pub-AKADEMIE-VERLAG, address = pub-AKADEMIE-VERLAG:adr, pages = "330", year = "1996", ISBN = "3-05-501732-3", ISBN-13 = "978-3-05-501732-2", LCCN = "QA281 .W349 1996", bibdate = "Thu Dec 1 10:25:13 MST 2011", bibsource = "catalog.princeton.edu:7090/voyager; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", series = "Mathematical research", acknowledgement = ack-nhfb, subject = "Extrapolation; Asymptotic expansions", tableofcontents = "Part I: Asymptotic Expansions \\ 1. Asymptotic Systems and Expansions / 15 \\ 2. Geometric Asymptotic Expansions / 21 \\ 3. Logarithmic Asymptotic Expansions / 39 \\ Part II: Linear Extrapolation Methods \\ 4. Fundamental Concepts and General Philosophy / 193 \\ 5. Error Bounds, Stopping Rules and Monotonicity / 244 \\ 6. Generalizations and Final Remarks / 289 \\ Historical Notes / 303 \\ References / 308\\ Index / 329", } @Article{Weniger:1996:CWF, author = "Ernst Joachim Weniger", title = "Computation of the {Whittaker} function of the second kind by summing its divergent asymptotic series with the help of nonlinear sequence transformations", journal = j-COMPUT-PHYS, volume = "10", number = "5", pages = "496--??", month = sep, year = "1996", CODEN = "CPHYE2", DOI = "https://doi.org/10.1063/1.168579", ISSN = "0894-1866 (print), 1558-4208 (electronic)", ISSN-L = "0894-1866", bibdate = "Wed Apr 10 08:46:03 MDT 2019", bibsource = "https://www.math.utah.edu/pub/tex/bib/computphys.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "https://aip.scitation.org/doi/10.1063/1.168579", acknowledgement = ack-nhfb, ajournal = "Comput. Phys", fjournal = "Computers in Physics", journal-URL = "https://aip.scitation.org/journal/cip", } @Article{Williams:1996:TMF, author = "K. B. Williams", title = "Testing Math Functions: When requirements are tight, we must carefully examine all potential sources of error. {Make} sure your math library isn't the weak link in the chain", journal = j-CCCUJ, volume = "14", number = "12", pages = "49--54, 58--65", month = dec, year = "1996", CODEN = "CCUJEX", ISSN = "1075-2838", bibdate = "Thu Nov 14 06:34:33 1996", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "Describes a package that extends the Cody-Waite-Plauger work on the ELEFUNT package for the testing of the elementary functions, including the inverse hyperbolic functions, cube root, and Bessel functions of the first and second kinds. The C++ package implements 192-bit extended precision versions of all of the functions, so that accurate results are available for comparison with the normal double-precision results.", acknowledgement = ack-nhfb, fjournal = "C/C++ Users Journal", } @Article{Zahle:1996:FDW, author = "M. Z{\"a}hle and H. Ziezold", title = "Fractional derivatives of {Weierstrass}-type functions", journal = j-J-COMPUT-APPL-MATH, volume = "76", number = "1--2", pages = "265--275", day = "17", month = dec, year = "1996", CODEN = "JCAMDI", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Sat Feb 25 12:35:58 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath1990.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0377042796001100", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @Book{Zhang:1996:CSF, author = "Shanjie Zhang and Jianming Jin", title = "Computation of Special Functions", publisher = pub-WILEY, address = pub-WILEY:adr, pages = "xxvi + 717", year = "1996", ISBN = "0-471-11963-6", ISBN-13 = "978-0-471-11963-0", LCCN = "QA351.C45 1996", bibdate = "Wed Mar 22 14:39:04 2000", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/master.bib; https://www.math.utah.edu/pub/tex/bib/mathcw.bib", price = "US\$94.00", acknowledgement = ack-nhfb, shorttableofcontents = "Preface / xi \\ Acknowledgments / xvii \\ List of Computer Programs / xix \\ 1: Bernoulli and Euler Numbers / 1 \\ 2: Orthogonal Polynomials / 12 \\ 3: Gamma, Beta, and Psi Functions / 44 \\ 4: Legendre Functions / 77 \\ 5: Bessel Functions / 126 \\ 6: Modified Bessel Functions / 202 \\ 7: Integrals of Bessel Functions / 252 \\ 8: Spherical Bessel Functions / 273 \\ 9: Kelvin Functions / 307 \\ 10: Airy Functions / 325 \\ 11: Struve Functions / 341 \\ 12: Hypergeometric and Confluent Hypergeometric / 366 \\ 13: Parabolic Cylinder Functions / 425 \\ 14: Mathieu Functions / 475 \\ 15: Spheroidal Wave Functions / 536 \\ 16: Error Function and Fresnel Integrals / 620 \\ 17: Cosine and Sine Integrals / 644 \\ 18: Elliptic Integrals and Jacobian Elliptic Functions 19: Exponential Integrals / 680 \\ 20: Summary of Methods for Computing Special Functions Appendix A: Derivation of Some Special Differential Appendix B: Root-Finding Methods / 704 \\ Reference / 706 \\ Appendix C: About the Software / 707 \\ Index / 709 \\ Index of Computer Programs / 715", tableofcontents = "Preface / xi \\ Acknowledgments / xvii \\ List of Computer Programs / xix \\ 1: Bernoulli and Euler Numbers / 1 \\ 1.1 Bernoulli Numbers / 1 \\ 1.2 Euler Numbers / 6 \\ 1.3 Mathematical Table / 10 \\ References / 11 \\ 2: Orthogonal Polynomials / 12 \\ 2.1 Introduction / 12 \\ 2.2 Chebyshev Polynomials / 13 \\ 2.3 Laguerre Polynomials / 18 \\ 2.4 Hermite Polynomials / 20 \\ 2.5 Numerical Computation / 23 \\ 2.6 Application in Numerical Integration / 27 \\ References / 43 \\ 3: Gamma, Beta, and Psi Functions / 44 \\ 3.1 Gamma Function / 44 \\ 3.2 Beta Function / 53 \\ 3.3 Psi Function / 55 \\ 3.4 Incomplete Gamma Function / 61 \\ 3.5 Incomplete Beta Function / 64 \\ 3.6 Mathematical Tables / 66 \\ References and Further Reading / 76 \\ 4: Legendre Functions / 77 \\ 4.1 Introduction / 77 \\ 4.2 Legendre Functions of the First Kind / 78 \\ 4.3 Legendre Functions of the Second Kind / 83 \\ 4.4 Associated Legendre Functions of the First Kind / 89 \\ 4.5 Associated Legendre Functions of the Second Kind / 96 \\ 4.6 Legendre Functions with an Arbitrary Degree / 104 \\ 4.7 Mathematical Tables / 113 \\ References and Further Reading / 125 \\ 5: Bessel Functions / 126 \\ 5.1 Introduction / 126 \\ 5.2 Computation of $J_0(x)$, $J_1(x)$, $Y_0(x)$, and $Y_1(x)$ / 131 \\ 5.3 Computation of $J_n(x)$ and $Y_n(x)$ with Real Arguments / 140 \\ 5.4 Computation of $Y_n(z)$ and$ Y_n(z)$ with Complex Arguments / 149 \\ 5.5 Computation of $J_\nu(z)$ and $J_\nu(z)$ with an Arbitrary Order / 161 \\ 5.6 Assessment of Validity and Accuracy of Computation / 175 \\ 5.7 Zeros of Bessel Functions / 180 \\ 5.8 Lambda Functions / 182 \\ 5.9 Mathematical Tables / 184 \\ References and Further Reading / 201 \\ 6: Modified Bessel Functions / 202 \\ 6.1 Introduction / 202 \\ 6.2 Computation of $I_0(x)$, $I_1(x)$, $K_0(x)$, and $K_1(x)$ / 207 \\ 6.3 Computation of $I_n(x)$ and $K_n(x)$ with Real Arguments / 213 \\ 6.4 Computation of $I_n(z)$ and $K_n(z)$ with Complex Arguments / 217 \\ 6.5 Computation of $I_\nu(z)$ and $K_\nu(z)$ with an Arbitrary Order / 225 \\ 6.6 Computation of $H_\nu^{(1)}(z)$ and $H_\nu^{(2)}(z)$ for Complex Arguments / 235 \\ 6.7 Mathematical Tables / 239 \\ References and Further Reading / 251 \\ 7: Integrals of Bessel Functions / 252 \\ 7.1 Simple Integrals of Bessel Functions / 252 \\ 7.2 Simple Integrals of Modified Bessel Functions / 261 \\ 7.3 Curves and Tables / 268 \\ References / 272 \\ 8: Spherical Bessel Functions / 273 \\ 8.1 Spherical Bessel Functions / 273 \\ 8.2 Riccati--Bessel Functions / 283 \\ 8.3 Modified Spherical Bessel Functions / 286 \\ 8.4 Mathematical Tables / 295 \\ References and Further Reading / 306 \\ 9: Kelvin Functions / 307 \\ 9.1 Introduction / 307 \\ 9.2 Mathematical Properties / 311 \\ 9.3 Asymptotic Expansions / 312 \\ 9.4 Numerical Computation / 315 \\ 9.5 Zeros of Kelvin Functions / 321 \\ 9.6 Mathematical Tables / 321 \\ Reference / 324 \\ 10: Airy Functions / 325 \\ 10.1 Introduction / 325 \\ 10.2 Numerical Computation / 329 \\ 10.3 Mathematical Tables / 336 \\ References / 340 \\ 11: Struve Functions / 341 \\ 11.1 Struve Functions / 341 \\ 11.2 Modified Struve Functions / 353 \\ 11.3 Mathematical Tables / 362 \\ References / 365 \\ 12: Hypergeometric and Confluent Hypergeometric Functions / 366 \\ 12.1 Definition of Hypergeometric Functions / 366 \\ 12.2 Properties of Hypergeometric Functions / 368 \\ 12.3 Linear Transformation Formulas / 369 \\ 12.4 Recurrence Relations for Hypergeometric Functions / 372 \\ 12.5 Special Functions Expressed as Hypergeometric Functions / 373 \\ 12.6 Numerical Computation of Hypergeometric Functions / 374 \\ 12.7 Definition of Confluent Hypergeometric Functions / 385 \\ 12.8 Properties of Confluent Hypergeometric Functions / 387 \\ 12.9 Recurrence Relations for Confluent Hypergeometric Functions / 389 \\ 12.10 Special Functions Expressed as Confluent Hypergeometric Functions / 394 \\ 12.11 Definition of Whittaker Functions / 395 \\ 12.12 Numerical Computation of Confluent Hypergeometric Functions / 398 \\ 12.13 Mathematical Tables / 411 \\ References and Further Reading / 424 \\ 13: Parabolic Cylinder Functions / 425 \\ 13.1 Introduction / 425 \\ 13.2 Definitions of Parabolic Cylinder Functions / 428 \\ 13.3 Basic Properties / 432 \\ 13.4 Series and Asymptotic Expansions / 437 \\ 13.5 Numerical Computation / 438 \\ 13.6 Mathematical Tables / 455 \\ References and Further Reading / 474 \\ 14: Mathieu Functions / 475 \\ 14.1 Definition of Mathieu Functions / 475 \\ 14.2 Determination of Expansion Coefficients and Characteristic Values / 477 \\ 14.3 Approximate Calculation of Characteristic Values / 482 \\ 14.4 Expansion of Mathieu Functions When $|q| < 1$ / 485 \\ 14.5 Properties of Mathieu Functions / 487 \\ 14.6 Definition of Modified Mathieu Functions / 489 \\ 14.7 Properties of Modified Mathieu Functions / 496 \\ 14.8 Numerical Computation: Algorithms and Computer Programs / 501 \\ 14.9 Mathematical Tables / 520 \\ References and Further Reading / 535 \\ 15: Spheroidal Wave Functions / 536 \\ 15.1 Spheroidal Coordinate Systems / 536 \\ 15.2 Wave Equation and Its Solution in Spheroidal Coordinates / 540 \\ 15.3 Definitions of Angular and Radial Prolate Spheroidal Wave Functions / 542 \\ 15.4 Determination of Characteristic Values and Expansion Coefficients / 550 \\ 15.5 Evaluation of Prolate Radial Wave Functions of the Second Kind for Small $c \xi$ / 556 \\ 15.6 Definitions of Angular and Radial Oblate Spheroidal Wave Functions / 559 \\ 15.7 Evaluation of Oblate Radial Wave Functions of the Second Kind for Small $c \xi$ / 561 \\ 15.8 Numerical Computation: Algorithms and Computer Programs / 569 \\ 15.9 Mathematical Tables / 594 \\ References / 619 \\ 16: Error Function and Fresnel Integrals / 620 \\ 16.1 Introduction to Error Function / 620 \\ 16.2 Numerical Computation of Error Function / 621 \\ 16.3 Gaussian Probability Integral / 624 \\ 16.4 Introduction to Fresnel Integrals / 625 \\ 16.5 Series and Asymptotic Expansions of Fresnel Integrals / 629 \\ 16.6 Numerical Computation of Fresnel Integrals / 630 \\ 16.7 Zeros of Error Function and Fresnel Integrals / 635 \\ 16.8 Mathematical Tables / 636 \\ References and Further Reading / 643 \\ 17: Cosine and Sine Integrals / 644 \\ 17.1 Introduction / 644 \\ 17.2 Series and Asymptotic Expansions / 646 \\ 17.3 Numerical Computation / 647 \\ 17.4 Mathematical Table / 651 \\ References and Further Readings / 653 \\ 18: Elliptic Integrals and Jacobian Elliptic Functions / 654 \\ 18.1 Introduction to Elliptic Integrals / 654 \\ 18.2 Series Expansion of Elliptic Integrals / 659 \\ 18.3 Numerical Computation of Elliptic Integrals / 661 \\ 18.4 Introduction to Jacobian Elliptic Functions / 666 \\ 18.5 Numerical Computation of Jacobian Elliptic Functions / 670 \\ 18.6 Mathematical Tables / 672 \\ References and Further Reading / 679 \\ 19: Exponential Integrals / 680 \\ 19.1 Introduction / 680 \\ 19.2 Series, Asymptotic, and Continued Fraction Expressions / 682 \\ 19.3 Rational Approximations / 683 \\ 19.4 Numerical Computation / 684 \\ 19.5 Mathematical Tables / 688 \\ References / 693 \\ 20: Summary of Methods for Computing Special Functions / 694 \\ Appendix A: Derivation of Some Special Differential Equations / 697 \\ A.1 Helmholtz Equation and Separation of Variables / 697 \\ A.2 Circular Cylindrical Coordinates / 698 \\ A.3 Elliptic Cylindrical Coordinates / 700 \\ A.4 Parabolic Cylindrical Coordinates / 700 \\ A.5 Spherical Coordinates / 701 \\ A.6 Prolate Spheroidal Coordinates / 701 \\ A.7 Oblate Spheroidal Coordinates / 702 \\ A.8 Parabolic Coordinates / 703 \\ References / 703 \\ Appendix B: Root-Finding Methods / 704 \\ B.1 Newton's Method / 704 \\ B.2 Modified Newton's Method / 706 \\ B.3 Secant Method / 706 \\ Reference / 706 \\ Appendix C: About the Software / 707 \\ Index / 709 \\ Index of Computer Programs / 715", xxauthor = "Shan-chieh Chang and Shanjie Zhang and Jianming Jin", xxnote = "There is online bookstore and library catalog confusion over the authors of this book. The publisher's Web page at http://catalog.wiley.com/remsrch.cgi has Shan-jie Zhang (Nanjing Univ., China) / Jianming Jin (Univ. of Illinois at Urbana-Champaign), and a price of US\$125.00. It looks like Shan-chieh Chang is merely a different English transcription of Shanjie Zhang. http://www.fatbrain.com/ lists this book for US\$99.95. My copy of the book lists the authors as Shanjie Zhang and Jianming Jin in four places.", } @Article{Zhang:1996:NTM, author = "J. Zhang", title = "A note on the tau-method approximations for the {Bessel} functions {$ Y_0 (z) $} and {$ Y_1 (z) $}", journal = j-COMPUT-MATH-APPL, volume = "31", number = "9", pages = "63--70", month = may, year = "1996", CODEN = "CMAPDK", DOI = "https://doi.org/10.1016/0898-1221(96)00043-0", ISSN = "0898-1221 (print), 1873-7668 (electronic)", ISSN-L = "0898-1221", bibdate = "Sun Jun 12 08:43:36 2011", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/l/lanczos-cornelius.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/0898122196000430", abstract = "This paper is to complete and improve the work reported in [1,2], using the Lanczos $ \tau $-method (in Coleman's version) to approximate the Bessel functions $ Y_0 (z) $ and $ Y_1 (z) $. We introduce symbolic representations of the scaled Faber polynomials on any fan-shaped section of the complex plane. These Faber polynomials are used as the perturbation terms in the $ \tau $-method. Numerical comparison among the power series, the Chebyshev series and the $ \tau $-method are conducted to show the accuracy improvement achieved by this new version of the $ \tau $-method. Some concluding remarks and suggestions on future research are given.", acknowledgement = ack-nhfb, fjournal = "Computers and Mathematics with Applications", journal-URL = "http://www.sciencedirect.com/science/journal/08981221", keywords = "Automated $\tau$-method; Bessel functions; Chebyshev series; Symbolic Faber polynomials", } @Article{Zhang:1996:SNC, author = "Jun Zhang", title = "Symbolic and numerical computation on {Bessel} functions of complex argument and large magnitude", journal = j-J-COMPUT-APPL-MATH, volume = "75", number = "1", pages = "99--118", day = "12", month = nov, year = "1996", CODEN = "JCAMDI", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Sat Feb 25 12:35:58 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath1990.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0377042796000635", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @Article{Abad:1997:NEC, author = "Julio Abad and Javier Sesma", title = "A new expansion of the confluent hypergeometric function in terms of modified {Bessel} functions", journal = j-J-COMPUT-APPL-MATH, volume = "78", number = "1", pages = "97--101", day = "3", month = feb, year = "1997", CODEN = "JCAMDI", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Sat Feb 25 12:35:59 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath1990.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0377042796001331", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @Article{Alzer:1997:HMI, author = "Horst Alzer", title = "A harmonic mean inequality for the gamma function", journal = j-J-COMPUT-APPL-MATH, volume = "87", number = "2", pages = "195--198", day = "23", month = dec, year = "1997", CODEN = "JCAMDI", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Sat Feb 25 12:36:06 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath1990.bib", note = "See corrigendum \cite{Alzer:1998:CHM}.", URL = "http://www.sciencedirect.com/science/article/pii/S0377042796001811", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @Article{Alzer:1997:SIG, author = "Horst Alzer", title = "On some inequalities for the gamma and psi functions", journal = j-MATH-COMPUT, volume = "66", number = "217", pages = "373--389", month = jan, year = "1997", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "33B15 (26D07)", MRnumber = "97e:33004", MRreviewer = "Peter Schroth", bibdate = "Fri Jul 16 10:38:40 MDT 1999", bibsource = "http://www.ams.org/mcom/1997-66-217; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.ams.org/jourcgi/jour-pbprocess?fn=110&arg1=S0025-5718-97-00807-7&u=/mcom/1997-66-217/", acknowledgement = ack-nhfb, fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Alzer:1997:SII, author = "Horst Alzer", title = "On some inequalities for the incomplete gamma function", journal = j-MATH-COMPUT, volume = "66", number = "218", pages = "771--778", month = apr, year = "1997", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "33B20 (26D07)", MRnumber = "97h:33004", bibdate = "Fri Jul 16 10:38:42 MDT 1999", bibsource = "http://www.ams.org/mcom/1997-66-218; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.ams.org/jourcgi/jour-pbprocess?fn=110&arg1=S0025-5718-97-00814-4&u=/mcom/1997-66-218/", acknowledgement = ack-nhfb, fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Bailey:1997:RCV, author = "David Bailey and Peter Borwein and Simon Plouffe", title = "On the rapid computation of various polylogarithmic constants", journal = j-MATH-COMPUT, volume = "66", number = "218", pages = "903--913", month = apr, year = "1997", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "11Yxx", MRnumber = "1 415 794", bibdate = "Fri Jul 16 10:38:42 MDT 1999", bibsource = "http://www.ams.org/mcom/1997-66-218; http://www.jstor.org/journals/00029890.htm; https://www.math.utah.edu/pub/tex/bib/amermathmonthly1990.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp1990.bib", URL = "http://www.ams.org/journals/mcom/1997-66-218/S0025-5718-97-00856-9/S0025-5718-97-00856-9.pdf", acknowledgement = ack-nhfb, fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "BBP formula", xxnote = "See \cite{Adamchik:1997:SF}.", } @Article{Blinn:1997:JBC, author = "James F. Blinn", title = "{Jim Blinn}'s Corner: Floating-Point Tricks", journal = j-IEEE-CGA, volume = "17", number = "4", pages = "80--84", month = jul # "\slash " # aug, year = "1997", CODEN = "ICGADZ", DOI = "https://doi.org/10.1109/38.595279", ISSN = "0272-1716 (print), 1558-1756 (electronic)", ISSN-L = "0272-1716", bibdate = "Sat Jul 16 08:40:52 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", note = "Discusses use of IEEE 754 single-precision floating-point bit patterns as integers for implementations of fast, but low-accuracy, functions useful in computer graphics.", acknowledgement = ack-nhfb, fjournal = "IEEE Computer Graphics and Applications", journal-URL = "http://www.computer.org/portal/web/csdl/magazines/cga", summary = "The author discusses IEEE floating point representation that stores numbers in what amounts to scientific notation. He considers the sign bit, the logarithm function, function approximations, errors and refinements \ldots{}", } @InCollection{Borwein:1997:AGMa, author = "J. M. Borwein and P. B. Borwein", title = "The Arithmetic--Geometric Mean and Fast Computation of Elementary Functions", crossref = "Berggren:1997:PSB", pages = "537--552", year = "1997", DOI = "https://doi.org/10.1007/978-1-4757-2736-4_56", bibdate = "Thu Aug 11 09:36:22 MDT 2016", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib; https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "Reprint of \cite{Borwein:1984:AGM}.", URL = "http://link.springer.com/chapter/10.1007/978-1-4757-2736-4_56", acknowledgement = ack-nhfb, author-dates = "Jonathan Michael Borwein (20 May 1951--2 August 2016)", ORCID-numbers = "Borwein, Jonathan/0000-0002-1263-0646", } @Article{Bshouty:1997:TBA, author = "Nader H. Bshouty and Yishay Mansour and Baruch Schieber and Prasoon Tiwari", title = "A tight bound for approximating the square root", journal = j-INFO-PROC-LETT, volume = "63", number = "4", pages = "211--213", day = "10", month = sep, year = "1997", CODEN = "IFPLAT", ISSN = "0020-0190 (print), 1872-6119 (electronic)", ISSN-L = "0020-0190", MRclass = "68Q25 (65B15 68Q40)", MRnumber = "1 477 306", bibdate = "Sat Nov 7 17:55:54 MST 1998", bibsource = "http://www.elsevier.com:80/inca/publications/store/5/0/5/6/1/2/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, fjournal = "Information Processing Letters", journal-URL = "http://www.sciencedirect.com/science/journal/00200190", } @Article{El-Gabali:1997:MTA, author = "Magdi A. El-Gabali", title = "Multiple-term approximations for {Appell}'s {$ F_1 $} function", journal = j-J-AUSTRAL-MATH-SOC-SER-B, volume = "39", number = "1", pages = "135--148", month = jul, year = "1997", CODEN = "JAMMDU", DOI = "https://doi.org/10.1017/S0334270000009267", ISSN = "0334-2700", ISSN-L = "0334-2700", bibdate = "Fri Apr 26 16:13:39 MDT 2019", bibsource = "https://www.math.utah.edu/pub/tex/bib/anziamj.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "https://www.cambridge.org/core/journals/anziam-journal/article/multipleterm-approximations-for-appells-f1-function/8E218109899D68FD0DC15B6E9D61E8BD", acknowledgement = ack-nhfb, ajournal = "J. Austral Math. Soc. Ser. B", fjournal = "Journal of the Australian Mathematical Society. Series B, Applied Mathematics", journal-URL = "http://journals.cambridge.org/action/displayJournal?jid=ANZ", onlinedate = "17 February 2009", } @Article{Fdil:1997:SRC, author = "A. Fdil", title = "Some results of convergence acceleration for a general {$ \Theta $}-type algorithm", journal = j-APPL-NUM-MATH, volume = "23", number = "2", pages = "219--240", day = "21", month = mar, year = "1997", CODEN = "ANMAEL", ISSN = "0168-9274 (print), 1873-5460 (electronic)", ISSN-L = "0168-9274", MRclass = "65B10 (65D32)", MRnumber = "1 437 884", bibdate = "Wed Jul 28 14:36:42 MDT 1999", bibsource = "http://www.elsevier.com/cgi-bin/cas/tree/store/apnum/cas_free/browse/browse.cgi?year=1997&volume=23&issue=2; https://www.math.utah.edu/pub/tex/bib/applnummath.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.elsevier.com/cas/tree/store/apnum/sub/1997/23/2/738.pdf", acknowledgement = ack-nhfb, fjournal = "Applied Numerical Mathematics: Transactions of IMACS", journal-URL = "http://www.sciencedirect.com/science/journal/01689274/", keywords = "convergence acceleration", } @Article{Forrey:1997:CHF, author = "Robert C. Forrey", title = "Computing the hypergeometric function", journal = j-J-COMPUT-PHYS, volume = "137", number = "1", pages = "79--100", month = oct, year = "1997", CODEN = "JCTPAH", DOI = "https://doi.org/10.1006/jcph.1997.5794", ISSN = "0021-9991 (print), 1090-2716 (electronic)", ISSN-L = "0021-9991", MRclass = "33C05 (33-04 65D20)", MRnumber = "MR1481885 (99g:33004)", bibdate = "Thu Dec 01 09:06:55 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputphys1990.bib", acknowledgement = ack-nhfb, fjournal = "Journal of Computational Physics", journal-URL = "http://www.sciencedirect.com/science/journal/00219991/", remark = "The author gives a FORTRAN program for computing $_2 F_1$ for real variable and parameters, using rapidly-convergent power series in six separate intervals.", } @Article{Ghanem:1997:SBF, author = "Riadh Ben Ghanem and Cl{\'e}ment Frappier", title = "Spherical {Bessel} functions and explicit quadrature formula", journal = j-MATH-COMPUT, volume = "66", number = "217", pages = "289--296", month = jan, year = "1997", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "33C10 (41A55 65D32)", MRnumber = "97c:33005", MRreviewer = "N. Hayek Calil", bibdate = "Fri Jul 16 10:38:40 MDT 1999", bibsource = "http://www.ams.org/mcom/1997-66-217; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.ams.org/jourcgi/jour-pbprocess?fn=110&arg1=S0025-5718-97-00794-1&u=/mcom/1997-66-217/", acknowledgement = ack-nhfb, fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Goano:1997:RA7, author = "Michele Goano", title = "Remark on {Algorithm 745}", journal = j-TOMS, volume = "23", number = "2", pages = "295--295", month = jun, year = "1997", CODEN = "ACMSCU", DOI = "https://doi.org/10.1145/264029.643581", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Tue Mar 9 10:19:38 1999", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "See \cite{Goano:1995:ACC}.", acknowledgement = ack-rfb # " and " # ack-kr # "\slash " # ack-nhfb, fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", } @Article{Hare:1997:CPB, author = "D. E. G. Hare", title = "Computing the Principal Branch of {log-Gamma}", journal = j-J-ALG, volume = "25", number = "2", pages = "221--236", month = nov, year = "1997", CODEN = "JOALDV", DOI = "https://doi.org/10.1006/jagm.1997.0881", ISSN = "0196-6774 (print), 1090-2678 (electronic)", ISSN-L = "0196-6774", bibdate = "Tue Dec 11 09:16:52 MST 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jalg.bib; https://www.math.utah.edu/pub/tex/bib/maple-extract.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0196677497908816", abstract = "The log-Gamma function is an important special function of mathematics, and its principal branch is required in many applications. We develop here the mathematics required to evaluate the principal branch to arbitrary precision, including a new bound for the error in Stirling's asymptotic series. We conclude with a discussion of the implementation of the principal branch of the log-Gamma function in the Maple symbolic algebra system, starting with version Maple V, Release 3.", acknowledgement = ack-nhfb, fjournal = "Journal of Algorithms", journal-URL = "http://www.sciencedirect.com/science/journal/01966774", } @Article{Harris:1997:NAC, author = "Frank E. Harris", title = "New Approach to Calculation of the Leaky Aquifer Function", journal = j-IJQC, volume = "63", number = "5", pages = "913--916", month = "????", year = "1997", CODEN = "IJQCB2", DOI = "https://doi.org/10.1002/(SICI)1097-461X(1997)63:5<913::AID-QUA1>3.0.CO%3B2-Z", ISSN = "0020-7608 (print), 1097-461X (electronic)", ISSN-L = "0020-7608", bibdate = "Tue Oct 4 06:59:09 MDT 2011", bibsource = "Compendex database; http://www.interscience.wiley.com/jpages/0020-7608; http://www3.interscience.wiley.com/journalfinder.html; https://www.math.utah.edu/pub/bibnet/authors/h/harris-frank-e.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/ijqc.bib; https://www.math.utah.edu/pub/tex/bib/ijqc1990.bib", URL = "http://www3.interscience.wiley.com/cgi-bin/abstract?ID=42641; http://www3.interscience.wiley.com/cgi-bin/fulltext?ID=42641&PLACEBO=IE.pdf", acknowledgement = ack-nhfb, ajournal = "Int. J. Quantum Chem.", fjournal = "International Journal of Quantum Chemistry", journal-URL = "http://www.interscience.wiley.com/jpages/0020-7608/", journalabr = "Int J Quant Chem", onlinedate = "6 Dec 1998", } @Article{Ito:1997:EIA, author = "M. Ito and N. Takagi and S. Yajima", title = "Efficient initial approximation for multiplicative division and square root by a multiplication with operand modification", journal = j-IEEE-TRANS-COMPUT, volume = "46", number = "4", pages = "495--498", month = apr, year = "1997", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/12.588066", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Wed Jul 6 10:06:22 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput1990.bib", URL = "https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=588066", acknowledgement = ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", summary = "An efficient initial approximation method for multiplicative division and square root is proposed. It is a modification of the piecewise linear approximation. The multiplication and the addition required for the linear approximation are replaced by \ldots{}", } @Article{Karp:1997:HPD, author = "Alan H. Karp and Peter Markstein", title = "High-Precision Division and Square Root", journal = j-TOMS, volume = "23", number = "4", pages = "561--589", month = dec, year = "1997", CODEN = "ACMSCU", DOI = "https://doi.org/10.1145/279232.279237", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Thu Nov 8 14:50:37 2007", bibsource = "https://www.math.utah.edu/pub/bibnet/subjects/acc-stab-num-alg-2ed.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib", URL = "http://www.acm.org/pubs/articles/journals/toms/forthcoming/a0-karp/a0-karp.ps; http://www.acm.org/pubs/citations/journals/toms/1997-23-4/p561-karp/", abstract = "We present division and square root algorithms for calculation with more bits than are handled by the floating-point hardware. These algorithms avoid the need to multiply two high-precision numbers, speeding up the last iteration by as much as a factor of 10. We also show how to produce the floating-point number closest to the exact result with relatively few additional operations.", accepted = "June 1997", acknowledgement = ack-rfb # " and " # ack-kr, fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", keywords = "algorithms, performance, division, quad precision, square root.", subject = "G.1.0 [Numerical Analysis]: General -- computer arithmetic. G.4 [Mathematics of Computing]: Mathematical Software.", } @Book{Khinchin:1997:CF, author = "Aleksandr Yakovlevich Khinchin and Herbert Eagle", title = "Continued Fractions", publisher = pub-DOVER, address = pub-DOVER:adr, pages = "xi + 95", year = "1997", ISBN = "0-486-69630-8 (paperback)", ISBN-13 = "978-0-486-69630-0 (paperback)", LCCN = "QA295 .K513 1997", bibdate = "Wed Apr 15 16:52:20 MDT 2009", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; z3950.loc.gov:7090/Voyager", URL = "http://www.loc.gov/catdir/description/dover032/97008056.html; http://www.loc.gov/catdir/toc/dover031/97008056.html", acknowledgement = ack-nhfb, author-dates = "1894--1959", remark = "Translated from the Russian by Scripta Technica, Inc. Originally published: Chicago : University of Chicago Press, 1964. Edited by Herbert Eagle.", subject = "Continued fractions", tableofcontents = "Chapter I. Properties of the Apparatus\\ 1. Introduction\\ 2. Convergents\\ 3. Infinite continued fractions\\ 4. Continued fractions with natural elements\\ Chapter II. The Representation of Numbers by Continued Fractions\\ 5. Continued fractions as an apparatus for representing real numbers\\ 6. Convergents as best approximations\\ 7. The order of approximation\\ 8. General approximation theorems\\ 9. The approximation of algebraic irrational numbers and Liouville's transcendental numbers\\ 10. Quadratic irrational numbers and periodic continued fractions\\ Chapter III. The Measure Theory of Continued Fractions\\ 11. Introduction\\ 12. The elements as functions of the number represented\\ 13. Measure-theoretic evaluation of the increase in the elements\\ 14. Measure-theoretic evaluation of the increase in the denominators of the convergents. The fundamental theorem of the measure theory of approximation\\ 15. Gauss's problem and Kuz'min's theorem\\ 16. Average values\\ Index", xxURL = "http://www.loc.gov/catdir/description/dover032/97008056.html; http://www.loc.gov/catdir/toc/dover031/97008056.html", } @Article{Kolbig:1997:TEH, author = "K. S. K{\"o}lbig", title = "Table errata: {{\booktitle{Handbook of elliptic integrals for engineers and scientists}} [Second edition, Springer, New York, 1971, MR {\bf 43} \#3506] by P. F. Byrd and M. D. Friedman}", journal = j-MATH-COMPUT, volume = "66", number = "220", pages = "1767--1767", month = oct, year = "1997", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "44-00 (33-00)", MRnumber = "1 434 945", bibdate = "Tue Dec 2 11:25:56 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp1990.bib", acknowledgement = ack-nhfb, fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Lee:1997:PRF, author = "M. Howard Lee", title = "Polylogarithms and {Riemann}'s $ \zeta $ function", journal = j-PHYS-REV-E, volume = "56", number = "4", pages = "3909--3912", month = oct, year = "1997", CODEN = "PLEEE8", DOI = "https://doi.org/10.1103/physreve.56.3909", ISSN = "1539-3755 (print), 1550-2376 (electronic)", ISSN-L = "1539-3755", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", fjournal = "Physical Review E (Statistical physics, plasmas, fluids, and related interdisciplinary topics)", journal-URL = "http://pre.aps.org/browse", remark = "The paper presents increasingly complicated closed forms of $ \Li_n(x) $ for negative $n$ to $ n = - 8$, and reports that no general form for negative $n$ is apparent. See https://oeis.org/A131758 for related functions and sequences. Maple and Mathematica can produce such formulas with code like simplify(expand(polylog(-13,x))) and PolyLog[-13, x].", } @Article{Lether:1997:CNM, author = "Frank G. Lether", title = "Constrained near-minimax rational approximations to {Dawson}'s integral", journal = j-APPL-MATH-COMP, volume = "88", number = "2--3", pages = "267--274", day = "30", month = dec, year = "1997", CODEN = "AMHCBQ", DOI = "https://doi.org/10.1016/S0096-3003(96)00330-X", ISSN = "0096-3003 (print), 1873-5649 (electronic)", ISSN-L = "0096-3003", bibdate = "Tue Nov 20 21:02:59 MST 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/applmathcomput1995.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/S009630039600330X", acknowledgement = ack-nhfb, fjournal = "Applied Mathematics and Computation", journal-URL = "http://www.sciencedirect.com/science/journal/00963003/", } @InProceedings{Li:1997:ISP, author = "Yamin Li and Wanming Chu", booktitle = "Proceedings of the 5th Annual {IEEE} Symposium on {FPGAs} for Custom Computing Machines, 16--18 April 1997", title = "Implementation of single precision floating point square root on {FPGAs}", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "226--232", year = "1997", CODEN = "????", ISSN = "????", bibdate = "Sat Jul 16 15:28:14 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, summary = "The square root operation is hard to implement on FPGAs because of the complexity of the algorithms. In this paper, we present a non-restoring square root algorithm and two very simple single precision floating point square root implementations \ldots{}", } @InProceedings{Li:1997:PAI, author = "Yamin Li and Wanming Chu", booktitle = "Proceedings of the 1997 {IEEE} International Conference on Computer Design: {VLSI} in Computers and Processors: {ICCD '97}", title = "Parallel-array implementations of a non-restoring square root algorithm", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "690--695", year = "1997", CODEN = "????", ISSN = "????", bibdate = "Sat Jul 16 15:28:14 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, summary = "In this paper we present a parallel-array implementation of a new non-restoring square root algorithm (PASQRT). The carry-save adder (CSA) is used in the parallel array. The PASQRT has several features unlike other implementations. First, it does \ldots{}", } @InCollection{Lozier:1997:PST, author = "Daniel W. Lozier", title = "A Proposed Software Test Service for Special Functions", crossref = "Boisvert:1997:QNS", pages = "167--178", year = "1997", bibdate = "Fri Jul 09 06:00:46 2004", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, remark = "See preprint \cite{Lozier:1996:PST}.", } @TechReport{Lozier:1997:TRN, author = "Daniel W. Lozier", title = "Toward a Revised {NBS} Handbook of Mathematical Functions", type = "Technical Report", number = "NISTIR 6072", institution = pub-NIST, address = pub-NIST:adr, pages = "8", month = sep, year = "1997", bibdate = "Fri Jul 09 06:35:07 2004", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://math.nist.gov/acmd/Staff/DLozier/publications/nistir6072.ps.gz", acknowledgement = ack-nhfb, } @Article{MacLeod:1997:AEE, author = "Allan J. MacLeod", title = "Accurate and efficient evaluation of the {Bose--Einstein} functions $ g_{3 / 2} $ and $ g_{5 / 2} $", journal = j-COMPUT-PHYS, volume = "11", number = "4", pages = "385--??", month = jul, year = "1997", CODEN = "CPHYE2", DOI = "https://doi.org/10.1063/1.168609", ISSN = "0894-1866 (print), 1558-4208 (electronic)", ISSN-L = "0894-1866", bibdate = "Wed Apr 10 08:46:08 MDT 2019", bibsource = "https://www.math.utah.edu/pub/tex/bib/computphys.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "https://aip.scitation.org/doi/10.1063/1.168609", acknowledgement = ack-nhfb, ajournal = "Comput. Phys", fjournal = "Computers in Physics", journal-URL = "https://aip.scitation.org/journal/cip", } @InProceedings{Matsubara:1997:LPZ, author = "G. Matsubara and N. Ide", booktitle = "Proceedings of the Third International Symposium on Advanced Research in Asynchronous Circuits and Systems, 7--10 April 1997", title = "A low power zero-overhead self-timed division and square root unit combining a single-rail static circuit with a dual-rail dynamic circuit", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "198--209", year = "1997", CODEN = "????", ISSN = "????", bibdate = "Sat Jul 16 15:28:14 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, summary = "An asynchronous pipeline scheme that combines a low power static circuit with a high-speed dual-rail dynamic circuit is proposed. The scheme utilizes a dual-rail circuit only in the critical path of an SRT division and square root calculation unit. \ldots{}", } @Book{Muller:1997:EFA, author = "Jean-Michel Muller", title = "Elementary Functions: Algorithms and Implementation", publisher = pub-BIRKHAUSER, address = pub-BIRKHAUSER:adr, pages = "xv + 204", year = "1997", ISBN = "0-8176-3990-X", ISBN-13 = "978-0-8176-3990-7", LCCN = "QA331.M866 1997", bibdate = "Fri Jul 25 12:00:55 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/maple-extract.bib; https://www.math.utah.edu/pub/tex/bib/master.bib", price = "US\$59.95", URL = "http://www.birkhauser.com/cgi-win/ISBN/0-8176-3990-X; http://www.ens-lyon.fr/~jmmuller/book_functions.html", abstract = "The elementary functions (sine, cosine, tan, exponentials, and logarithms) are the most commonly used mathematical functions in science and engineering. Computing these functions quickly and accurately is a major goal in computer arithmetic. This new book gives the concepts and background necessary to understand and build algorithms for computing these functions, presenting and structuring the algorithms (hardware-oriented as well as software-oriented), and discusses issues related to the accurate floating-point implementation. The purpose is not to give ``cookbook recipes'' that allow one to implement some given function, but to provide the reader with the knowledge that is necessary to build, or adapt, algorithms to their specific computing environment. The book provides an up-to-date presentation of the information needed to understand and accurately use mathematical functions and algorithms in computational work and design. Graduates, professionals and researchers in scientific computing, software engineering and computer engineering will find the book a useful reference and resource.", acknowledgement = ack-nhfb, shorttableofcontents = "1: Introduction \\ 2: Computer arithmetic \\ I: Algorithms based on polynomial approximation and/or table lookup \\ 3: Polynomial approximations \\ 4: Table-based methods \\ II: Shift-and-Add algorithms \\ 5: Shift-and-Add algorithms 6: The CORDIC algorithm \\ 7: Other shift-and-add algorithms \\ III: Range reduction, final rounding and exceptions \\ 8: Range reduction \\ 9: Final rounding \\ 10: Miscellaneous", tableofcontents = "1 Introduction / 1 \\ 2 Computer Arithmetic / 9 \\ 2.1 Floating-Point Arithmetic / 9 \\ 2.1.1 Floating-point formats / 9 \\ 2.1.2 Rounding modes / 10 \\ 2.1.3 Subnormal numbers and exceptions / 12 \\ 2.1.4 ULPs / 13 \\ 2.1.5 Testing your computational environment / 13 \\ 2.2 Redundant Number Systems / 13 \\ 2.2.1 Signed-digit number systems / 14 \\ 2.2.2 Radix-2 redundant number systems / 15 \\ I Algorithms Based on Polynomial Approximation and/or Table Lookup / 19 \\ 3 Polynomial Approximations / 21 \\ 3.1 Least Squares Polynomial Approximations / 22 \\ 3.1.1 Legendre polynomials / 23 \\ 3.1.2 Chebyshev polynomials / 23 \\ 3.1.3 Jacobi polynomials / 23 \\ 3.2 Least Maximum Approximations / 24 \\ 3.3 Speed of Convergence / 31 \\ 3.4 Rational Approximations / 34 \\ 3.5 Actual Computation / 38 \\ 3.6 Example: the Cyrix FasMath Processor / 41 \\ 3.7 Algorithms and Architectures / 43 \\ 3.7.1 The E-Method / 45 \\ 3.7.2 Estrin's Method / 47 \\ 3.8 Miscellaneous / 47 \\ 4 Table-Based Methods / 51 \\ 4.1 Introduction / 51 \\ 4.2 Table-Driven Algorithms / 53 \\ 4.2.1 Tang's algorithm for $\exp(x)$ in IEEE floating-point arithmetic / 55 \\ 4.2.2 $\ln(x)$ on $[1,2]$ / 57 \\ 4.2.3 $\sin(x)$ on $[0,\pi/4]$ / 58 \\ 4.3 Gal's Accurate Tables Method / 58 \\ 4.4 Methods Requiring Specialized Hardware / 62 \\ 4.4.1 Wong and Goto, logarithm / 62 \\ 4.4.2 Wong and Goto, exponential / 65 \\ II Shift-and-Add Algorithms / 69 \\ 5 Shift-and-Add algorithms / 71 \\ 5.1 The Restoring and Nonrestoring Algorithms / 73 \\ 5.2 Simple Algorithms for Exponentials and Logarithms / 77 \\ 5.2.1 The restoring algorithm for exponentials / 77 \\ 5.2.2 The restoring algorithm for logarithms / 79 \\ 5.3 Faster Algorithms / 81 \\ 5.3.1 Faster computation of exponentials / 81 \\ 5.3.2 Faster computation of logarithms / 87 \\ 5.4 Baker's Predictive Algorithm / 90 \\ 5.5 Bibliographic notes / 98 \\ 6 The CORDIC Algorithm / 101 \\ 6.1 Introduction / 101 \\ 6.2 The Conventional Iteration / 101 \\ 6.3 Scale Factor Compensation / 107 \\ 6.4 CORDIC With Redundant Number Systems / 109 \\ 6.4.1 Signed-digit implementation / 111 \\ 6.4.2 Carry-save implementation / 111 \\ 6.4.3 The variable scale factor problem / 112 \\ 6.5 The Double Rotation Method / 112 \\ 6.6 Branching CORDIC / 115 \\ 6.7 Differential CORDIC / 118 \\ 6.8 Computation of $\cos^{-1}$ and $\sin^{-1}$ / 122 \\ 6.9 Variations on CORDIC / 124 \\ 7 Other Shift-and-Add Algorithms / 127 \\ 7.1 High-Radix Algorithms / 127 \\ 7.1.1 Ercegovac's radix-16 algorithms / 127 \\ 7.2 The BKM Algorithm / 131 \\ 7.2.1 The BKM iteration / 133 \\ 7.2.2 Computation of the exponential function (E-mode) / 133 \\ 7.2.3 Computation of the logarithm function (L-mode) / 137 \\ 7.2.4 Application to the computation of elementary functions / 138 \\ III Range Reduction, Final Rounding and Exceptions / 141 \\ 8 Range Reduction / 143 \\ 8.1 Introduction / 143 \\ 8.2 Cody and Waite's Method for Range Reduction / 148 \\ 8.3 Worst Cases for Range Reduction / 149 \\ 8.3.1 A few basic notions on continued fractions / 149 \\ 8.3.2 Finding worst cases using continued fractions / 151 \\ 8.4 The Payne and Hanek Algorithm / 154 \\ 8.5 The Modular Algorithm / 158 \\ 8.5.1 Fixed-point reduction / 158 \\ 8.5.2 Floating-point reduction / 161 \\ 8.5.3 Architectures for Modular Reduction / 161 \\ 9 Final Rounding / 163 \\ 9.1 Introduction / 163 \\ 9.2 Monotonicity / 164 \\ 9.3 Exact Rounding: Presentation of the Problem / 165 \\ 9.4 Some Experiments / 168 \\ 9.5 A ``Probabilistic'' Approach / 168 \\ 9.6 Upper Bounds on $m$ / 171 \\ 9.6.1 Frequency of failures / 173 \\ 9.6.2 Computing with one million bits / 173 \\ 10 Miscellaneous / 175 \\ 10.1 Exceptions / 175 \\ 10.1.1 NaNs / 176 \\ 10.1.2 Exact results / 177 \\ 10.2 Notes on $x^y$ / 178 \\ 10.3 Multiple Precision / 180", } @InProceedings{Schulte:1997:AFA, author = "M. J. Schulte and James E. Stine", title = "Accurate Function Approximations by Symmetric Table Lookup and Addition", crossref = "Thiele:1997:IIC", pages = "144--153", year = "1997", bibdate = "Sun Mar 04 10:55:40 2007", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://mesa.ece.wisc.edu/publications/cp_1997-02.pdf", acknowledgement = ack-nhfb, } @InProceedings{Schulte:1997:SBT, author = "M. Schulte and J. Stine", title = "Symmetric Bipartite Tables for Accurate Function Approximation", crossref = "Lang:1997:ISC", pages = "175--183", year = "1997", bibdate = "Mon May 20 05:45:32 MDT 2002", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; OCLC Proceedings database", URL = "http://mesa.ece.wisc.edu/publications/cp_1997-01.pdf", acknowledgement = ack-nhfb, } @Article{Segura:1997:CEM, author = "J. Segura and P. Fern{\'a}ndez de C{\'o}rdoba and Yu. L. Ratis", title = "A code to evaluate modified {Bessel} functions based on the continued fraction method", journal = j-COMP-PHYS-COMM, volume = "105", number = "2--3", pages = "263--272", day = "1", month = oct, year = "1997", CODEN = "CPHCBZ", DOI = "https://doi.org/10.1016/S0010-4655(97)00069-6", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", bibdate = "Mon Feb 13 21:30:19 MST 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/compphyscomm1990.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0010465597000696", abstract = "We present an algorithm to evaluate the modified Bessel functions $ I \_ n u $ and $ K_\nu $ of integral and half-integral order based on the calculation of the continued fraction for the $ I \_ n u $'s, the Wronskian and the application of forward recurrence relations for the $ K_\nu $'s and backward recurrence for the $ I \_ n u $'s. The main feature of the algorithm is that it does not require recalculations using normalization relations nor trial values to start the recurrences; the code evaluates in each step (already normalized) Bessel functions. The accuracy of the method ($ 10^{-16} $ for half-integral order and better than $ 2 \times 10^{-7} $ for integral order in our code) is limited only by the precision in the initial values for the recurrence and the maximum order available for a given value of the argument is restricted only by the maximum real number available in the computer.", acknowledgement = ack-nhfb, fjournal = "Computer Physics Communications", journal-URL = "http://www.sciencedirect.com/science/journal/00104655", } @Article{Soderquist:1997:DSR, author = "Peter Soderquist and Miriam Leeser", title = "Division and Square Root: Choosing the Right Implementation: Exploring the major design choices for microprocessor implementations of floating-point division and square root", journal = j-IEEE-MICRO, volume = "17", number = "4", pages = "56--66", month = jul # "\slash " # aug, year = "1997", CODEN = "IEMIDZ", DOI = "https://doi.org/10.1109/40.612224", ISSN = "0272-1732 (print), 1937-4143 (electronic)", ISSN-L = "0272-1732", bibdate = "Thu Dec 14 06:08:58 MST 2000", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeemicro.bib; Science Citation Index database (1980--2000)", URL = "http://pascal.computer.org/mi/books/mi1997/pdf/m4056.pdf", acknowledgement = ack-nhfb, ajournal = "IEEE Micro", fjournal = "IEEE Micro", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=40", } @Book{Thompson:1997:ACMa, author = "William J. (William Jackson) Thompson", title = "Atlas for Computing Mathematical Functions: an Illustrated Guidebook for Practitioners: with Programs in {C} and {Mathematica}", publisher = pub-WILEY, address = pub-WILEY:adr, pages = "xiv + 903", year = "1997", ISBN = "0-471-00260-7 (cloth)", ISBN-13 = "978-0-471-00260-4 (cloth)", LCCN = "QA331.T385 1997", bibdate = "Fri May 21 07:11:19 MDT 1999", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, annote = "A Wiley-Interscience publication. Includes CD-ROM.", keywords = "C (Computer program language); Functions -- Computer programs; Mathematica (Computer program language); Science -- Mathematics -- Computer programs", tableofcontents = "Preface / xiii \\ INTRODUCTION \\ The Atlas of Functions / 1 \\ What This Atlas Contains / 1 \\ How to Use the Atlas / 2 \\ About the Production of the Atlas / 2 \\ The Computer Interface / 3 \\ What the CD-ROM Contains / 3 \\ How to Locate a Function / 3 \\ Exploring Functions with Mathematica / 4 \\ The C Functions: No Assembly Required / 7 \\ Hints for Fortran and Pascal Programmers / 3 \\ File Names for PC-Based Systems / 12 \\ Reliability of Programs: Disclaimer / 12 \\ References on the Computer Interface / 12 \\ PART I. THE FUNCTIONS \\ 1 Introduction to the Functions / 15 \\ How the Function Descriptions Are Organized / 16 \\ 2 A Visual Tour of the Atlas / 17 \\ 3 Computing Strategies / 25 \\ 3.1 General Computing Strategies / 25 \\ 3.2 Iteration and Recursion / 27 \\ 3.3 Continued Fractions and Rational Approximations / 30 \\ 3.4 Using Asymptotic Expansions / 31 \\ 3.5 Euler--Maclaurin Summation Formula / 32 \\ 3.6 Accuracy and Precision of the Functions / 33 \\ 3.7 Mathematical Constants Used in the Atlas / 34 \\ References on Computing Strategies / 34 \\ 4 Elementary Transcendental Functions / 35 \\ 4.1 Exponential and Logarithmic Functions / 35 \\ 4.1.1 Exponentials / 36 \\ 4.1.2 Logarithms / 38 \\ 4.2 Circular and Inverse Circular Functions / 40 \\ 42.1 Circular Functions / 40 \\ 4.2.2 Inverse Circular Functions / 44 \\ 4.3 Hyperbolic and Inverse Hyperbolic Functions / 49 \\ 4.3.1 Hyperbolic Functions / 49 \\ 4.3.2 Inverse Hyperbolic Functions / 53 \\ References on Elementary Transcendental Functions / 58 \\ S Exponential Integrals and Related Functions / 59 \\ 5.1 Exponential and Logarithmic Integrals / 59 \\ 5.1.1 Exponential Integral of the First Kind / 59 \\ 5.1.2 Exponential Integral of the Second Kind / 64 \\ 5.1.3 Logarithmic Integral / 69 \\ 5.2 Cosine and Sine Integrals / 72 \\ References on Exponential Integrals and Related Functions / 78 \\ 6 Gamma and Beta Functions / 79 \\ 6.1 Gamma Function and Beta Function / 79 \\ 6.1.1 Gamma Function / 79 \\ 6.1.2 Beta Function / 84 \\ 6.2 Psi (Digamma) and Polygamma Functions / 86 \\ 6.2.1 Psi Function / 87 \\ 6.2.2 Polygamma Functions / 91 \\ 6.3 Incomplete Gamma and Beta Functions / 97 \\ 6.3.1 Incomplete Gamma Function / 97 \\ 6.3.2 Incomplete Beta Function / 102 \\ References on Gamma and Beta Functions / 106 \\ 7 Combinatorial Functions / 109 \\ 7.1 Factorials and Rising Factorials / 109 \\ 7.1.1 Factorial Function / 110 \\ 7.1.2 Rising Factorial Function / 113 \\ 7.2 Binomial and Multinomial Coefficients / 115 \\ 7.2.1 Binomial Coefficients / 115 \\ 7.2.2 Multinomial Coefficients / 118 \\ 7.3 Stirling Numbers of First and Second Kinds / 121 \\ 7.3.1 Stirling Numbers of the First Kind / 121 \\ 7.3.2 Stirling Numbers of the Second Kind / 124 \\ 7.4 Fibonacci and Lucas Polynomials / 126 \\ 7.4.1 Fibonacci Polynomials and Fibonacci Numbers / 126 \\ 7.4.2 Lucas Polynomials and Lucas Numbers / 128 \\ References on Combinatorial Functions / 130 \\ 8 Number Theory Functions / 133 \\ 8.1 Bernoulli Numbers and Bernoulli Polynomials / 133 \\ 8.1.1 Bernoulli Numbers / 133 \\ 8.1.2 Bernoulli Polynomials / 137 \\ 8.2 Euler Numbers and Euler Polynomials / 139 \\ 8.2.1 Euler Numbers / 139 \\ 8.2.2 Euler Polynomials / 142 \\ 8.3 Riemann Zeta Function / 144 \\ 8.4 Other Sums of Reciprocal Powers / 147 \\ 8.5 Polylogarithms / 151 \\ References on Number Theory Functions / 156 \\ 9 Probability Distributions / 159 \\ 9.1 Overview of Probability Distribution Functions / 160 \\ 9.2 Discrete Probability Distributions / 160 \\ 9.2.1 Binomial Distribution / 162 \\ 9.2.2 Negative Binomial (Pascal) Distribution / 164 \\ 9.2.3 Geometric Distribution / 166 \\ 9.2.4 Hypergeometric Distribution / 168 \\ 9.2.5 Logarithmic Series Distribution / 171 \\ 9.2.6 Poisson Distribution / 173 \\ 9.3 Normal Probability Distributions / 175 \\ 9.3.1 Gauss (Normal) Probability Function / 177 \\ 9.3.2 Bivariate Normal Probability Function / 179 \\ 9.3.3 Chi-Square Probability Functions / 182 \\ 9.3.4 $F$-(Variance-Ratio) Distribution Functions / 188 \\ 9.3.5 Student's $t$-Distribution Functions / 192 \\ 9.3.6 Lognormal Distribution / 196 \\ 9.4 Other Continuous Probability Distributions / 199 \\ 9.4.1 Cauchy (Lorentz) Distribution / 200 \\ 9.4.2 Exponential Distribution / 203 \\ 9.4.3 Pareto Distribution / 205 \\ 9.4.4 Weibull Distribution / 208 \\ 9.4.5 Logistic Distribution / 211 \\ 9.4.6 Laplace Distribution / 213 \\ 9.4.7 Kolmogorov--Smirnov Distribution / 215 \\ 9.4.8 Beta Distribution / 218 \\ References on Probability Distribution Functions / 221 \\ 10 Error Function, Fresnel and Dawson Integrals / 223 \\ 10.1 Error Function / 223 \\ 10.2 Fresnel Integrals / 226 \\ 10.3 Dawson Integral / 234 \\ References on Error Functions, Fresnel and Dawson Integrals / 238 \\ 11 Orthogonal Polynomials / 239 \\ 11.1 Overview of Orthogonal Polynomials / 239 \\ 11.2 Chebyshev Polynomials / 244 \\ 11.2.1 Chebyshev Polynomials of the First Kind / 245 \\ 11.2.2 Chebyshev Polynomials of the Second Kind / 247 \\ 11.3 Gegenbauer (Ultraspherical) Polynomials / 251 \\ 11.4 Hermite Polynomials / 254 \\ 11.5 Laguerre Polynomials / 257 \\ 11.6 Legendre Polynomials / 260 \\ 11.7 Jacobi Polynomials / 263 \\ References on Orthogonal Polynomials / 267 \\ 12 Legendre Functions / 269 \\ 12.1 Overview of Legendre Functions / 269 \\ 12.1.1 Visualizing Legendre Functions of the First Kind / 270 \\ 12.1.2 Visualizing Legendre Functions of the Second Kind / 273 \\ 12.1.3 Legendre Functions and Coordinate Systems / 277 \\ 12.2 Spherical Legendre Functions / 278 \\ 12.2.1 Spherical Polar Coordinates / 278 \\ 12.2.2 Legendre Functions of the First Kind for Integer $m$ and $n$ / 279 \\ 12.2.3 Legendre Functions of the Second Kind for Integer $m$ and $n$ / 284 \\ 12.3 Toroidal Legendre Functions / 291 \\ 12.3.1 Toroidal Coordinates / 291 \\ 12.3.2 Toroidal Functions of the First Kind / 293 \\ 12.3.3 Toroidal Functions of the Second Kind / 296 \\ 12.4 Conical Legendre Functions / 300 \\ 12.4.1 Laplace Equation on a Cone / 300 \\ 12.4.2 Conical Functions / 301 \\ References on Legendre Functions / 304 \\ 13 Spheroidal Wave Functions / 307 \\ 13.1 Overview of Spheroidal Wave Functions / 307 \\ 13.1.1 Spheroidal Coordinates / 308 \\ 13.1.2 Scalar Wave Equation in Spheroidal Coordinates / 309 \\ 13.1.3 Eigenvalues for Spheroidal Equations / 310 \\ 13.1.4 Auxiliary Functions for Eigenvalues / 320 \\ 13.2 Spheroidal Angular Functions / 321 \\ 13.2.1 Expansion Coefficients for Angular Functions / 321 \\ 13.2.2 Spheroidal Angular Functions / 331 \\ 13.3 Spheroidal Radial Functions / 336 \\ 13.3.1 Expansion Coefficients for Radial Functions / 336 \\ 13.3.2 Spheroidal Radial Functions / 340 \\ References on Spheroidal Wave Functions / 344 \\ 14 Bessel Functions / 345 \\ 14.1 Overview of Bessel Functions / 345 \\ 14.2 Bessel Functions of Integer Order / 350 \\ 14.2.1 Regular Cylindrical Bessel Function / 351 \\ 14.2.2 Irregular Cylindrical Bessel Function / 357 \\ 14.2.3 Regular Hyperbolic Bessel Function / 361 \\ 14.2.4 Irregular Hyperbolic Bessel Function / 368 \\ 14.3 Kelvin Functions / 375 \\ 14.3.1 Regular Kelvin Functions / 375 \\ 14.3.2 Irregular Kelvin Functions / 385 \\ 14.4 Bessel Functions of Half-Integer Order / 394 \\ 14.4.1 Regular Spherical Bessel Function / 394 \\ 14.4.2 Irregular Spherical Bessel Function / 401 \\ 14.4.3 Regular Modified Spherical Bessel Function / 405 \\ 14.4.4 Irregular Modified Spherical Bessel Function / 412 \\ 14.5 Airy Functions / 416 \\ 14.5.1 Airy Functions / 416 \\ 14.5.2 Derivatives of Airy Functions / 425 \\ References on Bessel Functions / 434 \\ 15 Struve, Anger, and Weber Functions / 435 \\ 15.1 Struve Functions / 435 \\ 15.1.1 Struve Function / 435 \\ 15.1.1 Modified Struve Function / 442 \\ 15.2 Anger and Weber Functions / 448 \\ 15.2.1 Overview of Anger and Weber Functions / 448 \\ 15.2.2 Anger Function / 449 \\ 15.2.3 Weber Function / 455 \\ References on Struve, Anger, and Weber Functions / 458 \\ 16 Hypergeometric Functions and Coulomb Wave Functions / 461 \\ 16.1 Hypergeometric Functions / 461 \\ 16.2 Confluent Hypergeometric Functions / 465 \\ 16.2.1 Regular Function / 465 \\ 16.2.2 Irregular Function / 471 \\ 16.3 Coulomb Wave Functions / 478 \\ 16.3.1 Regular Functions and Derivatives / 478 \\ 16.3.2 Irregular Functions and Derivatives / 487 \\ References on Hypergeometric Functions and Coulomb Wave Functions / 493 \\ 17 Elliptic Integrals and Elliptic Functions / 495 \\ 17.1 Overview of Elliptic Integrals and Elliptic Functions / 495 \\ 17.2 Elliptic Integrals / 496 \\ 17.2.1 Elliptic Integrals of the First Kind / 496 \\ 17.2.2 Elliptic Integrals of the Second Kind / 502 \\ 17.2.3 Jacobi Zeta Function / 506 \\ 17.2.4 Heuman Lambda Function / 510 \\ 17.2.5 Elliptic Integrals of the Third Kind / 513 \\ 17.3 Jacobi Elliptic Functions and Theta Functions / 517 \\ 17.3.1 Jacobi Elliptic Functions / 517 \\ 17.3.2 Theta Functions / 525 \\ 17.3.3 Logarithmic Derivatives of Theta Functions / 531 \\ References on Elliptic Integrals and Elliptic Functions / 536 \\ 18 Parabolic Cylinder Functions / 539 \\ 18.1 Parabolic Cylinder Coordinates / 539 \\ 18.2 Parabolic Cylinder Functions / 540 \\ 18.2.1 Parabolic Cylinder Functions U / 540 \\ 18.2.2 Parabolic Cylinder Functions V / 546 \\ References on Parabolic Cylinder Functions / 550 \\ 19 Miscellaneous Functions for Science and Engineering / 551 \\ 19.1 Debye Functions / 551 \\ 19.2 Sievert Integral / 554 \\ 19.3 Abramowitz Function / 557 \\ 19.4 Spence Integeral / 562 \\ 19.5 Clausen Integral / 565 \\ 19.6 Voigt (Plasma Dispersion) Function / 570 \\ 19.7 Angular Momentum Coupling Coefficients 576 / 539 \\ 19.7.1 3-j Coefficients / 578 \\ 19.7.2 6-j Coefficients / 582 \\ 19.7.3 9-j Coefficients / 586 \\ References on Miscellaneous Functions for Science and Engineering \\ PART II. THE COMPUTER INTERFACE \\ 20 The Mathematica Notebooks / 593 \\ 20.1 Introduction to the Notebooks / 593 \\ 20.2 Exploring with the Notebook Cells / 594 \\ 20.3 The Annotated Notebooks / 594 \\ 20.4 Elementary Transcendental Functions / 595 \\ 20.5 Exponential Integrals and Related Functions / 602 \\ 20.6 Gamma and Beta Functions / 607 \\ 20.7 Combinatorial Functions / 618 \\ 20.8 Number Theory Functions / 627 \\ 20.9 Probability Distributions / 633 \\ 20.10 Error Function, Fresnel and Dawson Integrals / 663 \\ 20.11 Orthogonal Polynomials / 667 \\ 20.12 Legendre Functions / 676 \\ 20.14 Bessel Functions / 708 \\ 20.15 Struve, Anger, and Weber Functions / 747 \\ 20.16 Hypergeometric Functions and Coulomb Wave Functions / 756 \\ 20.17 Elliptic Integrals and Elliptic Functions / 765 \\ 20.18 Parabolic Cylinder Functions / 782 \\ 20.19 Miscellaneous Functions for Science and Engineering / 788 \\ 21 The C Driver Programs / 797 \\ 21.1 Introduction to the C Driver Programs / 797 \\ 21.2 How the C Drivers are Organized / 797 \\ 21.3 Annotations to the C Driver Programs / 798 \\ 21.4 Elementary Transcendental Functions / 798 \\ 21.4.1 Exponential and Logarithmic Functions / 798 \\ 21.4.2 Circular and Inverse Circular Functions / 799 \\ 21.4.3 Hyperbolic and Inverse Hyperbolic Functions / 800 \\ 21.5 Exponential Integrals and Related Functions / 802 \\ 21.5.1 Exponential and Logarithmic Integrals / 802 \\ 21.5.2 Cosine and Sine Integrals / 803 \\ 21.6 Gamma and Beta Functions / 804 \\ 21.6.1 Gamma Function and Beta Function / 804 \\ 21.6.2 Psi (Digamma) and Polygamma Functions / 805 \\ 21.6.3 Incomplete Gamma and Beta Functions / 807 \\ 21.7 Combinatorial Functions / 808 \\ 21.7.1 Factorials and Rising Factorials / 808 \\ 21.7.2 Binomial and Multinomial Coefficients / 809 \\ 21.7.3 Stirling Numbers of the First and Second Kinds / 811 \\ 21.7.4 Fibonacci and Lucas Polynomials / 811 \\ 21.8 Number Theory Functions / 813 \\ 21.8.1 Bernoulli Numbers and Bernoulli Polynomials / 813 \\ 21.8.2 Euler Numbers and Euler Polynomials / 814 \\ 21.8.3 Riemann Zeta Function / 814 \\ 21.8.4 Other Sums of Reciprocal Powers / 815 \\ 21.8.5 Polylogarithms / 816 \\ 21.9 Probability Distributions / 816 \\ 21.9.1 Organization of the PDFs / 816 \\ 21.9.2 Discrete Probability Distributions / 816 \\ 21.9.3 Normal Probability Distributions / 820 \\ 21.9.4 Other Continuous Probability Distributions / 825 \\ 21.10 Error Function, Fresnel and Dawson Integrals / 830 \\ 21.10.1 Error Function / 830 \\ 21.10.2 Fresnel Integrals / 831 \\ 21.10.3 Dawson Integral / 832 \\ 21.11 Orthogonal Polynomials / 833 \\ 21.11.1 Orthogonal Polynomial Functions / 833 \\ 21.11.2 Chebyshev Polynomials / 833 \\ 21.11.3 Gegenbauer (Ultraspherical) Polynomials / 834 \\ 21.11.4 Hermite Polynomials / 835 \\ 21.11.5 Laguerre Polynomials / 835 \\ 21.11.6 Legendre Polynomials / 836 \\ 21.11.7 Jacobi Polynomials / 837 \\ 21.12 Legendre Functions / 837 \\ 21.12.1 Overview of Legendre Functions / 838 \\ 21.12.2 Spherical Legendre Functions / 838 \\ 21.12.3 Toroidal Legendre Functions / 840 \\ 21.12.4 Conical Legendre Functions / 841 \\ 21.13 Spheroidal Wave Functions / 842 \\ 21.13.1 Overview of Spheroidal Wave Functions / 842 \\ 21.13.2 Spheroidal Angular Functions / 843 \\ 21.13.3 Spheroidal Radial Functions / 844 \\ 21.14 Bessel Functions / 846 \\ 21.14.1 Overview of Bessel Functions / 846 \\ 21.14.2 Bessel Functions of Integer Order / 846 \\ 21.14.3 Kelvin Functions / 850 \\ 21.14.4 Bessel Functions of Half-Integer Order / 852 \\ 21.14.5 Airy Functions / 856 \\ 21.15 Struve, Anger, and Weber Functions / 857 \\ 21.15.1 Struve Functions / 857 \\ 21.15.2 Anger and Weber Functions / 859 \\ 21.16 Hypergeometric Functions and Coulomb Wave Functions / 861 \\ 21.16.1 Hypergeometric Functions / 861 \\ 21.16.2 Confluent Hypergeometric Functions / 862 \\ 21.16.3 Coulomb Wave Functions / 863 \\ 21.17 Elliptic Integrals and Elliptic Functions / 864 \\ 21.17.1 Overview of Elliptic Integrals and Elliptic Functions / 864 \\ 21.17.2 Elliptic Integrals / 864 \\ 21.17.3 Jacobi Elliptic Functions and Theta Functions / 867 \\ 21.18 Parabolic Cylinder Functions / 870 \\ 21.18.1 Parabolic Cylinder Functions / 870 \\ 21.19 Miscellaneous Functions for Science and Engineering / 872 \\ 21.19.1 Debye Functions / 872 \\ 21.19.2 Sievert Integral / 873 \\ 21.19.3 Abramowitz Function / 873 \\ 21.19.4 Spence Integral / 874 \\ 21.19.5 Clausen Integral / 875 \\ 21.19.6 Voigt (Plasma Dispersion) Function / 876 \\ 21.19.7 Angular Momentum Coupling Coefficients / 876 \\ APPENDIX: File Names for PC-Based Systems / 879 \\ INDEXES \\ Index of Function Notations / 883 \\ Index of Programs and Dependencies / 886 \\ Index of Subjects and Authors / 889", } @Book{Thompson:1997:ACMb, author = "William J. (William Jackson) Thompson", title = "Atlas for Computing Mathematical Functions: an Illustrated Guide for Practitioners with Programs in {Fortran 90} and {Mathematica}", publisher = pub-WILEY, address = pub-WILEY:adr, pages = "xiv + 888", year = "1997", ISBN = "0-471-18171-4 (cloth)", ISBN-13 = "978-0-471-18171-2 (cloth)", LCCN = "QA331.T386 1997", bibdate = "Fri May 21 07:11:19 MDT 1999", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "Includes CD-ROM.", acknowledgement = ack-nhfb, annote = "A Wiley-Interscience publication. System requirements for accompanying computer disc: Windows; Macintosh compatible.", keywords = "FORTRAN (Computer program language); Functions -- Computer programs; Mathematica (Computer program language); Science -- Mathematics -- Computer programs", } @Misc{Underwood:1997:HDC, author = "J. Underwood and B. Edwards", title = "How do calculators calculate trigonometric functions?", howpublished = "Educational Resources Information Center (ERIC) document ED461519", year = "1997", bibdate = "Tue Nov 11 19:58:53 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.math.ufl.edu/~haven/papers/paper.pdf; https://eric.ed.gov/?q=ED461519&id=ED461519", abstract = "How does your calculator quickly produce values of trigonometric functions? You might be surprised to learn that it does not use series or polynomial approximations, but rather the so-called CORDIC method. This paper will focus on the geometry of the CORDIC method, as originally developed by Volder in 1959. This algorithm is a wonderful application of sequences and will be demonstrated on the TI-86 graphing calculator. A rigorous convergence proof for the CORDIC method is also provided.", acknowledgement = ack-nhfb, remark = "URL no longer resolvable at ufl.edu, and not found at archive.org; PDF file under reconstruction at ERIC site.", } @Book{Yoshida:1997:HFM, author = "Masaaki Yoshida", title = "Hypergeometric Functions, My Love: Modular Interpretations of Configuration Spaces", volume = "E 32", publisher = pub-VIEWEG, address = pub-VIEWEG:adr, pages = "xvi + 292", year = "1997", ISBN = "3-528-06925-2", ISBN-13 = "978-3-528-06925-4", ISSN = "0179-2156", LCCN = "QA353.H9 Y67 1997", bibdate = "Sat Oct 30 21:12:24 MDT 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; z3950.loc.gov:7090/Voyager", series = "Aspects of mathematics", acknowledgement = ack-nhfb, subject = "Hypergeometric functions; Configuration space", tableofcontents = "Part 1: The Story of the Configuration Space $X(2,4)$ of Four Points on the Projective Line \\ I. Configuration Spaces --- The Simplest Case \\ II. Elliptic Curves \\ III. Modular Interpretations of X(2,4) \\ IV. Hypergeometric Integrals and Loaded Cycles \\ 2 The Story of the Configuration Space X(2,n) of n Points on the Projective Line \\ V. The Configuration Space X(2,5) \\ VI. Modular Interpretation of the Configuration Space X(2,n) \\ 3 The Story of the Configuration Space X(3,6) of Six Lines on the Projective Plane \\ VII The Configuration Space X(3,6) \\ VIII. Hypergeometric Functions of Type (3,6) \\ IX. Modular Interpretation of the Configuration Space X(3,6)", } @Article{Yousif:1997:BFF, author = "Hashim A. Yousif and Richard Melka", title = "{Bessel} function of the first kind with complex argument", journal = j-COMP-PHYS-COMM, volume = "106", number = "3", pages = "199--206", month = nov, year = "1997", CODEN = "CPHCBZ", DOI = "https://doi.org/10.1016/S0010-4655(97)00087-8", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", bibdate = "Mon Feb 13 21:30:21 MST 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/compphyscomm1990.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0010465597000878", abstract = "A new method of computing integral order Bessel functions of the first kind $ J_n(z) $ when either the absolute value of the real part or the imaginary part of the argument $ z = x + i y $ is small, is described. This method is based on computing the Bessel functions from asymptotic expressions when $ x \sim 0 $ (or $ y \sim 0 $ ). These expansions are derived from the integral definition of Bessel functions. This method is necessary because some existing algorithms and methods fail to give correct results for small $x$ or small $y$. In addition, our overall method of computing Bessel functions of any order and argument is discussed and the logarithmic derivative is used in computing these functions. The starting point of the backward recurrence relations needed to evaluate the Bessel function and their logarithmic derivatives are investigated in order to obtain accurate numerical results. Our numerical method, together with established techniques of computing the Bessel functions, is easy to implement, efficient, and produces reliable results for all $z$.", acknowledgement = ack-nhfb, fjournal = "Computer Physics Communications", journal-URL = "http://www.sciencedirect.com/science/journal/00104655", } @Article{Zhang:1997:CSA, author = "Jun Zhang and John A. Belward", title = "{Chebyshev} series approximations for the {Bessel} function {$ Y_n(z) $} of complex argument", journal = j-APPL-MATH-COMP, volume = "88", number = "2--3", pages = "275--286", day = "30", month = dec, year = "1997", CODEN = "AMHCBQ", DOI = "https://doi.org/10.1016/S0096-3003(96)00335-9", ISSN = "0096-3003 (print), 1873-5649 (electronic)", ISSN-L = "0096-3003", bibdate = "Tue Nov 20 21:02:59 MST 2012", bibsource = "http://www.sciencedirect.com/science/journal/00963003; https://www.math.utah.edu/pub/tex/bib/applmathcomput1995.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0096300396003359", acknowledgement = ack-nhfb, fjournal = "Applied Mathematics and Computation", journal-URL = "http://www.sciencedirect.com/science/journal/00963003", } @Article{Aberbour:1998:PMF, author = "M. Aberbour and A. Houelle and H. Mehrez and N. Vaucher and A. Guyot", title = "On portable macrocell {FPU} generators for division and square root operators complying to the full {IEEE-754} standard", journal = j-IEEE-TRANS-VLSI-SYST, volume = "6", number = "1", pages = "114--121", month = mar, year = "1998", CODEN = "IEVSE9", DOI = "https://doi.org/10.1109/92.661253", ISSN = "1063-8210 (print), 1557-9999 (electronic)", ISSN-L = "1063-8210", bibdate = "Sat Jul 16 15:28:13 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "IEEE Transactions on Very Large Scale Integration (VLSI) Systems", summary = "In this paper, we investigate the design of macrocell generators of division and square root floating-point operators. The number representation used in our operators is the IEEE-754-1985 standard for binary floating-point numbers. The design and \ldots{}", } @Article{Adamchik:1998:PFN, author = "Victor S. Adamchik", title = "{Polygamma} functions of negative order", journal = j-J-COMPUT-APPL-MATH, volume = "100", number = "2", pages = "191--199", day = "21", month = dec, year = "1998", CODEN = "JCAMDI", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Sat Feb 25 12:39:42 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath1990.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0377042798001927", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @Article{Alzer:1998:CHM, author = "Horst Alzer", title = "Corrigendum: {A harmonic mean inequality for the gamma function [J. Comput. Appl. Math. {\bf 87} (1997) 195--198]}", journal = j-J-COMPUT-APPL-MATH, volume = "90", number = "2", pages = "265--265", day = "17", month = apr, year = "1998", CODEN = "JCAMDI", DOI = "https://doi.org/10.1016/S0377-0427(98)00040-5", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Sat Feb 25 12:36:08 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath1990.bib", note = "See \cite{Alzer:1997:HMI}.", URL = "http://www.sciencedirect.com/science/article/pii/S0377042798000405", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @Book{Andrews:1998:SFM, author = "Larry C. Andrews", title = "Special functions of mathematics for engineers", publisher = pub-OXFORD, address = pub-OXFORD:adr, edition = "Second", pages = "xvii + 479", year = "1998", ISBN = "0-19-856558-5 (Oxford hardcover), 0-8194-2616-4 (SPIE Press hardcover)", ISBN-13 = "978-0-19-856558-1 (Oxford hardcover), 978-0-8194-2616-1 (SPIE Press)", LCCN = "QA351 .A75 1998", bibdate = "Sat Oct 30 16:44:00 MDT 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcw.bib; library.ox.ac.uk:210/ADVANCE; z3950.loc.gov:7090/Voyager", acknowledgement = ack-nhfb, remark = "Originally published: New York : McGraw-Hill, c1992.", subject = "Functions, Special", tableofcontents = "Preface to the Second Edition \\ Preface to the First Edition \\ Notation for Special Functions \\ Infinite Series, Improper Integrals, and Infinite Products \\ The Gamma Function and Related Functions \\ Other Functions Defined by Integrals \\ Legendre Polynomials and Related Functions \\ Other Orthogonal Polynomials \\ Bessel Functions \\ Bessel Functions of Other Kinds \\ Applications Involving Bessel Functions \\ The Hypergeometric Function \\ The Confluent Hypergeometric Functions \\ Generalized Hypergeometric Functions \\ Applications Involving Hypergeometric-Type Functions \\ Bibliography \\ Appendix: A List of Special Function Formulas \\ Selected Answers to Exercises \\ Index", } @Book{Anonymous:1998:AMS, editor = "Anonymous", title = "Analytical methods and special functions", publisher = "Gordon and Breach Science Publishers", address = "Amsterdam, The Netherlands", pages = "????", year = "1998", ISSN = "1027-0264", LCCN = "A299.6 A533 v. 1 1998", bibdate = "Sat Oct 30 19:02:18 2010", bibsource = "http://cat.cisti-icist.nrc-cnrc.gc.ca/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, } @Article{Antelo:1998:CVH, author = "E. Antelo and T. Lang and J. D. Bruguera", title = "Computation of $ \sqrt {(x / d)} $ in a very high radix combined division\slash square-root unit with scaling and selection by rounding", journal = j-IEEE-TRANS-COMPUT, volume = "47", number = "2", pages = "152--161", month = feb, year = "1998", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/12.663761", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Wed Jul 6 09:35:53 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput1990.bib", URL = "https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=663761", acknowledgement = ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", summary = "A very-high radix digit-recurrence algorithm for the operation {\surd}(x/d) is developed, with residual scaling and digit selection by rounding. This is an extension of the division and square-root algorithms presented previously, and for which a \ldots{}", } @Article{BenGhanem:1998:QFU, author = "Riadh {Ben Ghanem}", title = "Quadrature formulae using zeros of {Bessel} functions as nodes", journal = j-MATH-COMPUT, volume = "67", number = "221", pages = "323--336", month = jan, year = "1998", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "65D32", MRnumber = "98c:65031", MRreviewer = "Kai Diethelm", bibdate = "Fri Jul 16 10:38:50 MDT 1999", bibsource = "http://www.ams.org/mcom/1998-67-221; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.ams.org/jourcgi/jour-pbprocess?fn=110&arg1=S0025-5718-98-00882-5&u=/mcom/1998-67-221/", acknowledgement = ack-nhfb, fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @TechReport{Borwein:1998:CSR, author = "Jonathan M. Borwein and David M. Bradley and Richard E. Crandall", title = "Computational Strategies for the {Riemann} Zeta Function", type = "Report", number = "CECM-98-118", institution = inst-CECM, address = inst-CECM:adr, pages = "68", day = "30", month = oct, year = "1998", bibdate = "Mon Oct 24 11:29:15 2011", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib; https://www.math.utah.edu/pub/bibnet/authors/c/crandall-richard-e.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "Published in \cite{Borwein:2000:CSR}.", URL = "http://docserver.carma.newcastle.edu.au/211; http://people.reed.edu/~crandall/papers/attach01.pdf", abstract = "We provide a compendium of evaluation methods for the Riemann zeta function, presenting formulae ranging from historical attempts to recently found convergent series to curious oddities old and new. We concentrate primarily on practical computational issues, such issues depending on the domain of the argument, the desired speed of computation, and the incidence of what we call ``value recycling.''", acknowledgement = ack-nhfb, author-dates = "Jonathan Michael Borwein (20 May 1951--2 August 2016); Richard Eugene Crandall (29 December 1947--20 December 2012)", ORCID-numbers = "Borwein, Jonathan/0000-0002-1263-0646", } @InCollection{Buhring:1998:ACG, author = "Wolfgang B{\"u}hring and H. M. Srivastava", editor = "Themistocles M. Rassias", booktitle = "Approximation theory and applications", title = "Analytic Continuation of the Generalized Hypergeometric Series Near Unit Argument with Emphasis on the Zero-Balanced Series", publisher = "Hadronic Press", address = "Palm Harbor, FL, USA", bookpages = "v + 193", pages = "17--35", year = "1998", ISBN = "1-57485-041-5", ISBN-13 = "978-1-57485-041-3", LCCN = "QA297.5 .A685 1998", MRclass = "33C20; 41-06 (00B15)", MRnumber = "MR1924838 (2003i:33006); MR1924835 (2003c:41003)", bibdate = "Thu Dec 01 10:08:00 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://catalog.hathitrust.org/api/volumes/oclc/42786578.html", acknowledgement = ack-nhfb, remark = "The paper treats $_{p + 1F}_p(z)$ for $ z \approx 1 $. Available as arxiv:math/0102032.", } @Article{Carsky:1998:IGF, author = "Petr C{\'a}rsky and Martin Pol{\'a}sek", title = "Incomplete Gamma {$ F_m(x) $} Functions for Real Negative and Complex Arguments", journal = j-J-COMPUT-PHYS, volume = "143", number = "1", pages = "259--265", day = "10", month = jun, year = "1998", CODEN = "JCTPAH", DOI = "https://doi.org/10.1006/jcph.1998.5975", ISSN = "0021-9991 (print), 1090-2716 (electronic)", ISSN-L = "0021-9991", bibdate = "Mon Jan 2 07:55:26 MST 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputphys1990.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0021999198959757", acknowledgement = ack-nhfb, fjournal = "Journal of Computational Physics", journal-URL = "http://www.sciencedirect.com/science/journal/00219991/", } @Article{Cornea-Hasegan:1998:PIC, author = "Marius Cornea-Hasegan", title = "Proving the {IEEE} Correctness of Iterative Floating-Point Square Root, Divide, and Remainder Algorithms", journal = j-INTEL-TECH-J, volume = "Q2", number = "Q2", pages = "11", year = "1998", ISSN = "1535-766X", bibdate = "Fri Jun 01 06:02:08 2001", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "http://developer.intel.com/technology/itj/q21998/articles/art_3.htm; http://developer.intel.com/technology/itj/q21998/pdf/ieee.pdf", acknowledgement = ack-nhfb, } @Article{Crenshaw:1998:ISR, author = "Jack W. Crenshaw", title = "Integer Square Roots", journal = j-EMBED-SYS-PROG, volume = "11", number = "2", pages = "15--32", month = feb, year = "1998", CODEN = "EYPRE4", ISSN = "1040-3272", bibdate = "Fri Nov 28 16:31:58 2003", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "http://www.embedded.com/98/9802fe2.htm", acknowledgement = ack-mfc # " and " # ack-nhfb, fjournal = "Embedded Systems Programming", } @Article{Dattoli:1998:GBF, author = "G. Dattoli and A. Torre and S. Lorenzutta and G. Maino", title = "Generalized {Bessel} functions and {Kapteyn} series", journal = j-COMPUT-MATH-APPL, volume = "35", number = "8", pages = "117--125", month = apr, year = "1998", CODEN = "CMAPDK", ISSN = "0898-1221 (print), 1873-7668 (electronic)", ISSN-L = "0898-1221", bibdate = "Wed Mar 1 21:48:48 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/computmathappl1990.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0898122198000509", acknowledgement = ack-nhfb, fjournal = "Computers and Mathematics with Applications", journal-URL = "http://www.sciencedirect.com/science/journal/08981221", } @Article{Deleglise:1998:C, author = "Marc Del{\'e}glise and Jo{\"e}l Rivat", title = "Computing $ \psi (x) $", journal = j-MATH-COMPUT, volume = "67", number = "224", pages = "1691--1696", month = oct, year = "1998", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "11Y35 (11N56)", MRnumber = "1 474 649", bibdate = "Fri Jul 16 10:38:58 MDT 1999", bibsource = "http://www.ams.org/mcom/1998-67-224; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.ams.org/jourcgi/jour-pbprocess?fn=110&arg1=S0025-5718-98-00977-6&u=/mcom/1998-67-224/", acknowledgement = ack-nhfb, fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Fowler:1998:SRA, author = "David Fowler and Eleanor Robson", title = "Square Root Approximations in Old {Babylonian} Mathematics: {YBC 7289} in Context", journal = j-HIST-MATH, volume = "25", number = "4", pages = "366--378", month = nov, year = "1998", CODEN = "HIMADS", ISSN = "0315-0860 (print), 1090-249X (electronic)", ISSN-L = "0315-0860", bibdate = "Wed Jun 26 06:19:31 MDT 2013", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/histmath.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0315086098922091", acknowledgement = ack-nhfb, fjournal = "Historia Mathematica", journal-URL = "http://www.sciencedirect.com/science/journal/03150860", } @InProceedings{Gautschi:1998:IGF, author = "Walter Gautschi", title = "The incomplete gamma functions since {Tricomi}", crossref = "Anonymous:1998:TIC", pages = "203--237", year = "1998", bibdate = "Fri May 31 16:38:24 2024", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/g/gautschi-walter.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "https://web.archive.org/web/20070503103643/http://citeseer.ist.psu.edu/gautschi98incomplete.html", acknowledgement = ack-nhfb, } @Article{Giordano:1998:UTG, author = "C. Giordano and A. Laforgia and J. Pecari{\'c}", title = "Unified treatment of {Gautschi--Kershaw} type inequalities for the gamma function", journal = j-J-COMPUT-APPL-MATH, volume = "99", number = "1--2", pages = "167--175", day = "16", month = nov, year = "1998", CODEN = "JCAMDI", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Sat Feb 25 12:36:13 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath1990.bib", URL = "http://www.sciencedirect.com/science/article/pii/S037704279800154X", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @Article{Harris:1998:MAL, author = "Frank E. Harris", title = "More About the Leaky Aquifer Function", journal = j-IJQC, volume = "70", number = "4--5", pages = "623--626", month = "????", year = "1998", CODEN = "IJQCB2", DOI = "https://doi.org/10.1002/(SICI)1097-461X(1998)70:4/5<623::AID-QUA8>3.0.CO%3B2-X", ISSN = "0020-7608 (print), 1097-461X (electronic)", ISSN-L = "0020-7608", bibdate = "Tue Oct 4 06:59:18 MDT 2011", bibsource = "http://www3.interscience.wiley.com/journalfinder.html; https://www.math.utah.edu/pub/bibnet/authors/h/harris-frank-e.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/ijqc.bib; https://www.math.utah.edu/pub/tex/bib/ijqc1990.bib", URL = "http://www3.interscience.wiley.com/cgi-bin/abstract?ID=75040; http://www3.interscience.wiley.com/cgi-bin/fulltext?ID=75040&PLACEBO=IE.pdf", acknowledgement = ack-nhfb, ajournal = "Int. J. Quantum Chem.", fjournal = "International Journal of Quantum Chemistry", journal-URL = "http://www.interscience.wiley.com/jpages/0020-7608/", onlinedate = "7 Dec 1998", } @Article{Homeier:1998:AHC, author = "Herbert H. H. Homeier", title = "An asymptotically hierarchy-consistent, iterative sequence transformation for convergence acceleration of {Fourier} series", journal = j-NUMER-ALGORITHMS, volume = "18", number = "1", pages = "1--30", month = sep, year = "1998", CODEN = "NUALEG", ISSN = "1017-1398 (print), 1572-9265 (electronic)", ISSN-L = "1017-1398", bibdate = "Mon Sep 29 08:36:54 MDT 2003", bibsource = "http://www.kluweronline.com/issn/1017-1398; http://www.math.psu.edu/dna/contents/na.html; https://www.math.utah.edu/pub/bibnet/authors/h/homeier-herbert-h-h.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://ipsapp007.kluweronline.com/content/getfile/5058/13/2/abstract.htm; http://ipsapp007.kluweronline.com/content/getfile/5058/13/2/fulltext.pdf; http://www.chemie.uni-regensburg.de/pub/preprint/preprint.html#TCNA972", ZMnumber = "914.65140", acknowledgement = ack-nhfb, fjournal = "Numerical Algorithms", journal-URL = "http://link.springer.com/journal/11075", keywords = "convergence acceleration", tech = "Technical Report TC-NA-97-2, Institut f{\"u}r {Physikalische} und {Theoretische Chemie, Universit{\"a}t Regensburg, D-93040 Regensburg}, 1997", } @Article{Homeier:1998:CAM, author = "H. H. H. Homeier", title = "On Convergence Acceleration of Multipolar and Orthogonal Expansions", journal = j-INTERNET-J-CHEM, volume = "1", number = "Article 28", pages = "????", year = "1998", CODEN = "IJCHFJ", ISSN = "1099-8292", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/h/homeier-herbert-h-h.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "Proceedings of the {4$^{th}$ Electronic Computational Chemistry Conference}.", URL = "http://www.ijc.com/articles/1998v1/28/", fjournal = "Internet Journal of Chemistry", keywords = "convergence acceleration", tech = "Technical Report TC-QM-97-5, Institut f{\"u}r {Physikalische} und {Theoretische Chemie, Universit{\"a}t Regensburg, D-93040 Regensburg}, 1997", } @Article{Jukic:1998:DTN, author = "D. Juki{\'c} and T. Maros{\v{s}}evi{\'c} and R. Scitovski", title = "Discrete total $ l_p $-norm approximation problem for the exponential function", journal = j-APPL-MATH-COMP, volume = "94", number = "2--3", pages = "137--143", day = "15", month = aug, year = "1998", CODEN = "AMHCBQ", DOI = "https://doi.org/10.1016/S0096-3003(97)10068-6", ISSN = "0096-3003 (print), 1873-5649 (electronic)", ISSN-L = "0096-3003", bibdate = "Tue Nov 20 21:03:11 MST 2012", bibsource = "http://www.sciencedirect.com/science/journal/00963003; https://www.math.utah.edu/pub/tex/bib/applmathcomput1995.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0096300397100686", acknowledgement = ack-nhfb, fjournal = "Applied Mathematics and Computation", journal-URL = "http://www.sciencedirect.com/science/journal/00963003", } @Article{Kiranon:1998:SRV, author = "W. Kiranon and N. Kumprasert", title = "Square-rooting and vector summation circuits using current conveyors", journal = "IEE Proceedings on Circuits, Devices and Systems [see also IEE Proceedings G - Circuits, Devices and Systems]", volume = "145", number = "2", pages = "139", month = apr, year = "1998", CODEN = "????", ISSN = "????", bibdate = "Sat Jul 16 15:28:13 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, summary = "Recently, Lui [1995] presented a square-rooting circuit using CCII, MOS transistors and a buffered unity-gain inverting amplifier. It is interesting since it finds various applications as described in his paper. However, an error occurred in the \ldots{}", } @InBook{Knuth:1998:EP, author = "Donald E. Knuth", title = "Evaluation of polynomials", crossref = "Knuth:1998:SA", chapter = "4", pages = "485--524", year = "1998", bibdate = "Fri Oct 20 11:29:58 2023", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, keywords = "number of multiplications to evaluate a polynomial", remark = "This is the definitive treatment of the rearrangement of polynomial coefficients to reduce the multiplication count. See \cite{Todd:1955:MWN} and references therein to early papers on the subject.", } @Article{Kramer:1998:PWC, author = "W. Kramer", title = "A priori worst case error bounds for floating-point computations", journal = j-IEEE-TRANS-COMPUT, volume = "47", number = "7", pages = "750--756", month = jul, year = "1998", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/12.709374", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Wed Jul 6 09:35:55 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput1990.bib", note = "See \cite{Tang:1992:TDI}.", URL = "https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=709374", abstract = "A new technique for the a priori calculation of rigorous error bounds for floating-point computations is introduced. The theorems given in the paper combined with interval arithmetic lead to the implementation of reliable software routines, which enable the user to compute the desired error bounds automatically by a suitable computer program. As a prominent example, a table-lookup algorithm for calculating the function $ e x p(x) - 1 $ that has been published by P. T. P. Tang (1992) is analyzed using these new tools. The result shows the high quality of the new approach", acknowledgement = ack-nhfb, ajournal = "IEEE Trans. Comput.", author-dates = "1952--2014 (WK)", fjournal = "IEEE Transactions on Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", } @Article{Kravanja:1998:ZMS, author = "P. Kravanja and O. Ragos and M. N. Vrahatis and F. A. Zafiropoulos", title = "{ZEBEC}: a mathematical software package for computing simple zeros of {Bessel} functions of real order and complex argument", journal = j-COMP-PHYS-COMM, volume = "113", number = "2--3", pages = "220--238", month = oct, year = "1998", CODEN = "CPHCBZ", DOI = "https://doi.org/10.1016/S0010-4655(98)00064-2", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", bibdate = "Mon Feb 13 21:30:30 MST 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/compphyscomm1990.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0010465598000642", acknowledgement = ack-nhfb, fjournal = "Computer Physics Communications", journal-URL = "http://www.sciencedirect.com/science/journal/00104655", } @InProceedings{Kuhlmann:1998:FLP, author = "M. Kuhlmann and K. K. Parhi", booktitle = "{Proceedings of the 1998 International Conference on Computer Design: VLSI in Computers and Processors. ICCD '98}", title = "Fast low-power shared division and square-root architecture", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "128--135", year = "1998", CODEN = "????", ISSN = "????", bibdate = "Sat Jul 16 15:28:14 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, summary = "This paper addresses a fast low-power implementation of a shared division and square-root architecture. Two approaches are considered in this paper; these include the SRT (Sweeney, Robertson and Tocher) approach which does not require prescaling and \ldots{}", } @Article{Lefevre:1998:TCR, author = "V. Lef{\`e}vre and J.-M. Muller and A. Tisserand", title = "Toward correctly rounded transcendentals", journal = j-IEEE-TRANS-COMPUT, volume = "47", number = "11", pages = "1235--1243", month = nov, year = "1998", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/12.736435", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Sat Jul 16 11:25:04 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", summary = "The Table Maker's Dilemma is the problem of always getting correctly rounded results when computing the elementary functions. After a brief presentation of this problem, we present new developments that have helped us to solve this problem for the \ldots{}", } @Article{Lopez:1998:SSC, author = "Jos{\'e}L. L{\'o}pez", title = "Several series containing gamma and polygamma functions", journal = j-J-COMPUT-APPL-MATH, volume = "90", number = "1", pages = "15--23", day = "6", month = apr, year = "1998", CODEN = "JCAMDI", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Sat Feb 25 12:36:07 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath1990.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0377042798000077", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @Article{Miller:1998:CGI, author = "Allen R. Miller and Ira S. Moskowitz", title = "On certain generalized incomplete gamma functions", journal = j-J-COMPUT-APPL-MATH, volume = "91", number = "2", pages = "179--190", day = "4", month = may, year = "1998", CODEN = "JCAMDI", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Sat Feb 25 12:36:08 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath1990.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0377042798000314", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @Article{Morozov:1998:NWR, author = "D. Kh. Morozov and V. V. Voitsekhovich", title = "A new wide-range approximation of modified {Bessel} functions in terms of elementary functions", journal = "Rev. Mexicana F\'\i s.", volume = "44", number = "3", pages = "231--234", year = "1998", CODEN = "RMXFAT", ISSN = "0035-001X", MRclass = "65D20", MRnumber = "MR1629601", bibdate = "Wed Apr 13 06:46:35 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Revista Mexicana de F\'\i sica", } @Article{Nguyen:1998:MLS, author = "Phong Nguyen", title = "A {Montgomery}-Like Square Root for the Number Field Sieve", journal = j-LECT-NOTES-COMP-SCI, volume = "1423", pages = "151--??", year = "1998", CODEN = "LNCSD9", ISSN = "0302-9743 (print), 1611-3349 (electronic)", ISSN-L = "0302-9743", bibdate = "Tue Feb 5 11:52:18 MST 2002", bibsource = "http://link.springer-ny.com/link/service/series/0558/tocs/t1423.htm; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "http://link.springer-ny.com/link/service/series/0558/bibs/1423/14230151.htm; http://link.springer-ny.com/link/service/series/0558/papers/1423/14230151.pdf", acknowledgement = ack-nhfb, fjournal = "Lecture Notes in Computer Science", journal-URL = "http://link.springer.com/bookseries/558", } @Article{Palumbo:1998:GSI, author = "Biagio Palumbo", title = "A generalization of some inequalities for the gamma function", journal = j-J-COMPUT-APPL-MATH, volume = "88", number = "2", pages = "255--268", day = "2", month = mar, year = "1998", CODEN = "JCAMDI", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Sat Feb 25 12:36:06 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath1990.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0377042797001878", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @Article{Qiu:1998:SIG, author = "S.-L. Qiu and M. K. Vamanamurthy and M. Vuorinen", title = "Some Inequalities for the Growth of Elliptic Integrals", journal = j-SIAM-J-MATH-ANA, volume = "29", number = "5", pages = "1224--1237", month = sep, year = "1998", CODEN = "SJMAAH", DOI = "https://doi.org/10.1137/S0036141096310491", ISSN = "0036-1410 (print), 1095-7154 (electronic)", ISSN-L = "0036-1410", bibdate = "Sat Dec 5 14:39:16 MST 1998", bibsource = "http://epubs.siam.org/sam-bin/dbq/toc/SIMA/29/5; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://epubs.siam.org/sam-bin/dbq/article/31049", acknowledgement = ack-nhfb, fjournal = "SIAM Journal on Mathematical Analysis", journal-URL = "http://epubs.siam.org/sima", } @Article{Rivolo:1998:CDR, author = "M. T. Rivolo and A. Simi", title = "Il Calcolo delle Radici Quadrate e Cubiche in {Italia} da {Fibonacci} a {Bombelli}. ({Italian}) [{The} calculation of square and cube roots in {Italy} from {Fibonacci} to {Bombelli}]", journal = j-ARCH-HIST-EXACT-SCI, volume = "52", number = "2", pages = "161--193", month = feb, year = "1998", CODEN = "AHESAN", DOI = "https://doi.org/10.1007/s004070050015", ISSN = "0003-9519 (print), 1432-0657 (electronic)", ISSN-L = "0003-9519", MRclass = "01A35 (01A40)", MRnumber = "1610136 (99d:01015)", MRreviewer = "Massimo Galuzzi", bibdate = "Fri Feb 4 21:50:33 MST 2011", bibsource = "http://springerlink.metapress.com/openurl.asp?genre=issue&issn=0003-9519&volume=52&issue=2; https://www.math.utah.edu/pub/tex/bib/archhistexactsci.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.springerlink.com/openurl.asp?genre=article&issn=0003-9519&volume=52&issue=2&spage=161", acknowledgement = ack-nhfb, fjournal = "Archive for History of Exact Sciences", journal-URL = "http://link.springer.com/journal/407", language = "Italian", MRtitle = "The computation of square and cube roots in {Italy} from {Fibonacci} to {Bombelli}", } @Article{Russinoff:1998:MCP, author = "David M. Russinoff", title = "A Mechanically Checked Proof of {IEEE} Compliance of the Floating Point Multiplication, Division and Square Root Algorithms of the {AMD-K7} Processor", journal = j-LMS-J-COMPUT-MATH, volume = "1", pages = "148--200", year = "1998", CODEN = "????", DOI = "https://doi.org/10.1112/S1461157000000176", ISSN = "1461-1570", ISSN-L = "1461-1570", MRclass = "68M07 (65Y99 68T15)", MRnumber = "99m:68015", MRreviewer = "J. Michel Muller", bibdate = "Fri Nov 29 08:13:48 2002", bibsource = "http://journals.cambridge.org/action/displayJournal?jid=JCM; https://www.math.utah.edu/pub/bibnet/subjects/acc-stab-num-alg-2ed.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/lms-j-comput-math.bib", note = "Appendices A and B available to subscribers electronically (http://www.lms.ac.uk/jcm/1/lms98001/appendix-a/ and http://www.lms.ac.uk/jcm/1/lms98001/appendix-b/)", URL = "http://www.lms.ac.uk/jcm/1/lms1998-001/", acknowledgement = ack-nhfb, ajournal = "LMS J. Comput. Math.", fjournal = "LMS Journal of Computation and Mathematics", journal-URL = "http://journals.cambridge.org/action/displayJournal?jid=JCM", onlinedate = "01 February 2010", } @Article{Segura:1998:PCF, author = "J. Segura and A. Gil", title = "Parabolic cylinder functions of integer and half-integer orders for nonnegative arguments", journal = j-COMP-PHYS-COMM, volume = "115", number = "1", pages = "69--86", day = "1", month = dec, year = "1998", CODEN = "CPHCBZ", DOI = "https://doi.org/10.1016/S0010-4655(98)00097-6", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", bibdate = "Mon Feb 13 21:30:32 MST 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/compphyscomm1990.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0010465598000976", acknowledgement = ack-nhfb, fjournal = "Computer Physics Communications", journal-URL = "http://www.sciencedirect.com/science/journal/00104655", } @Article{Sidi:1998:UBC, author = "Avram Sidi and Yair Shapira", title = "Upper bounds for convergence rates of acceleration methods with initial iterations", journal = j-NUMER-ALGORITHMS, volume = "18", number = "2", pages = "113--132", month = sep, year = "1998", CODEN = "NUALEG", ISSN = "1017-1398 (print), 1572-9265 (electronic)", ISSN-L = "1017-1398", bibdate = "Mon Sep 29 08:36:55 MDT 2003", bibsource = "http://www.kluweronline.com/issn/1017-1398; http://www.math.psu.edu/dna/contents/na.html; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://ipsapp007.kluweronline.com/content/getfile/5058/14/1/abstract.htm; http://ipsapp007.kluweronline.com/content/getfile/5058/14/1/fulltext.pdf", acknowledgement = ack-nhfb, fjournal = "Numerical Algorithms", journal-URL = "http://link.springer.com/journal/11075", keywords = "convergence acceleration", } @Article{Wei:1998:NFS, author = "Liqiang Wei", title = "New formula for $9$--$j$ symbols and their direct calculation", journal = j-COMPUT-PHYS, volume = "12", number = "6", pages = "632--??", month = nov, year = "1998", CODEN = "CPHYE2", DOI = "https://doi.org/10.1063/1.168745", ISSN = "0894-1866 (print), 1558-4208 (electronic)", ISSN-L = "0894-1866", bibdate = "Wed Apr 10 08:46:17 MDT 2019", bibsource = "https://www.math.utah.edu/pub/tex/bib/computphys.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "https://aip.scitation.org/doi/10.1063/1.168745", acknowledgement = ack-nhfb, ajournal = "Comput. Phys", fjournal = "Computers in Physics", journal-URL = "https://aip.scitation.org/journal/cip", } @InProceedings{Agarwal:1999:SAM, author = "R. C. Agarwal and F. G. Gustavson and M. S. Schmookler", title = "Series approximation methods for divide and square root in the {Power3{\TM}} processor", crossref = "Koren:1999:ISC", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "116--123", year = "1999", CODEN = "????", ISSN = "????", bibdate = "Sat Jul 16 15:28:14 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://euler.ecs.umass.edu/paper/final/paper-144.pdf; http://euler.ecs.umass.edu/paper/final/paper-144.ps", acknowledgement = ack-nhfb, keywords = "ARITH; computer arithmetic; IEEE", summary = "The Power3 processor is a 64-bit implementation of the PowerPC TM architecture and is the successor to the Power2 TM processor for workstations and servers which REQUIRE high performance floating point capability. The previous \ldots{}", } @Article{Alzer:1999:SPP, author = "Horst Alzer and O. G. Ruehr", title = "A submultiplicative property of the psi function", journal = j-J-COMPUT-APPL-MATH, volume = "101", number = "1--2", pages = "53--60", day = "15", month = jan, year = "1999", CODEN = "JCAMDI", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Sat Feb 25 12:39:42 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath1990.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0377042798001903", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @Book{Andrews:1999:SF, author = "George E. Andrews and Richard Askey and Ranjan Roy", title = "Special Functions", volume = "71", publisher = pub-CAMBRIDGE, address = pub-CAMBRIDGE:adr, pages = "xvi + 664", year = "1999", DOI = "https://doi.org/10.1017/CBO9781107325937", ISBN = "0-521-62321-9 (hardcover), 0-521-78988-5 (paperback), 1-107-32593-5 (e-book)", ISBN-13 = "978-0-521-62321-6 (hardcover), 978-0-521-78988-2 (paperback), 978-1-107-32593-7 (e-book)", LCCN = "QA351 .A74 1999", bibdate = "Mon Sep 17 18:52:30 2001", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; z3950.loc.gov:7090/Voyager", price = "US\$90.00 (hardcover), US\$34.95 (paperback)", series = "Encyclopedia of mathematics and its applications", acknowledgement = ack-nhfb, subject = "Functions, Special", tableofcontents = "Frontmatter / i--vi \\ Contents / vii--xii \\ Preface / xiii--xvi \\ 1: The Gamma and Beta Functions / 1--60 \\ 2: The Hypergeometric Functions / 61--123 \\ 3: Hypergeometric Transformations and Identities / 124--186 \\ 4: Bessel Functions and Confluent Hypergeometric Functions / 187--239 \\ 5: Orthogonal Polynomials / 240--276 \\ 6: Special Orthogonal Polynomials / 277--354 \\ 7: Topics in Orthogonal Polynomials / 355--400 \\ 8: The Selberg Integral and Its Applications / 401--444 \\ 9: Spherical Harmonics / 445--480 \\ 10: Introduction to $q$-Series / 481--552 \\ 11: Partitions / 553--576 \\ 12: Bailey Chains / 577--594 \\ A: Infinite Products / 595--598 \\ B: Summability and Fractional Integration / 599--610 \\ C: Asymptotic Expansions / 611--616 \\ D: Euler--Maclaurin Summation Formula / 617--628 \\ E: Lagrange Inversion Formula / 629--636 \\ F: Series Solutions of Differential Equations / 637--640 \\ Bibliography / 641--654 \\ Index / 655--658 \\ Subject Index / 659--662 \\ Symbol Index / 663--664", xxURL = "http://www.loc.gov/catdir/toc/cam024/98025757.html; http://www.loc.gov/catdir/description/cam029/98025757.html", } @Article{Bach:1999:NTS, author = "E. Bach and K. Huber", title = "Note on taking square-roots modulo {$N$}", journal = j-IEEE-TRANS-INF-THEORY, volume = "45", number = "2", pages = "807--809", month = mar, year = "1999", CODEN = "IETTAW", DOI = "https://doi.org/10.1109/18.749034", ISSN = "0018-9448 (print), 1557-9654 (electronic)", ISSN-L = "0018-9448", bibdate = "Sat Jul 16 15:28:13 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "IEEE Transactions on Information Theory", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=18", summary = "In this article it is shown how Gauss' (1981) famous cyclotomic sum formula can be used for extracting square-roots modulo \ldots{}", } @InProceedings{Batten:1999:IBO, author = "D. Batten and S. Jinturkar and J. Glossner and M. Schulte and R. Peri and P. D'arcy", editor = "????", booktitle = "Proceedings of the International Conference on Signal Processing Applications and Technologies, Orlando, Florida, November, 1999", title = "Interactions Between Optimizations and a New Type of {DSP} Intrinsic Function", publisher = "????", address = "????", year = "1999", ISBN = "????", ISBN-13 = "????", LCCN = "????", bibdate = "Sun Mar 04 11:05:23 2007", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "Shortened version in \cite{Batten:1999:IFB}.", URL = "http://mesa.ece.wisc.edu/publications/cp_1999-09.pdf", acknowledgement = ack-nhfb, } @Article{Batten:1999:IFB, author = "D. Batten and P. D'arcy", title = "Intrinsic Functions Boost Compilers", journal = "Electrical Engineering Times", volume = "1085", pages = "104--104", month = nov, year = "1999", bibdate = "Sun Mar 04 11:06:22 2007", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, } @TechReport{Beebe:1999:FAE, author = "Nelson H. F. Beebe", title = "Fast Approximate Exponential Functions", type = "Report", institution = inst-CSC, address = inst-CSC:adr, day = "7", month = dec, year = "1999", bibdate = "Sat Feb 02 15:08:59 2019", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/b/beebe-nelson-h-f.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", abstract = "This package contains software conforming to 1989 ANSI/ISO Standard C (ANSI X3.159-1989, ISO/IEC 9899-1990) and 1998 ISO Standard C++ (ISO/IEC 14882:1998) for testing an interesting algorithm for fast approximate exp() functions, published in \cite{Schraudolph:1999:FCA}. There is a font error in figure 2 of that paper: all carets should be replaced by underscore.", acknowledgement = ack-nhfb, remark = "From the report: ``Schraudolph's formula for the approximate exponential function computes $ a \times x + b - c $ in floating-point arithmetic, then converts it to a 32-bit integer which is stored in the appropriate integer word overlaying the floating-point representation. The entire cost is thus a floating-point multiply and add (one instruction on some RISC architectures), a conversion to an integer, and a storage to memory.''", } @InCollection{Brezinski:1999:EEC, author = "C. Brezinski", booktitle = "Error control and adaptivity in scientific computing ({Antalya}, 1998)", title = "Error estimates and convergence acceleration", volume = "536", publisher = pub-KLUWER, address = pub-KLUWER:adr, pages = "87--94", year = "1999", MRclass = "65B05 (65D15)", MRnumber = "1735125", bibdate = "Thu Dec 1 17:36:45 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", series = "NATO Sci. Ser. C Math. Phys. Sci.", acknowledgement = ack-nhfb, keywords = "convergence acceleration", } @InProceedings{Bui:1999:DSI, author = "H. Bui and S. Tahar", booktitle = "1999 {IEEE} Canadian Conference on Electrical and Computer Engineering, 9--12 May 1999", title = "Design and synthesis of an {IEEE-754} exponential function", volume = "1", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "450--455", year = "1999", CODEN = "????", ISSN = "????", bibdate = "Sat Jul 16 17:14:11 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, summary = "We have designed a floating-point exponential function using the table-driven method. The algorithm was first implemented using sequential VHDL and later translated to Concurrent Verilog. The main part of the work consisted of creating modules that \ldots{}", } @Article{Cappuccino:1999:HSS, author = "G. Cappuccino and G. Cocorullo and P. Corsonello and S. Perri", title = "High speed self-timed pipelined datapath for square rooting", journal = "IEE Proceedings on Circuits, Devices and Systems [see also IEE Proceedings G --- Circuits, Devices and Systems]", volume = "146", number = "1", pages = "16--22", month = feb, year = "1999", CODEN = "????", ISSN = "????", bibdate = "Sat Jul 16 15:28:13 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, summary = "The authors describe a new high-performance self-timed circuit for asynchronous square rooting. The new architecture is based on a modified nonrestoring algorithm. An asynchronous pipelined cellular array without auxiliary system for the \ldots{}", } @Article{Carlson:1999:TSI, author = "B. C. Carlson", title = "Toward Symbolic Integration of Elliptic Integrals", journal = j-J-SYMBOLIC-COMP, volume = "28", number = "6", pages = "739--753", month = dec, year = "1999", CODEN = "JSYCEH", DOI = "https://doi.org/10.1006/jsco.1999.0336", ISSN = "0747-7171 (print), 1095-855X (electronic)", ISSN-L = "0747-7171", bibdate = "Tue Mar 7 11:48:04 MST 2000", bibsource = "http://www.idealibrary.com/cgi-bin/links/toc/sy; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.idealibrary.com/links/doi/10.1006/jsco.1999.0336/production; http://www.idealibrary.com/links/doi/10.1006/jsco.1999.0336/production/pdf; http://www.idealibrary.com/links/doi/10.1006/jsco.1999.0336/production/ref", acknowledgement = ack-nhfb, fjournal = "Journal of Symbolic Computation", journal-URL = "http://www.sciencedirect.com/science/journal/07477171", } @InProceedings{Cornea-Hasegan:1999:CPO, author = "M. A. Cornea-Hasegan and R. A. Golliver and P. Markstein", title = "Correctness proofs outline for {Newton--Raphson} based floating-point divide and square root algorithms", crossref = "Koren:1999:ISC", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "96--105", year = "1999", CODEN = "????", ISSN = "????", bibdate = "Sat Jul 16 15:28:14 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://euler.ecs.umass.edu/paper/final/paper-121.pdf; http://euler.ecs.umass.edu/paper/final/paper-121.ps", acknowledgement = ack-nhfb, keywords = "ARITH; computer arithmetic; IEEE", summary = "This paper describes a study of a class of algorithms for the floating-point divide and square root operations, based on the Newton--Raphson iterative method. The two main goals were. (1) Proving the IEEE correctness of these iterative floating-point \ldots{}", } @Article{Corsonello:1999:HPS, author = "P. Corsonello and S. Perri", title = "High performance square rooting circuit using hybrid radix-$2$ adders", journal = j-ELECT-LETTERS, volume = "35", number = "3", pages = "185--186", day = "4", month = feb, year = "1999", CODEN = "ELLEAK", DOI = "https://doi.org/10.1049/el:19990178", ISSN = "0013-5194 (print), 1350-911X (electronic)", ISSN-L = "0013-5194", bibdate = "Sat Jul 16 15:28:13 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Electronics Letters", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=2220; https://digital-library.theiet.org/journal/el", summary = "A new high performance bit parallel architecture for computing square roots is proposed. The architecture implements a non-restoring algorithm and is structured as a pipelined cellular array. To improve the performance, hybrid radix-$2$ adders are \ldots{}", } @TechReport{DiDonato:1999:TFC, author = "Armido R. DiDonato and Russ Gnoffo", title = "Testing a {Fortran 90} Compiler Using the {NSWC Fortran 77 Mathematics Library}", type = "Technical Report", number = "NSWCDD/TR-98/75", institution = "Naval Surface Warfare Center", address = "Dahlgren, VA 22448-5100, USA", pages = "v + 64", month = feb, year = "1999", bibdate = "Tue Jun 13 11:49:57 2023", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fortran3.bib; https://www.math.utah.edu/pub/tex/bib/maple-extract.bib", URL = "https://apps.dtic.mil/sti/pdfs/ADA360604.pdf", abstract = "This report describes the analysis and associated Fortran program (TEST90) that were developed to aid in establishing the validity of a new Fortran 90 mainframe compiler. The FORTRAN 77 Naval Surface Warfare Center (NSWC) Mathematics library (MLJB) is used as a source of routines for checking the Fortran 90 compiler. At the same time, this study can be considered as an aid to determine whether MLIB can operate in a Fortran 90 environment. The inputs for the routines were chosen so that many of the different possible paths of the routines were executed. Seventy-four directly callable routines, with 293 supporting routines, were chosen for testing. All but 17, and their supporting routines, were taken from MLIB. The ones not belonging to MLIB, are double-precision versions of routines in MLIB. Thirteen hundred and twenty five numerical cases were submitted for testing. A true value for each test was obtained independently and given correctly to 35 digits by using MAPLE software. If the difference in the test output and the corresponding true value exceeds a prespecified error tolerance, an error message is printed identifying the routine and the input Additional test cases were also prepared to check the bit and string instructions, since these do not appear in MLIB.\par TEST90 has been used to test the latest Fortran 90 compilers of the CRAY EL98 and IBM PC machines. No errors were found; however, TEST90 did reveal a complex arithmetic error in an earlier version of the Cray EL98 compiler. MLIB routines ran under TEST90 without any problems on both machines.\par The transportability of MLIB allows TEST90 to be used as an aid in testing Fortran 90 compilers on a variety of computers, with a single-precision word length no larger than 64 bits.", acknowledgement = ack-nhfb, } @Article{Elbert:1999:SFZ, author = "{\'A}rp{\'a}d Elbert and Panayiotis D. Siafarikas", title = "On the Square of the First Zero of the {Bessel} Function {$ J_\nu (z) $}", journal = j-CAN-MATH-BULL, volume = "42", number = "1", pages = "56--77", month = mar, year = "1999", CODEN = "CMBUA3", DOI = "https://doi.org/10.4153/CMB-1999-007-4", ISSN = "0008-4395 (print), 1496-4287 (electronic)", ISSN-L = "0008-4395", MRclass = "33A40", bibdate = "Thu Sep 8 10:22:25 MDT 2011", bibsource = "http://cms.math.ca/cmb/v42/; https://www.math.utah.edu/pub/tex/bib/canmathbull.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", abstract = "Let $ j_{\nu, 1} $ be the smallest (first) positive zero of the Bessel function $ J_{\nu }(z) $, $ \nu > - 1 $, which becomes zero when $ \nu $ approaches $ - 1 $. Then $ j_{\nu, 1}^2 $ can be continued analytically to $ - 2 < \nu < - 1 $, where it takes on negative values. We show that $ j_{\nu, 1}^2 $ is a convex function of $ \nu $ in the interval $ - 2 < \nu \leq 0 $, as an addition to an old result [{\'A}. Elbert and A. Laforgia, SIAM J. Math. Anal. {\bf 15}(1984), 206--212], stating this convexity for $ \nu > 0 $. Also the monotonicity properties of the functions $ \frac {j_{\nu, 1}^24 (\nu + 1)} $, $ \frac {j_{\nu, 1}^24(\nu + 1) \sqrt {\nu + 2}} $ are determined. Our approach is based on the series expansion of Bessel function $ J_{\nu }(z) $ and it turned out to be effective, especially when $ - 2 < \nu < - 1 $.", acknowledgement = ack-nhfb, ams-subject-primary = "33A40", fjournal = "Canadian mathematical bulletin = Bulletin canadien de math{\'e}matiques", journal-URL = "http://cms.math.ca/cmb/", journalabbrev = "CMB", refnum = "7139", xxpages = "56--67", } @TechReport{Ercegovac:1999:IGD, author = "Milo{\v{s}} D. Ercegovac and Laurent Imbert and David W. Matula and Jean-Michel Muller and Guoheng Wei", title = "Improving {Goldschmidt} Division, Square Root, and Square Root Reciprocal", type = "Research Report", number = "99-41", institution = "Laboratoire de l'Informatique du Parall{\'e}lisme", address = "Lyon, France", pages = "ii + 17", month = sep, year = "1999", bibdate = "Mon Dec 11 07:53:15 2023", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "https://inria.hal.science/inria-00072909/file/RR1999-41.pdf", abstract = "The aim of this paper is to accelerate division, square root and square root reciprocal computations, when Goldschmidt method is used on a pipelined multiplier. This is done by replacing the last iteration by the addition of a correcting term that can be looked up during the early iterations. We describe several variants of the Goldschmidt algorithm assuming 4-cycle pipelined multiplier and discuss obtained number of cycles and error achieved. Extensions to other than 4-cycle multipliers are given", acknowledgement = ack-nhfb, keywords = "Computer Arithmetic; Convergence division; Division; Goldschmidt iteration; Square root; Square root reciprocal", } @Article{Fabijonas:1999:RAE, author = "Bruce R. Fabijonas and F. W. J. Olver", title = "On the Reversion of an Asymptotic Expansion and the Zeros of the {Airy} Functions", journal = j-SIAM-REVIEW, volume = "41", number = "4", pages = "762--773", month = dec, year = "1999", CODEN = "SIREAD", DOI = "https://doi.org/10.1137/S0036144598349538", ISSN = "0036-1445 (print), 1095-7200 (electronic)", ISSN-L = "0036-1445", bibdate = "Fri Jun 21 11:25:02 MDT 2013", bibsource = "http://epubs.siam.org/toc/siread/41/4; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/siamreview.bib", URL = "http://epubs.siam.org/sam-bin/dbq/article/34953", acknowledgement = ack-nhfb, fjournal = "SIAM Review", journal-URL = "http://epubs.siam.org/sirev", onlinedate = "Dec-1999", } @Article{Fuller:1999:HVH, author = "A. Thomas Fuller", title = "{Horner} versus {Holdred}: an Episode in the History of Root Computation", journal = j-HIST-MATH, volume = "26", number = "1", pages = "29--51", day = "1", month = feb, year = "1999", CODEN = "HIMADS", ISSN = "0315-0860 (print), 1090-249X (electronic)", ISSN-L = "0315-0860", bibdate = "Wed Jun 26 06:19:37 MDT 2013", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/histmath.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0315086098922145", acknowledgement = ack-nhfb, fjournal = "Historia Mathematica", journal-URL = "http://www.sciencedirect.com/science/journal/03150860", } @Article{Gautschi:1999:NRC, author = "Walter Gautschi", title = "A Note on the Recursive Calculation of Incomplete Gamma Functions", journal = j-TOMS, volume = "25", number = "1", pages = "101--107", month = mar, year = "1999", CODEN = "ACMSCU", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Thu Jul 15 19:01:02 1999", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://doi.acm.org/10.1145/305658.305717; http://www.acm.org/pubs/citations/journals/toms/cgi-bin/TOMSbibget?Gautschi:1999:NRC; http://www.acm.org:80/pubs/citations/journals/toms/1999-25-1/p101-gautschi/", abstract = "It is known that the recurrence relation for incomplete gamma functions $ \Gamma (a + n, x), 0 \le a < 1 $, $ n = 0, 1, 2 \ldots $, when $x$ is positive, is unstable---more so the larger $x$. Nevertheless, the recursion can be used in the range $ 0 \le n \le x $ practically without error growth, and in larger ranges $ 0 \le n \le N $ with a loss of accuracy that can be controlled by suitably limiting $N$.", acknowledgement = ack-nhfb, fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", keywords = "algorithms; reliability", subject = "{\bf G.1.0} Mathematics of Computing, NUMERICAL ANALYSIS, General, Stability (and instability). {\bf G.1.2} Mathematics of Computing, NUMERICAL ANALYSIS, Approximation.", } @TechReport{Gourdon:1999:NCC, author = "Xavier Gourdon and Pascal Sebah", title = "Numbers, constants, and computation", institution = "????", address = "Paris, France", year = "1999", bibdate = "Sat Mar 15 16:28:07 2003", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "World-Wide Web site.", URL = "http://numbers.computation.free.fr/Constants/home.html", acknowledgement = ack-nhfb, annote = "Although this site concentrates mainly on computation of particular mathematical constants, it also treats high-precision computation of inverse and square root.", } @Article{Harrison:1999:CTF, author = "John Harrison and Ted Kubaska and Shane Story and Ping Tak Peter Tang", title = "The Computation of Transcendental Functions on the {IA-64} Architecture", journal = j-INTEL-TECH-J, number = "Q4", pages = "7", day = "22", month = nov, year = "1999", bibdate = "Fri Jun 01 06:02:08 2001", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://developer.intel.com/technology/itj/q41999/articles/art_5.htm; http://developer.intel.com/technology/itj/q41999/pdf/transendental.pdf", acknowledgement = ack-nhfb, } @Article{Hayashi:1999:SRR, author = "Takao Hayashi", title = "A set of rules for the root-extraction prescribed by the sixteenth-century {Indian} mathematicians, {N{\=\i}laka{\d{n}}{\d{t}}ha Somastuvan} and {{\'S}a{\.n}kara V{\=a}riyar}", journal = j-HIST-SCI-2, volume = "9", number = "2", pages = "135--153", month = nov, year = "1999", CODEN = "HISCDU", ISSN = "0285-4821", ISSN-L = "0285-4821", MRclass = "01A32", MRnumber = "1762168", MRreviewer = "A. I. Volodarski{\u\i}", bibdate = "Sat Oct 6 17:22:25 2018", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/histscijpn.bib", acknowledgement = ack-nhfb, fjournal = "Historia Scientiarum. Second Series. International Journal of the History of Science Society of Japan", journal-URL = "http://hssj.info/", } @Article{Homeier:1999:CAL, author = "H. H. H. Homeier", title = "Convergence acceleration of logarithmically convergent series avoiding summation", journal = j-APPL-MATH-LETT, volume = "12", number = "3", pages = "29--32", year = "1999", CODEN = "AMLEEL", DOI = "https://doi.org/10.1016/S0893-9659(98)00167-0", ISSN = "0893-9659 (print), 1873-5452 (electronic)", ISSN-L = "0893-9659", MRclass = "65B05", MRnumber = "1749733", bibdate = "Thu Dec 1 17:36:45 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Applied Mathematics Letters", journal-URL = "http://www.sciencedirect.com/science/journal/08939659", keywords = "convergence acceleration", } @InProceedings{Hyogo:1999:LVF, author = "A. Hyogo and Y. Fukutomi and K. Sekine", booktitle = "Proceedings of the 1999 {IEEE} International Symposium on Circuits and Systems: {ISCAS '99}, 2 June 1999", title = "Low voltage four-quadrant analog multiplier using square-root circuit based on {CMOS} pair", volume = "2", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "274--277", year = "1999", CODEN = "????", ISSN = "????", bibdate = "Sat Jul 16 15:28:14 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, summary = "We proposed a square-root circuit based on CMOS pairs. In this paper, we propose a low voltage four-quadrant analog multiplier using the square-root circuit. Also we confirmed this operation by PSpice \ldots{}", } @Article{Iordache:1999:ARS, author = "Cristina Iordache and David W. Matula", title = "Analysis of Reciprocal and Square Root Reciprocal Instructions in the {AMD K6-2} Implementation of {3DNow!}", journal = j-ELECT-NOTES-THEOR-COMP-SCI, volume = "24", pages = "34--62", year = "1999", CODEN = "????", DOI = "https://doi.org/10.1016/S1571-0661(05)80621-8", ISSN = "1571-0661", bibdate = "Fri Jun 24 20:23:13 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", abstract = "Reciprocal and root reciprocal functions at ``half'' and IEEE single precision formats are specified in the AMD 3DNow! instruction set. Implementations in the recently released AMD K6-2 microprocessor are analyzed herein by exhaustive computation and timing loops to ascertain the accuracy and monotonicity properties of the output and throughput\slash latency cycle counts. Periodicities in stepwise function output were observed and employed to construct an underlying bipartite table that can serve as the core of the respective reciprocal function outputs. The recommended RISC instruction macros generated single precision reciprocals and root reciprocals accurate to a unit in the last place. However, the root reciprocal functions failed to satisfy the desirable monotonicity property typically implemented as an industry standard for elementary functions on x86 floating point units. Reasons for the failure are provided and an adjusted table is shown to satisfy the monotonicity standard. Results are summarized in Table 1 and described in the body of this report.", acknowledgement = ack-nhfb, fjournal = "Electronic Notes in Theoretical Computer Science", journal-URL = "http://www.sciencedirect.com/science/journal/15710661", } @InProceedings{Iordache:1999:IPR, author = "Cristina Iordache and David W. Matula", title = "On Infinitely Precise Rounding for Division, Square Root, Reciprocal and Square Root Reciprocal", crossref = "Koren:1999:ISC", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "233--240", year = "1999", CODEN = "????", ISSN = "????", bibdate = "Sat Jul 16 15:28:13 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "http://euler.ecs.umass.edu/paper/final/paper-164.pdf; http://euler.ecs.umass.edu/paper/final/paper-164.ps; http://www.acsel-lab.com/arithmetic/arith14/papers/ARITH14_Iordache.pdf", abstract = "Quotients, reciprocals, square roots and square root reciprocals all have the property that infinitely precise p-bit rounded results for p-bit input operands can be obtained from approximate results of bounded accuracy. We investigate lower bounds on the number of bits of an approximation accurate to a unit in the last place sufficient to guarantee that correct round and sticky bits can be determined. Known lower bounds for quotients and square roots are given and/or sharpened, and a new lower bound for root reciprocals is proved. Specifically for reciprocals, quotients and square roots, tight bounds of order $ 2 p + O(1) $ are presented. For infinitely precise rounding of the root reciprocal a lower bound can be found at $ 3 p + O(1) $, but exhaustive testing for small sizes of the operand suggests that in practice $ (2 + \epsilon)p $ for small $ \epsilon $ is usually sufficient. Algorithms can be designed for obtaining the round and sticky bits based on the bit pattern of an approximation computed to the required accuracy. We show that some improvement of the known lower bound for reciprocals and division is achievable at the cost of somewhat more complex hardware for rounding. Tests for the exactness of the quotient and square root are also provided.", acknowledgement = ack-nhfb, keywords = "ARITH-14; computer arithmetic; IEEE", summary = "Quotients, reciprocals, square roots and square root reciprocals all have the property that infinitely precise p-bit rounded results for p-bit input operands can be obtained from approximate results of bounded accuracy. We investigate lower bounds \ldots{}", } @Article{Jamieson:1999:NRF, author = "M. J. Jamieson", title = "Notes: On rational function approximations to square roots", journal = j-AMER-MATH-MONTHLY, volume = "106", number = "1", pages = "50--52", month = jan, year = "1999", CODEN = "AMMYAE", ISSN = "0002-9890 (print), 1930-0972 (electronic)", ISSN-L = "0002-9890", MRclass = "11Yxx", MRnumber = "1 674 202", bibdate = "Tue Jun 22 10:29:34 MDT 1999", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, fjournal = "American Mathematical Monthly", journal-URL = "https://www.jstor.org/journals/00029890.htm", } @Book{Jeffreys:1999:MMP, author = "Harold Jeffreys and Bertha {Swirles Jeffreys}", title = "Methods of Mathematical Physics", publisher = pub-CAMBRIDGE, address = pub-CAMBRIDGE:adr, edition = "Third", pages = "viii + 718", year = "1999", ISBN = "0-521-66402-0 (paperback)", ISBN-13 = "978-0-521-66402-8 (paperback)", LCCN = "QA401 .J4 1999", bibdate = "Thu Aug 17 10:48:45 2023", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/h/hartree-douglas-r.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/numana1990.bib", note = "Reprint of \cite{Jeffreys:1956:MMP}.", URL = "https://en.wikipedia.org/wiki/Bertha_Swirles; https://en.wikipedia.org/wiki/Harold_Jeffreys", acknowledgement = ack-nhfb, author-dates = "Sir Harold Jeffreys (22 April 1891--18 March 1989); Lady Bertha Swirles Jeffreys (22 May 1903--18 December 1999)", remark = "First edition 1946, second edition 1950, third edition 1956, first paperback edition 1972, reprinted 1978, 1980, 1988, 1992, 1999, 2001. Third edition preface is dated April 1953. Second edition preface is dated 15 November 1948. First edition preface is dated 1946.", subject-dates = "Douglas Rayner Hartree (27 March 1897--12 February 1958)", tableofcontents = "Preface \\ Authors' Notes \\ 1: The Real Variable \\ 2: Scalars and Vectors \\ 3: Tensors \\ 4: Matrices \\ 5: Multiple Integrals \\ 6: Potential Theory \\ 7: Operational Methods \\ 8: Physical Applications of the Operational Method \\ 9: Numerical Methods \\ 10: Calculus of Variations \\ 11: Functions of a Complex Variable \\ 12: Contour Integration and Bromwich's Integral \\ 13: Conformal Representation \\ 14: Fourier's Theorem \\ 15: The Factorial and Related Functions \\ 16: Solution of Linear Differential Equation \\ 17: Asymptotic Expansions \\ 18: The Equations of Potential, Waves, and Heat Conduction \\ 19: Waves in One Dimension and Waves With Spherical Symmetry \\ 20: Conduction of Heat in One and Three Dimensions \\ 21: Bessel Functions \\ 22: Applications of Bessel Functions \\ 23: The Confluent Hypergeometric Function \\ 24: Legendre Functions and Associated Functions \\ 25: Elliptic Functions \\ Notes \\ Appendix on Notation \\ Index", } @Misc{Kahan:1999:SRD, author = "W. Kahan", title = "Square Root Without Division", howpublished = "World-Wide Web document", pages = "3", day = "23", month = feb, year = "1999", bibdate = "Mon Apr 25 18:01:49 2005", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/k/kahan-william-m.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "http://www.cs.berkeley.edu/~wkahan/ieee754status/reciprt.pdf", acknowledgement = ack-nhfb, } @Article{Krukier:1999:CAT, author = "L. A. Krukier", title = "Convergence acceleration of triangular iterative methods based on the skew-symmetric part of the matrix", journal = j-APPL-NUM-MATH, volume = "30", number = "2--3", pages = "281--290", day = "10", month = jun, year = "1999", CODEN = "ANMAEL", ISSN = "0168-9274 (print), 1873-5460 (electronic)", ISSN-L = "0168-9274", bibdate = "Wed Jul 28 14:37:31 MDT 1999", bibsource = "http://www.elsevier.com/cgi-bin/cas/tree/store/apnum/cas_free/browse/browse.cgi?year=1999&volume=30&issue=2-3; https://www.math.utah.edu/pub/tex/bib/applnummath.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.elsevier.com/cgi-bin/cas/tree/store/apnum/cas_sub/browse/browse.cgi?year=1999&volume=30&issue=2-3&aid=981", acknowledgement = ack-nhfb, fjournal = "Applied Numerical Mathematics: Transactions of IMACS", journal-URL = "http://www.sciencedirect.com/science/journal/01689274/", keywords = "convergence acceleration", } @Article{Kzaz:1999:CAG, author = "M. Kzaz", title = "Convergence acceleration of the {Gauss--Laguerre} quadrature formula", journal = j-APPL-NUM-MATH, volume = "29", number = "2", pages = "201--220", day = "1", month = feb, year = "1999", CODEN = "ANMAEL", ISSN = "0168-9274 (print), 1873-5460 (electronic)", ISSN-L = "0168-9274", bibdate = "Wed Jul 28 14:37:22 MDT 1999", bibsource = "http://www.elsevier.com/cgi-bin/cas/tree/store/apnum/cas_free/browse/browse.cgi?year=1999&volume=29&issue=2; https://www.math.utah.edu/pub/tex/bib/applnummath.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.elsevier.com/cas/tree/store/apnum/sub/1999/29/2/940.pdf", acknowledgement = ack-nhfb, fjournal = "Applied Numerical Mathematics: Transactions of IMACS", journal-URL = "http://www.sciencedirect.com/science/journal/01689274/", keywords = "convergence acceleration", } @Article{Lang:1999:VHR, author = "T. Lang and P. Montuschi", title = "Very high radix square root with prescaling and rounding and a combined division\slash square root unit", journal = j-IEEE-TRANS-COMPUT, volume = "48", number = "8", pages = "827--841", month = aug, year = "1999", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/12.795124", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Sat Jul 16 15:28:13 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", summary = "An algorithm for square root with prescaling and selection by rounding is developed and combined with a similar scheme for division. Since division is usually more frequent than square root, the main concern of the combined implementation is to \ldots{}", } @InProceedings{Lee:1999:STS, author = "Young-Sang Lee and Jun-Woo Kang and Lee-Sup Kim and Seung-Ho Hwang", booktitle = "6th International Conference on {VLSI} and {CAD}: {ICVC '99}", title = "Self-timed shared division and square-root implementation using full redundant signed digit numbers", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "541--544", year = "1999", CODEN = "????", ISSN = "????", bibdate = "Sat Jul 16 15:28:14 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, summary = "A radix-$2$ square root implementation for self-timed dividers using redundant signed-digit (RSD) adders is presented. In this method, two self-timed RSD adder stages are used for each result bit selection. A very efficient and simple result bit \ldots{}", } @InProceedings{Lozier:1999:DDM, author = "Daniel W. Lozier and B. R. Miller and B. V. Saunders", title = "Design of a Digital Mathematical Library for Science, Technology and Education", crossref = "IEEE:1999:PIF", pages = "118--128", year = "1999", bibdate = "Fri Jul 09 06:33:35 2004", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://dlmf.nist.gov/about/publications/nistir6297.ps.gz", acknowledgement = ack-nhfb, remark = "Preprint: NISTIR 6297, Feb. 1999, 13 pages", } @Article{Morita:1999:CEI, author = "T. Morita", title = "Calculation of the elliptic integrals of the first and second kinds with complex modulus", journal = j-NUM-MATH, volume = "82", number = "4", pages = "677--688", month = jun, year = "1999", CODEN = "NUMMA7", ISSN = "0029-599X (print), 0945-3245 (electronic)", ISSN-L = "0029-599X", bibdate = "Mon Oct 18 10:45:11 MDT 1999", bibsource = "http://link.springer-ny.com/link/service/journals/00211/tocs/t9082004.htm; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://link.springer-ny.com/link/service/journals/00211/bibs/9082004/90820677.htm; http://link.springer-ny.com/link/service/journals/00211/papers/9082004/90820677.pdf", acknowledgement = ack-nhfb, fjournal = "Numerische Mathematik", journal-URL = "http://link.springer.com/journal/211", } @Article{Muller:1999:CAT, author = "J. M{\"u}ller", title = "Convergence acceleration of {Taylor} sections by convolution", journal = j-CONST-APPROX, volume = "15", number = "4", pages = "523--536", year = "1999", DOI = "https://doi.org/10.1007/s003659900120", ISSN = "0176-4276 (print), 1432-0940 (electronic)", ISSN-L = "0176-4276", MRclass = "41A58 (30E10)", MRnumber = "1702803 (2000i:41040)", bibdate = "Thu Dec 1 17:36:45 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Constructive Approximation", journal-URL = "http://link.springer.com/journal/365", keywords = "convergence acceleration", } @Article{Muroi:1999:ESR, author = "Kazuo Muroi", title = "Extraction of square roots in {Babylonian} mathematics", journal = j-HIST-SCI-2, volume = "9", number = "2", pages = "127--133", month = nov, year = "1999", CODEN = "HISCDU", ISSN = "0285-4821", ISSN-L = "0285-4821", MRclass = "01A17", MRnumber = "1762167", MRreviewer = "Bruno Poizat", bibdate = "Sat Oct 6 17:22:25 2018", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/histscijpn.bib", acknowledgement = ack-nhfb, fjournal = "Historia Scientiarum. Second Series. International Journal of the History of Science Society of Japan", journal-URL = "http://hssj.info/", } @InProceedings{Nannarelli:1999:LPR, author = "A. Nannarelli and T. Lang", booktitle = "{(ICCD '99)} International Conference on Computer Design", title = "Low-power radix-$4$ combined division and square root", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "236--242", year = "1999", CODEN = "????", ISSN = "????", bibdate = "Sat Jul 16 15:28:14 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, summary = "Because of the similarities in the algorithm it is quite common to implement division and square root in the same unit. The purpose of this work is to implement a low-power combined radix-$4$ division and square root floating-point double precision \ldots{}", } @InProceedings{Oberman:1999:FPD, author = "S. F. Oberman", title = "Floating point division and square root algorithms and implementation in the {AMD-K7{\TM}} microprocessor", crossref = "Koren:1999:ISC", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "106--115", year = "1999", CODEN = "????", ISSN = "????", bibdate = "Sat Jul 16 15:28:14 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://euler.ecs.umass.edu/paper/final/paper-139.pdf; http://euler.ecs.umass.edu/paper/final/paper-139.ps", acknowledgement = ack-nhfb, keywords = "ARITH; computer arithmetic; IEEE", summary = "This paper presents the AMD-K7 IEEE 754 and $\times$87 compliant floating point division and square root algorithms and implementation. The AMD-K7 processor employs an iterative implementation of a series expansion to converge quadratically to the \ldots{}", } @InProceedings{Parhami:1999:ALT, author = "B. Parhami", booktitle = "Conference Record of the Thirty-Third Asilomar Conference on Signals, Systems, and Computers, 1999", title = "Analysis of the lookup table size for square-rooting", volume = "2", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "1327--1330", year = "1999", CODEN = "????", ISSN = "????", bibdate = "Sat Jul 16 15:28:14 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, summary = "Convergence methods are widely used for division, reciprocation, and square-rooting. With such methods, it is common to use an initial table lookup step for obtaining an approximate result that leads to faster convergence. In the case of division \ldots{}", } @Article{Russinoff:1999:MCP, author = "David M. Russinoff", title = "A mechanically checked proof of correctness of the {AMD K5} floating point square root microcode", journal = j-FORM-METHODS-SYST-DES, volume = "14", number = "1", pages = "75--125", month = jan, year = "1999", CODEN = "FMSDE6", ISSN = "0925-9856 (print), 1572-8102 (electronic)", ISSN-L = "0925-9856", bibdate = "Sat Jun 02 07:51:51 2001", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", note = "Special issue on arithmetic circuits.", URL = "http://www.wkap.nl/jrnltoc.htm/0925-9856; http://www.wkap.nl/oasis.htm/194808", acknowledgement = ack-nhfb, fjournal = "Formal Methods in System Design", } @Article{Schraudolph:1999:FCA, author = "N. N. Schraudolph", title = "A Fast, Compact Approximation of the Exponential Function", journal = j-NEURAL-COMP, volume = "11", number = "4", pages = "853--862", day = "1", month = may, year = "1999", CODEN = "NEUCEB", DOI = "https://doi.org/10.1162/089976699300016467", ISSN = "0899-7667 (print), 1530-888x (electronic)", ISSN-L = "0899-7667", bibdate = "Fri Nov 8 05:39:32 MST 2002", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; Ingenta database", URL = "https://www.mitpressjournals.org/doi/abs/10.1162/089976699300016467", acknowledgement = ack-nhfb, fjournal = "Neural Computation", journal-URL = "http://www.mitpressjournals.org/loi/neco", pagecount = "10", } @Article{Schulte:1999:AEF, author = "M. Schulte and J. Stine", title = "Approximating Elementary Functions with Symmetric Bipartite Tables", journal = j-IEEE-TRANS-COMPUT, volume = "48", number = "8", pages = "842--847", year = "1999", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/12.795125", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Fri Jun 24 20:20:58 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "http://mesa.ece.wisc.edu/publications/cp_1999-10.pdf", acknowledgement = ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", } @InProceedings{Schulte:1999:ESO, author = "M. J. Schulte and K. E. Wires", title = "Efficient Second Order Approximations for Reciprocals and Square Roots", crossref = "Luk:1999:PSA", volume = "3807", pages = "10--18", year = "1999", bibdate = "Sun Mar 04 11:10:48 2007", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://mesa.ece.wisc.edu/publications/cp_1999-05.pdf", acknowledgement = ack-nhfb, } @InProceedings{Schulte:1999:HSI, author = "Michael J. Schulte and Kent E. Wires", title = "High-Speed Inverse Square Roots", crossref = "Koren:1999:ISC", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "124--131", year = "1999", CODEN = "????", ISSN = "????", bibdate = "Sat Jul 16 15:28:14 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://euler.ecs.umass.edu/paper/final/paper-109.pdf; http://euler.ecs.umass.edu/paper/final/paper-109.ps", acknowledgement = ack-nhfb, keywords = "ARITH; computer arithmetic; IEEE", summary = "Inverse square roots are used in several digital signal processing, multimedia, and scientific computing applications. This paper presents a high-speed method for computing inverse square roots. This method uses a table lookup, operand modification, \ldots{}", } @Article{Segura:1999:EGT, author = "J. Segura and A. Gil", title = "{ELF} and {GNOME}: Two tiny codes to evaluate the real zeros of the {Bessel} functions of the first kind for real orders", journal = j-COMP-PHYS-COMM, volume = "117", number = "3", pages = "250--262", day = "11", month = mar, year = "1999", CODEN = "CPHCBZ", DOI = "https://doi.org/10.1016/S0010-4655(98)00193-3", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", bibdate = "Mon Feb 13 21:30:36 MST 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/compphyscomm1990.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0010465598001933", acknowledgement = ack-nhfb, fjournal = "Computer Physics Communications", journal-URL = "http://www.sciencedirect.com/science/journal/00104655", } @Article{Seidel:1999:HSR, author = "P.-M. Seidel", title = "High-speed redundant reciprocal approximation", journal = j-INTEGRATION-VLSI-J, volume = "28", number = "1", pages = "1--12", month = sep, year = "1999", CODEN = "IVJODL", DOI = "https://doi.org/10.1016/S0167-9260(99)00008-5", ISSN = "0167-9260 (print), 1872-7522 (electronic)", ISSN-L = "0167-9260", bibdate = "Fri Nov 8 05:39:32 MST 2002", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; Ingenta database", URL = "https://www.sciencedirect.com/science/article/pii/S0167926099000085", acknowledgement = ack-nhfb, fjournal = "Integration, the VLSI journal", journal-URL = "https://www.sciencedirect.com/journal/integration/issues", keywords = "Booth recoding; floating point; multiplicative division; reciprocal approximation; redundant compression", pagecount = "12", } @Article{Stine:1999:STA, author = "E. Stine and M. J. Schulte", title = "The Symmetric Table Addition Method for Accurate Function Approximation", journal = j-J-VLSI-SIGNAL-PROC, volume = "21", number = "2", pages = "167--177", month = jun, year = "1999", CODEN = "JVSPED", ISSN = "0922-5773 (print), 1573-109x (electronic)", ISSN-L = "0922-5773", bibdate = "Sun Mar 04 11:02:59 2007", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://mesa.ece.wisc.edu/publications/cp_1999-11.pdf", acknowledgement = ack-nhfb, fjournal = "Journal of VLSI Signal Processing", } @InProceedings{Story:1999:NAI, author = "S. Story and P. T. P. Tang", title = "New Algorithms for Improved Transcendental Functions on {IA-64}", crossref = "Koren:1999:ISC", pages = "4--11", year = "1999", bibdate = "Mon Feb 7 07:28:26 MST 2000", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "http://euler.ecs.umass.edu/paper/final/paper-118.pdf; http://euler.ecs.umass.edu/paper/final/paper-118.ps", acknowledgement = ack-nhfb, keywords = "ARITH; computer arithmetic; IEEE", } @Book{Suetin:1999:OPT, author = "P. K. (Pavel Kondratevich) Suetin", title = "Orthogonal Polynomials in Two Variables", volume = "3", publisher = "Gordon and Breach Science Publishers", address = "Amsterdam, The Netherlands", pages = "xx + 348", year = "1999", ISBN = "90-5699-167-1", ISBN-13 = "978-90-5699-167-8", LCCN = "QA404.5 .S8813 1999", bibdate = "Sat Oct 30 17:21:54 MDT 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; prodorbis.library.yale.edu:7090/voyager", note = "Translated from the Russian by E. V. Pankratiev.", series = "Analytical methods and special functions", acknowledgement = ack-nhfb, remark = "Originally published in Russian in 1988 by Nauka, Moscow.", subject = "Orthogonal polynomials", tableofcontents = "Preface / ix \\ Preface to the English Edition / xv \\ Notation / xix \\ I. General properties of polynomials orthogonal over a domain / 1 \\ 1. Polynomials in two variables orthogonal over a domain / 1 \\ 2. The existence theorem and criteria of orthogonality / 6 \\ 3. Algebraic properties / 10 \\ 4. Monic orthogonal polynomials / 18 \\ 5. Normal biorthogonal Systems / 24 \\ 6. Fourier series of orthogonal polynomials in two variables / 28 \\ 7. Fourier series for differentiable functions / 31 \\ II. Some typical examples and special cases of orthogonality over a domain / 37 \\ 1. Different products of classical orthogonal polynomials / 37 \\ 2. Various cases of connection between orthogonality over a domain and orthogonality on an interval / 42 \\ 3. Some theorems in the case of a weight function with separating variables / 48 \\ 4. Conditions of interconnection between the weight function and the domain of orthogonality / 52 \\ 5. Examples of computations of weight function moments / 57 \\ III. Classical Appell's orthogonal polynomials / 63 \\ 1. Rodrigues formula for Appell's polynomials / 63 \\ 2. Representation of the Appell polynomials via the hypergeometric function of two variables / 69 \\ 3. Differential equation for the Appell polynomials / 72 \\ 4. Orthogonality of eigenfunctions of the Appell equation / 75 \\ 5. Normal biorthogonal Appell System / 79 \\ 6. Series in the Appell polynomials / 83 \\ IV. Admissible differential equation for polynomials orthogonal over a domain / 87 \\ 1. The main differential Operator and a theorem on orthogonality / 87 \\ 2. Admissibility conditions for the main differential equation / 92 \\ 3. Some examples and properties of admissible differential equations / 97 \\ 4. Affine transformations of the arguments of the main differential equation / 101 \\ 5. Transformation of the coefficients of the characteristic polynomial / 105 \\ 6. Normal forms of the admissible differential equation / 115 \\ 7. Normal forms when reducing the degree of the characteristic polynomial / 123 \\ V. Potentially self-adjoint equation and Rodrigues formula / 131 \\ 1. Potentially self-adjoint Operators / 131 \\ 2. Admissible and potentially self-adjoint equations / 135 \\ 3. Rodrigues formula for polynomials orthogonal over a domain / 146 \\ 4. Weight functions and the Rodrigues formula in the most typical cases / 153 \\ VI. Harmonic polynomials orthogonal over a domain / 163 \\ 1. Homogeneous harmonic polynomials / 163 \\ 2. An analogue of the Christoffel--Darboux formula / 169 \\ 3. Harmonic polynomials orthogonal in the unit disk / 173 \\ 4. Harmonic polynomials orthogonal over a domain in the general case / 176 \\ 5. Harmonic polynomials superorthogonal over a domain / 179 \\ VII. Polynomials in two variables orthogonal on a contour / 187 \\ 1. Main definitions and the simplest properties / 187 \\ 2. Polynomials in two variables orthogonal on an algebraic curve / 191 \\ 3. Harmonic polynomials orthogonal on a contour / 196 \\ 4. Fourier series in harmonic polynomials orthogonal on a contour / 200 \\ 5. Harmonic polynomials superorthogonal on a contour / 206 \\ 6. Examples of superorthogonal Systems of harmonic polynomials / 213 \\ VIII. Generalized orthogonal polynomials in two variables / 223 \\ 1. Main definitions and the simplest properties / 223 \\ 2. The existence theorem in the most general form / 228 \\ 3. Fourier series in generalized orthogonal polynomials in two variables / 233 \\ 4. Monic orthogonal polynomials under minimal conditions / 241 \\ 5. Generalized generating functions for monic orthogonal polynomials / 247 \\ IX. Other results concerning the connection between orthogonal polynomials and differential equations / 253 \\ 1. Canonical admissible differential equation and monic orthogonal polynomials / 253 \\ 2. Necessary consistency conditions of the canonical Operator and the functional / 258 \\ 3. Sufficient conditions of consistency of the canonical Operator and the functional / 262 \\ 4. The deduction of the differential equation from the Pearson equation System / 268 \\ 5. An admissible partial differential equation of an arbitrary order / 276 \\ X. Original results of T. Koornwinder / 285 \\ 1. Examples of the representation of polynomials orthogonal over a domain via the Jacobi polynomials / 285 \\ 2. Orthogonal polynomials in two conjugate complex variables / 291 \\ 3. The Chebyshev polynomials in two conjugate complex variables for the Steiner domain / 296 \\ 4. Another generalization of the Jacobi polynomials onto the case of two variables / 308 \\ XI. Some recent results / 313 \\ 1. A new generalization of the Appell polynomials / 313 \\ 2. Some properties of the Koornwinder--Steiner polynomials / 318 \\ 3. A two-dimensional analogue of the Chebyshev--Laguerre polynomials / 319 \\ Comments and Supplements / 323 \\ References / 329 \\ Author Index / 343 \\ Subject Index / 345", } @Article{Vrahatis:1999:ESP, author = "M. N. Vrahatis and O. Ragos and T. Skiniotis and F. A. Zafiropoulos and T. N. Grapsa", title = "Erratum to: {{\booktitle{RFSFNS: a portable package for the numerical determination of the number and the calculation of roots of Bessel functions}} [Comput. Phys. Commun. {\bf 92} (1995) 252--266]}", journal = j-COMP-PHYS-COMM, volume = "117", number = "3", pages = "290--290", day = "11", month = mar, year = "1999", CODEN = "CPHCBZ", DOI = "https://doi.org/10.1016/S0010-4655(98)00109-X", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", bibdate = "Mon Feb 13 21:30:36 MST 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/compphyscomm1990.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "See \cite{Vrahatis:1995:RPP}.", URL = "http://www.sciencedirect.com/science/article/pii/S001046559800109X", acknowledgement = ack-nhfb, fjournal = "Computer Physics Communications", journal-URL = "http://www.sciencedirect.com/science/journal/00104655", } @Article{Vyridis:1999:ICA, author = "D. G. Vyridis and S. D. Panteliou and I. N. Katz", title = "An inverse convergence approach for arguments of {Jacobian} elliptic functions", journal = j-COMPUT-MATH-APPL, volume = "37", number = "2", pages = "21--26", month = jan, year = "1999", CODEN = "CMAPDK", ISSN = "0898-1221 (print), 1873-7668 (electronic)", ISSN-L = "0898-1221", bibdate = "Wed Mar 1 21:48:56 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/computmathappl1990.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0898122198002508", acknowledgement = ack-nhfb, fjournal = "Computers and Mathematics with Applications", journal-URL = "http://www.sciencedirect.com/science/journal/08981221", } @Article{Wieder:1999:ANH, author = "Thomas Wieder", title = "{Algorithm 794}: Numerical {Hankel} transform by the {Fortran} program {HANKEL}", journal = j-TOMS, volume = "25", number = "2", pages = "240--250", month = jun, year = "1999", CODEN = "ACMSCU", DOI = "https://doi.org/10.1145/317275.317284", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Wed Oct 20 18:21:35 MDT 1999", bibsource = "http://www.acm.org/pubs/toc/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "ftp://netlib.bell-labs.com/netlib/toms/794.gz; http://phase.etl.go.jp/netlib/toms/794; http://www.acm.org/pubs/citations/journals/toms/1999-25-2/p240-wieder/; http://www.acm.org/pubs/citations/journals/toms/cgi-bin/TOMSbibget?Wieder:1999:ANH; http://www.hensa.ac.uk/netlib/toms/794.gz; http://www.netlib.no/netlib/toms/794; http://www.netlib.org/toms/794", abstract = "The numerical evaluation of the Hankel transform poses the problems of both infinite integration and Bessel function calculation. Using the corresponding numerical program routines from the literature, a Fortran program has been written to perform the Hankel transform for real functions, given either in analytical form as subroutines or in discrete form as tabulated data.", accepted = "February 1999", acknowledgement = ack-nhfb, fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", keywords = "Hankel transform; numerical analysis", subject = "Software --- Programming Languages --- Language Classifications (D.3.2): FORTRAN 77; Theory of Computation --- Analysis of Algorithms and Problem Complexity --- Numerical Algorithms and Problems (F.2.1): Computation of transforms", } @TechReport{Zimmermann:1999:KSR, author = "Paul Zimmermann", title = "{Karatsuba} Square Root", type = "Research Report", number = "3805", institution = inst-LORIA-INRIA-LORRAINE, address = inst-LORIA-INRIA-LORRAINE:adr, pages = "8", year = "1999", bibdate = "Sun Sep 10 08:56:48 2006", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "ftp://ftp.inria.fr/INRIA/publication/publi-pdf/RR/RR-3805.pdf; ftp://ftp.inria.fr/INRIA/publication/publi-ps-gz/RR/RR-3805.ps.gz; http://www.inria.fr/rrrt/rr-3805.html", abstract = "We exhibit an algorithm to compute the square-root with remainder of a $n$-word number in $ 3 / 2 $ word operations, where $ K(n) $ is the number of words operations to multiply two $n$-word numbers using Karatsuba's algorithm. If the remainder is not needed, the cost can be reduced to $ K(n) $ on average. This algorithm can be used for floating-point or polynomial computations too; although not optimal asymptotically, its simplicity gives a wide range of use, from about 50 to 1,000,000 digits, as shown by computer experiments.", acknowledgement = ack-nhfb, } @Article{Ziv:1999:SUR, author = "Abraham Ziv", title = "Sharp {ULP} rounding error bound for the hypotenuse function", journal = j-MATH-COMPUT, volume = "68", number = "227", pages = "1143--1148", month = jul, year = "1999", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Fri Jul 16 10:39:05 MDT 1999", bibsource = "http://www.ams.org/mcom/1999-68-227; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp1990.bib", URL = "http://www.ams.org/jourcgi/jour-pbprocess?fn=110&arg1=S0025-5718-99-01103-5&u=/mcom/1999-68-227/", acknowledgement = ack-nhfb, ajournal = "Math. Comput.", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Alhargan:2000:ACA, author = "Fayez A. Alhargan", title = "Algorithms for the Computation of all {Mathieu} Functions of Integer Orders", journal = j-TOMS, volume = "26", number = "3", pages = "390--407", month = sep, year = "2000", CODEN = "ACMSCU", DOI = "https://doi.org/10.1145/358407.358420", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Wed Feb 6 16:43:42 MST 2002", bibsource = "http://www.acm.org/pubs/contents/journals/toms/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib", abstract = "The paper presents methods for the computation of all Mathieu functions of integer order, which cover a large range of $n$ and $h$; previous algorithms were limited to small values of $n$. The algorithms are given in sufficient details to enable straightforward implementation. The algorithms can handle a large range of the order $n$ (0-200) and the parameter $h$ (0-4$n$ ).", accepted = "19 May 2000", acknowledgement = ack-nhfb, fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", } @Article{Alhargan:2000:ASC, author = "Fayez A. Alhargan", title = "{Algorithm 804}: subroutines for the computation of {Mathieu} functions of integer orders", journal = j-TOMS, volume = "26", number = "3", pages = "408--414", month = sep, year = "2000", CODEN = "ACMSCU", DOI = "https://doi.org/10.1145/358407.358422", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Wed Feb 6 16:43:42 MST 2002", bibsource = "http://www.acm.org/pubs/contents/journals/toms/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib", abstract = "Computer subroutines in C++ for computing Mathieu functions of integer orders are described. The core routines for computing Mathieu characteristic numbers and Mathieu coefficients are described in details, the rest of the subroutines are standard implementation of the series summations for each function. The routines can handle a large range of the order $n$ and the parameter $h$.", accepted = "19 May 2000", acknowledgement = ack-nhfb, fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", } @Article{Anderson:2000:RAF, author = "Stuart Anderson", title = "Remark on {Algorithm 723}: {Fresnel} integrals", journal = j-TOMS, volume = "26", number = "4", pages = "617--617", month = dec, year = "2000", CODEN = "ACMSCU", DOI = "https://doi.org/10.1145/365723.365737", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Wed Feb 6 16:43:42 MST 2002", bibsource = "http://www.acm.org/pubs/contents/journals/toms/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib", accepted = "16 October 2000", acknowledgement = ack-nhfb, fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", } @Article{Ball:2000:ACZ, author = "James S. Ball", title = "Automatic Computation of Zeros of {Bessel} Functions and Other Special Functions", journal = j-SIAM-J-SCI-COMP, volume = "21", number = "4", pages = "1458--1464", month = jul, year = "2000", CODEN = "SJOCE3", DOI = "https://doi.org/10.1137/S1064827598339074", ISSN = "1064-8275 (print), 1095-7197 (electronic)", ISSN-L = "1064-8275", bibdate = "Fri Oct 27 13:32:22 MDT 2000", bibsource = "http://epubs.siam.org/toc/sjoce3/21/4; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/siamjscicomput.bib", URL = "http://epubs.siam.org/sam-bin/dbq/article/33907", acknowledgement = ack-nhfb, fjournal = "SIAM Journal on Scientific Computing", journal-URL = "http://epubs.siam.org/sisc", } @Article{Barry:2000:AAR, author = "D. A. Barry and J.-Y Parlange and L. Li and H. Prommer and C. J. Cunningham and F. Stagnitti", title = "Analytical approximations for real values of the {Lambert} {$W$}-function", journal = j-MATH-COMPUT-SIMUL, volume = "53", number = "1--2", pages = "95--103", day = "15", month = aug, year = "2000", CODEN = "MCSIDR", DOI = "https://doi.org/10.1016/S0378-4754(00)00172-5", ISSN = "0378-4754 (print), 1872-7166 (electronic)", ISSN-L = "0378-4754", bibdate = "Sat Aug 16 16:59:35 MDT 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcomputsimul2000.bib", URL = "https://www.sciencedirect.com/science/article/pii/S0378475400001725", acknowledgement = ack-nhfb, ajournal = "Math. Comput. Simul.", fjournal = "Mathematics and Computers in Simulation", journal-URL = "https://www.sciencedirect.com/science/journal/03784754", } @InProceedings{Batten:2000:NAD, author = "D. Batten and S. Jinturkar and J. Glossner and M. Schulte and P. D'arcy", title = "A New Approach to {DSP} Intrinsic Functions", crossref = "Sprague:2000:PAH", pages = "2892--2901", year = "2000", bibdate = "Sun Mar 04 11:18:38 2007", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://mesa.ece.wisc.edu/publications/cp_2000-01.pdf", acknowledgement = ack-nhfb, } @Article{Becken:2000:ACG, author = "W. Becken and P. Schmelcher", title = "The analytic continuation of the {Gaussian} hypergeometric function {$_2 F_1 (a, b; c; z)$} for arbitrary parameters", journal = j-J-COMPUT-APPL-MATH, volume = "126", number = "1--2", pages = "449--478", day = "30", month = dec, year = "2000", CODEN = "JCAMDI", DOI = "https://doi.org/10.1016/S0377-0427(00)00267-3", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", MRclass = "33C05 (33F05)", MRnumber = "MR1806771 (2002e:33003)", bibdate = "Sat Feb 25 12:43:38 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2000.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0377042700002673", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @InCollection{Borwein:2000:AGM, author = "J. M. Borwein and P. B. Borwein", title = "The Arithmetic--Geometric Mean and Fast Computation of Elementary Functions", crossref = "Berggren:2000:PSB", pages = "537--552", year = "2000", DOI = "https://doi.org/10.1007/978-1-4757-3240-5_56", bibdate = "Thu Aug 11 09:36:22 MDT 2016", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib; https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "Reprint of \cite{Borwein:1984:AGM}.", URL = "http://link.springer.com/chapter/10.1007/978-1-4757-3240-5_56", acknowledgement = ack-nhfb, author-dates = "Jonathan Michael Borwein (20 May 1951--2 August 2016)", ORCID-numbers = "Borwein, Jonathan/0000-0002-1263-0646", } @Article{Borwein:2000:CSR, author = "Jonathan M. Borwein and David M. Bradley and Richard E. Crandall", title = "Computational Strategies for the {Riemann} Zeta Function", journal = j-J-COMPUT-APPL-MATH, volume = "121", number = "1--2", pages = "247--296", month = sep, year = "2000", CODEN = "JCAMDI", DOI = "https://doi.org/10.1016/S0377-0427(00)00336-8", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", MRclass = "11M06 (11Y35 33F05)", MRnumber = "1780051", MRreviewer = "Cem Y. Y{\i}ld{\i}r{\i}m", bibdate = "Mon Oct 24 11:41:28 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://people.reed.edu/~crandall/papers/attach01.pdf", abstract = "We provide a compendium of evaluation methods for the Riemann zeta function, presenting formulae ranging from historical attempts to recently found convergent series to curious oddities old and new. We concentrate primarily on practical computational issues, such issues depending on the domain of the argument, the desired speed of computation, and the incidence of what we call ``value recycling''.", acknowledgement = ack-nhfb, author-dates = "Jonathan Michael Borwein (20 May 1951--2 August 2016)", fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", ORCID-numbers = "Borwein, Jonathan/0000-0002-1263-0646", remark = "CECM Preprint 98:118.", } @Book{Bressoud:2000:CCN, author = "David M. Bressoud and S. Wagon", title = "A Course in Computational Number Theory", publisher = "Key College Publishers in cooperation with Springer", address = "New York, NY, USA", pages = "xii + 367", year = "2000", ISBN = "1-930190-10-7", ISBN-13 = "978-1-930190-10-8", LCCN = "QA241 .B788 2000", bibdate = "Fri Sep 26 14:29:31 MDT 2008", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; z3950.loc.gov:7090/Voyager", URL = "http://www.loc.gov/catdir/enhancements/fy0818/99016037-d.html; http://www.loc.gov/catdir/enhancements/fy0818/99016037-t.html", acknowledgement = ack-nhfb, shorttableofcontents = "Fundamentals \\ Congruences, equations, and powers \\ Euler's function \\ Prime numbers \\ Some applications \\ Quadratic residues \\ Continued fractions \\ Prime testing with Lucas sequences \\ Prime imaginaries and imaginary primes \\ Appendix A. Mathematica basics \\ Appendix B. Lucas certificates exist", subject = "Number theory; Algorithms", tableofcontents = "Preface Vv \\ Notation Xi \\ Chapter 1. Fundamentals / 1 \\ 1.0 Introduction / 1 \\ 1.1 A Famous Sequence of Numbers / 2 \\ 1.2 The Euclidean Algorithm / 6 \\ The Oldest Algorithm \\ Reversing the Euclidean Algorithm \\ The Extended GCD Algorithm \\ The Fundamental Theorem of Arithmetic \\ Two Applications \\ 1.3 Modular Arithmetic / 25 \\ 1.4 Fast Powers / 30 \\ A Fast Algorithm for Exponentiation \\ Powers of Matrices, Big-O Notation \\ Chapter 2. Congruences, Equations, and Powers / 41 \\ 2.0 Introduction / / 41 \\ 2.1 Solving Linear Congruences / 41 \\ Linear Diophantine Equations in Two Variables \\ Linear Equations in Several Variables \\ Linear Congruences \\ The Conductor \\ An Important Quadratic Congruence \\ 2.2 The Chinese Remainder Theorem / 49 \\ 2.3 PowerMod Patterns / 55 \\ Fermat's Little Theorem \\ More Patterns in Powers \\ 2.4 Pseudoprimes / 59 \\ Using the Pseudoprime Test \\ Chapter 3. Euler's $\phi$ Function / 65 \\ 3.0 Introduction / 65 \\ 3.1 Euler's $\phi$ Function / 656 \\ 3.2 Perfect Numbers and Their Relatives / 12 \\ The Sum of Divisors Function \\ Perfect Numbers \\ Amicable, Abundant, and Deficient Numbers \\ 3.3 Euler's Theorem / 81 \\ 3.4 Primitive Roots for Primes / 84 \\ The Order of an Integer \\ Primes Have Primitive Roots \\ Repeating Decimals \\ 3.5 Primitive Roots for Composites / 70 \\ 3.6 The Universal Exponent / 73 \\ Universal Exponents \\ Power Towers \\ The Form of Carmichael Numbers \\ Chapter 4. Prime Numbers / 99 \\ 4.0 Introduction / 99 \\ 4.1 The number of Primes / 100 \\ We'll Never Run Out of Primes \\ The Sieve of Eratosthenes \\ Chebyshev's Theorem and Bertrand's Postulate \\ 4.2 Prime Testing and Certification / 114 \\ Strong Pseudoprimes \\ Industrial-Grade Primes \\ Prime Certification Via Primitive Roots \\ An Improvement \\ Pratt Certificates \\ 4.3 Refinements and Other Directions / 131 \\ Other Primality Tests \\ Strong Liars Are Scarce \\ Finding the $n$th Prime \\ 4.4 A Dozen Prime Mysteries / 141 \\ 5.0 Introduction / 145 \\ 5.1 Coding Secrets / 145 \\ Tossing a Coin into a Well \\ The RSA Cryptosystem \\ Digital Signatures \\ 5.2 The Yao Millionaire Problem / 155 \\ 5.3 Check Digits .158 \\ Basic Check Digit Schemes \\ A Perfect Check Digit Method \\ Beyond Perfection: Correcting Errors \\ 5.4 Factoring Algorithms / 167 \\ Trial Division \\ Fermat's Algorithm \\ Pollard Rho \\ Pollard $p - 1$ \\ The Current Scene \\ Chapter 6. Quadratic Residues / 179 \\ 6.0 Introduction / 179 \\ 6.1 P{\'e}pin's Test / 119 \\ Quadratic Residues \\ P{\'e}pin's Test \\ Primes Congruent to 1 (Mod 4) \\ 6.2 Proof of Quadratic Reciprocity / 185 \\ Gauss's Lemma \\ Proof of Quadratic Reciprocity \\ Jacobi's Extension \\ An Application to Factoring \\ 6.3 Quadratic Equations / 194 \\ Chapter 7. Continued Fractions / 201 \\ 7.0 Introduction / 201 \\ 7.1 Finite Continued Fractions / 202 \\ 7.2 Infinite Continued Fractions / 207 \\ 7.3 Periodic Continued Fractions / 213 \\ 7.4 Pell's Equation / 227 \\ 7.5 Archimedes and the Sun God's Cattle / 232 \\ Wurm's Version: Using Rectangular Bulls \\ The Real Cattle Problem \\ 7.6 Factoring via Continued Fractions / 238 \\ Chapter 8. Prime Testing with Lucas Sequences / 247 \\ 8.0 Introduction / 247 \\ 8.1 Divisibility Properties of Lucas Sequences / 248 \\ 8.2 Prime Tests Using Lucas Sequences / 259 \\ Lucas Certification \\ The Lucas--Lehmer Algorithm Explained \\ Lucas Pseudoprimes \\ Strong Quadratic Pseudoprimes \\ Primality Testing's Holy Grail \\ Chapter 9. Prime Imaginaries and Imaginary Primes / 279 \\ 9.0 Introduction / 279 \\ 9.1 Sums of Two Squares / 279 \\ Primes \\ The General Problem \\ How Many Ways \\ Number Theory and Salt \\ 9.2 The Gaussian Integers / 302 \\ Complex Number Theory \\ Gaussian Primes \\ The Moat Problem \\ The Gaussian Zoo \\ 9.3 Higher Reciprocity / 325 \\ Appendix A. Mathematica Basics / 333 \\ A.0 Introduction / 333 \\ A.1 Plotting / 336 \\ A.2 Typesetting / 338 \\ Sending Files By E-Mail \\ A.3 Types of Functions / 341 \\ A.4 Programs / 345 \\ A.6 Solving Equations / 347 \\ A.7 Symbolic Algebra / 349 \\ Appendix B. Lucas Certificates Exist / 351 \\ References / 355 \\ Index of Mathematica Objects / 359 \\ Subject Index / 363", } @Article{Brezinski:2000:CAD, author = "C. Brezinski", title = "Convergence acceleration during the 20th century", journal = j-J-COMPUT-APPL-MATH, volume = "122", number = "1--2", pages = "1--21", month = oct, year = "2000", CODEN = "JCAMDI", DOI = "https://doi.org/10.1016/S0377-0427(00)00360-5", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", MRclass = "65-03 (01A60)", MRnumber = "1794649", bibdate = "Thu Dec 1 17:36:45 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "Numerical analysis 2000, Vol. II: Interpolation and extrapolation", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", keywords = "convergence acceleration", } @Article{Carlson:2000:RTE, author = "B. C. Carlson and James FitzSimons", title = "Reduction theorems for elliptic integrands with the square root of two quadratic factors", journal = j-J-COMPUT-APPL-MATH, volume = "118", number = "1--2", pages = "71--85", day = "1", month = jun, year = "2000", CODEN = "JCAMDI", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Sat Feb 25 12:43:35 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2000.bib", URL = "http://www.sciencedirect.com/science/article/pii/S037704270000282X", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @Article{Cawley:2000:FCA, author = "G. C. Cawley", title = "On a Fast, Compact Approximation of the Exponential Function", journal = j-NEURAL-COMP, volume = "12", number = "9", pages = "2009--2012", day = "1", month = sep, year = "2000", CODEN = "NEUCEB", ISSN = "0899-7667 (print), 1530-888x (electronic)", ISSN-L = "0899-7667", bibdate = "Fri Nov 8 05:39:32 MST 2002", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; Ingenta database", acknowledgement = ack-nhfb, fjournal = "Neural Computation", journal-URL = "http://www.mitpressjournals.org/loi/neco", pagecount = "4", } @Article{Cherri:2000:PCC, author = "A. K. Cherri and M. S. Alam", title = "Parallel computation of complex elementary functions using quaternary signed-digit arithmetic", journal = "Optics and Laser Technology", volume = "32", number = "6", pages = "391--399", year = "2000", CODEN = "????", ISSN = "0030-3992", bibdate = "Sat Dec 7 09:21:28 MST 2002", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; Ingenta database", acknowledgement = ack-nhfb, pagecount = "9", } @InProceedings{Chu:2000:CPT, author = "Wanming Chu and Yamin Li", booktitle = "{ACAC 2000}: 5th Australasian Computer Architecture Conference", title = "Cost\slash performance tradeoff of $n$-select square root implementations", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "9--16", year = "2000", CODEN = "????", ISSN = "????", bibdate = "Sat Jul 16 15:28:14 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, summary = "Hardware square-root units require large numbers of gates even for iterative implementations. In this paper we present four low-cost high-performance fully-pipelined n-select implementations (nS-Root) based on a non-restoring-remainder square root \ldots{}", } @Article{Cohen:2000:CAA, author = "Henri Cohen and Fernando {Rodriguez Villegas} and Don Zagier", title = "Convergence Acceleration of Alternating Series", journal = j-EXP-MATH, volume = "9", number = "1", pages = "3--12", year = "2000", ISSN = "1058-6458 (print), 1944-950X (electronic)", ISSN-L = "1058-6458", MRclass = "11Y55 (65B05)", MRnumber = "1758796 (2001m:11222)", bibdate = "Thu Dec 1 17:36:45 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://projecteuclid.org/euclid.em/1046889587; http://www.math.u-bordeaux.fr/~cohen/sumalt2new.ps", ZMnumber = "0972.11115", abstract = "We discuss some linear acceleration methods for alternating series which are in theory and in practice much better than that of Euler--Van Wijngaarden. One of the algorithms, for instance, allows one to calculate $ \sum ( - 1)^k a_k $ with an error of about $ 17.93^{-n} $ from the first $n$ terms for a wide class of sequences $ \{ a_k \} $. Such methods are useful for high precision calculations frequently appearing in number theory.", acknowledgement = ack-nhfb, fjournal = "Experimental Mathematics", journal-URL = "http://www.tandfonline.com/loi/uexm20", keywords = "convergence acceleration", } @Article{Corless:2000:AAS, author = "Robert M. Corless and James H. Davenport and David J. Jeffrey and Stephen M. Watt", title = "{``According} to {Abramowitz and Stegun}'' or $ \arccoth $ needn't be uncouth", journal = j-SIGSAM, volume = "34", number = "2", pages = "58--65", month = jun, year = "2000", CODEN = "SIGSBZ", ISSN = "0163-5824 (print), 1557-9492 (electronic)", ISSN-L = "0163-5824", bibdate = "Fri Feb 8 18:27:07 MST 2002", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/sigsam.bib", acknowledgement = ack-nhfb, ajournal = "SIGSAM Bull.", fjournal = "SIGSAM Bulletin (ACM Special Interest Group on Symbolic and Algebraic Manipulation)", issue = "132", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J1000", } @Article{Ercegovac:2000:IGD, author = "Milos D. Ercegovac and Laurent Imbert and David W. Matula and Jean-Michel Muller and Guoheng Wei", title = "Improving {Goldschmidt} Division, Square Root, and Square Root Reciprocal", journal = j-IEEE-TRANS-COMPUT, volume = "49", number = "7", pages = "759--763", month = jul, year = "2000", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/12.863046", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Sat Jul 16 15:28:13 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; OCLC Proceedings database", acknowledgement = ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", remark = "Selected papers from ARITH'14 \cite{Koren:1999:ISC}.", summary = "The aim of this paper is to accelerate division, square root, and square root reciprocal computations when the Goldschmidt method is used on a pipelined multiplier. This is done by replacing the last iteration by the addition of a correcting term \ldots{}", } @Article{Ercegovac:2000:RSR, author = "Milos D. Ercegovac and Tom{\'a}s Lang and Jean-Michel Muller and Arnaud Tisserand", title = "Reciprocation, Square Root, Inverse Square Root, and Some Elementary Functions Using Small Multipliers", journal = j-IEEE-TRANS-COMPUT, volume = "49", number = "7", pages = "628--637", month = jul, year = "2000", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/12.863031", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", MRclass = "68M07 (65B15)", MRnumber = "MR1783602 (2001e:68016)", bibdate = "Sat Jul 16 15:28:13 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; OCLC Proceedings database", acknowledgement = ack-nhfb, fjournal = "Institute of Electrical and Electronics Engineers. Transactions on Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", remark = "Selected papers from ARITH'14 \cite{Koren:1999:ISC}.", summary = "This paper deals with the computation of reciprocals, square roots, inverse square roots, and some elementary functions using small tables, small multipliers, and, for some functions, a final ``large'' (almost full-length) multiplication. \ldots{}", } @Article{Favati:2000:SAC, author = "P. Favati and G. Lotti and O. Menchi and F. Romani", title = "Separable asymptotic cost of evaluating elementary functions", journal = j-NUMER-ALGORITHMS, volume = "24", number = "3", pages = "255--274", year = "2000", CODEN = "NUALEG", ISSN = "1017-1398 (print), 1572-9265 (electronic)", ISSN-L = "1017-1398", MRclass = "65Y20", MRnumber = "MR1780414 (2001d:65174)", bibdate = "Wed Apr 13 06:46:35 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; Ingenta database", acknowledgement = ack-nhfb, fjournal = "Numerical Algorithms", journal-URL = "http://link.springer.com/journal/11075", pagecount = "20", } @Article{Galue:2000:MTG, author = "L. Galu{\'e} and H. G. Khajah and Shyam L. Kalla", title = "Multiplication theorems for generalized and double-index {Bessel} functions", journal = j-J-COMPUT-APPL-MATH, volume = "118", number = "1--2", pages = "143--150", day = "1", month = jun, year = "2000", CODEN = "JCAMDI", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Sat Feb 25 12:43:35 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2000.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0377042700002855", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @InProceedings{Gisuthan:2000:FCU, author = "B. Gisuthan and T. Srikanthan", booktitle = "{Proceedings of the 43rd IEEE Midwest Symposium on Circuits and Systems (Cat. No. CH37144)}", title = "Flat {CORDIC}: a unified architecture for high-speed generation of trigonometric and hyperbolic functions", volume = "3", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "1414--1417 (vol. 3)", year = "2000", DOI = "https://doi.org/10.1109/MWSCAS.2000.951478", bibdate = "Mon Oct 27 10:32:44 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, keywords = "Arithmetic; Circuits; CMOS process; Computer architecture; Delay; Equations; Iterative algorithms; Libraries; Silicon; Very large scale integration", } @Article{Harris:2000:SBE, author = "Frank E. Harris", title = "Spherical {Bessel} expansions of sine, cosine, and exponential integrals", journal = j-APPL-NUM-MATH, volume = "34", number = "1", pages = "95--98", month = jun, year = "2000", CODEN = "ANMAEL", DOI = "https://doi.org/10.1016/S0168-9274(99)00031-8", ISSN = "0168-9274 (print), 1873-5460 (electronic)", ISSN-L = "0168-9274", MRclass = "33B10; 65D30 (33F05 65D20)", MRnumber = "1755696 (2001a:65027)", bibdate = "Sat Oct 21 13:09:35 MDT 2000", bibsource = "http://www.elsevier.com/locate/issn/01689274; https://www.math.utah.edu/pub/bibnet/authors/h/harris-frank-e.bib; https://www.math.utah.edu/pub/tex/bib/applnummath.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.elsevier.nl/gej-ng/29/17/21/62/27/32/abstract.html; http://www.elsevier.nl/gej-ng/29/17/21/62/27/32/article.pdf; http://www.sciencedirect.com/science/article/pii/S0168927499000318", ZMnumber = "Zbl 0951.33002", acknowledgement = ack-nhfb, fjournal = "Applied Numerical Mathematics: Transactions of IMACS", journal-URL = "http://www.sciencedirect.com/science/journal/01689274/", } @InProceedings{Hasan:2000:FPI, author = "M. A. Hasan and A. A. Hasan and S. Rahman", booktitle = "Proceedings of the 39th {IEEE} Conference on Decision and Control", title = "Fixed point iterations for computing square roots and the matrix sign function of complex matrices", volume = "5", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "4253--4258", year = "2000", CODEN = "????", ISSN = "????", bibdate = "Sat Jul 16 15:28:14 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, summary = "The purpose of this work has been the development of new set of rational iterations for computing square roots and the matrix sign function of complex matrices. Given any positive integer r{\&}ges;2, we presented a systematic way of deriving rth order \ldots{}", } @InProceedings{Hassibi:2000:ESR, author = "B. Hassibi", booktitle = "Proceedings. 2000 {IEEE} International Conference on Acoustics, Speech, and Signal Processing: {ICASSP '00}, 5--9 June 2000", title = "An efficient square-root algorithm for {BLAST}", volume = "2", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "II737--II740", year = "2000", CODEN = "????", ISSN = "????", bibdate = "Sat Jul 16 15:28:14 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, summary = "Bell Labs Layered Space-Time (BLAST) is a scheme for transmitting information over a rich-scattering wireless environment using multiple receive and transmit antennas. The main computational bottleneck in the BLAST algorithm is a ``nulling and \ldots{}", } @InProceedings{Hassibi:2000:FSR, author = "B. Hassibi", booktitle = "{Conference Record of the Thirty-Fourth Asilomar Conference on Signals, Systems and Computers, 2000}", title = "A fast square-root implementation for {BLAST}", volume = "2", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "1255--1259", year = "2000", CODEN = "????", ISSN = "????", bibdate = "Sat Jul 16 15:28:14 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, summary = "Bell Labs Layered Space-Time (BLAST) is a scheme for transmitting information over a rich-scattering wireless environment using multiple receive and transmit antennas. The main computational bottleneck in the BLAST algorithm is a ``nulling and \ldots{}", } @Article{Holmgren:2000:CAL, author = "Sverker Holmgren and Henrik Brand{\'e}n and Erik Sterner", title = "Convergence Acceleration for the Linearized {Navier--Stokes} Equations using Semicirculant Approximations", journal = j-SIAM-J-SCI-COMP, volume = "21", number = "4", pages = "1524--1550", month = jul, year = "2000", CODEN = "SJOCE3", DOI = "https://doi.org/10.1137/S1064827597317983", ISSN = "1064-8275 (print), 1095-7197 (electronic)", ISSN-L = "1064-8275", bibdate = "Fri Oct 27 13:32:22 MDT 2000", bibsource = "http://epubs.siam.org/sam-bin/dbq/toc/SISC/21/4; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://epubs.siam.org/sam-bin/dbq/article/31798", acknowledgement = ack-nhfb, fjournal = "SIAM Journal on Scientific Computing", journal-URL = "http://epubs.siam.org/sisc", keywords = "convergence acceleration", } @Misc{Intel:2000:DSR, author = "{Intel}", title = "Divide, Square Root, and Remainder Algorithms for the {Itanium} Architecture", howpublished = "Intel Software Development Products", month = jul, year = "2000", bibdate = "Fri Sep 22 17:06:23 2023", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "https://studylib.net/doc/7921762/divide--square-root-and-remainder-algorithms-for-the-ia-64", acknowledgement = ack-nhfb, } @Book{Jeffrey:2000:HMF, author = "Alan Jeffrey", title = "Handbook of Mathematical Formulas and Integrals", publisher = pub-ACADEMIC, address = pub-ACADEMIC:adr, edition = "Second", pages = "xxvi + 433", year = "2000", ISBN = "0-12-382251-3", ISBN-13 = "978-0-12-382251-2", LCCN = "QA47 .J38 2000", bibdate = "Wed Jun 12 14:47:49 MDT 2024", bibsource = "fsz3950.oclc.org:210/WorldCat; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, shorttableofcontents = "0: Quick Reference List of Frequently Used Data \\ 1: Numerical, Algebraic, and Analytical Results for Series and Calculus \\ 2: Functions and Identities \\ 3: Derivatives of Elementary Functions \\ 4: Indefinite Integrals of Algebraic Functions \\ 5: Indefinite Integrals of Exponential Functions \\ 6: Indefinite Integrals of Logarithmic Functions \\ 7: Indefinite Integrals of Hyperbolic Functions \\ 8: Indefinite Integrals Involving Inverse Hyperbolic Functions \\ 9: Indefinite Integrals of Trigonometric Functions \\ 10: Indefinite Integrals of Inverse Trigonometric Functions \\ 11: The Gamma, Beta, Pi, and Psi Functions \\ 12: Elliptic Integrals and Functions \\ 13: Probability Integrals and the Error Function \\ 14: Fresnel Integrals \\ 15: Definite Integrals \\ 16: Different Forms of Fourier Series \\ 17: Bessel Functions \\ 18: Orthogonal Polynomials \\ 19: Laplace Transformation \\ 20: Fourier Transforms \\ 21: Numerical Integration \\ 22: Solutions of Standard Ordinary Differential Equations \\ 23: Vector Analysis \\ 24: Systems of Orthogonal Coordinates \\ 25: Partial Differential Equations and Special Functions", subject = "Mathematics; Tables; Formulae; Mathematics.", tableofcontents = "0: Quick Reference List of Frequently Used Data \\ 0.1: Useful Identities / 1 \\ 0.2: Complex Relationships / 2 \\ 0.3: Constants / 2 \\ 0.4: Derivatives of Elementary Functions / 3 \\ 0.5: Rules of Differentiation and Integration / 3 \\ 0.6: Standard Integrals / 4 \\ 0.7: Standard Series / 11 \\ 0.8: Geometry / 13 \\ 1: Numerical, Algebraic, and Analytical Results for Series and Calculus \\ 1.1: Algebraic Results Involving Real and Complex Numbers / 25 \\ 1.2: Finite Sums / 29 \\ 1.3: Bernoulli and Euler Numbers and Polynomials / 37 \\ 1.4: Determinants / 47 \\ 1.5: Matrices / 55 \\ 1.6: Permutations and Combinations / 62 \\ 1.7: Partial Fraction Decomposition / 63 \\ 1.8: Convergence of Series / 66 \\ 1.9: Infinite Products / 71 \\ 1.10: Functional Series / 73 \\ 1.11: Power Series / 74 \\ 1.12: Taylor Series / 79 \\ 4.13: Fourier Series / 81 \\ 4.14: Asymptotic Expansions / 85 \\ 1.15: Basic Results from the Calculus / 86 \\ 2: Functions and Identities \\ 2.1: Complex Numbers and Trigonometric and Hyperbolic Functions / 101 \\ 2.2: Logarithms and Exponentials / 112 \\ 2.3: Exponential Function / 114 \\ 2.4: Trigonometric Identities / 115 \\ 2.5: Hyperbolic Identities / 121 \\ 2.6: Logarithm / 126 \\ 2.7: Inverse Trigonometric and Hyperbolic Functions 128 \\ 2.8: Series Representations of Trigonometric and Hyperbolic Functions / 133 \\ 2.9: Useful Limiting Values and Inequalities Involving Elementary Functions / 136 \\ 3: Derivatives of Elementary Functions \\ 3.1: Derivatives of Algebraic, Logarithmic, and Exponential Functions / 139 \\ 3.2: Derivatives of Trigonometric Functions / 140 \\ 3.3: Derivatives of Inverse Trigonometric Functions 140 \\ 3.4: Derivatives of Hyperbolic Functions / 141 \\ 3.5: Derivatives of Inverse Hyperbolic Functions 142 \\ 4: Indefinite Integrals of Algebraic Functions \\ 4.1: Algebraic and Transcendental Functions / 145 \\ 4.2: Indefinite Integrals of Rational Functions / 146 \\ 4.3: Nonrational Algebraic Functions / 158 \\ 5: Indefinite Integrals of Exponential Functions \\ 5.1: Basic Results / 167 \\ 6: Indefinite Integrals of Logarithmic Functions \\ 6.1: Combinations of Logarithms and Polynomials / 173 \\ 7: Indefinite Integrals of Hyperbolic Functions \\ 7.1: Basic Results / 179 \\ 7.2: Integrands Involving Powers of sinh(bx) or cosh(bx) / 180 \\ 7.3: Integrands Involving (a [plus or minus] bx)[superscript m] sinh(cx) or (a + bx)[superscript m] cosh(cx) / 181 \\ 7.4: Integrands Involving x[superscript m] sinh[superscript n] x or x[superscript m] cosh[superscript n] x / 183 \\ 7.5: Integrands Involving x[superscript m] sinh[superscript -n] x or x[superscript m] cosh[superscript -n] x / 183 \\ 7.6: Integrands Involving (1 [plus or minus] cosh x)[superscript -m] / 185 \\ 7.7: Integrands Involving sinh(ax)cosh[superscript -n] x or cosh(ax)sinh[superscript -n] x / 185 \\ 7.8: Integrands Involving sinh(ax + b) and cosh(cx + d) / 186 \\ 7.9: Integrands Involving tanh kx and coth kx / 188 \\ 7.10: Integrands Involving (a + bx)[superscript m] sinh kx or (a + bx)[superscript m] cosh kx / 189 \\ 8: Indefinite Integrals Involving Inverse Hyperbolic Functions \\ 8.1: Basic Results / 191 \\ 8.2: Integrands Involving x[superscript -n] arcsinh(x/a) or x[superscript -n] arccosh(x/a) / 193 \\ 8.3: Integrands Involving x[superscript n] arctanh(x/a) or x[superscript n] arccoth(x/a) / 194 \\ 8.4: Integrands Involving x[superscript -n] arctanh(x/a) or x[superscript -n] arccoth(x/a) / 195 \\ 9: Indefinite Integrals of Trigonometric Functions \\ 9.1: Basic Results / 197 \\ 9.2: Integrands Involving Powers of x and Powers of sin x or cos x / 197 \\ 9.3: Integrands Involving tan x and/or cot x / 205 \\ 9.4: Integrands Involving sin x and cos x / 207 \\ 9.5: Integrands Involving Sines and Cosines with Linear Arguments and Powers of x / 211 \\ 10: Indefinite Integrals of Inverse Trigonometric Functions \\ 10.1: Integrands Involving Powers of x and Powers of Inverse Trigonometric Functions 215 --. 11: Gamma, Beta, Pi, and Psi Functions \\ 11.1: Euler Integral and Limit and Infinite Product Representations for [Gamma] (x) / 221 \\ 12: Elliptic Integrals and Functions \\ 12.1: Elliptic Integrals / 229 \\ 12.2: Jacobian Elliptic Functions / 235 \\ 12.3: Derivatives and Integrals / 237 \\ 12.4: Inverse Jacobian Elliptic Functions / 237 \\ 13: Probability Integrals and the Error Function \\ 13.1: Normal Distribution / 239 \\ 13.2: Error Function / 242 \\ 14: Fresnel Integrals, Sine and Cosine Integrals \\ 14.1: Definitions, Series Representations, and Values at Infinity / 245 \\ 14.2: Definitions, Series Representations, and Values at Infinity / 247 \\ 15: Definite Integrals \\ 15.1: Integrands Involving Powers of x / 249 \\ 15.2: Integrands Involving Trigonometric Functions 251 \\ 15.3: Integrands Involving the Exponential Function 254 \\ 15.4: Integrands Involving the Hyperbolic Function 256 \\ 15.5: Integrands Involving the Logarithmic Function 256 \\ 16: Different Forms of Fourier Series \\ 16.1: Fourier Series for f(x) on -[pi] [less than or equal] x [less than or equal] [pi] / 257 \\ 16.2: Fourier Series for f(x) on -L [less than or equal] x [less than or equal] L / 258 \\ 16.3: Fourier Series for f(x) on a [less than or equal] x [less than or equal] b / 258 \\ 16.4: Half-Range Fourier Cosine Series for f(x) on 0 [less than or equal] x [less than or equal] [pi] 259 \\ 16.5: Half-Range Fourier Cosine Series for f(x) on 0 [less than or equal] x [less than or equal] L / 259 \\ 16.6: Half-Range Fourier Sine Series for f(x) on 0 [less than or equal] x [less than or equal] [pi] 260 \\ 16.7: Half-Range Fourier Sine Series for f(x) on 0 [less than or equal] x [less than or equal] L / 260 \\ 16.8: Complex (Exponential) Fourier Series for f(x) on -[pi] [less than or equal] x [less than or equal] [pi] / 260 \\ 16.9: Complex (Exponential) Fourier Series for f(x) on -L [less than or equal] x [less than or equal] L 261 \\ 16.10: Representative Examples of Fourier Series 261 \\ 16.11: Fourier Series and Discontinuous Functions 265 \\ 17: Bessel Functions \\ 17.1: Bessel's Differential Equation / 269 \\ 17.2: Series Expansions for J[subscript v](x) and Y[subscript v](x) / 270 \\ 17.3: Bessel Functions of Fractional Order / 272 \\ 17.4: Asymptotic Representations for Bessel Functions / 273 \\ 17.5: Zeros of Bessel Functions / 273 \\ 17.6: Bessel's Modified Equation / 274 \\ 17.7: Series Expansions for I[subscript v](x) and K[subscript v](x) / 276 \\ 17.8: Modified Bessel Functions of Fractional Order 277 \\ 17.9: Asymptotic Representations of Modified Bessel Functions / 278 \\ 17.10: Relationships between Bessel Functions / 278 \\ 17.11: Integral Representations of J[subscript n](x), I[subscript n](x), and K[subscript n](x) / 281 \\ 17.12: Indefinite Integrals of Bessel Functions / 281 \\ 17.13: Definite Integrals Involving Bessel Functions / 282 \\ 17.14: Spherical Bessel Functions / 283 \\ 18: Orthogonal Polynomials \\ 18.2: Legendre Polynomials P[subscript n](x) / 286 \\ 18.3: Chebyshev Polynomials T[subscript n](x) and U[subscript n](x) / 290 \\ 18.4: Laguerre Polynomials L[subscript n](x) / 294 \\ 18.5: Hermite Polynomials H[subscript n](x) / 296 \\ 19: Laplace Transformation \\ 20: Fourier Transforms \\ 21: Numerical Integration \\ 21.1: Classical Methods / 315 \\ 22: Solutions of Standard Ordinary Differential Equations \\ 22.2: Separation of Variables / 323 \\ 22.3: Linear First-Order Equations / 323 \\ 22.4: Bernoulli's Equation / 324 \\ 22.5: Exact Equations / 325 \\ 22.6: Homogeneous Equations / 325 \\ 22.7: Linear Differential Equations / 326 \\ 22.8: Constant Coefficient Linear Differential Equations--Homogeneous Case / 327 \\ 22.9: Linear Homogeneous Second-Order Equation / 330 \\ 22.10: Constant Coefficient Linear Differential Equations--Inhomogeneous Case / 331 \\ 22.11: Linear Inhomogeneous Second-Order Equation 333 \\ 22.12: Determination of Particular Integrals by the Method of- Undetermined Coefficients / 334 \\ 22.13: Cauchy-Euler Equation / 336 \\ 22.14: Legendre's Equation / 337 \\ 22.15: Bessel's Equations / 337 \\ 22.16: Power Series and Frobenius Methods / 339 \\ 22.17: Hypergeometric Equation / 344 \\ 22.18: Numerical Methods / 345 \\ 23: Vector Analysis \\ 23.1: Scalars and Vectors / 353 \\ 23.2: Scalar Products / 358 \\ 23.3: Vector Products / 359 \\ 23.4: Triple Products / 360 \\ 23.5: Products of Four Vectors / 361 \\ 23.6: Derivatives of Vector Functions of a Scalar t 361 \\ 23.7: Derivatives of Vector Functions of Several Scalar Variables / 362 \\ 23.8: Integrals of Vector Functions of a Scalar Variable t / 363 \\ 23.9: Line Integrals / 364 \\ 23.10: Vector Integral Theorems / 366 \\ 23.11: A Vector Rate of Change Theorem / 368 \\ 23.12: Useful Vector Identities and Results / 368 \\ 24: Systems of Orthogonal Coordinates \\ 24.1: Curvilinear Coordinates / 369 \\ 24.2: Vector Operators in Orthogonal Coordinates 371 \\ 24.3: Systems of Orthogonal Coordinates / 371 \\ 25: Partial Differential Equations and Special Functions \\ 25.1: Fundamental Ideas / 381 \\ 25.2: Method of Separation of Variables / 385 \\ 25.3: Sturm--Liouville Problem and Special Functions 387 \\ 25.4: A First-Order System and the Wave Equation 390 \\ 25.5: Conservation Equations (Laws) / 391 \\ 25.6: Method of Characteristics / 392 \\ 25.7: Discontinuous Solutions (Shocks) / 396 \\ 25.8: Similarity Solutions / 398 \\ 25.9: Burgers's Equation, the KdV Equation, and the KdVB Equation / 400 \\ 26: Z-Transform \\ 26.1: Z-Transform and Transform Pairs / 403 \\ 27: Numerical Approximation \\ 27.2: Economization of Series / 411 \\ 27.3: Pade Approximation / 413 \\ 27.4: Finite Difference Approximations to Ordinary and Partial Derivatives / 415", } @Article{Kilbas:2000:CFI, author = "A. A. Kilbas and J. J. Trujillo", title = "Computation of fractional integrals via functions of hypergeometric and {Bessel} type", journal = j-J-COMPUT-APPL-MATH, volume = "118", number = "1--2", pages = "223--239", day = "1", month = jun, year = "2000", CODEN = "JCAMDI", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Sat Feb 25 12:43:35 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2000.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0377042700002910", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @Article{Lang:2000:CBC, author = "Tom{\'a}s Lang and Elisardo Antelo", title = "{CORDIC}-Based Computation of {ArcCos} and $ \sqrt {1 - t^2} $", journal = j-J-VLSI-SIGNAL-PROC-SSIVT, volume = "25", number = "1", pages = "19--38", month = may, year = "2000", DOI = "https://doi.org/10.1023/a:1008121502359", ISSN = "1387-5485 (print), 1573-0506 (electronic)", ISSN-L = "1387-5485", bibdate = "Tue Oct 28 07:04:09 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, ajournal = "J. VLSI Signal Process. Syst. Signal, Image Video Tech.", fjournal = "Journal of VLSI Signal Processing Systems for Signal, Image and Video Technology", journal-URL = "https://link.springer.com/journal/11265", } @InProceedings{Lefevre:2000:CRF, author = "V. D. Lefevre and J.-M. Muller", booktitle = "Conference Record of the Thirty-Fourth Asilomar Conference on Signals, Systems and Computers, 2000", title = "Correctly rounded functions for better arithmetic", volume = "2", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "875--878", year = "2000", CODEN = "????", ISSN = "????", bibdate = "Sat Jul 16 11:25:05 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, summary = "The IEEE 754 standard for floating-point arithmetic requires that the four arithmetic operations and the square root should be correctly rounded. This has improved the accuracy, reliability and portability of numerical software. Unfortunately, such \ldots{}", } @Article{Lopez:2000:AES, author = "Jos{\'e} L. L{\'o}pez", title = "Asymptotic Expansions of Symmetric Standard Elliptic Integrals", journal = j-SIAM-J-MATH-ANA, volume = "31", number = "4", pages = "754--775", year = "2000", CODEN = "SJMAAH", DOI = "https://doi.org/10.1137/S0036141099351176", ISSN = "0036-1410 (print), 1095-7154 (electronic)", ISSN-L = "0036-1410", bibdate = "Fri Oct 27 08:17:04 MDT 2000", bibsource = "http://epubs.siam.org/sam-bin/dbq/toc/SIMA/31/4; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://epubs.siam.org/sam-bin/dbq/article/35117", acknowledgement = ack-nhfb, fjournal = "SIAM Journal on Mathematical Analysis", journal-URL = "http://epubs.siam.org/sima", } @InProceedings{Lozier:2000:DPN, author = "Daniel W. Lozier", title = "The {DLMF Project}: a New Initiative in Classical Special Functions", crossref = "Dunkl:2000:PIW", pages = "207--220", year = "2000", bibdate = "Fri Jul 09 06:31:32 2004", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, } @Book{Markstein:2000:IEF, author = "Peter Markstein", title = "{IA-64} and elementary functions: speed and precision", publisher = pub-PH, address = pub-PH:adr, pages = "xix + 298", year = "2000", ISBN = "0-13-018348-2", ISBN-13 = "978-0-13-018348-4", LCCN = "QA76.9.A73 M365 2000", bibdate = "Fri Jan 5 08:00:52 MST 2001", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/intel-ia-64.bib; https://www.math.utah.edu/pub/tex/bib/master.bib; https://www.math.utah.edu/pub/tex/bib/microchip.bib; University of California MELVYL catalog.", series = "Hewlett--Packard professional books", URL = "http://www.markstein.org/", acknowledgement = ack-nhfb, keywords = "IA-64 (computer architecture)", remark = "Besides recipes for accurate computation of elementary functions, this book also contains algorithms for the correctly-rounded computation of floating-point division and square-root, and of integer division, starting from low-precision reciprocal approximations. There is also a wealth of information on the tradeoffs between integer and floating-point instruction use in a pipelined parallel architecture.", tableofcontents = "IA-64 Architecture \\ New Architecture Objectives \\ VLIW \\ Memory Enhancements \\ Software Pipelining \\ Floating Point Enhancements \\ Summary \\ IA-64 Instructions And Registers \\ Instructions \\ Register Sets \\ Accessing Memory \\ Assembly Language \\ Problems \\ Increasing Instruction Level Parallelism \\ Branching \\ Speculation \\ Problems \\ Floating Point Architecture \\ Floating Point Status Register \\ Precision \\ Fused Multiply-Add \\ Division and Square Root Assists \\ Floating Comparisons \\ Communication between Floating Point and General Purpose Registers \\ Fixed Point Multiplication \\ SIMD Arithmetic \\ Problems \\ Programming For IA-64 \\ Compiler Options \\ Pragmas \\ Floating Point Data Types \\ In-Line Assembly \\ The fenv.h Header \\ Extended Examples \\ Quad Precision \\ Problems \\ Computation of Elementary Functions \\ Mathematical Preliminaries \\ Floating Point \\ Approximation and Error Analysis \\ The Exclusion Theorem \\ Ulps \\ Problems \\ Approximation Of Functions \\ Taylor Series \\ Lagrangian Interpolation \\ Chebychev Approximation \\ Remez Approximation \\ Practical Considerations \\ Function Evaluation \\ Table Construction \\ Problems \\ Division \\ Approximations for the Reciprocal \\ Computing the Quotient \\ Division Using Only Final Precision Results \\ Fast Variants of Division \\ Remainder \\ Integer Division \\ An Implementation of Division \\ Problems \\ Square Root \\ Approximations \\ Rounding the Square Root \\ Computing the Square Root \\ Calculating the Reciprocal Square Root \\ An Implementation of Square Root \\ Problems \\ Exponential Functions \\ Definitions and Formulas \\ Argument Reduction \\ Error Containment \\ Computing the Exponential \\ The Function expm \\ Problems \\ Logarithmic Functions \\ General Relations \\ Argument Reductions \\ Error Analysis \\ The Function log1p \\ Computing the Logarithm \\ Problems \\ The Power Function \\ Definition \\ Single Precision \\ Double Precision \\ Double-Extended Precision \\ Quad Precision \\ Computing the Power Function \\ Problems \\ Trigonometric Functions \\ Formulas and Identities \\ Argument Reduction \\ Error Analysis \\ Computing the Trigonometric Functions \\ Problems \\ Inverse Sine And Cosine \\ Definitions and Formulas \\ Argument Reduction \\ Error Analysis \\ Computing the arcsin \\ Problems \\ Inverse Tangent Functions \\ Definitions and Formulas \\ Argument Reduction \\ Error Analysis \\ Computing the arctan \\ Problems \\ Hyperbolic Functions \\ Definitions and Formulas \\ Argument Reduction \\ Error Analysis \\ Computing the Hyperbolic Functions \\ Problems \\ Inverse Hyperbolic Functions \\ Definitions and Formulas. arcsinh. arccosh. arctanh \\ Problems \\ Odds And Ends \\ Correctly Rounded Functions \\ Monotonicity \\ Alternative Algorithms \\ Testing \\ New Architectural Directions \\ Problems \\ In-Line Assembly \\ Solutions To Problems \\ Bibliography \\ Subject Index", } @Article{Paliouras:2000:FPP, author = "V. Paliouras and K. Karagianni and T. Stouraitis", title = "A floating-point processor for fast and accurate sine\slash cosine evaluation", journal = j-IEEE-TRANS-CIRCUITS-SYST-2, volume = "47", number = "5", pages = "441--451", month = may, year = "2000", CODEN = "ICSPE5", DOI = "https://doi.org/10.1109/82.842112", ISSN = "1057-7130 (print), 1558-125X (electronic)", ISSN-L = "1057-7130", bibdate = "Sat Jul 16 08:40:52 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "IEEE Transactions on Circuits and Systems. 2, Analog and Digital Signal Processing", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=82", summary = "A VLSI architecture for fast and accurate floating-point sine/cosine evaluation is presented, combining floating-point and simple fixed-point arithmetic. The algorithm implemented by the architecture is based on second order polynomial interpolation \ldots{}", } @Book{Simon:2000:DCF, author = "Marvin Kenneth Simon and Mohamed-Slim Alouini", title = "Digital Communication over Fading Channels: a Unified Approach to Performance Analysis", publisher = pub-WILEY, address = pub-WILEY:adr, pages = "xix + 544", year = "2000", ISBN = "0-471-31779-9", ISBN-13 = "978-0-471-31779-1", LCCN = "TK5103.7 .S523 2000", bibdate = "Sat Dec 16 17:34:06 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; z3950.loc.gov:7090/Voyager", series = "Wiley series in telecommunications and signal processing", URL = "http://www.loc.gov/catdir/bios/wiley043/99056352.html; http://www.loc.gov/catdir/description/wiley035/99056352.html; http://www.loc.gov/catdir/toc/onix06/99056352.html", acknowledgement = ack-nhfb, author-dates = "1939--", subject = "Digital communications; Reliability; Mathematics; Radio; Transmitters and transmission; Fading", tableofcontents = "1.1: System Performance Measures 4 \\ 1.1.1: Average Signal-to-Noise Ratio 4 \\ 1.1.2: Outage Probability 5 \\ 1.1.3: Average Bit Error Probability 6 \\ Chapter 2: Fading Channel Characterization and Modeling 15 \\ 2.1: Main Characteristics of Fading Channels 15 \\ 2.1.1: Envelope and Phase Fluctuations 15 \\ 2.1.2: Slow and Fast Fading 16 \\ 2.1.3: Frequency-Flat and Frequency-Selective Fading 16 \\ 2.2: Modeling of Flat Fading Channels 17 \\ 2.2.1: Multipath Fading 18 \\ 2.2.2: Log-Normal Shadowing 23 \\ 2.2.3: Composite Multipath/Shadowing 24 \\ 2.2.4: Combined (Time-Shared) Shadowed/Unshadowed Fading 25 \\ 2.3: Modeling of Frequency-Selective Fading Channels 26 \\ Chapter 3: Types of Communication 31 \\ 3.1: Ideal Coherent Detection 31 \\ 3.1.1: Multiple Amplitude-Shift-Keying or Multiple Amplitude Modulation 33 \\ 3.1.2: Quadrature Amplitude-Shift-Keying or Quadrature Amplitude Modulation 34 \\ 3.1.3: M-ary Phase-Shift-Keying 35 \\ 3.1.4: Differentially Encoded M-ary Phase-Shift-Keying 39 \\ 3.1.5: Offset QPSK or Staggered QPSK 41 \\ 3.1.6: M-ary Frequency-Shift-Keying 43 \\ 3.1.7: Minimum-Shift-Keying 45 \\ 3.2: Nonideal Coherent Detection 47 \\ 3.3: Noncoherent Detection 53 \\ 3.4: Partially Coherent Detection 55 \\ 3.4.1: Conventional Detection: One-Symbol Observation 55 \\ 3.4.2: Multiple Symbol Detection 57 \\ 3.5: Differentially Coherent Detection 59 \\ 3.5.1: M-ary Differential Phase Shift Keying 59 \\ 3.5.2: [pi]/4-Differential QPSK 65 \\ Part 2: Mathematical Tools \\ Chapter 4: Alternative Representations of Classical Functions 69 \\ 4.1: Gaussian $Q$-Function 70 \\ 4.1.1: One-Dimensional Case 70 \\ 4.1.2: Two-Dimensional Case 72 \\ 4.2: Marcum $Q$-Function 74 \\ 4.2.1: First-Order Marcum $Q$-Function 74 \\ 4.2.2: Generalized ($m$th-Order) Marcum $Q$-Function 81 \\ 4.3: Other Functions 90 \\ Appendix 4A: Derivation of Eq. (4.2) 95 \\ Chapter 5: Useful Expressions for Evaluating Average Error Probability Performance 99 \\ 5.1: Integrals Involving the Gaussian $Q$-Function 99 \\ 5.1.1: Rayleigh Fading Channel 101 \\ 5.1.2: Nakagami-q (Hoyt) Fading Channel 101 \\ 5.1.3: Nakagami-n (Rice) Fading Channel 102 \\ 5.1.4: Nakagami-m Fading Channel 102 \\ 5.1.5: Log-Normal Shadowing Channel 104 \\ 5.1.6: Composite Log-Normal Shadowing/Nakagami-m Fading Channel 104 \\ 5.2: Integrals Involving the Marcum $Q$-Function 107 \\ 5.2.1: Rayleigh Fading Channel 108 \\ 5.2.2: Nakagami-q (Hoyt) Fading Channel 109 \\ 5.2.3: Nakagami-n (Rice) Fading Channel 109 \\ 5.2.4: Nakagami-m Fading Channel 109 \\ 5.2.5: Log-Normal Shadowing Channel 109 \\ 5.2.6: Composite Log-Normal Shadowing/Nakagami-m Fading Channel 110 \\ 5.3: Integrals Involving the Incomplete Gamma Function 111 \\ 5.3.1: Rayleigh Fading Channel 112 \\ 5.3.2: Nakagami-q (Hoyt) Fading Channel 112 \\ 5.3.3: Nakagami-n (Rice) Fading Channel 112 \\ 5.3.4: Nakagami-m Fading Channel 113 \\ 5.3.5: Log-Normal Shadowing Channel 114 \\ 5.3.6: Composite Log-Normal Shadowing/Nakagami-m Fading Channel 114 \\ 5.4: Integrals Involving Other Functions 114 \\ 5.4.1: M-PSK Error Probability Integral 114 \\ 5.4.2: Arbitrary Two-Dimensional Signal Constellation Error Probability Integral 116 \\ 5.4.3: Integer Powers of the Gaussian $Q$-Function 117 \\ 5.4.4: Integer Powers of M-PSK Error Probability Integrals 121 \\ Appendix 5A: Evaluation of Definite Integrals Associated with Rayleigh and Nakagami-$m$ Fading 124 \\ Chapter 6: New Representations of Some PDF's and CDF's for Correlative Fading Applications 141 \\ 6.1: Bivariate Rayleigh PDF and CDF 142 \\ 6.2: PDF and CDF for Maximum of Two Rayleigh Random Variables 146 \\ 6.3: PDF and CDF for Maximum of Two Nakagami-m Random Variables 149 \\ Part 3: Optimum Reception and Performance Evaluation \\ Chapter 7: Optimum Receivers for Fading Channels 157 \\ 7.1: Case of Known Amplitudes, Phases, and Delays: Coherent, Detection 159 \\ 7.2: Case of Known Phases and Delays, Unknown Amplitudes 163 \\ 7.2.1: Rayleigh Fading 163 \\ 7.2.2: Nakagami-m Fading 164 \\ 7.3: Case of Known Amplitudes and Delays, Unknown Phases 166 \\ 7.4: Case of Known Delays and Unknown Amplitudes and Phases 168 \\ 7.4.1: One-Symbol Observation: Noncoherent Detection 168 \\ 7.4.2: Two-Symbol Observation: Conventional Differentially Coherent Detection 181 \\ 7.4.3: N-Symbol Observation: Multiple Symbol Differentially Coherent Detection 186 \\ 7.5: Case of Unknown Amplitudes, Phases, and Delays 188 \\ 7.5.1: One-Symbol Observation: Noncoherent Detection 188 \\ 7.5.2: Two-Symbol Observation: Conventional Differentially Coherent Detection 190 \\ Chapter 8: Performance of Single Channel Receives 193 \\ 8.1: Performance Over the AWGN Channel 193 \\ 8.1.1: Ideal Coherent Detection 194 \\ 8.1.2: Nonideal Coherent Detection 206 \\ 8.1.3: Noncoherent Detection 209 \\ 8.1.4: Partially Coherent Detection 210 \\ 8.1.5: Differentially Coherent Detection 213 \\ 8.1.6: Generic Results for Binary Signaling 218 \\ 8.2: Performance Over Fading Channels 219 \\ 8.2.1: Ideal Coherent Detection 220 \\ 8.2.2: Nonideal Coherent Detection 234 \\ 8.2.3: Noncoherent Detection 239 \\ 8.2.4: Partially Coherent Detection 242 \\ 8.2.5: Differentially Coherent Detection 243 \\ Appendix 8A: Stein's Unified Analysis of the Error Probability Performance of Certain Communication Systems 253 \\ Chapter 9: Performance of Multichannel Receivers 259 \\ 9.1: Diversity Combining 260 \\ 9.1.1: Diversity Concept 260 \\ 9.1.2: Mathematical Modeling 260 \\ 9.1.3: Brief Survey of Diversity Combining Techniques 261 \\ 9.1.4: Complexity-Performance Trade-offs 264 \\ 9.2: Maximal-Ratio Combining 265 \\ 9.2.1: Receiver Structure 265 \\ 9.2.2: PDF-Based Approach 267 \\ 9.2.3: MGF-Based Approach 268 \\ 9.2.4: Bounds and Asymptotic SER Expressions 275 \\ 9.3: Coherent Equal Gain Combining 278 \\ 9.3.1: Receiver Structure 279 \\ 9.3.2: Average Output SNR 279 \\ 9.3.3: Exact Error Rate Analysis 281 \\ 9.3.4: Approximate Error Rate Analysis 288 \\ 9.3.5: Asymptotic Error Rate Analysis 289 \\ 9.4: Noncoherent Equal-Gain Combining 290 \\ 9.4.1: DPSK, DQPSK, and BFSK: Exact and Bounds 290 \\ 9.4.2: M-ary Orthogonal FSK 304 \\ 9.5: Outage Probability Performance 311 \\ 9.5.1: MRC and Noncoherent EGC 312 \\ 9.5.2: Coherent EGC 313 \\ 9.5.3: Numerical Examples 314 \\ 9.6: Impact of Fading Correlation 316 \\ 9.6.1: Model A: Two Correlated Branches with Nonidentical Fading 320 \\ 9.6.2: Model B: D Identically Distributed Branches with Constant Correlation 323 \\ 9.6.3: Model C: D Identically Distributed Branches with Exponential Correlation 324 \\ 9.6.4: Model D: D Nonidentically Distributed Branches with Arbitrary Correlation 325 \\ 9.6.5: Numerical Examples 329 \\ 9.7: Selection Combining 333 \\ 9.7.1: MGF of Output SNR 335 \\ 9.7.2: Average Output SNR 336 \\ 9.7.3: Outage Probability 338 \\ 9.7.4: Average Probability of Error 340 \\ 9.8: Switched Diversity 348 \\ 9.8.1: Performance of SSC over Independent Identically Distributed Branches 348 \\ 9.8.2: Effect of Branch Unbalance 362 \\ 9.8.3: Effect of Branch Correlation 366 \\ 9.9: Performance in the Presence of Outdated or Imperfect Channel Estimates 370 \\ 9.9.1: Maximal-Ratio Combining 370 \\ 9.9.2: Noncoherent EGC over Rician Fast Fading 371 \\ 9.9.3: Selection Combining 373 \\ 9.9.4: Switched Diversity 374 \\ 9.9.5: Numerical Results 377 \\ 9.10: Hybrid Diversity Schemes 378 \\ 9.10.1: Generalized Selection Combining 378 \\ 9.10.2: Generalized Switched Diversity 403 \\ 9.10.3: Two-Dimensional Diversity Schemes 408 \\ Appendix 9A: Alternative Forms of the Bit Error Probability for a Decision Statistic that is a Quadratic Form of Complex Gaussian Random Variables 421 \\ Appendix 9B: Simple Numerical Techniques for thee Inversion of the Laplace Transform of Cumulative Distribution Functions 427 \\ 9B.1: Euler Summation-Based Technique 427 \\ 9B.2: Gauss-Chebyshev Quadrature-Based Technique 428 \\ Appendix 9C: Proof of Theorem 1 430 \\ Appendix 9D: Direct Proof of Eq. (9.331) 431 \\ Appendix 9E: Special Definite Integrals 432 \\ Part 4: Application in Practical Communication Systems \\ Chapter 10: Optimum Combining: A Diversity Technique for Communication Over Fading Channels in the Presence of Interference 437 \\ 10.1: Performance of Optimum Combining Receivers 438 \\ 10.1.1: Single Interferer, Independent Identically Distributed Fading 438 \\ 10.1.2: Multiple Interferers, Independent Identically Distributed Fading 454 \\ 10.1.3: Comparison with Results for MRC in the Presence of Interference 466 \\ Chapter 11: Direct-Sequence Code-Division Multiple Access 473 \\ 11.1: Single-Carrier DS-CDMA Systems 474 \\ 11.1.1: System and Channel Models 474 \\ 11.1.2: Performance Analysis 477 \\ 11.2: Multicarrier DS-CDMA Systems 479 \\ 11.2.1: System and Channel Models 480 \\ 11.2.2: Performance Analysis 483 \\ 11.2.3: Numerical Examples 489 \\ Part 5: Further Extensions \\ Chapter 12: Coded Communication Over Fading Channels 497 \\ 12.1: Coherent Detection 499 \\ 12.1.1: System Model 499 \\ 12.1.2: Evaluation of Pairwise Error Probability 502 \\ 12.1.3: Transfer Function Bound on Average Bit Error Probability 510 \\ 12.1.4: Alternative Formulation of the Transfer Function Bound 513 \\ 12.2: Differentially Coherent Detection 520 \\ 12.2.1: System Model 520 \\ 12.2.2: Performance Evaluation 522 \\ 12.3: Numerical Results: Comparison of the True Upper Bounds and Union-Chernoff Bounds 526 \\ Appendix 12A: Evaluation of a Moment Generating Function Associated with Differential Detection of M-PSK Sequences 532", } @InProceedings{Takahashi:2000:IMP, author = "D. Takahashi", booktitle = "Proceedings of the 2000 International Workshops on Parallel Processing", title = "Implementation of multiple-precision parallel division and square root on distributed-memory parallel computers", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "229--235", year = "2000", CODEN = "????", ISSN = "????", bibdate = "Sat Jul 16 15:28:14 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, summary = "We present efficient parallel algorithms for multiple-precision division and square root operation of more than several million decimal digits on distributed-memory parallel computers. It is well known that multiple-precision division and square \ldots{}", } @InProceedings{Tchoumatchenko:2000:FBS, author = "V. Tchoumatchenko and T. Vassileva and P. Gurov", booktitle = "{Proceedings of the 22nd EUROMICRO Conference EUROMICRO 96. 'Beyond 2000: Hardware and Software Design Strategies'}", title = "A {FPGA} based square-root coprocessor", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "520--525", year = "2000", CODEN = "????", ISSN = "????", bibdate = "Sat Jul 16 15:28:14 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, summary = "We present an FPGA implementation of a non-restoring integer square-root algorithm, that uses estimates for result-digit selection and radix-$2$ redundant addition in recurrence. On-the-fly conversion of the result-digit and signed-digit adder/ \ldots{}", } @Article{Temme:2000:NAA, author = "Nico M. Temme", title = "Numerical and asymptotic aspects of parabolic cylinder functions", journal = j-J-COMPUT-APPL-MATH, volume = "121", number = "1--2", pages = "221--246", day = "1", month = sep, year = "2000", CODEN = "JCAMDI", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Sat Feb 25 12:43:36 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2000.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0377042700003472", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @InProceedings{Tommiska:2000:AEI, author = "M. T. Tommiska", booktitle = "Proceedings of the 2000 Third {IEEE} International Caracas Conference on Devices, Circuits and Systems, 15--17 March 2000", title = "Area-efficient implementation of a fast square root algorithm", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "S18/1--S18/4", year = "2000", CODEN = "????", ISSN = "????", bibdate = "Sat Jul 16 15:28:14 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, summary = "An area-efficient implementation of a fast-converging square-root algorithm is presented. The design of special arithmetic operations differs in many ways from the traditional tasks that digital designers are used to, and the role of \ldots{}", } @Article{Wachspress:2000:EEF, author = "E. L. Wachspress", title = "Evaluating elliptic functions and their inverses", journal = j-COMPUT-MATH-APPL, volume = "39", number = "3--4", pages = "131--136", month = feb, year = "2000", CODEN = "CMAPDK", DOI = "https://doi.org/10.1016/S0898-1221(99)00339-9", ISSN = "0898-1221 (print), 1873-7668 (electronic)", ISSN-L = "0898-1221", bibdate = "Wed Mar 1 21:49:06 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/computmathappl2000.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0898122199003399", acknowledgement = ack-nhfb, fjournal = "Computers and Mathematics with Applications", journal-URL = "http://www.sciencedirect.com/science/journal/08981221", keywords = "arithmetic-geometric mean (AGM)", } @Book{Wall:2000:ATC, author = "H. S. Wall", title = "Analytic Theory of Continued Fractions", publisher = pub-AMS, address = pub-AMS:adr, pages = "xiii + 433", year = "2000", ISBN = "0-8218-2106-7", ISBN-13 = "978-0-8218-2106-0", LCCN = "QA295 .W28 2000", bibdate = "Thu Apr 3 20:24:06 MDT 2008", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcw.bib; z3950.loc.gov:7090/Voyager", note = "This is a reprint of the definitive, and widely cited, treatise first published in 1948.", URL = "https://archive.org/details/dli.ernet.16804/", acknowledgement = ack-nhfb, remark = "Originally published: New York: D. Van Nostrand Co., 1948.", shorttableofcontents = "Preface \\ Introduction \\ Part I: Convergence Theory \\ I: The Continued fraction as a Product of Linear Fractional Transformations \\ II: Convergence Theorems \\ III: Convergence of Continued Fractions Whose Partial Denominators Are Equal to Unity \\ IV: Introduction to the Theory of Positive Definite Continued Fractions \\ V: Some General Convergence Theorems \\ VI: Stieltjes Type Continued Fractions \\ VII: Extensions of the Parabola Theorem \\ VIII: The Value Region Problem \\ Part II: Function Theory \\ IX: J-Fraction Expansions for Rational Functions \\ X: Theory of Equations \\ XI: J-Fraction Expansions for Power Series \\ XII: Matrix Theory of Continued Fractions \\ XIII: Continued Fractions and Definite Integrals \\ XIV: The Moment Problem for a Finite Interval \\ XV: Bounded Analytic Functions \\ XVI: Hausdorff Summability \\ XVII: The Moment Problem for an Infinite Interval \\ XVIII: The Continued Fraction of Gauss \\ XIX: Stieltjes Summability \\ XX: The Pad{\'e} Table \\ Bibliography \\ Index", subject = "Continued fractions", tableofcontents = "Preface / vi \\ Introduction / 1 \\ Part I: Convergence Theory \\ \\ I: The Continued fraction as a Product of Linear Fractional Transformations \\ 1. Definitions and Formulas / 13 \\ 2. Continued Fractions and Series / 17 \\ 3. Equivalence Transformations / 19 \\ 4. Even and Odd Parts of a Continued Fraction / 20 \\ \\ II: Convergence Theorems \\ 5. Some General Remarks on the Convergence Problem / 25 \\ 6. Necessary Conditions for Convergence / 27 \\ 7. A Sufficient Condition for Convergence / 33 \\ 8. Convergence of Periodic Continued Fractions / 35 \\ \\ III: Convergence of Continued Fractions Whose Partial Denominators Are Equal to Unity \\ 9. The First Interpretation of the Fundamental Inequalities / 40 \\ 10. Worpitzky's Theorem / 42 \\ 11. Convergence of Continued Fractions Whose Partial Quotients Are of the Form $(1 - g_{p - 1}) g_p x_p / 1$ / 45 \\ 12. A Convergence Theorem of von Koch / 50 \\ 13. Second Interpretation of the Fundamental Inequalities / 52 \\ 14. The Parabola Theorem / 56 \\ 15. ``Convergence Neighborhoods'' of a Point (1) / 62 \\ \\ IV: Introduction to the Theory of Positive Definite Continued Fractions \\ 16. Definition of a Positive Definite Continued Fraction / 64 \\ 17. The Nest of Circles / 170 \\ 18. Positive Definite Continued Fractions and the Parabola Theorem / 175 \\ 19. Chain Sequences / 19 \\ 20. Quadratic Forms and Chain Sequences / 86 \\ \\ V: Some General Convergence Theorems \\ 21. Schwarz's Inequality / 95 \\ 22. The Theorem of Invariability / 96 \\ 23. The Indeterminate Case / 99 \\ 24. Convergence Continuation Theorem / 104 \\ 25. The Determinate Case / 109 \\ 26. Bounded J-fractions / 110 \\ 27. Real J-fractions / 114 \\ \\ VI: Stieltjes Type Continued Fractions \\ 28. Convergence and Divergence of the Continued Fraction of Stieltjes / 118 \\ 29. The Condition (H) / 122 \\ 30. Three Convergence Theorems / 131 \\ \\ VII: Extensions of the Parabola Theorem \\ 31. A Family of Parabolic Domains / 135 \\ 32. ``Convergence Neighborhoods'' of a Point (2) / 137 \\ 33. A Theorem of Van Vleck / 138 \\ 34. The Cardioid Theorem / 140 \\ 35. An Extension of a Theorem of Sz{\'a}sz / 143 \\ \\ VIII: The Value Region Problem \\ 36. A Sufficient Condition / 147 \\ 37. The Two-Circle Theorem / 148 \\ 38. Circular Element Regions with Centers at the Origin / 150 \\ 39. A Family of Parabolic Element Regions / 152 \\ \\ Part II: Function Theory \\ \\ IX: J-Fraction Expansions for Rational Functions \\ 40. The Expansion Algorithm / 161 \\ 41. Conditions Involving Determinants / 164 \\ 42. Relationship Between the J-fraction and the Power Series for $f_1 / f_0$ / 166 \\ 43. Rational Fractions with Simple Poles and Positive Residues / 167 \\ 44, Expansion of Rational Functions into Stieltjes Type Continued Fractions / 170 \\ \\ X: Theory of Equations \\ 45. The Test-Fraction / 174 \\ 46. Polygonal Bounds for the Roots of a Polynomial / 176 \\ 47. Polynomials Whose Roots Are in a Given Half-Plane / 178 \\ 48. Determination of the Number of Roots of P(z) in Each of the Half Planes R(z) 20 / 182 \\ 49. Computation of the Roots of Polynomials / 185 \\ \\ XI: J-Fraction Expansions for Power Series \\ 50. Polynomials Orthogonal Relative to a Sequence / 192 \\ 51. Algorithm for Expanding a Power Series into a J-fraction / 196 \\ 52. Stieltjes Type Continued Fraction Expansions for Power Series / 200 \\ 53. Stieltjes' Expansion Theorem / 202 \\ 54. Convergence Questions 208 \\ \\ XII: Matrix Theory of Continued Fractions \\ 55. Linear Forms / 214 \\ 56. Bilinear Forms / 216 \\ 57. Bounded Matrices / 218 \\ 58. Bounded Reciprocals of Bounded Matrices / 223 \\ 59. The Bounded Reciprocal of a Bounded J-matrix / 226 \\ 60. Reciprocals of an Arbitrary J-matrix / 228 \\ 61. Reciprocals of the J-matrix Associated with a Positive Definite J-fraction / 230 \\ 62. Estimates for the Equivalent Functions / 235 \\ XIII: Continued Fractions and Definite Integrals \\ 63. The Stieltjes Integral / 239 \\ 64. Sequences of Stieltjes Integrals / 245 \\ 65. The Stieltjes Inversion Formula / 247 \\ 66. Representation of an Equivalent Function of a Positive Definite J-fraction as a Stieltjes Transform / 250 \\ 67. Proper Equivalent Functions / 254 \\ \\ XIV: The Moment Problem for a Finite Interval \\ 68. Formulation of the Problem / 258 \\ 69. Solution of the Moment Problem by Means of S-fractions / 260 \\ 70. Some Geometry / 263 \\ 71. Totally Monotone Sequences / 267 \\ 72. Composition of Moment Sequences / 269 \\ \\ XV: Bounded Analytic Functions \\ 73. Integral Formulas for Bounded Analytic Functions / 275 \\ 74. Continued Fraction Expansions for Real Analytic Functions / 278 \\ 75. Continued Fraction Expansions for $1/G(z)$ and for $G[-z / (1 + z)]$ in Terms of the Expansions for $G(z)$ / 280 \\ 76. Condition for $G(z)/\sqrt{1 + z}$ to Be Bounded in the Unit Circle / 283 \\ 77. Analytic Functions Bounded in the Unit Circle / 285 \\ 78. Continued Fraction Expansions for Arbitrary Functions Which Are Analytic and Have Positive Real Parts in $\Ext(-1, -\infty)$ / 288 \\ \\ XVI: Hausdorff Summability \\ 79. Hausdorff Matrices / 302 \\ 80. A Theorem on $(A, d_n)$-Transformations / 304 \\ 81. Hausdorff Means / 306 \\ 82. Examples of Hausdorff Means / 309 \\ 83. The Hausdorff Inclusion Problem / 310 \\ \\ XVII: The Moment Problem for an Infinite Interval \\ 84. Asymptotic Expressions for J-fractions / 316 \\ 85. A Theorem of Hamburger / 321 \\ 86. The Moment Problem for the Interval $(-\infty, +\infty)$ / 325 \\ 87. The Stieltjes Moment Problem / 327 \\ 88. A Theorem of Carleman / 330 \\ \\ XVIII: The Continued Fraction of Gauss \\ 89. General Properties / 335 \\ 90. Elementary Functions / 342 \\ 91. Certain Meromorphic Functions / 347 \\ 92. A Class of Divergent Series / 349 \\ \\ XIX: Stieltjes Summability \\ 93. Definition and Illustrative Examples / 362 \\ 94. List of Expansion Formulas / 369 \\ \\ XX: The Pad{\'e} Table \\ 95. Definitions / 37 \\ 96. The Normal Pad{\'e} Table / 379 \\ 97. The Pad{\'e} Table for the Series of Stieltjes / 389 \\ 98. General Theorems on the Pad{\'e} Table / 393 \\ 99. C-fractions / 399 \\ 100. Regular C-fractions and Power Series / 405 \\ 101. $\alpha$-regular C-fractions / 409 \\ 102. Concluding Remarks on the Pad{\'e} Table / 410 \\ \\ Bibliography / 417 \\ Index / 427", xxauthor = "H. S. (Hubert Stanley) Wall", } @TechReport{Zimmermann:2000:PGF, author = "Paul Zimmermann", title = "A proof of {GMP} fast division and square root implementations", type = "Technical report", institution = inst-LORIA-INRIA-LORRAINE, address = inst-LORIA-INRIA-LORRAINE:adr, pages = "14", month = sep, year = "2000", bibdate = "Sun Sep 10 08:48:46 2006", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "http://www.loria.fr/~zimmerma/papers/proof-div-sqrt.ps.gz", abstract = "This short note gives a detailed correctness proof of fast (i.e., subquadratic) versions of the GNU MP {\tt mpn\_bz\_divrem\_n} and {\tt mpn\_sqrtrem} functions, together with complete GMP code. The {\tt mpn\_bz\_divrem\_n} function divides (with remainder) a number of $ 2 n $ limbs by a divisor of $n$ limbs in $ 2 K(n) $, where $ K(n) $ is the time spent in a $ (n \times n) $ multiplication, using the Moenck--Borodin--Jebelean--Burnikel--Ziegler algorithm. The {\tt mpn\_sqrtrem} computes the square root and the remainder of a number of $ 2 n $ limbs (square root and remainder have about $n$ limbs each) in time $ 3 K(n) / 2 $; it uses Karatsuba Square Root.", acknowledgement = ack-nhfb, } @Book{Arfken:2001:MMP, author = "George B. (George Brown) Arfken and Hans-Jurgen Weber", title = "Mathematical methods for physicists", publisher = "Harcourt/Academic Press", address = "San Diego, CA, USA", edition = "Fifth", pages = "xiv + 1112", year = "2001", ISBN = "0-12-059825-6, 0-12-059826-4", ISBN-13 = "978-0-12-059825-0, 978-0-12-059826-7", LCCN = "QA37.3 .A74 2001", MRclass = "00A06, 15-01, 26-01, 30-01, 34-01, 35-01, 65-01", bibdate = "Wed Mar 15 06:50:49 MDT 2017", bibsource = "fsz3950.oclc.org:210/WorldCat; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://catalog.hathitrust.org/api/volumes/oclc/45705658.html", acknowledgement = ack-nhfb, author-dates = "1922--", subject = "Mathematics; Mathematical physics; Matem{\'a}ticas; F{\'i}sica matem{\'a}tica; Mathematical physics; Mathematics; Wiskundige methoden; Natuurkunde; Matem{\'a}tica; F{\'i}sica; Math{\'e}matiques; Physique math{\'e}matique", tableofcontents = "1: Vector Analysis / 1 \\ 1.2: Rotation of the Coordinate Axes / 8 \\ 1.3: Scalar or Dot Product / 13 \\ 1.4: Vector or Cross Product / 19 \\ 1.5: Triple Scalar Product, Triple Vector Product / 27 \\ 1.6: Gradient, [down triangle, open] / 35 \\ 1.7: Divergence, [down triangle, open] / 40 \\ 1.8: Curl, [down triangle, open] x / 44 \\ 1.9: Successive Applications of [down triangle, open] / 51 \\ 1.10: Vector Integration / 55 \\ 1.11: Gauss's Theorem / 61 \\ 1.12: Stokes's Theorem / 65 \\ 1.13: Potential Theory / 69 \\ 1.14: Gauss's Law, Poisson's Equation / 80 \\ 1.15: Dirac Delta Function / 84 \\ 1.16: Helmholtz's Theorem / 96 \\ 2: Curved Coordinates, Tensors / 103 \\ 2.1: Orthogonal Coordinates / 103 \\ 2.2: Differential Vector Operators / 108 \\ 2.3: Special Coordinate Systems: Introduction / 113 \\ 2.4: Circular Cylindrical Coordinates / 114 \\ 2.5: Spherical Polar Coordinates / 121 \\ 2.6: Tensor Analysis / 131 \\ 2.7: Contraction, Direct Product / 137 \\ 2.8: Quotient Rule / 139 \\ 2.9: Pseudotensors, Dual Tensors / 141 \\ 2.10: Non-Cartesian Tensors / 150 \\ 2.11: Tensor Derivative Operators / 160 \\ 3: Determinants and Matrices / 165 \\ 3.1: Determinants / 165 \\ 3.2: Matrices / 174 \\ 3.3: Orthogonal Matrices / 192 \\ 3.4: Hermitian Matrices, Unitary Matrices / 206 \\ 3.5: Diagonalization of Matrices / 213 \\ 3.6: Normal Matrices / 227 \\ 4: Group Theory / 237 \\ 4.1: Introduction to Group Theory / 237 \\ 4.2: Generators of Continuous Groups / 242 \\ 4.3: Orbital Angular Momentum / 258 \\ 4.4: Angular Momentum Coupling / 263 \\ 4.5: Homogeneous Lorentz Group / 275 \\ 4.6: Lorentz Covariance of Maxwell's Equations / 278 \\ 4.7: Discrete Groups / 286 \\ 5: Infinite Series / 303 \\ 5.2: Convergence Tests / 306 \\ 5.3: Alternating Series / 322 \\ 5.4: Algebra of Series / 325 \\ 5.5: Series of Functions / 329 \\ 5.6: Taylor's Expansion / 334 \\ 5.7: Power Series / 346 \\ 5.8: Elliptic Integrals / 354 \\ 5.9: Bernoulli Numbers, Euler--Maclaurin Formula / 360 \\ 5.10: Asymptotic Series / 373 \\ 5.11: Infinite Products / 381 \\ 6: Functions of a Complex Variable I / 389 \\ 6.1: Complex Algebra / 390 \\ 6.2: Cauchy--Riemann Conditions / 399 \\ 6.3: Cauchy's Integral Theorem / 404 \\ 6.4: Cauchy's Integral Formula / 411 \\ 6.5: Laurent Expansion / 416 \\ 6.6: Mapping / 425 \\ 6.7: Conformal Mapping / 434 \\ 7: Functions of a Complex Variable II / 439 \\ 7.1: Singularities / 439 \\ 7.2: Calculus of Residues / 444 \\ 7.3: Dispersion Relations / 469 \\ 7.4: Method of Steepest Descents / 477 \\ 8: Differential Equations / 487 \\ 8.1: Partial Differential Equations / 487 \\ 8.2: First-Order Differential Equations / 496 \\ 8.3: Separation of Variables / 506 \\ 8.4: Singular Points / 516 \\ 8.5: Series Solutions--Frobenius's Method / 518 \\ 8.6: A Second Solution / 533 \\ 8.7: Nonhomogeneous Equation--Green's Function / 548 \\ 8.8: Numerical Solutions / 567 \\ 9: Sturm--Liouville Theory / 575 \\ 9.1: Self-Adjoint ODEs / 575 \\ 9.2: Hermitian Operators / 588 \\ 9.3: Gram--Schmidt Orthogonalization / 596 \\ 9.4: Completeness of Eigenfunctions / 604 \\ 9.5: Green's Function--Eigenfunction Expansion / 616 \\ 10: Gamma-Factorial Function / 631 \\ 10.1: Definitions, Simple Properties / 631 \\ 10.2: Digamma and Polygamma Functions / 643 \\ 10.3: Stirling's Series / 649 \\ 10.4: Beta Function / 654 \\ 10.5: Incomplete Gamma Function / 660 \\ 11: Bessel Functions / 669 \\ 11.1: Bessel Functions of the First Kind J[subscript v](x) / 669 \\ 11.2: Orthogonality / 688 \\ 11.3: Neumann Functions, Bessel Functions of the Second Kind / 694 \\ 11.4: Hankel Functions / 702 \\ 11.5: Modified Bessel Functions I[subscript v](x) and K[subscript v](x) / 709 \\ 11.6: Asymptotic Expansions / 716 \\ 11.7: Spherical Bessel Functions / 722 \\ 12: Legendre Functions / 739 \\ 12.1: Generating Function / 739 \\ 12.2: Recurrence Relations / 748 \\ 12.3: Orthogonality / 755 \\ 12.4: Alternate Definitions / 767 \\ 12.5: Associated Legendre Functions / 771 \\ 12.6G: Spherical Harmonics / 786 \\ 12.7: Orbital Angular Momentum Operators / 792 \\ 12.8: Addition Theorem for Spherical Harmonics / 796 \\ 12.9: Integrals of Three Ys / 802 \\ 12.10: Legendre Functions of the Second Kind / 806 \\ 12.11: Vector Spherical Harmonics / 813 \\ 13: Special Functions / 817 \\ 13.1: Hermite Functions / 817 \\ 13.2: Laguerre Functions / 828 \\ 13.3: Chebyshev Polynomials / 839 \\ 13.4: Hypergeometric Functions / 850 \\ 13.5: Confluent Hypergeometric Functions / 855 \\ 14: Fourier Series / 863 \\ 14.1: General Properties / 863 \\ 14.2: Advantages, Uses of Fouries Series / 870 \\ 14.3: Applications of Fourier Series / 874 \\ 14.4: Properties of Fourier Series / 886 \\ 14.5: Gibbs Phenomenon / 893 \\ 14.6: Discrete Fourier Transform / 898 \\ 15: Integral Transforms / 905 \\ 15.1: Integral Transforms / 905 \\ 15.2: Development of the Fourier Integral / 909 \\ 15.3: Fourier Transforms--Inversion Theorem / 911 \\ 15.4: Fourier Transform of Derivatives / 920 \\ 15.5: Convolution Theorem / 924 \\ 15.6: Momentum Representation / 928 \\ 15.7: Transfer Functions / 935 \\ 15.8: Laplace Transforms / 938 \\ 15.9: Laplace Transform of Derivatives / 946 \\ 15.10: Other Properties / 953 \\ 15.11: Convolution or Faltungs Theorem / 965 \\ 15.12: Inverse Laplace Transform / 969 \\ 16: Integral Equations / 983 \\ 16.2: Integral Transforms, Generating Functions / 991 \\ 16.3: Neumann Series, Separable Kernels / 997 \\ 16.4: Hilbert--Schmidt Theory / 1009 \\ 17: Calculus of Variations / 1017 \\ 17.1: A Dependent and an Independent Variable / 1018 \\ 17.2: Applications of the Euler Equation / 1023 \\ 17.3: Several Dependent Variables / 1031 \\ 17.4: Several Independent Variables / 1036 \\ 17.5: Several Dependent and Independent Variables / 1038 \\ 17.6: Lagrangian Multipliers / 1039 \\ 17.7: Variation With Constraints / 1045 \\ 17.8: Rayleigh--Ritz Variational Technique / 1052 \\ 18: Nonlinear Methods and Chaos / 1059 \\ 18.2: Logistic Map / 1060 \\ 18.3: Sensitivity to Initial Conditions / 1064 \\ 18.4: Nonlinear Differential Equations / 1068 \\ Appendix 1: Real Zeros of a Function / 1085 \\ Appendix 2: Gaussian Quadrature / 1089", } @Book{Askey:2001:SFG, editor = "R. A. Askey and Tom H. Koornwinder and Walter J. Schempp", title = "Special Functions: Group Theoretical Aspects and Applications", publisher = pub-SV, address = pub-SV:adr, pages = "xxxiv + 311", year = "2001", ISBN = "90-277-1822-9", ISBN-13 = "978-90-277-1822-8", LCCN = "QA1 M428 v. 18 c.2", bibdate = "Sat Oct 30 17:58:21 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", series = "Mathematics and Its Applications", acknowledgement = ack-nhfb, tableofcontents = "Editor's Preface / M. Hazewinkel / vii \\ Preface / R. Askey / xi \\ Jacobi functions and analysis on noncompact semisimple Lie groups / Tom H. Koornwinder / 1 \\ Orthogonal polynomials and Chevalley groups / Dennis Stanton /87 \\ Special functions and group theory in theoretical physics / L. C. Biedenharn, R. A. Gustafson, M. A. Lohe, J. D. Louck, S. C. Milne / 129 \\ Lattice gauge theory, orthogonal polynomials and q-Hypergeometric functions / George E. Andrews, Enrico Onofri / 163 \\ The Laguerre calculus on the Heisenberg group / R. W. Beals, P. C. Greiner, J. Vauthier / 189 \\ Radar ambiguity functions, nilpotent harmonic analysis, and holomorphic theta series / Walter Schempp / 217 \\ A factorization theorem for the Fourier transform of a separable locally compact Abelian group / Louis Auslander / 261 \\ Band and time limiting, recursion relations and some nonlinear evolution equations / F. Alberto Gr{\"u}nbaum / 271 \\ Harmonics and combinatorics / J. J. Seidel / 287 \\ Subject Index / / 305", } @Article{Bashagha:2001:NRS, author = "A. E. Bashagha", title = "Novel radix-$2$ $k$ square root module", journal = "Circuits, Devices and Systems, IEE Proceedings [see also IEE Proceedings G-Circuits, Devices and Systems]", volume = "148", number = "4", pages = "190--196", month = aug, year = "2001", CODEN = "????", ISSN = "????", bibdate = "Sat Jul 16 15:28:13 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, summary = "The conventional two's complement radix-$2$ $k$ square root algorithm requires a set of $2^k$ full precision comparisons to generate all the $2^k$ possible values of the partial remainder. The correct remainder is the minimum \ldots{}", } @Article{Berg:2001:CMF, author = "Christian Berg and Henrik L. Pedersen", title = "A completely monotone function related to the Gamma function", journal = j-J-COMPUT-APPL-MATH, volume = "133", number = "1--2", pages = "219--230", day = "1", month = aug, year = "2001", CODEN = "JCAMDI", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Sat Feb 25 12:45:19 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2000.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0377042700006440", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @Article{Berry:2001:WSF, author = "Michael Berry", title = "Why are special functions special?", journal = j-PHYS-TODAY, volume = "54", number = "4", pages = "11--12", month = apr, year = "2001", CODEN = "PHTOAD", DOI = "https://doi.org/10.1063/1.1372098", ISSN = "0031-9228 (print), 1945-0699 (electronic)", ISSN-L = "0031-9228", bibdate = "Sat Feb 19 13:23:33 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.physicstoday.org/resource/1/phtoad/v54/i4/p11_s1", acknowledgement = ack-nhfb, fjournal = "Physics Today", journal-URL = "http://www.physicstoday.org/", } @Article{Boisvert:2001:MM, author = "Ronald F. Boisvert and M. J. Donahue and Daniel W. Lozier and R. McMichael and B. W. Rust", title = "Mathematics and Measurement", journal = "NIST Journal of Research", volume = "106", number = "1", pages = "293--313", month = jan # "\slash " # feb, year = "2001", bibdate = "Fri Jul 09 06:26:11 2004", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, } @Book{Boyd:2001:CFS, author = "John Philip Boyd", title = "{Chebyshev} and {Fourier} spectral methods", publisher = pub-DOVER, address = pub-DOVER:adr, edition = "Second (revised).", pages = "1375", year = "2001", ISBN = "0-486-41183-4 (paperback), 0-486-14192-6 (e-book)", ISBN-13 = "978-0-486-41183-5 (paperback), 978-0-486-14192-3 (e-book)", LCCN = "QA404.5 .B69 2001", bibdate = "Sat Feb 17 14:05:46 MST 2024", bibsource = "fsz3950.oclc.org:210/WorldCat; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", series = "Dover Books on Mathematics", URL = "http://www.freading.com/ebooks/details/r:download/ZnJlYWQ6OTc4MDQ4NjE0MTkyMzpl", abstract = "Completely revised text focuses on use of spectral methods to solve boundary value, eigenvalue, and time-dependent problems, but also covers Hermite, Laguerre, rational Chebyshev, sinc, and spherical harmonic functions, as well as cardinal functions, linear eigenvalue problems, matrix-solving methods, coordinate transformations, spherical and cylindrical geometry, and more. Includes 7 appendices and over 160 text figures.", acknowledgement = ack-nhfb, author-dates = "1951--", subject = "Chebyshev polynomials; Fourier analysis; Spectral theory (Mathematics); Polyn{\'y}omes de Tchebychev; Analyse de Fourier; Spectre (Math{\'y}ematiques); MATHEMATICS; General.; Chebyshev polynomials; Fourier analysis; Spectral theory (Mathematics)", tableofcontents = "1. Introduction \\ 2. Chebyshev and Fourier Series \\ 3. Galerkin and Weighted Residual Methods \\ 4. Interpolation, Collocation and All That \\ 5. Cardinal Functions \\ 6. Pseudospectral Methods for BVPs \\ 7. Linear Eigenvalue Problems \\ 8. Symmetry and Parity \\ 9. Explicit Time-Integration Methods \\ 10. Partial Summation, the FFT and MMT \\ 11. Aliasing, Spectral Blocking, and Blow-Up \\ 12. Implicit Schemes and the Slow Manifold \\ 13. Splitting and its Cousins \\ 14. Semi-Lagrangian Advection \\ 15. Matrix-Solving Methods \\ 16. Coordinate Transformations \\ 17. Methods for Unbounded Intervals \\ 18. Spherical and Cylindrical Geometry \\ 19. Special Tricks \\ 20. Symbolic Calculations \\ 21. The Tau Method \\ 22. Domain Decomposition Methods \\ 23. Books and Reviews", } @InProceedings{Burgess:2001:DIR, author = "N. Burgess and C. Hinds", booktitle = "Conference Record of the Thirty-Fifth Asilomar Conference on Signals, Systems and Computers, 2001", title = "Design issues in radix-$4$ {SRT} square root {\&} divide unit", volume = "2", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "1646--1650", year = "2001", CODEN = "????", ISSN = "????", bibdate = "Sat Jul 16 15:28:14 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, summary = "This paper introduces a number of design issues not covered in the open literature that arose during the design of a radix-$4$ SRT divide/square root unit for a vector processing chip. These include compression of the partial remainder's m.s.b.'s, \ldots{}", } @Book{Bustoz:2001:SFC, editor = "Joaquin Bustoz and Mourad Ismail and S. K. (Sergei Konstantinovich) Suslov", title = "Special functions 2000: current perspective and future directions", volume = "30", publisher = pub-KLUWER, address = pub-KLUWER:adr, pages = "xi + 520", year = "2001", ISBN = "0-7923-7119-4, 0-7923-7120-8", ISBN-13 = "978-0-7923-7119-9, 978-0-7923-7120-5", LCCN = "QA351 .S694 2001", bibdate = "Sat Oct 30 17:31:39 MDT 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; prodorbis.library.yale.edu:7090/voyager", series = "NATO science series. Series II, Mathematics, physics, and chemistry", acknowledgement = ack-nhfb, subject = "functions, special; congresses", tableofcontents = "Preface \\ Foreword \\ Bailey's transform, lemma, chains and tree / George E. Andrews 1 \\ Riemann--Hilbert problems for multiple orthogonal polynomials / Walter Van Assche, Jeffrey S. Geronimo, Arno B. J. Kuijlaars 23 \\ Flowers which we cannot yet see growing in Ramanujan's garden of hypergeometric series, elliptic functions and $q$'s / Bruce C. Berndt 61 \\ Orthogonal rational functions and continued fractions [et al.] 87 \\ Orthogonal polynomials and reflection groups / Charles F. Dunkl 111 \\ The bispectral problem: an overview / F. Alberto Grunbaum 129 \\ The Bochner--Krall problem: some new perspectives / Luc Haine 141 \\ Lectures on $q$-orthogonal polynomials / Mourad E. H. Ismail 179 \\ The Askey--Wilson function transform scheme / Erik Koelink, Jasper V. Stokman 221 \\ Arithmetic of the partition function / Ken Ono 243 \\ The associated classical orthogonal polynomials / Mizan Rahman 255 \\ Special functions defined by analytic difference equations / S. N. M. Ruijsenaars 281 \\ The factorization method, self-similar potentials and quantum algebras / V. P. Spiridonov 335 \\ Generalized eigenvalue problem and a new family of rational functions biorthogonal on elliptic grids / V. P. Spiridonov, A. S. Zhedanov 365 \\ Orthogonal polynomials and combinatorics / Dennis Stanton 389 \\ Basic exponential functions on a $q$-quadratic grid / Sergei K. Suslov 411 \\ Projection operator method for quantum groups / V. N. Tolstoy 457 \\ Uniform asymptotic expansions / R. Wong 489 \\ Exponential asymptotics / R. Wong 505 \\ Index 519", } @InCollection{Corless:2001:RAE, author = "Robert M. Corless and James H. Davenport and David J. Jeffrey and Gurjeet Litt and Stephen M. Watt", booktitle = "Artificial intelligence and symbolic computation (Madrid, 2000)", title = "Reasoning about the elementary functions of complex analysis", volume = "1930", publisher = pub-SV, address = pub-SV:adr, pages = "115--126", year = "2001", MRclass = "68W30 (30C35)", MRnumber = "MR1882755 (2002m:68126)", bibdate = "Wed Apr 13 06:46:35 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", series = "Lecture Notes in Comput. Sci.", acknowledgement = ack-nhfb, } @Book{Dunkl:2001:OPS, author = "Charles F. Dunkl and Yuan Xu", title = "Orthogonal Polynomials of Several Variables", publisher = pub-CAMBRIDGE, address = pub-CAMBRIDGE:adr, pages = "xv + 390", year = "2001", DOI = "https://doi.org/10.1017/CBO9780511565717", ISBN = "0-511-56571-2 (e-book), 0-521-80043-9 (hardcover), 1-107-09582-4", ISBN-13 = "978-0-511-56571-7 (e-book), 978-0-521-80043-3 (hardcover), 978-1-107-09582-3", LCCN = "QA404.5 .D86 2001", bibdate = "Sat Nov 11 07:30:57 2023", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", abstract = "This is the first modern book on orthogonal polynomials of several variables, which are valuable tools used in multivariate analysis, including approximations and numerical integration. The book presents the theory in elegant form and with modern concepts and notation. It introduces the general theory and emphasizes the classical types of orthogonal polynomials whose weight functions are supported on standard domains such as the cube, the simplex, the sphere and the ball. It also focuses on those of Gaussian type, for which fairly explicit formulae exist. The authors' approach blends classical analysis and symmetry-group-theoretic methods.", acknowledgement = ack-nhfb, remark = "See also second edition \cite{Dunkl:2014:OPS}", tableofcontents = "1. Background \\ 2. Examples of Orthogonal Polynomials in Several Variables \\ 3. General Properties of Orthogonal Polynomials in Several Variables \\ 4. Root Systems and Coxeter groups \\ 5. Spherical Harmonics Associated with Reflection Groups \\ 6. Classical and Generalized Classical Orthogonal Polynomials \\ 7. Summability of Orthogonal Expansions \\ 8. Orthogonal Polynomials Associated with Symmetric Groups \\ 9. Orthogonal Polynomials Associated with Octahedral Groups and Applications", } @Article{Eklund:2001:CEF, author = "Neil Eklund", title = "{CORDIC}: Elementary Function Computation Using Recursive Sequences", journal = j-COLLEGE-MATH-J, volume = "32", number = "5", pages = "330--333", month = nov, year = "2001", CODEN = "????", DOI = "https://doi.org/10.1080/07468342.2001.11921899", ISSN = "0746-8342 (print), 1931-1346 (electronic)", ISSN-L = "0746-8342", bibdate = "Thu Feb 14 09:53:12 MST 2019", bibsource = "https://www.math.utah.edu/pub/tex/bib/collegemathj.bib; https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.tandfonline.com/doi/abs/10.1080/07468342.2001.11921899", acknowledgement = ack-nhfb, fjournal = "College Mathematics Journal", journal-URL = "https://maa.tandfonline.com/loi/ucmj20; https://www.jstor.org/journal/collmathj", onlinedate = "30 Jan 2018", } @Article{Elbert:2001:CZB, author = "{\'A}rp{\'a}d Elbert and Andrea Laforgia", title = "A conjecture on the zeros of {Bessel} functions", journal = j-J-COMPUT-APPL-MATH, volume = "133", number = "1--2", pages = "683--683", day = "1", month = aug, year = "2001", CODEN = "JCAMDI", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Sat Feb 25 12:45:19 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2000.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0377042700007172", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @Article{Elbert:2001:SRR, author = "{\'A}. Elbert", title = "Some recent results on the zeros of {Bessel} functions and orthogonal polynomials", journal = j-J-COMPUT-APPL-MATH, volume = "133", number = "1--2", pages = "65--83", day = "1", month = aug, year = "2001", CODEN = "JCAMDI", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Sat Feb 25 12:45:19 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2000.bib", URL = "http://www.sciencedirect.com/science/article/pii/S037704270000635X", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @Article{Giordano:2001:IMP, author = "C. Giordano and A. Laforgia", title = "Inequalities and monotonicity properties for the gamma function", journal = j-J-COMPUT-APPL-MATH, volume = "133", number = "1--2", pages = "387--396", day = "1", month = aug, year = "2001", CODEN = "JCAMDI", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Sat Feb 25 12:45:19 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2000.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0377042700006592", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @Unpublished{Godfrey:2001:NCC, author = "P. Godfrey", title = "A Note on the Computation of the Convergent {Lanczos} Complex Gamma Approximation", year = "2001", bibdate = "Mon Nov 24 21:04:40 2008", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "Unpublished Web file.", URL = "http://my.fit.edu/~gabdo/gamma.txt", acknowledgement = ack-nhfb, } @Article{Harris:2001:KFL, author = "Frank E. Harris", title = "On {Kryachko}'s formula for the leaky aquifer function", journal = j-IJQC, volume = "81", number = "5", pages = "332--334", month = "????", year = "2001", CODEN = "IJQCB2", DOI = "https://doi.org/10.1002/1097-461X(2001)81:5<332::AID-QUA1002>3.0.CO%3B2-W", ISSN = "0020-7608 (print), 1097-461X (electronic)", ISSN-L = "0020-7608", bibdate = "Wed Apr 4 11:48:33 MDT 2001", bibsource = "http://www.interscience.wiley.com/jpages/0020-7608; http://www3.interscience.wiley.com/journalfinder.html; https://www.math.utah.edu/pub/bibnet/authors/h/harris-frank-e.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/ijqc2000.bib", URL = "http://www3.interscience.wiley.com/cgi-bin/abstract/76507286/START; http://www3.interscience.wiley.com/cgi-bin/fulltext/76507286/FILE?TPL=ftx_start; http://www3.interscience.wiley.com/cgi-bin/fulltext?ID=76507286&PLACEBO=IE.pdf", acknowledgement = ack-nhfb, ajournal = "Int. J. Quantum Chem.", fjournal = "International Journal of Quantum Chemistry", journal-URL = "http://www.interscience.wiley.com/jpages/0020-7608/", } @Unpublished{Kahan:2001:PDA, author = "William Kahan", title = "Pseudo-Division Algorithms for Floating-Point Logarithms and Exponentials", pages = "8", day = "20", month = may, year = "2001", bibdate = "Sat Aug 23 06:17:04 2025", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/k/kahan-william-m.bib; https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "https://ieeemilestones.ethw.org/w/images/3/30/Wk_pseudo_division_log_exp_may01.pdf", abstract = "Among the CORDIC-like algorithms for computing elementary transcendental functions like log and exp, certain pseudo-division algorithms are peculiarly well suited to implementation in microcode or in conjunction with software-implemented floating-point arithmetic. These algorithms need tables of comparatively modest size; they are almost as fast as the fastest digit-by-digit algorithms known; and they can achieve accuracy to within a unit or two in the last sig. bit carried. Algorithms like these are used by the Intel 8087 family of numeric coprocessors. This document is for people who wish to imitate or surpass them.", acknowledgement = ack-nhfb, } @Article{Karatsuba:2001:ARE, author = "Ekatherina A. Karatsuba", title = "On the asymptotic representation of the {Euler} gamma function by {Ramanujan}", journal = j-J-COMPUT-APPL-MATH, volume = "135", number = "2", pages = "225--240", day = "15", month = oct, year = "2001", CODEN = "JCAMDI", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Sat Feb 25 12:45:20 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2000.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0377042700005860", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @InProceedings{Lang:2001:CRR, author = "Tom{\'a}s Lang and Elisardo Antelo", title = "Correctly Rounded Reciprocal Square-Root by Digit Recurrence and Radix-$4$ Implementation", crossref = "Burgess:2001:ISC", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "83--93", year = "2001", CODEN = "????", ISSN = "????", bibdate = "Sat Jul 16 15:28:14 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; OCLC Proceedings database", acknowledgement = ack-nhfb, summary = "We present a reciprocal square-root algorithm by digit recurrence and selection by a staircase function, and the radix-$4$ implementation. As similar algorithms for division and square-root, the results are obtained correctly rounded in a \ldots{}", } @Article{Lether:2001:VPA, author = "F. G. Lether", title = "Variable Precision Algorithm for the Numerical Computation of the {Fermi--Dirac} Function {$ F_j(x) $} of Order $ j = - 3 / 2 $", journal = j-J-SCI-COMPUT, volume = "16", number = "1", pages = "69--79", month = mar, year = "2001", CODEN = "JSCOEB", DOI = "https://doi.org/10.1023/A:1011150530703", ISSN = "0885-7474 (print), 1573-7691 (electronic)", ISSN-L = "0885-7474", bibdate = "Sat Dec 22 13:05:47 MST 2012", bibsource = "http://springerlink.metapress.com/openurl.asp?genre=issue&issn=0885-7474&volume=16&issue=1; https://www.math.utah.edu/pub/bibnet/authors/d/dirac-p-a-m.bib; https://www.math.utah.edu/pub/bibnet/authors/f/fermi-enrico.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jscicomput.bib", URL = "http://link.springer.com/content/pdf/10.1023/A%3A1011150530703; http://www.springerlink.com/openurl.asp?genre=article&issn=0885-7474&volume=16&issue=1&spage=69-79", acknowledgement = ack-nhfb, fjournal = "Journal of Scientific Computing", journal-URL = "http://link.springer.com/journal/10915", } @TechReport{Li:2001:LLF, author = "Ren-Cang Li and Peter Markstein and Jon P. Okada and James W. Thomas", title = "The {\tt libm} library and floating-point arithmetic for {HP-UX} on {Itanium}", type = "Technical report", institution = pub-HP, address = pub-HP:adr, pages = "??", month = apr, year = "2001", bibdate = "Fri Jun 24 20:12:09 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://h21007.www2.hp.com/dspp/ddl/ddl_Download_File_TRX/1,1249,942,00.pdf; http://h21007.www2.hp.com/dspp/tech/tech_TechDocumentDetailPage_IDX/1,1701,981,00.html", acknowledgement = ack-nhfb, } @Article{Loenko:2001:CEF, author = "M. Yu. Loenko", title = "Computation of elementary functions with guaranteed accuracy", journal = j-PROGRAMMIROVANIE, volume = "2", pages = "68--80", year = "2001", CODEN = "PROGD3", ISSN = "0132-3474, 0361-7688", MRclass = "65D15 (65G20)", MRnumber = "MR1867584", bibdate = "Wed Apr 13 06:46:35 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Rossi\u\i skaya Akademiya Nauk. Programmirovanie", } @Article{Meyer:2001:JEF, author = "Kenneth R. Meyer", title = "{Jacobi} Elliptic Functions from a Dynamical Systems Point of View", journal = j-AMER-MATH-MONTHLY, volume = "108", number = "8", pages = "729--737", month = oct, year = "2001", CODEN = "AMMYAE", ISSN = "0002-9890 (print), 1930-0972 (electronic)", ISSN-L = "0002-9890", bibdate = "Mon Jan 30 12:00:14 MST 2012", bibsource = "http://www.jstor.org/journals/00029890.html; http://www.jstor.org/stable/i346008; https://www.math.utah.edu/pub/tex/bib/amermathmonthly2000.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.jstor.org/stable/2695616", acknowledgement = ack-nhfb, fjournal = "American Mathematical Monthly", journal-URL = "https://www.jstor.org/journals/00029890.htm", } @Article{Muller:2001:CCH, author = "Keith E. Muller", title = "Computing the confluent hypergeometric function, {$ M(a, b, x) $}", journal = j-NUM-MATH, volume = "90", number = "1", pages = "179--196", month = nov, year = "2001", CODEN = "NUMMA7", DOI = "https://doi.org/10.1007/s002110100285", ISSN = "0029-599X (print), 0945-3245 (electronic)", ISSN-L = "0029-599X", bibdate = "Sun Feb 3 10:07:57 MST 2002", bibsource = "http://link.springer-ny.com/link/service/journals/00211/tocs/t1090001.htm; http://springerlink.metapress.com/openurl.asp?genre=journal&issn=0029-599X; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/nummath2000.bib", URL = "http://link.springer-ny.com/link/service/journals/00211/bibs/1090001/10900179.htm; http://link.springer-ny.com/link/service/journals/00211/papers/1090001/10900179.pdf", acknowledgement = ack-nhfb, fjournal = "Numerische Mathematik", journal-URL = "http://link.springer.com/journal/211", } @Article{Nagel:2001:EHF, author = "Bengt Nagel", title = "An expansion of the hypergeometric function in {Bessel} functions", journal = j-J-MATH-PHYS, volume = "42", number = "12", pages = "5910--5914", month = dec, year = "2001", CODEN = "JMAPAQ", ISSN = "0022-2488 (print), 1089-7658 (electronic), 1527-2427", ISSN-L = "0022-2488", bibdate = "Thu Mar 28 19:47:21 MST 2002", bibsource = "http://www.aip.org/ojs/jmp.html; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jmathphys2000.bib", acknowledgement = ack-nhfb, fjournal = "Journal of Mathematical Physics", journal-URL = "http://jmp.aip.org/", } @Article{Plagianakos:2001:LCP, author = "V. P. Plagianakos and N. K. Nousis and M. N. Vrahatis", title = "Locating and computing in parallel all the simple roots of special functions using {PVM}", journal = j-J-COMPUT-APPL-MATH, volume = "133", number = "1--2", pages = "545--554", day = "1", month = aug, year = "2001", CODEN = "JCAMDI", DOI = "https://doi.org/10.1016/S0377-0427(00)00675-0", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Sat Feb 25 12:45:19 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2000.bib; https://www.math.utah.edu/pub/tex/bib/pvm.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0377042700006750", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @Article{Plofker:2001:EIT, author = "Kim Plofker", title = "The {``Error''} in the {Indian} ``{Taylor} Series Approximation'' to the Sine", journal = j-HIST-MATH, volume = "28", number = "4", pages = "283--295", month = nov, year = "2001", CODEN = "HIMADS", DOI = "https://doi.org/10.1006/hmat.2001.2331", ISSN = "0315-0860 (print), 1090-249X (electronic)", ISSN-L = "0315-0860", bibdate = "Wed Jun 26 06:20:02 MDT 2013", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/histmath.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0315086001923316", acknowledgement = ack-nhfb, fjournal = "Historia Mathematica", journal-URL = "http://www.sciencedirect.com/science/journal/03150860", } @Article{Rabi:2001:OCA, author = "J. A. Rabi and M. J. S. de Lemos", title = "Optimization of convergence acceleration in multigrid numerical solutions of conductive-convective problems", journal = j-APPL-MATH-COMP, volume = "124", number = "2", pages = "215--226", day = "25", month = oct, year = "2001", CODEN = "AMHCBQ", ISSN = "0096-3003 (print), 1873-5649 (electronic)", ISSN-L = "0096-3003", bibdate = "Sun Nov 18 09:58:00 MST 2001", bibsource = "http://www.elsevier.com/locate/issn/00963003; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.elsevier.com/gej-ng/10/9/12/113/31/31/abstract.html", acknowledgement = ack-nhfb, fjournal = "Applied Mathematics and Computation", journal-URL = "http://www.sciencedirect.com/science/journal/00963003", keywords = "convergence acceleration", } @Article{Rahavachary:2001:LSS, author = "Saty Rahavachary", title = "Letters: Setting the {\tt sqrt()} record straight", journal = j-DDJ, volume = "26", number = "4", pages = "12--12", month = apr, year = "2001", CODEN = "DDJOEB", ISSN = "1044-789X", bibdate = "Tue Mar 13 15:22:36 MST 2001", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.ddj.com/", acknowledgement = ack-nhfb, fjournal = "Dr. Dobb's Journal of Software Tools", } @Article{Rappoport:2001:CVP, author = "J. M. Rappoport", title = "Canonical vector polynomials for the computation of complex order {Bessel} functions with the tau method", journal = j-COMPUT-MATH-APPL, volume = "41", number = "3--4", pages = "399--406", month = feb, year = "2001", CODEN = "CMAPDK", ISSN = "0898-1221 (print), 1873-7668 (electronic)", ISSN-L = "0898-1221", bibdate = "Wed Mar 1 21:49:14 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/computmathappl2000.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0898122100002820", acknowledgement = ack-nhfb, fjournal = "Computers and Mathematics with Applications", journal-URL = "http://www.sciencedirect.com/science/journal/08981221", } @Article{Rump:2001:RPS, author = "Siegfried M. Rump", title = "Rigorous and Portable Standard Functions", journal = j-BIT-NUM-MATH, volume = "41", number = "3", pages = "540--562", month = jun, year = "2001", CODEN = "BITTEL, NBITAB", DOI = "https://doi.org/10.1023/A:1021971313412", ISSN = "0006-3835 (print), 1572-9125 (electronic)", ISSN-L = "0006-3835", bibdate = "Wed Jan 4 15:06:04 MST 2006", bibsource = "http://springerlink.metapress.com/openurl.asp?genre=issue&issn=0006-3835&volume=41&issue=3; http://www.mai.liu.se/BIT/contents/bit41.html; https://www.math.utah.edu/pub/tex/bib/bit.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "http://www.springerlink.com/openurl.asp?genre=article&issn=0006-3835&volume=41&issue=3&spage=540", abstract = "Today's floating point implementations of elementary transcendental functions are usually very accurate. However, with few exceptions, the actual accuracy is not known. In the present paper we describe a rigorous, accurate, fast and portable implementation of the elementary standard functions based on some existing approximate standard functions. The scheme is outlined for IEEE 754, but not difficult to adapt to other floating point formats. A Matlab implementation is available on the net. Accuracy of the proposed algorithms can be rigorously estimated. As an example we prove that the relative accuracy of the exponential function is better than 2.07 eps in a slightly reduced argument range (eps denoting the relative rounding error unit). Otherwise, extensive computational tests suggest for all elementary functions and all suitable arguments an accuracy better than about 3 eps.", acknowledgement = ack-nhfb, journal-URL = "http://link.springer.com/journal/10543", keywords = "elementary functions; floating-point arithmetic", } @Article{Smith:2001:AFS, author = "David M. Smith", title = "{Algorithm 814}: {Fortran 90} software for floating-point multiple precision arithmetic, gamma and related functions", journal = j-TOMS, volume = "27", number = "4", pages = "377--387", month = dec, year = "2001", CODEN = "ACMSCU", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Wed Mar 13 08:49:29 MST 2002", bibsource = "http://www.acm.org/pubs/contents/journals/toms/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib", acknowledgement = ack-nhfb, fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", } @Book{Srivastava:2001:SAZ, author = "H. M. Srivastava and Choi Junesang", title = "Series Associated with the Zeta and Related Functions", publisher = pub-KLUWER, address = pub-KLUWER:adr, pages = "ix + 388", year = "2001", DOI = "https://doi.org/10.1007/978-94-015-9672-5", ISBN = "0-7923-7054-6", ISBN-13 = "978-0-7923-7054-3", LCCN = "QA351 .S74 2001", bibdate = "Wed Jun 10 16:22:26 MDT 2015", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; z3950.loc.gov:7090/Voyager", URL = "http://www.loc.gov/catdir/enhancements/fy0822/2001035764-d.html; http://www.loc.gov/catdir/enhancements/fy0822/2001035764-t.html", acknowledgement = ack-nhfb, subject = "Functions, Zeta; Series", tableofcontents = "Acknowledgements / ix \\ 1. Introduction and Preliminaries \\ 1.1. Gamma and Beta functions / 1 \\ 1.2. Polygamma functions / 13 \\ 1.3. The multiple Gamma functions / 24 \\ 1.4. The Gaussian hypergeometric function and its generalization / 44 \\ 1.5. Stirling numbers of the first and second kind / 56 \\ 1.6. Bernoulli and Euler polynomials and numbers / 59 \\ Problems / 67 \\ 2. The Zeta and Related Functions \\ 2.1. Multiple Hurwitz Zeta functions / 77 \\ 2.2. The Hurwitz (or generalized) Zeta function / 88 \\ 2.3. The Riemann Zeta function / 96 \\ 2.4. Polylogarithm functions / 106 \\ 2.5. Hurwitz--Lerch Zeta functions / 121 \\ Problems / 128 \\ 3. Series Involving Zeta Functions \\ 3.1. Historical introduction / 142 \\ 3.2. Use of the Binomial theorem / 143 \\ 3.3. Use of generating functions / 152 \\ 3.4. Use of multiple Gamma functions / 159 \\ 3.5. Use of hypergeometric identities / 250 \\ 3.6. Other methods and their applications / 260 \\ Problems / 269 \\ 4. Evaluations and Series Representations \\ 4.1. Evaluation of $\zeta(2n)$ / 275 \\ 4.2. Rapidly convergent series for $\zeta(2n + 1)$ / 280 \\ 4.3. Further series representations / 289 \\ 4.4. Computational results / 295 \\ Problems / 304 \\ 5. Determinants of the Laplacians \\ 5.1. The $n$-dimensional problem / 315 \\ 5.2. Computations using the simple and multiple Gamma functions / 318 \\ 5.3. Computations using series of Zeta functions / 325 \\ 5.4. Remarks and observations / 328 \\ Problems / 329 \\ 6. Miscellaneous Results \\ 6.1. Bernoulli and Euler polynomials at rational arguments / 335 \\ 6.2. Closed-form summation of trigonometric series / 341 \\ 6.3. Integrals associated with the use of the Euler--Maclaurin summation formula / 344 \\ Problems / 350 \\ Bibliography / 353 \\ Author Index / 379 \\ Subject Index / 383", } @InProceedings{Takagi:2001:HAC, author = "N. Takagi", title = "A Hardware Algorithm for Computing Reciprocal Square Root", crossref = "Burgess:2001:ISC", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "94--100", year = "2001", CODEN = "????", ISSN = "????", bibdate = "Sat Jul 16 15:28:14 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; OCLC Proceedings database", acknowledgement = ack-nhfb, summary = "A hardware algorithm for computing the reciprocal square root which appears frequently in multimedia and graphics applications is proposed. The reciprocal square root is computed by iteration of carry-propagation-free additions, shifts, and \ldots{}", } @Article{Thorsley:2001:AEH, author = "Michael D. Thorsley and Marita C. Chidichimo", title = "An asymptotic expansion for the hypergeometric function {$_2 F_1 (a, b; c; x)$}", journal = j-J-MATH-PHYS, volume = "42", number = "4", pages = "1921--1930", month = apr, year = "2001", CODEN = "JMAPAQ", ISSN = "0022-2488 (print), 1089-7658 (electronic), 1527-2427", ISSN-L = "0022-2488", bibdate = "Wed Apr 18 05:33:53 MDT 2001", bibsource = "http://www.aip.org/ojs/jmp.html; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jmathphys2000.bib", acknowledgement = ack-nhfb, fjournal = "Journal of Mathematical Physics", journal-URL = "http://jmp.aip.org/", } @Article{Verdonk:2001:PRIa, author = "Brigitte Verdonk and Annie Cuyt and Dennis Verschaeren", title = "A precision- and range-independent tool for testing floating-point arithmetic {I}: {Basic} operations, square root, and remainder", journal = j-TOMS, volume = "27", number = "1", pages = "92--118", month = mar, year = "2001", CODEN = "ACMSCU", DOI = "https://doi.org/10.1145/382043.382404", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Wed Feb 6 16:43:42 MST 2002", bibsource = "http://www.acm.org/pubs/contents/journals/toms/; https://www.math.utah.edu/pub/bibnet/subjects/acc-stab-num-alg-2ed.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib", URL = "http://doi.acm.org/10.1145/382043.382404; http://www.win.ua.ac.be/~cant/ieeecc754.html", abstract = "This paper introduces a precision- and range-independent tool for testing the compliance of hardware or software implementations of (multiprecision) floating-point arithmetic with the principles of the IEEE standards 754 and 854. The tool consists of a driver program, offering many options to test only specific aspects of the IEEE standards, and a large set of test vectors, encoded in a precision-independent syntax to allow the testing of basic and extended hardware formats as well as multiprecision floating-point implementations. The suite of test vectors stems on one hand from the integration and fully precision- and range-independent generalization of existing hardware test sets, and on the other hand from the systematic testing of exact rounding for all combinations of round and sticky bits that can occur. The former constitutes only 50\% of the resulting test set. In the latter we especially focus on hard-to-round cases. In addition, the test suite implicitly tests properties of floating-point operations, following the idea of Paranoia, and it reports which of the three IEEE-compliant underflow mechanisms is used by the floating-point implementation under consideration. We also check whether that underflow mechanism is used consistently. The tool is backward compatible with the UCBTEST package and with Coonen's test syntax.", accepted = "23 February 2001", acknowledgement = ack-nhfb, fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", keywords = "arithmetic; floating-point; floating-point testing; IEEE floating-point standard; multiprecision; validation; Verification", subject = "Primary Classification: G. Mathematics of Computing G.1 NUMERICAL ANALYSIS G.1.0 General Subjects: Computer arithmetic\\ Additional Classification: D. Software D.3 PROGRAMMING LANGUAGES D.3.0 General Subjects: Standards", } @Article{Weniger:2001:IID, author = "Ernst Joachim Weniger", title = "Irregular input data in convergence acceleration and summation processes: {General} considerations and some special {Gaussian} hypergeometric series as model problems", journal = j-COMP-PHYS-COMM, volume = "133", number = "2--3", pages = "202--228", day = "15", month = jan, year = "2001", CODEN = "CPHCBZ", DOI = "https://doi.org/10.1016/S0010-4655(00)00175-2", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", bibdate = "Thu Dec 01 09:12:48 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/compphyscomm2000.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Computer Physics Communications", journal-URL = "http://www.sciencedirect.com/science/journal/00104655", keywords = "convergence acceleration", remark = "This paper concentrates on $_2 F_1 (a, b; c; z)$.", } @InProceedings{Zheng:2001:ARE, author = "Liang Zheng and Shen Xu-Bang and Peng Zuo-Hui", booktitle = "Proceedings of the 4th International Conference on {ASIC}", title = "The application of redundant encoding in iterative implementation of division and square root", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "603--606", year = "2001", CODEN = "????", ISSN = "????", bibdate = "Sat Jul 16 15:28:14 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, summary = "The purpose of this paper is to discuss the speed improvement in division and square root computation with small area penalty. The digit recurrence SRT algorithm and functional iteration Newton--Raphson algorithm are generally used in modern \ldots{}", } @Misc{Ziv:2001:APM, author = "Abraham Ziv and Moshe Olshansky and Ealan Henis and Anna Reitman", title = "Accurate Portable Mathematical Library ({IBM APMathLib})", howpublished = "World-Wide Web document", publisher = pub-IBM, address = pub-IBM:adr, day = "20", month = dec, year = "2001", bibdate = "Wed Nov 24 08:06:54 2004", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "ftp://www-126.ibm.com/pub/mathlib/mathlib12.20.2001.tar.gz; http://oss.software.ibm.com/mathlib/", acknowledgement = ack-nhfb, } @Article{Al-Jarrah:2002:GSB, author = "A. Al-Jarrah and K. M. Dempsey and M. L. Glasser", title = "Generalized series of {Bessel} functions", journal = j-J-COMPUT-APPL-MATH, volume = "143", number = "1", pages = "1--8", day = "1", month = jun, year = "2002", CODEN = "JCAMDI", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Sat Feb 25 12:52:28 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2000.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0377042701005052", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @Article{Bertot:2002:PGS, author = "Yves Bertot and Nicolas Magaud and Paul Zimmermann", title = "A Proof of {GMP} Square Root", journal = j-J-AUTOM-REASON, volume = "29", number = "3--4", pages = "225--252", month = sep, year = "2002", CODEN = "JAREEW", DOI = "https://doi.org/10.1023/A:1021987403425", ISSN = "0168-7433 (print), 1573-0670 (electronic)", ISSN-L = "0168-7433", bibdate = "Sat Feb 08 08:59:09 2020", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/gnu.bib; https://www.math.utah.edu/pub/tex/bib/jautomreason.bib", URL = "https://link.springer.com/article/10.1023/A:1021987403425", acknowledgement = ack-nhfb, ajournal = "J. Autom. Reason.", fjournal = "Journal of Automated Reasoning", journal-URL = "http://link.springer.com/journal/10817", keywords = "GNU Multiple Precision library", } @InCollection{Boisvert:2002:HMF, author = "Ronald F. Boisvert and Daniel W. Lozier", title = "Handbook of Mathematical Functions", crossref = "Lide:2002:CEM", pages = "135--139", year = "2002", bibdate = "Fri Jul 09 06:28:13 2004", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "Also printed as NIST Special Publication 958, Jan. 2001.", URL = "https://nvlpubs.nist.gov/nistpubs/sp958-lide/135-139.pdf; https://nvlpubs.nist.gov/nistpubs/sp958-lide/html/135-139.html", acknowledgement = ack-nhfb, remark = "This article describes the history of the creation of the famous 1964 book by Milton Abramowitz and Irene Stegun named in the title.", } @Article{Bradford:2002:RAE, author = "Russell Bradford and Robert M. Corless and James H. Davenport and David J. Jeffrey and Stephen M. Watt", title = "Reasoning about the elementary functions of complex analysis", journal = j-ANN-MATH-ARTIF-INTELL, volume = "36", number = "3", pages = "303--318", year = "2002", CODEN = "AMAIEC", ISSN = "1012-2443 (print), 1573-7470 (electronic)", ISSN-L = "1012-2443", MRclass = "30-01 (03B35 68W30)", MRnumber = "MR1950025 (2003m:30001)", bibdate = "Wed Apr 13 06:46:35 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "Artificial intelligence and symbolic computation (Madrid, 2000)", acknowledgement = ack-nhfb, fjournal = "Annals of Mathematics and Artificial Intelligence", journal-URL = "http://link.springer.com/journal/10472", } @InProceedings{Bradford:2002:TBS, author = "Russell Bradford and James H. Davenport", booktitle = "Proceedings of the 2002 International Symposium on Symbolic and Algebraic Computation", title = "Towards better simplification of elementary functions", publisher = pub-ACM, address = pub-ACM:adr, pages = "16--22 (electronic)", year = "2002", MRclass = "68W30 (33B10)", MRnumber = "MR2035228", bibdate = "Wed Apr 13 06:46:35 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, } @Article{Bryc:2002:UAR, author = "W. Bryc", title = "A uniform approximation to the right normal tail integral", journal = j-APPL-MATH-COMP, volume = "127", number = "2--3", pages = "365--374", day = "15", month = apr, year = "2002", CODEN = "AMHCBQ", DOI = "https://doi.org/10.1016/S0096-3003(01)00015-7", ISSN = "0096-3003 (print), 1873-5649 (electronic)", ISSN-L = "0096-3003", bibdate = "Wed Feb 27 08:48:29 MST 2002", bibsource = "http://www.elsevier.com/locate/issn/00963003; https://www.math.utah.edu/pub/tex/bib/applmathcomput2000.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.elsevier.com/gej-ng/10/9/12/123/27/44/abstract.html; http://www.sciencedirect.com/science/article/pii/S0096300301000157", acknowledgement = ack-nhfb, fjournal = "Applied Mathematics and Computation", journal-URL = "http://www.sciencedirect.com/science/journal/00963003/", } @Article{Ceberio:2002:HRI, author = "M. Ceberio and L. Granvilliers", title = "{Horner}'s Rule for Interval Evaluation Revisited", journal = j-COMPUTING, volume = "69", number = "1", pages = "51--81", month = mar, year = "2002", CODEN = "CMPTA2", DOI = "https://doi.org/10.1007/s00607-002-1448-y", ISSN = "0010-485X (print), 1436-5057 (electronic)", ISSN-L = "0010-485X", bibdate = "Tue Nov 5 07:12:39 MST 2002", bibsource = "http://link.springer-ny.com/link/service/journals/00607/tocs/t2069001.htm; http://springerlink.metapress.com/openurl.asp?genre=journal&issn=0010-485X; https://www.math.utah.edu/pub/tex/bib/computing.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://link.springer.de/link/service/journals/00607/bibs/2069001/20690051.htm; http://link.springer.de/link/service/journals/00607/papers/2069001/20690051.pdf", acknowledgement = ack-nhfb, fjournal = "Computing", journal-URL = "http://link.springer.com/journal/607", keywords = "interval arithmetic; number of multiplications to evaluate a polynomial", } @InProceedings{Chiani:2002:IEB, author = "M. Chiani and D. Dardari", booktitle = "Global Telecommunications Conference, 2002. {GLOBECOM '02}. {IEEE}", title = "Improved exponential bounds and approximation for the {$Q$}-function with application to average error probability computation", publisher = pub-IEEE, address = pub-IEEE:adr, year = "2002", DOI = "https://doi.org/10.1109/glocom.2002.1188428", bibdate = "Sat Dec 16 16:54:47 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "https://ieeexplore.ieee.org/document/1188428/", acknowledgement = ack-nhfb, } @Article{Fabijonas:2002:LMC, author = "Bruce R. Fabijonas", title = "{Laplace}'s method on a computer algebra system with an application to the real valued modified {Bessel} functions", journal = j-J-COMPUT-APPL-MATH, volume = "146", number = "2", pages = "323--342", day = "15", month = sep, year = "2002", CODEN = "JCAMDI", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Sat Feb 25 12:52:30 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2000.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0377042702003643", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @Article{Gautschi:2002:GQA, author = "W. Gautschi", title = "{Gauss} quadrature approximations to hypergeometric and confluent hypergeometric functions", journal = j-J-COMPUT-APPL-MATH, volume = "139", number = "1", pages = "173--187", day = "1", month = feb, year = "2002", CODEN = "JCAMDI", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", MRclass = "33F05 (33C05 33C15 65D20)", MRnumber = "MR1876879 (2002m:33029)", bibdate = "Thu Dec 01 09:11:13 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", remark = "The paper treats ordinary and confluent hypergeometric functions $_2 F_1$ and $_1 F_1$, using their integral representations to obtain Gaussian quadrature rules.", } @Article{Gil:2002:AAB, author = "Amparo Gil and Javier Segura and Nico M. Temme", title = "{Algorithm 819}: {AIZ}, {BIZ}: two {Fortran 77} routines for the computation of complex {Airy} functions", journal = j-TOMS, volume = "28", number = "3", pages = "325--336", month = sep, year = "2002", CODEN = "ACMSCU", DOI = "https://doi.org/10.1145/569147.569150", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Sat Nov 9 11:16:50 MST 2002", bibsource = "http://www.acm.org/pubs/contents/journals/toms/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib", abstract = "Two Fortran 77 routines for the evaluation of Airy functions of complex arguments $ A i(z) $, $ B i(z) $ and their derivatives are presented. The routines are based on the use of Gaussian quadrature, Maclaurin series and asymptotic expansions. Comparison with a previous code by D. E. Amos (ACM TOMS 12 (1986)) is provided.", acknowledgement = ack-nhfb, fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", } @Article{Gil:2002:AGH, author = "Amparo Gil and Javier Segura and Nico M. Temme", title = "{Algorithm 822}: {GIZ}, {HIZ}: two {Fortran} 77 routines for the computation of complex {Scorer} functions", journal = j-TOMS, volume = "28", number = "4", pages = "436--447", month = dec, year = "2002", CODEN = "ACMSCU", DOI = "https://doi.org/10.1145/592843.592847", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Fri Mar 28 08:17:55 MST 2003", bibsource = "http://www.acm.org/pubs/contents/journals/toms/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib", abstract = "Two Fortran 77 routines for the evaluation of Scorer functions of complex arguments $ G i(z) $, $ H i(z) $ and their derivatives are presented. The routines are based on the use of quadrature, Maclaurin series and asymptotic expansions. For real $z$ comparison with a previous code by A. J. Macleod (J. Comput. Appl. Math. 53 (1994)) is provided.", acknowledgement = ack-nhfb, fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", } @TechReport{Gil:2002:CSF, author = "A. Gil and J. Segura and N. M. Temme", title = "Computing special functions by using quadrature rules", type = "Report", number = "MAS-R0230", institution = pub-CWI, address = pub-CWI:adr, pages = "11", year = "2002", LCCN = "QA9.A1 R426 MAS-R0230", bibdate = "Sat Oct 30 19:13:12 2010", bibsource = "http://cat.cisti-icist.nrc-cnrc.gc.ca/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "Published in \cite{Gil:2003:CSF}.", acknowledgement = ack-nhfb, } @Article{Gil:2002:EMB, author = "Amparo Gil and Javier Segura and Nico M. Temme", title = "Evaluation of the Modified {Bessel} Function of the Third Kind of Imaginary Orders", journal = j-J-COMPUT-PHYS, volume = "175", number = "2", pages = "398--411", day = "20", month = jan, year = "2002", CODEN = "JCTPAH", DOI = "https://doi.org/10.1006/jcph.2001.6894", ISSN = "0021-9991 (print), 1090-2716 (electronic)", ISSN-L = "0021-9991", bibdate = "Mon Jan 2 22:12:13 MST 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputphys2000.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0021999101968949", abstract = "The evaluation of the modified Bessel function of the third kind of purely imaginary order $ \mathrm {K}_{ia}(x) $ is discussed; we also present analogous results for the derivative. The methods are based on the use of Maclaurin series, nonoscillatory integral representations, asymptotic expansions, and a continued fraction method, depending on the ranges of x and a. We discuss the range of applicability of the different approaches considered and conclude that power series, the continued fraction method, and the nonoscillatory integral representation can be used to accurately compute the function $ \mathrm {K}_{ia}(x) $ in the range $ 0 \leq a \leq 200 $, $ 0 \leq x \leq 100 $; using a similar scheme the derivative $ \mathrm {K}'_{ia(x)} $ can also be computed within these ranges.", acknowledgement = ack-nhfb, fjournal = "Journal of Computational Physics", journal-URL = "http://www.sciencedirect.com/science/journal/00219991/", } @Article{Gisuthan:2002:PFC, author = "Bimal Gisuthan and Thambipillai Srikanthan", title = "Pipelining flat {CORDIC} based trigonometric function generators", journal = j-MICROELECT-J, volume = "33", number = "1", pages = "77--89", year = "2002", CODEN = "MICEB9", DOI = "https://doi.org/10.1016/S0026-2692(01)00107-0", ISSN = "1879-2391", ISSN-L = "0026-2692", bibdate = "Wed Oct 29 14:21:46 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "https://www.sciencedirect.com/science/article/pii/S0026269201001070", acknowledgement = ack-nhfb, fjournal = "Microelectronics Journal", journal-URL = "http://www.sciencedirect.com/science/journal/00262692", keywords = "CORDIC; Flat CORDIC; Hyperbolic functions; Pipelining; Trigonometric functions", } @Article{Gray:2002:ARE, author = "Norman Gray", title = "Automatic reduction of elliptic integrals using {Carlson}'s relations", journal = j-MATH-COMPUT, volume = "71", number = "237", pages = "311--318", month = jan, year = "2002", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Thu Jan 31 06:16:28 MST 2002", bibsource = "http://www.ams.org/mcom/2002-71-237; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.ams.org/journal-getitem?pii=S0025-5718-01-01333-3; http://www.ams.org/mcom/2002-71-237/S0025-5718-01-01333-3/S0025-5718-01-01333-3.dvi; http://www.ams.org/mcom/2002-71-237/S0025-5718-01-01333-3/S0025-5718-01-01333-3.pdf; http://www.ams.org/mcom/2002-71-237/S0025-5718-01-01333-3/S0025-5718-01-01333-3.ps; http://www.ams.org/mcom/2002-71-237/S0025-5718-01-01333-3/S0025-5718-01-01333-3.tex", acknowledgement = ack-nhfb, fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Hassenpflug:2002:EAS, author = "W. C. Hassenpflug", title = "Error analysis in the series evaluation of the exponential type integral {$ e^z E_1 (z) $}", journal = j-COMPUT-MATH-APPL, volume = "43", number = "1--2", pages = "207--266", month = jan, year = "2002", CODEN = "CMAPDK", ISSN = "0898-1221 (print), 1873-7668 (electronic)", ISSN-L = "0898-1221", bibdate = "Wed Mar 1 21:49:20 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/computmathappl2000.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0898122101002838", acknowledgement = ack-nhfb, fjournal = "Computers and Mathematics with Applications", journal-URL = "http://www.sciencedirect.com/science/journal/08981221", } @Article{Kilbas:2002:ARH, author = "Anatoly A. Kilbas and Luis Rodr{\'\i}guez and Juan J. Trujillo", title = "Asymptotic representations for hypergeometric-{Bessel} type function and fractional integrals", journal = j-J-COMPUT-APPL-MATH, volume = "149", number = "2", pages = "469--487", day = "15", month = dec, year = "2002", CODEN = "JCAMDI", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Sat Feb 25 12:52:32 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2000.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0377042702005629", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @Book{Korenev:2002:BFT, author = "B. G. (Boris Grigorevich) Korenev", title = "{Bessel} Functions and Their Applications", publisher = pub-TAYLOR-FRANCIS, address = pub-TAYLOR-FRANCIS:adr, pages = "ix + 276", year = "2002", ISBN = "0-415-28130-X (hardcover)", ISBN-13 = "978-0-415-28130-0 (hardcover)", LCCN = "QA408 .K67 2002", bibdate = "Sat Oct 30 17:01:51 MDT 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; prodorbis.library.yale.edu:7090/voyager", series = "Analytical methods and special functions", acknowledgement = ack-nhfb, subject = "Bessel functions", tableofcontents = "Part 1. Foundation of the theory of Bessel functions \\ 1. The Bessel equation \\ Properties of Bessel functions \\ 2. Definite and improper integrals \\ Series in Bessel functions \\ Part 2. Applications of Bessel functions \\ 3. Problems of the theory of plates and shells \\ 4. Problems of the theory of oscillations, hydrodynamics and heat transfer \\ Appendix A. Brief information on gamma functions", xxaddress = pub-CRC:adr, xxpublisher = pub-CRC, } @Book{Li:2002:SWF, author = "Le-Wei Li and Xiao-Kang Kang and Mook-Seng Leong", title = "Spheroidal Wave Functions in Electromagnetic Theory", publisher = pub-WILEY, address = pub-WILEY:adr, pages = "xiii + 295", year = "2002", ISBN = "0-471-03170-4 (hardcover)", ISBN-13 = "978-0-471-03170-3 (hardcover)", LCCN = "QC670 .L49 2002", bibdate = "Sat Apr 1 14:32:29 MDT 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; z3950.loc.gov:7090/Voyager", series = "Wiley series in microwave and optical engineering", URL = "http://www.loc.gov/catdir/bios/wiley043/2001045399.html; http://www.loc.gov/catdir/description/wiley036/2001045399.html; http://www.loc.gov/catdir/toc/onix07/2001045399.html", acknowledgement = ack-nhfb, subject = "Electromagnetic theory; Spheroidal functions", tableofcontents = "Preface \\ Acknowledgments \\ Introduction \\ Spheroidal Coordinates and Wave Functions \\ Dyadic Green's Functions in Spheroidal Systems \\ EM Scattering by a Conducting Spheroid \\ EM Scattering by a Coated Dielectric Spheroid \\ Spheroidal Antennas \\ SAR Distributions in a Spheroidal Head Model \\ Analysis of Rainfall Attenuation Using Oblate Raindrops \\ EM Eigenfrequencies in a Spheroidal Cavity \\ Appendix A: Expressions of Spheroidal Vector Wave Functions \\ Appendix B: Intermediates $I_{t,\ell}^{mn}(c)$ in Closed Form \\ Appendix C: ${\cal U}^{q(i),t}$ and ${\cal V}^{(i),t}$ Used in the Matrix Equation System \\ References \\ Index", } @Article{McCluskey:2002:MLF, author = "Glen McCluskey", title = "Math Library Functions in {C9X}", journal = j-LOGIN, volume = "27", number = "2", pages = "8--13", month = apr, year = "2002", CODEN = "LOGNEM", ISSN = "1044-6397", bibdate = "Tue Apr 11 10:52:14 MDT 2006", bibsource = "http://www.usenix.org/publications/login/2002-04/index.html; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.usenix.org/publications/login/2002-04/pdfs/mccluskey.pdf", acknowledgement = ack-nhfb, fjournal = ";login: the USENIX Association newsletter", remark = "This is a short tutorial on some of the new math library functions in C99.", } @Article{Paris:2002:EBU, author = "R. B. Paris", title = "Error bounds for the uniform asymptotic expansion of the incomplete gamma function", journal = j-J-COMPUT-APPL-MATH, volume = "147", number = "1", pages = "215--231", day = "1", month = oct, year = "2002", CODEN = "JCAMDI", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Sat Feb 25 12:52:30 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2000.bib", URL = "http://www.sciencedirect.com/science/article/pii/S037704270200434X", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @Article{Paris:2002:UAE, author = "R. B. Paris", title = "A uniform asymptotic expansion for the incomplete gamma function", journal = j-J-COMPUT-APPL-MATH, volume = "148", number = "2", pages = "323--339", month = nov, year = "2002", CODEN = "JCAMDI", DOI = "https://doi.org/10.1016/s0377-0427(02)00553-8", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Sat Feb 18 09:18:08 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "See related work \cite{Paris:2016:UAE}.", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @InProceedings{Pineiro:2002:HRL, author = "J.-A. Pineiro and M. D. Ercegovac and J. D. Bruguera", booktitle = "{The IEEE International Conference on Application-Specific Systems, Architectures and Processors, 2002. Proceedings. 17--19 July 2002}", title = "High-radix logarithm with selection by rounding", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "101--110", year = "2002", CODEN = "????", ISSN = "????", bibdate = "Sat Jul 16 11:25:05 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, summary = "A high-radix digit-recurrence algorithm or the computation of the logarithm is presented in this paper. Selection by rounding is used in iterations j/spl ges/2, and selection by table in the first iteration is combined with a restricted digit-set \ldots{}", } @Article{Pineiro:2002:HSD, author = "J. A. Pi{\~n}eiro and J. D. Bruguera", title = "High-Speed Double Precision Computation of Reciprocal, Division, Square Root, and Inverse Square Root", journal = j-IEEE-TRANS-COMPUT, volume = "51", number = "12", pages = "1377--1388", month = dec, year = "2002", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/TC.2002.1146704", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Sat Jul 16 15:28:13 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput2000.bib", URL = "https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=1146704", acknowledgement = ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", summary = "A new method for the high-speed computation of double-precision floating-point reciprocal, division, square root, and inverse square root operations is presented in this paper. This method employs a second-degree minimax polynomial approximation to \ldots{}", } @Book{Samko:2002:HIT, author = "S. G. (Stefan Grigorevich) Samko", title = "Hypersingular Integrals and Their Applications", volume = "5", publisher = pub-TAYLOR-FRANCIS, address = pub-TAYLOR-FRANCIS:adr, pages = "xvii + 359", year = "2002", DOI = "https://doi.org/10.1201/9781482264968", ISBN = "0-415-27268-8", ISBN-13 = "978-0-415-27268-1", LCCN = "QA403.5 .S26 2002", bibdate = "Sat Oct 30 17:22:10 MDT 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; prodorbis.library.yale.edu:7090/voyager", series = "Analytical methods and special functions", acknowledgement = ack-nhfb, subject = "singular integrals", tableofcontents = "Part 1: Hypersingular integrals \\ 1: Some basics from the theory of special functions and operator theory \\ 2: The Riesz potential operator and Lizorkin type invariant spaces $\Phi_v$ \\ 3: Hypersingular integrals with constant characteristics \\ 4: Potentials and hypersingular integrals with homogeneous characteristics \\ 5: Hypersingular integrals with non-homogeneous characteristics \\ 6: Hypersingular integrals on the unit sphere \\ Part 2: Applications of hypersingular integrals \\ 7: Characterization of some function spaces in terms of hypersingular integrals \\ 8: Solution of multidimensional integral equations of the first kind with a potential type kernel \\ 9: Hypersingular operators as positive fractional powers of some operators of mathematical physics \\ 10: Regularization of multidimensional integral equations of the first kind with a potential type kernel \\ 11: Some modifications of hypersingular integrals and their applications", } @InProceedings{Sawada:2002:FVD, author = "J. Sawada", title = "Formal verification of divide and square root algorithms using series calculation", crossref = "Borrione:2002:TIW", pages = "31--49", year = "2002", bibdate = "Fri Jun 24 15:14:00 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, } @Article{Sawada:2002:MVS, author = "Jun Sawada and Ruben Gamboa", title = "Mechanical Verification of a Square Root Algorithm Using {Taylor}'s Theorem", journal = j-LECT-NOTES-COMP-SCI, volume = "2517", pages = "274--??", year = "2002", CODEN = "LNCSD9", ISSN = "0302-9743 (print), 1611-3349 (electronic)", ISSN-L = "0302-9743", bibdate = "Sat Nov 30 20:58:00 MST 2002", bibsource = "http://link.springer-ny.com/link/service/series/0558/tocs/t2517.htm; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://link.springer.de/link/service/series/0558/bibs/2517/25170274.htm; http://link.springer.de/link/service/series/0558/papers/2517/25170274.pdf", acknowledgement = ack-nhfb, fjournal = "Lecture Notes in Computer Science", journal-URL = "http://link.springer.com/bookseries/558", } @Misc{Sebah:2002:IGF, author = "Pascal Sebah and Xavier Gourdon", title = "Introduction to the Gamma Function", howpublished = "World-Wide Web document", day = "4", month = feb, year = "2002", bibdate = "Sat May 01 16:07:51 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://numbers.computation.free.fr/Constants/constants.html; http://numbers.computation.free.fr/Constants/Miscellaneous/gammaFunction.ps", acknowledgement = ack-nhfb, } @Article{Shore:2002:RMM, author = "Haim Shore", title = "Response Modeling Methodology ({RMM})-Exploring the Properties of the Implied Error Distribution", journal = j-COMMUN-STAT-THEORY-METH, volume = "31", number = "12", pages = "2225--2249", year = "2002", CODEN = "CSTMDC", DOI = "https://doi.org/10.1081/STA-120017223", ISSN = "0361-0926 (print), 1532-415X (electronic)", ISSN-L = "0361-0926", bibdate = "Wed Jan 27 05:41:30 MST 2016", bibsource = "https://www.math.utah.edu/pub/tex/bib/communstattheorymeth2000.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Communications in Statistics: Theory and Methods", journal-URL = "http://www.tandfonline.com/loi/lsta20", } @Article{Tornaria:2002:SRM, author = "Gonzalo Tornar{\'\i}a", title = "Square Roots Modulo $p$", journal = j-LECT-NOTES-COMP-SCI, volume = "2286", pages = "430--??", year = "2002", CODEN = "LNCSD9", ISSN = "0302-9743 (print), 1611-3349 (electronic)", ISSN-L = "0302-9743", bibdate = "Tue Sep 10 19:09:12 MDT 2002", bibsource = "http://link.springer-ny.com/link/service/series/0558/tocs/t2286.htm; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "http://link.springer-ny.com/link/service/series/0558/bibs/2286/22860430.htm; http://link.springer-ny.com/link/service/series/0558/papers/2286/22860430.pdf", acknowledgement = ack-nhfb, fjournal = "Lecture Notes in Computer Science", journal-URL = "http://link.springer.com/bookseries/558", } @Book{Vladimirov:2002:MTG, author = "V. S. (Vasilii Sergeevich) Vladimirov", title = "Methods of the Theory of Generalized Functions", volume = "6", publisher = pub-TAYLOR-FRANCIS, address = pub-TAYLOR-FRANCIS:adr, pages = "xiv + 311", year = "2002", DOI = "https://doi.org/10.1201/9781482288162", ISBN = "0-415-27356-0", ISBN-13 = "978-0-415-27356-5", LCCN = "QC20.7.T45 V53 2002", bibdate = "Sat Oct 30 17:22:15 MDT 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; prodorbis.library.yale.edu:7090/voyager", series = "Analytical methods and special functions", acknowledgement = ack-nhfb, subject = "Theory of distributions (Functional analysis); Integral transforms; Mathematical physics", tableofcontents = "Preface / xi \\ Symbols and Definitions / 1 \\ \\ 1. Generalized Functions and their Properties / 5 \\ 1. Test and Generalized Functions / 5 \\ 1.1. Introduction / 5 \\ 1.2. The space of test functions $\mathcal{D}(\mathcal{O})$ / 6 \\ 1.3. The space of generalized functions $\mathcal{D}'(\mathcal{O})$ / 10 \\ 1.4. The completeness of the space of generalized functions $\mathcal{D}'(\mathcal{O})$ / 12 \\ 1.5. The support of a generalized function / 13 \\ 1.6. Regular generalized functions / 15 \\ 1.7. Measures / 16 \\ 1.8. Sochozki formulae / 19 \\ 1.9. Change of variables in generalized functions / 21 \\ 1.10. Multiplication of generalized functions / 23 \\ \\ 2. Differentiation of Generalized Functions / 25 \\ 2.1. Derivatives of generalized functions / 25 \\ 2.2. The antiderivative (primitive) of a generalized function / 27 \\ 2.3. Examples / 29 \\ 2.4. The local structure of generalized functions / 35 \\ 2.5. Generalized functions with compact support / 36 \\ 2.6. Generalized functions with point support / 37 \\ 2.7. Generalized functions $\mathcal{P}(\pi_nu|x|^{\alpha - 1})$ / 39 \\ \\ 3. Direct Product of Generalized Functions / 41 \\ 3.1. The definition of a direct product / 41 \\ 3.2. The properties of a direct product / 43 \\ 3.3. Some applications / 56 \\ 3.4. Generalized functions that are smooth with respect to some of the variables / 48 \\ \\ 4. The Convolution of Generalized Functions / 50 \\ 4.1. The definition of convolution / 50 \\ 4.2. The properties of a convolution / 53 \\ 4.3. The existence of a convolution / 57 \\ 4.4. Cones in $\mathcal{R}^n$ / 59 \\ 4.5. Convolution algebras $\mathcal{D}'(\Gamma+)$ and $\mathcal{D}'(\Gamma)$ / 63 \\ 4.6. Mean functions of generalized functions / 64 \\ 4.7. Multiplication of generalized functions / 66 \\ 4.8. Convolution as a continuous linear translation invariant operator / 66 \\ 4.9. Some applications / 68 \\ \\ 5. Tempered Generalized Functions / 74 \\ 5.1. The space $S$ of test (rapidly decreasing) functions / 74 \\ 5.2. The space $S'$ of tempered generalized functions / 77 \\ 5.3. Examples of tempered generalized functions and elementary operations in $S$ /' 78 \\ 5.4. The structure of tempered generalized functions / 80 \\ 5.5. The direct product of tempered generalized functions / 81 \\ 5.6. The convolution of tempered generalized functions / 82 \\ 5.7. Homogeneous generalized functions / 85 \\ 2. Integral Transformations of Generalized Functions / 89 \\ \\ 6. The Fourier Transform of Tempered Generalized Functions / 89 \\ 6.1. The Fourier transform of test functions in $S$ / 89 \\ 6.2. The Fourier transform of tempered generalized functions / 90 \\ 6.3. Properties of the Fourier transform / 92 \\ 6.4. The Fourier transform of generalized functions with compact support / 93 \\ 6.5. The Fourier transform of a convolution / 94 \\ 6.6. Examples / 96 \\ 6.7. The Mellin transform / 109 \\ \\ 7. Fourier Series of Periodic Generalized Functions / 113 \\ 7.1. The definition and elementary properties of periodic generalized functions / 113 \\ 7.2. Fourier series of periodic generalized functions / 116 \\ 7.3. The convolution algebra $\mathcal{D}'_T$ / 117 \\ 7.4. Examples / 119 \\ \\ 8. Positive Definite Generalized Functions / 121 \\ 8.1. The definition and elementary properties of positive definite generalized functions / 121 \\ 8.2. The Bochner--Schwartz theorem / 123 \\ 8.3. Examples / 125 \\ \\ 9. The Laplace Transform of Tempered Generalized Functions / 126 \\ 9.1. Definition of the Laplace transform / 126 \\ 9.2. Properties of the Laplace transform / 128 \\ 9.3. Examples / 130 \\ \\ 10. The Cauchy Kernel and the Transforms of Cauchy--Bochner and Hilbert / 133 \\ 10.1. The space $\mathcal{H}_s$ / 133 \\ 10.2. The Cauchy kernel $\mathcal{K}_(z)$ / 138 \\ 10.3. The Cauchy--Bochner transform / 144 \\ 10.4. The Hilbert transform / 146 \\ 10.5. Holomorphic functions of the class $\mathcal{H}_a^{(s)}(C)$ / 147 \\ 10.6. The generalized Cauchy--Bochner representation / 151 \\ \\ 11. Poisson Kernel and Poisson Transform / 152 \\ 11.1. The definition and properties of the Poisson kernel / 152 \\ 11.2. The Poisson transform and Poisson representation / 155 \\ 11.3. Boundary values of the Poisson integral / 157 \\ \\ 12. Algebras of Holomorphic Functions / 159 \\ 12.1. The definition of the $H_+(C)$ and $H(C)$ algebras / 160 \\ 12.2. Isomorphism of the algebras $S'(C*+) \sim H_+(C)$ and $S'(C*) \sim H(C)$ / 160 \\ 12.3. The Paley--Wiener--Schwartz theorem and its generalizations / 165 \\ 12.4. The space $H_a(C)$ is the projective limit of the spaces $H_{a'}(C)$ / 166 \\ 12.5. The Schwartz representation / 168 \\ 12.6. A generalization of the Phragmen--Lindelof theorem / 171 \\ \\ 13. Equations in Convolution Algebras / 171 \\ 13.1. Divisors of unity in the $H_+(C)$ and $H(C)$ algebras / 171 \\ 13.2. On division by a polynomial in the $H(C)$ algebra / 172 \\ 13.3. Estimates for holomorphic functions with nonnegative imaginary part in $T^C$ / 174 \\ 13.4. Divisors of unity in the algebra $W(C)$ / 177 \\ 13.5. Example / 177 \\ \\ 14. Tauberian Theorems for Generalized Functions / 179 \\ 14.1. Preliminary results / 179 \\ 14.2. General Tauberian theorem / 183 \\ 14.3. One-dimensional Tauberian theorems / 186 \\ 14.4. Tauberian and Abelian theorems for nonnegative measures / 187 \\ 14.5. Tauberian theorems for holomorphic functions of bounded argument / 188 \\ 3. Some Applications in Mathematical Physics / 191 \\ \\ 15. Differential Operators with Constant Coefficients / 191 \\ 15.1. Fundamental solutions in $\mathcal{D}'$ / 191 \\ 15.2. Tempered fundamental solutions / 194 \\ 15.3. A descent method / 196 \\ 15.4. Examples / 199 \\ 15.5. A comparison of differential operators / 207 \\ 15.6. Elliptic and hypoelliptic operators / 210 \\ 15.7. Hyperbolic operators / 212 \\ 15.8. The sweeping principle / 212 \\ \\ 16. The Cauchy Problem / 213 \\ 16.1. The generalized Cauchy problem for a hyperbolic equation / 213 \\ 16.2. Wave potential / 216 \\ 16.3. Surface wave potentials / 220 \\ 16.4. The Cauchy problem for the wave equation / 222 \\ 16.5. A statement of the generalized Cauchy problem for the heat equation / 224 \\ 16.6. Heat potential / 224 \\ 16.7. Solution of the Cauchy problem for the heat equation / 228 \\ \\ 17. Holomorphic Functions with Nonnegative Imaginary Part in $T^C$ / 229 \\ 17.1. Preliminary remarks / 229 \\ 17.2. Properties of functions of the class $\mathcal{P}_+(T^C)$ / 231 \\ 17.3. Estimates of the growth of functions of the class $H_+(T^C)$ / 238 \\ 17.4. Smoothness of the spectral function / 240 \\ 17.5. Indicator of growth of functions of the class $\mathcal{P}_+T^C$ / 242 \\ 17.6. An integral representation of functions of the class $H_+(T^C)$ / 245 \\ \\ 18. Holomorphic Functions with Nonnegative Imaginary Part in $T^n$ / 249 \\ 18.1. Lemmas / 249 \\ 18.2. Functions of the classes $H_+(T^1)$ and $\mathcal{P}_+(T^1)$ / 254 \\ 18.3. Functions of the class $\mathcal{P}_+(T^n)$ / 258 \\ 18.4. Functions of the class $H_+(T^n)$ / 263 \\ \\ 19. Positive Real Matrix Functions in $T^C$ / 266 \\ 19.1. Positive real functions in $T^C$ / 267 \\ 19.2. Positive real matrix functions in $T^C$ / 269 \\ \\ 20. Linear Passive Systems / 271 \\ 20.1. Introduction / 271 \\ 20.2. Corollaries to the condition of passivity / 273 \\ 20.3. The necessary and sufficient conditions for passivity / 277 \\ 20.4. Multidimensional dispersion relations / 282 \\ 20.5. The fundamental solution and the Cauchy problem / 285 \\ 20.6. What differential and difference operators are passive operators? / 287 \\ 20.7. Examples / 290 \\ 20.8. Quasiasymptotics of the solutions of systems of equations in convolutions / 294 \\ \\ 21. Abstract Scattering Operator / 295 \\ 21.1. The definition and properties of an abstract scattering matrix / 295 \\ 21.2. A description of abstract scattering matrices / 298 \\ 21.3. The relationship between passive operators and scattering operators / 299 \\ \\ Bibliography / 303 \\ Index / 309", } @Article{Aarts:2003:ASF, author = "Ronald M. Aarts and Augustus J. E. M. Janssen", title = "Approximation of the {Struve} function {$ H_1 $} occurring in impedance calculations", journal = j-J-ACOUST-SOC-AM, volume = "113", number = "5", pages = "2635--2637", month = may, year = "2003", CODEN = "JASMAN", DOI = "https://doi.org/10.1121/1.1564019", ISSN = "0001-4966", ISSN-L = "0001-4966", bibdate = "Tue Mar 28 07:23:10 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Journal of the Acoustical Society of America", journal-URL = "http://scitation.aip.org/content/asa/journal/jasa", } @Article{Abad:2003:AEQ, author = "J. Abad and J. Sesma", title = "Asymptotic expansion of the quasiconfluent hypergeometric function", journal = j-J-MATH-PHYS, volume = "44", number = "4", pages = "1723--1729", month = apr, year = "2003", CODEN = "JMAPAQ", ISSN = "0022-2488 (print), 1089-7658 (electronic), 1527-2427", ISSN-L = "0022-2488", bibdate = "Tue Dec 16 11:36:01 MST 2003", bibsource = "http://www.aip.org/ojs/jmp.html; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jmathphys2000.bib", acknowledgement = ack-nhfb, fjournal = "Journal of Mathematical Physics", journal-URL = "http://jmp.aip.org/", } @Article{Agou:2003:SPR, author = "Simon Joseph Agou and Marc Del{\'e}glise and Jean-Louis Nicolas", title = "Short Polynomial Representations for Square Roots Modulo $p$", journal = j-DESIGNS-CODES-CRYPTOGR, volume = "28", number = "1", pages = "33--44", month = jan, year = "2003", CODEN = "DCCREC", ISSN = "0925-1022 (print), 1573-7586 (electronic)", ISSN-L = "0925-1022", bibdate = "Thu Dec 11 06:27:20 MST 2003", bibsource = "http://www.wkap.nl/jrnltoc.htm/0925-1022; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "http://ipsapp007.kluweronline.com/content/getfile/4630/45/2/abstract.htm; http://ipsapp007.kluweronline.com/content/getfile/4630/45/2/fulltext.pdf", acknowledgement = ack-nhfb, fjournal = "Designs, codes, and cryptography", journal-URL = "http://link.springer.com/journal/10623", } @Article{Alzer:2003:GHM, author = "Horst Alzer", title = "On {Gautschi}'s harmonic mean inequality for the gamma function", journal = j-J-COMPUT-APPL-MATH, volume = "157", number = "1", pages = "243--249", day = "1", month = aug, year = "2003", CODEN = "JCAMDI", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Sat Feb 25 12:52:37 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2000.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0377042703004564", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @InProceedings{Beaumont:2003:BSE, author = "James Beaumont and Russell Bradford and James H. Davenport", booktitle = "Proceedings of the 2003 International Symposium on Symbolic and Algebraic Computation", title = "Better simplification of elementary functions through power series", publisher = pub-ACM, address = pub-ACM:adr, pages = "30--36 (electronic)", year = "2003", MRclass = "33F10 (68W30)", MRnumber = "MR2035192 (2005e:33018)", MRreviewer = "Ekatherina A. Karatsuba", bibdate = "Wed Apr 13 06:46:35 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, remark = "A new algorithm testing the correctness of simplifications of elementary functions in the presence of branch cuts is proposed.", } @Article{Buhring:2003:PSH, author = "Wolfgang B{\"u}hring", title = "Partial sums of hypergeometric functions of unit argument", journal = j-PROC-AM-MATH-SOC, volume = "132", number = "2", pages = "407--415", month = "????", year = "2003", CODEN = "PAMYAR", ISSN = "0002-9939 (print), 1088-6826 (electronic)", ISSN-L = "0002-9939", MRclass = "33C20", MRnumber = "MR2022363 (2005f:33011)", bibdate = "Thu Dec 01 09:53:54 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Proceedings of the American Mathematical Society", journal-URL = "http://www.ams.org/journals/proc", } @Article{Chiani:2003:NEB, author = "M. Chiani and D. Dardari and M. K. Simon", title = "New exponential bounds and approximations for the computation of error probability in fading channels", journal = j-IEEE-TRANS-WIREL-COMMUN, volume = "24", number = "5", pages = "840--845", month = may, year = "2003", CODEN = "ITWCAX", DOI = "https://doi.org/10.1109/twc.2003.814350", ISSN = "1536-1276 (print), 1558-2248 (electronic)", ISSN-L = "1536-1276", bibdate = "Sat Dec 16 15:47:42 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "https://ieeexplore.ieee.org/document/1210748/", acknowledgement = ack-nhfb, fjournal = "IEEE Transactions on Wireless Communications", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=7693", } @TechReport{Cornea:2003:DSR, author = "M. Cornea and J. Harrison and C. Iordache and B. Norin and S. Story", title = "Division, Square Root and Remainder Algorithms for the {Intel Itanium} Architecture", type = "Report", institution = pub-INTEL, address = pub-INTEL:adr, month = nov, year = "2003", bibdate = "Fri Jun 24 12:05:58 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, } @Article{Coussement:2003:AMO, author = "Els Coussement and Walter {Van Assche}", title = "Asymptotics of multiple orthogonal polynomials associated with the modified {Bessel} functions of the first kind", journal = j-J-COMPUT-APPL-MATH, volume = "153", number = "1--2", pages = "141--149", day = "1", month = apr, year = "2003", CODEN = "JCAMDI", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Sat Feb 25 12:52:34 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2000.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0377042702005964", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @Article{Dominici:2003:NDS, author = "Diego Dominici", title = "Nested derivatives: a simple method for computing series expansions of inverse functions", journal = j-INT-J-MATH-MATH-SCI, volume = "58", pages = "3699--3715", year = "2003", CODEN = "????", ISSN = "0161-1712 (print), 1687-0425 (electronic)", ISSN-L = "0161-1712", MRclass = "41A58 (33F10)", MRnumber = "MR2031140 (2005f:41079)", MRreviewer = "Tord H. Ganelius", bibdate = "Mon Oct 24 11:37:20 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www2.newpaltz.edu/~dominicd/NESTED14.pdf", abstract = "We give an algorithm to compute the series expansion for the inverse of a given function. The algorithm is extremely easy to implement and gives the first $N$ terms of the series. We show several examples of its application in calculating the inverses of some special functions.", acknowledgement = ack-nhfb, fjournal = "International Journal of Mathematics and Mathematical Sciences", journal-URL = "https://www.hindawi.com/journals/ijmms/", keywords = "error function, erf(x); incomplete beta function, B(nu,mu,x); incomplete gamma function, gamma(nu,x); logarithm integral, li(x); Maple; sine integral, Si(x)", } @InProceedings{Ercegovac:2003:DRA, author = "M. D. Ercegovac and J.-M. Muller", booktitle = "Conference Record of the Thirty-Seventh Asilomar Conference on Signals, Systems and Computers, 2003", title = "Digit-recurrence algorithms for division and square root with limited precision primitives", volume = "2", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "1440--1444", year = "2003", CODEN = "????", ISSN = "????", bibdate = "Sat Jul 16 15:28:13 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, summary = "We propose a digit-recurrence algorithm for square root using limited-precision multipliers, adders, and table-lookups. The algorithm, except in the initialization, uses the digit-recurrence algorithm for division with limited-precision primitives \ldots{}", } @Article{Fabijonas:2003:ACM, author = "B. R. Fabijonas and Daniel W. Lozier and J. M. Rappoport", title = "Algorithms and Codes for the {Macdonald} Function: Recent Progress and Comparisons", journal = j-J-COMPUT-APPL-MATH, volume = "161", number = "1", pages = "179--192", month = "????", year = "2003", CODEN = "JCAMDI", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", MRclass = "33F05 (33C10 65D20)", MRnumber = "MR2018582", bibdate = "Fri Jul 09 06:21:51 2004", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://math.nist.gov/acmd/Staff/DLozier/publications/nistir6596.ps", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @Article{Galue:2003:FRG, author = "L. Galu{\'e} and A. Al-Zamel and Shyam L. Kalla", title = "Further results on generalized hypergeometric functions", journal = j-APPL-MATH-COMP, volume = "136", number = "1", pages = "17--25", day = "25", month = mar, year = "2003", CODEN = "AMHCBQ", ISSN = "0096-3003 (print), 1873-5649 (electronic)", ISSN-L = "0096-3003", bibdate = "Fri Jan 9 08:40:52 MST 2004", bibsource = "https://www.math.utah.edu/pub/tex/bib/applmathcomput2000.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Applied Mathematics and Computation", journal-URL = "http://www.sciencedirect.com/science/journal/00963003", } @Article{Gautschi:2003:EEI, author = "W. Gautschi and F. E. Harris and N. M. Temme", title = "Expansions of the exponential integral in incomplete gamma functions", journal = j-APPL-MATH-LETT, volume = "16", number = "7", pages = "1095--1099", month = oct, year = "2003", CODEN = "AMLEEL", DOI = "https://doi.org/10.1016/S0893-9659(03)90100-5", ISSN = "0893-9659 (print), 1873-5452 (electronic)", ISSN-L = "0893-9659", MRclass = "33B20", bibdate = "Wed Dec 4 10:29:43 2013", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/h/harris-frank-e.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", ZMnumber = "1058.33002", acknowledgement = ack-nhfb, fjournal = "Applied Mathematics Letters", journal-URL = "http://www.sciencedirect.com/science/journal/08939659", } @Article{Gil:2003:CMB, author = "Amparo Gil and Javier Segura and Nico M. Temme", title = "Computation of the modified {Bessel} function of the third kind of imaginary orders: uniform {Airy}-type asymptotic expansion", journal = j-J-COMPUT-APPL-MATH, volume = "153", number = "1--2", pages = "225--234", day = "1", month = apr, year = "2003", CODEN = "JCAMDI", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Sat Feb 25 12:52:34 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2000.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0377042702006088", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @Article{Gil:2003:CSF, author = "Amparo Gil and Javier Segura and Nico M. Temme", title = "Computing Special Functions by Using Quadrature Rules", journal = j-NUMER-ALGORITHMS, volume = "33", number = "1--4", pages = "265--275", month = aug, year = "2003", CODEN = "NUALEG", ISSN = "1017-1398 (print), 1572-9265 (electronic)", ISSN-L = "1017-1398", bibdate = "Mon Sep 29 08:37:11 MDT 2003", bibsource = "http://www.kluweronline.com/issn/1017-1398; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://ipsapp007.kluweronline.com/content/getfile/5058/46/22/abstract.htm; http://ipsapp007.kluweronline.com/content/getfile/5058/46/22/fulltext.pdf", acknowledgement = ack-nhfb, fjournal = "Numerical Algorithms", journal-URL = "http://link.springer.com/journal/11075", } @Article{Harrison:2003:FVS, author = "John Harrison", title = "Formal verification of square root algorithms", journal = j-FORM-METHODS-SYST-DES, volume = "22", number = "2", pages = "143--153", month = mar, year = "2003", CODEN = "FMSDE6", DOI = "https://doi.org/10.1023/A:1022973506233", ISSN = "0925-9856 (print), 1572-8102 (electronic)", ISSN-L = "0925-9856", bibdate = "Sat Feb 08 08:47:21 2020", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/intel-ia-64.bib", URL = "https://dl.acm.org/doi/abs/10.1023/A:1022973506233", abstract = "We discuss the formal verification of some low-level mathematical software for the Intel Itanium architecture. A number of important algorithms have been proven correct using the HOL Light theorem prover. After briefly surveying some of our formal verification work, we discuss in more detail the verification of a square root algorithm, which helps to illustrate why some features of HOL Light, in particular programmability, make it especially suitable for these applications.", acknowledgement = ack-nhfb, fjournal = "Formal Methods in System Design", journal-URL = "https://dl.acm.org/loi/fmsd", } @Misc{Intel:2003:DSR, author = "{Intel}", title = "Divide, Square Root, and Remainder Algorithms for the {Itanium} Architecture", howpublished = "Intel Software Development Products", day = "18", month = dec, year = "2003", bibdate = "Tue Nov 18 16:23:36 2008", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "http://www.intel.com/cd/software/products/asmo-na/eng/219863.htm", acknowledgement = ack-nhfb, } @Misc{Intel:2003:NID, author = "{Intel}", title = "Non-{IEEE} Division, Square Root, Reciprocal, and Reciprocal Square Root Algorithms for the {Intel Itanium} Architecture", howpublished = "Intel Software Development Products", day = "18", month = dec, year = "2003", bibdate = "Tue Nov 18 16:23:36 2008", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "http://www.intel.com/cd/software/products/asmo-na/eng/219864.htm", acknowledgement = ack-nhfb, } @Article{Kzaz:2003:CAG, author = "M. Kzaz and M. Pr{\'e}vost", title = "Convergence Acceleration of {Gauss--Chebyshev} Quadrature Formulae", journal = j-NUMER-ALGORITHMS, volume = "34", number = "2--4", pages = "379--391", month = dec, year = "2003", CODEN = "NUALEG", ISSN = "1017-1398 (print), 1572-9265 (electronic)", ISSN-L = "1017-1398", bibdate = "Tue Jan 13 17:32:50 MST 2004", bibsource = "http://www.kluweronline.com/issn/1017-1398; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://ipsapp007.kluweronline.com/content/getfile/5058/48/6/abstract.htm; http://ipsapp007.kluweronline.com/content/getfile/5058/48/6/fulltext.pdf", acknowledgement = ack-nhfb, fjournal = "Numerical Algorithms", journal-URL = "http://link.springer.com/journal/11075", keywords = "convergence acceleration", } @Article{Lang:2003:RRS, author = "Tom{\'a}s Lang and Elisardo Antelo", title = "Radix-$4$ Reciprocal Square-root and Its Combination with Division and Square Root", journal = j-IEEE-TRANS-COMPUT, volume = "52", number = "9", pages = "1100--1114", month = sep, year = "2003", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/TC.2003.1228508", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Sat Jul 16 15:28:13 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", abstract = "In this work, we present a reciprocal square root algorithm by digit recurrence and selection by a staircase function and the radix-$4$ implementation. As in similar algorithms for division and square root, the results are obtained correctly rounded in a straightforward manner (in contrast to existing methods to compute the reciprocal square root). Although, apparently, a single selection function can only be used for $ j \geq 2 $ (the selection constants are different for $ j = 0 $, $ j = 1 $, and $ j \geq 2 $ ), we show that it is possible to use a single selection function for all iterations. We perform a rough comparison with existing methods and we conclude that our implementation is a low hardware complexity solution with moderate latency, especially for exactly rounded results. We also extend the unit to support division and square root with the same selection function and with slight modifications in the initialization of the reciprocal square root unit.", acknowledgement = ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", } @TechReport{Lefevre:2003:WCC, author = "Vincent Lef{\`e}vre and Jean-Michel Muller", title = "Worst Cases for Correct Rounding for the Elementary Functions in Double Precision", type = "Technical report", institution = "INRIA, Projet Spaces, LORIA, Campus Scientifique", address = "B.P. 239, 54506 Vandoeuvre-l{\`e}s-Nancy Cedex, France", day = "14", month = aug, year = "2003", bibdate = "Thu Jul 08 08:27:53 2004", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://perso.ens-lyon.fr/jean-michel.muller/TMDworstcases.pdf", abstract = "We give the results of our search for the worst cases for correct rounding of the major elementary functions in double precision floating-point arithmetic. These results allow the design of reasonably fast routines that will compute these functions with correct rounding, at least in some interval, for any of the four rounding modes specified by the IEEE-754 standard. They will also allow one to easily test libraries that are claimed to provide correctly rounded functions.", acknowledgement = ack-nhfb, keywords = "computer arithmetic; elementary functions; floating-point arithmetic; Table Maker's Dilemma", } @Article{Lozier:2003:NDL, author = "Daniel W. Lozier", title = "{NIST Digital Library of Mathematical Functions}", journal = j-ANN-MATH-ARTIF-INTELL, volume = "38", number = "1--3", pages = "105--119", month = may, year = "2003", CODEN = "AMAIEC", ISSN = "1012-2443 (print), 1573-7470 (electronic)", ISSN-L = "1012-2443", bibdate = "Fri Jul 09 06:23:08 2004", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://math.nist.gov/acmd/Staff/DLozier/publications/Linz01.ps", acknowledgement = ack-nhfb, fjournal = "Annals of Mathematics and Artificial Intelligence", journal-URL = "http://link.springer.com/journal/10472", } @InProceedings{Markstein:2003:FQP, author = "Peter Markstein", title = "A fast quad precision elementary function library for {Itanium}", crossref = "Anonymous:2003:CRN", pages = "5--12", year = "2003", bibdate = "Fri Jun 24 20:14:39 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", abstract = "This talk will describe Itanium's floating point architecture and how it has been used to produce a high performance, highly accurate quad precision elementary function library.\par Itanium's floating-point features will first be described, from the point of view of a computer architect. Many conflicting requirements vie for consideration during the design of a new computer architecture. These include instruction word size, number of registers, the set of operations, arithmetic precisions supported, and memory access. Some of the trade-offs during the design phase will be discussed.\par One of the objectives of the original Itanium design was to accelerate quad precision arithmetic. The talk will describe how the Itanium elementary function library was constructed, with attention to performance and accuracy. Because a pair of double-extended floating point words are used for internal operations involving quad precision numbers, intermediate results, holding 128 bits, provide 15 guard bits during intermediate calculations, resulting in a very low percentage of misrounded results.", acknowledgement = ack-nhfb, } @Book{Mason:2003:CP, author = "J. C. Mason and D. C. Handscomb", title = "{Chebyshev} Polynomials", publisher = pub-CHAPMAN-HALL-CRC, address = pub-CHAPMAN-HALL-CRC:adr, pages = "xiii + 341", year = "2003", ISBN = "0-8493-0355-9", ISBN-13 = "978-0-8493-0355-5", LCCN = "QA404.5 .M37 2003", bibdate = "Fri Apr 17 09:45:35 MDT 2009", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcw.bib; z3950.loc.gov:7090/Voyager", acknowledgement = ack-nhfb, subject = "Chebyshev polynomials", tableofcontents = "1 Definitions / 1 \\ 1.1 Preliminary remarks / 1 \\ 1.2 Trigonometric definitions and recurrences / 1 \\ 1.2.1 The first-kind polynomial $T_n$ / 2 \\ 1.2.2 The second-kind polynomial $U_n$ / 3 \\ 1.2.3 The third- and fourth-kind polynomials $V_n$ and $W_n$ (the airfoil polynomials) / 5 \\ 1.2.4 Connections between the four kinds of polynomial / 7 \\ 1.3 Shifted Chebyshev polynomials / 9 \\ 1.3.1 The shifted polynomials $T_n^*$, $U_n^*$, $V_n^*$, $W_n^*$ / 9 \\ 1.3.2 Chebyshev polynomials for the general range $[a, b]$ / 11 \\ 1.4 Chebyshev polynomials of a complex variable / 11 \\ 1.4.1 Conformal mapping of a circle to and from an ellipse / 12 \\ 1.4.2 Chebyshev polynomials in $z$ / 14 \\ 1.4.3 Shabat polynomials / 17 \\ 1.5 Problems for Chapter 1 / 17 \\ 2 Basic Properties and Formulae / 19 \\ 2.1 Introduction / 19 \\ 2.2 Chebyshev polynomial zeros and extrema / 19 \\ 2.3 Relations between Chebyshev polynomials and powers of x / 22 \\ 2.3.1 Powers of $x$ in terms of $\{T_n(x)\}$ / 22 \\ 2.3.2 $T_n(x)$ in terms of powers of $x$ / 23 \\ 2.3.3 Ratios of coefficients in $T_n(x)$ / 25 \\ 2.4 Evaluation of Chebyshev sums, products, integrals and derivatives / 25 \\ 2.4.1 Evaluation of a Chebyshev sum / 25 \\ 2.4.2 Stability of the evaluation of a Chebyshev sum / 29 \\ 2.4.3 Evaluation of a product / 31 \\ 2.4.4 Evaluation of an integral / 32 \\ 2.4.5 Evaluation of a derivative / 34 \\ 6.3.1 Aliasing / 152 \\ 6.3.2 Second-kind interpolation / 155 \\ 6.3.3 Third- and fourth-kind interpolation / 156 \\ 6.3.4 Conditioning / 158 \\ 6.4 Best $\mathcal{L}_1$ approximation by Chebyshev interpolation / 158 \\ 6.5 Near-minimax approximation by Chebyshev interpolation / 160 \\ 6.6 Problems for Chapter 6 / 162 \\ 7 Near-Best $\mathcal{L}_\infty$, $\mathcal{L}_1$ and $\mathcal{L}_p$ Approximations / 165 \\ 7.1 Near-best $\mathcal{L}_\infty$ (near-minimax) approximations / 165 \\ 7.1.1 Second-kind expansions in $\mathcal{L}_\infty$ / 165 \\ 7.1.2 Third-kind expansions in $\mathcal{L}_\infty$ / 167 \\ 7.2 Near-best $\mathcal{L}_1$ approximations / 169 \\ 7.3 Best and near-best $\mathcal{L}_p$ approximations / 170 \\ 7.3.1 Complex variable results for elliptic-type regions / 172 \\ 7.4 Problems for Chapter 7 / 173 \\ 8 Integration Using Chebyshev Polynomials / 177 \\ 8.1 Indefinite integration with Chebyshev series / 177 \\ 8.2 Gauss--Chebyshev quadrature / 180 \\ 8.3 Quadrature methods of Clenshaw--Curtis type / 186 \\ 8.3.1 Introduction / 186 \\ 8.3.2 First-kind formulae / 187 \\ 8.3.3 Second-kind formulae / 189 \\ 8.3.4 Third-kind formulae / 191 \\ 8.3.5 General remark on methods of Clenshaw--Curtis type / 192 \\ 8.4 Error estimation for Clenshaw--Curtis methods / 192 \\ 8.4.1 First-kind polynomials / 193 \\ 8.4.2 Fitting an exponential curve / 195 \\ 8.4.3 Other abscissae and polynomials / 196 \\ 8.5 Some other work on Clenshaw--Curtis methods / 200 \\ 8.6 Problems for Chapter 8 / 201 \\ 9 Solution of Integral Equations / 203 \\ 9.1 Introduction / 203 \\ 9.2 Fredholm equations of the second kind / 204 \\ 9.3 Fredholm equations of the third kind / 206 \\ 9.4 Fredholm equations of the first kind / 207 \\ 9.5 Singular kernels / 209 \\ 9.5.1 Hilbert-type kernels and related kernels / 209 \\ 9.5.2 Symm's integral equation / 212 \\ 9.6 Regularisation of integral equations / 214 \\ 9.6.1 Discrete data with second derivative regularisation / 214 \\ 9.6.2 Details of a smoothing algorithm (second derivative regularisation) / 215 \\ 9.6.3 A smoothing algorithm with weighted function regularisation / 217 \\ 9.6.4 Evaluation of $V(\lambda)$ / 220 \\ 9.6.5 Other basis functions / 221 \\ 9.7 Partial differential equations and boundary integral equation methods / 222 \\ 9.7.1 A hypersingular integral equation derived from a mixed boundary value problem for Laplace's equation / 222 \\ 9.8 Problems for Chapter 9 / 227 \\ 10 Solution of Ordinary Differential Equations / 231 \\ 10.1 Introduction / 231 \\ 10.2 A simple example / 232 \\ 10.2.1 Collocation methods / 234 \\ 10.2.2 Error of the collocation method / 237 \\ 10.2.3 Projection (tau) methods / 239 \\ 10.2.4 Error of the preceding projection method / 241 \\ 10.3 The original Lanczos tau ($\tau$) method / 242 \\ 10.4 A more general linear equation / 244 \\ 10.4.1 Collocation method / 244 \\ 10.4.2 Projection method / 245 \\ 10.5 Pseudospectral methods --- another form of collocation / 245 \\ 10.5.1 Differentiation matrices / 246 \\ 10.5.2 Differentiation matrix for Chebyshev points / 247 \\ 10.5.3 Collocation using differentiation matrices / 249 \\ 10.6 Nonlinear equations / 251 \\ 10.7 Eigenvalue problems / 252 \\ 10.7.1 Collocation methods / 252 \\ 10.7.2 Collocation using the differentiation matrix / 254 \\ 10.8 Differential equations in one space and one time dimension / 256 \\ 10.8.1 Collocation methods / 257 \\ 10.8.2 Collocation using the differentiation matrix / 258 \\ 10.9 Problems for Chapter 10 / 259 \\ 11 Chebyshev and Spectral Methods for Partial Differential Equations / 261 \\ 11.1 Introduction / 261 \\ 11.2 Interior, boundary and mixed methods / 262 \\ 11.2.1 Interior methods / 262 \\ 11.2.2 Boundary methods / 263 \\ 11.2.3 Mixed methods / 265 \\ 11.3 Differentiation matrices and nodal representation / 265 \\ 11.4 Method of weighted residuals / 265 \\ 11.4.1 Continuous MWR / 265 \\ 11.4.2 Discrete MWR --- a new nomenclature / 266 \\ 11.5 Chebyshev series and Galerkin methods / 267 \\ 11.6 Collocation/interpolation and related methods / 269 \\ 11.7 PDE methods / 271 \\ 11.7.1 Error analysis / 272 \\ 11.8 Some PDE problems and various methods / 272 \\ 11.8.1 Power basis: collocation for Poisson problem / 273 \\ 11.8.2 Power basis: interior collocation for the L-membrane / 275 \\ 11.8.3 Chebyshev basis and discrete orthogonalisation / 278 \\ 11.8.4 Differentiation matrix approach: Poisson problem / 281 \\ 11.8.5 Explicit collocation for the quasilinear Dirichlet problem: Chebyshev basis / 283 \\ 11.9 Computational fluid dynamics / 295 \\ 11.10 Particular issues in spectral methods / 296 \\ 11.11 More advanced problems / 297 \\ 11.12 Problems for Chapter 11 / 298 \\ 12 Conclusion / 303 \\ Bibliography / 305 \\ Appendices: \\ A Biographical Note / 321 \\ B Summary of Notations, Definitions and Important Properties / 323 \\ B.I Miscellaneous notations / 323 \\ B.2 The four kinds of Chebyshev polynomial / 325 \\ C Tables of Coefficients / 329 \\ Index / 335", xxauthor = "J. C. Mason and D. C. (David Christopher) Handscomb", xxURL = "http://www.loc.gov/catdir/enhancements/fy0646/2002073578-d.html", } @InProceedings{Meunier:2003:EAG, author = "Ludovic Meunier and Bruno Salvy", editor = "Hoon Hong", booktitle = "Proceedings of the 2003 International Symposium on Symbolic and Algebraic Computation: {Philadelphia, PA, USA, August 3--6, 2003}", title = "{ESF}: an automatically generated encyclopedia of special functions", publisher = pub-ACM, address = pub-ACM:adr, month = aug, year = "2003", DOI = "https://doi.org/10.1145/860854.860898", ISBN = "1-58113-641-2 (paperback)", ISBN-13 = "978-1-58113-641-8 (paperback)", LCCN = "QA76.5 S98 2003; QA76.95.I59 2003", bibdate = "Sat Nov 11 06:21:45 2023", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", abstract = "We present our on-going work on the automatic generation of an encyclopedia of special functions on the web, called The Encyclopedia of Special Functions (ESF) (\url{http://algo.inria.fr/esf}). All mathematical formulae in the ESF are computed, typeset and displayed without any human intervention. This is achieved by exploiting a collection of computer algebra algorithms in a systematic way, on top of a specially designed data structure for a class of special functions.", acknowledgement = ack-nhfb, book-DOI = "https://doi.org/10.1145/860854", book-URL = "https://dl.acm.org/doi/proceedings/10.1145/860854", } @Article{Ovtchinnikov:2003:CEGb, author = "E. Ovtchinnikov", title = "Convergence Estimates for the Generalized {Davidson} Method for Symmetric Eigenvalue Problems {II}: The Subspace Acceleration", journal = j-SIAM-J-NUMER-ANAL, volume = "41", number = "1", pages = "272--286", month = feb, year = "2003", CODEN = "SJNAAM", DOI = "https://doi.org/10.1137/S0036142902411768", ISSN = "0036-1429 (print), 1095-7170 (electronic)", ISSN-L = "0036-1429", bibdate = "Fri Aug 15 05:57:09 MDT 2003", bibsource = "http://epubs.siam.org/sam-bin/dbq/toc/SINUM/41/1; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://epubs.siam.org/sam-bin/dbq/article/41176", acknowledgement = ack-nhfb, fjournal = "SIAM Journal on Numerical Analysis", journal-URL = "http://epubs.siam.org/sinum", keywords = "convergence acceleration", } @Article{Paris:2003:AEG, author = "R. B. Paris", title = "The asymptotic expansion of a generalised incomplete gamma function", journal = j-J-COMPUT-APPL-MATH, volume = "151", number = "2", pages = "297--306", day = "15", month = feb, year = "2003", CODEN = "JCAMDI", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Sat Feb 25 12:52:33 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2000.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0377042702008099", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @Article{Paszkowski:2003:CAS, author = "Stefan Paszkowski", title = "Convergence Acceleration of Some Continued Fractions", journal = j-NUMER-ALGORITHMS, volume = "32", number = "2--4", pages = "193--247", month = apr, year = "2003", CODEN = "NUALEG", ISSN = "1017-1398 (print), 1572-9265 (electronic)", ISSN-L = "1017-1398", bibdate = "Mon Sep 29 08:37:11 MDT 2003", bibsource = "http://www.kluweronline.com/issn/1017-1398; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://ipsapp007.kluweronline.com/content/getfile/5058/45/5/abstract.htm; http://ipsapp007.kluweronline.com/content/getfile/5058/45/5/fulltext.pdf", acknowledgement = ack-nhfb, fjournal = "Numerical Algorithms", journal-URL = "http://link.springer.com/journal/11075", keywords = "convergence acceleration", } @Article{Pedersen:2003:DGF, author = "Henrik L. Pedersen", title = "The double gamma function and related {Pick} functions", journal = j-J-COMPUT-APPL-MATH, volume = "153", number = "1--2", pages = "361--369", day = "1", month = apr, year = "2003", CODEN = "JCAMDI", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Sat Feb 25 12:52:34 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2000.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0377042702006040", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @Article{Petropoulou:2003:CZB, author = "Eugenia N. Petropoulou and Panayiotis D. Siafarikas and Ioannis D. Stabolas", title = "On the common zeros of {Bessel} functions", journal = j-J-COMPUT-APPL-MATH, volume = "153", number = "1--2", pages = "387--393", day = "1", month = apr, year = "2003", CODEN = "JCAMDI", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Sat Feb 25 12:52:34 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2000.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0377042702006416", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @InProceedings{Pineiro:2003:LHR, author = "J.-A. Pineiro and J. D. Bruguera and M. D. Ercegovac", booktitle = "{ISCAS '03. Proceedings of the 2003 International Symposium on Circuits and Systems. 25--28 May 2003}", title = "On-line high-radix exponential with selection by rounding", volume = "4", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "IV-121--IV-124", year = "2003", CODEN = "????", ISSN = "????", bibdate = "Sat Jul 16 11:25:05 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, summary = "An on-line high-radix algorithm for computing the exponential function (e/sup x/) with arbitrary precision n is presented. Selection by rounding and a redundant digit-set for the digits e/sub j/ are used, with selection by table in the first \ldots{}", } @Book{Sidi:2003:PEM, author = "Avram Sidi", title = "Practical Extrapolation Methods: Theory and Applications", volume = "10", publisher = pub-CAMBRIDGE, address = pub-CAMBRIDGE:adr, pages = "xxii + 519", year = "2003", DOI = "https://doi.org/10.1017/CBO9780511546815", ISBN = "0-521-66159-5 (hardcover), 0-511-54681-5 (e-book), 0-511-06018-1 (Adobe Reader), 0-511-06649-X (e-book)", ISBN-13 = "978-0-521-66159-1 (hardcover), 978-0-511-54681-5 (e-book), 978-0-511-06018-2 (Adobe Reader), 978-0-511-06649-8 (e-book)", LCCN = "QA281 .S555 2003", bibdate = "Mon Jul 5 16:49:09 MDT 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; z3950.loc.gov:7090/Voyager", series = "Cambridge monographs on applied and computational mathematics", acknowledgement = ack-nhfb, subject = "extrapolation", tableofcontents = "Preface / xix--xxii \\ Introduction / 1--18 \\ I: The Richardson Extrapolation Process and Its Generalizations / 19--20 \\ 1: The Richardson Extrapolation Process / 21--41 \\ 2: Additional Topics in Richardson Extrapolation / 42--56 \\ 3: First Generalization of the Richardson Extrapolation Process / 57--80 \\ 4: GREP: Further Generalization of the Richardson Extrapolation Process / 81--94 \\ 5: The $D$-Transformation: A GREP for Infinite-Range Integrals / 95--120 \\ 6: The $d$-Transformation: A GREP for Infinite Series and Sequences / 121--157 \\ 7: Recursive Algorithms for GREP / 158--175 \\ 8: Analytic Study of GREP(1): Slowly Varying $A(y) \in F^{(1)}$ / 176--202 \\ 9: Analytic Study of GREP(1): Quickly Varying $A(y) \in F^{(1)}$ / 203--211 \\ 10: Efficient Use of GREP(1): Applications to the $D(1)$-, $d(1)$-, and $d(m)$-Transformations / 212--217 \\ 11: Reduction of the $D$-Transformation for Oscillatory Infinite-Range Integrals: The $\bar{D}$-, $D'$-, $W$-, and $mW$-Transformations / 218--237 \\ 12: Acceleration of Convergence of Power Series by the $d$-Transformation: Rational $d$-Approximants / 238--252 \\ 13: Acceleration of Convergence of Fourier and Generalized Fourier Series by the $d$-Transformation: The Complex Series Approach with APS / 253--262 \\ 14: Special Topics in Richardson Extrapolation / 263--276 \\ II: Sequence Transformations / 277--278 \\ 15: The Euler Transformation, Aitken 2-Process, and Lubkin W-Transformation / 279--296 \\ 16: The Shanks Transformation / 297--322 \\ 17: The Pad{\'e} Table / 323--347 \\ 18: Generalizations of Pad{\'e} Approximants / 348--362 \\ 19: The Levin- and Sidi $S$-Transformations / 363--374 \\ 20: The Wynn- and Brezinski-Algorithms / 375--383 \\ 21: The $G$-Transformation and Its Generalizations / 384--389 \\ 22: The Transformations of Overholt and Wimp / 390--395 \\ 23: Confluent Transformations / 396--406 \\ 24: Formal Theory of Sequence Transformations / 407--412 \\ III: Further Applications / 413--414 \\ 25: Further Applications of Extrapolation Methods and Sequence Transformations / 415--456 \\ IV: Appendices / 457--458 \\ A: Review of Basic Asymptotics / 459--462 \\ B: The Laplace Transform and Watson's Lemma / 463--464 \\ C: The Gamma Function / 465--466 \\ D: Bernoulli Numbers and Polynomials and the Euler Maclaurin Formula / 467--476 \\ E: The Riemann Zeta Function and the Generalized Zeta Function / 477--479 \\ F: Some Highlights of Polynomial Approximation Theory / 480--482 \\ G: A Compendium of Sequence Transformations / 483--487 \\ H: Efficient Application of Sequence Transformations: Summary / 488--492 \\ I: FORTRAN 77 Program for the $d(m)$-Transformation / 493--500 \\ Bibliography / 501--514 \\ Index / 515--519", xxURL = "http://www.loc.gov/catdir/samples/cam033/2002024669.html; http://www.loc.gov/catdir/description/cam022/2002024669.html; http://www.loc.gov/catdir/toc/cam024/2002024669.html", } @Misc{Tkachev:2003:EFI, author = "Vladimir G. Tkachev", title = "Elliptic functions: Introduction course", howpublished = "Web lecture notes.", day = "25", month = nov, year = "2003", bibdate = "Wed Mar 15 08:43:21 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://users.mai.liu.se/vlatk48/teaching/lect2-agm.pdf", acknowledgement = ack-nhfb, tableofcontents = "Chapter 1. Elliptic integrals and Jacobi's theta functions / 5 \\ 1.1. Elliptic integrals and the AGM: real case / 5 \\ 1.1.3. The arithmetic-geometric mean iteration / 7 \\ 1.2. Lemniscates and elastic curves / 11 \\ 1.3. Euler's addition theorem / 18 \\ 1.4. Theta functions: preliminaries 5 / 24 \\ Chapter 2. General theory of doubly periodic functions / 31 \\ 2.1. Preliminaries / 31 \\ 2.2. Periods of analytic functions / 33 \\ 2.3. Existence of doubly periodic functions / 36 \\ 2.4. Liouville's theorems / 38 \\ 2.5. The Weierstrass function $\wp(z)$ / 43 \\ 2.6. Modular forms / 51 \\ Bibliography / 61", } @InProceedings{Wang:2003:TDF, author = "Xiaojun Wang and B. E. Nelson", booktitle = "{FCCM 2003}: 11th Annual {IEEE} Symposium on Field-Programmable Custom Computing Machines, 9--11 April 2003", title = "Tradeoffs of designing floating-point division and square root on {Virtex FPGAs}", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "195--203", year = "2003", CODEN = "????", ISSN = "????", bibdate = "Sat Jul 16 15:28:14 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, summary = "Low latency, high throughput and small area are three major design considerations of an FPGA (field programmable gate array) design. In this paper, we present a high radix SRT division algorithm and a binary restoring square root algorithm. We \ldots{}", } @Article{Yousif:2003:CBF, author = "Hashim A. Yousif and Richard Melka", title = "Computing {Bessel} functions of the second kind in extreme parameter regimes", journal = j-COMP-PHYS-COMM, volume = "151", number = "1", pages = "25--34", day = "1", month = mar, year = "2003", CODEN = "CPHCBZ", DOI = "https://doi.org/10.1016/S0010-4655(02)00697-5", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", bibdate = "Mon Feb 13 23:41:27 MST 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/compphyscomm2000.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathematica.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0010465502006975", abstract = "A useful method of computing the integral order Bessel functions of the second kind $ Y_n(x + i y) $ when either, the absolute value of the real part, or the imaginary part of the argument $ z = x + i y $ is small, is described. This method is based on computing the Bessel functions for extreme parameter regimes when $ x \sim 0 $ (or $ y \sim 0 $ ) and is useful because a number existing algorithms and methods fail to give correct results for small $x$ or small $y$. The approximating equations are derived by expanding the Bessel function in Taylor series, are tested and discussed. The present work is a continuation of the previous one conducted in regard to the Bessel function of the first kind. The results of our formalism are compared to the available existing numerical methods used in Mathematica, IMSL, MATLAB, and the Amos library. Our numerical method is easy to implement, efficient, and produces reliable results. In addition, this method reduces the computation of the Bessel functions of the second complex argument to that of real argument which simplify the computation considerably.", acknowledgement = ack-nhfb, fjournal = "Computer Physics Communications", journal-URL = "http://www.sciencedirect.com/science/journal/00104655", } @Book{Zwillinger:2003:CSM, editor = "Daniel Zwillinger", title = "{CRC} Standard Mathematical Tables and Formulae", publisher = pub-CHAPMAN-HALL-CRC, address = pub-CHAPMAN-HALL-CRC:adr, edition = "31st", pages = "xiv + 910", year = "2003", ISBN = "1-58488-291-3 (hardcover), 1-4200-3534-7 (e-book)", ISBN-13 = "978-1-58488-291-6 (hardcover), 978-1-4200-3534-6 (e-book)", LCCN = "QA47 .M315 2003", bibdate = "Thu Nov 25 11:07:20 MST 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcw.bib; prodorbis.library.yale.edu:7090/voyager", acknowledgement = ack-nhfb, subject = "mathematics; tables", tableofcontents = "Preface \\ Contributors \\ Table of Contents \\ 1 Analysis \\ 2 Algebra \\ 3 Discrete Mathematics \\ 4 Geometry \\ 5 Continuous Mathematics \\ 6 Special Functions \\ 7 Probability and Statistics \\ 8 Scientific Computing \\ 9 Financial Analysis \\ 10 Miscellaneous \\ List of References \\ List of Figures \\ List of Notation \\ Index", } @Book{Bell:2004:SFS, author = "W. W. (William Wallace) Bell", title = "Special Functions for Scientists and Engineers", publisher = pub-DOVER, address = pub-DOVER:adr, pages = "xiv + 247", year = "2004", ISBN = "0-486-43521-0", ISBN-13 = "978-0-486-43521-3", LCCN = "QA351 .B4 2004", bibdate = "Sat Oct 30 16:30:44 MDT 2010", bibsource = "fsz3950.oclc.org:210/WorldCat; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", series = "Dover books on mathematics", acknowledgement = ack-nhfb, remark = "Reprinting of \cite{Bell:1968:SFS}.", subject = "Functions, Special", tableofcontents = "Preface / v List of Symbols / ix Series Solution of Differential Equations Method of Frobenius / 1 \\ Examples / 8 \\ Problems / 21 \\ Gamma and Beta Functions Definitions / 23 \\ Properties of the beta and gamma functions / 24 \\ Definition of the gamma function for negative values of the argument / 30 \\ Examples / 37 \\ Problems / 40 \\ Legendre Polynomials and Functions Legendre's equation and its solution / 42 \\ Generating function for the Legendre polynomials / 46 \\ Further expressions for the Legendre polynomials / 48 \\ Explicit expressions for and special values of the Legendre polynomials / 50 \\ Orthogonality properties of the Legendre polynomials / 52 \\ Legendre series / 55 \\ Relations between the Legendre polynomials and their derivatives; / 58 \\ recurrence relations Associated Legendre functions / 62 \\ Properties of the associated Legendre functions / 65 \\ Legendre functions of the second kind / 70 \\ Spherical harmonics / 78 \\ Graphs of the Legendre functions / 82 \\ Examples / 85 \\ Problems / 90 \\ Bessel Functions Bessel's equation and its solutions; Bessel functions of the first and second kind / 92 \\ Generating function for the Bessel functions / 99 \\ Integral representations for Bessel functions / 101 \\ Recurrence relations / 104 \\ Hankel functions / 107 \\ Equations reducible to Bessel's equation / 108 \\ Modified Bessel functions / 110 \\ Recurrence relations for the modified Bessel functions / 113 \\ Integral representations for the modified Bessel functions / 116 \\ Kelvin's functions / 120 \\ Spherical Bessel functions / 121 \\ Behaviour of the Bessel functions for large and small values of the argument / 127 \\ Graphs of the Bessel functions / 133 \\ Orthonormality of the Bessel functions; Bessel series / 137 \\ Integrals involving Bessel functions / 141 \\ Examples / 148 \\ Problems / 154 \\ Hermite Polynomials Hermite's equation and its solution / 156 \\ Generating function / 157 \\ Other expressions for the Hermite polynomials / 158 \\ Explicit expressions for, and special values of, the Hermite polynomials / 160 \\ Orthogonality properties of the Hermite polynomials / 161 \\ Relations between Hermite polynomials and their derivatives; recurrence relations / 162 \\ Weber--Hermite functions / 163 \\ Examples / 164 \\ Problems / 166 \\ Laguerre Polynomials Laguerre's equation and its solution / 168 \\ Generating function / 169 \\ Alternative expression for the Laguerre polynomials / 170 \\ Explicit expressions for, and special values of, the Laguerre polynomials / 171 \\ Orthogonality properties of the Laguerre polynomials / 172 \\ Relations between Laguerre polynomials and their derivatives; recurrence relations / 173 \\ Associated Laguerre polynomials / 176 \\ Properties of the associated Laguerre polynomials / 177 \\ Notation / 182 \\ Examples / 182 \\ Problems / 185 \\ Chebyshev Polynomials Definition of Chebyshev polynomials; Chebyshev's equation / 187 \\ Generating function / 190 \\ Orthogonality properties / 192 \\ Recurrence relations / 193 \\ Examples / 194 \\ Problems / 196 \\ Gegenbauer and Jacobi Polynomials Gegenbauer polynomials / 197 \\ Jacobi polynomials / 198 \\ Examples / 200 \\ Problems / 201 \\ Hypergeometric Functions Definition of hypergeometric functions / 203 \\ Properties of the hypergeometric function / 207 \\ Properties of the confluent hypergeometric function / 210 \\ Examples / 212 \\ Problem / 216 \\ Other Special Functions Incomplete gamma functions / 218 \\ Exponential integral and related functions / 218 \\ The error function and related functions / 221 \\ Riemann's zeta function / 223 \\ Debye functions / 224 \\ Elliptic integrals / 224 \\ Examples / 225 \\ Problems / 228 \\ Appendices Convergence of Legendre series / 230 \\ Euler's constant / 231 \\ Differential equations / 233 \\ Orthogonality relations / 234 \\ Generating functions / 236 \\ Hints and Solutions to Problems / 237 \\ Bibliography / 243 \\ Index / 245", } @Article{Berrut:2004:APS, author = "Jean-Paul Berrut and Hans D. Mittelmann", title = "Adaptive Point Shifts in Rational Approximation with Optimized Denominator", journal = j-J-COMPUT-APPL-MATH, volume = "164--165", number = "??", pages = "81--92", day = "1", month = mar, year = "2004", CODEN = "JCAMDI", DOI = "https://doi.org/10.1016/S0377-0427(03)00485-0", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Tue Mar 24 21:10:48 2009", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "Proceedings of the 10th International Congress on Computational and Applied Mathematics University of Leuven, Belgium, 22--26 July 2002. Edited by M. J. Goovaerts, S. Vandewalle, and L. Wuytack.", abstract = "Classical rational interpolation is known to suffer from several drawbacks, such as unattainable points and randomly located poles for a small number of nodes, as well as an erratic behavior of the error as this number grows larger. In a former article, we have suggested to obtain rational interpolants by a procedure that attaches optimally placed poles to the interpolating polynomial, using the barycentric representation of the interpolants. In order to improve upon the condition of the derivatives in the solution of differential equations, we have then experimented with a conformal point shift suggested by Kosloff and Tal-Ezer. As it turned out, such shifts can achieve a spectacular improvement in the quality of the approximation itself for functions with a large gradient in the center of the interval. This leads us to the present work which combines the pole attachment method with shifts optimally adjusted to the interpolated function. Such shifts are also constructed for functions with several shocks away from the extremities of the interval.", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", keywords = "interpolation; optimal interpolation; point shifts; rational approximation", } @InCollection{Borwein:2004:AGMa, author = "J. M. Borwein and P. B. Borwein", title = "The Arithmetic--Geometric Mean and Fast Computation of Elementary Functions", crossref = "Berggren:2004:PSB", pages = "537--552", year = "2004", DOI = "https://doi.org/10.1007/978-1-4757-4217-6_56", bibdate = "Thu Aug 11 09:36:22 MDT 2016", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib; https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "Reprint of \cite{Borwein:1984:AGM}.", URL = "http://link.springer.com/chapter/10.1007/978-1-4757-4217-6_56", acknowledgement = ack-nhfb, author-dates = "Jonathan Michael Borwein (20 May 1951--2 August 2016)", ORCID-numbers = "Borwein, Jonathan/0000-0002-1263-0646", } @InProceedings{Boudabous:2004:IHF, author = "A. Boudabous and F. Ghozzi and M. W. Kharrat and N. Masmoudi", booktitle = "{Proceedings. The 16th International Conference on Microelectronics, 2004. ICM 2004.}", title = "Implementation of hyperbolic functions using {CORDIC} algorithm", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "738--741", year = "2004", DOI = "https://doi.org/10.1109/ICM.2004.1434772", bibdate = "Mon Oct 27 10:32:44 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, keywords = "Algorithm design and analysis; Calculators; Calculus; Computational modeling; Convergence; Design engineering; Equations; Field programmable gate arrays; Hardware; Information technology", } @Article{Brisebarre:2004:ACR, author = "N. Brisebarre and J.-M. Muller and Saurabh Kumar Raina", title = "Accelerating correctly rounded floating-point division when the divisor is known in advance", journal = j-IEEE-TRANS-COMPUT, volume = "53", number = "8", pages = "1069--1072", month = aug, year = "2004", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/TC.2004.37", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Sat Jul 16 08:40:52 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", abstract = "We present techniques for accelerating the floating-point computation of $ x / y $ when $y$ is known before $x$. The proposed algorithms are oriented toward architectures with available fused-mac operations. The goal is to get exactly the same result as with \ldots{}", acknowledgement = ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", summary = "We present techniques for accelerating the floating-point computation of x/y when y is known before x. The proposed algorithms are oriented toward architectures with available fused-mac operations. The goal is to get exactly the same result as with \ldots{}", } @Article{Chaudhry:2004:EHC, author = "M. Aslam Chaudhry and Asghar Qadir and H. M. Srivastava and R. B. Paris", title = "Extended hypergeometric and confluent hypergeometric functions", journal = j-APPL-MATH-COMP, volume = "159", number = "2", pages = "589--602", day = "6", month = dec, year = "2004", CODEN = "AMHCBQ", ISSN = "0096-3003 (print), 1873-5649 (electronic)", ISSN-L = "0096-3003", bibdate = "Mon Jul 4 09:15:38 MDT 2005", bibsource = "http://www.sciencedirect.com/science/journal/00963003; https://www.math.utah.edu/pub/tex/bib/applmathcomput2000.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Applied Mathematics and Computation", journal-URL = "http://www.sciencedirect.com/science/journal/00963003", } @Article{Croot:2004:ACC, author = "Ernie Croot and Ren-Cang Li and H. J. Hui June Zhu", title = "The {\em abc\/} conjecture and correctly rounded reciprocal square roots", journal = j-THEOR-COMP-SCI, volume = "315", number = "2--3", pages = "405--417", day = "6", month = may, year = "2004", CODEN = "TCSCDI", ISSN = "0304-3975 (print), 1879-2294 (electronic)", ISSN-L = "0304-3975", bibdate = "Thu Nov 4 10:19:15 MST 2004", bibsource = "http://www.sciencedirect.com/science/journal/03043975; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/tcs2000.bib", abstract = "The reciprocal square root calculation $ \alpha = 1 / \sqrt {x} $ is very common in scientific computations. Having a correctly rounded implementation of it is of great importance in producing numerically predictable code among today's heterogeneous computing environment. Existing results suggest that to get the correctly rounded $ \alpha $ in a floating point number system with $p$ significant bits, we may have to compute up to $ 3 p + 1 $ leading bits of $ \alpha $. However, numerical evidence indicates the actual number may be as small as $ 2 p $ plus a few more bits. This paper attempts to bridge the gap by showing that this is indeed true, assuming the {\em abc\/} conjecture which is widely purported to hold. (But our results do not tell exactly how many more bits beyond the $ 2 p $ bits, due to the fact that the constants involved in the conjecture are ineffective.) Along the way, rough bounds which are comparable to the existing ones are also proven. The technique used here is a combination of the classical Liouville's estimation and contemporary number theory.", acknowledgement = ack-nhfb, fjournal = "Theoretical Computer Science", journal-URL = "http://www.sciencedirect.com/science/journal/03043975", } @InProceedings{deDinechin:2004:PCR, author = "Florent de Dinechin and Catherine Loirat and Jean-Michel Muller", title = "A proven correctly rounded logarithm in double-precision", crossref = "Frougny:2004:RCR", pages = "71--85", year = "2004", bibdate = "Fri Nov 17 07:00:31 2023", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "http://www.informatik.uni-trier.de/Reports/TR-08-2004/rnc6_07_dinechin.pdf", abstract = "This article is a case study in the implementation of a proven, portable, and efficient correctly rounded elementary function in double-precision. We describe the methodology used in the implementation of the natural logarithm in the crlibm library. The discipline required to prove a tight bound on the overall evaluation error allows to design a very efficient implementation with moderate effort.", acknowledgement = ack-nhfb, keywords = "arithmetic; correct rounding; elementary functions; floating-point; libm; logarithm", } @Article{Defour:2004:PSM, author = "David Defour and Guillaume Hanrot and Vincent Lef{\`e}vre and Jean-Michel Muller and Nathalie Revol and Paul Zimmermann", title = "Proposal for a Standardization of Mathematical Function Implementation in Floating-Point Arithmetic", journal = j-NUMER-ALGORITHMS, volume = "37", number = "1--4", pages = "367--375", month = dec, year = "2004", CODEN = "NUALEG", ISSN = "1017-1398 (print), 1572-9265 (electronic)", ISSN-L = "1017-1398", bibdate = "Mon Dec 6 07:00:28 MST 2004", bibsource = "http://www.kluweronline.com/issn/1017-1398; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://ipsapp009.kluweronline.com/IPS/content/ext/x/J/5058/I/58/A/30/abstract.htm; http://perso.ens-lyon.fr/jean-michel.muller/NumAlg04.pdf; http://www.loria.fr/~zimmerma/papers/PropStandFunctions.pdf", acknowledgement = ack-nhfb, fjournal = "Numerical Algorithms", journal-URL = "http://link.springer.com/journal/11075", remark = "Special Issue: SCAN'2002 International Conference (Guest Editors: Ren {\'e} Alt and Jean-Luc Lamotte)", } @InProceedings{Doss:2004:FBI, author = "C. C. Doss and R. L. {Riley, Jr.}", booktitle = "{FCCM 2004}. 12th Annual {IEEE} Symposium on Field-Programmable Custom Computing Machines, 20--23 April 2004", title = "{FPGA}-based implementation of a robust {IEEE-754} exponential unit", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "229--238", year = "2004", CODEN = "????", ISSN = "????", bibdate = "Sat Jul 16 17:14:11 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, summary = "This work explores the feasibility of implementing a floating-point exponentiation unit on reconfigurable computing systems. A table-driven exponentiation unit was implemented using synthesizable VHDL. The project included creating pipelined \ldots{}", } @TechReport{Ercegovac:2004:CSRa, author = "Milo{\v{s}} Ercegovac and Jean-Michel Muller", title = "Complex Square Root with Operand Prescaling", type = "Research Report", number = "RR2004-42", institution = "{\'E}cole Normale Sup{\'e}rieure de Lyon", address = "69364 Lyon Cedex 07, France", pages = "2 + 12", month = sep, year = "2004", bibdate = "Mon Dec 06 11:07:40 2004", bibsource = "http://www.ens-lyon.fr/LIP/Pub/rr2004.php; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "http://www.ens-lyon.fr/LIP/Pub/Rapports/RR/RR2004/RR2004-42.pdf", abstract = "We propose a radix-$r$ digit-recurrence algorithm for complex square-root. The operand is prescaled to allow the selection of square-root digits by rounding of the residual. This leads to a simple hardware implementation. Moreover, the use of digit recurrence approach allows correct rounding of the result. The algorithm, compatible with the complex division, and its design are described at a high-level. We also give rough comparisons of its latency and cost with respect to implementation based on standard floating-point instructions as used in software routines for complex square root.", acknowledgement = ack-nhfb, keywords = "complex square-root; Computer arithmetic; digit-recurrence algorithm; operand prescaling.", } @InProceedings{Ercegovac:2004:CSRb, author = "Milo{\v{s}} Ercegovac and Jean-Michel Muller", booktitle = "{Proceedings of the 15th IEEE International Conference on Application-Specific Systems, Architectures and Processors, 2004}", title = "Complex square root with operand prescaling", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "52--62", year = "2004", CODEN = "????", DOI = "https://doi.org/10.1109/ASAP.2004.1342458", ISBN = "0-7695-2226-2", ISBN-13 = "978-0-7695-2226-5", ISSN = "1063-6862", bibdate = "Sat Jul 16 15:28:13 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, summary = "We propose a radix-r digit-recurrence algorithm for complex square-root. The operand is prescaled to allow the selection of square-root digits by rounding of the residual. This leads to a simple hardware implementation. Moreover, the use of digit \ldots{}", } @Article{Fabijonas:2004:AAF, author = "B. R. Fabijonas", title = "{Algorithm 838}: {Airy} Functions", journal = j-TOMS, volume = "30", number = "4", pages = "491--501", month = dec, year = "2004", CODEN = "ACMSCU", DOI = "https://doi.org/10.1145/1039813.1039819", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Tue Apr 12 06:34:31 MDT 2005", bibsource = "http://www.acm.org/pubs/contents/journals/toms/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib", abstract = "We present a Fortran 90 module, which computes the solutions and their derivatives of Airy's differential equation, both on the real line and in the complex plane. The module also computes the zeros and associated values of the solutions and their derivatives, and the modulus and phase functions on the negative real axis. The computational methods are numerical integration of the differential equation and summation of asymptotic expansions for large argument. These methods were chosen because they are simple, adaptable to any precision, and amenable to rigorous error analysis. The module can be used to validate other codes or as a component in programs that require Airy functions.", acknowledgement = ack-nhfb, fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", } @Article{Fabijonas:2004:CCA, author = "B. R. Fabijonas and D. W. Lozier and F. W. J. Olver", title = "Computation of complex {Airy} functions and their zeros using asymptotics and the differential equation", journal = j-TOMS, volume = "30", number = "4", pages = "471--490", month = dec, year = "2004", CODEN = "ACMSCU", DOI = "https://doi.org/10.1145/1039813.1039818", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Tue Apr 12 06:34:31 MDT 2005", bibsource = "http://www.acm.org/pubs/contents/journals/toms/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib", abstract = "We describe a method by which one can compute the solutions of Airy's differential equation, and their derivatives, both on the real line and in the complex plane. The computational methods are numerical integration of the differential equation and summation of asymptotic expansions for large argument. We give details involved in obtaining all of the parameter values, and we control the truncation errors rigorously. Using the same computational methods, we describe an algorithm that computes the zeros and associated values of the Airy functions and their derivatives, and the modulus and phase functions on the negative real axis.", acknowledgement = ack-nhfb, fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", } @InProceedings{Gebali:2004:EAF, author = "Fayez Gebali and Mohamed Watheq El-Kharashi", title = "{ERL}: an algorithm for fast evaluation of exponential, reciprocal, and logarithmic functions", crossref = "Wahdan:2004:IHE", pages = "269--272", year = "2004", bibdate = "Sat Jul 16 18:04:58 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", abstract = "A fast algorithm (ERL) is proposed for evaluating Exponential, Reciprocal, and Logarithmic functions. The algorithm requires two to three iterations to complete using simple operations such as multiply, accumulate, and table lookup. The algorithm is independent of the number format used by the machine. Thus it can be implemented using the IEEE 754 floating-point standard or any other special format used by special-purpose processors. The dynamic range of the algorithm is limited only by the dynamic range of the machine on which it is implemented Numerical simulations are performed which verifies the speed and accuracy of the algorithm.", acknowledgement = ack-nhfb, } @Article{Gil:2004:AMB, author = "Amparo Gil and Javier Segura and Nico M. Temme", title = "{Algorithm 831}: {Modified} {Bessel} functions of imaginary order and positive argument", journal = j-TOMS, volume = "30", number = "2", pages = "159--164", month = jun, year = "2004", CODEN = "ACMSCU", DOI = "https://doi.org/10.1145/992200.992204", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Thu Jun 10 07:24:58 MDT 2004", bibsource = "http://www.acm.org/pubs/contents/journals/toms/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib", abstract = "Fortran 77 programs for the computation of modified Bessel functions of purely imaginary order are presented. The codes compute the functions $ K_{ia}(x) $, $ L_{ia}(x) $ and their derivatives for real $a$ and positive $x$; these functions are independent solutions of the differential equation $ x^2 w'' + x w' + (a^2 - x^2)w = 0 $. The code also computes exponentially scaled functions. The range of computation is $ (x, a) \in (0, 1500] \times [ - 1500, 1500] $ when scaled functions are considered and it is larger than $ (0, 500] \times [ - 400, 400] $ for standard IEEE double precision arithmetic. The relative accuracy is better than $ 10^{-13} $ in the range $ (0, 200] \times [ - 200, 200] $ and close to $ 10^{-12} $ in $ (0, 1500] \times [ - 1500, 1500] $.", acknowledgement = ack-nhfb, fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", } @Article{Gil:2004:CRZ, author = "Amparo Gil and Wolfram Koepf and Javier Segura", title = "Computing the Real Zeros of Hypergeometric Functions", journal = j-NUMER-ALGORITHMS, volume = "36", number = "2", pages = "113--134", month = jun, year = "2004", CODEN = "NUALEG", ISSN = "1017-1398 (print), 1572-9265 (electronic)", ISSN-L = "1017-1398", bibdate = "Mon Dec 6 07:00:32 MST 2004", bibsource = "http://www.kluweronline.com/issn/1017-1398; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/numeralgorithms.bib", URL = "http://ipsapp009.kluweronline.com/IPS/content/ext/x/J/5058/I/54/A/2/abstract.htm", acknowledgement = ack-nhfb, fjournal = "Numerical Algorithms", journal-URL = "http://link.springer.com/journal/11075", } @Article{Gil:2004:CSM, author = "Amparo Gil and Javier Segura and Nico M. Temme", title = "Computing solutions of the modified {Bessel} differential equation for imaginary orders and positive arguments", journal = j-TOMS, volume = "30", number = "2", pages = "145--158", month = jun, year = "2004", CODEN = "ACMSCU", DOI = "https://doi.org/10.1145/992200.992203", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Thu Jun 10 07:24:58 MDT 2004", bibsource = "http://www.acm.org/pubs/contents/journals/toms/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib", abstract = "We describe a variety of methods to compute the functions $ K_{ia}(x) $, $ L_{ia}(x) $ and their derivatives for real $a$ and positive $x$. These functions are numerically satisfactory independent solutions of the differential equation $ x^2 w'' + x w' + (a^2 - x^2)w = 0 $. In the accompanying paper [Gil et al. 2004], we describe the implementation of these methods in Fortran 77 codes.", acknowledgement = ack-nhfb, fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", } @Article{Guseinov:2004:EIG, author = "I. I. Guseinov and B. A. Mamedov", title = "Evaluation of Incomplete Gamma Functions Using Downward Recursion and Analytical Relations", journal = j-J-MATH-CHEM, volume = "36", number = "4", pages = "341--346", month = aug, year = "2004", CODEN = "JMCHEG", DOI = "https://doi.org/10.1023/B:JOMC.0000044521.18885.d3", ISSN = "0259-9791 (print), 1572-8897 (electronic)", ISSN-L = "0259-9791", bibdate = "Thu Apr 9 18:14:03 MDT 2015", bibsource = "http://link.springer.com/journal/10910/36/4; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jmathchem.bib", note = "See correction \cite{Guseinov:2025:CEI}.", URL = "http://link.springer.com/article/10.1023/B:JOMC.0000044521.18885.d3", acknowledgement = ack-nhfb, ajournal = "J. Math. Chem.", fjournal = "Journal of Mathematical Chemistry", journal-URL = "http://link.springer.com/journal/10910", journalabr = "J. Math. Chem.", } @Book{Jeffrey:2004:HMF, author = "Alan Jeffrey", title = "Handbook of Mathematical Formulas and Integrals", publisher = pub-ELSEVIER-ACADEMIC, address = pub-ELSEVIER-ACADEMIC:adr, edition = "Third", pages = "xxvi + 453", year = "2004", ISBN = "0-12-382256-4", ISBN-13 = "978-0-12-382256-7", LCCN = "QA47 .J38 2004", bibdate = "Thu May 8 16:02:52 MDT 2008", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcw.bib; z3950.loc.gov:7090/Voyager", acknowledgement = ack-nhfb, subject = "mathematics; tables; formulae", tableofcontents = "0. Quick Reference List of Frequently Used Data 1 \\ 1. Numerical, Algebraic, and Analytical Results for Series and Calculus 25 \\ 2. Functions and Identities 101 \\ 3. Derivatives of Elementary Functions 139 \\ 4. Indefinite Integrals of Algebraic Functions 145 \\ 5. Indefinite Integrals of Exponential Functions 167 \\ 6. Indefinite Integrals of logarithmic Functions 173 \\ 7. Indefinite Integrals of Hyperbolic Functions 179 \\ 8. Indefinite Integrals Involving Inverse Hyperbolic Functions 191 \\ 9. Indefinite Integrals of Trigonometric Functions 197 \\ 10. Indefinite Integrals of Inverse Trigonometric Functions 215 \\ 11. The Gamma, Beta, Pi, and Psi Functions 221 \\ 12. Elliptic Integrals and Functions 229 \\ 13. Probability Integrals and the Error Function 239 \\ 14. Fresnel Integrals, Sine and Cosine Integrals 245 \\ 15. Definite Integrals 249 \\ 16. Different Forms of Fourier Series 257 \\ 17. Bessel Functions 269 \\ 18. Orthogonal Polynomials 285 \\ 19. Laplace Transformation 299 \\ 20. Fourier Transforms 307 \\ 21. Numerical Integration 315 \\ 22. Solutions of Standard Ordinary Differential Equations 321 \\ 23. Vector Analysis 353 \\ 24. Systems of Orthogonal Coordinates 369 \\ 25. Partial Differential Equations and Special Functions 381 \\ 26. the z-Transform 403 \\ 27. Numerical Approximation 409 \\ 28. Solutions of Elliptic, Parabolic, and Hyperbolic Equations 419 \\ 29. Qualitative Properties of the Heat and Laplace Equation 437", xxURL = "http://www.loc.gov/catdir/description/els041/2003049507.html; http://www.loc.gov/catdir/toc/els041/2003049507.html; http://www.e-streams.com/es0710/es0710_3628.html", } @Misc{Kahan:2004:LTC, author = "W. Kahan", title = "A Logarithm Too Clever by Half", howpublished = "World-Wide Web document", pages = "9", day = "9", month = aug, year = "2004", bibdate = "Mon Apr 25 17:39:08 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.cs.berkeley.edu/~wkahan/LOG10HAF.TXT", acknowledgement = ack-nhfb, remark = "Careful analysis of the problem of computing {\tt log10(x)} accurately from {\tt log(x)}.", } @Article{Kalmykov:2004:SEH, author = "M. Y. Kalmykov", title = "Series and $ \epsilon $-expansion of the hypergeometric functions", journal = j-NUCL-PHYS-B-PROC-SUPPL, volume = "135", number = "??", pages = "280--284", month = "????", year = "2004", CODEN = "NPBSE7", ISSN = "0920-5632 (print), 1873-3832 (electronic)", ISSN-L = "0920-5632", bibdate = "Thu Dec 01 09:14:29 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Nuclear Physics B, Proceedings Supplements", journal-URL = "http://www.sciencedirect.com/science/journal/09205632", } @Book{Kilbas:2004:TTA, author = "A. A. (Anatolii Aleksandrovich) Kilbas and Megumi Saigo", title = "{$H$}-transforms: theory and applications", volume = "9", publisher = pub-CHAPMAN-HALL-CRC, address = pub-CHAPMAN-HALL-CRC:adr, pages = "xii + 389", year = "2004", ISBN = "0-203-48737-0, 0-415-29916-0, 1-58488-116-X", ISBN-13 = "978-0-203-48737-2, 978-1-58488-116-2, 978-0-415-29916-9", LCCN = "????", bibdate = "Sat Oct 30 17:20:21 MDT 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; z3950.bibsys.no:2100/BIBSYS", series = "Analytical methods and special functions", acknowledgement = ack-nhfb, subject = "$H$-functions; integral transforms", tableofcontents = "1. Definition, representations and expansions of the H-function \\ 2. Properties of the H-function \\ 3. H-transform on the space [symbol] \\ 4. H-transform on the space [symbol] \\ 5. Modified H-transforms on the space [symbol] \\ 6. G-transform and modified G-transforms on the space [symbol] \\ 7. Hypergeometric type integral transforms on the space [symbol] \\ 8. Bessel type integral transforms on the space [symbol] \\ Bibliography \\ Author Index \\ Subject Index \\ Symbol Index", } @InProceedings{Kucukkabak:2004:DIR, author = "U. Kucukkabak and A. Akkas", editor = "Henry Selvaraj", booktitle = "{DSD 2004: Euromicro Symposium on Digital System Design: Architectures, Methods and Tools. Rennes, France, August 31--September 2, 2004}", title = "Design and implementation of reciprocal unit using table look-up and {Newton--Raphson} iteration", publisher = pub-IEEE, address = pub-IEEE:adr, bookpages = "xiii + 631", pages = "249--253", year = "2004", DOI = "https://doi.org/10.1109/dsd.2004.1333284", ISBN = "0-7695-2203-3", ISBN-13 = "978-0-7695-2203-6", LCCN = "QA76.9.S88 E97 2004; TK7868.D5 E93 2004", bibdate = "Thu Apr 10 14:57:55 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, } @Article{Kyurkchiev:2004:FCN, author = "N. Kyurkchiev and A. Iliev", title = "Failure of convergence of the {Newton--Weierstrass} iterative method for simultaneous root finding of generalized polynomials", journal = j-COMPUT-MATH-APPL, volume = "47", number = "2--3", pages = "441--446", month = jan # "\slash " # feb, year = "2004", CODEN = "CMAPDK", ISSN = "0898-1221 (print), 1873-7668 (electronic)", ISSN-L = "0898-1221", bibdate = "Wed Mar 1 21:49:35 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/computmathappl2000.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0898122104900363", acknowledgement = ack-nhfb, fjournal = "Computers and Mathematics with Applications", journal-URL = "http://www.sciencedirect.com/science/journal/08981221", } @Book{Lau:2004:NLJ, author = "H. T. (Hang Tong) Lau", title = "A Numerical Library in {Java} for Scientists and Engineers", publisher = pub-CHAPMAN-HALL-CRC, address = pub-CHAPMAN-HALL-CRC:adr, pages = "xxiii + 1063", year = "2004", DOI = "https://doi.org/10.1201/9780203507643", ISBN = "1-58488-430-4", ISBN-13 = "978-1-58488-430-9", LCCN = "QA76.73.J38 L363 2004", bibdate = "Fri Sep 26 14:28:47 MDT 2008", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; z3950.loc.gov:7090/Voyager", URL = "http://www.loc.gov/catdir/enhancements/fy0646/2003055149-d.html", acknowledgement = ack-nhfb, subject = "Java (Computer program language)", tableofcontents = "1: Elementary Procedures \\ 1.1: Real vector and matrix \\ Initialization \\ 1.2: Real vector and matrix \\ Duplication \\ 1.3: Real vector and matrix \\ Multiplication \\ 1.4: Real vector vector products \\ 1.5: Real matrix vector products \\ 1.6: Real matrix matrix products \\ 1.7: Real vector and matrix \\ Elimination \\ 1.8: Real vector and matrix \\ Interchanging \\ 1.9: Real vector and matrix \\ Rotation \\ 1.10: Real vector and matrix \\ Norms \\ 1.11: Real vector and matrix \\ Scaling \\ 1.12: Complex vector and matrix \\ Multiplication \\ 1.13: Complex vector and matrix \\ Scalar products \\ 1.14: Complex vector and matrix \\ Elimination \\ 1.15: Complex vector and matrix \\ Rotation \\ 1.16: Complex vector and matrix \\ Norms \\ 1.17: Complex vector and matrix \\ Scaling \\ 1.18: Complex monadic operations \\ 1.19: Complex dyadic operations \\ 1.20: Long integer arithmetic \\ 2: Algebraic Evaluations \\ 2.1: Evaluation of polynomials in Grunert form \\ 2.2: Evaluation of general orthogonal polynomials \\ 2.3: Evaluation of Chebyshev polynomials \\ 2.4: Evaluation of Fourier series \\ 2.5: Evaluation of continued fractions \\ 2.6: Transformation of polynomial representation \\ 2.7: Operations on orthogonal polynomials \\ 3: Linear Algebra \\ 3.1: Full real general matrices \\ 3.2: Real symmetric positive definite matrices \\ 3.3: General real symmetric matrices \\ 3.4: Real full rank overdetermined systems \\ 3.5: Other real matrix problems \\ 3.6: Real sparse nonsymmetric band matrices \\ 3.7: Real sparse nonsymmetric tridiagonal matrices \\ 3.8: Sparse symmetric positive definite band matrices \\ 3.9: Symmetric positive definite tridiagonal matrices \\ 3.10: Sparse real matrices \\ Iterative methods \\ 3.11: Similarity transformation \\ 3.12: Other transformations \\ 3.13: The (ordinary) eigenvalue problem \\ 3.14: The generalized eigenvalue problem \\ 3.15: Singular values \\ 3.16: Zeros of polynomials \\ 4: Analytic Evaluations \\ 4.1: Evaluation of an infinite series \\ 4.2: Quadrature \\ 4.3: Numerical differentiation \\ 5: Analytic Problems \\ 5.1: Nonlinear equations \\ 5.2: Unconstrained optimization \\ 5.3: Overdetermined nonlinear systems \\ 5.4: Differential equations \\ Initial value problems \\ 5.5: Two point boundary value problems \\ 5.6: Two-dimensional boundary value problems \\ 5.7: Parameter estimation in differential equations \\ 6: Special Functions \\ 6.1: Elementary functions \\ 6.2: Exponential Integral \\ 6.3: Gamma function \\ 6.4: Error function \\ 6.5: Bessel functions of integer order \\ 6.6: Bessel functions of real order \\ 7: Interpolation and Approximation \\ 7.1: Real data in one dimension \\ I: Fast Fourier transforms \\ II: Time series analysis \\ Worked Examples \\ Examples for chapter 1 procedures \\ Examples for chapter 2 procedures \\ Examples for chapter 3 procedures \\ Examples for chapter 4 procedures \\ Examples for chapter 5 procedures \\ Examples for chapter 6 procedures \\ Examples for chapter 7 procedures \\ App. B: Procedures Description", } @InProceedings{Markstein:2004:SDS, author = "Peter Markstein", title = "Software Division and Square Root Using {Goldschmidt}'s Algorithms", crossref = "Frougny:2004:RCR", pages = "146--157", year = "2004", bibdate = "Fri Nov 17 07:00:31 2023", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "http://www.informatik.uni-trier.de/Reports/TR-08-2004/rnc6_12_markstein.pdf", abstract = "Goldschmidt's Algorithms for division and square root are often characterized as being useful for hardware implementation, and lacking self-correction. A reexamination of these algorithms show that there are good software counterparts that retain the speed advantage of Goldschmidt's Algorithm over the Newton--Raphson iteration. A final step is needed, however, to get the last bit rounded correctly.", acknowledgement = ack-nhfb, keywords = "division; floating-point; Goldschmidt; square root", } @Article{Marsaglia:2004:END, author = "George Marsaglia", title = "Evaluating the Normal Distribution", journal = j-J-STAT-SOFT, volume = "11", number = "4", pages = "1--7", month = "????", year = "2004", CODEN = "JSSOBK", ISSN = "1548-7660", bibdate = "Sat Dec 04 09:18:40 2004", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.jstatsoft.org/counter.php?id=100&url=v11/i04/cphi.pdf&ct=1", accepted = "2004-07-18", acknowledgement = ack-nhfb, fjournal = "Journal of Statistical Software", journal-URL = "http://www.jstatsoft.org/", remark = "This article exhibits accurate, compact, and fast algorithms for computation of the normal distribution function and the complementary normal distribution, which have a simple relation to the error function and the complementary error function. They appear to be improvements on almost all previously-published algorithms for these functions. However, closer study shows that the complementary normal distribution function has an unchecked out-of-bounds array access for $ |x| \geq 17 $, and its Taylor series sum has poor convergence because the tabulated intervals are twice too wide. The Taylor series sum for the normal distribution function is expanded around $ x = 0 $, and thus has poor convergence for large $ |x| $. Neither function takes into account the accuracy loss when the computed result is the larger of the two (their sum is one, and their range is $ [ - \infty, + \infty] $ ), although the text discusses the problem. The article also discusses the historical origin of the term ``error function'', tracing it to J. W. Glaisher in 1871.", submitted = "2004-06-05", } @Article{Mathar:2004:NRI, author = "Richard J. Mathar", title = "Numerical Representations of the Incomplete Gamma Function of Complex-Valued Argument", journal = j-NUMER-ALGORITHMS, volume = "36", number = "3", pages = "247--264", month = jul, year = "2004", CODEN = "NUALEG", ISSN = "1017-1398 (print), 1572-9265 (electronic)", ISSN-L = "1017-1398", bibdate = "Mon Dec 6 07:00:32 MST 2004", bibsource = "http://www.kluweronline.com/issn/1017-1398; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://ipsapp009.kluweronline.com/IPS/content/ext/x/J/5058/I/57/A/4/abstract.htm", acknowledgement = ack-nhfb, fjournal = "Numerical Algorithms", journal-URL = "http://link.springer.com/journal/11075", } @Misc{Miller:2004:AMF, author = "Alan Miller", title = "{Alan Miller}'s {Fortran} Software", howpublished = "Web site", day = "4", month = feb, year = "2004", bibdate = "Tue Jun 13 12:03:37 2023", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fortran3.bib", note = "From the Web site: All code written by Alan Miller is released into the public domain.", URL = "https://jblevins.org/mirror/amiller/", acknowledgement = ack-nhfb, remark = "The Web site contains a section ``Code converted from the Naval Surface Warfare Center Math. Library'' with links to individual Fortran 90 source files.", } @Article{Moore:2004:PSW, author = "Ian C. Moore and Michael Cada", title = "Prolate spheroidal wave functions, an introduction to the {Slepian} series and its properties", journal = j-APPL-COMPUT-HARMON-ANAL, volume = "16", number = "3", pages = "208--230", month = may, year = "2004", DOI = "https://doi.org/10.1016/j.acha.2004.03.004", ISSN = "1063-5203 (print), 1096-603x (electronic)", ISSN-L = "1063-5203", bibdate = "Sun Oct 31 09:58:00 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", abstract = "For decades mathematicians, physicists, and engineers have relied on various orthogonal expansions such as Fourier, Legendre, and Chebyschev to solve a variety of problems. In this paper we exploit the orthogonal properties of prolate spheroidal wave functions (PSWF) in the form of a new orthogonal expansion which we have named the Slepian series. We empirically show that the Slepian series is potentially optimal over more conventional orthogonal expansions for discontinuous functions such as the square wave among others. With regards to interpolation, we explore the connections the Slepian series has to the Shannon sampling theorem. By utilizing Euler's equation, a relationship between the even and odd ordered PSWFs is investigated. We also establish several other key advantages the Slepian series has such as the presence of a free tunable bandwidth parameter.", acknowledgement = ack-nhfb, fjournal = "Applied and Computational Harmonic Analysis. Time-Frequency and Time-Scale Analysis, Wavelets, Numerical Algorithms, and Applications", journal-URL = "http://www.sciencedirect.com/science/journal/10635203", keywords = "Interpolation; Orthogonal expansion; Prolate spheroidal wave function", } @Article{Muller:2004:CSR, author = "Siguna M{\"u}ller", title = "On the Computation of Square Roots in Finite Fields", journal = j-DESIGNS-CODES-CRYPTOGR, volume = "31", number = "3", pages = "301--312", month = mar, year = "2004", CODEN = "DCCREC", ISSN = "0925-1022 (print), 1573-7586 (electronic)", ISSN-L = "0925-1022", bibdate = "Tue Aug 3 16:38:18 MDT 2004", bibsource = "http://www.wkap.nl/jrnltoc.htm/0925-1022; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "http://ipsapp008.kluweronline.com/IPS/content/ext/x/J/4630/I/61/A/8/abstract.htm", acknowledgement = ack-nhfb, fjournal = "Designs, codes, and cryptography", journal-URL = "http://link.springer.com/journal/10623", } @Article{Nagel:2004:CEG, author = "Bengt Nagel", title = "Confluence expansions of the generalized hypergeometric function", journal = j-J-MATH-PHYS, volume = "45", number = "1", pages = "495--508", month = jan, year = "2004", CODEN = "JMAPAQ", DOI = "https://doi.org/10.1063/1.1629777", ISSN = "0022-2488 (print), 1089-7658 (electronic), 1527-2427", ISSN-L = "0022-2488", bibdate = "Tue Oct 25 18:16:52 MDT 2011", bibsource = "http://www.aip.org/ojs/jmp.html; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jmathphys2000.bib", URL = "http://jmp.aip.org/resource/1/jmapaq/v45/i1/p495_s1", acknowledgement = ack-nhfb, fjournal = "Journal of Mathematical Physics", journal-URL = "http://jmp.aip.org/", onlinedate = "19 December 2003", pagecount = "14", } @InProceedings{Ortiz:2004:SPI, author = "I. Ortiz and M. Jimenez", booktitle = "{MWSCAS '04. The 2004 47th Midwest Symposium on Circuits and Systems. 25--28 July 2004}", title = "Scalable pipeline insertion in floating-point division and square root units", volume = "2", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "II-225--II-228", year = "2004", CODEN = "????", ISSN = "????", bibdate = "Sat Jul 16 15:28:14 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", abstract = "Division and square root are important operations in a number of data processing algorithms. They are inherently time consuming operations and can require a significant amount of resources when implemented in hardware. This work reports the development of scalable, floating-point (FP) division and square root operators with adjustable precision, range, and pipeline granularity. An algorithm for pipeline insertion was used for both operators, enabling speeds up to 204MFLOPS when implemented on a Xilinx Virtex II FPGA.", acknowledgement = ack-nhfb, summary = "Division and square root are important operations in a number of data processing algorithms. They are inherently time consuming operations and can require a significant amount of resources when implemented in hardware. This work reports the \ldots{}", } @Article{Petkovic:2004:GCS, author = "M. S. Petkovi{\'c} and L. Ranci{\'c}", title = "On the guaranteed convergence of the square-root iteration method", journal = j-J-COMPUT-APPL-MATH, volume = "170", number = "1", pages = "169--179", day = "1", month = sep, year = "2004", CODEN = "JCAMDI", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Sat Feb 25 13:00:00 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2000.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0377042704000184", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @Article{Pineiro:2004:AAL, author = "J. A. Pi{\~n}eiro and M. D. Ercegovac and J. D. Bruguera", title = "Algorithm and Architecture for Logarithm, Exponential and Powering Computation", journal = j-IEEE-TRANS-COMPUT, volume = "53", number = "9", pages = "1085--1096", year = "2004", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/TC.2004.53", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Fri Jun 24 10:05:48 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.ac.usc.es/arquivos/articulos/2004/gac2004-j05.ps", acknowledgement = ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", } @PhdThesis{Pugh:2004:ALG, author = "Glendon Ralpha Pugh", title = "An Analysis of the {Lanczos} Gamma Approximation", type = "Ph.D. thesis", school = "Department of Mathematics, University of British Columbia", address = "Vancouver, BC, Canada", pages = "viii + 154", month = "????", year = "2004", ISBN = "0-612-99536-4", ISBN-13 = "978-0-612-99536-9", LCCN = "AW5 .B7 2005-995364", bibdate = "Mon Nov 24 20:55:30 2008", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, } @Book{Raade:2004:MHS, author = "Lennart R{\aa}de and Bertil Westergren", title = "Mathematics Handbook for Science and Engineering", publisher = pub-SV, address = pub-SV:adr, edition = "Fifth", pages = "562", year = "2004", ISBN = "3-540-21141-1 (hardcover)", ISBN-13 = "978-3-540-21141-9 (hardcover)", LCCN = "QA41 .R34 2004", bibdate = "Sat May 15 09:15:39 MDT 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/master.bib; z3950.loc.gov:7090/Voyager", URL = "http://www.loc.gov/catdir/enhancements/fy0818/2006286513-d.html; http://www.loc.gov/catdir/toc/fy0704/2006286513.html", acknowledgement = ack-nhfb, libnote = "Not in my library.", subject = "mathematics; formulae; tables; handbooks, manuals, etc.", tableofcontents = "1. Fundamentals \\ Discrete Mathematics / 9 \\ 1.1 Logic / 9 \\ 1.2 Set Theory / 14 \\ 1.3 Binary Relations and Functions / 17 \\ 1.4 Algebraic Structures / 21 \\ 1.5 Graph Theory / 33 \\ 1.6 Codes / 37 \\ 2: Algebra / 43 \\ 2.1 Basic Algebra of Real Numbers / 43 \\ 2.2 Number Theory / 49 \\ 2.3 Complex Numbers / 61 \\ 2.4 Algebraic Equations / 63 \\ 3: Geometry and Trigonometry / 66 \\ 3.1 Plane Figures / 66 \\ 3.2 Solids / 71 \\ 3.3 Spherical Trigonometry / 75 \\ 3.4 Geometrical Vectors / 77 \\ 3.5 Plane Analytic Geometry / 79 \\ 3.6 Analytic Geometry in Space / 83 \\ 3.7 Fractals / 87 \\ 4: Linear Algebra / 90 \\ 4.1 Matrices / 90 \\ 4.2 Determinants / 93 \\ 4.3 Systems of Linear Equations / 95 \\ 4.4 Linear Coordinate Transformations / 97 \\ 4.5 Eigenvalues. Diagonalization / 98 \\ 4.6 Quadratic Forms / 103 \\ 4.7 Linear Spaces / 106 \\ 4.8 Linear Mappings / 108 \\ 4.9 Tensors / 114 \\ 4.10 Complex matrices / 114 \\ 5: The Elementary Functions / 118 \\ 5.1 A Survey of the Elementary Functions / 118 \\ 5.2 Polynomials and Rational Functions / 119 \\ 5.3 Logarithmic, Exponential, Power and Hyperbolic Functions / 121 \\ 5.4 Trigonometric and Inverse Trigonometric Functions / 125 \\ 6: Differential Calculus (one variable) / 132 \\ 6.1 Some Basic Concepts / 132 \\ 6.2 Limits and Continuity / 133 \\ 6.3 Derivatives / 136 \\ 6.4 Monotonicity. Extremes of Functions / 139 \\ 7: Integral Calculus / 141 \\ 7.1 Indefinite Integrals / 141 \\ 7.2 Definite Integrals / 146 \\ 7.3 Applications of Differential and Integral Calculus / 148 \\ 7.4 Table of Indefinite Integral / 153 \\ 7.5 Tables of Definite Integrals / 178 \\ 8: Sequences and Series / 183 \\ 8.1 Sequences of Numbers / 183 \\ 8.2 Sequences of Functions / 184 \\ 8.3 Series of Constant Terms / 185 \\ 8.4 Series of Functions / 187 \\ 8.5 Taylor Series / 189 \\ 8.6 Special Sums and Series / 192 \\ 9: Ordinary Differential Equations (ODE) / 200 \\ 9.1 Differential Equations of the First Order / 200 \\ 9.2 Differential Equations of the Second Order / 202 \\ 9.3 Linear Differential Equations / 205 \\ 9.4 Autonomous systems / 2313 \\ 9.5 General Concepts and Results / 216 \\ 9.6 Linear Difference Equations / 218 \\ 10: Multidimensional Calculus / 221 \\ 10.1 The Space Rn / 221 \\ 10.2 Surfaces. Tangent Planes / 222 \\ 10.3 Limits and Continuity / 223 \\ 10.4 Partial Derivatives / 224 \\ 10.5 Extremes of Functions / 227 \\ 10.6 Functions $f: R^n \to R^m (R^n \to R^n)$ / 229 \\ 10.7 Double Integrals / 231 \\ 10.8 Triple Integrals / 234 \\ 10.9 Partial Differential Equations / 239 \\ 11: Vector Analysis / 246 \\ 11.1 Curves / 246 \\ 11.2 Vector Fields / 248 \\ 11.3 Line Integrals / 253 \\ 11.4 Surface Integrals / 256 \\ 12: Orthogonal Series and Special Functions / 259 \\ 12.1 Orthogonal Systems / 259 \\ 12.2 Orthogonal Polynomials / 263 \\ 12.3 Bernoulli and Euler Polynomials / 269 \\ 12.4 Bessel Functions / 270 \\ 12.5 Functions Defined by Transcendental Integrals / 287 \\ 12.6 Step and Impulse Functions / 297 \\ 12.7 Functional Analysis / 298 \\ 12.8 Lebesgue Integrals / 303 \\ 12.9 Generalized functions (Distributions) / 308 \\ 13: Transforms / 310 \\ 13.1 Trigonometric Fourier Series / 310 \\ 13.2 Fourier Transforms / 315 \\ 13.3 Discrete Fourier Transforms / 325 \\ 13.4 The $z$-transform / 327 \\ 13.5 Laplace Transforms / 330 \\ 13.6 Dynamical Systems (Filters) / 338 \\ 13.7 Hankel and Hilbert transforms / 341 \\ 13.8 Wavelets / 344 \\ 14: Complex Analysis / 349 \\ 14.1 Functions of a Complex Variable / 349 \\ 14.2 Complex Integration / 352 \\ 14.3 Power Series Expansions / 354 \\ 14.4 Zeros and Singularities / 355 \\ 14.5 Conformal Mappings / 356 \\ 15: Optimization / 365 \\ 15.1 Calculus of Variations / 365 \\ 15.2 Linear Optimization / 371 \\ 15.3 Integer and Combinatorial Optimization / 379 \\ 15.4 Nonlinear Optimization / 383 \\ 15.5 Dynamic Optimization / 389 \\ 16: Numerical Analysis / 391 \\ 16.1 Approximations and Errors / 391 \\ 16.2 Numerical Solution of Equations / 392 \\ 16.3 Perturbation analysis / 397 \\ 16.4 Interpolation / 398 \\ 16.5 Numerical Integration and Differentiation / 404 \\ 16.6 Numerical Solutions of Differential Equations / 412 \\ 16.7 Numerical summation / 421 \\ 17: Probability Theory / 424 \\ 17.1 Basic Probability Theory / 424 \\ 17.2 Probability Distributions / 434 \\ 17.3 Stochastic Processes / 439 \\ 17.4 Algorithms for Calculation of Probability Distributions / 443 \\ 17.5 Simulation / 445 \\ 17.6 Queueing Systems / 449 \\ 17.7 Reliability / 452 \\ 17.8 Tables / 459 \\ 18: Statistics / 479 \\ 18.1 Descriptive Statistics / 479 \\ 18.2 Point Estimation / 488 \\ 18.3 Confidence Intervals / 491 \\ 18.4 Tables for Confidence Intervals / 495 \\ 18.5 Tests of Significance / 501 \\ 18.6 Linear Models / 507 \\ 18.7 Distribution-free Methods / 512 \\ 18.8 Statistical Quality Control / 518 \\ 18.9 Factorial Experiments / 522 \\ 18.10 Analysis of life time (failure time) data / 525 \\ 18.11 Statistical glossary / 526 \\ 19: Miscellaneous / 530", } @Article{Skorokhodov:2004:STP, author = "S. L. Skorokhodov", title = "Symbolic transformations in the problem of analytic continuation of the hypergeometric function {$_p F_{p - 1}(z)$} in the neighborhood of the point $ z = 1 $ in the logarithmic case", journal = j-PROG-COMP-SOFT, volume = "30", number = "3", pages = "150--156", month = "????", year = "2004", CODEN = "PCSODA", ISSN = "0361-7688 (print), 1608-3261 (electronic)", ISSN-L = "0361-7688", MRclass = "3C20 (33B15 33C05 33F05 33F10)", MRnumber = "MR2082811 (2005f:33013)", bibdate = "Thu Dec 01 09:18:16 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Programming and Computer Software; translation of Programmirovaniye (Moscow, USSR) Plenum", journal-URL = "http://link.springer.com/journal/11086", } @Article{Thompson:2004:EBB, author = "I. J. Thompson", title = "Erratum to {{\booktitle{Modified Bessel functions $ I \_ n u(z) $ and $ K_\nu (z) $ of real order and complex argument}} [Comput. Phys. Commun. {\bf 47} (1987) 245--257]}", journal = j-COMP-PHYS-COMM, volume = "159", number = "3", pages = "243--244", day = "1", month = jun, year = "2004", CODEN = "CPHCBZ", DOI = "https://doi.org/10.1016/j.cpc.2004.02.007", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", bibdate = "Thu Apr 24 10:35:27 2008", bibsource = "https://www.math.utah.edu/pub/tex/bib/compphyscomm1980.bib; https://www.math.utah.edu/pub/tex/bib/compphyscomm2000.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "See \cite{Thompson:1987:MBF}.", URL = "http://www.sciencedirect.com/science/article/pii/S0010465504001067", acknowledgement = ack-nhfb, fjournal = "Computer Physics Communications", journal-URL = "http://www.sciencedirect.com/science/journal/00104655", } @Book{Vallee:2004:AFA, author = "Olivier Vall{\'e}e and Manuel Soares", title = "{Airy} Functions and Applications to Physics", publisher = pub-WORLD-SCI, address = pub-WORLD-SCI:adr, pages = "x + 194", year = "2004", ISBN = "1-86094-478-7 (hardcover)", ISBN-13 = "978-1-86094-478-9 (hardcover)", LCCN = "QA351 .V35 2004", bibdate = "Tue Dec 5 10:16:05 2023", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", abstract = "The use of special functions, and in particular Airy functions, is rather common in physics. The reason may be found in the need, and even in the necessity, to express a physical phenomenon in terms of an effective and comprehensive analytical form for the whole scientific community. However, for the past twenty years, many physical problems have been resolved by computers. This trend is now becoming the norm as the importance of computers continues to grow. As a last resort, the special functions employed in physics will have to be calculated numerically, even if the analytic formulation of physics is of primary importance.\par Airy functions have periodically been the subject of many review articles, but no noteworthy compilation on this subject has been published since the 1950s. In this work, we provide an exhaustive compilation of the current knowledge on the analytical properties of Airy functions, developing with care the calculus implying the Airy functions.", acknowledgement = ack-nhfb, remark = "See also second edition \cite{Vallee:2010:AFA}.", shorttableofcontents = "1: A historical introduction: Sir George Biddell Airy / 1 \\ 2: Definitions and properties / 5 \\ 3: Primitives and integrals of Airy functions / 37 \\ 4: Transformations of Airy functions / 71 \\ 5: The uniform approximation / 91 \\ 6: Generalisation of Airy functions / 101 \\ 7: Applications to classical physics / 115 \\ 8: Applications to quantum physics / 137 \\ Appendix A: Numerical computation of the Airy functions / 177 \\ Bibliography / 183 \\ Index / 193", tableofcontents = "Preface / v \\ 1. A Historical Introduction: Sir George Biddell Airy / 1 \\ 2. Definitions and Properties / 5 \\ 2.1 The Homogeneous Airy Functions / 5 \\ 2.1.1 The Airy's equation / 5 \\ 2.1.2 Elementary properties / 8 \\ 2.1.2.1 Wronskians of homogeneous Airy functions / 8 \\ 2.1.2.2 Particular values of Airy functions / 8 \\ 2.1.2.3 Relations between Airy functions / 9 \\ 2.1.3 Integral representations / 9 \\ 2.1.4 Ascending and asymptotic series / 11 \\ 2.1.4.1 Expansion of $\Ai$ near the origin / 11 \\ 2.1.4.2 Ascending series of $\Ai$ and $\Bi$ / 12 \\ 2.1.4.3 Asymptotic series of $\Ai$ and $\Bi$ / 13 \\ 2.2 Properties of Airy Functions / 15 \\ 2.2.1 Zeros of Airy functions / 15 \\ 2.2.2 The spectral zeta function / 18 \\ 2.2.3 Inequalities / 20 \\ 2.2.4 Connection with Bessel functions / 20 \\ 2.2.5 Modulus and phase of Airy functions / 21 \\ 2.2.5.1 Definitions / 21 \\ 2.2.5.2 Differential equations / 22 \\ 2.2.5.3 Asymptotic expansions / 23 \\ 2.2.5.4 Functions of positive arguments / 24 \\ 2.3 The Inhomogeneous Airy Functions / 25 \\ 2.3.1 Definitions / 25 \\ 2.3.2 Properties of inhomogeneous Airy functions / 27 \\ 2.3.2.1 Values at the origin / 27 \\ 2.3.2.2 Other integral representations / 27 \\ 2.3.3 Ascending and asymptotic series / 28 \\ 2.3.3.1 Ascending series / 28 \\ 2.3.3.2 Asymptotic series / 29 \\ 2.3.4 Zeros of the Scorer functions / 29 \\ 2.4 Squares and Products of Airy Functions / 30 \\ 2.4.1 Differential equation and integral representation / 30 \\ 2.4.2 A remarkable identity / 32 \\ 2.4.3 The product $\Ai(x) \Ai(-x)$: Airy wavelets / 32 \\ 3. Primitives and Integrals of Airy Functions / 37 \\ 3.1 Primitives Containing One Airy Function / 37 \\ 3.1.1 In terms of Airy functions / 37 \\ 3.1.2 Ascending series / 38 \\ 3.1.3 Asymptotic series / 38 \\ 3.1.4 Primitive of Scorer functions / 39 \\ 3.1.5 Repeated primitives / 40 \\ 3.2 Product of Airy Functions / 40 \\ 3.2.1 The method of Albright / 41 \\ 3.2.2 Some primitives / 43 \\ 3.3 Other Primitives / 48 \\ 3.4 Miscellaneous / 49 \\ 3.5 Elementary Integrals / 50 \\ 3.5.1 Particular integrals / 50 \\ 3.5.2 Integrals containing a single Airy function / 51 \\ 3.5.2.1 Integrals involving algebraic functions / 51 \\ 3.5.2.2 Integrals involving transcendental functions / 54 \\ 3.5.3 Integrals of products of two Airy functions / 56 \\ 3.6 Other Integrals / 60 \\ 3.6.1 Integrals involving the Volterra $\mu$-function / 60 \\ 3.6.2 Canonisation of cubic form / 64 \\ 3.6.3 Integrals with three Airy functions / 65 \\ 3.6.4 Integrals with four Airy functions / 67 \\ 3.6.5 Double integrals / 68 \\ 4. Transformations of Airy Functions / 71 \\ 4.1 Causal Properties of Airy Functions / 71 \\ 4.1.1 Causal relations / 71 \\ 4.1.2 Green function of the Airy equation / 73 \\ 4.2 The Airy Transform / 74 \\ 4.2.1 Definitions and elementary properties / 74 \\ 4.2.2 Some examples / 77 \\ 4.2.3 Airy polynomials / 82 \\ 4.2.4 Summary of Airy transform / 84 \\ 4.2.5 Airy averaging / 85 \\ 4.3 Other Kinds of Transformations / 85 \\ 4.3.1 Laplace transform of Airy functions / 85 \\ 4.3.2 Mellin transform of Airy function / 86 \\ 4.3.3 Fourier transform of Airy functions / 87 \\ 4.4 Expansion into Fourier-Airy Series / 88 \\ 5. The Uniform Approximation / 91 \\ 5.1 Oscillating Integrals / 91 \\ 5.1.1 The method of stationary phase / 91 \\ 5.1.2 The uniform approximation of oscillating integrals / 93 \\ 5.1.3 The Airy uniform approximation / 94 \\ 5.2 Differential Equation of the Second Order / 95 \\ 5.2.1 The JWKB method / 95 \\ 5.2.2 The generalisation of Langer / 97 \\ 5.3 Inhomogeneous Differential Equations / 98 \\ 6. Generalisation of Airy Functions / 101 \\ 6.1 Generalisation of the Airy Integral / 101 \\ 6.2 Third Order Differential Equations / 105 \\ 6.2.1 The linear third order differential equation / 105 \\ 6.2.2 Asymptotic solutions / 106 \\ 6.2.3 The comparison equation / 107 \\ 6.3 Differential Equation of the Fourth Order / 111 \\ 7. Applications to Classical Physics / 115 \\ 7.1 Optics and Electromagnetism / 115 \\ 7.2 Fluid Mechanics / 119 \\ 7.2.1 The Tricomi equation / 119 \\ 7.2.2 The Orr--Sommerfeld equation / 121 \\ 7.3 Elasticity / 124 \\ 7.4 The Heat Equation / 127 \\ 7.5 Nonlinear Physics / 129 \\ 7.5.1 Korteweg--de Vries equation / 129 \\ 7.5.1.1 The linearised Korteweg--de Vries equation / 129 \\ 7.5.1.2 Similarity solutions / 131 \\ 7.5.2 The second Painlev{\'e} equation / 132 \\ 7.5.2.1 The Painlev{\'e} equations / 132 \\ 7.5.2.2 An integral equation / 134 \\ 7.5.2.3 Rational solutions 135 \\ 8. Applications to Quantum Physics / 137 \\ 8.1 The Schr{\"o}dinger Equation / 137 \\ 8.1.1 Particle in a uniform field / 137 \\ 8.1.2 The $|x|$ potential / 140 \\ 8.1.3 Uniform approximation of the Schr{\"o}dinger equation / 144 \\ 8.1.3.1 The JWKB approximation / 145 \\ 8.1.3.2 The Airy uniform approximation / 146 \\ 8.1.3.3 Exact vs approximate wave functions / 148 \\ 8.2 Evaluation of the Franck--Condon Factors / 152 \\ 8.2.1 The Franck--Condon principle / 153 \\ 8.2.2 The JWKB approximation / 154 \\ 8.2.3 The uniform approximation / 157 \\ 8.3 The Semiclassical Wigner Distribution / 162 \\ 8.3.1 The Weyl--Wigner formalism / 163 \\ 8.3.2 The one-dimensional Wigner distribution / 164 \\ 8.3.3 The two-dimensional Wigner distribution / 166 \\ 8.3.4 Configuration of the Wigner distribution / 169 \\ 8.4 Airy Transform of the Schr{\"o}dinger Equation / 173 \\ Appendix A: Numerical Computation of the Airy Functions / 177 \\ A.1 The Homogeneous Functions / 177 \\ A.2 The Inhomogeneous Functions / 180 \\ Bibliography / 183 \\ Index / 193", } @Article{VanDeun:2004:IAO, author = "J. {Van Deun} and A. Bultheel", title = "An Interpolation Algorithm for Orthogonal Rational Functions", journal = j-J-COMPUT-APPL-MATH, volume = "164--165", number = "??", pages = "749--762", month = mar, year = "2004", CODEN = "JCAMDI", DOI = "https://doi.org/10.1016/S0377-0427(03)00493-X", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Tue Mar 24 21:14:11 2009", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "Proceedings of the 10th International Congress on Computational and Applied Mathematics University of Leuven, Belgium, 22--26 July 2002. Edited by M. J. Goovaerts, S. Vandewalle, and L. Wuytack.", abstract = "Let $ A = \{ \alpha_1, \alpha_2, \ldots {} \} $ be a sequence of numbers on the extended real line $ \mathcal {R} = \mathcal {R} \union \{ \infty \} $ and $ \mu $ a positive bounded Borel measure with support in (a subset of) $ \mathcal {R} $. We introduce rational functions n with poles $ \{ \alpha_1, \ldots {}, \alpha_n \} $ that are orthogonal with respect to $ \mu $ (if all poles are at infinity, we recover the polynomial situation). It is well known that under certain conditions on the location of the poles, the system $ \{ \phi_n \} $ is regular such that the orthogonal functions satisfy a three-term recurrence relation similar to the one for orthogonal polynomials. To compute the recurrence coefficients one can use explicit formulas involving inner products. We present a theoretical alternative to these explicit formulas that uses certain interpolation properties of the Riesz--Herglotz--Nevanlinna transform $ \Omega_\mu $ of the measure $ \mu $. Error bounds are derived and some examples serve as illustration.", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", keywords = "interpolation; orthogonal polynomials; orthogonal rational functions; three-term recurrence", } @Article{Wang:2004:CHP, author = "Ren-Hong Wang and Cheng-de Zheng", title = "Cubic {Hermite--Pad{\'e}} approximation to the exponential function", journal = j-J-COMPUT-APPL-MATH, volume = "163", number = "1", pages = "259--268", day = "1", month = feb, year = "2004", CODEN = "JCAMDI", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Sat Feb 25 12:59:56 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2000.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0377042703008124", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @Article{Zeng:2004:AMM, author = "Zhonggang Zeng", title = "Algorithm 835: {MultRoot}---a {Matlab} package for computing polynomial roots and multiplicities", journal = j-TOMS, volume = "30", number = "2", pages = "218--236", month = jun, year = "2004", CODEN = "ACMSCU", DOI = "https://doi.org/10.1145/992200.992209", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", MRclass = "65H05", MRnumber = "MR2075984 (2005c:65041)", bibdate = "Tue Mar 30 16:16:28 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", abstract = "MultRoot is a collection of Matlab modules for accurate computation of polynomial roots, especially roots with non-trivial multiplicities. As a blackbox-type software, MultRoot requires the polynomial coefficients as the only input, and outputs the computed roots, multiplicities, backward error, estimated forward error, and the structure-preserving condition number. The most significant features of MultRoot are the multiplicity identification capability and high accuracy on multiple roots without using multiprecision arithmetic, even if the polynomial coefficients are inexact. A comprehensive test suite of polynomials that are collected from the literature is included for numerical experiments and performance comparison.", acknowledgement = ack-nhfb, fjournal = "Association for Computing Machinery. Transactions on Mathematical Software", journal-URL = "https://dl.acm.org/loi/toms", } @Article{Zhu:2004:ISR, author = "Hufei Zhu and Zhongding Lei and F. P. S. Chin", title = "An improved square-root algorithm for {BLAST}", journal = j-IEEE-SIGNAL-PROCESS-LETT, volume = "11", number = "9", pages = "772--775", month = sep, year = "2004", CODEN = "ISPLEM", DOI = "https://doi.org/10.1109/LSP.2004.833483", ISSN = "1070-9908 (print), 1558-2361 (electronic)", ISSN-L = "1070-9908", bibdate = "Sat Jul 16 15:28:13 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "IEEE Signal Processing Letters", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=97", summary = "In this letter, an improved square-root algorithm for Bell Labs Layered Space-Time (BLAST) system is proposed. It speeds up the original square-root algorithm by 36\% in terms of the number of multiplications and additions. Compared with the \ldots{}", } @Article{Abad:2005:TNA, author = "Julio Abad and Javier Sesma", title = "Two new asymptotic expansions of the ratio of two gamma functions", journal = j-J-COMPUT-APPL-MATH, volume = "173", number = "2", pages = "359--363", day = "15", month = jan, year = "2005", CODEN = "JCAMDI", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Sat Feb 25 13:00:02 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2005.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0377042704001669", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @InProceedings{Antelo:2005:LLD, author = "Elisardo Antelo and Tom{\'a}s Lang and Paolo Montuschi and Alberto Nannarelli", title = "Low Latency Digit-Recurrence Reciprocal and Square-Root Reciprocal Algorithm and Architecture", crossref = "IEEE:2005:PIS", pages = "??--??", year = "2005", bibdate = "Wed Jun 22 07:02:55 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://arith17.polito.it/final/paper-116.pdf", abstract = "The reciprocal and square-root reciprocal operations are important in several applications. For the operations, we present algorithms that combine a digit-by-digit module and one iteration of a quadratic-convergence approximation. The latter is implemented by a digit-recurrence, which uses the digits produced by the digit-by-digit part. In this way, both parts execute in an overlapped manner, so that the total number of cycles is about half the number that would be required by the digit-by-digit part alone. Because of the approximation, correct rounding of the result cannot be obtained directly in all cases; we propose a variable-time implementation that produces the correctly rounded result with a small average overhead. Radix-4 implementations are described and have been synthesized. They achieve the same cycle time as the standard digit-by-digit implementation, resulting in a speed-up of about 2 and, because of the approximation part, the area factor is also about 2. We also show a combined implementation for both operations that has essentially the same complexity as that for square-root reciprocal alone.", acknowledgement = ack-nhfb, pagecount = "8", } @Book{Arfken:2005:MMP, author = "George B. Arfken and Hans-J{\"u}rgen Weber", title = "Mathematical Methods for Physicists", publisher = pub-ELSEVIER, address = pub-ELSEVIER:adr, edition = "Sixth", pages = "xii + 1182", year = "2005", ISBN = "0-12-059876-0, 0-12-088584-0 (paperback)", ISBN-13 = "978-0-12-059876-2, 978-0-12-088584-8 (paperback)", LCCN = "QA37.3 .A74 2005", bibdate = "Tue Feb 17 18:23:45 MST 2009", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/master.bib; z3950.loc.gov:7090/Voyager", acknowledgement = ack-nhfb, subject = "Mathematics; Mathematical physics", tableofcontents = "1. Vector Analysis \\ 2. Vector Analysis in Curved Coordinates and Tensors \\ 3. Determinants and Matrices \\ 4. Group Theory \\ 5. Infinite Series \\ 6. Functions of a Complex Variable I: Analytic Properties, Mapping \\ 7. Functions of a Complex Variable II \\ 8. The Gamma Function (Factorial Function) \\ 9. Differential Equations \\ 10. Sturm--Liouville Theory-Orthogonal Functions \\ 11. Bessel Functions \\ 12. Legendre Functions \\ 13. More Special Functions \\ 14. Fourier Series \\ 15. Integral Transforms \\ 16. Integral Equations \\ 17. Calculus of Variations \\ 18. Nonlinear Methods and Chaos \\ 19. Probability", xxauthor = "George B. (George Brown) Arfken and Hans-J{\"u}rgen Weber", xxURL = "http://www.loc.gov/catdir/enhancements/fy0625/2005049844-d.html; http://www.loc.gov/catdir/enhancements/fy0625/2005049844-t.html", } @Article{Bonan-Hamada:2005:SCF, author = "Catherine M. Bonan-Hamada and William B. Jones", title = "{Stieltjes} continued fractions for polygamma functions; speed of convergence", journal = j-J-COMPUT-APPL-MATH, volume = "179", number = "1--2", pages = "47--55", day = "1", month = jul, year = "2005", CODEN = "JCAMDI", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Sat Feb 25 13:00:05 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2005.bib", URL = "http://www.sciencedirect.com/science/article/pii/S037704270400442X", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @Article{Brisebarre:2005:NRR, author = "Nicolas Brisebarre and David Defour and Peter Kornerup and Jean-Michel Muller and Nathalie Revol", title = "A New Range-Reduction Algorithm", journal = j-IEEE-TRANS-COMPUT, volume = "54", number = "3", pages = "331--339", month = mar, year = "2005", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/TC.2005.36", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Wed Apr 27 18:04:38 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://csdl.computer.org/comp/trans/tc/2005/03/t0331abs.htm; http://csdl.computer.org/dl/trans/tc/2005/03/t0331.htm; http://csdl.computer.org/dl/trans/tc/2005/03/t0331.pdf; http://ieeexplore.ieee.org/iel5/12/30205/01388197.pdf; http://ieeexplore.ieee.org/iel5/12/30205/01388197.pdf?isnumber=30205&prod=JNL&arnumber=1388197&arSt=+331&ared=+339&arAuthor=Brisebarre%2C+N.%3B+Defour%2C+D.%3B+Kornerup%2C+P.%3B+Muller%2C+J.-M.%3B+Revol%2C+N.; http://ieeexplore.ieee.org/xpls/abs_all.jsp?isnumber=30205&arnumber=1388197&count=13&index=8; http://ieeexplore.ieee.org/xpls/references.jsp?arnumber=1388197", abstract = "Range-reduction is a key point for getting accurate elementary function routines. We introduce a new algorithm that is fast for input arguments belonging to the most common domains, yet accurate over the full double-precision range.", acknowledgement = ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", } @Article{Carlson:2005:JEF, author = "B. C. Carlson", title = "{Jacobian} elliptic functions as inverses of an integral", journal = j-J-COMPUT-APPL-MATH, volume = "174", number = "2", pages = "355--359", day = "15", month = feb, year = "2005", CODEN = "JCAMDI", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Sat Feb 25 13:00:02 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2005.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0377042704002201", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @Article{Cheng:2005:SEEa, author = "Howard Cheng and Barry Gergel and Ethan Kim and Eugene Zima", title = "Space-efficient evaluation of hypergeometric series", journal = j-SIGSAM, volume = "39", number = "2", pages = "41--52", month = jun, year = "2005", CODEN = "SIGSBZ", ISSN = "0163-5824 (print), 1557-9492 (electronic)", ISSN-L = "0163-5824", bibdate = "Tue Nov 29 06:11:40 MST 2005", bibsource = "http://portal.acm.org/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/sigsam.bib", acknowledgement = ack-nhfb, fjournal = "SIGSAM Bulletin (ACM Special Interest Group on Symbolic and Algebraic Manipulation)", issue = "152", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J1000", } @Article{Cheng:2005:SEEb, author = "Howard Cheng and Barry Gergel and Ethan Kim and Eugene Zima", title = "Space-efficient evaluation of hypergeometric series", journal = j-SIGSAM, volume = "39", number = "3", pages = "81--83", year = "2005", CODEN = "SIGSBZ", ISSN = "0163-5824 (print), 1557-9492 (electronic)", ISSN-L = "0163-5824", bibdate = "Sat Feb 4 09:52:36 MST 2006", bibsource = "http://portal.acm.org/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/sigsam.bib", note = "ISSAC 2005 poster abstract.", acknowledgement = ack-nhfb, fjournal = "SIGSAM Bulletin (ACM Special Interest Group on Symbolic and Algebraic Manipulation)", issue = "153", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J1000", } @InProceedings{deDinechin:2005:TPU, author = "Florent de Dinechin and Alexey Ershov and Nicolas Gast", title = "Towards the Post-ultimate {\tt libm}", crossref = "IEEE:2005:PIS", pages = "??--??", year = "2005", bibdate = "Wed Jun 22 07:02:55 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://arith17.polito.it/final/paper-165.pdf", abstract = "This article presents advances on the subject of correctly rounded elementary functions since the publication of the {\tt libultim} mathematical library developed by Ziv at IBM. This library showed that the average performance and memory overhead of correct rounding could be made negligible. However, the worst-case overhead was still a factor 1000 or more. It is shown here that, with current processor technology, this worst-case overhead can be kept within a factor of 2 to 10 of current best libms. This low overhead has very positive consequences on the techniques for implementing and proving correctly rounded functions, which are also studied. These results lift the last technical obstacles to a generalisation of (at least some) correctly rounded double precision elementary functions.", acknowledgement = ack-nhfb, pagecount = "8", } @InProceedings{Detrey:2005:TBP, author = "J{\'e}r{\'e}mie Detrey and Florent de Dinechin", booktitle = "2005 {IEEE} International Conference on Application-Specific Systems, Architecture Processors ({ASAP'05})", title = "Table-based polynomials for fast hardware function evaluation", publisher = pub-IEEE, address = pub-IEEE:adr, year = "2005", DOI = "https://doi.org/10.1109/asap.2005.61", bibdate = "Sat Jan 3 08:52:30 2026", bibsource = "https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, } @Article{Freitas:2005:IPF, author = "Pedro Freitas", title = "Integrals of polylogarithmic functions, recurrence relations, and associated {Euler} sums", journal = j-MATH-COMPUT, volume = "74", number = "251", pages = "1425--1440", month = jul, year = "2005", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Aug 2 10:37:19 MDT 2005", bibsource = "http://www.ams.org/mcom/2005-74-251; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp2000.bib", URL = "http://www.ams.org/mcom/2005-74-251/S0025-5718-05-01747-3/home.html; http://www.ams.org/mcom/2005-74-251/S0025-5718-05-01747-3/S0025-5718-05-01747-3.dvi; http://www.ams.org/mcom/2005-74-251/S0025-5718-05-01747-3/S0025-5718-05-01747-3.pdf; http://www.ams.org/mcom/2005-74-251/S0025-5718-05-01747-3/S0025-5718-05-01747-3.ps; http://www.ams.org/mcom/2005-74-251/S0025-5718-05-01747-3/S0025-5718-05-01747-3.tex", acknowledgement = ack-nhfb, fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Hernandez:2005:ACN, author = "M. A. Hern{\'a}ndez and N. Romero", title = "Accelerated convergence in {Newton}'s method for approximating square roots", journal = j-J-COMPUT-APPL-MATH, volume = "177", number = "1", pages = "225--229", day = "1", month = may, year = "2005", CODEN = "JCAMDI", DOI = "https://doi.org/10.1016/j.cam.2004.09.025", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Sat Feb 25 13:00:04 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2005.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0377042704004315", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @Misc{IBM:2005:MAS, author = "{IBM Corporation}", title = "{Mathematical Acceleration Subsystem} for {Linux}", howpublished = "World Wide Web document", year = "2005", bibdate = "Mon Dec 05 18:59:35 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www-306.ibm.com/software/awdtools/mass/linux/mass-linux.html", abstract = "Mathematical Acceleration Subsystem (MASS) for Linux consists of libraries of mathematical intrinsic functions tuned specifically for optimum performance on POWER architectures.", acknowledgement = ack-nhfb, keywords = "Mathematical Acceleration Subsystem (MASS)", remark = "Scalar library functions: atan, atan2, cos, cosh, dnint, exp, log, pow [Fortran **], rsqrt, sin, sinh, sqrt, tan, and tanh.\par Vector library double-precision function: vacos, vasin, vatan2, vcbrt, vcos, vcosh, vcosisin, vdint, vdiv, vdnint, vexp, vexpm1, vlog, vlog10, vlog1p, vpow, vrcbrt, vrec, vrsqrt, vsin, vsincos, vsinh, vsqrt, vtan, and vtanh.\par Vector library single-precision functions: vsacos, vsasin, vsatan2, vscbrt, vscos, vscosh, vscosisin, vsdiv, vsexp, vsexpm1, vslog, vslog10, vslog1p, vspow, vsrcbrt, vsrec, vsrsqrt, vssin, vssincos, vssinh, vssqrt, vstan, and vstanh.", } @Article{Kornerup:2005:DSS, author = "Peter Kornerup", title = "Digit Selection for {SRT} Division and Square Root", journal = j-IEEE-TRANS-COMPUT, volume = "54", number = "3", pages = "294--303", month = mar, year = "2005", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/TC.2005.47", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Tue Jul 19 09:20:54 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://csdl.computer.org/comp/trans/tc/2005/03/t0294abs.htm; http://csdl.computer.org/dl/trans/tc/2005/03/t0294.htm; http://csdl.computer.org/dl/trans/tc/2005/03/t0294.pdf; http://ieeexplore.ieee.org/iel5/12/30205/01388194.pdf?isnumber=30205&prod=JNL&arnumber=1388194&arSt=+294&ared=+303&arAuthor=Kornerup%2C+P.; http://ieeexplore.ieee.org/xpls/abs_all.jsp?isnumber=30205&arnumber=1388194&count=13&index=5; http://ieeexplore.ieee.org/xpls/references.jsp?arnumber=1388194", acknowledgement = ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", summary = "The quotient digit selection in the SRT division algorithm is based on a few most significant bits of the remainder and divisor, where the remainder is usually represented in a redundant representation. The number of leading bits needed depends on \ldots{}", } @Article{Ledoux:2005:CME, author = "V. Ledoux and M. {Van Daele} and G. {Vanden Berghe}", title = "{CP} methods and the evaluation of negative energy {Coulomb} {Whittaker} functions", journal = j-J-COMPUT-APPL-MATH, volume = "183", number = "1", pages = "168--176", day = "1", month = nov, year = "2005", CODEN = "JCAMDI", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Sat Feb 25 13:00:34 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2005.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0377042705000233", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @Article{Lee:2005:OHF, author = "Dong-U. Lee and Altaf Abdul Gaffar and Oskar Mencer and Wayne Luk", title = "Optimizing hardware function evaluation", journal = j-IEEE-TRANS-COMPUT, volume = "54", number = "12", pages = "1520--1531", month = dec, year = "2005", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/TC.2005.201", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Tue May 30 12:04:26 2006", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", abstract = "We present a methodology and an automated system for function evaluation unit generation. Our system selects the best function evaluation hardware for a given function, accuracy requirements, technology mapping, and optimization metrics, such as area, throughput, and latency. Function evaluation $ f(x) $ typically consists of range reduction and the actual evaluation on a small convenient interval such as $ [0, \pi / 2) $ for $ \sin (x) $. We investigate the impact of hardware function evaluation with range reduction for a given range and precision of $x$ and $ f(x) $ on area and speed. An automated bit-width optimization technique for minimizing the sizes of the operators in the data paths is also proposed. We explore a vast design space for fixed-point $ \sin (x) $, $ \log (x) $, and $ \sqrt {x} $ accurate to one unit in the last place using MATLAB and ASC, a stream compiler for field-programmable gate arrays (FPGAs). In this study, we implement over 2,000 placed-and-routed FPGA designs, resulting in over 100 million application-specific integrated circuit (ASIC) equivalent gates. We provide optimal function evaluation results for range and precision combinations between 8 and 48 bits.", acknowledgement = ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", keywords = "application specific integrated circuits; application-specific integrated circuit equivalent gates; ASC; ASIC; automated bit-width optimization technique; circuit optimisation; computer arithmetic; elementary function approximation; field programmable gate arrays; field-programmable gate arrays; fixed point arithmetic; fixed-point arithmetic; FPGA; hardware function evaluation optimisation; logic design; MATLAB; minimax approximation; range reduction; stream compiler", } @InProceedings{Lefevre:2005:NRD, author = "Vincent Lef{\`e}vre", title = "New Results on the Distance Between a Segment and {$ Z^2 $}. {Application} to the Exact Rounding", crossref = "IEEE:2005:PIS", pages = "??--??", year = "2005", bibdate = "Wed Jun 22 07:02:55 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://arith17.polito.it/final/paper-147.pdf", abstract = "This paper presents extensions to Lef{\'e}vre's algorithm that computes a lower bound on the distance between a segment and a regular grid $ Z^2 $. This algorithm and, in particular, the extensions are useful in the search for worst cases for the exact rounding of unary elementary functions or base-conversion functions. The proof that is presented here is simpler and less technical than the original proof. This paper also gives benchmark results with various optimization parameters, explanations of these results, and an application to base conversion.", acknowledgement = ack-nhfb, pagecount = "8", } @InProceedings{Markstein:2005:FSM, author = "Peter Markstein", title = "A Fast-Start Method for Computing the Inverse Tangent", crossref = "IEEE:2005:PIS", pages = "??--??", year = "2005", bibdate = "Wed Jun 22 07:02:55 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://arith17.polito.it/final/paper-112.pdf", abstract = "In a search for an algorithm to compute $ \atan (x) $ which has both low latency and few floating point instructions, an interesting variant of familiar trigonometry formulas was discovered that allow the start of argument reduction to commence before any references to tables stored in memory are needed. Low latency makes the method suitable for a closed subroutine, and few floating point operations make the method advantageous for a software-pipelined implementation.", acknowledgement = ack-nhfb, keywords = "IA-64; Itanium-2", pagecount = "6", } @Article{Merkle:2005:GRG, author = "M. Merkle", title = "{Gurland}'s ratio for the gamma function", journal = j-COMPUT-MATH-APPL, volume = "49", number = "2--3", pages = "389--406", month = jan # "\slash " # feb, year = "2005", CODEN = "CMAPDK", ISSN = "0898-1221 (print), 1873-7668 (electronic)", ISSN-L = "0898-1221", bibdate = "Wed Mar 1 21:49:42 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/computmathappl2000.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0898122105000416", acknowledgement = ack-nhfb, fjournal = "Computers and Mathematics with Applications", journal-URL = "http://www.sciencedirect.com/science/journal/08981221", } @Article{Perram:2005:EFW, author = "John W. Perram and Edgar R. Smith", title = "Elliptic Functions of the Worst Kind: Non-linear Quantisation of the Classical Spherical Pendulum", journal = j-ADV-QUANTUM-CHEM, volume = "48", pages = "111--125", year = "2005", CODEN = "AQCHA9", DOI = "https://doi.org/10.1016/S0065-3276(05)48008-9", ISSN = "0065-3276", ISSN-L = "0065-3276", bibdate = "Thu Oct 13 11:45:04 MDT 2011", bibsource = "http://www.sciencedirect.com/science/bookseries/00653276; https://www.math.utah.edu/pub/tex/bib/advquantumchem.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0065327605480089", acknowledgement = ack-nhfb, ajournal = "Adv. Quantum Chem.", fjournal = "Advances in Quantum Chemistry", journal-URL = "http://www.sciencedirect.com/science/bookseries/00653276", } @Article{Pineiro:2005:HSF, author = "Jose-Alejandro Pi{\~n}eiro and Stuart F. Oberman and Jean-Michel Muller and Javier D. Bruguera", title = "High-Speed Function Approximation Using a Minimax Quadratic Interpolator", journal = j-IEEE-TRANS-COMPUT, volume = "54", number = "3", pages = "304--318", month = mar, year = "2005", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/TC.2005.52", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Tue Jul 19 09:20:54 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://csdl.computer.org/comp/trans/tc/2005/03/t0304abs.htm; http://csdl.computer.org/dl/trans/tc/2005/03/t0304.htm; http://csdl.computer.org/dl/trans/tc/2005/03/t0304.pdf; http://ieeexplore.ieee.org/iel5/12/30205/01388195.pdf?isnumber=30205&prod=JNL&arnumber=1388195&arSt=+304&ared=+318&arAuthor=Pineiro%2C+J.-A.%3B+Oberman%2C+S.F.%3B+Muller%2C+J.-M.%3B+Bruguera%2C+J.D.; http://ieeexplore.ieee.org/xpls/abs_all.jsp?isnumber=30205&arnumber=1388195&count=13&index=6; http://ieeexplore.ieee.org/xpls/references.jsp?arnumber=1388195", abstract = "A table-based method for high-speed function approximation in single-precision floating-point format is presented in this paper. Our focus is the approximation of reciprocal, square root, square root reciprocal, exponentials, logarithms, trigonometric functions, powering (with a fixed exponent $p$ ), or special functions. The algorithm presented here combines table look-up, an enhanced minimax quadratic approximation, and an efficient evaluation of the second-degree polynomial (using a specialized squaring unit, redundant arithmetic, and multioperand addition). The execution times and area costs of an architecture implementing our method are estimated, showing the achievement of the fast execution times of linear approximation methods and the reduced area requirements of other second-degree interpolation algorithms. Moreover, the use of an enhanced minimax approximation which, through an iterative process, takes into account the effect of rounding the polynomial coefficients to a finite size allows for a further reduction in the size of the look-up tables to be used, making our method very suitable for the implementation of an elementary function generator in state-of-the-art DSPs or graphics processing units (GPUs).", acknowledgement = ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", } @Article{Shore:2005:ARB, author = "Haim Shore", title = "Accurate {RMM}-Based Approximations for the {CDF} of the Normal Distribution", journal = j-COMMUN-STAT-THEORY-METH, volume = "34", number = "3", pages = "507--513", year = "2005", CODEN = "CSTMDC", DOI = "https://doi.org/10.1081/STA-200052102", ISSN = "0361-0926 (print), 1532-415X (electronic)", ISSN-L = "0361-0926", bibdate = "Wed Jan 27 05:42:00 MST 2016", bibsource = "https://www.math.utah.edu/pub/tex/bib/communstattheorymeth2000.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Communications in Statistics: Theory and Methods", journal-URL = "http://www.tandfonline.com/loi/lsta20", } @Book{Simon:2005:DCF, author = "Marvin Kenneth Simon and Mohamed-Slim Alouini", title = "Digital Communication over Fading Channels", publisher = pub-WI, address = pub-WI:adr, edition = "Second", pages = "xxxiv + 900", year = "2005", DOI = "https://doi.org/10.1002/0471715220", ISBN = "0-471-64953-8 (hardcover)", ISBN-13 = "978-0-471-64953-3 (hardcover)", LCCN = "TK5103.7 .S523 2005", bibdate = "Sat Dec 16 17:34:06 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; z3950.loc.gov:7090/Voyager", series = "Wiley series in telecommunications and signal processing", URL = "http://www.loc.gov/catdir/description/wiley042/2004042040.html; http://www.loc.gov/catdir/enhancements/fy0617/2004042040-b.html; http://www.loc.gov/catdir/toc/wiley041/2004042040.html", acknowledgement = ack-nhfb, author-dates = "1939--", subject = "Digital communications; Reliability; Mathematics; Radio; Transmitters and transmission; Fading", tableofcontents = "Preface \\ Nomenclature \\ Part 1: Fundamentals \\ 1. Introduction \\ 2. Fading Channel Characterization and Modeling \\ 3. Types of Communication \\ Part 2: Mathematical TOOLS \\ 4. Alternative Representations of Classical Functions \\ 5. Some Useful Expressions for Evaluating Average Error Probability Performance \\ 6. New Representations of Some Probability Density and Cumulative Distribution Functions for Correlative Fading Applications \\ Part 3: Optimum Reception and Performance Evaluation \\ 7. Optimum Receivers for Fading Channels. \\ 8. Performance of Single-Channel Receivers. \\ 9. Performance of Multichannel Receivers. \\ Part 4: Multiuser Communication Systems \\ 10. Outage Performance of Multiuser Communication Systems \\ 11. Optimum Combining --- A Diversity Technique for Communication Over Fading Channels in the Presence of Interference \\ 12. Direct-Sequence Code-Division Multiple Access (DS-CDMA) \\ Part 5: Coded Communication Systems \\ 13. Coded Communicatuion over Fading Channels. \\ 14. Multichannel Transmission-Transmit Diversity and Space-Time Coding \\ 15. Capacity of Fading Channels \\ Index", } @Article{Skorokhodov:2005:MCG, author = "S. L. Skorokhodov", title = "A method for computing generalized hypergeometric function {$_p F_{p - 1}(a_1, \ldots {}, a_p; b_1, \ldots {}, b_{p - 1}; 1)$} in terms of the {Riemann} zeta function", journal = j-COMPUT-MATH-MATH-PHYS, volume = "45", number = "4", pages = "550--562", month = "????", year = "2005", CODEN = "????", ISSN = "0965-5425 (print), 1555-6662 (electronic)", ISSN-L = "0965-5425", bibdate = "Thu Dec 01 09:31:40 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", ZMnumber = "1077.33008", acknowledgement = ack-nhfb, classmath = "33C20 (Generalized hypergeometric series, ${}_pF_q$)", fjournal = "Computational Mathematics and Mathematical Physics", keywords = "generalized hypergeometric functions; Hurwitz zeta function; hypergeometric function; Riemann zeta function", xxnote = "Is the journal name correct??", } @InProceedings{Stehle:2005:GAT, author = "Damien Stehl{\'e} and Paul Zimmermann", title = "{Gal}'s Accurate Tables Method Revisited", crossref = "IEEE:2005:PIS", pages = "??--??", year = "2005", bibdate = "Wed Jun 22 07:02:55 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://arith17.polito.it/final/paper-152.pdf", abstract = "Gal's accurate tables algorithm aims at providing an efficient implementation of mathematical functions with correct rounding as often as possible. This method requires an expensive pre-computation of the values taken by the function or by several related functions at some distinguished points. Our improvements of Gal's method are two-fold: on the one hand we describe what is the arguably best set of distinguished values and how it improves the efficiency and accuracy of the function implementation, and on the other hand we give an algorithm which drastically decreases the cost of the pre-computation. These improvements are related to the worst cases for the correct rounding of mathematical functions and to the algorithms for finding them. We demonstrate how the whole method can be turned into practice for $ 2^x $ and $ \sin x $ for $ x \in [1 / 2, 1) $, in double precision.", acknowledgement = ack-nhfb, pagecount = "8", } @Article{Stehle:2005:SWC, author = "Damien Stehl{\'e} and Vincent Lef{\`e}vre and Paul Zimmermann", title = "Searching Worst Cases of a One-Variable Function Using Lattice Reduction", journal = j-IEEE-TRANS-COMPUT, volume = "54", number = "3", pages = "340--346", month = mar, year = "2005", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/TC.2005.55", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Tue Jul 19 09:20:54 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://csdl.computer.org/comp/trans/tc/2005/03/t0340abs.htm; http://csdl.computer.org/dl/trans/tc/2005/03/t0340.htm; http://csdl.computer.org/dl/trans/tc/2005/03/t0340.pdf; http://ieeexplore.ieee.org/iel5/12/30205/01388198.pdf?isnumber=30205&prod=JNL&arnumber=1388198&arSt=+340&ared=+346&arAuthor=Stehle%2C+D.%3B+Lefevre%2C+V.%3B+Zimmermann%2C+P.; http://ieeexplore.ieee.org/xpls/abs_all.jsp?isnumber=30205&arnumber=1388198&count=13&index=9; http://ieeexplore.ieee.org/xpls/references.jsp?arnumber=1388198", abstract = "We propose a new algorithm to find worst cases for the correct rounding of a mathematical function of one variable. We first reduce this problem to the real small value problem---i.e., for polynomials with real coefficients. Then, we show that this second problem can be solved efficiently by extending Coppersmith's work on the integer small value problem---for polynomials with integer coefficients---using lattice reduction. For floating-point numbers with a mantissa less than $N$ and a polynomial approximation of degree $d$, our algorithm finds all worst cases at distance less than $ N^{\frac {-d^2}{2d + 1}} $ from a machine number in time $ O(N^{{\frac {d + 12d + 1}} + \varepsilon }) $. For $ d = 2 $, a detailed study improves on the $ O(N^{2 / 3 + \varepsilon }) $ complexity from Lef{\`e}vre's algorithm to $ O(N^{4 / 7 + \varepsilon }) $. For larger $d$, our algorithm can be used to check that there exist no worst cases at distance less than $ N^{-k} $ in time $ O(N^{1 / 2 + \varepsilon }) $.", acknowledgement = ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", keywords = "Computer arithmetic; correct rounding; multiple precision arithmetic; special function approximations", } @Article{Uzer:2005:CAS, author = "A. Uzer and T. Ege", title = "On the Convergence Acceleration of Slowly Convergent Sums Involving Oscillating Terms", journal = j-COMPUTING, volume = "75", number = "4", pages = "311--318", month = aug, year = "2005", CODEN = "CMPTA2", DOI = "https://doi.org/10.1007/s00607-005-0126-2", ISSN = "0010-485X (print), 1436-5057 (electronic)", ISSN-L = "0010-485X", MRclass = "65F05; 65F30; 65F50", bibdate = "Tue Jul 8 22:32:46 MDT 2008", bibsource = "http://springerlink.metapress.com/openurl.asp?genre=issue&issn=0010-485X&volume=75&issue=4; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.springerlink.com/openurl.asp?genre=article&issn=0010-485X&volume=75&issue=4&spage=311", acknowledgement = ack-nhfb, fjournal = "Computing", journal-URL = "http://link.springer.com/journal/607", keywords = "convergence acceleration; Fourier series; infinite sums; slowly convergent sums; zeta functions", } @InProceedings{Walters:2005:EFA, author = "George Walters and Michael Schulte", title = "Efficient Function Approximation Using Truncated Multipliers and Squarers", crossref = "IEEE:2005:PIS", pages = "??--??", year = "2005", bibdate = "Wed Jun 22 07:02:55 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://arith17.polito.it/final/paper-190.pdf", abstract = "This paper presents a technique for designing linear and quadratic interpolators for function approximation using truncated multipliers and squarers. Initial coefficient values are found using a Chebyshev series approximation, and then adjusted through exhaustive simulation to minimize the maximum absolute error of the interpolator output. This technique is suitable for any function and any precision up to 24-bits (IEEE single precision). Designs for linear and quadratic interpolators that implement the reciprocal function, $ f(x) = 1 / x, $ are presented and analyzed as an example. We show that a 24-bit truncated reciprocal quadratic interpolator with a design specification of $ \pm 1 $ ulp error requires 24.1\% fewer partial products to implement than a comparable standard interpolator with the same error specification.", acknowledgement = ack-nhfb, pagecount = "8", } @InProceedings{Wang:2005:DFPa, author = "L.-K. Wang and M. J. Schulte", title = "Decimal Floating-Point Square Root Using {Newton--Raphson} Iteration", crossref = "Vassiliadis:2005:IIC", pages = "309--315", year = "2005", bibdate = "Sun Mar 04 10:19:28 2007", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "http://mesa.ece.wisc.edu/publications/cp_2005-05.pdf", abstract = "With continued reductions in feature size, additional functionality may be added to future microprocessors to boost the performance of important application domains. Due to growth in commercial, financial, and Internet-based applications, decimal floating point arithmetic is now attracting more attention and hardware support for decimal operations is being considered by various computer manufacturers. In order to standardize decimal number formats and operations, specifications for decimal floating-point arithmetic have been added to the draft revision of the IEEE-754 Standard for Floating-Point Arithmetic (IEEE-754R). This paper presents an efficient arithmetic algorithm and hardware design for decimal floating-point square root. This design uses an optimized piecewise linear approximation, a modified Newton--Raphson iteration, a specialized rounding technique, and a modified decimal multiplier. Synthesis results show that a 64-bit (16-digit) implementation of decimal square root, which is compliant with IEEE-754R, has an estimated critical path delay of 0.95 ns and a maximum latency of 210 clock cycles when implemented using a sequential multiplier and LSI Logic's 0.11 micron Gflx-P standard cell library.", acknowledgement = ack-nhfb, keywords = "decimal floating-point arithmetic", } @Article{Weber:2005:MIG, author = "Kenneth Weber and Vilmar Trevisan and Luiz Felipe Martins", title = "A modular integer {GCD} algorithm", journal = j-J-ALG, volume = "54", number = "2", pages = "152--167", month = feb, year = "2005", CODEN = "JOALDV", DOI = "https://doi.org/10.1016/j.jalgor.2004.06.006", ISSN = "0196-6774 (print), 1090-2678 (electronic)", ISSN-L = "0196-6774", bibdate = "Tue Dec 11 09:21:34 MST 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jalg.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0196677404001075", acknowledgement = ack-nhfb, fjournal = "Journal of Algorithms", journal-URL = "http://www.sciencedirect.com/science/journal/01966774", } @Article{West:2005:BAC, author = "G. West", title = "Better approximations to cumulative normal functions", journal = "Wilmott Magazine", volume = "??", number = "??", pages = "70--76", month = "????", year = "2005", ISSN = "1540-6962 (print), 1541-8286 (electronic)", ISSN-L = "1540-6962", bibdate = "Sat Dec 16 17:59:43 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, journal-URL = "http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)1541-8286; https://www.wilmott.com/category/magazine/", remark = "No issues online at Wiley before year 2011, or at Wilmott before 2006.", } @TechReport{Zimmermann:2005:XXX, author = "Paul Zimmermann", title = "5,341,321", type = "Technical report", institution = inst-LORIA-INRIA-LORRAINE, address = inst-LORIA-INRIA-LORRAINE:adr, pages = "2", day = "8", month = jun, year = "2005", bibdate = "Sun Sep 10 07:32:04 2006", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "http://www.loria.fr/~zimmerma/papers/5341321.ps.gz", abstract = "This short note shows the nasty effects of patents for the development of free software, even for patents that were not written with software applications in mind.", acknowledgement = ack-nhfb, keywords = "floating-point division; Karp--Markstein patent on modified Newton--Raphson iteration", remark = "The title is the number of the U.S. Patent on the algorithm described in the article, which is a completely trivial modification of Newton--Raphson iteration, published in \cite{Karp:1997:HPD}. The patent itself is \cite{Karp:1994:FPA}, and it expired on 5 May 2013.", } @InProceedings{Anderson:2006:AMF, author = "Cristina S. Anderson and Shane Story and Nikita Astafiev", title = "Accurate Math Functions on the {Intel IA-32} Architecture: a Performance-Driven Design", crossref = "Anonymous:2006:PCR", pages = "??--??", year = "2006", bibdate = "Tue Jun 27 10:28:05 2006", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, keywords = "elementary functions", } @TechReport{Bertot:2006:PGS, author = "Yves Bertot and Nicolas Magaud and Paul Zimmermann", title = "A proof of {GMP} square root using the {Coq} assistant", type = "Research Report", number = "RR-4475", institution = inst-LORIA-INRIA-LORRAINE, address = inst-LORIA-INRIA-LORRAINE:adr, pages = "28", year = "2006", bibdate = "Sun Sep 10 08:34:35 2006", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "ftp://ftp.inria.fr/INRIA/publication/publi-pdf/RR/RR-4475.pdf; ftp://ftp.inria.fr/INRIA/publication/publi-ps-gz/RR/RR-4475.ps.gz; http://www.inria.fr/rrrt/rr-4475.html", abstract = "We present a formal proof (at the implementation level) of an efficient algorithm proposed in to compute square roots of arbitrarily large integers. This program, which is part of the GNU Multiple Precision Arithmetic Library (GMP), is completely proven within the system. Proofs are developed using the Correctness tool to deal with imperative features of the program. The formalization is rather large (more than 13000 lines) and requires some advanced techniques for proof management and reuse.", acknowledgement = ack-nhfb, } @Article{Bogolubsky:2006:FEH, author = "A. I. Bogolubsky and S. L. Skorokhodov", title = "Fast evaluation of the hypergeometric function {$_p F_{p - 1}(a; b; z)$} at the singular point $ z = 1 $ by means of the {Hurwitz} zeta function $ \zeta (\alpha, s) $", journal = j-PROG-COMP-SOFT, volume = "32", number = "??", pages = "145--153", month = "????", year = "2006", CODEN = "PCSODA", ISSN = "0361-7688 (print), 1608-3261 (electronic)", ISSN-L = "0361-7688", bibdate = "Thu Dec 01 09:34:31 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Programming and Computer Software; translation of Programmirovaniye (Moscow, USSR) Plenum", journal-URL = "http://link.springer.com/journal/11086", } @Article{Boldo:2006:PFF, author = "Sylvie Boldo", editor = "Ulrich Furbach and Natarajan Shankar", booktitle = "{Automated Reasoning: Third International Joint Conference, IJCAR 2006, Seattle, WA, USA, August 17--20, 2006, Proceedings}", title = "Pitfalls of a full floating-point proof: Example on the formal proof of the {Veltkamp\slash Dekker} algorithms", journal = j-LECT-NOTES-COMP-SCI, bookpages = "xv + 680", pages = "52--66", year = "2006", CODEN = "LNCSD9", DOI = "https://doi.org/10.1007/11814771_6", ISBN = "3-540-37187-7 (paperback), 3-540-37188-5", ISBN-13 = "978-3-540-37187-8 (paperback), 978-3-540-37188-5", ISSN = "0302-9743 (print), 1611-3349 (electronic)", ISSN-L = "0302-9743", LCCN = "QA76.9.A96 I33 2006eb", MRnumber = "MR2354672", bibdate = "Mon Jun 12 16:14:21 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, book-DOI = "https://doi.org/10.1007/11814771", book-URL = "http://www.springer.com/us/book/9783540371878", fjournal = "Lecture Notes in Computer Science", journal-URL = "http://link.springer.com/bookseries/558", } @TechReport{Brent:2006:FAH, author = "Richard P. Brent", title = "Fast Algorithms for High-Precision Computation of Elementary Functions", type = "Report", number = "??", institution = "Australian National University", address = "Canberra, ACT 0200, Australia", pages = "61", day = "12", month = jul, year = "2006", bibdate = "Fri Sep 04 16:33:10 2020", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://rnc7.loria.fr/brent_invited.pdf; https://maths-people.anu.edu.au/~brent/pd/RNC7t.pdf", acknowledgement = ack-nhfb, keywords = "arithmetic-geometric mean", remark = "From page 57: ``This talk is based on a chapter of a book that Paul Zimmermann and I are writing''. That book is entry \cite{Brent:2011:MCA}.", } @TechReport{Crandall:2006:NFP, author = "Richard E. Crandall", title = "Note on fast polylogarithm computation", type = "Report", institution = "Reed College", address = "Portland, OR, USA", pages = "6", month = jan, year = "2006", bibdate = "Tue Mar 19 09:03:09 2013", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/c/crandall-richard-e.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://people.reed.edu/~crandall/papers/Polylog.pdf; https://web.archive.org/web/20120916145721/http://people.reed.edu/~crandall/papers/Polylog.pdf", abstract = "The polylogarithm function $ \Li_n(z) = \sum_{k = 1}^\infty z^k / k^n $, manifestly convergent for $ |z| \eq 1 $, integer $ n > 1 $, is sometimes numerically\slash symbolically relevant for $ |z| > 1 $, i.e., the analytic continuation may be required. By exploiting analytic symmetry relations, we give, for integer $n$, simple and efficient algorithms for complete continuation in complex $z$.", acknowledgement = ack-nhfb, } @Article{Cuyt:2006:ERM, author = "Annie Cuyt and Brigitte Verdonk and Haakon Waadeland", title = "Efficient and Reliable Multiprecision Implementation of Elementary and Special Functions", journal = j-SIAM-J-SCI-COMP, volume = "28", number = "4", pages = "1437--1462", month = jan, year = "2006", CODEN = "SJOCE3", DOI = "https://doi.org/10.1137/050629203", ISSN = "1064-8275 (print), 1095-7197 (electronic)", ISSN-L = "1064-8275", bibdate = "Wed May 19 10:43:41 MDT 2010", bibsource = "http://epubs.siam.org/sam-bin/dbq/toc/SISC/28/4; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "SIAM Journal on Scientific Computing", journal-URL = "http://epubs.siam.org/sisc", } @InProceedings{deDinechin:2006:STP, author = "Florent de Dinechin and Sergey Maidanov", title = "Software techniques for perfect elementary functions in floating-point interval arithmetic", crossref = "Anonymous:2006:PCR", pages = "??--??", year = "2006", bibdate = "Tue Jun 27 10:28:05 2006", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, keywords = "elementary functions", } @Book{ElAttar:2006:SFO, author = "Refaat A. {El Attar}", title = "Special functions and orthogonal polynomials", volume = "3", publisher = "Lulu Press", address = "Morrisville, NC, USA", pages = "vi + 302", year = "2006", ISBN = "1-4116-6690-9 (paperback)", ISBN-13 = "978-1-4116-6690-0 (paperback)", LCCN = "QA404.5 .E5 2006; QA351 .E5 2006", bibdate = "Sat Oct 30 17:42:31 MDT 2010", bibsource = "fsz3950.oclc.org:210/WorldCat; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcw.bib", series = "Mathematical series", acknowledgement = ack-nhfb, subject = "Functions, Special; Orthogonal polynomials; Polinomios ortogonales; Series ortogonales", tableofcontents = "Series solutions of differential equations \\ Gamma and beta functions and others \\ Legendre polynomials \\ Hermite polynomials \\ Laguerre and other orthogonal polynomials \\ Bessel functions", } @Article{Ferreira:2006:GHF, author = "C. Ferreira and J. L. L{\'o}pez and E. P. Sinus{\'\i}a", title = "The {Gauss} hypergeometric function {$ F(a; b; c; z) $} for large $c$", journal = j-J-COMPUT-APPL-MATH, volume = "197", number = "2", pages = "568--577", day = "15", month = jan, year = "2006", CODEN = "JCAMDI", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", MRclass = "33C05 (33F05 41A60)", MRnumber = "MR2260426 (2007i:33012)", bibdate = "Thu Dec 01 09:20:59 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", remark = "$ F(a; b; c; z) = {}_2 F_1 (a, b + 1; c + 2; z) $", } @Article{Gil:2006:ARP, author = "Amparo Gil and Javier Segura and Nico M. Temme", title = "{Algorithm 850}: {Real} parabolic cylinder functions {$ U(a, x) $, $ V(a, x) $}", journal = j-TOMS, volume = "32", number = "1", pages = "102--112", month = mar, year = "2006", CODEN = "ACMSCU", DOI = "https://doi.org/10.1145/1132973.1132978", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Fri May 26 06:32:19 MDT 2006", bibsource = "http://www.acm.org/pubs/contents/journals/toms/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib", abstract = "Fortran 90 programs for the computation of real parabolic cylinder functions are presented. The code computes the functions $ U(a, x) $, $ V(a, x) $ and their derivatives for real $a$ and $ x (x \geq 0) $. The code also computes scaled functions. The range of computation for scaled PCFs is practically unrestricted. The aimed relative accuracy for scaled functions is better than $ 5 \times 10^{14} $. Exceptions to this accuracy are the evaluation of the functions near their zeros and the error caused by the evaluation of trigonometric functions of large arguments when $ |a| > x $. The routines always give values for which the Wronskian relation for scaled functions is verified with a relative accuracy better than $ 5 \times 10^{14} $. The accuracy of the unscaled functions is also better than $ 5 \times 10^{14} $ for moderate values of $x$ and $a$ (except close to the zeros), while for large $x$ and $a$ the error is dominated by exponential and trigonometric function evaluations. For IEEE standard double precision arithmetic, the accuracy is better than $ 5 \times 10^{13} $ in the computable range of unscaled PCFs (except close to the zeros).", acknowledgement = ack-nhfb, fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", } @Article{Gil:2006:CRP, author = "Amparo Gil and Javier Segura and Nico M. Temme", title = "Computing the real parabolic cylinder functions {$ U(a, x) $, $ V(a, x) $}", journal = j-TOMS, volume = "32", number = "1", pages = "70--101", month = mar, year = "2006", CODEN = "ACMSCU", DOI = "https://doi.org/10.1145/1132973.1132977", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Fri May 26 06:32:19 MDT 2006", bibsource = "http://www.acm.org/pubs/contents/journals/toms/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib", abstract = "Methods for the computation of real parabolic cylinder functions $ U(a, x) $, and $ V(a, x) $ and their derivatives are described. We give details on power series, asymptotic series, recursion and quadrature. A combination of these methods can be used for computing parabolic cylinder functions for unrestricted values of the order $a$ and the variable $x$ except for the overflow\slash underflow limitations. By factoring the dominant exponential factor, scaled functions can be computed without practical overflow\slash underflow limitations. In an accompanying article we describe the precise domains for these methods and we present the Fortran 90 codes for the computation of these functions.", acknowledgement = ack-nhfb, fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", } @Article{Jones:2006:PCF, author = "D. S. Jones", title = "Parabolic cylinder functions of large order", journal = j-J-COMPUT-APPL-MATH, volume = "190", number = "1--2", pages = "453--469", day = "1", month = jun, year = "2006", CODEN = "JCAMDI", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Sat Feb 25 13:11:58 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2005.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0377042705002463", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @Article{Kong:2006:IGA, author = "Fanyu Kong and Zhun Cai and Jia Yu and Daxing Li", title = "Improved generalized {Atkin} algorithm for computing square roots in finite fields", journal = j-INFO-PROC-LETT, volume = "98", number = "1", pages = "1--5", day = "15", month = apr, year = "2006", CODEN = "IFPLAT", ISSN = "0020-0190 (print), 1872-6119 (electronic)", ISSN-L = "0020-0190", bibdate = "Thu Mar 31 18:41:08 MDT 2011", bibsource = "http://www.sciencedirect.com/science/journal/00200190; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, fjournal = "Information Processing Letters", journal-URL = "http://www.sciencedirect.com/science/journal/00200190", } @Article{Kornerup:2006:CSV, author = "Peter Kornerup and Jean-Michel Muller", title = "Choosing starting values for certain {Newton--Raphson} iterations", journal = j-THEOR-COMP-SCI, volume = "351", number = "1", pages = "101--110", day = "14", month = feb, year = "2006", CODEN = "TCSCDI", ISSN = "0304-3975 (print), 1879-2294 (electronic)", ISSN-L = "0304-3975", bibdate = "Tue Mar 29 06:48:55 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/tcs2005.bib", abstract = "We aim at finding the best possible seed values when computing $ a^{1 / p} $ using the Newton--Raphson iteration in a given interval. A natural choice of the seed value would be the one that best approximates the expected result. It turns out that in most cases, the best seed value can be quite far from this natural choice. When we evaluate a monotone function $ f(a) $ in the interval $ [a_\mathrm {min}, a_\mathrm {max}] $, by building the sequence $ x_n $ defined by the Newton--Raphson iteration, the natural choice consists in choosing $ x_0 $ equal to the arithmetic mean of the endpoint values. This minimizes the maximum possible distance between $ x_0 $ and $ f(a) $. And yet, if we perform $n$ iterations, what matters is to minimize the maximum possible distance between $ x_n $ and $ f(a) $. In several examples, the value of the best starting point varies rather significantly with the number of iterations.", acknowledgement = ack-nhfb, ajournal = "Theor. Comput. Sci.", fjournal = "Theoretical Computer Science", journal-URL = "http://www.sciencedirect.com/science/journal/03043975/", } @Book{Muller:2006:EFA, author = "Jean-Michel Muller", booktitle = "Elementary Functions: Algorithms and Implementation", title = "Elementary Functions: Algorithms and Implementation", publisher = pub-BIRKHAUSER, address = pub-BIRKHAUSER:adr, edition = "Second", pages = "xxii + 266", year = "2006", ISBN = "0-8176-4372-9", ISBN-13 = "978-0-8176-4372-0", LCCN = "QA331 .M866 2006", bibdate = "Fri Jul 25 12:00:55 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/maple-extract.bib; https://www.math.utah.edu/pub/tex/bib/master.bib; z3950.loc.gov:7090/Voyager", price = "US\$59.95", URL = "http://perso.ens-lyon.fr/jean-michel.muller/SecondEdition.html; http://www.springer.com/sgw/cda/frontpage/0,,4-40109-22-72377986-0,00.html", acknowledgement = ack-nhfb, subject = "Functions; Data processing; Algorithms", tableofcontents = "Preface to the second edition \\ Preface to the first edition \\ Introduction / 1--7 \\ Some basic things about computer arithmetic / 9--24 \\ Part I. Algorithms based on polynomial approximation and/or table lookup, multiple-precision evaluation of functions / 25--25 \\ Polynomial or rational approximations / 27--66 \\ Table-based methods / 67--87 \\ Multiple-precision evaluation of functions / 89--100 \\ Part II. Shift-and-add algorithms / 101--101 \\ Introduction to shift-and-add algorithms / 103--131 \\ The CORDIC algorithm / 133--156 \\ Some other shift-and-add algorithms / 157--169 \\ Part III. Range reduction, final rounding and exceptions / 171--171 \\ Range reduction / 173--191 \\ Final rounding / 193--216 \\ Miscellaneous / 217--223 \\ Examples of implementation / 225--232 \\ Bibliography / 233--259 \\ Index / 261--265", } @InProceedings{Muller:2006:GFA, author = "Jean-Michel Muller", editor = "Michael B. Matthews", booktitle = "{2006 Fortieth Asilomar Conference on Signals, Systems and Computers. October 29--November 1, 2006. Pacific Grove, California}", title = "Generating function approximations at compile time", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "328--331", year = "2006", DOI = "https://doi.org/10.1109/ACSSC.2006.354761", ISBN = "1-4244-0785-0", ISBN-13 = "978-1-4244-0785-9", bibdate = "Fri Sep 29 10:57:58 2023", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, } @Article{Nowak:2006:MCA, author = "Rafal Nowak", title = "A method of convergence acceleration of some continued fractions", journal = j-NUMER-ALGORITHMS, volume = "41", number = "3", pages = "297--317", month = mar, year = "2006", CODEN = "NUALEG", DOI = "https://doi.org/10.1007/s11075-005-9013-3", ISSN = "1017-1398 (print), 1572-9265 (electronic)", ISSN-L = "1017-1398", MRclass = "subject classification; 30B70; 40A15; 65B99", bibdate = "Tue Jul 8 19:14:28 MDT 2008", bibsource = "http://springerlink.metapress.com/openurl.asp?genre=issue&issn=1017-1398&volume=41&issue=3; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.springerlink.com/openurl.asp?genre=article&issn=1017-1398&volume=41&issue=3&spage=297", acknowledgement = ack-nhfb, fjournal = "Numerical Algorithms", journal-URL = "http://link.springer.com/journal/11075", keywords = "continued fraction; convergence acceleration; modified approximant; tail", } @Article{Ozban:2006:NMA, author = "Ahmet Ya{\c{s}}ar {\"O}zban", title = "New methods for approximating square roots", journal = j-APPL-MATH-COMP, volume = "175", number = "1", pages = "532--540", day = "1", month = apr, year = "2006", CODEN = "AMHCBQ", ISSN = "0096-3003 (print), 1873-5649 (electronic)", ISSN-L = "0096-3003", bibdate = "Sat Jul 12 09:02:54 MDT 2008", bibsource = "http://www.sciencedirect.com/science/journal/00963003; https://www.math.utah.edu/pub/tex/bib/applmathcomput2005.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, fjournal = "Applied Mathematics and Computation", journal-URL = "http://www.sciencedirect.com/science/journal/00963003", } @InProceedings{Parks:2006:UTS, author = "Michael Parks", title = "Unifying Tests for Square Root", crossref = "Anonymous:2006:PCR", pages = "??--??", year = "2006", bibdate = "Tue Jun 27 10:28:05 2006", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, keywords = "elementary functions", } @Article{Qian:2006:HMP, author = "Jianbo Qian and Cao An Wang", title = "How much precision is needed to compare two sums of square roots of integers?", journal = j-INFO-PROC-LETT, volume = "100", number = "5", pages = "194--198", day = "16", month = dec, year = "2006", CODEN = "IFPLAT", ISSN = "0020-0190 (print), 1872-6119 (electronic)", ISSN-L = "0020-0190", bibdate = "Thu Mar 31 15:52:31 MDT 2011", bibsource = "http://www.sciencedirect.com/science/journal/00200190; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, fjournal = "Information Processing Letters", journal-URL = "http://www.sciencedirect.com/science/journal/00200190", } @Article{Shi:2006:NAS, author = "Xiquan Shi and Fengshan Liu and Minghan Hu", title = "A new asymptotic series for the Gamma function", journal = j-J-COMPUT-APPL-MATH, volume = "195", number = "1--2", pages = "134--154", day = "15", month = oct, year = "2006", CODEN = "JCAMDI", DOI = "https://doi.org/10.1016/j.cam.2005.03.081", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Sat Feb 25 13:12:01 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2005.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0377042705004802", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @Article{Sidi:2006:CTC, author = "Avram Sidi", title = "A challenging test for convergence accelerators: Summation of a series with a special sign pattern", journal = "App. Math. E-Notes", volume = "6", number = "??", pages = "225--234", month = "????", year = "2006", CODEN = "????", ISSN = "????", MRclass = "40A99 (11M41 40A05 65B10)", MRnumber = "MR2231748 (2007h:40009)", bibdate = "Thu Dec 01 10:33:54 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, keywords = "convergence acceleration; Shanks transformation", } @InProceedings{Thakkar:2006:PDP, author = "Anuja J. Thakkar and Abdel Ejnioui", title = "Pipelining of double precision floating point division and square root operations", crossref = "Menezes:2006:PAS", pages = "488--493", year = "2006", DOI = "https://doi.org/10.1145/1185448.1185555", bibdate = "Sat Oct 9 13:04:49 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", abstract = "Space applications rely increasingly on high data rate DSP algorithms. These algorithms use double precision floating point arithmetic operations. While most DSP applications can be compiled on DSP processors, high data rate DSP computations require novel implementation technologies to support their high throughputs. Only recently, gate densities in FPGAs have reached a level which makes them attractive platforms to implement compute-intensive DSP applications. In this context, this paper presents the sequential and pipelined designs of a double precision floating point divider and square root unit on FPGAs. Contrary to pipelined parallel implementations, the pipelining of these units is based on unrolling the iterations in low-radix digit recurrence algorithms. These units are mapped on generic FPGA reconfigurable fabric without taking advantage of any advanced architectural components available in high capacity FPGAs. The implementations of these designs show that their performances are comparable to, and sometimes higher than, the performances of non-iterative designs based of high radix numbers. The iterative divider and square root unit occupy less than 1\% of an XC2V6000 FPGA chip while their pipelined counterparts can produce throughputs that reach the 100 MFLOPS mark by consuming a modest 8\% of the chip area.", acknowledgement = ack-nhfb, } @Article{VanDeun:2006:ACI, author = "Joris {Van Deun} and Ronald Cools", title = "{Algorithm 858}: {Computing} infinite range integrals of an arbitrary product of {Bessel} functions", journal = j-TOMS, volume = "32", number = "4", pages = "580--596", month = dec, year = "2006", CODEN = "ACMSCU", DOI = "https://doi.org/10.1145/1186785.1186790", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Sat Apr 14 09:48:57 MDT 2007", bibsource = "http://www.acm.org/pubs/contents/journals/toms/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib", abstract = "We present an algorithm to compute integrals of the form $ \int_0^\infty x^m \prod^k_i = 1 J_{\nu_i}(a_i x)d x $ with $ J_{\nu_i}(x) $ the Bessel function of the first kind and (real) order $ \nu_i $. The parameter $m$ is a real number such that $ \sum_i \nu_i + m > - 1 $ and the coefficients $ a_i $ are strictly positive real numbers. The main ingredients in this algorithm are the well-known asymptotic expansion for $ J_{\nu_i}(x) $ and the observation that the infinite part of the integral can be approximated using the incomplete Gamma function $ \Gamma (a, z) $. Accurate error estimates are included in the algorithm, which is implemented as a MATLAB program.", acknowledgement = ack-nhfb, fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", } @Article{VanDeun:2006:SRI, author = "Joris {Van Deun} and Ronald Cools", title = "A stable recurrence for the incomplete gamma function with imaginary second argument", journal = j-NUM-MATH, volume = "104", number = "4", pages = "445--456", month = oct, year = "2006", CODEN = "NUMMA7", DOI = "https://doi.org/10.1007/s00211-006-0026-1", ISSN = "0029-599X (print), 0945-3245 (electronic)", ISSN-L = "0029-599X", bibdate = "Tue Jul 8 10:28:23 MDT 2008", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Numerische Mathematik", journal-URL = "http://link.springer.com/journal/211", } @InProceedings{Wang:2006:EAR, author = "Laixiong Wang and Yangping Chen and Shitan Huang", editor = "{IEEE}", booktitle = "{Sixth World Congress on Intelligent Control and Automation, June 21--23, 2006, Dalian, China}", title = "Efficient Argument Range Reduction for Implementation of Double-Precision Floating-Point Exponential Function", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "6800--6803", year = "2006", DOI = "https://doi.org/10.1109/wcica.2006.1714401", bibdate = "Mon Nov 10 06:50:26 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, } @Article{Wires:2006:RRS, author = "Kent E. Wires and Michael J. Schulte", title = "Reciprocal and Reciprocal Square Root Units with Operand Modification and Multiplication", journal = j-J-VLSI-SIGNAL-PROC, volume = "42", number = "3", pages = "257--272", month = mar, year = "2006", CODEN = "JVSPED", DOI = "https://doi.org/10.1007/s11265-006-4186-0", ISSN = "0922-5773 (print), 1573-109x (electronic)", ISSN-L = "0922-5773", bibdate = "Mon Mar 05 08:26:23 2007", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://springerlink.metapress.com/content/t6027p6713727606/fulltext.pdf", acknowledgement = ack-nhfb, fjournal = "Journal of VLSI Signal Processing", } @InProceedings{Barnett:2007:HPV, author = "Ross Barnett and J. A. Youngman", booktitle = "{1st Joint Meeting of the American Mathematical Society and the New Zealand Mathematical Society, Victoria University of Wellington, Wellington, New Zealand, December 12--15, 2007}", title = "High-Precision Values of the Gamma Function of real argument", publisher = pub-AMS, address = pub-AMS:adr, pages = "????", year = "2007", ISBN = "????", ISBN-13 = "????", LCCN = "????", bibdate = "Mon Jul 14 11:57:00 2008", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://homepages.mcs.vuw.ac.nz/~mathmeet/amsnzms2007/abstracts.pdf", abstract = "A method is described to calculate values of $ \Gamma (\nu) $, $ 0 \leq \nu \leq 1 $ to arbitrary precision by combining a Bessel function with a $_0 F_1$ function. Steed's algorithm is used to compute the regular Bessel function $ J_\nu (x) $, for a suitable argument $x$ and real $ \nu $, to arbitrary accuracy. Hence the gamma function is obtained. Example values are given to 200D. Verification is by the 80D-results of Frans{\'e}n and Wrigge, by the use of the duplication formula, and by computing the closed form results of Borwein and Zucker. A caveat is offered concerning the coding of the Bessel functions in Numerical Recipes and in the GSL library.", acknowledgement = ack-nhfb, } @Article{Batterman:2007:SSF, author = "Robert W. Batterman", title = "On the Specialness of Special Functions (The Nonrandom Effusions of the Divine Mathematician)", journal = j-BRITISH-J-PHILOS-SCI, volume = "58", number = "2", pages = "263--286", month = jun, year = "2007", CODEN = "BJPIA5", DOI = "https://doi.org/10.1093/bjps/axm007", ISSN = "0007-0882 (print), 1464-3537 (electronic)", ISSN-L = "0007-0882", bibdate = "Thu Oct 7 14:03:55 MDT 2010", bibsource = "http://bjps.oxfordjournals.org/content/58/2.toc; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://bjps.oxfordjournals.org/content/58/2/263.full.pdf+html", acknowledgement = ack-nhfb, fjournal = "British Journal for the Philosophy of Science", journal-URL = "http://www.jstor.org/journals/00070882.html", onlinedate = "May 18, 2007", } @InProceedings{Brisebarre:2007:EPA, author = "Nicolas Brisebarre and Sylvain Chevillard", title = "Efficient polynomial {$ L^\infty $}-approximations", crossref = "Kornerup:2007:PIS", pages = "169--176", year = "2007", DOI = "https://doi.org/10.1109/ARITH.2007.17", bibdate = "Tue Oct 9 17:16:03 MDT 2007", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", abstract = "We address the problem of computing a good floating-point-coefficient polynomial approximation to a function, with respect to the supremum norm. This is a key step in most processes of evaluation of a function. We present a fast and efficient method, based on lattice basis reduction, that often gives the best polynomial possible and most of the time returns a very good approximation.", acknowledgement = ack-nhfb, keywords = "ARITH-18", } @InProceedings{Burgess:2007:DAV, author = "Neil Burgess and Chris N. Hinds", title = "Design of the {ARM VFP11} Divide and Square Root Synthesisable Macrocell", crossref = "Kornerup:2007:PIS", pages = "87--96", year = "2007", DOI = "https://doi.org/10.1109/ARITH.2007.15", bibdate = "Tue Oct 9 16:32:41 MDT 2007", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", abstract = "This paper presents the detailed design of the ARM VFP11 Divide and Square Root synthesisable macrocell. The macrocell was designed using the minimum-redundancy radix-4 SRT digit recurrence algorithm, and this paper describes a novel acceleration technique employed to achieve the required processor clock frequency of up to 750MHz in 90nm CMOS. Logical Effort theory is used to provide a delay analysis of the unit, which demonstrates the balanced nature of the two critical paths therein.", acknowledgement = ack-nhfb, keywords = "ARITH-18", } @Article{Cerone:2007:SFA, author = "Pietro Cerone", title = "Special functions: approximations and bounds", journal = "Applicable Analysis and Discrete Mathematics", volume = "1", number = "1", pages = "72--91", year = "2007", DOI = "https://doi.org/10.2298/AADM0701072C", ISSN = "1452-8630 (print), 2406-100X (electronic)", ISSN-L = "1452-8630", MRclass = "26D15 (26D20 33B15 33C05)", MRnumber = "2316589", MRreviewer = "Pierpaolo Natalini", bibdate = "Thu Jul 29 07:41:55 2021", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://pefmath.etf.rs/accepted/AADM-Vol1-No1-72-91.pdf", abstract = "The Steffensen inequality and bounds for the {\v{C}}eby{\v{s}}ev functional are utilised to obtain bounds for some classical special functions. The technique relies on determining bounds on integrals of products of functions. The above techniques are used to obtain novel and useful bounds for the Bessel function of the first kind, the Beta function, and the Zeta function.", acknowledgement = ack-nhfb, ajournal = "Appl. Anal. Discrete Math.", fjournal = "Applicable Analysis and Discrete Mathematics", } @Book{Chakraborty:2007:VSF, author = "Kalyan Chakraborty and Shigeru Kanemitsu and Haruo Tsukada", title = "Vistas of special functions {II}", publisher = pub-WORLD-SCI, address = pub-WORLD-SCI:adr, pages = "xii + 215", year = "2007", ISBN = "981-270-774-3", ISBN-13 = "978-981-270-774-1", LCCN = "QA351 .K35 2007", bibdate = "Sat Oct 30 17:02:07 MDT 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcw.bib; prodorbis.library.yale.edu:7090/voyager", abstract = "This is a unique book for studying special functions through zeta-functions. Many important formulas of special functions scattered throughout the literature are located in their proper positions and readers get enlightened access to them in this book. The areas covered include: Bernoulli polynomials, the gamma function (the beta and the digamma function), the zeta-functions (the Hurwitz, the Lerch, and the Epstein zeta-function), Bessel functions, an introduction to Fourier analysis, finite Fourier series, Dirichlet L-functions, the rudiments of complex functions and summation formulas. The Fourier series for the (first) periodic Bernoulli polynomial is effectively used, familiarizing the reader with the relationship between special functions and zeta-functions.", acknowledgement = ack-nhfb, subject = "Functions, Special; Bernoulli polynomials", tableofcontents = "The theory of Bernoulli and allied polynomials \\ The theory of the gamma and related functions \\ The theory of the Hurwitz--Lerch zeta-functions \\ The theory of Bernoulli polynomials via zeta-functions \\ The theory of the gamma and related functions via zeta-functions \\ The theory of Bessel functions and the Epstein zeta-functions \\ Fourier series and Fourier transforms \\ Around Dirichlet's $L$-functions \\ Appendix A: Complex functions \\ Appendix B: Summation formulas and convergence theorems", } @InProceedings{Detrey:2007:FPT, author = "J{\'e}r{\'e}mie Detrey and Florent de Dinechin", booktitle = "2007 International Conference on Field Programmable Logic and Applications", title = "Floating-Point Trigonometric Functions for {FPGAs}", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "29--34", month = aug, year = "2007", DOI = "https://doi.org/10.1109/fpl.2007.4380621", bibdate = "Sat Jan 3 08:30:19 2026", bibsource = "https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, } @Article{Detrey:2007:PFP, author = "J{\'e}r{\'e}mie Detrey and Florent de Dinechin", title = "Parameterized floating-point logarithm and exponential functions for {FPGAs}", journal = j-MICROPROC-MICROSYS, volume = "31", number = "8", pages = "537--545", day = "3", month = dec, year = "2007", CODEN = "MIMID5", DOI = "https://doi.org/10.1016/j.micpro.2006.02.008", ISSN = "0141-9331 (print), 1872-9436 (electronic)", ISSN-L = "0141-9331", bibdate = "Sat Nov 8 14:03:10 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.ens-lyon.fr/LIP/Arenaire/", acknowledgement = ack-nhfb, fjournal = "Microprocessors and Microsystems", keywords = "Elementary functions; Exponential; Floating-point; FPGA; FPLibrary; Logarithm; Parameterized operators", } @Article{Dyer:2007:AEF, author = "Stephen Dyer and Justin Dyer", title = "Approximations to Error Functions", journal = "IEEE Instrumentation \& Measurement Magazine", volume = "10", number = "6", pages = "45--48", month = dec, year = "2007", DOI = "https://doi.org/10.1109/mim.2007.4428581", bibdate = "Sat Dec 16 16:26:38 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "https://ieeexplore.ieee.org/abstract/document/4428581/", acknowledgement = ack-nhfb, } @Article{Ercegovac:2007:CSR, author = "Milo{\v{s}} D. Ercegovac and Jean-Michel Muller", title = "Complex Square Root with Operand Prescaling", journal = j-J-VLSI-SIGNAL-PROC, volume = "49", number = "1", pages = "19--30", month = oct, year = "2007", CODEN = "JVSPED", DOI = "https://doi.org/10.1007/s11265-006-0029-2", ISSN = "0922-5773 (print), 1573-109x (electronic)", ISSN-L = "0922-5773", bibdate = "Mon Nov 05 19:24:36 2007", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", abstract = "We propose a radix-$r$ digit-recurrence algorithm for complex square-root. The operand is prescaled to allow the selection of square-root digits by rounding of the residual. This leads to a simple hardware implementation of digit selection. Moreover, the use of digit recurrence approach allows correct rounding of the result if needed. The algorithm, compatible with the complex division presented in Ercegovac and Muller (``Complex Division with Prescaling of the Operands,'' in Proc. Application-Specific Systems, Architectures, and Processors (ASAP'03), The Hague, The Netherlands, June 24---26, 2003), and its design are described. We also give rough estimates of its latency and cost with respect to implementation based on standard floating-point instructions as used in software routines for complex square root.", acknowledgement = ack-nhfb, fjournal = "Journal of VLSI Signal Processing", } @Article{Ferraro:2007:FAG, author = "Giovanni Ferraro", title = "The foundational aspects of {Gauss}'s work on the hypergeometric, factorial and digamma functions", journal = j-ARCH-HIST-EXACT-SCI, volume = "61", number = "5", pages = "457--518", month = sep, year = "2007", CODEN = "AHESAN", DOI = "https://doi.org/10.1007/s00407-007-0004-8", ISSN = "0003-9519 (print), 1432-0657 (electronic)", ISSN-L = "0003-9519", MRclass = "33-03 (01A50 33B15 33C05)", MRnumber = "2329096 (2009d:33002)", MRreviewer = "M. E. Muldoon", bibdate = "Fri Feb 4 21:50:42 MST 2011", bibsource = "http://springerlink.metapress.com/openurl.asp?genre=issue&issn=0003-9519&volume=61&issue=5; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.springerlink.com/openurl.asp?genre=article&issn=0003-9519&volume=61&issue=5&spage=457", acknowledgement = ack-nhfb, fjournal = "Archive for History of Exact Sciences", journal-URL = "http://link.springer.com/journal/407", MRtitle = "The foundational aspects of {Gauss}'s work on the hypergeometric, factorial and digamma functions", } @Article{Fousse:2007:MMP, author = "Laurent Fousse and Guillaume Hanrot and Vincent Lef{\`e}vre and Patrick P{\'e}lissier and Paul Zimmermann", title = "{MPFR}: a multiple-precision binary floating-point library with correct rounding", journal = j-TOMS, volume = "33", number = "2", pages = "1--15", month = jun, year = "2007", CODEN = "ACMSCU", DOI = "https://doi.org/10.1145/1236463.1236468", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", MRclass = "65G99", MRnumber = "MR2326955", bibdate = "Thu Jul 26 17:36:59 MDT 2007", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib", abstract = "This article presents a multiple-precision binary floating-point library, written in the ISO C language, and based on the GNU MP library. Its particularity is to extend to arbitrary-precision, ideas from the IEEE 754 standard, by providing correct rounding and exceptions. We demonstrate how these strong semantics are achieved---with no significant slowdown with respect to other arbitrary-precision tools---and discuss a few applications where such a library can be useful.", acknowledgement = ack-nhfb, articleno = "13", fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", } @Book{Gil:2007:NMS, author = "Amparo Gil and Javier Segura and N. M. Temme", title = "Numerical Methods for Special Functions", publisher = pub-SIAM, address = pub-SIAM:adr, pages = "xvi + 415", year = "2007", DOI = "https://doi.org/10.1137/1.9780898717822", ISBN = "0-89871-634-9", ISBN-13 = "978-0-89871-634-4", LCCN = "QA351 .G455 2007", bibdate = "Fri Sep 14 10:24:22 MDT 2007", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; z3950.loc.gov:7090/Voyager", price = "US\$99.00", acknowledgement = ack-nhfb, shorttableofcontents = "Preface \\ 1. Introduction \\ I. Basic methods \\ 2. Convergent and divergent series \\ 3. Chebyshev expansions \\ 4. Linear recurrence relations and associated continued fractions \\ 5. Quadrature methods \\ II. Further tools and methods \\ 6. Numerical aspects of continued fractions \\ 7. Computation of the zeros of special functions \\ 8. Uniform asymptotic expansions \\ 9. Other methods \\ III. Related topics and examples \\ 10. Inversion of cumulative distribution functions \\ 11. Further examples \\ 12. Associated algorithms \\ List of algorithms \\ Bibliography \\ Index", subject = "functions, special; data processing; numerical analysis; asymptotic expansions; approximation theory", tableofcontents = "Preface xiii 1 Introduction / 1 \\ I Basic Methods / 13 \\ 2 Convergent and Divergent Series / 15 \\ 2.1 Introduction / 15 \\ 2.1.1 Power series: First steps / 15 \\ 2.1.2 Further practical aspects / 17 \\ 2.2 Differential equations and Frobenius series solutions / 18 \\ 2.2.1 Singular points / 19 \\ 2.2.2 The solution near a regular point / 20 \\ 2.2.3 Power series expansions around a regular singular point / 22 \\ 2.2.4 The Liouville transformation / 25 \\ 2.3 Hypergeometric series / 26 \\ 2.3.1 The Gauss hypergeometric function / 28 \\ 2.3.2 Other power series for the Gauss hypergeometric function / 30 \\ 2.3.3 Removable singularities / 33 \\ 2.4 Asymptotic expansions / 34 \\ 2.4.1 Watson's lemma / 36 \\ 2.4.2 Estimating the remainders of asymptotic expansions / 38 \\ 2.4.3 Exponentially improved asymptotic expansions / 39 \\ 2.4.4 Alternatives of asymptotic expansions / 40 \\ 3 Chebyshev Expansions / 51 \\ 3.1 Introduction / 51 \\ 3.2 Basic results on interpolation / 52 \\ 3.2.1 The Runge phenomenon and the Chebyshev nodes / 54 \\ 3.3 Chebyshev polynomials: Basic properties / 56 \\ 3.3.1 Properties of the Chebyshev polynomials $T_n(x)$ / 56 \\ 3.3.2 Chebyshev polynomials of the second, third, and fourth kinds / 60 \\ vii 3.4 Chebyshev interpolation / 62 \\ 3.4.1 Computing the Chebyshev interpolation polynomial / 64 \\ 3.5 Expansions in terms of Chebyshev polynomials / 66 \\ 3.5.1 Convergence properties of Chebyshev expansions / 68 \\ 3.6 Computing the coefficients of a Chebyshev expansion / 69 \\ 3.6.1 Clenshaw's method for solutions of linear differential equations with polynomial coefficients / 70 \\ 3.7 Evaluation of a Chebyshev sum / 75 \\ 3.7.1 Clenshaw's method for the evaluation of a Chebyshev sum / 75 \\ 3.8 Economization of power series / 80 \\ 3.9 Example: Computation of Airy functions of real variable / 80 \\ 3.10 Chebyshev expansions with coefficients in terms of special functions / 83 \\ 4 Linear Recurrence Relations and Associated Continued Fractions / 87 \\ 4.1 Introduction / 87 \\ 4.2 Condition of three-term recurrence relations / 88 \\ 4.2.1 Minimal solutions / 89 \\ 4.3 Perron's theorem / 92 \\ 4.3.1 Scaled recurrence relations / 94 \\ 4.4 Minimal solutions of TTRRs and continued fractions / 95 \\ 4.5 Some notable recurrence relations / 96 \\ 4.5.1 The confluent hypergeometric family / 96 \\ 4.5.2 The Gauss hypergeometric family / 102 \\ 4.6 Computing the minimal solution of a TTRR / 105 \\ 4.6.1 Miller's algorithm when a function value is known / 105 \\ 4.6.2 Miller's algorithm with a normalizing sum / 107 \\ 4.6.3 ``Anti-Miller'' algorithm / 110 \\ 4.7 Inhomogeneous linear difference equations / 112 \\ 4.7.1 Inhomogeneous first order difference equations. Examples / 112 \\ 4.7.2 Inhomogeneous second order difference equations / 115 \\ 4.7.3 Olver's method / 116 \\ 4.8 Anomalous behavior of some second order homogeneous and first order inhomogeneous recurrences / 118 \\ 4.8.1 A canonical example: Modified Bessel function / 118 \\ 4.8.2 Other examples: Hypergeometric recursions / 120 \\ 4.8.3 A first order inhomogeneous equation / 121 \\ 4.8.4 A warning / 122 \\ 5 Quadrature Methods / 123 \\ 5.1 Introduction / 123 \\ 5.2 Newton--Cotes quadrature: The trapezoidal and Simpson's rule / 124 \\ 5.2.1 The compound trapezoidal rule / 126 \\ 5.2.2 The recurrent trapezoidal rule / 129 \\ 5.2.3 Euler's summation formula and the trapezoidal rule / 130 \\ 5.3 Gauss quadrature / 132 \\ 5.3.1 Basics of the theory of orthogonal polynomials and Gauss quadrature / 133 \\ 5.3.2 The Golub--Welsch algorithm / 141 \\ 5.3.3 Example: The Airy function in the complex plane / 145 \\ 5.3.4 Further practical aspects of Gauss quadrature / 146 \\ 5.4 The trapezoidal rule on $\mathbb{R}$ / 147 \\ 5.4.1 Contour integral formulas for the truncation errors / 148 \\ 5.4.2 Transforming the variable of integration / 153 \\ 5.5 Contour integrals and the saddle point method / 157 \\ 5.5.1 The saddle point method / 158 \\ 5.5.2 Other integration contours / 163 \\ 5.5.3 Integrating along the saddle point contours and examples / 165 \\ II Further Tools and Methods / 171 \\ 6 Numerical Aspects of Continued Fractions / 173 \\ 6.1 Introduction / 173 \\ 6.2 Definitions and notation / 173 \\ 6.3 Equivalence transformations and contractions / 175 \\ 6.4 Special forms of continued fractions / 178 \\ 6.4.1 Stieltjes fractions / 178 \\ 6.4.2 Jacobi fractions / 179 \\ 6.4.3 Relation with Pad{\'e} approximants / 179 \\ 6.5 Convergence of continued fractions / 179 \\ 6.6 Numerical evaluation of continued fractions / 181 \\ 6.6.1 Steed's algorithm / 181 \\ 6.6.2 The modified Lentz algorithm / 183 \\ 6.7 Special functions and continued fractions / 185 \\ 6.7.1 Incomplete gamma function / 186 \\ 6.7.2 Gauss hypergeometric functions / 187 \\ 7 Computation of the Zeros of Special Functions / 191 \\ 7.1 Introduction / 191 \\ 7.2 Some classical methods / 193 \\ 7.2.1 The bisection method / 193 \\ 7.2.2 The fixed point method and the Newton--Raphson method / 193 \\ 7.2.3 Complex zeros / 197 \\ 7.3 Local strategies: Asymptotic and other approximations / 197 \\ 7.3.1 Asymptotic approximations for large zeros / 199 \\ 7.3.2 Other approximations / 202 \\ 7.4 Global strategies I: Matrix methods / 205 \\ 7.4.1 The eigenvalue problem for orthogonal polynomials / 206 \\ 7.4.2 The eigenvalue problem for minimal solutions of TTRRs / 207 \\ 7.5 Global strategies II: Global fixed point methods / 213 \\ 7.5.1 Zeros of Bessel functions / 213 \\ 7.5.2 The general case / 219 \\ 7.6 Asymptotic methods: Further examples / 224 \\ 7.6.1 Airy functions / 224 \\ 7.6.2 Scorer functions / 227 \\ 7.6.3 The error functions / 229 \\ 7.6.4 The parabolic cylinder function / 233 \\ 7.6.5 Bessel functions / 233 \\ 7.6.6 Orthogonal polynomials / 234 \\ 8 Uniform Asymptotic Expansions / 237 \\ 8.1 Asymptotic expansions for the incomplete gamma functions / 237 \\ 8.2 Uniform asymptotic expansions / 239 \\ 8.3 Uniform asymptotic expansions for the incomplete gamma functions / 240 \\ 8.3.1 The uniform expansion / 242 \\ 8.3.2 Expansions for the coefficients / 244 \\ 8.3.3 Numerical algorithm for small values of $\eta$ / 245 \\ 8.3.4 A simpler uniform expansion / 247 \\ 8.4 Airy-type expansions for Bessel functions / 249 \\ 8.4.1 The Airy-type asymptotic expansions / 250 \\ 8.4.2 Representations of $a_s()$, $b_s()$, $c_s()$, $d_s()$ / 253 \\ 8.4.3 Properties of the functions $A_\nu$, $B_\nu$, $C_\nu$, $D_\nu$ / 254 \\ 8.4.4 Expansions for $a_s()$, $b_s()$, $c_s()$, $d_s()$ / 256 \\ 8.4.5 Evaluation of the functions $A_\nu()$, $B_\nu()$ by iteration / 258 \\ 8.5 Airy-type asymptotic expansions obtained from integrals / 263 \\ 8.5.1 Airy-type asymptotic expansions / 264 \\ 8.5.2 How to compute the coefficients $\alpha_n$, $\beta_n$ / 267 \\ 8.5.3 Application to parabolic cylinder functions / 270 \\ 9 Other Methods / 275 \\ 9.1 Introduction / 275 \\ 9.2 Pad{\'e} approximations / 276 \\ 9.2.1 Pad{\'e} approximants and continued fractions / 278 \\ 9.2.2 How to compute the Pad{\'e} approximants / 278 \\ 9.2.3 Pad{\'e} approximants to the exponential function / 280 \\ 9.2.4 Analytic forms of Pad{\'e} approximations / 283 \\ 9.3 Sequence transformations / 286 \\ 9.3.1 The principles of sequence transformations / 286 \\ 9.3.2 Examples of sequence transformations / 287 \\ 9.3.3 The transformation of power series / 288 \\ 9.3.4 Numerical examples / 288 \\ 9.4 Best rational approximations / 290 \\ 9.5 Numerical solution of ordinary differential equations: Taylor expansion method / 291 \\ 9.5.1 Taylor-series method: Initial value problems / 292 \\ 9.5.2 Taylor-series method: Boundary value problem / 293 \\ 9.6 Other quadrature methods / 294 \\ 9.6.1 Romberg quadrature / 294 \\ 9.6.2 Fej{\'e}r and Clenshaw--Curtis quadratures / 296 \\ 9.6.3 Other Gaussian quadratures / 298 \\ 9.6.4 Oscillatory integrals / 301 \\ III Related Topics and Examples / 307 \\ 10 Inversion of Cumulative Distribution Functions / 309 \\ 10.1 Introduction / 309 \\ 10.2 Asymptotic inversion of the complementary error function / 309 \\ 10.3 Asymptotic inversion of incomplete gamma functions / 312 \\ 10.3.1 The asymptotic inversion method / 312 \\ 10.3.2 Determination of the coefficients i / 314 \\ 10.3.3 Expansions of the coefficients i / 316 \\ 10.3.4 Numerical examples / 316 \\ 10.4 Generalizations / 317 \\ 10.5 Asymptotic inversion of the incomplete beta function / 318 \\ 10.5.1 The nearly symmetric case / 319 \\ 10.5.2 The general error function case / 322 \\ 10.5.3 The incomplete gamma function case / 324 \\ 10.5.4 Numerical aspects / 326 \\ 10.6 High order Newton-like methods / 327 \\ 11 Further Examples / 331 \\ 11.1 Introduction / 331 \\ 11.2 The Euler summation formula / 331 \\ 11.3 Approximations of Stirling numbers / 336 \\ 11.3.1 Definitions / 337 \\ 11.3.2 Asymptotics for Stirling numbers of the second kind / 338 \\ 11.3.3 Stirling numbers of the first kind / 343 \\ 11.4 Symmetric elliptic integrals / 344 \\ 11.4.1 The standard forms in terms of symmetric integrals / 345 \\ 11.4.2 An algorithm / 346 \\ 11.4.3 Other elliptic integrals / 347 \\ 11.5 Numerical inversion of Laplace transforms / 347 \\ 11.5.1 Complex Gauss quadrature / 348 \\ 11.5.2 Deforming the contour / 349 \\ 11.5.3 Using Pad{\'e} approximations / 352 \\ IV Software / 353 \\ 12 Associated Algorithms / 355 \\ 12.1 Introduction / 355 \\ 12.1.1 Errors and stability: Basic terminology / 356 \\ 12.1.2 Design and testing of software for computing functions: General philosophy / 357 \\ 12.1.3 Scaling the functions / 358 \\ 12.2 Airy and Scorer functions of complex arguments / 359 \\ 12.2.1 Purpose / 359 \\ 12.2.2 Algorithms / 359 \\ 12.3 Associated Legendre functions of integer and half-integer degrees / 363 \\ 12.3.1 Purpose / 363 \\ 12.3.2 Algorithms / 364 \\ 12.4 Bessel functions / 369 \\ 12.4.1 Modified Bessel functions of integer and half-integer orders / 370 \\ 12.4.2 Modified Bessel functions of purely imaginary orders / 372 \\ 12.5 Parabolic cylinder functions / 377 \\ 12.5.1 Purpose / 377 \\ 12.5.2 Algorithm / 378 \\ 12.6 Zeros of Bessel functions / 385 \\ 12.6.1 Purpose / 385 \\ 12.6.2 Algorithm / 385 \\ List of Algorithms / 387 \\ Bibliography / 389 \\ Index / 405", } @Article{Gil:2007:NSS, author = "Amparo Gil and Javier Segura and Nico M. Temme", title = "Numerically satisfactory solutions of hypergeometric recursions", journal = j-MATH-COMPUT, volume = "76", number = "259", pages = "1449--1468", month = jul, year = "2007", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Jul 8 06:24:22 MDT 2008", bibsource = "http://www.ams.org/mcom/2007-76-259; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp2000.bib", URL = "http://www.ams.org/mcom/2007-76-259/S0025-5718-07-01918-7/home.html; http://www.ams.org/mcom/2007-76-259/S0025-5718-07-01918-7/S0025-5718-07-01918-7.dvi; http://www.ams.org/mcom/2007-76-259/S0025-5718-07-01918-7/S0025-5718-07-01918-7.pdf; http://www.ams.org/mcom/2007-76-259/S0025-5718-07-01918-7/S0025-5718-07-01918-7.ps", acknowledgement = ack-nhfb, fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Glaser:2007:FAC, author = "Andreas Glaser and Xiangtao Liu and Vladimir Rokhlin", title = "A Fast Algorithm for the Calculation of the Roots of Special Functions", journal = j-SIAM-J-SCI-COMP, volume = "29", number = "4", pages = "1420--1438", month = "????", year = "2007", CODEN = "SJOCE3", DOI = "https://doi.org/10.1137/06067016X", ISSN = "1064-8275 (print), 1095-7197 (electronic)", ISSN-L = "1064-8275", bibdate = "Wed May 19 10:43:53 MDT 2010", bibsource = "http://epubs.siam.org/sam-bin/dbq/toc/SISC/29/4; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", abstract = "We describe a procedure for the determination of the roots of functions satisfying second-order ordinary differential equations, including the classical special functions. The scheme is based on a combination of the Pr{\"u}fer transform with the classical Taylor series method for the solution of ordinary differential equations and requires $ O(1) $ operations for the determination of each root. When the functions in question are classical orthogonal polynomials (Legendre, Hermite, etc.), the techniques presented here also provide tools for the evaluation of the weights for the associated Gaussian quadratures. The performance of the scheme for several classical special functions (prolate spheroidal wave functions, Bessel functions, and Legendre, Hermite, and Laguerre polynomials) is illustrated with numerical examples.", acknowledgement = ack-nhfb, fjournal = "SIAM Journal on Scientific Computing", journal-URL = "http://epubs.siam.org/sisc", } @Book{Gradshteyn:2007:TIS, author = "I. S. Gradshteyn and I. M. Ryzhik and Alan Jeffrey and Daniel Zwillinger", title = "Table of Integrals, Series and Products", publisher = pub-ACADEMIC, address = pub-ACADEMIC:adr, edition = "Seventh", pages = "xlv + 1171", year = "2007", ISBN = "0-12-373637-4 (hardcover)", ISBN-13 = "978-0-12-373637-6 (hardcover)", LCCN = "QA55 .G6613 2007", bibdate = "Thu Feb 18 12:04:10 MST 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/master.bib; https://www.math.utah.edu/pub/tex/bib/mathcw.bib; prodorbis.library.yale.edu:7090/voyager", acknowledgement = ack-nhfb, remark = "Previous edition 2000. Includes CD-ROM.", subject = "Mathematics; Tables", tableofcontents = "0 Introduction \\ 1 Elementary Functions \\ 2 Indefinite Integrals of Elementary Functions \\ 3 Definite Integrals of Elementary Functions \\ 4.Combinations involving trigonometric and hyperbolic functions and power \\ 5 Indefinite Integrals of Special Functions \\ 6 Definite Integrals of Special Functions \\ 7.Associated Legendre Functions \\ 8 Special Functions \\ 9 Hypergeometric Functions \\ 10 Vector Field Theory \\ 11 Algebraic Inequalities \\ 12 Integral Inequalities \\ 13 Matrices and related results \\ 14 Determinants \\ 15 Norms \\ 16 Ordinary differential equations \\ 17 Fourier, Laplace, and Mellin Transforms \\ 18 The z-transform", xxauthor = "I. S. (Izrail Solomonovich) Gradshteyn and I. M. (Iosif Moiseevich) Ryzhik and Alan Jeffrey and Daniel Zwillinger", } @Article{Guseinov:2007:UTE, author = "I. I. Guseinov and B. A. Mamedov", title = "Unified treatment for the evaluation of generalized complete and incomplete gamma functions", journal = j-J-COMPUT-APPL-MATH, volume = "202", number = "2", pages = "435--439", day = "15", month = may, year = "2007", CODEN = "JCAMDI", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Sat Feb 25 13:13:14 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2005.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0377042706001506", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @Article{Hernandez:2007:MPO, author = "M. A. Hern{\'a}ndez and N. Romero", title = "Methods with prefixed order for approximating square roots with global and general convergence", journal = j-APPL-MATH-COMP, volume = "194", number = "2", pages = "346--353", day = "15", month = dec, year = "2007", CODEN = "AMHCBQ", ISSN = "0096-3003 (print), 1873-5649 (electronic)", ISSN-L = "0096-3003", bibdate = "Sat Jul 12 09:03:09 MDT 2008", bibsource = "http://www.sciencedirect.com/science/journal/00963003; https://www.math.utah.edu/pub/tex/bib/applmathcomput2005.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, fjournal = "Applied Mathematics and Computation", journal-URL = "http://www.sciencedirect.com/science/journal/00963003", } @Article{Kalmykov:2007:AOEa, author = "M. Y. Kalmykov and B. F. L. Ward and Y. Yost", title = "All order $ \epsilon $-expansion of {Gauss} hypergeometric functions with integer and half\slash integer values of parameters", journal = j-J-HIGH-ENERGY-PHYS, volume = "2007", number = "02", pages = "040--??", month = "????", year = "2007", CODEN = "JHEPAB", ISSN = "1126-6708", ISSN-L = "1029-8479", MRclass = "33C05 (33B30)", MRnumber = "MR2318011 (2009g:33004)", bibdate = "Thu Dec 01 09:16:04 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, ajournal = "JHEP", fjournal = "Journal of High Energy Physics", pagecount = "21", } @Article{Kalmykov:2007:AOEb, author = "M. Y. Kalmykov and B. F. L. Ward and Y. Yost", title = "On the all-order $ \epsilon $-expansion of generalized hypergeometric functions with integer values of parameters", journal = j-J-HIGH-ENERGY-PHYS, volume = "2007", number = "11", pages = "009", month = "????", year = "2007", CODEN = "JHEPAB", ISSN = "1126-6708", ISSN-L = "1029-8479", MRclass = "33C20 (33B30 41A58)", MRnumber = "MR2362140 (2008m:33016)", bibdate = "Thu Dec 01 09:16:04 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, ajournal = "JHEP", fjournal = "Journal of High Energy Physics", pagecount = "13", } @Article{Karagiannidis:2007:IAG, author = "George Karagiannidis and Athanasios Lioumpas", title = "An Improved Approximation for the {Gaussian} {$Q$}-Function", journal = j-IEEE-COMMUN-LET, volume = "11", number = "8", pages = "644--646", month = aug, year = "2007", CODEN = "ICLEF6", DOI = "https://doi.org/10.1109/lcomm.2007.070470", ISSN = "1089-7798 (print), 1558-2558 (electronic)", ISSN-L = "1089-7798", bibdate = "Sat Dec 16 16:49:58 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "See corrections and comments \cite{Dyer:2008:CCI}.", acknowledgement = ack-nhfb, fjournal = "IEEE Communications Letters", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=4234", } @Book{King:2007:DNC, author = "Louis Vessot King", title = "On the Direct Numerical Calculation of Elliptic Functions and Integrals", publisher = "Mellon Press", address = "????", pages = "56", year = "2007", ISBN = "1-4067-4226-0", ISBN-13 = "978-1-4067-4226-8", LCCN = "", bibdate = "Wed Feb 03 08:53:04 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcw.bib", note = "Dedicated to the memory of James Harkness, Peter Redpath Professor of Pure Mathematics, McGill University 1903--1923.", URL = "https://www.google.com/books/edition/On_the_Direct_Numerical_Calculation_of_E/CMM4AAAAMAAJ", acknowledgement = ack-nhfb, remark = "The AGM method for Jacobian elliptic functions was discovered by this book's author at McGill University in 1913, first published in \cite{King:1921:SNF}, and then in a 1924 monograph, of which this is a reprint.", tableofcontents = "Preface / v \\ Introduction / 1 \\ II. Historical Note on Landen's Transformation and the Various Scales of Moduli and Amplitudes / 2 \\ III. On the Scale of Arithmetico-Geometrical Means / 6 \\ IV. Landen's Scale of Increasing Amplitudes: $\tan(\phi_{n + 1} - \phi_n) = (b_n / a_n) \tan \phi_n$ / 7 \\ (i) Calculation of $F(\phi, k)$, $E(\phi, k)$, $K$, and $E$ / 7 \\ (ii) Calculation of $\sn(u,k)$, $\cn(u,k)$, $\dn(u,k)$ in terms of the argument $u$ / 9 \\ V. The Hyperbolic Scale of Increasing Amplitudes: $\tanh(\phi_{n + 1} - \phi_n) = (b_n / a_n) \tanh(\phi_n)$ / 10 \\ (i) Calculation of $F(\phi, k')$, $E(\phi, k')$, etc. / 10 \\ (ii) Calculation of $\sn(u,k')$, $\cn(u,k')$, $\dn(u,k')$ / 11 \\ (iii) Calculation of $\sn(i u,k)$, $\cn(i u,k)$, $\dn(i u,k)$ / 12 \\ VI: Gauss' Scale of Increasing Amplitudes \\ VII: Landen's Scale of Decreasing Amplitudes: $\sin(2\psi_{n + 1} - \psi_n) = (b_n / a_n) \sin \psi_n$ / 14 \\ (i) Calculation of $F(\psi, k')$, $E(\psi, k')$, etc. / 14 \\ (ii) Calculation of $\sn(u,k')$, $\cn(u,k')$, $\dn(u,k')$ / 16 \\ VIII. The Hyperbolic Scale of Decreasing Amplitudes: $\sinh(2\psi_{n + 1} - \psi_n) = (b_n / a_n) \sinh \psi_n$ / 17 \\ (i) Calculation of $F(\psi, k)$, $E(\psi, k)$, etc. / 17 \\ (ii) Calculation of $\sn(u,k)$, $\cn(u,k)$, $\dn(u,k)$ / 18 \\ IX. On the Numerical Computation of the Third Elliptic Integral / 19 \\ Case I. $n$ negative, between $0$ and $-k^2$ (Hyperbolic case) / 19 \\ Case II. $n$ negative, between $-1$ and $-\infty$ (Hyperbolic case) / 20 \\ Case III. $n$ negative, between $-k^2$ and $-1$ (Circular case) / 21 \\ First Method. (i) Use of circular recurrence formulae / 22 \\ (ii) Use of hyperbolic recurrence formulae / 23 \\ Second Method. (i) Use of circular recurrence formulae / 25 \\ (ii) Use of hyperbolic recurrence formulae / 25 \\ Case IV. $n$ positive, between $0$ and $\infty$ (Circular case) / 26 \\ First Method. (i) Use of circular recurrence formulae / 26 \\ (ii) Use of hyperbolic recurrence formulae / 27 \\ Second Method. (i) Use of circular recurrence formulae / 28 \\ (ii) Use of hyperbolic recurrence formulae / 29 \\ X. Note on the Calculation of the Third Elliptic Integral in Terms of the Complementary A.G.M. Scale / 30 \\ Summary of Formulae / 31 \\ Appendix, Examples 1--35 / 35", } @Article{Kodama:2007:RA, author = "Masao Kodama", title = "Remark on {Algorithm 644}", journal = j-TOMS, volume = "33", number = "4", pages = "28:1--28:3", month = aug, year = "2007", CODEN = "ACMSCU", DOI = "https://doi.org/10.1145/1268776.1268783", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Mon Dec 17 18:09:13 MST 2007", bibsource = "http://www.acm.org/pubs/contents/journals/toms/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib", note = "See \cite{Amos:1986:APP,Amos:1990:RPP,Amos:1995:RAP}.", abstract = "This remark details correction for errors in the functions which compute the modified Bessel function of the second kind and the log of the gamma function. In both cases these errors cause a loss of precision for a small range of values of the $ \nu $ argument. These routines are used in the calculation of a number of other functions within the package whose accuracy is thus similarly affected.", acknowledgement = ack-nhfb, articleno = "28", fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", } @Article{Kuijlaars:2007:TIH, author = "A. B. J. Kuijlaars and H. Stahl and W. {Van Assche} and F. Wielonsky", title = "{Type II} {Hermite--Pad{\'e}} approximation to the exponential function", journal = j-J-COMPUT-APPL-MATH, volume = "207", number = "2", pages = "227--244", day = "15", month = oct, year = "2007", CODEN = "JCAMDI", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Sat Feb 25 13:13:18 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2005.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0377042706005978", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @Article{Kuliamin:2007:STI, author = "V. V. Kuliamin", title = "Standardization and testing of implementations of mathematical functions in floating point numbers", journal = j-PROG-COMP-SOFT, volume = "33", number = "3", pages = "154--173", year = "2007", CODEN = "PCSODA", DOI = "https://doi.org/10.1134/S036176880703005X", ISSN = "0361-7688 (print), 1608-3261 (electronic)", ISSN-L = "0361-7688", bibdate = "Fri Aug 08 09:01:30 2008", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", abstract = "Requirements definition and test suites development for implementations of mathematical functions in floating point arithmetic in the framework of the IEEE 754 standard are considered. A method based on this standard is proposed for defining requirements for such functions. This method can be used for the standardization of implementations of such functions; this kind of standardization extends IEEE 754. A method for designing test suites for the verification of those requirements is presented. The proposed methods are based on specific properties of the representation of floating point numbers and on some features of the functions under examination.", acknowledgement = ack-nhfb, fjournal = "Programming and Computer Software; translation of Programmirovaniye (Moscow, USSR) Plenum", journal-URL = "http://link.springer.com/journal/11086", keywords = "floating-point function testing and verification", } @TechReport{Lefevre:2007:SNP, author = "Vincent Lef{\'e}vre and Jean-Michel Muller", title = "Some notes on the possible under\slash overflow of the most common elementary functions", type = "Report", institution = "LIP, {\'E}cole Normale Sup{\'e}rieure de Lyon", address = "Lyon, France", pages = "7", year = "2007", bibdate = "Fri May 25 16:18:32 2007", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://prunel.ccsd.cnrs.fr/ensl-00149414", abstract = "The purpose of this short note is not to describe when underflow or overflow must be signalled (it is quite clear that the rules are the same as for the basic arithmetic operations). We just want to show that for some of the most common functions and floating-point formats, in many cases, we can know in advance that the results will always lie in the range of the numbers that are representable by normal floating-point numbers, so that in these cases there is no need to worry about underflow or overflow. Note that when it is not the case, an implementation is still possible using a run-time test.", acknowledgement = ack-nhfb, keywords = "elementary functions; floating-point arithmetic; overflow; underflow", } @Article{Neher:2007:CSF, author = "Markus Neher", title = "Complex standard functions and their implementation in the {CoStLy} library", journal = j-TOMS, volume = "33", number = "1", pages = "2:1--2:27", month = mar, year = "2007", CODEN = "ACMSCU", DOI = "https://doi.org/10.1145/1206040.1206042", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Sat Apr 14 09:48:58 MDT 2007", bibsource = "http://www.acm.org/pubs/contents/journals/toms/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib", abstract = "The practical calculation of range bounds for some complex standard functions is addressed in this article. The functions under consideration are root and power functions, the exponential, trigonometric and hyperbolic functions, and their inverse functions. For such a function $f$ and a given rectangular complex interval $z$, some interval $w$ is computed that contains all function values of $f$ in $z$. This is done by expressing the real and the imaginary part of $f$ as compositions of real standard functions and then estimating the ranges of these compositions. In many cases, the inclusions are optimal, such that $w$ is the smallest rectangular interval containing the range of $f$. The algorithms presented in this article have been implemented in a C++ class library called CoStLy (Complex Standard Functions License), which is distributed under the conditions of the GNU General Public License.", acknowledgement = ack-nhfb, articleno = "2", fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", } @Book{Press:2007:NRA, author = "William H. Press and Saul A. Teukolsky and William T. Vetterling and Brian P. Flannery", title = "Numerical Recipes --- The Art of Scientific Computing", publisher = pub-CAMBRIDGE, address = pub-CAMBRIDGE:adr, edition = "Third", pages = "xxi + 1235", year = "2007", ISBN = "0-521-88068-8 (hardcover), 0-521-88407-1 (with source code CD ROM), 0-521-70685-8 (source code CD ROM)", ISBN-13 = "978-0-521-88068-8 (hardcover), 978-0-521-88407-5 (with source code CD ROM), 978-0-521-70685-8 (source code CD ROM)", LCCN = "QA297 .N866 2007", bibdate = "Wed Dec 15 10:40:52 1993", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/master.bib; https://www.math.utah.edu/pub/tex/bib/numana2000.bib", URL = "http://www.cambridge.org/numericalrecipes", acknowledgement = ack-nhfb, subject = "numerical analysis; computer programs; science; mathematics; C++ (computer program language)", tableofcontents = "1. Preliminaries \\ 2. Solution of linear algebraic equations \\ 3. Interpolation and extrapolation \\ 4. Integration of functions \\ 5. Evaluation of functions \\ 6. Special functions \\ 7. Random numbers \\ 8. Sorting and selection \\ 9. Root finding and nonlinear sets of equations \\ 10. Minimization or maximization of functions \\ 11. Eigensystems \\ 12. Fast Fourier Transform \\ 13. Fourier and spectral applications \\ 14. Statistical description of data \\ 15. Modeling of data \\ 16. Classification and inference \\ 17. Integration of ordinary differential equations \\ 18. Two-point boundary value problems \\ 19. Integral equations and inverse theory \\ 20. Partial differential equations \\ 21. Computational geometry \\ 22. Less-numerical algorithms", } @Article{Ren:2007:CFA, author = "C. Ren and A. R. MacKenzie", title = "Closed-form approximations to the error and complementary error functions and their applications in atmospheric science", journal = j-ATMOS-SCI-LETT, volume = "8", number = "3", pages = "70--73", month = "????", year = "2007", DOI = "https://doi.org/10.1002/asl.154", ISSN = "1530-261X", ISSN-L = "1530-261X", bibdate = "Sat Dec 16 17:25:42 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://onlinelibrary.wiley.com/doi/10.1002/asl.154/full", acknowledgement = ack-nhfb, fjournal = "Atmospheric Science Letters", journal-URL = "http://www.sciencedirect.com/science/journal/1530261X; http://rmets.onlinelibrary.wiley.com/hub/journal/10.1002/(ISSN)1530-261X/", } @Article{Rokhlin:2007:AFC, author = "Vladimir Rokhlin and Hong Xiao", title = "Approximate formulae for certain prolate spheroidal wave functions valid for large values of both order and band-limit", journal = j-APPL-COMPUT-HARMON-ANAL, volume = "22", number = "1", pages = "105--123", month = jan, year = "2007", DOI = "https://doi.org/10.1016/j.acha.2006.05.004", ISSN = "1063-5203 (print), 1096-603x (electronic)", ISSN-L = "1063-5203", bibdate = "Sun Oct 31 10:00:51 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", abstract = "We construct asymptotic formulae for the approximation of certain prolate spheroidal wave functions and of the corresponding eigenvalues. We investigate two regimes: when the ratio $ c / m $ decays, and when both $c$ and $m$ grow, but the ratio $ c / m $ stays bounded. Both the regions of validity and the accuracies of the obtained expansions are illustrated with numerical examples.", acknowledgement = ack-nhfb, fjournal = "Applied and Computational Harmonic Analysis. Time-Frequency and Time-Scale Analysis, Wavelets, Numerical Algorithms, and Applications", journal-URL = "http://www.sciencedirect.com/science/journal/10635203", keywords = "approximation; asymptotic; band-limit; prolate spheroidal wave functions", } @Article{Schmelzer:2007:CGF, author = "Thomas Schmelzer and Lloyd N. Trefethen", title = "Computing the Gamma Function Using Contour Integrals and Rational Approximations", journal = j-SIAM-J-NUMER-ANAL, volume = "45", number = "2", pages = "558--571", month = "????", year = "2007", CODEN = "SJNAAM", DOI = "https://doi.org/10.1137/050646342", ISSN = "0036-1429 (print), 1095-7170 (electronic)", ISSN-L = "0036-1429", bibdate = "Mon Nov 24 18:03:07 MST 2008", bibsource = "http://siamdl.aip.org/dbt/dbt.jsp?KEY=SJNAAM&Volume=45&Issue=2; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "SIAM Journal on Numerical Analysis", journal-URL = "http://epubs.siam.org/sinum", } @Book{Slavjanov:2007:SFU, author = "Sergej J. Slavjanov and Wolfgang Lay", title = "Special Functions: a Unified Theory Based on Singularities", publisher = pub-OXFORD, address = pub-OXFORD:adr, pages = "xvi + 293", year = "2007", ISBN = "0-19-850573-6", ISBN-13 = "978-0-19-850573-0", LCCN = "????", bibdate = "Tue Dec 5 11:27:46 MST 2023", bibsource = "fsz3950.oclc.org:210/WorldCat; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", series = "Oxford mathematical monographs; Oxford science publications", acknowledgement = ack-nhfb, shorttableofcontents = "1: Linear second-order ODEs with polynomial coefficients \\ 2: The hypergeometric class of equations \\ 3: The Heun class of equations \\ 4: Application to physical sciences \\ 5: The Painlev{\'e} class of equations \\ Appendix A: The gamma function and related functions \\ Appendix B: CTCPs Heun equations in general form \\ Appendix C: Multipole Matrix elements \\ Appendix D: SFTools \\ Database of the special functions", subject = "Functions, Special; Fonctions sp{\'e}ciales; Functions, Special", tableofcontents = "1: Linear second-order ODEs with polynomial coefficients \\ Regular singularities and Fuchsian equations \\ Regular and Fuchsian singularities \\ Fuchsian equations and their transformations \\ Characteristic exponents \\ Frobenius solutions \\ Irregular singularities and confluent equations \\ The $s$-rank of a singularity \\ Confluent and reduced confluent equations \\ The $s$-homotopic transformation \\ Asymptotic solutions at irregular singularities \\ Canonical forms \\ A generalization of Fuchs's theorem \\ Confluence and reduction processes \\ Strong and weak confluence. A confluence theorem \\ A confluence principle \\ Reduction of an equation \\ Classes and types of equations \\ Standard forms of equations \\ Invariants of $s$-homotopic transformations \\ Types of solutions \\ Eigenfunctions of singular Sturm--Liouville problems \\ Central and lateral connection problems \\ Stokes lines at singularities. Stokes matrices \\ Generalized Riemann scheme \\ Applications \\ Central two-point connection problems (CTCPs) \\ Two regular singularities as relevant endpoints \\ One regular singularity and one irregular singularity as the endpoints \\ A proof \\ Two irregular singularities \\ 2: The hypergeometric class of equations \\ Classification scheme \\ General presentation \\ Hypergeometric equation \\ Confluent equations \\ Reduced confluent equations \\ Difference equations \\ General consideration \\ Difference equations for hypergeometric functions \\ Confluent hypergeometric functions \\ \ldots{}", } @Article{Srinivasan:2007:GFE, author = "Gopala Krishna Srinivasan", title = "The Gamma Function: An Eclectic Tour", journal = j-AMER-MATH-MONTHLY, volume = "114", number = "4", pages = "297--315", month = apr, year = "2007", CODEN = "AMMYAE", ISSN = "0002-9890 (print), 1930-0972 (electronic)", ISSN-L = "0002-9890", bibdate = "Mon Jan 30 12:00:28 MST 2012", bibsource = "http://www.jstor.org/journals/00029890.html; http://www.jstor.org/stable/i27642189; https://www.math.utah.edu/pub/tex/bib/amermathmonthly2000.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.jstor.org/stable/27642193", acknowledgement = ack-nhfb, fjournal = "American Mathematical Monthly", journal-URL = "https://www.jstor.org/journals/00029890.htm", } @Article{Temme:2007:NAS, author = "Nico M. Temme", title = "Numerical aspects of special functions", journal = j-ACTA-NUMERICA, volume = "16", pages = "379--478", year = "2007", CODEN = "ANUMFU", DOI = "https://doi.org/10.1017/S0962492906330012", ISBN = "0-521-87743-1", ISBN-13 = "978-0-521-87743-5", ISSN = "0962-4929 (print), 1474-0508 (electronic)", ISSN-L = "0962-4929", MRclass = "33F05 (65D20)", MRnumber = "2417932 (2009g:33027)", MRreviewer = "Amparo Gil", bibdate = "Sat Sep 24 11:37:18 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/actanumerica.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", abstract = "This paper describes methods that are important for the numerical evaluation of certain functions that frequently occur in applied mathematics, physics and mathematical statistics. This includes what we consider to be the basic methods, such as recurrence relations, series expansions (both convergent and asymptotic), and numerical quadrature. Several other methods are available and some of these will be discussed in less detail. Examples will be given on the use of special functions in certain problems from mathematical physics and mathematical statistics (integrals and series with special functions).", acknowledgement = ack-nhfb, ajournal = "Acta Numer.", fjournal = "Acta Numerica", journal-URL = "http://journals.cambridge.org/action/displayJournal?jid=ANU", onlinedate = "24 April 2007", } @InCollection{Weniger:2007:AAT, author = "Ernst Joachim Weniger", title = "Asymptotic approximations to truncation errors of series representations for special functions", crossref = "Iske:2007:AAP", pages = "331--348", year = "2007", MRclass = "33F05", MRnumber = "MR2335174 (2008h:33051)", bibdate = "Thu Dec 01 09:38:02 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, keywords = "Bernoulli numbers; Euler--MacLaurin formula; exponential integer $E_1(z)$; Gaussian hypergeometric series $_2F_1(a / b / c / z)$; Riemann zeta function", remark = "Available as math.CA/0511074.", } @Book{Agarwal:2008:OPD, author = "Ravi P. Agarwal and Donal O'Regan", title = "Ordinary and Partial Differential Equations: with Special Functions, {Fourier} Series, and Boundary Value Problems", publisher = pub-SV, address = pub-SV:adr, pages = "xiv + 410", year = "2008", ISBN = "0-387-79145-0 (paperback)", ISBN-13 = "978-0-387-79145-6 (paperback)", LCCN = "????", bibdate = "Sat Oct 30 17:22:04 MDT 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; prodorbis.library.yale.edu:7090/voyager", series = "Universitext", acknowledgement = ack-nhfb, subject = "differential equations; differential equations, partial; Fourier analysis; boundary value problems", tableofcontents = "Preface vii l. Solvable Differential Equations / 1 \\ 2. Second-Order Differential Equations / 8 \\ 3. Preliminaries to Series Solutions / 15 \\ 4. Solution at an Ordinary Point / 23 \\ 5. Solution at a Singular Point / 31 \\ 6. Solution at a Singular Point (Cont'd.) / 37 \\ 7. Legendre Polynomials and Functions / 47 \\ 8. Chebyshev, Hermite and Laguerre Polynomials / 57 \\ 9. Bessel Functions / 64 \\ 10. Hypergeometrie Functions / 75 \\ 11. Piecewise Continuous and Periodic Functions / 83 \\ 12. Orthogonal Functions and Polynomials / 90 \\ 13. Orthogonal Functions and Polynomials (Cont'd.) / 95 \\ 14. Boundary Value Problems / 104 \\ 15. Boundary Value Problems (Cont'd.) / 109 \\ 16. Green's Functions / 119 \\ 17. Regular Perturbations / 129 \\ 18. Singular Perturbations / 138 \\ 19. Sturm--Liouville Problems / 145 \\ 20. Eigenfunction Expansions / 157 \\ 21. Eigenfunction Expansions (Cont'd.) / 163 \\ 22. Convergence of the Fourier Series / 171 \\ 23. Convergence of the Fourier Series (Cont'd.) / 176 \\ 24. Fourier Series Solutions of Ordinary Differential Equations / 187 \\ 25. Partial Differential Equations / 194 \\ 26. First-Order Partial Differential Equations / 202 \\ 27. Solvable Partial Differential Equations / 210 \\ 28. The Canonical Forms / 219 \\ 29. The Method of Separation of Variables / 227 \\ 30. The One-Dimensional Heat Equation / 234 \\ 31. The One-Dimensional Heat Equation (Cont'd.) / 241 \\ 32. The One-Dimensional Wave Equation / 249 \\ 33. The One-Dimensional Wave Equation (Cont'd.) / 256 \\ 34. Laplace Equation in Two Dimensions / 266 \\ 35. Laplace Equation in Polar Coordinates / 275 \\ 36. Two-Dimensional Heat Equation / 284 \\ 37. Two-Dimensional Wave Equation / 292 \\ 38. Laplace Equation in Three Dimensions / 300 \\ 39. Laplace Equation in Three Dimensions (Cont'd.) / 306 \\ 40. Nonhomogeneous Equations / 316 \\ 41. Fourier Integral and Transforms / 323 \\ 42. Fourier Integral and Transforms (Cont'd.) / 330 \\ 43. Fourier Transform Method for Partial DEs / 338 \\ 44. Fourier Transform Method for Partial DEs (Cont'd.) / 344 \\ 45. Laplace Transforms / 354 \\ 46. Laplace Transforms (Cont'd.) / 361 \\ 47. Laplace Transform Method for Ordinary DEs / 374 \\ 48. Laplace Transform Method for Partial DEs / 384 \\ 49. Well-Posed Problems / 394 \\ 50. Verification of Solutions / 399 \\ References for Further Reading / 405 \\ Index / 407", } @Article{Alzer:2008:GFI, author = "Horst Alzer", title = "Gamma function inequalities", journal = j-NUMER-ALGORITHMS, volume = "49", number = "1--4", pages = "53--84", month = dec, year = "2008", CODEN = "NUALEG", ISSN = "1017-1398 (print), 1572-9265 (electronic)", ISSN-L = "1017-1398", bibdate = "Mon May 17 14:08:26 MDT 2010", bibsource = "http://springerlink.metapress.com/openurl.asp?genre=issue&issn=1017-1398&volume=49&issue=1; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.springerlink.com/openurl.asp?genre=article&issn=1017-1398&volume=49&issue=1&spage=53", acknowledgement = ack-nhfb, fjournal = "Numerical Algorithms", journal-URL = "http://link.springer.com/journal/11075", } @Article{Ancarani:2008:DOC, author = "L. U. Ancarani and G. Gasaneo", title = "Derivatives of any order of the confluent hypergeometric function {$_1 F_1 (a, b, z)$} with respect to the parameter $a$ or $b$", journal = j-J-MATH-PHYS, volume = "49", number = "6", pages = "063508", month = jun, year = "2008", CODEN = "JMAPAQ", DOI = "https://doi.org/10.1063/1.2939395", ISSN = "0022-2488 (print), 1089-7658 (electronic), 1527-2427", ISSN-L = "0022-2488", bibdate = "Wed Oct 26 09:06:03 MDT 2011", bibsource = "http://www.aip.org/ojs/jmp.html; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jmathphys2005.bib", URL = "http://jmp.aip.org/resource/1/jmapaq/v49/i6/p063508_s1", acknowledgement = ack-nhfb, fjournal = "Journal of Mathematical Physics", journal-URL = "http://jmp.aip.org/", onlinedate = "20 June 2008", pagecount = "16", } @Article{Borwein:2008:EBF, author = "David Borwein and Jonathan M. Borwein and O-Yeat Chan", title = "The evaluation of {Bessel} functions via exp--arc integrals", journal = j-J-MATH-ANAL-APPL, volume = "341", number = "1", pages = "478--500", month = may, year = "2008", CODEN = "JMANAK", DOI = "https://doi.org/10.1016/j.jmaa.2007.10.003", ISSN = "0022-247X (print), 1096-0813 (electronic)", ISSN-L = "0022-247X", MRclass = "33C10 (33F05 65D20)", MRnumber = "2394100", MRreviewer = "Richard B. Paris", bibdate = "Thu Aug 11 10:27:38 2016", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://adsabs.harvard.edu/abs/2008JMAA..341..478B; http://docserver.carma.newcastle.edu.au/1231/; http://www.sciencedirect.com/science/article/pii/S0022247X07012346", abstract = "A standard method for computing values of Bessel functions has been to use the well-known ascending series for small argument, and to use an asymptotic series for large argument; with the choice of the series changing at some appropriate argument magnitude, depending on the number of digits required. In a recent paper, D. Borwein, J. Borwein, and R. Crandall [D. Borwein, J. M. Borwein, R. Crandall, Effective Laguerre asymptotics, preprint at http://locutus.cs.dal.ca:8088/archive/00000334/] derived a series for an ``exp-arc'' integral which gave rise to an absolutely convergent series for the J and I Bessel functions with integral order. Such series can be rapidly evaluated via recursion and elementary operations, and provide a viable alternative to the conventional ascending-asymptotic switching. In the present work, we extend the method to deal with Bessel functions of general (non-integral) order, as well as to deal with the Y and K Bessel functions.", acknowledgement = ack-nhfb, fjournal = "Journal of Mathematical Analysis and Applications", journal-URL = "http://www.sciencedirect.com/science/journal/0022247X", keywords = "Bessel function; Exponential-hyperbolic expansions; Uniform series expansion", ORCID-numbers = "Borwein, Jonathan/0000-0002-1263-0646", } @Article{Borwein:2008:ELA, author = "David Borwein and Jonathan M. Borwein and Richard E. Crandall", title = "Effective {Laguerre} asymptotics", journal = j-SIAM-J-NUMER-ANAL, volume = "46", number = "6", pages = "3285--3312", year = "2008", CODEN = "SJNAAM", DOI = "https://doi.org/10.1137/07068031X", ISSN = "0036-1429 (print), 1095-7170 (electronic)", ISSN-L = "0036-1429", MRclass = "33C65 (30E20 34E05)", MRnumber = "2448665", MRreviewer = "Yu-Qiu Zhao", bibdate = "Wed Aug 10 11:09:47 2016", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://docserver.carma.newcastle.edu.au/334/", abstract = "It is known that the generalized Laguerre polynomials can enjoy subexponential growth for large primary index. In particular, for certain fixed parameter pairs (a, z) one has the large-n asymptotic behavior L-n((-a)) (-z) similar to C(a, z)(n)(-a)/2-1/ (4)e(2) root nz. We introduce a computationally motivated contour integral that allows efficient numerical Laguerre evaluations yet also leads to the complete asymptotic series over the full parameter domain of subexponential behavior. We present a fast algorithm for symbolic generation of the rather formidable expansion coefficients. Along the way we address the difficult problem of establishing effective (i. e., rigorous and explicit) error bounds on the general expansion. A primary tool for these developments is an ``exp-arc'' method giving a natural bridge between converging series and effective asymptotics.", acknowledgement = ack-nhfb, fjournal = "SIAM Journal on Numerical Analysis", journal-URL = "http://epubs.siam.org/sinum", ORCID-numbers = "Borwein, Jonathan/0000-0002-1263-0646", researcherid-numbers = "Borwein, Jonathan/A-6082-2009", unique-id = "Borwein:2008:ELA", } @InProceedings{Brisebarre:2008:EME, author = "Nicolas Brisebarre and Sylvain Chevillard and Milo{\v{s}} D. Ercegovac and Jean-Michel Muller and Serge Torres", title = "An Efficient Method for Evaluating Polynomial and Rational Function Approximations", crossref = "IEEE:2008:ICA", pages = "233--238", year = "2008", DOI = "https://doi.org/10.1109/ASAP.2008.4580185", bibdate = "Mon Feb 10 07:28:25 2020", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, } @Article{Chatterjee:2008:CNT, author = "S. Chatterjee and D. Roy", title = "A class of new transforms tailored for the hypergeometric series", journal = j-COMP-PHYS-COMM, volume = "179", number = "8", pages = "555--561", day = "15", month = oct, year = "2008", CODEN = "CPHCBZ", DOI = "https://doi.org/10.1016/j.cpc.2008.05.001", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", bibdate = "Thu Dec 01 09:09:57 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/compphyscomm2000.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Computer Physics Communications", journal-URL = "http://www.sciencedirect.com/science/journal/00104655", xxauthor = "S. Charterjee and D. Roy", } @Book{Cuyt:2008:HCF, author = "Annie Cuyt and Vigdis B. Petersen and Brigitte Verdonk and Haakon Waadeland and William B. Jones", title = "Handbook of Continued Fractions for Special Functions", publisher = pub-SV, address = pub-SV:adr, pages = "xx + 440", year = "2008", DOI = "https://doi.org/10.1007/978-1-4020-6949-9", ISBN = "1-4020-6948-0", ISBN-13 = "978-1-4020-6948-2", LCCN = "QA295 .H275 2008", bibdate = "Tue Jun 24 07:17:37 MDT 2008", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/master.bib; z3950.loc.gov:7090/Voyager", acknowledgement = ack-nhfb, keywords = "applying the limit process; associated continued fraction; asymptotic series expansion; basic hypergeometric functions; canonical contraction; combination with property; complementary incomplete gamma function; complex error function; confluent hypergeometric series; continued fraction converges; continued fraction representations; fraction approximants; modified approximant; monic orthogonal polynomial sequence; normed field; nth approximant; nth denominator; nth numerator; nth tail; oval sequence theorem; parabola theorem; partial numerators; strong moment distribution function; successive approximants; truncation error bounds", shorttableofcontents = "General considerations \\ Part 1, Basic Theory \\ 1. Basics \\ 2. Continued fraction representation of functions \\ 3. Convergence criteria \\ 4. Pade approximants \\ 5. Moment theory and orthogonal functions \\ Part 2, Numerics \\ 6. Continued fraction construction \\ 7. Truncation error bounds \\ 8. Continued fraction evaluation \\ Part 3, Special Functions \\ 9. On tables and graphs \\ 10. Mathematical constants \\ 11. Elementary functions \\ 12. Gamma function and related functions \\ 13. Error function and related integrals \\ 14. Exponential integrals and related functions \\ 15. Hypergeometric functions \\ 16. Confluent hypergeometric functions, \\ 17. Bessel functions \\ 18. Probability functions \\ 19. Basic hypergeometric functions", tableofcontents = "Preface / xi \\ Notation / xiii \\ 0 General considerations \\ / 1 \\ 0.1 Part one / 1 \\ 0.2 Part two / 2 \\ 0.3 Part three / 2 \\ \\ Part I: Basic Theory \\ \\ 1 Basics / 9 \\ 1.1 Symbols and notation \\ 1.2 Definitions / 10 \\ 1.3 Recurrence relations / 13 \\ 1.4 Equivalence transformations / 15 \\ 1.5 Contractions and extensions / 16 \\ 1.6 Continued fractions with prescribed approximants / 18 \\ 1.7 Connection between continued fractions and series / 19 \\ 1.8 Periodic and limit periodic continued fractions / 21 \\ 1.9 Tails of continued fractions / 23 \\ 1.10 Continued fractions over normed fields / 26 \\ 1.11 Generalisations of continued fractions / 28 \\ \\ 2 Continued fraction representation of functions / 29 \\ 2.1 Symbols and notation / 29 \\ 2.2 Correspondence / 30 \\ 2.3 Families of continued fractions / 35 \\ 2.4 Correspondence of C-fractions / 39 \\ 2.5 Correspondence of P-fractions / 40 \\ 2.6 Correspondence of J-fractions and T-fractions / 41 \\ 2.7 Correspondence and three-term recurrences / 42 \\ \\ 3 Convergence criteria / 45 \\ 3.1 Some classical theorems / 45 \\ 3.2 Convergence sets and value sets / 47 \\ 3.3 Parabola and oval theorems / 49 \\ 3.4 Correspondence and uniform convergence / 52 \\ 3.5 Periodic and limit periodic continued fractions / 53 \\ 3.6 Convergence and minimal solutions / 56 \\ \\ 4 Pad{\'e} approximants / 59 \\ 4.1 Definition and notation / 59 \\ 4.2 Fundamental properties / 60 \\ 4.3 Connection with regular C-fractions / 64 \\ 4.4 Connection with P-fractions / 65 \\ 4.5 Extension of the Pad{\'e} table / 67 \\ 4.6 Connection with M-fractions and the M-table / 68 \\ 4.7 Convergence of Pad{\'e} approximants / 70 \\ 4.8 Formal orthogonality property / 72 \\ \\ 5 Moment theory and orthogonal functions / 77 \\ 5.1 Moment theory / 77 \\ 5.2 Stieltjes transforms / 85 \\ 5.3 Construction of solutions / 90 \\ 5.4 Orthogonal polynomials / 91 \\ 5.5 Monic orthogonal polynomials on $\mathbb{R}$ and J-fractions / 92 \\ 5.6 Szeg{\H{o}} polynomials and PPC-fractions / 100 \\ 5.7 Orthogonal Laurent polynomials and APT-fractions / 102 \\ \\ Part II: Numerics \\ \\ 6 Continued fraction construction / 107 \\ 6.1 Regular C-fractions / 107 \\ 6.2 C-fractions / 113 \\ 6.3 S-fractions / 114 \\ 6.4 P-fractions / 114 \\ 6.5 J-fractions / 120 \\ 6.6 M-fractions / 122 \\ 6.7 Positive T-fractions / 124 \\ 6.8 Thiele fractions / 125 \\ \\ 7 Truncation error bounds / 129 \\ 7.1 Parabola theorems / 129 \\ 7.2 The oval sequence theorem / 131 \\ 7.3 The interval sequence theorem / 136 \\ 7.4 Specific a priori bounds for S-fractions / 138 \\ 7.5 A posteriori truncation error bounds / 140 \\ 7.6 Tails and truncation error bounds / 143 \\ 7.7 Choice of modification / 143 \\ \\ 8 Continued fraction evaluation / 149 \\ 8.1 The effect of finite precision arithmetic / 149 \\ 8.2 Evaluation of approximants / 152 \\ 8.3 The forward recurrence and minimal solutions / 154 \\ 8.4 Round-off error in the backward recurrence / 156 \\ \\ Part III: Special Functions \\ \\ 9 On tables and graphs / 163 \\ 9.1 Introduction / 163 \\ 9.2 Comparative tables / 163 \\ 9.3 Reliable graphs / 168 \\ \\ 10 Mathematical constants / 175 \\ 10.1 Regular continued fractions / 175 \\ 10.2 Archimedes' constant, symbol $\pi$ / 176 \\ 10.3 Euler's number, base of the natural logarithm / 178 \\ 10.4 Integer powers and roots of $\pi$ and $e$ / 180 \\ 10.5 The natural logarithm, $\ln(2)$ / 181 \\ 10.6 Pythagoras' constant, the square root of two / 183 \\ 10.7 The cube root of two / 183 \\ 10.8 Euler's constant, symbol $\gamma$ / 185 \\ 10.9 Golden ratio, symbol $\phi$ / 185 \\ 10.10 The rabbit constant, symbol $\rho$ / 186 \\ 10.11 Ap{\'e}ry's constant, $\zeta(3)$ / 188 \\ 10.12 Catalan's constant, symbol $C$ / 189 \\ 10.13 Gompertz' constant, symbol $G$ / 190 \\ 10.14 Khinchin's constant, symbol $K$ / 190 \\ \\ 11 Elementary functions / 193 \\ 11.1 The exponential function / 193 \\ 11.2 The natural logarithm / 196 \\ 11.3 Trigonometric functions / 200 \\ 11.4 Inverse trigonometric functions / 204 \\ 11.5 Hyperbolic functions / 210 \\ 11.6 Inverse hyperbolic functions / 213 \\ 11.7 The power function / 217 \\ \\ 12 Gamma function and related functions / 221 \\ 12.1 Gamma function / 221 \\ 12.2 Binet function / 224 \\ 12.3 Polygamma functions / 229 \\ 12.4 Trigamma function / 232 \\ 12.5 Tetragamma function / 235 \\ 12.6 Incomplete gamma functions / 238 \\ \\ 13 Error function and related integrals / 253 \\ 13.1 Error function and Dawson's integral / 253 \\ 13.2 Complementary and complex error function / 261 \\ 13.3 Repeated integrals / 268 \\ 13.4 Fresnel integrals / 269 \\ \\ 14 Exponential integrals and related functions / 275 \\ 14.7 Exponential integrals / 275 \\ 14.2 Related functions / 285 \\ \\ 15 Hypergeometric functions / 291 \\ 15.1 Definition and basic properties / 291 \\ 15.2 Stieltjes transform / 295 \\ 15.3 Continued fraction representations / 295 \\ 15.4 Pad{\'e} approximants / 309 \\ 15.5 Monotonicity properties / 313 \\ 15.6 Hypergeometric series $_pF_q$ / 315 \\ \\ 16 Confluent hypergeometric functions / 319 \\ 16.1 Kummer functions / 319 \\ 16.2 Confluent hypergeometric series $_2F_0$ / 330 \\ 16.3 Confluent hypergeometric limit function / 333 \\ 16.4 Whittaker functions / 334 \\ 16.5 Parabolic cylinder functions / 337 \\ \\ 17 Bessel functions / 334 \\ 17.7 Bessel functions / 334 \\ 17.2 Modified Bessel functions / 356 \\ \\ 18 Probability functions / 371 \\ 18.1 Definitions and elementary properties / 371 \\ 18.2 Normal and log-normal distributions / 373 \\ 18.3 Repeated integrals / 377 \\ 18.4 Gamma and chi-square distribution / 378 \\ 18.5 Beta, $F$- and Student's $t$-distributions / 382 \\ \\ 19 Basic hypergeometric functions / 391 \\ 19.1 Definition and basic properties / 391 \\ 19.2 Continued fraction representations / 395 \\ 19.3 Higher order basic hypergeometric functions / 399 \\ \\ Bibliography / 401 \\ \\ Index / 421", } @Article{Dyer:2008:CCI, author = "J. S. Dyer and S. A. Dyer", title = "Corrections to, and comments on, {``An improved approximation for the Gaussian $Q$-Function''}", journal = j-IEEE-COMMUN-LET, volume = "12", number = "4", pages = "231--231", month = apr, year = "2008", CODEN = "ICLEF6", DOI = "https://doi.org/10.1109/lcomm.2008.080009", ISSN = "1089-7798 (print), 1558-2558 (electronic)", ISSN-L = "1089-7798", bibdate = "Sat Dec 16 18:08:34 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "See \cite{Karagiannidis:2007:IAG}.", URL = "https://ieeexplore.ieee.org/document/4489650/", acknowledgement = ack-nhfb, fjournal = "IEEE Communications Letters", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=4234", } @Article{Elbert:2008:ZCE, author = "{\'A}rp{\'a}d Elbert and Andrea Laforgia", title = "The zeros of the complementary error function", journal = j-NUMER-ALGORITHMS, volume = "49", number = "1--4", pages = "153--157", month = dec, year = "2008", CODEN = "NUALEG", ISSN = "1017-1398 (print), 1572-9265 (electronic)", ISSN-L = "1017-1398", bibdate = "Mon May 17 14:08:26 MDT 2010", bibsource = "http://springerlink.metapress.com/openurl.asp?genre=issue&issn=1017-1398&volume=49&issue=1; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.springerlink.com/openurl.asp?genre=article&issn=1017-1398&volume=49&issue=1&spage=153", acknowledgement = ack-nhfb, fjournal = "Numerical Algorithms", journal-URL = "http://link.springer.com/journal/11075", } @Article{Gabutti:2008:EQG, author = "Bruno Gabutti and Giampietro Allasia", title = "Evaluation of $q$-gamma function and $q$-analogues by iterative algorithms", journal = j-NUMER-ALGORITHMS, volume = "49", number = "1--4", pages = "159--168", month = dec, year = "2008", CODEN = "NUALEG", ISSN = "1017-1398 (print), 1572-9265 (electronic)", ISSN-L = "1017-1398", bibdate = "Mon May 17 15:07:19 MDT 2010", bibsource = "http://springerlink.metapress.com/openurl.asp?genre=issue&issn=1017-1398&volume=49&issue=1; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.springerlink.com/openurl.asp?genre=article&issn=1017-1398&volume=49&issue=1&spage=159", acknowledgement = ack-nhfb, fjournal = "Numerical Algorithms", journal-URL = "http://link.springer.com/journal/11075", } @Article{Gautschi:2008:LGW, author = "Walter Gautschi and Carla Giordano", title = "{Luigi Gatteschi}'s work on asymptotics of special functions and their zeros", journal = j-NUMER-ALGORITHMS, volume = "49", number = "1--4", pages = "11--31", month = dec, year = "2008", CODEN = "NUALEG", ISSN = "1017-1398 (print), 1572-9265 (electronic)", ISSN-L = "1017-1398", bibdate = "Mon May 17 14:08:26 MDT 2010", bibsource = "http://springerlink.metapress.com/openurl.asp?genre=issue&issn=1017-1398&volume=49&issue=1; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.springerlink.com/openurl.asp?genre=article&issn=1017-1398&volume=49&issue=1&spage=11", acknowledgement = ack-nhfb, fjournal = "Numerical Algorithms", journal-URL = "http://link.springer.com/journal/11075", } @Article{Harris:2008:IBG, author = "Frank E. Harris", title = "Incomplete {Bessel}, generalized incomplete gamma, or leaky aquifer functions", journal = j-J-COMPUT-APPL-MATH, volume = "215", number = "1", pages = "260--269", year = "2008", CODEN = "JCAMDI", DOI = "https://doi.org/10.1016/j.cam.2007.04.008", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", MRclass = "33B10 (33C10 41A58); 33B20", MRnumber = "2400632 (2009d:33003)", MRreviewer = "Necdet Batir", bibdate = "Wed Dec 4 07:03:09 2013", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/h/harris-frank-e.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0377042707002014", ZMnumber = "Zbl 1135.33002", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @Article{Isukapalli:2008:ATA, author = "Yogananda Isukapalli and Bhaskar D. Rao", title = "An Analytically Tractable Approximation for the {Gaussian} {$Q$}-Function", journal = j-IEEE-COMMUN-LET, volume = "12", number = "9", pages = "669--671", month = sep, year = "2008", CODEN = "ICLEF6", DOI = "https://doi.org/10.1109/lcomm.2008.080815", ISSN = "1089-7798 (print), 1558-2558 (electronic)", ISSN-L = "1089-7798", bibdate = "Sat Dec 16 16:44:42 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "IEEE Communications Letters", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=4234", } @Book{Jeffrey:2008:HMF, author = "Alan Jeffrey and Hui-Hui Dai", title = "Handbook of Mathematical Formulas and Integrals", publisher = pub-ELSEVIER-ACADEMIC, address = pub-ELSEVIER-ACADEMIC:adr, edition = "Fourth", pages = "xlv + 541", year = "2008", ISBN = "0-12-374288-9 (paperback), 0-08-055684-1 (e-book)", ISBN-13 = "978-0-12-374288-9 (paperback), 978-0-08-055684-0 (e-book)", LCCN = "QA47 .J38 2008", bibdate = "Thu May 8 16:02:52 MDT 2008", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/master.bib; https://www.math.utah.edu/pub/tex/bib/mathcw.bib; melvyl.cdlib.org:210/CDL90", URL = "https://shop.elsevier.com/books/handbook-of-mathematical-formulas-and-integrals/jeffrey/978-0-12-374288-9", acknowledgement = ack-nhfb, subject = "mathematics; tables; formulae", tableofcontents = "Preface \\ Preface to the Fourth Edition \\ Notes for Handbook Users \\ Index of Special Functions and Notations \\ 0. Quick Reference List of Frequently Used Data \\ 1. Numerical, Algebraic, and Analytical Results for Series and Calculus \\ 2. Functions and Identities \\ 3. Derivatives of Elementary Functions \\ 4. Indefinite Integrals of Algebraic Functions \\ 5. Indefinite Integrals of Exponential Functions \\ 6. Indefinite Integrals of Logarithmic Functions \\ 7. Indefinite Integrals of Hyperbolic Functions \\ 8. Indefinite Integrals Involving Inverse Hyperbolic Functions \\ 9. Indefinite Integrals of Trigonometric Functions \\ 10. Indefinite Integrals of Inverse Trigonometric Functions \\ 11. The Gamma, Beta, Pi, and Psi Functions, and the Incomplete Gamma Functions \\ 12. Elliptic Integrals and Functions \\ 13. Probability Distributions and Integrals, and the Error Function \\ 14. Fresnel Integrals, Sine and Cosine Integrals \\ 15. Definite Integrals \\ 16. Different Forms of Fourier Series \\ 17. Bessel Functions \\ 18. Orthogonal Polynomials \\ 19. Laplace Transformation \\ 20. Fourier Transforms \\ 21. Numerical Integration \\ 22. Solutions of Standard Ordinary Differential Equations \\ 23. Vector Analysis \\ 24. Systems of Orthogonal Coordinates \\ 25. Partial Differential Equations and Special Functions \\ 26. Qualitative Properties of the Heat and Laplace Equation \\ 27. Solutions of Elliptic, Parabolic, and Hyperbolic Equations \\ 28. The z-Transform \\ 29. Numerical Approximation \\ 30. Conformal Mapping and Boundary Value Problems \\ Short Classified Reference List \\ Index", } @Article{Kiani:2008:AND, author = "M. Kiani and J. Panaretos and S. Psarakis and M. Saleem", title = "Approximations to the normal distribution function and an extended table for the mean range of the normal variables", journal = "J. Iran. Stat. Soc.", volume = "7", number = "1", pages = "57--72", month = "????", year = "2008", DOI = "", bibdate = "Sat Dec 16 16:56:03 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "", acknowledgement = ack-nhfb, } @Article{Kodama:2008:ASP, author = "Masao Kodama", title = "{Algorithm 877}: a Subroutine Package for Cylindrical Functions of Complex Order and Nonnegative Argument", journal = j-TOMS, volume = "34", number = "4", pages = "22:1--22:21", month = jul, year = "2008", CODEN = "ACMSCU", DOI = "https://doi.org/10.1145/1377596.1377602", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Wed Jul 16 11:30:01 MDT 2008", bibsource = "http://www.acm.org/pubs/contents/journals/toms/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib", abstract = "The algorithm presented provides a package of subroutines for calculating the cylindrical functions $ J_\nu (x) $, $ N_\nu (x) $, $ H_\nu^1 (x) $, $ H_\nu^2 (x) $ where the order $ \nu $ is complex and the real argument $x$ is nonnegative. The algorithm is written in Fortran 95 and calculates the functions using single, double, or quadruple precision according to the value of a parameter defined in the algorithm. The methods of calculating the functions are based on a series expansion, Debye's asymptotic expansions, Olver's asymptotic expansions, and recurrence methods (Miller's algorithms). The relative errors of the functional values computed by this algorithm using double precision are less than $ 2.4 \times 10^{-13} $ in the region $ 0 \leq \mbox {Re}(\nu) \leq 64 $, $ 0 \leq \mbox {Im}(\nu) \leq 63 $, $ 0.024 \leq x \leq 97 $.", acknowledgement = ack-nhfb, articleno = "22", fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", keywords = "Bessel functions; complex order; Cylindrical functions; Debye's asymptotic expansions; Hankel functions; Miller's algorithms; Neumann functions; nonnegative argument; numerical calculation; Olver's asymptotic expansions", } @Article{Lefevre:2008:WCE, author = "Vincent Lef{\`e}vre and Damien Stehl{\'e} and Paul Zimmermann", title = "Worst Cases for the Exponential Function in the {IEEE 754r decimal64} Format", journal = j-LECT-NOTES-COMP-SCI, volume = "5045", pages = "114--126", year = "2008", CODEN = "LNCSD9", DOI = "https://doi.org/10.1007/978-3-540-85521-7_7", ISSN = "0302-9743 (print), 1611-3349 (electronic)", ISSN-L = "0302-9743", bibdate = "Thu Oct 1 11:29:36 MDT 2015", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/lncs2008a.bib", URL = "http://link.springer.com/content/pdf/10.1007/978-3-540-85521-7_7.pdf", acknowledgement = ack-nhfb, book-DOI = "https://doi.org/10.1007/978-3-540-85521-7", book-URL = "http://www.springerlink.com/content/978-3-540-85521-7", fjournal = "Lecture Notes in Computer Science", journal-URL = "http://link.springer.com/bookseries/558", remark = "From the abstract: ``the worst case for $ |x| \geq 3 \times 10^{-11} $ is exp(9.407822313572878e-2) = 1.09864568206633850000000000000000278.''", } @Article{Lorch:2008:MSR, author = "Lee Lorch and Martin E. Muldoon", title = "Monotonic sequences related to zeros of {Bessel} functions", journal = j-NUMER-ALGORITHMS, volume = "49", number = "1--4", pages = "221--233", month = dec, year = "2008", CODEN = "NUALEG", ISSN = "1017-1398 (print), 1572-9265 (electronic)", ISSN-L = "1017-1398", bibdate = "Mon May 17 15:07:19 MDT 2010", bibsource = "http://springerlink.metapress.com/openurl.asp?genre=issue&issn=1017-1398&volume=49&issue=1; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.springerlink.com/openurl.asp?genre=article&issn=1017-1398&volume=49&issue=1&spage=221", acknowledgement = ack-nhfb, fjournal = "Numerical Algorithms", journal-URL = "http://link.springer.com/journal/11075", } @Book{Mathai:2008:SFA, author = "A. M. Mathai and H. J. Haubold", title = "Special Functions for Applied Scientists", publisher = "Springer Science+Business Media", address = "New York, NY, USA", pages = "xxv + 464", year = "2008", DOI = "https://doi.org/10.1007/978-0-387-75894-7", ISBN = "0-387-75893-3", ISBN-13 = "978-0-387-75893-0", LCCN = "QA351 .M37X 2008", bibdate = "Sat Oct 30 17:02:02 MDT 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; prodorbis.library.yale.edu:7090/voyager", acknowledgement = ack-nhfb, shorttableofcontents = "Basic ideas of special functions and statistical distributions \\ Mittag-Leffler functions and fractional calculus \\ An introduction to q-series \\ Ramanujan's theories of theta and elliptic functions \\ Lie group and special functions \\ Applications to stochastic process and time series \\ Applications to density estimation \\ Applications to order statistics \\ Applications to astrophysics problems \\ An introduction to wavelet analysis \\ Jacobians of matrix formations \\ Special functions of matrix argument", subject = "Functions, Special; Fractional calculus; Wavelets (Mathematics)", tableofcontents = "1 Basic Ideas of Special Functions and Statistical Distributions / 1 \\ 1.0 Introduction / 1 \\ 1.1 Gamma Function / 3 \\ 1.1.1 Some basic properties of gamma functions / 4 \\ 1.1.2 Wedge product and Jacobians of transformations / 6 \\ 1.1.3 Multiplication formula for gamma functions / 8 \\ 1.1.4 Asymptotic formula for a gamma function / 9 \\ 1.1.5 Bernoulli polynomials / 9 \\ 1.1.6 Some basic properties of generalized Bernoulli polynomials / 9 \\ 1.1.7 The first three generalized Bernoulli polynomials / 10 \\ Exercises 1.1 / 11 \\ 1.2 The Psi and Zeta Functions / 12 \\ 1.2.1 Generalized zeta function / 12 \\ Exercises 1.2 / 13 \\ 1.3 Integral Transforms / 14 \\ 1.3.1 Mellin transform / 14 \\ 1.3.2 Laplace transform / 17 \\ Exercises 1.3 / 18 \\ 1.4 Some Statistical Preliminaries / 19 \\ Exercises 1.4 / 26 \\ 1.5 Some Properties of Random Variables / 28 \\ 1.5.1 Multivariate analogues / 30 \\ 1.5.2 Marginal and conditional densities / 31 \\ Exercises 1.5 / 33 \\ 1.6 Beta and Related Functions / 34 \\ 1.6.1 Dirichlet integrals and Dirichlet densities / 37 \\ Exercises 1.6 / 42 \\ 1.7 Hypergeometric Series / 42 \\ 1.7.1 Evaluation of some contour integrals / 45 \\ 1.7.2 Residues when several gammas are involved / 46 \\ Exercises 1.7 / 48 \\ 1.8 Meijer's $G$-function / 49 \\ Exercises 1.8 / 54 \\ 1.9 The $H$-function / 54 \\ Exercises 1.9 / 57 \\ 1.10 Lauricella Functions and Appell's Functions / 58 \\ 1.10.1 Some properties of $f_A$ / 59 \\ 1.10.2 Some properties of $f_B$ / 60 \\ 1.10.3 Some properties of $f_C$ / 60 \\ 1.10.4 Some properties of $f_D$ / 61 \\ Exercises 1.10 / 64 \\ 1.11 Special Functions as Solutions of Differential Equations and Applications / 64 \\ 1.11.0 Introduction / 64 \\ 1.11.1 Sine, cosine and exponential functions / 64 \\ Exercises 1.11 / 66 \\ 1.11.2 Linear second order differential equations / 66 \\ 1.11.3 Hypergeometric function / 67 \\ Exercises 1.11 / 69 \\ 1.11.4 Confluent hypergeometric function / 70 \\ Exercises 1.11 / 71 \\ 1.11.5 Hermite polynomials / 71 \\ Exercises 1.11 / 72 \\ 1.11.6 Bessel functions / 72 \\ Exercises 1.11 / 73 \\ 1.11.7 Laguerre polynomial / 74 \\ 1.11.8 Legendre polynomial / 74 \\ Exercises 1.11 / 74 \\ 1.11.9 Generalized hypergeometric function / 75 \\ 1.11.10 $G$-function / 76 \\ Exercises 1.11 / 77 \\ References / 77 \\ 2 Mittag-Leffler Functions and Fractional Calculus / 79 \\ 2.0 Introduction / 79 \\ 2.1 Mittag-Leffler Function / 79 \\ Revision Exercises 2.1 / 81 \\ 2.2 Basic Properties of Mittag-Leffler Function / 82 \\ 2.2.1 Mittag-Leffler functions of rational order / 84 \\ 2.2.2 Euler transform of Mittag-Leffler function / 84 \\ 2.2.3 Laplace transform of Mittag-Leffler function / 85 \\ 2.2.4 Application of Laplace transform / 87 \\ 2.2.5 Mittag-Leffler functions and the $H$-function / 88 \\ Exercises 2.2 / 90 \\ 2.3 Generalized Mittag-Leffler Function / 91 \\ 2.3.1 Special cases of $E_{\beta, \gamma}^\delta (z)$ / 92 \\ 2.3.2 Mellin--Barnes integral representation / 92 \\ 2.3.3 Relations with the H-function and Wright function / 92 \\ 2.3.4 Cases of reducibility / 93 \\ 2.3.5 Differentiation of generalized Mittag-Leffler function / 94 \\ 2.3.6 Integral property of generalized Mittag-Leffler function / 94 \\ 2.3.7 Integral transform of $E_{\beta, \gamma}^\delta (z)$ / 95 \\ Exercises 2.3 / 96 \\ 2.4 Fractional Integrals / 97 \\ 2.4.1 Riemann--Liouville fractional integrals of arbitrary order / 98 \\ 2.4.2 Riemann--Liouville fractional integrals of order $\alpha$ / 99 \\ 2.4.3 Basic properties of fractional integrals / 100 \\ 2.4.4 A useful integral / 101 \\ 2.4.5 The Weyl integral / 103 \\ 2.4.6 Basic properties of Weyl integral / 104 \\ Exercises 2.4 / 105 \\ 2.4.7 Laplace transform of the fractional integral / 107 \\ 2.4.8 Laplace transform of the fractional derivative / 107 \\ 2.4.9 Laplace transform of Caputo derivative / 108 \\ Exercises 2.4 / 109 \\ 2.5 Mellin Transform of the Fractional Integrals and the Fractional Derivatives / 109 \\ 2.5.1 Mellin transform / 109 \\ 2.5.2 Mellin transform of the fractional integral / 110 \\ 2.5.3 Mellin transform of the fractional derivative / 111 \\ Exercises 2.5 / 111 \\ 2.6 Kober Operators / 111 \\ Exercises 2.6 / 113 \\ 2.7 Generalized Kober Operators / 114 \\ Exercises 2.7 / 118 \\ 2.8 Compositions of Riemann--Liouville Fractional Calculus Operators and Generalized Mittag-Leffler Functions / 121 \\ 2.8.1 Composition relations between R-L operators and $E_{\beta, \gamma}^\delta (z)$ / 121 \\ Exercises 2.8 / 126 \\ 2.9 Fractional Differential Equations / 126 \\ 2.9.1 Fractional relaxation / 127 \\ Exercises 2.9 / 130 \\ 2.9.2 Fractional diffusion / 131 \\ Exercises 2.9 / 132 \\ References / 133 \\ 3 An Introduction to $q$-Series / 135 \\ 3.0 Introduction / 135 \\ 3.1 Hypergeometric Series / 135 \\ Exercises 3.1 / 136 \\ 3.2 Basic Hypergeometric Series ($q$-Series) / 138 \\ Exercises 3.2 / 140 \\ 3.2.1 The $q$-binomial theorem / 142 \\ Exercises 3.2 / 146 \\ 3.2.2 The $q$-binomial coefficients / 146 \\ Exercises 3.2 / 148 \\ 3.3 $q$-Calculus / 149 \\ Exercises 3.3 / 151 \\ 3.4 The $q$-Gamma and $q$-Beta Functions / 151 \\ 3.5 Transformation and Summation Formulas for $q$-Series / 152 \\ Exercises 3.5 / 154 \\ 3.6 Jacobi's Triple Product and Rogers--Ramanujan Identities / 154 \\ References / 157 \\ 4 Ramanujan's Theories of Theta and Elliptic Functions / 159 \\ 4.0 Introduction / 159 \\ 4.1 Ramanujan's Theory of Classical Theta Functions / 159 \\ 4.1.1 Series definition and additive results / 159 \\ 4.2 Ramanujan's $_1\Psi_1$ Summation Formula and Multiplicative Results for Theta Functions / 163 \\ 4.3 Modular Equations / 170 \\ 4.4 Inversion Formulas and Evaluations / 172 \\ 4.5 Modular Identities (Classical Theory) / 175 \\ 4.6 Ramanujan's Theory of Cubic Theta Functions / 177 \\ 4.6.1 The cubic theta functions / 177 \\ 4.6.2 Inversion formulas and evaluations (cubic theory) / 179 \\ 4.6.3 Triplication and trimediation formulas / 182 \\ 4.6.4 Further evaluations / 183 \\ 4.6.5 Evaluations of Ramanujan--Eisenstein series ($L$, $M$, $N$ or $P$, $Q$, $R$) / 185 \\ 4.6.6 The cubic analogue of the Jacobian elliptic functions / 186 \\ Test on Ramanujan's work / 187 \\ 4.7 The One-variable Cubic Theta Functions / 188 \\ 4.7.1 Cubic theta functions and some properties / 189 \\ 4.7.2 Product representations for $b(q)$ and $c(q)$ / 190 \\ 4.7.3 The cubic analogue of Jacobi's quartic modular equations / 191 \\ 4.8 The Two-variable Cubic Theta Functions / 193 \\ 4.8.1 Series definitions and some properties / 193 \\ 4.8.2 Product representations for $b(q,z)$ and $c(q,z)$ / 195 \\ 4.8.3 A two-variable cubic counterpart of Jacobi's quartic modular equation / 200 \\ 4.9 The Three-variable Cubic Theta Functions / 200 \\ 4.9.1 Unification of one and two-variable cubic theta functions / 200 \\ Exercises 4.9 / 200 \\ 4.9.2 Generalization of Hirschhorn--Garvan--Borwein identity / 202 \\ 4.9.3 Laurent's expansions for two-parameter cubic theta functions / 205 \\ References / 209 \\ 5 Lie Group and Special Functions / 211 \\ 5.1 General Introduction to Group Theory / 211 \\ 5.1.1 Isomorphisms / 213 \\ 5.1.2 Symmetry groups / 213 \\ 5.1.3 Isometries of the Euclidean plane / 214 \\ 5.1.4 Finite groups of motion / 215 \\ 5.1.5 Discrete groups of motions / 216 \\ 5.2 Lie Group and Special Functions / 217 \\ 5.2.1 Subspace of a vector space / 218 \\ 5.3 Lie Algebra / 219 \\ 5.4 Representations of Lie Algebra / 222 \\ Exercises 5.4 / 225 \\ 5.5 Special Functions / 226 \\ 5.5.1 Gauss hypergeometric function / 226 \\ 5.5.2 Differential equation satisfied by $_2F_1$ / 228 \\ 5.5.3 Integral representation of $_F_q(\alpha_1,\alpha_2, \ldots{}, \alpha_p \beta_1, \beta_2, \ldots{}, \beta_p; z)$ / 230 \\ 5.6 Laguerre Polynomial $L^{(\alpha)}_n(x)$ / 231 \\ 5.6.1 Laguerre polynomial and Lie algebra / 232 \\ Exercises 5.6 / 232 \\ 5.7 Helmholtz Equation / 234 \\ 5.7.1 Helmholtz equation in three variables / 238 \\ 5.8 Lie Group / 241 \\ 5.8.1 Basis of the Lie algebra of the Lie group SL(2) / 242 \\ Exercises 5.8 / 243 \\ Test on Lie Theory and Special Functions / 244 \\ 6 Applications to Stochastic Process and Time Series / 247 \\ 6.0 Introduction / 247 \\ 6.1 Stochastic Processes / 247 \\ 6.1.1 Classical types of stochastic processes / 252 \\ 6.1.2 Processes with stationary independent increments / 252 \\ 6.1.3 Stationary processes / 253 \\ 6.1.4 Gaussian processes and stationarity / 253 \\ 6.1.5 Brownian processes / 257 \\ 6.1.6 Markov chains / 258 \\ Exercises 6.1 / 261 \\ 6.2 Modern Concepts in Distribution Theory / 262 \\ 6.2.1 Introduction / 262 \\ 6.2.2 Geometric infinite divisibility / 263 \\ 6.2.3 Bernstein functions / 264 \\ 6.2.4 Self-decomposability / 265 \\ 6.2.5 Stable distributions / 265 \\ 6.2.6 Geometrically strictly stable distributions / 266 \\ 6.2.7 Mittag-Leffler distribution / 266 \\ 6.2.8 $\alpha$-Laplace distribution / 267 \\ 6.2.9 Semi-Pareto distribution / 268 \\ Exercises 6.2 / 268 \\ 6.3 Stationary Time Series Models / 269 \\ 6.3.1 Introduction / 269 \\ 6.3.2 Autoregressive models / 270 \\ 6.3.3 A general solution / 270 \\ 6.3.4 Extension to a $k$-th order autoregressive model / 272 \\ 6.3.5 Mittag-Leffler autoregressive structure / 273 \\ Exercises 6.3 / 274 \\ 6.4 A Structural Relationship and New Processes / 275 \\ 6.4.1 The TMLAR(1) process / 276 \\ 6.4.2 The NEAR(1) model / 277 \\ 6.4.3 New Mittag-Leffler autoregressive models / 278 \\ 6.4.4 The NSMLAR(1) process / 280 \\ Exercises 6.4 / 281 \\ 6.5 Tailed Processes / 281 \\ 6.5.1 The exponential tailed autoregressive process [ETAR(1)] / 282 \\ 6.5.2 The Mittag-Leffler tailed autoregressive process [MLTAR(1)] / 283 \\ Exercises 6.5 / 286 \\ 6.6 Marshall--Olkin Weibull Time Series Models / 286 \\ 6.6.1 Introduction / 286 \\ 6.6.2 Marshall--Olkin semi-Weibull distribution / 287 \\ 6.6.3 An AR(1) model with MOSW marginal distribution / 289 \\ 6.6.4 Marshall--Olkin generalized Weibull distribution / 291 \\ 6.6.5 An AR(1) model with MOGW marginal distribution / 292 \\ 6.6.6 Case study / 293 \\ Exercises 6.6 / 293 \\ References / 294 \\ 7 Applications to Density Estimation / 297 \\ 7.0 Density Estimation and Orthogonal Polynomials / 297 \\ 7.1 Introduction / 297 \\ 7.2 Approximants Based on Legendre Polynomials / 299 \\ 7.3 Approximants Based on Laguerre Polynomials / 301 \\ 7.4 A Unified Methodology / 304 \\ 7.5 Approximants Expressed in Terms of Orthogonal Polynomials / 305 \\ 7.5.1 Approximants expressed in terms of Laguerre polynomials / 306 \\ 7.5.2 Approximants expressed in terms of Legendre polynomials / 306 \\ 7.5.3 Approximants expressed in terms of Jacobi polynomials / 307 \\ 7.5.4 Approximants expressed in terms of Hermite polynomials / 307 \\ References / 308 \\ 8 Applications to Order Statistics / 311 \\ 8.0 Introduction / 311 \\ 8.1 Distribution Function / 311 \\ 8.1.1 Density of the $r$-th order statistic / 313 \\ 8.1.2 Joint distribution function of two order statistics / 314 \\ 8.1.3 Joint density of two order statistics / 314 \\ 8.1.4 Moments of order statistics / 315 \\ 8.1.5 Recurrence relations for moments / 317 \\ 8.1.6 Recurrence relations on the product moments / 318 \\ 8.1.7 Order statistics from symmetric distributions / 319 \\ 8.2 Discrete Order Statistics / 319 \\ 8.2.1 Probability function of discrete order statistics / 320 \\ 8.2.2 Joint probability function of two order statistics / 321 \\ 8.2.3 Bernoulli order statistics / 321 \\ 8.3 Independent Random Variables / 322 \\ 8.3.1 Distribution of a single order statistic / 322 \\ 8.3.2 Joint distribution of two order statistics / 324 \\ Test on Order Statistics / 325 \\ 8.4 On Concomitants of Order Statistics / 326 \\ 8.4.1 Application of concomitants of order statistics / 326 \\ 8.4.2 Application in estimation / 328 \\ 8.4.3 Concomitants of record values and estimation problems / 333 \\ References / 339 \\ 9 Applications to Astrophysics Problems / 341 \\ 9.0 Introduction / 341 \\ 9.1 Entropy: Boltzmann, Planck, and Einstein on W / 342 \\ 9.1.1 Entropic functional / 342 \\ 9.1.2 Entropy and probability / 342 \\ 9.1.3 Boltzmann--Gibbs / 343 \\ 9.2 Gravitationally Stabilized Fusion Reactor: The Sun / 345 \\ 9.2.1 Internal solar structure / 345 \\ 9.2.2 Solar fusion plasma / 349 \\ 9.2.3 Estimation of central temperature in the Sun / 349 \\ 9.3 Crucial Astrophysical Experiments: Data Analysis / 351 \\ 9.3.1 The experiments / 351 \\ 9.3.2 Analysis of the time series / 352 \\ 9.4 Fundamental Equations for Nonequilibrium Processes / 355 \\ 9.4.1 Chapman--Kolmogorov equation / 355 \\ 9.4.2 Master equation / 355 \\ 9.4.3 Fokker--Planck equation / 356 \\ 9.4.4 Langevin equation / 357 \\ 9.4.5 Reaction-diffusion equation / 357 \\ 9.5 Fractional Calculus / 360 \\ 9.6 Nonextensive Statistical Mechanics / 362 \\ 9.7 Standard and Fractional Reaction / 363 \\ 9.7.1 Boltzmann--Gibbs statistical mechanics / 363 \\ 9.7.2 Generalized Boltzmann--Gibbs statistical mechanics / 366 \\ 9.7.3 Fractional reaction / 369 \\ 9.7.4 Thermonuclear reaction coefficient / 371 \\ 9.8 Standard and Fractional Diffusion / 380 \\ 9.8.1 Fick's first law of diffusion / 380 \\ 9.8.2 Einstein's approach to diffusion / 381 \\ 9.8.3 Fractional diffusion / 381 \\ 9.8.4 Spatio-temporal pattern formation / 383 \\ References / 384 \\ 10 An Introduction to Wavelet Analysis / 389 \\ 10.0 Introduction / 389 \\ 10.1 Fourier Analysis to Wavelet Analysis / 390 \\ 10.2 Construction of Orthonormal Wavelets / 393 \\ 10.3 Classification of Wavelets and Multiresolution Analysis / 400 \\ 10.4 Spline Wavelets / 405 \\ 10.5 A Variant of Construction of Orthonormal Wavelets / 407 \\ Exercises 10.5 / 408 \\ References / 408 \\ 11 Jacobians of Matrix Transformations / 409 \\ 11.0 Introduction / 409 \\ 11.1 Jacobians of Linear Matrix Transformations / 410 \\ Exercises 11.1 / 415 \\ 11.2 Jacobians in Some Nonlinear Transformations / 417 \\ Exercises 11.2 / 421 \\ 11.3 Transformations Involving Orthonormal Matrices / 423 \\ Exercises 11.3 / 426 \\ References / 428 \\ 12 Special Functions of Matrix Argument / 429 \\ 12.0 Introduction / 429 \\ 12.1 Real Matrix-Variate Scalar Functions / 429 \\ 12.1.1 Real matrix-variate gamma / 430 \\ 12.1.2 Real matrix-variate gamma density / 430 \\ Exercises 12.1 / 434 \\ 12.2 The Laplace Transform in the Matrix Case / 435 \\ 12.2.1 A convolution property for Laplace transforms / 436 \\ Exercises 12.2 / 439 \\ 12.3 Hypergeometric Functions with Matrix Argument / 439 \\ 12.3.1 Hypergeometric function through Laplace transform / 440 \\ 12.3.2 Hypergeometric function through zonal polynomials / 441 \\ 12.3.3 Hypergeometric functions through $M$-transforms / 443 \\ 12.3.4 A convolution theorem for $M$-transforms / 445 \\ Exercises 12.3 / 446 \\ 12.4 A Pathway Model / 448 \\ 12.4.1 The pathway density / 448 \\ 12.4.2 A general density / 451 \\ 12.4.3 Arbitrary moments / 451 \\ 12.4.4 Quadratic forms / 452 \\ 12.4.5 Generalized quadratic form / 452 \\ 12.4.6 Applications to random volumes / 453 \\ Exercises 12.4 / 453 \\ References / 454 \\ Author Index / 457 \\ Subject Index / 461", } @Article{Nam:2008:PAE, author = "Byeong-Gyu Nam and Hyejung Kim and Hoi-Jun Yoo", title = "Power and Area-Efficient Unified Computation of Vector and Elementary Functions for Handheld {$3$D} Graphics Systems", journal = j-IEEE-TRANS-COMPUT, volume = "57", number = "4", pages = "490--504", month = apr, year = "2008", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/TC.2008.12", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Mon Jul 4 12:17:40 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", URL = "https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4432232", abstract = "A unified computation method of vector and elementary functions is proposed for handheld 3D graphics systems. It unifies vector operations like vector multiply, multiply-and-add, divide, divide-by-square-root, and dot product and elementary functions like trigonometric, inverse trigonometric, hyperbolic, inverse hyperbolic, power ($ x^y $ with two variables), and logarithm to an arbitrary base into a single four-way arithmetic platform. A number system called the fixed-point hybrid number system (FXP-HNS), which combines the fixed-point number system (FXP) and the logarithmic number system (LNS), is proposed for the power and area-efficient unification. Power and area-efficient logarithmic and antilogarithmic conversion schemes are also proposed for the data conversions between fixed-point and logarithmic numbers in the FXP-HNS and achieve 0.41 percent and 0.08 percent maximum conversion errors, respectively. The unified arithmetic unit based on the proposed schemes is presented with less than 6.3 percent operation error. Its fully pipelined architecture achieves single-cycle throughput with maximum four-cycle latency for all of the supported operations. Comparison results show that the proposed arithmetic unit achieves 30 percent power and 10.9 percent area reductions and runs two times faster than the previous approach.", acknowledgement = ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", } @Article{Paszkowski:2008:CAO, author = "Stefan Paszkowski", title = "Convergence acceleration of orthogonal series", journal = j-NUMER-ALGORITHMS, volume = "47", number = "1", pages = "35--62", month = jan, year = "2008", CODEN = "NUALEG", DOI = "https://doi.org/10.1007/s11075-007-9146-7", ISSN = "1017-1398 (print), 1572-9265 (electronic)", ISSN-L = "1017-1398", MRclass = "subject classification (2000); 33C45; 42A32; 42C10; 42C20; 65B10", bibdate = "Tue Jul 8 19:14:30 MDT 2008", bibsource = "http://springerlink.metapress.com/openurl.asp?genre=issue&issn=1017-1398&volume=47&issue=1; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.springerlink.com/openurl.asp?genre=article&issn=1017-1398&volume=47&issue=1&spage=35", acknowledgement = ack-nhfb, fjournal = "Numerical Algorithms", journal-URL = "http://link.springer.com/journal/11075", keywords = "Convergence acceleration; convergence acceleration; Orthogonal polynomials; Orthogonal series; Trigonometric series", } @Article{Pinchon:2008:NEL, author = "Didier Pinchon and Philip E. Hoggan and Frank E. Harris", title = "A new expansion of the leaky aquifer function", journal = j-IJQC, volume = "108", number = "15", pages = "3042--3046", month = "????", year = "2008", CODEN = "IJQCB2", DOI = "https://doi.org/10.1002/qua.21448; https://doi.org/10.1002/qua.21835", ISSN = "0020-7608 (print), 1097-461X (electronic)", ISSN-L = "0020-7608", MRclass = "86A05 80A20 33C10 82B80", bibdate = "Sat Oct 1 14:02:23 MDT 2011", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/h/harris-frank-e.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/ijqc2000.bib", ZMnumber = "Zbl 1189.86005", acknowledgement = ack-nhfb, ajournal = "Int. J. Quantum Chem.", fjournal = "International Journal of Quantum Chemistry", journal-URL = "http://www.interscience.wiley.com/jpages/0020-7608/", onlinedate = "4 Aug 2008", } @Article{Pineiro:2008:RDD, author = "J.-A. Pineiro and J. D. Bruguera and F. Lamberti and P. Montuschi", title = "A Radix-2 Digit-by-Digit Architecture for Cube Root", journal = j-IEEE-TRANS-COMPUT, volume = "57", number = "4", pages = "562--566", month = apr, year = "2008", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/TC.2007.70848", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Mon Jul 4 12:17:41 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", URL = "https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4407683", abstract = "A radix-2 digit-recurrence algorithm and architecture for the computation of the cube root are presented in this paper. The original recurrence based on the concept of completing the cube is modified to allow an efficient implementation of the algorithm and the cycle time and area cost of the resulting architecture are estimated as 7.5 times the delay of a full adder and around 9,000 nand2 cells, respectively, for double-precision computations.", acknowledgement = ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", } @InProceedings{Piso:2008:NRA, author = "D. Piso and J. D. Bruguera", editor = "Luca Fanucci", booktitle = "Proceedings: {11th Euromicro Symposium on Digital Systems Design: Architectures, Methods and Tools (DSD 2008), Parma, Italy, September 3--5, 2008}", title = "A New Rounding Algorithm for Variable Latency Division and Square Root Implementations", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "760--767", year = "2008", DOI = "https://doi.org/10.1109/DSD.2008.28.", ISBN = "0-7695-3277-2", ISBN-13 = "978-0-7695-3277-6", bibdate = "Sun Dec 10 13:55:38 2023", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", abstract = "The aim of this work is to present a method for rounding quadratically converging algorithms that improves their performance. This method is able to reduce significantly the number of cases where the remainder calculation is necessary. It is based on previous methods and incorporates additional bits of the result approximation to be checked. This work includes the result of exhaustive simulations that permit us to measure exactly how many calculations are avoided. Using these simulations, it is concluded that the presented method is able to reduce by half the number of remainder calculations. Using adequate result approximations the remainder calculation is necessary in only 5\% of the total cases", acknowledgement = ack-nhfb, } @Article{Rodriguez-Henriquez:2008:LCB, author = "F. Rodriguez-Henriquez and G. Morales-Luna and J. Lopez", title = "Low-Complexity Bit-Parallel Square Root Computation over {$ \mathrm {GF}(2^m) $} for All Trinomials", journal = j-IEEE-TRANS-COMPUT, volume = "57", number = "4", pages = "472--480", month = apr, year = "2008", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/TC.2007.70822", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Mon Jul 4 12:17:40 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput2000.bib", URL = "https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4358282", acknowledgement = ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", } @Article{Sablonniere:2008:BSH, author = "Paul Sablonni{\`e}re", title = "{B}-splines and {Hermite--Pad{\'e}} approximants to the exponential function", journal = j-J-COMPUT-APPL-MATH, volume = "219", number = "2", pages = "509--517", day = "1", month = oct, year = "2008", CODEN = "JCAMDI", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Sat Feb 25 13:13:26 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2005.bib", URL = "http://www.sciencedirect.com/science/article/pii/S037704270700252X", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @Article{Schreier:2008:OIR, author = "Franz Schreier and Dieter Kohlert", title = "Optimized implementations of rational approximations --- a case study on the {Voigt} and complex error function", journal = j-COMP-PHYS-COMM, volume = "179", number = "7", pages = "457--465", day = "1", month = oct, year = "2008", CODEN = "CPHCBZ", DOI = "https://doi.org/10.1016/j.cpc.2008.04.012", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", bibdate = "Mon Feb 13 23:42:36 MST 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/compphyscomm2000.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0010465508001495", acknowledgement = ack-nhfb, fjournal = "Computer Physics Communications", journal-URL = "http://www.sciencedirect.com/science/journal/00104655", } @Article{Segura:2008:IZC, author = "Javier Segura", title = "Interlacing of the zeros of contiguous hypergeometric functions", journal = j-NUMER-ALGORITHMS, volume = "49", number = "1--4", pages = "387--407", month = dec, year = "2008", CODEN = "NUALEG", ISSN = "1017-1398 (print), 1572-9265 (electronic)", ISSN-L = "1017-1398", bibdate = "Mon May 17 15:07:19 MDT 2010", bibsource = "http://springerlink.metapress.com/openurl.asp?genre=issue&issn=1017-1398&volume=49&issue=1; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/numeralgorithms.bib", URL = "http://www.springerlink.com/openurl.asp?genre=article&issn=1017-1398&volume=49&issue=1&spage=387", acknowledgement = ack-nhfb, fjournal = "Numerical Algorithms", journal-URL = "http://link.springer.com/journal/11075", } @InProceedings{Sima:2008:RAT, author = "Mihai Sima and Michael McGuire and Scott Miller", booktitle = "Proceedings of the {2005 International Conference on Engineering of Reconfigurable Systems and Algorithms (ERSA 2005)}", title = "Reconfigurable Array for Transcendental Functions Calculation", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "49--56", month = dec, year = "2008", DOI = "https://doi.org/10.1109/fpt.2008.4762365", bibdate = "Mon Nov 10 06:57:23 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, keywords = "acos; asin; cosine; exp; log; sine; sqrt", } @Misc{Steele:2008:FPSb, author = "Guy L. {Steele Jr.}", title = "Floating point square root provider with embedded status information", howpublished = "US Patent 7430576", day = "30", month = sep, year = "2008", bibdate = "Tue Dec 23 15:06:43 2008", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "http://www.patentstorm.us/patents/7430576/fulltext.html", abstract = "A system for providing a floating point square root comprises an analyzer circuit configured to determine a first status of a first floating point operand based upon data within the first floating point operand. In addition, the system comprises a results circuit coupled to the analyzer circuit. The results circuit is configured to assert a resulting floating point operand containing the square root of the first floating point operand and a resulting status embedded within the resulting floating point operand.", acknowledgement = ack-nhfb, } @Article{Vepstas:2008:EAA, author = "Linas Vepstas", title = "An efficient algorithm for accelerating the convergence of oscillatory series, useful for computing the polylogarithm and {Hurwitz} zeta functions", journal = j-NUMER-ALGORITHMS, volume = "47", number = "3", pages = "211--252", month = mar, year = "2008", CODEN = "NUALEG", DOI = "https://doi.org/10.1007/s11075-007-9153-8", ISSN = "1017-1398 (print), 1572-9265 (electronic)", ISSN-L = "1017-1398", bibdate = "Mon May 17 15:49:34 MDT 2010", bibsource = "http://springerlink.metapress.com/openurl.asp?genre=issue&issn=1017-1398&volume=47&issue=3; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.springerlink.com/openurl.asp?genre=article&issn=1017-1398&volume=47&issue=3&spage=211", abstract = "This paper sketches a technique for improving the rate of convergence of a general oscillatory sequence, and then applies this series acceleration algorithm to the polylogarithm and the Hurwitz zeta function. As such, it may be taken as an extension of the techniques given by Borwein's ``An efficient algorithm for computing the Riemann zeta function'' by Borwein for computing the Riemann zeta function, to more general series. The algorithm provides a rapid means of evaluating $ \operatorname {Li}_s(z) $ for general values of complex $s$ and a kidney-shaped region of complex $z$ values given by $ |z^2 / (z - 1)| < 4 $. By using the duplication formula and the inversion formula, the range of convergence for the polylogarithm may be extended to the entire complex $z$-plane, and so the algorithms described here allow for the evaluation of the polylogarithm for all complex $s$ and $z$ values. Alternatively, the Hurwitz zeta can be very rapidly evaluated by means of an Euler Maclaurin series. The polylogarithm and the Hurwitz zeta are related, in that two evaluations of the one can be used to obtain a value of the other; thus, either algorithm can be used to evaluate either function. The Euler Maclaurin series is a clear performance winner for the Hurwitz zeta, while the Borwein algorithm is superior for evaluating the polylogarithm in the kidney-shaped region. Both algorithms are superior to the simple Taylor's series or direct summation. The primary, concrete result of this paper is an algorithm allows the exploration of the Hurwitz zeta in the critical strip, where fast algorithms are otherwise unavailable. A discussion of the monodromy group of the polylogarithm is included.", acknowledgement = ack-nhfb, fjournal = "Numerical Algorithms", journal-URL = "http://link.springer.com/journal/11075", keywords = "convergence acceleration", } @Book{Ware:2008:RIE, author = "Willis H. Ware", title = "{RAND} and the information evolution: a history in essays and vignettes", publisher = "Rand Corporation", address = "Santa Monica, CA", pages = "xxvi + 201", year = "2008", DOI = "https://doi.org/10.7249/cp537rc", ISBN = "0-8330-4513-X, 0-8330-4816-3, 1-282-45123-5", ISBN-13 = "978-0-8330-4513-3, 978-0-8330-4816-5, 978-1-282-45123-0", LCCN = "QA76.27", bibdate = "Tue Jun 2 19:14:18 MDT 2020", bibsource = "fsz3950.oclc.org:210/WorldCat; https://www.math.utah.edu/pub/bibnet/authors/v/von-neumann-john.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/prng.bib; https://www.math.utah.edu/pub/tex/bib/unix.bib", URL = "http://www.jstor.org/stable/10.7249/cp537rc; https://www.rand.org/content/dam/rand/pubs/corporate_pubs/2008/RAND_CP537.pdf", abstract = "This professional memoir describes RAND's contributions to the evolution of computer science, particularly during the first decades following World War II, when digital computers succeeded slide rules, mechanical desk calculators, electric accounting machines, and analog computers. The memoir includes photographs and vignettes that reveal the collegial, creative, and often playful spirit in which the groundbreaking research was conducted at RAND.", acknowledgement = ack-nhfb, keywords = "JOHNNIAC; JOSS; JOSS-1; JOSS-2; RAND tablet", remark-1 = "Page 13 has a photograph of the JOHNNIAC, and on the wall of its room, a photograph of John von Neumann.", remark-2 = "From page 15: ``\ldots{} the JOHNNIAC, which nonetheless was the basis of a continuing series of engineering advances, each making important contributions to the art of the time. Among them were the first commercially produced magnetic core memory, which, for a while, was the largest in existence [4096 40-bit words]; a transistor-based adder and logic which caused the JOHNNIAC to become a hybrid transistor-vacuum tube device; the first high-speed impact printer 140 columns wide (manufactured by Anderson--Nichols, an engineering contracting firm); and the first machine with extensive trouble-diagnostic capability from the operating console.''", remark-3 = "From page 53: ``the only bright spot was the Princeton development at IAS, and thus it was that a working alliance between RAND and IAS came into being. RAND would build a machine patterned in the likeness of the Princeton one. So JOHNNIAC came from an illustrious ancestor --- the so-called von Neumann machine developed at Princeton's IAS.''", remark-4 = "Page 57 has a photograph of the JOHNNIAC's 256-word Selectron high-speed memory. Page 59, a picture of its 140-column drum printer. Page 61 has an inside view of the JOHNNIAC. Page 73 shows a step in the installation of the JOHNNIAC. Page 162 has a photograph of the JOHNNIAC console.", remark-5 = "From page 66: ``RAND purchased the first commercially available license for UNIX.''", remark-6 = "Page 84 has a photo of a young Cecil Hastings, an early pioneer of function approximation on digital computers, and a few paragraphs about his work and its influence.", remark-7 = "Pages 87--90 discuss the preparation of RAND's famous book of one million random digits, computed in Spring 1947, tested for two years after that before publication in 1955. About 7000 copies of the book were sold over three printings and fifteen years, and the book was reprinted in 1966 and 2001.", remark-8 = "From page 138: ``In the 1950s, RAND was involved in designing and building one of the first stored-program digital computers, the JOHNNIAC (named after John von Neumann, a RAND consultant in the late 1940s and early 1950s). It was in operation from 1953 to 1966, \ldots{}.''", shorttableofcontents = "Introduction \\ The department \\ RAND's first computer people \\ RAND's early computers \\ A building for people with computers \\ Project essays \\ Lore, snippets, and snapshots \\ Epilogue", tableofcontents = "Dedication / v \\ Preface / vii \\ Figures / xiii \\ Photographs / xv \\ Tables / xvii \\ Acknowledgments / xix \\ Abbreviations / xxiii \\ CHAPTER ONE \\ Introduction / 1 \\ Purpose and Scope / 1 \\ Organization of the Document / 3 \\ CHAPTER TWO \\ The Department / 5 \\ The Genesis of RAND / 5 \\ The Need for a New Kind of Organization / 6 \\ The Douglas Years / 7 \\ An Independent, Private Nonprofit Organization / 8 \\ The Nature of RAND's Contributions / 9 \\ RAND Contributions to the Development of Computing / 10 \\ In the Beginning / 10 \\ An Early Computing Success / 11 \\ The Move to Electronic Machines / 11 \\ The Middle Years / 14 \\ The JOHNNIAC Open-Shop System / 15 \\ The Tablet / 16 \\ Videographic System / 16 \\ The Later Years / 17 \\ RAND and the USAF Computing Evolution / 18 \\ The Bottom Line / 19 \\ CHAPTER THREE \\ RAND's First Computer People / 21 \\ The Legacy of Wartime Collaboration / 21 \\ Early RAND Leaders / 22 \\ Early Technical Staff / 24 \\ The Douglas Thread / 24 \\ The Wartime Thread / 26 \\ The University Thread / 28 \\ The Recruiting Thread / 30 \\ Departmental Growth / 36 \\ CHAPTER FOUR \\ RAND's Early Computers / 45 \\ Mid-20th Century Computation / 45 \\ Reeves Electronic Analog Computer / 47 \\ Plug-Board Interconnections / 50 \\ Chopper-Stabilized Amplifiers / 50 \\ Arbitrary Function Input / 51 \\ The JOHNNIAC Digital Computer / 53 \\ JOHNNIAC's ``Obituary'' / 63 \\ IBM Mainframes / 64 \\ Other Machinery. / 66 \\ CHAPTER FIVE \\ A Building for People with Computers / 67 \\ A New Building and Campus. / 68 \\ The Machine Room. / 72 \\ Two-Story Installation / 72 \\ REAC Installation. / 73 \\ Raised-Floor Installation / 73 \\ Air Conditioning. / 74 \\ Configurations of the Machine Room / 75 \\ Open House. / 75 \\ Later Enhancements / 79 \\ The Camera / 79 \\ Kevershan's Trough / 80 \\ Programmer-Alert Lights / 80 \\ CHAPTER SIX \\ Project Essays / 83 \\ Approximations / 83 \\ Random Digits and Normal Deviates / 87 \\ The Bombing Simulator (aka Pinball Machine) / 90 \\ The Air-Combat Room / 94 \\ System Research Laboratory / 94 \\ The RAND Tablet, Videographics, and Related Projects / 98 \\ The RAND Tablet / 98 \\ Handwriting Recognition / 99 \\ Chinese-Character Lookup / 100 \\ Map Annotation / 100 \\ Videographic System / 103 \\ GRAIL / 105 \\ BIOMOD / 105 \\ CLINFO / 107 \\ Time-Shared Computing: JOSS / 109 \\ JOSS-1 / 110 \\ JOSS-2 / 113 \\ Networked Computing: Packet Switching and Distributed Communications / 115 \\ The Beginnings of Packet Switching: Some Underlying Concepts / 116 \\ Text Editors (NED and e) / 122 \\ Word Processing / 126 \\ The Mail Handler / 128 \\ The Original MH-Proposal Memorandum / 129 \\ Implementation / 132 \\ Another Perspective / 134 \\ A User's Perspective / 135 \\ The Developers' Present Views / 137 \\ Artificial-Intelligence Research / 138 \\ The Beginnings of Artificial Intelligence / 138 \\ Newell, Shaw, and Simon: The Development of List-Processing Languages / 138 \\ Expert Systems / 140 \\ Knowledge-Based Simulation / 142 \\ Computational Linguistics / 143 \\ The Perfect Buddy / 144 \\ Department of Defense Computer Institute / 147 \\ Officer Career Paths / 149 \\ Software / 150 \\ Security and Privacy / 152 \\ Security / 152 \\ Privacy / 154 \\ Fair Information Practices / 155 \\ CHAPTER SEVEN \\ Lore, Snippets, and Snapshots / 159 \\ The Great Machine Fire / 159 \\ The Gavel Caper / 159 \\ Department-Head-Office Decor / 161 \\ Oliver Alfred Gross and JOSS-1 / 162 \\ The Soviet ``Threat'' / 163 \\ Social Events / 164 \\ The One-Way Wire / 166 \\ Soviet Cybernetics / 166 \\ Inter/Exhume / 167 \\ The RAND Computer Symposia / 168 \\ Professional Societies / 169 \\ Microvignettes / 170 \\ The Marchant March / 170 \\ Getting Out the Documents / 171 \\ Hero of the Week / 171 \\ The Chiquita Banana War / 171 \\ The Mengel Joint / 171 \\ John Williams' Jaguar / 172 \\ Programmer Sweepstakes / 173 \\ CHAPTER EIGHT \\ Epilogue / 175 \\ Bibliography / 177 \\ Index / 191", } @Article{Zhu:2008:SNR, author = "Ling Zhu and Jinju Sun", title = "Six new {Redheffer}-type inequalities for circular and hyperbolic functions", journal = j-COMPUT-MATH-APPL, volume = "56", number = "2", pages = "522--529", month = jul, year = "2008", CODEN = "CMAPDK", ISSN = "0898-1221 (print), 1873-7668 (electronic)", ISSN-L = "0898-1221", bibdate = "Wed Mar 1 21:50:15 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/computmathappl2000.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0898122108000813", acknowledgement = ack-nhfb, fjournal = "Computers and Mathematics with Applications", journal-URL = "http://www.sciencedirect.com/science/journal/08981221", } @Article{Zou:2008:FIE, author = "X. Zou", title = "{FPGA} Implementation of Exponent Function Based on {CORDIC}", journal = "Public Technology", volume = "10", number = "??", pages = "36--37", month = "", year = "2008", DOI = "", bibdate = "Tue Nov 11 20:12:21 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "", acknowledgement = ack-nhfb, } @Article{Anand:2009:OCS, author = "C. K. Anand and W. Kahl", title = "An Optimized {Cell BE} Special Function Library Generated by {Coconut}", journal = j-IEEE-TRANS-COMPUT, volume = "58", number = "8", pages = "1126--1138", month = aug, year = "2009", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/TC.2008.223", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Mon Jul 4 11:37:43 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", URL = "https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4731241", abstract = "Coconut, a tool for developing high-assurance, high-performance kernels for scientific computing, contains an extensible domain-specific language (DSL) embedded in Haskell. The DSL supports interactive prototyping and unit testing, simplifying the process of designing efficient implementations of common patterns. Unscheduled C and scheduled assembly language output are supported. Using the patterns, even nonexpert users can write efficient function implementations, leveraging special hardware features. A production-quality library of elementary functions for the cell BE SPU compute engines has been developed. Coconut-generated and -scheduled vector functions were more than four times faster than commercially distributed functions written in C with intrinsics (a nicer syntax for in-line assembly), wrapped in loops and scheduled by {\tt spuxlc}. All Coconut functions were faster, but the difference was larger for hard-to-approximate functions for which register-level SIMD lookups made a bigger difference. Other helpful features in the language include facilities for translating interval and polynomial descriptions between GHCi, a Haskell interpreter used to prototype in the DSL, and Maple, used for exploration and minimax polynomial generation. This makes it easier to match mathematical properties of the functions with efficient calculational patterns in the SPU ISA. By using single, literate source files, the resulting functions are remarkably readable.", acknowledgement = ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", } @Article{Backeljauw:2009:ACF, author = "Franky Backeljauw and Annie Cuyt", title = "{Algorithm 895}: a continued fractions package for special functions", journal = j-TOMS, volume = "36", number = "3", pages = "15:1--15:20", month = jul, year = "2009", CODEN = "ACMSCU", DOI = "https://doi.org/10.1145/1527286.1527289", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Tue Jul 21 14:09:07 MDT 2009", bibsource = "http://www.acm.org/pubs/contents/journals/toms/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib", abstract = "The continued fractions for special functions package (in the sequel abbreviated as CFSF package) complements a systematic study of continued fraction representations for special functions. It provides all the functionality to create continued fractions, in particular $k$-periodic or limit $k$-periodic fractions, to compute approximants, make use of continued fraction tails, perform equivalence transformations and contractions, and much more. The package, developed in Maple, includes a library of more than 200 representations of special functions, of which only 10\% can be found in the 1964 NBS {\em Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables\/} by M. Abramowitz and I. Stegun.", acknowledgement = ack-nhfb, articleno = "15", fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", keywords = "CAS software; continued fractions; Maple; special functions", } @Article{Blomquist:2009:MSC, author = "Frithjof Blomquist and Werner Hofschuster and Walter Kr{\"a}mer", title = "A Modified Staggered Correction Arithmetic with Enhanced Accuracy and Very Wide Exponent Range", journal = j-LECT-NOTES-COMP-SCI, volume = "5492", pages = "41--67", year = "2009", CODEN = "LNCSD9", DOI = "https://doi.org/10.1007/978-3-642-01591-5_4", ISSN = "0302-9743 (print), 1611-3349 (electronic)", ISSN-L = "0302-9743", bibdate = "Tue Apr 10 08:32:19 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "http://www.springerlink.com/content/k038294004403504/", acknowledgement = ack-nhfb, author-dates = "1952--2014 (WK)", fjournal = "Lecture Notes in Computer Science", journal-URL = "http://link.springer.com/bookseries/558", keywords = "C-XSC; complex interval functions; interval computation; multiple precision; reliable numerical computations; staggered correction; wide exponent range", remark = "Conference on Numerical Validation in Current Hardware Architectures", remark-2 = "Includes algorithms for division, $\exp(x)$, $(1 + x)^n$, $\log(x)$, $\log(1 + x)$, and $\sqrt{x}$. Staggered arithmetic represents numbers with tuples $(e, x_1, x_2, \ldots{}, x_n)$ where $e$ is either integer or a floating-point whole number, the $x_k$ are floating-point, and a number has the value $2^e \sum_{k = 1}^n x_k$. For interval arithmetic, the last element is a pair of lower and upper bounds.", } @Article{Boldo:2009:FVA, author = "S. Boldo and M. Daumas and Ren-Cang Li", title = "Formally Verified Argument Reduction with a Fused Multiply-Add", journal = j-IEEE-TRANS-COMPUT, volume = "58", number = "8", pages = "1139--1145", month = aug, year = "2009", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/TC.2008.216", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Mon Jul 4 11:37:43 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", URL = "https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4711042", abstract = "The Cody and Waite argument reduction technique works perfectly for reasonably large arguments, but as the input grows, there are no bits left to approximate the constant with enough accuracy. Under mild assumptions, we show that the result computed with a fused multiply-add provides a fully accurate result for many possible values of the input with a constant almost accurate to the full working precision. We also present an algorithm for a fully accurate second reduction step to reach full double accuracy (all the significand bits of two numbers are accurate) even in the worst cases of argument reduction. Our work recalls the common algorithms and presents proofs of correctness. All the proofs are formally verified using the Coq automatic proof checker.", acknowledgement = ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", } @Article{Bowling:2009:LAC, author = "Shannon R. Bowling and Mohammad T. Khasawneh and Sittichai Kaewkuekool and Byung Rae Cho", title = "A logistic approximation to the cumulative normal distribution", journal = "Journal of Industrial Engineering and Management", volume = "2", number = "1", pages = "114--127", month = "", year = "2009", DOI = "https://doi.org/10.3926/jiem..v2n1.p114-127", bibdate = "Sat Dec 16 15:22:05 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.jiem.org/index.php/jiem/article/view/60", acknowledgement = ack-nhfb, ajournal = "J. Ind. Eng. Manage.", fjournal = "Journal of Industrial Engineering and Management", journal-URL = "http://www.jiem.org/index.php/jiem/", } @Article{Boyd:2009:AAC, author = "John P. Boyd", title = "Acceleration of algebraically-converging {Fourier} series when the coefficients have series in powers of $ 1 / n $", journal = j-J-COMPUT-PHYS, volume = "228", number = "5", pages = "1404--1411", day = "20", month = mar, year = "2009", CODEN = "JCTPAH", DOI = "https://doi.org/10.1016/j.jcp.2008.10.039", ISSN = "0021-9991 (print), 1090-2716 (electronic)", ISSN-L = "0021-9991", bibdate = "Thu Dec 01 10:35:35 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputphys2000.bib", acknowledgement = ack-nhfb, fjournal = "Journal of Computational Physics", journal-URL = "http://www.sciencedirect.com/science/journal/00219991/", keywords = "Bernoulli polynomials; Clausen functions; convergence acceleration; Lanczos--Krylov (LK) functions", } @Article{Bunck:2009:FAE, author = "Benjamin F. Bunck", title = "A fast algorithm for evaluation of normalized {Hermite} functions", journal = j-BIT-NUM-MATH, volume = "49", number = "2", pages = "281--295", month = jun, year = "2009", CODEN = "BITTEL, NBITAB", ISSN = "0006-3835 (print), 1572-9125 (electronic)", ISSN-L = "0006-3835", bibdate = "Mon May 24 15:36:43 MDT 2010", bibsource = "http://springerlink.metapress.com/openurl.asp?genre=issue&issn=0006-3835&volume=49&issue=2; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.springerlink.com/openurl.asp?genre=article&issn=0006-3835&volume=49&issue=2&spage=281", abstract = "An algorithm for computing the normalized Hermite Functions, $ h_n(x) $, in floating point arithmetic is presented. The algorithm is based on an efficient numerical evaluation of certain closed contour integrals in the complex plane. For large degree $n$, the algorithm is significantly faster than the $ O(n) $ complexity of the well known three-term recurrence relation. Comparable accuracy is achieved in no more $ O(\sqrt {n}) $ than operations, and for arguments bounded away from $ \pm \sqrt {2n} $, only $ O(\sqrt {\log n}) $ operations.", acknowledgement = ack-nhfb, fjournal = "BIT. Numerical Mathematics", journal-URL = "http://link.springer.com/journal/10543", keywords = "fast algorithm; Hermite functions; numerical integration; recursion", } @Article{Chen:2009:SPA, author = "Yunfei Chen and Norman C. Beaulieu", title = "A simple polynomial approximation to the {Gaussian} {$Q$}-function and its application", journal = j-IEEE-COMMUN-LET, volume = "13", number = "2", pages = "124--126", month = feb, year = "2009", CODEN = "ICLEF6", DOI = "https://doi.org/10.1109/lcomm.2009.081754", ISSN = "1089-7798 (print), 1558-2558 (electronic)", ISSN-L = "1089-7798", bibdate = "Sat Dec 16 15:46:17 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "https://ieeexplore.ieee.org/document/4783779/", acknowledgement = ack-nhfb, fjournal = "IEEE Communications Letters", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=4234", } @InProceedings{Chevillard:2009:CFC, author = "Sylvain Chevillard and Mioara Joldes and Christoph Lauter", title = "Certified and Fast Computation of Supremum Norms of Approximation Errors", crossref = "Bruguera:2009:PIS", pages = "169--176", year = "2009", bibdate = "Fri Jun 12 12:34:25 2009", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", abstract = "In many numerical programs there is a need for a high-quality floating-point approximation of useful functions $f$, such as such as $ \exp $, $ \sin $, $ \erf $. In the actual implementation, the function is replaced by a polynomial $p$, which leads to an approximation error (absolute or relative) $ \epsilon = p - f $ or $ \epsilon = p / f - 1 $. The tight yet certain bounding of this error is an important step towards safe implementations. The problem is difficult mainly because that approximation error is very small and the difference $ p - f $ is subject to high cancellation. Previous approaches for computing the supremum norm in this degenerate case, have proven to be unsafe, not sufficiently tight or too tedious in manual work. We present a safe and fast algorithm that computes a tight lower and upper bound for the supremum norms of approximation errors. The algorithm is based on a combination of several techniques, including enhanced interval arithmetic, automatic differentiation and isolation of the roots of a polynomial. We have implemented our algorithm and give timings on several examples.", acknowledgement = ack-nhfb, keywords = "approximation error; ARITH-19; automatic/algorithmic differentiation; certified computation; elementary function; interval arithmetic; roots isolation technique.; supremum/infinity norm", } @TechReport{Chevillard:2009:FEE, author = "S. Chevillard", title = "The functions {ERF} and {ERFC} computed with arbitrary precision", type = "Report", number = "RRLIP2009-04", institution = "HAL", address = "????", pages = "32", year = "2009", bibdate = "Mon Jun 12 16:09:53 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, } @Article{Daumas:2009:VRN, author = "M. Daumas and D. Lester and C. Muoz", title = "Verified Real Number Calculations: a Library for Interval Arithmetic", journal = j-IEEE-TRANS-COMPUT, volume = "58", number = "2", pages = "226--237", month = feb, year = "2009", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/TC.2008.213", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Fri Jun 12 08:51:00 MDT 2009", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", abstract = "Real number calculations on elementary functions are remarkably difficult to handle in mechanical proofs. In this paper, we show how these calculations can be performed within a theorem prover or proof assistant in a convenient and highly automated as well as interactive way. First, we formally establish upper and lower bounds for elementary functions. Then, based on these bounds, we develop a rational interval arithmetic where real number calculations take place in an algebraic setting. In order to reduce the dependency effect of interval arithmetic, we integrate two techniques: interval splitting and Taylor series expansions. This pragmatic approach has been developed, and formally verified, in a theorem prover. The formal development also includes a set of customizable strategies to automate proofs involving explicit calculations over real numbers. Our ultimate goal is to provide guaranteed proofs of numerical properties with minimal human theorem-prover interaction.", acknowledgement = ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", keywords = "interval arithmetic; proof checking; real number calculations; theorem proving", remark = "Extended version of ARITH-18 article \cite{Daumas:2009:VRN}.", } @Article{Deano:2009:MAS, author = "Alfredo Dea{\~n}o and Nico M. Temme", title = "On modified asymptotic series involving confluent hypergeometric functions", journal = j-ELECTRON-TRANS-NUMER-ANAL, volume = "35", pages = "88--103", year = "2009", CODEN = "????", ISSN = "1068-9613 (print), 1097-4067 (electronic)", ISSN-L = "1068-9613", bibdate = "Mon Sep 6 12:28:30 MDT 2010", bibsource = "http://etna.mcs.kent.edu/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://etna.mcs.kent.edu/vol.35.2009/pp88-103.dir/pp88-103.pdf", acknowledgement = ack-nhfb, fjournal = "Electronic Transactions on Numerical Analysis", journal-URL = "http://etna.mcs.kent.edu/", } @Article{FreitasDeAbreu:2009:JCU, author = "Giuseppe Thadeu {Freitas De Abreu}", title = "{Jensen--Cotes} upper and lower bounds on the {Gaussian} {$Q$}-function and related functions", journal = j-IEEE-TRANS-COMM, volume = "57", number = "11", pages = "3328--3338", month = nov, year = "2009", CODEN = "IECMBT", DOI = "https://doi.org/10.1109/tcomm.2009.11.080479", ISSN = "0090-6778 (print), 1558-0857 (electronic)", ISSN-L = "0090-6778", bibdate = "Sat Dec 16 15:12:46 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "IEEE Transactions on Communications", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=26", } @Article{Fukushima:2009:FCC, author = "Toshio Fukushima", title = "Fast computation of complete elliptic integrals and {Jacobian} elliptic functions", journal = j-CELEST-MECH-DYN-ASTR, volume = "105", number = "4", pages = "305--328", month = dec, year = "2009", CODEN = "CLMCAV", DOI = "https://doi.org/10.1007/s10569-009-9228-z", ISSN = "0923-2958 (print), 1572-9478 (electronic)", ISSN-L = "0923-2958", MRclass = "33E05 (33F05 65E05)", MRnumber = "2559416", MRreviewer = "Mehdi Hassani", bibdate = "Wed Oct 20 21:29:31 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.springerlink.com/content/0923-2958/", abstract = "As a preparation step to compute Jacobian elliptic functions efficiently, we created a fast method to calculate the complete elliptic integral of the first and second kinds, $ K(m) $ and $ E(m) $, for the standard domain of the elliptic parameter, $ 0 < m < 1 $. For the case $ 0 < m < 0.9 $, the method utilizes $ 10 $ pairs of approximate polynomials of the order of $9$--$ 19 $ obtained by truncating Taylor series expansions of the integrals. Otherwise, the associate integrals, $ K(1 - m) $ and $ E(1 - m) $, are first computed by a pair of the approximate polynomials and then transformed to $ K(m) $ and $ E(m) $ by means of Jacobi's nome, $q$, and Legendre's identity relation. In average, the new method runs more-than-twice faster than the existing methods including Cody's Chebyshev polynomial approximation of Hastings type and Innes' formulation based on $q$-series expansions. Next, we invented a fast procedure to compute simultaneously three Jacobian elliptic functions, {\tt sn(u|m)}, {\tt cn(u|m)}, and {\tt dn(u|m)}, by repeated usage of the double argument formulae starting from the Maclaurin series expansions with respect to the elliptic argument, $u$, after its domain is reduced to the standard range, $ 0 \leq u < K(m) / 4 $, with the help of the new method to compute K(m). The new procedure is 25--70\% faster than the methods based on the Gauss transformation such as Bulirsch's algorithm, sncndn, quoted in the Numerical Recipes even if the acceleration of computation of $ K(m) $ is not taken into account.", acknowledgement = ack-nhfb, fjournal = "Celestial Mechanics \& Dynamical Astronomy. An International Journal of Space Dynamics", keywords = "complete elliptic integrals; Encke's method; Innes' method; Jacobian elliptic functions; nome expansion; numerical methods", } @Article{Fukushima:2009:FCJ, author = "Toshio Fukushima", title = "Fast computation of {Jacobian} elliptic functions and incomplete elliptic integrals for constant values of elliptic parameter and elliptic characteristic", journal = j-CELEST-MECH-DYN-ASTR, volume = "105", number = "1--3", pages = "245--260", year = "2009", CODEN = "CLMCAV", DOI = "https://doi.org/10.1007/s10569-008-9177-y", ISSN = "0923-2958 (print), 1572-9478 (electronic)", ISSN-L = "0923-2958", MRclass = "33E05 (33F05 65D20 70M20)", MRnumber = "2551836", bibdate = "Mon Oct 24 11:37:20 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.springerlink.com/content/0923-2958/", abstract = "In order to accelerate the numerical evaluation of torque-free rotation of triaxial rigid bodies, we present a fast method to compute various kinds of elliptic functions for a series of the elliptic argument when the elliptic parameter and the elliptic characteristic are fixed. The functions we evaluate are the Jacobian elliptic functions and the incomplete elliptic integral of the second and third kinds regarded as a function of that of the first kind. The key technique is the utilization of the Maclaurin series expansion and the addition theorems with respect to the elliptic argument. The new method is around 25 times faster than the method using the incomplete elliptic integral of general kind and around 70 times faster than the method using mathematical libraries given in the latest version of Numerical Recipes.", acknowledgement = ack-nhfb, fjournal = "Celestial Mechanics \& Dynamical Astronomy. An International Journal of Space Dynamics", keywords = "elliptic integrals; extended body dynamics; Jacobian elliptic functions; numerical method; rotation", } @Article{Guo:2009:CLC, author = "Senlin Guo and Feng Qi", title = "A class of logarithmically completely monotonic functions associated with the gamma function", journal = j-J-COMPUT-APPL-MATH, volume = "224", number = "1", pages = "127--132", day = "1", month = feb, year = "2009", CODEN = "JCAMDI", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Sat Feb 25 13:13:29 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2005.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0377042708001829", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @InProceedings{Guralnik:2009:ISV, author = "Elena Guralnik and Ariel J. Birnbaum and Anatoly Koyfman and Avi Kaplan", title = "Implementation Specific Verification of Divide and Square Root Instructions", crossref = "Bruguera:2009:PIS", pages = "114--121", year = "2009", bibdate = "Fri Jun 12 12:34:25 2009", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", abstract = "Floating point operations such as divide and square root are typically implemented in microcode rather than dedicated logic. Bugs in these operations missed by generic black-box verification tools, were analyzed. This led to the conclusion that the corner cases, in addition to being implementation dependent, could not be characterized in terms of special input or output values in a straightforward manner.\par However, many of those cases can be easily generalized for many known implementations. The typical implementation uses a known iterative approximation algorithm, such as the Newton--Raphson method, to calculate the desired result; thus, it is sufficient to produce the corner cases associated with the specific algorithm.\par We investigated the following problem: given an iterative algorithm to compute a binary floating point operation, the iteration number, and an interval, find random inputs for the operation that, after the requested iteration, yield a relative error within the specified interval. This paper describes a method to solve this problem. This method was implemented in a floating-point test generator and is currently being used to verify the floating-point units of several processors.", acknowledgement = ack-nhfb, keywords = "ARITH-19", } @Article{Han:2009:ICS, author = "Dong-Guk Han and Dooho Choi and Howon Kim", title = "Improved Computation of Square Roots in Specific Finite Fields", journal = j-IEEE-TRANS-COMPUT, volume = "58", number = "2", pages = "188--196", month = feb, year = "2009", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/TC.2008.201", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Mon Jul 4 11:37:39 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/cryptography2000.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput2000.bib", URL = "https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4663058", abstract = "In this paper, we study exponentiation in the specific finite fields $ F_q $ with very special exponents such as those that occur in algorithms for computing square roots. Here, $q$ is a prime power, $ q = p^k $, where $ k > 1 $, and $k$ is odd. Our algorithmic approach improves the corresponding exponentiation resulted from the better rewritten exponent. To the best of our knowledge, it is the first major improvement to the Tonelli--Shanks algorithm, for example, the number of multiplications can be reduced to at least 60 percent on the average when $ p \equiv 1 \pmod 16 $. Several numerical examples are given that show the speedup of the proposed methods.", acknowledgement = ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", keywords = "cryptography; efficient computation; finite fields; square roots", } @Article{Harris:2009:MIB, author = "Frank E. Harris and J. G. Fripiat", title = "Methods for incomplete {Bessel} function evaluation", journal = j-IJQC, volume = "109", number = "8", pages = "1728--1740", month = feb, year = "2009", CODEN = "IJQCB2", DOI = "https://doi.org/10.1002/qua.21972", ISSN = "0020-7608 (print), 1097-461X (electronic)", ISSN-L = "0020-7608", bibdate = "Fri Mar 27 07:41:18 2009", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", abstract = "Presented here are detailed methods for evaluating the incomplete Bessel functions arising when Gaussian-type orbitals are used for systems periodic in one spatial dimension. The scheme is designed to yield these incomplete Bessel functions with an absolute accuracy of $ \pm 1 \times 10^{-10} $, for the range of integer orders $ 0 \leq n \leq 12 $ [a range sufficient for a basis whose members have angular momenta of up to three units ($s$, $p$, $d$, or $f$ atomic functions)]. To reach this accuracy level within acceptable computation times, new rational approximations were developed to compute the special functions involved, namely, the exponential integral $ E_1 (x) $ and the modified Bessel functions $ K_0 (x) $ and $ K_1 (x) $, to absolute accuracy $ \pm 1 \times 10^{-15} $.", acknowledgement = ack-nhfb, fjournal = "International Journal of Quantum Chemistry", journal-URL = "http://www.interscience.wiley.com/jpages/0020-7608/", keywords = "E1(x); exponential integral; incomplete Bessel function; K0(x); K1(x); leaky aquifer function; modified Bessel function; numerical methods", } @InProceedings{Harrison:2009:DTB, author = "John Harrison", title = "Decimal Transcendentals via Binary", crossref = "Bruguera:2009:PIS", pages = "187--194", year = "2009", DOI = "https://doi.org/10.1109/ARITH.2009.32", bibdate = "Fri Jun 12 12:34:25 2009", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", abstract = "We describe the design and implementation of a comprehensive library of transcendental functions for the new IEEE decimal floating-point formats. In principle, such functions are very much analogous to their binary counterparts, though with a few additional subtleties connected with `scale' (preferred exponent). But our approach has been not to employ direct techniques, but rather to re-use existing binary functions as much as possible, both for greater efficiency and ease of implementation. For some functions the most straightforward approach (convert from decimal to binary, perform binary operation, convert back) works well. In many cases, however, these are insufficiently accurate, and subtler approaches must be used.", acknowledgement = ack-nhfb, keywords = "ARITH-19", } @InProceedings{Harrison:2009:FAB, author = "John Harrison", title = "Fast and Accurate {Bessel} Function Computation", crossref = "Bruguera:2009:PIS", pages = "104--113", year = "2009", bibdate = "Fri Jun 12 12:34:25 2009", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", abstract = "The Bessel functions are considered relatively difficult to compute. Although they have a simple power series expansion that is everywhere convergent, they exhibit approximately periodic behavior which makes the direct use of the power series impractically slow and numerically unstable. We describe an alternative method based on systematic expansion around the zeros, refining existing techniques based on Hankel expansions, which mostly avoids the use of multiprecision arithmetic while yielding accurate results.", acknowledgement = ack-nhfb, keywords = "$J0(x), J1(1), Y0(x), Y1(1)$; ARITH-19; ordinary Bessel functions of the first and second kinds", } @Book{Henner:2009:MMP, author = "Victor Henner and Tatyana Belozerova and Kyle Forinash", title = "Mathematical Methods in Physics: Partial Differential Equations, {Fourier} Series, and Special Functions", publisher = pub-A-K-PETERS, address = pub-A-K-PETERS:adr, pages = "xviii + 841", year = "2009", DOI = "https://doi.org/10.1201/b10695", ISBN = "1-56881-335-X", ISBN-13 = "978-1-56881-335-6", LCCN = "QC20 .H487 2009", bibdate = "Sat Oct 30 17:39:29 MDT 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcw.bib; z3950.loc.gov:7090/Voyager", acknowledgement = ack-nhfb, subject = "Mathematical physics; Textbooks", tableofcontents = "1: Fourier Series \\ 2: Sturm--Liouville Theory \\ 3: One-Dimensional Hyperbolic Equations \\ 4: Two-Dimensional Hyperbolic Equations \\ 5: One-Dimensional Parabolic Equations \\ 6: Parabolic Equations for Higher-Dimensional Problems \\ 7: Elliptic Equations \\ 8: Bessel Functions \\ 9: Legendre Functions \\ A: Eigenvalues and Eigen functions of the Sturm--Liouville Problem \\ B: Auxiliary Functions for Different Types of Boundary Conditions \\ C: The Sturm--Liouville Problem and the Laplace Equation \\ D: Vector Calculus \\ E: How to Use the Software Associated with this Book", } @Article{Lauter:2009:ERB, author = "C. Q. Lauter and V. Lefevre", title = "An Efficient Rounding Boundary Test for {\tt pow(x, y)} in Double Precision", journal = j-IEEE-TRANS-COMPUT, volume = "58", number = "2", pages = "197--207", month = feb, year = "2009", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/TC.2008.202", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Fri Jun 12 08:51:00 MDT 2009", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", abstract = "The correct rounding of the function $ \textrm {pow} : (x, y) \rightarrow x^y $ is currently based on Ziv's iterative approximation process. In order to ensure its termination, cases when $ x^y $ falls on a rounding-boundary must be filtered out. Such rounding-boundaries are floating-point numbers and midpoints between two consecutive floating-point numbers. Detecting rounding-boundaries for pow is a difficult problem. Previous approaches use repeated square root extraction followed by repeated square and multiply. This paper presents a new rounding-boundary test for pow in double precision, which reduces this to a few comparisons with precomputed constants. These constants are deduced from worst cases for the Table Maker's Dilemma, searched over a small subset of the input domain. This is a novel use of such worst-case bounds. The resulting algorithm has been designed for a fast-on-average correctly rounded implementation of pow, considering the scarcity of rounding-boundary cases. It does not stall average computations for rounding-boundary detection. This paper includes its correctness proof and experimental results.", acknowledgement = ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", keywords = "correct rounding; floating-point arithmetic; power function.", } @Article{Linhart:2009:ACL, author = "Jean Marie Linhart", title = "{Algorithm 885}: Computing the Logarithm of the Normal Distribution", journal = j-TOMS, volume = "35", number = "3", pages = "20:1--20:10", month = oct, year = "2009", CODEN = "ACMSCU", DOI = "https://doi.org/10.1145/1391989.1391993", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Sat Nov 1 19:57:00 MDT 2008", bibsource = "http://www.acm.org/pubs/contents/journals/toms/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib", abstract = "We present and compare three C functions to compute the logarithm of the cumulative standard normal distribution. The first is a new algorithm derived from Algorithm 304's calculation of the standard normal distribution via a series or continued fraction approximation, and it is good to the accuracy of the machine. The second is based on Algorithm 715's calculation of the standard normal distribution via rational Chebyshev approximation. This is related to, and an improvement on, the algorithm for the logarithm of the normal distribution available in the software package R. The third is a new and simple algorithm that uses the compiler's implementation of the error function, and complement of the error function, to compute the log of the normal distribution.", acknowledgement = ack-nhfb, articleno = "20", fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", keywords = "error function; logarithm of the standard normal distribution; Normal distribution; normal integral", } @Article{Loskot:2009:PPA, author = "P. Loskot and N. C. Beaulieu", title = "{Prony} and Polynomial Approximations for Evaluation of the Average Probability of Error Over Slow-Fading Channels", journal = j-IEEE-TRANS-VEH-TECHNOL, volume = "58", number = "3", pages = "1269--1280", month = mar, year = "2009", CODEN = "ITUTAB", DOI = "https://doi.org/10.1109/tvt.2008.926072", ISSN = "0018-9545 (print), 1939-9359 (electronic)", ISSN-L = "0018-9545", bibdate = "Sat Dec 16 18:08:41 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "https://ieeexplore.ieee.org/document/4529094/", acknowledgement = ack-nhfb, fjournal = "IEEE Transactions on Vehicular Technology", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=25", } @Article{Nagayama:2009:CGB, author = "S. Nagayama and T. Sasao", title = "Complexities of Graph-Based Representations for Elementary Functions", journal = j-IEEE-TRANS-COMPUT, volume = "58", number = "1", pages = "106--119", month = jan, year = "2009", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/TC.2008.134", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Mon Jul 4 11:37:39 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", URL = "https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4599569", abstract = "This paper analyzes complexities of decision diagrams for elementary functions such as polynomial, trigonometric, logarithmic, square root, and reciprocal functions. These real functions are converted into integer-valued functions by using fixed-point representation. This paper presents the numbers of nodes in decision diagrams representing the integer-valued functions. First, complexities of decision diagrams for polynomial functions are analyzed, since elementary functions can be approximated by polynomial functions. A theoretical analysis shows that binary moment diagrams (BMDs) have low complexity for polynomial functions. Second, this paper analyzes complexity of edge-valued binary decision diagrams (EVBDDs) for monotone functions, since many common elementary functions are monotone. It introduces a new class of integer functions, Mp-monotone increasing function, and derives an upper bound on the number of nodes in an EVBDD for the Mp-monotone increasing function. A theoretical analysis shows that EVBDDs have low complexity for Mp-monotone increasing functions. This paper also presents the exact number of nodes in the smallest EVBDD for the n-bit multiplier function, and a variable order for the smallest EVBDD.", acknowledgement = ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", keywords = "binary moment diagram; decision diagrams; edge-valued binary decision diagram; elementary function; elementary function approximation; fixed-point representation; general representations; graph-based representation; integer-valued function; monotone function; polynomial function; trees", } @Book{Oldham:2009:AF, editor = "Keith B. Oldham and Jan Myland and Jerome Spanier", title = "An Atlas of Functions: With Equator, the Atlas Function Calculator", publisher = pub-SV, address = pub-SV:adr, edition = "Second", pages = "xi + 748", year = "2009", DOI = "https://doi.org/10.1007/978-0-387-48807-3", ISBN = "0-387-48807-3 (softcover), 0-387-48806-5 (hardcover)", ISBN-13 = "978-0-387-48807-3 (softcover), 978-0-387-48806-6 (hardcover)", LCCN = "QA331 .S685 2009", bibdate = "Fri Aug 31 16:20:13 MDT 2007", bibsource = "https://www.math.utah.edu/pub/tex/bib/canjstat.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; z3950.loc.gov:7090/Voyager", price = "US\$129.95", acknowledgement = ack-nhfb, subject = "Functions", tableofcontents = "Front Matter / i--xi \\ General Considerations / 1--11 \\ The Constant Function $c$ / 13--19 \\ The Factorial Function $n!$ / 21--27 \\ The Zeta Numbers and Related Functions / 29--38 \\ The Bernoulli Numbers $B_n$ / 39--44 \\ The Euler Numbers $E_n$ / 45--48 \\ The Binomial Coefficients $\binom{\nu}{m}$ / 49--56 \\ The Linear Function $b x + c$ and Its Reciprocal / 57--65 \\ Modifying Functions / 67--74 \\ The Heaviside $u(x - a)$ And Dirac $\delta(x - a)$ Functions / 75--80 \\ The Integer Powers $x^n$ and $(b x + c)^n$ / 81--93 \\ The Square-Root Function $\sqrt{b x + c}$ and Its Reciprocal / 95--102 \\ The Noninteger Powers $x^\nu$ / 103--112 \\ The Semielliptic Function $(b / a)\sqrt{a^2 - x^2}$ and Its Reciprocal / 113--120 \\ The Semihyperbolic Functions $(b / a) \sqrt{x^2 \pm a^2}$ and Their Reciprocals / 121--130 \\ The Quadratic Function $a x^2 + b x + c$ and Its Reciprocal / 131--138 \\ The Cubic Function $x^3 + a x^2 + b x + c$ / 139--146 \\ Polynomial Functions / 147--158 \\ The Pochhammer Polynomials $(x)_n$ / 159--174 \\ The Bernoulli Polynomials $B_n(x)$ / 175--180 \\ The Euler Polynomials $E_n(x)$ / 181--186 \\ The Legendre Polynomials $P_n(x)$ / 187--196 \\ The Chebyshev Polynomials $T_n(x)$ and $U_n(x)$ / 197--208 \\ The Laguerre Polynomials $L_n(x)$ / 209--216 \\ The Hermite Polynomials $H_n(x)$ / 217--227 \\ The Logarithmic Function $\ln(x)$ / 229--239 \\ The Exponential Function $\exp(\pm x)$ / 241--253 \\ Exponentials of Powers $\exp(\pm x^\nu)$ / 255--267 \\ The Hyperbolic Cosine $\cosh(x)$ and Sine $\sinh(x)$ Functions / 269--279 \\ The Hyperbolic Secant $\sech(x)$ and Cosecant $\csch(x)$ Functions / 281--288 \\ The Hyperbolic Tangent $\tanh(x)$ and Cotangent $\coth(x)$ Functions / 289--296 \\ The Inverse Hyperbolic Functions / 297--307 \\ The Cosine $\cos(x)$ and Sine $\sin(x)$ Functions / 309--328 \\ The Secant $\sec(x)$ and Cosecant $\csc(x)$ Functions / 329--338 \\ The Tangent $\tan(x)$ and Cotangent $\cot(x)$ Functions / 339--350 \\ The Inverse Circular Functions / 351--366 \\ Periodic Functions / 367--374 \\ The Exponential Integrals $\Ei(x)$ and $\Ein(x)$ / 375--383 \\ Sine and Cosine Integrals / 385--394 \\ The Fresnel Integrals $C(x)$ and $S(x)$ / 395--404 \\ The Error Function $\erf(x)$ and Its Complement $\erfc(x)$ / 405--415 \\ The $\exp(x)\erfc(\sqrt{x})$ and Related Functions / 417--426 \\ Dawson's Integral $\daw(x)$ / 427--433 \\ The Gamma Function $\Gamma(\nu)$ / 435--448 \\ The Digamma Function $\psi(\nu)$ / 449--460 \\ The Incomplete Gamma Functions / 461--470 \\ The Parabolic Cylinder Function $D_\nu(x)$ / 471--484 \\ The Kummer Function $M(a, c, x)$ / 485--496 \\ The Tricomi Function $U(a, c, x)$ / 497--506 \\ The Modified Bessel Functions $I_n(x)$ of Integer Order / 507--517 \\ The Modified Bessel Function $I_\nu(x)$ of Arbitrary Order / 519--526 \\ The Macdonald Function $K_\nu(x)$ / 527--536 \\ The Bessel Functions $J_n(x)$ of Integer Order / 537--552 \\ The Bessel Function $J_\nu(x)$ of Arbitrary Order / 553--565 \\ The Neumann Function $Y_\nu(x)$ / 567--576 \\ The Kelvin Functions / 577--584 \\ The Airy Functions $\Ai(x)$ and $\Bi(x)$ / 585--592 \\ The Struve Function $h_\nu(x)$ / 593--601 \\ The Incomplete Beta Function $B(\nu, \mu, x)$ / 603--609 \\ The Legendre Functions $P_\nu(x)$ and $Q_\nu(x)$ / 611--626 \\ The Gauss Hypergeometric Function $F(a, b, c, x)$ / 627--636 \\ The Complete Elliptic Integrals $K(k)$ and $E(k)$ / 637--651 \\ The Incomplete Elliptic Integrals $F(k, \phi)$ and $E(k, \phi)$ / 653--669 \\ The Jacobian Elliptic Functions / 671--684 \\ The Hurwitz Function $\zeta(\nu, u)$ / 685--695 \\ Back Matter / 697--748", } @Article{Opps:2009:RFA, author = "Sheldon B. Opps and Nasser Saad and H. M. Srivastava", title = "Recursion formulas for {Appell}'s hypergeometric function {$ F_2 $} with some applications to radiation field problems", journal = j-APPL-MATH-COMP, volume = "207", number = "2", pages = "545--558", day = "15", month = jan, year = "2009", CODEN = "AMHCBQ", ISSN = "0096-3003 (print), 1873-5649 (electronic)", ISSN-L = "0096-3003", bibdate = "Fri Sep 3 10:53:24 MDT 2010", bibsource = "http://www.sciencedirect.com/science/journal/00963003; https://www.math.utah.edu/pub/tex/bib/applmathcomput2005.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Applied Mathematics and Computation", journal-URL = "http://www.sciencedirect.com/science/journal/00963003", } @Article{Paris:2009:HPE, author = "R. B. Paris", title = "High-precision evaluation of the {Bessel} functions via {Hadamard} series", journal = j-J-COMPUT-APPL-MATH, volume = "224", number = "1", pages = "84--100", day = "1", month = feb, year = "2009", CODEN = "JCAMDI", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Sat Feb 25 13:13:29 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2005.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0377042708001799", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @MastersThesis{Pearson:2009:CHF, author = "J. Pearson", title = "Computation of hypergeometric functions", type = "{Master}'s thesis", school = "Oxford University", address = "Oxford, UK", year = "2009", bibdate = "Thu Dec 01 09:05:26 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, } @Article{Talman:2009:NSC, author = "J. D. Talman", title = "{NumSBT}: a subroutine for calculating spherical {Bessel} transforms numerically", journal = j-COMP-PHYS-COMM, volume = "180", number = "2", pages = "332--338", month = feb, year = "2009", CODEN = "CPHCBZ", DOI = "https://doi.org/10.1016/j.cpc.2008.10.003", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", bibdate = "Mon Feb 13 23:42:39 MST 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/compphyscomm2000.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0010465508003329", acknowledgement = ack-nhfb, fjournal = "Computer Physics Communications", journal-URL = "http://www.sciencedirect.com/science/journal/00104655", } @Article{Temme:2009:AER, author = "Nico M. Temme and Vladimir Varlamov", title = "Asymptotic expansions for {Riesz} fractional derivatives of {Airy} functions and applications", journal = j-J-COMPUT-APPL-MATH, volume = "232", number = "2", pages = "201--215", day = "15", month = oct, year = "2009", CODEN = "JCAMDI", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Sat Feb 25 13:24:17 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2005.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0377042709003410", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @InProceedings{Vazquez:2009:CDT, author = "{\'A}lvaro V{\'a}zquez and Julio Villalba and Elisardo Antelo", title = "Computation of Decimal Transcendental Functions Using the {CORDIC} Algorithm", crossref = "Bruguera:2009:PIS", pages = "179--186", year = "2009", bibdate = "Fri Jun 12 12:34:25 2009", bibsource = "https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", abstract = "In this work we propose new decimal floating-point CORDIC algorithms for transcendental function evaluation. We show how these algorithms are mapped to a state of the art Decimal Floating-Point Unit (DFPU), both considering the use of a carry-propagate adder or a carry-save redundant adder. We compared with previous decimal CORDIC proposals and with table-driven algorithms, and we concluded that our approach have significant potential advantages for transcendental function evaluation in state of the art DFPUs with minor modifications of the hardware.", acknowledgement = ack-nhfb, keywords = "ARITH-19", } @Article{Weniger:2009:SHF, author = "Ernst Joachim Weniger", title = "The strange history of {$B$} functions or how theoretical chemists and mathematicians do (not) interact", journal = j-IJQC, volume = "109", number = "8", pages = "1728--1740", month = feb, year = "2009", CODEN = "IJQCB2", DOI = "https://doi.org/10.1002/qua.22014", ISSN = "0020-7608 (print), 1097-461X (electronic)", ISSN-L = "0020-7608", bibdate = "Fri Mar 27 07:47:31 2009", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", abstract = "$B$ functions are a class of relatively complicated exponentially decaying basis functions. Because the molecular multicenter integrals of the much simpler Slater-type functions are notoriously difficult, it is not at all obvious why $B$ functions should offer any advantages. However, $B$ functions have Fourier transforms of exceptional simplicity, which greatly simplifies many of their molecular multicenter integrals. This article discusses the historical development of $B$ functions from the perspective of the interaction between mathematics and theoretical chemistry, which traditionally has not been very good. Nevertheless, future progress in theoretical chemistry depends very much on a fertile interaction with neighboring disciplines.", acknowledgement = ack-nhfb, fjournal = "International Journal of Quantum Chemistry", journal-URL = "http://www.interscience.wiley.com/jpages/0020-7608/", keywords = "B functions; electronic structure theory; exponentially decaying basis functions; interdisciplinary collaboration; multicenter integrals", } @Article{Wozny:2009:MSS, author = "Pawe{\l} Wo{\'z}ny and Rafa{\l} Nowak", title = "Method of summation of some slowly convergent series", journal = j-APPL-MATH-COMP, volume = "215", number = "4", pages = "1622--1645", month = "????", year = "2009", CODEN = "AMHCBQ", ISSN = "0096-3003 (print), 1873-5649 (electronic)", ISSN-L = "0096-3003", MRclass = "65B10", MRnumber = "MR2571650", bibdate = "Thu Dec 01 09:25:02 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Applied Mathematics and Computation", journal-URL = "http://www.sciencedirect.com/science/journal/00963003", keywords = "convergence acceleration", } @Article{Yun:2009:ACN, author = "Beong In Yun", title = "Approximation to the cumulative normal distribution using hyperbolic tangent based functions", journal = "Journal of the Korean Mathematical Society", volume = "46", number = "6", pages = "1267--1276", year = "2009", DOI = "https://doi.org/10.4134/JKMS.2009.46.6.1267", ISSN = "0304-9914", MRclass = "62E17 (65C60)", MRnumber = "2572515", bibdate = "Sat Dec 16 18:04:41 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, ajournal = "J. Korean Math. Soc.", fjournal = "Journal of the Korean Mathematical Society", } @Article{Ahmadi:2010:LCC, author = "O. Ahmadi and F. R. Henr{\'\i}quez", title = "Low Complexity Cubing and Cube Root Computation over {$ F_3^m $} in Polynomial Basis", journal = j-IEEE-TRANS-COMPUT, volume = "59", number = "10", pages = "1297--1308", month = oct, year = "2010", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/TC.2009.183", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Sun Jul 3 11:52:32 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput2010.bib", URL = "https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5374372", acknowledgement = ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", } @Article{Akbarpour:2010:VSI, author = "Behzad Akbarpour and Amr T. Abdel-Hamid and Sofi{\`e}ne Tahar and John Harrison", title = "Verifying a Synthesized Implementation of {IEEE-754} Floating-Point Exponential Function using {HOL}", journal = j-COMP-J, volume = "53", number = "4", pages = "465--488", month = may, year = "2010", CODEN = "CMPJA6", DOI = "https://doi.org/10.1093/comjnl/bxp023", ISSN = "0010-4620 (print), 1460-2067 (electronic)", ISSN-L = "0010-4620", bibdate = "Wed Apr 28 14:33:36 MDT 2010", bibsource = "http://comjnl.oxfordjournals.org/content/vol53/issue4/index.dtl; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://comjnl.oxfordjournals.org/cgi/content/abstract/53/4/465; http://comjnl.oxfordjournals.org/cgi/reprint/53/4/465", acknowledgement = ack-nhfb, fjournal = "The Computer Journal", journal-URL = "http://comjnl.oxfordjournals.org/", } @Article{Alimohammad:2010:UAA, author = "A. Alimohammad and S. F. Fard and B. F. Cockburn", title = "A Unified Architecture for the Accurate and High-Throughput Implementation of Six Key Elementary Functions", journal = j-IEEE-TRANS-COMPUT, volume = "59", number = "4", pages = "449--456", month = "????", year = "2010", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/TC.2009.169", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Sun Jul 3 11:52:27 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", URL = "https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5313801", abstract = "This paper presents a unified architecture for the compact implementation of several key elementary functions, including reciprocal, square root, and logarithm, in single-precision floating-point arithmetic. The proposed high-throughput design is based on uniform domain segmentation and curve fitting techniques. Numerically accurate least-squares regression is utilized to calculate the polynomial coefficients. The architecture is optimized by analyzing the trade-off between the size of the required memory and the precision of intermediate variables to achieve the minimum 23-bit accuracy required for single-precision floating-point representation. The efficiency of the proposed unified data path is demonstrated on a common field-programmable gate array.", acknowledgement = ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", } @Article{Alzer:2010:EFI, author = "Horst Alzer", title = "Error function inequalities", journal = j-ADV-COMPUT-MATH, volume = "33", number = "3", pages = "349--379", month = oct, year = "2010", CODEN = "ACMHEX", DOI = "https://doi.org/10.1007/s10444-009-9139-2", ISSN = "1019-7168 (print), 1572-9044 (electronic)", ISSN-L = "1019-7168", MRclass = "33B20 (26D07 26D15)", MRnumber = "2718103", MRreviewer = "Feng Qi", bibdate = "Sat Feb 3 18:22:50 MST 2018", bibsource = "https://www.math.utah.edu/pub/tex/bib/advcomputmath.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://link.springer.com/article/10.1007/s10444-009-9139-2", acknowledgement = ack-nhfb, fjournal = "Advances in Computational Mathematics", journal-URL = "http://link.springer.com/journal/10444", } @Article{Anand:2010:UTE, author = "Christopher Kumar Anand and Anuroop Sharma", title = "Unified Tables for Exponential and Logarithm Families", journal = j-TOMS, volume = "37", number = "3", pages = "28:1--28:23", month = sep, year = "2010", CODEN = "ACMSCU", DOI = "https://doi.org/10.1145/1824801.1824806", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Mon Sep 27 10:15:50 MDT 2010", bibsource = "http://www.acm.org/pubs/contents/journals/toms/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib", abstract = "Accurate table methods allow for very accurate and efficient evaluation of elementary functions. We present new single-table approaches to logarithm and exponential evaluation, by which we mean that a single table of values works for both $ \log (x) $ and $ l o g(1 + x) $, and a single table for $ e^x $ and $ e^x - 1 $. This approach eliminates special cases normally required to evaluate $ \log (1 + x) $ and $ e^x - 1 $ accurately near zero, which will significantly improve performance on architectures which use SIMD parallelism, or on which data-dependent branching is expensive.\par We have implemented it on the Cell/B.E. SPU (SIMD compute engine) and found the resulting functions to be up to twice as fast as the conventional implementations distributed in the IBM Mathematical Acceleration Subsystem (MASS). We include the literate code used to generate all the variants of exponential and log functions in the article, and discuss relevant language and hardware features.", acknowledgement = ack-nhfb, articleno = "28", fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", keywords = "Accurate tables method; Cell/B.E; IEEE arithmetic; SIMD; vector library", } @Book{Baricz:2010:GBF, author = "{\'A}rp{\'a}d Baricz", title = "Generalized {Bessel} Functions of the First Kind", volume = "1994", publisher = pub-SV, address = pub-SV:adr, pages = "xiv + 206", year = "2010", CODEN = "LNMAA2", DOI = "https://doi.org/10.1007/978-3-642-12230-9", ISBN = "3-642-12229-9 (print), 3-642-12230-2 (e-book)", ISBN-13 = "978-3-642-12229-3 (print), 978-3-642-12230-9 (e-book)", ISSN = "0075-8434 (print), 1617-9692 (electronic)", ISSN-L = "0075-8434", LCCN = "QA3 .L28 no. 1994", MRclass = "33C10 (33-02 33C05 33C75)", MRnumber = "2656410 (2011f:33007)", MRreviewer = "Matti Vuorinen", bibdate = "Tue May 6 14:56:34 MDT 2014", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/lnm2010.bib", series = ser-LECT-NOTES-MATH, URL = "http://link.springer.com/book/10.1007/978-3-642-12230-9; http://www.springerlink.com/content/978-3-642-12230-9", acknowledgement = ack-nhfb, series-URL = "http://link.springer.com/bookseries/304", tableofcontents = "1 Introduction and Preliminary Results / 1 \\ 1.1 Overview / 1 \\ 1.2 Generalized Bessel Functions of the First Kind / 7 \\ 1.3 Classical Inequalities / 21 \\ 2 Geometric Properties of Generalized Bessel Functions / 23 \\ 2.1 Univalence of Generalized Bessel Functions / 23 \\ 2.1.1 Sufficient Conditions Involving Jack's Lemma / 26 \\ 2.1.2 Sufficient Conditions Involving the Admissible Function Method / 28 \\ 2.1.3 Sufficient Conditions Involving the Alexander Transform / 31 \\ 2.1.4 Sufficient Conditions Involving Results of L. Fej{\'e}r / 36 \\ 2.2 Starlikeness and Convexity Properties of Generalized Bessel Functions / 39 \\ 2.2.1 Sufficient Conditions Involving Jack's Lemma / 39 \\ 2.2.2 Sufficient Conditions Involving the Admissible Function Method / 41 \\ 2.2.3 Sufficient Conditions Involving Results of H. Silverman / 50 \\ 2.2.4 Close-to-Convexity with Respect to Certain Functions / 55 \\ 2.3 Applications Involving Bessel Functions Associated with Hardy Space of Analytic Functions / 57 \\ 2.3.1 Bessel Transforms and Hardy Space of Generalized Bessel Functions / 58 \\ 2.3.2 A Monotonicity Property of Generalized Bessel Functions / 62 \\ 3 Inequalities Involving Bessel and Hypergeometric Functions / 71 \\ 3.1 Functional Inequalities Involving Quotients of Some Special Functions / 73 \\ 3.1.1 Preliminary Results / 77 \\ 3.1.2 Inequalities Involving Ratios of Generalized Bessel Functions / 80 \\ 3.1.3 Inequalities Involving Ratios of Hypergeometric Functions / 82 \\ 3.1.4 Inequalities Involving Ratios of General Power Series / 83 \\ 3.2 Functional Inequalities Involving Special Functions / 85 \\ 3.2.1 Inequalities Involving Gaussian Hypergeometric Functions / 85 \\ 3.2.2 Inequalities Involving Generalized Bessel Functions / 91 \\ 3.2.3 Inequalities Involving Confluent Hypergeometric Functions / 93 \\ 3.2.4 Inequalities Involving General Power Series and Concluding Remarks / 94 \\ 3.3 Landen-Type Inequality for Bessel Functions / 99 \\ 3.3.1 Landen-Type Inequality for Generalized Bessel Functions / 100 \\ 3.3.2 Landen-Type Inequality for General Power Series / 102 \\ 3.4 Convexity of Hypergeometric Functions with Respect to H{\"o}lder Means / 103 \\ 3.4.1 Introduction and Preliminaries / 103 \\ 3.4.2 Convexity of Hypergeometric Functions with Respect to H{\"o}lder Means / 104 \\ 3.4.3 Convexity of General Power Series with Respect to H{\"o}lder Means / 108 \\ 3.4.4 Concluding Remarks / 110 \\ 3.5 Askey's and Gr{\"u}nbaum's Inequality for Generalized Bessel Functions / 112 \\ 3.5.1 Askey's and Gr{\"u}nbaum's Inequality for Generalized Bessel Functions / 113 \\ 3.5.2 Lower and Upper Bounds for Generalized Bessel Functions / 115 \\ 3.6 Inequalities Involving Modified Bessel Functions / 118 \\ 3.7 Miscellaneous Inequalities Involving the Generalized Bessel Functions / 128 \\ 3.7.1 Mitrinovic's Inequality and Mahajan's Inequality / 129 \\ 3.7.2 Redheffer's Inequality / 132 \\ 3.7.3 Cusa's Inequality and Related Inequalities / 135 \\ 3.7.4 Extensions of Jordan's Inequality / 139 \\ 3.7.5 Sharp Jordan Type Inequalities for Bessel Functions / 144 \\ 3.7.6 The Sine and Hyperbolic Sine Integral / 159 \\ 3.8 Redheffer Type Inequalities for Bessel Functions / 161 \\ 3.8.1 An Extension of Redheffer's Inequality and Its Hyperbolic Analogue / 162 \\ 3.8.2 Sharp Exponential Redheffer-Type Inequalities for Bessel Functions / 165 \\ 3.8.3 A Lower Bound for the Gamma Function / 183 \\ Appendix A / 187 \\ A. 1 Conjectures / 187 \\ A.2 Open Problems / 187 \\ A.3 Matlab Programs for Graphs / 188 \\ References / 193 \\ Index / 203", } @Article{Baricz:2010:GPG, author = "{\'A}rp{\'a}d Baricz", title = "Geometric Properties of Generalized {Bessel} Functions", journal = j-LECT-NOTES-MATH, volume = "1994", pages = "23--69", year = "2010", CODEN = "LNMAA2", DOI = "https://doi.org/10.1007/978-3-642-12230-9_2", ISBN = "3-642-12229-9 (print), 3-642-12230-2 (e-book)", ISBN-13 = "978-3-642-12229-3 (print), 978-3-642-12230-9 (e-book)", ISSN = "0075-8434 (print), 1617-9692 (electronic)", ISSN-L = "0075-8434", bibdate = "Fri May 9 19:06:58 MDT 2014", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/lnm2010.bib", URL = "http://link.springer.com/content/pdf/10.1007/978-3-642-12230-9_2.pdf", acknowledgement = ack-nhfb, book-DOI = "https://doi.org/10.1007/978-3-642-12230-9", book-URL = "http://www.springerlink.com/content/978-3-642-12230-9", fjournal = "Lecture Notes in Mathematics", journal-URL = "http://link.springer.com/bookseries/304", } @Book{Beals:2010:SFG, author = "Richard Beals and R. (Roderick) Wong", title = "Special functions: a graduate text", volume = "126", publisher = pub-CAMBRIDGE, address = pub-CAMBRIDGE:adr, pages = "ix + 456", year = "2010", ISBN = "0-521-19797-X", ISBN-13 = "978-0-521-19797-7", LCCN = "QA351 .B34 2010; QA351 BEA 2010", bibdate = "Sat Oct 30 16:43:46 MDT 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcw.bib; library.ox.ac.uk:210/ADVANCE", series = "Cambridge studies in advanced mathematics", URL = "http://assets.cambridge.org/97805211/97977/cover/9780521197977.jpg; http://www.loc.gov/catdir/enhancements/fy1009/2010017815-b.html; http://www.loc.gov/catdir/enhancements/fy1009/2010017815-d.html; http://www.loc.gov/catdir/enhancements/fy1009/2010017815-t.html", abstract = "From the Publisher: The subject of special functions is often presented as a collection of disparate results, which are rarely organised in a coherent way. This book answers the need for a different approach to the subject. The authors' main goals are to emphasise general unifying principles coherently and to provide clear motivation, efficient proofs, and original references for all of the principal results. The book covers standard material, but also much more, including chapters on discrete orthogonal polynomials and elliptic functions. The authors show how a very large part of the subject traces back to two equations --- the hypergeometric equation and the confluent hypergeometric equation --- and describe the various ways in which these equations are canonical and special. Providing ready access to theory and formulas, this book serves as an ideal graduate-level textbook as well as a convenient reference.", acknowledgement = ack-nhfb, subject = "Functions, Special; Textbooks", tableofcontents = "Preface; 1. Orientation\\ 2. Gamma, beta, zeta\\ 3. Second order differential equations\\ 4. Orthogonal polynomials\\ 5. Discrete orthogonal polynomials\\ 6. Confluent hypergeometric functions\\ 7. Cylinder functions\\ 8. Hypergeometric functions\\ 9. Spherical functions\\ 10. Asymptotics\\ 11. Elliptic functions\\ References\\ Index", } @InProceedings{Benoit:2010:DDM, author = "Alexandre Benoit and Fr{\'e}d{\'e}ric Chyzak and Alexis Darrasse and Stefan Gerhold and Marc Mezzarobba and Bruno Salvy", title = "The Dynamic Dictionary of Mathematical Functions {(DDMF)}", crossref = "Fukuda:2010:MSI", pages = "35--41", year = "2010", DOI = "https://doi.org/10.1007/978-3-642-15582-6_7", bibdate = "Sat Sep 23 06:20:46 2023", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, } @Article{Borghi:2010:AFE, author = "Riccardo Borghi", title = "Asymptotic and factorial expansions of {Euler} series truncation errors via exponential polynomials", journal = j-APPL-NUM-MATH, volume = "60", number = "12", pages = "1242--1250", month = dec, year = "2010", CODEN = "ANMAEL", ISSN = "0168-9274 (print), 1873-5460 (electronic)", ISSN-L = "0168-9274", MRclass = "65B10 (33F05 40A25)", MRnumber = "MR2735157", bibdate = "Thu Dec 01 09:47:34 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/applnummath.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Applied Numerical Mathematics: Transactions of IMACS", journal-URL = "http://www.sciencedirect.com/science/journal/01689274/", } @Article{Brent:2010:UAE, author = "Richard P. Brent", title = "Unrestricted algorithms for elementary and special functions", journal = "arxiv.org", volume = "??", number = "??", pages = "1--13", month = apr, year = "2010", bibdate = "Sat Feb 25 10:56:45 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "https://arxiv.org/abs/1004.3621", abstract = "We describe some ``unrestricted'' algorithms which are useful for the computation of elementary and special functions when the precision required is not known in advance. Several general classes of algorithms are identified and illustrated by examples. The topics include: power series methods, use of halving identities, asymptotic expansions, continued fractions, recurrence relations, Newton's method, numerical contour integration, and the arithmetic-geometric mean. Most of the algorithms discussed are implemented in the MP package.", acknowledgement = ack-nhfb, } @Article{Celledoni:2010:AFF, author = "Elena Celledoni and Antonella Zanna", title = "{Algorithm 903}: {FRB} --- {Fortran} routines for the exact computation of free rigid body motions", journal = j-TOMS, volume = "37", number = "2", pages = "23:1--23:24", month = apr, year = "2010", CODEN = "ACMSCU", DOI = "https://doi.org/10.1145/1731022.1731033", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Wed Apr 21 11:39:57 MDT 2010", bibsource = "http://www.acm.org/pubs/contents/journals/toms/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib", abstract = "We present two algorithms and their corresponding Fortran routines for the exact computation of free rigid body motions. The methods use the same description of the angular momentum part $m$ by Jacobi elliptic functions, and suitably chosen frames for the attitude matrix\slash quaternion $ Q / q $, respectively. The frame transformation requires the computation of elliptic integrals of the third kind. Implementation and usage of the routines are described, and some examples of drivers are included. Accuracy and performance are also tested to provide reliable numerical results.", acknowledgement = ack-nhfb, articleno = "23", fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", keywords = "attitude rotation; Jacobi elliptic integrals; numerical methods; Rigid body; splitting methods", } @InProceedings{Chevillard:2010:SED, author = "Sylvain Chevillard and Mioara Jolde and Christoph Lauter", title = "{Sollya}: An Environment for the Development of Numerical Codes", crossref = "Fukuda:2010:MSI", pages = "28--31", year = "2010", DOI = "https://doi.org/10.1007/978-3-642-15582-6_5", bibdate = "Tue Sep 24 14:50:44 2024", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, } @Article{Cuyt:2010:VSF, author = "Annie Cuyt and Franky Backeljauw and Stefan Becuwe and Joris {Van Deun}", title = "Validated Special Functions Software", journal = j-LECT-NOTES-COMP-SCI, volume = "6327", pages = "32--34", year = "2010", CODEN = "LNCSD9", DOI = "https://doi.org/10.1007/978-3-642-15582-6_6", ISSN = "0302-9743 (print), 1611-3349 (electronic)", ISSN-L = "0302-9743", bibdate = "Sat Aug 9 15:34:11 MDT 2014", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/lncs2010a.bib", URL = "http://link.springer.com/content/pdf/10.1007/978-3-642-15582-6_6.pdf", acknowledgement = ack-nhfb, book-DOI = "https://doi.org/10.1007/978-3-642-15582-6", book-URL = "http://www.springerlink.com/content/978-3-642-15582-6", fjournal = "Lecture Notes in Computer Science", journal-URL = "http://link.springer.com/bookseries/558", } @InProceedings{deDinechin:2010:AGP, author = "Florent de Dinechin and Mioara Joldes and Bogdan Pasca", booktitle = "{ASAP 2010} --- {21st IEEE International Conference on Application-specific Systems, Architectures and Processors}", title = "Automatic generation of polynomial-based hardware architectures for function evaluation", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "216--222", month = jul, year = "2010", DOI = "https://doi.org/10.1109/asap.2010.5540952", bibdate = "Thu Apr 10 13:04:38 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, } @InProceedings{deDinechin:2010:FPE, author = "Florent de Dinechin and Bogdan Pasca", editor = "Jinian Bian and Qiang Zhou and Kang Zhao", booktitle = "{Proceedings 2010 International Conference on Field-Programmable Technology, 8--10 December 2010, Beijing, China}", title = "Floating-point exponential functions for {DSP}-enabled {FPGAs}", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "110--117", month = dec, year = "2010", DOI = "https://doi.org/10.1109/FPT.2010.5681764", bibdate = "Sat Feb 08 09:35:06 2020", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, } @InProceedings{deDinechin:2010:MSR, author = "Florent de Dinechin and Mioara Joldes and Bogdan Pasca and Guillaume Revy", editor = "Fabrizio Ferrandi and Jari Nurmi and Marco D. Santambrogio", booktitle = "2010 International Conference on Field Programmable Logic and Applications: {FPL 2010, 31 August--2 September 2010, Milano, Italy}", title = "Multiplicative Square Root Algorithms for {FPGAs}", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "547--577", year = "2010", DOI = "https://doi.org/10.1109/FPL.2010.112", ISBN = "0-7695-4179-8", ISBN-13 = "978-0-7695-4179-2", bibdate = "Sat Nov 08 11:18:48 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "https://ens-lyon.hal.science/ensl-00475779v2/document", acknowledgement = ack-nhfb, remark = "LIP Research Report RR2010-17", } @Article{Dumbgen:2010:BSG, author = "Lutz D{\"u}mbgen", title = "Bounding standard {Gaussian} tail probabilities", journal = "arxiv.org", volume = "??", number = "??", pages = "??--??", day = "9", month = dec, year = "2010", bibdate = "Sat Dec 16 16:24:48 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "https://arxiv.org/abs/1012.2063", abstract = "We review various inequalities for Mills' ratio $ (1 - \Phi) / \phi $, where $ \phi $ and $ \Phi $ denote the standard Gaussian density and distribution function, respectively. Elementary considerations involving finite continued fractions lead to a general approximation scheme which implies and refines several known bounds.", acknowledgement = ack-nhfb, } @InProceedings{Erocal:2010:SPU, author = "Bur{\c{c}}in Er{\"o}cal and William Stein", title = "The {Sage Project}: Unifying Free Mathematical Software to Create a Viable Alternative to {Magma}, {Maple}, {Mathematica} and {MATLAB}", crossref = "Fukuda:2010:MSI", pages = "12--27", year = "2010", DOI = "https://doi.org/10.1007/978-3-642-15582-6_4", bibdate = "Sat Sep 23 06:20:46 2023", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/magma.bib; https://www.math.utah.edu/pub/tex/bib/maple-extract.bib; https://www.math.utah.edu/pub/tex/bib/mathematica.bib; https://www.math.utah.edu/pub/tex/bib/matlab.bib", acknowledgement = ack-nhfb, } @Article{Fukushima:2010:FCI, author = "Toshio Fukushima", title = "Fast computation of incomplete elliptic integral of first kind by half argument transformation", journal = j-NUM-MATH, volume = "116", number = "4", pages = "687--719", month = oct, year = "2010", CODEN = "NUMMA7", DOI = "https://doi.org/10.1007/s00211-010-0321-8", ISSN = "0029-599X (print), 0945-3245 (electronic)", ISSN-L = "0029-599X", bibdate = "Sat Oct 16 16:02:41 MDT 2010", bibsource = "http://springerlink.metapress.com/openurl.asp?genre=issue&issn=0029-599X&volume=116&issue=4; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.springerlink.com/openurl.asp?genre=article&issn=0029-599X&volume=116&issue=4&spage=687", abstract = "We developed a new method to calculate the incomplete elliptic integral of the first kind, $ F(\varphi |m) $, by using the half argument formulas of Jacobian elliptic functions. The method reduces the magnitude of $ \varphi $ by repeated usage of the formulas while fixing $m$. The method is sufficiently precise in the sense that the maximum relative error is $3$--$5$ machine epsilons at most. Thanks to the simplicity of the half argument formulas, the new procedure is significantly faster than the existing procedures. For example, it runs 20--60\% faster than Bulirsch's function, {\tt el1}, and 1.9--2.2 times faster than the method using Carlson's function, $ R_F $.", acknowledgement = ack-nhfb, fjournal = "Numerische Mathematik", journal-URL = "http://link.springer.com/journal/211", } @InProceedings{Habegger:2010:EHI, author = "Andreas Habegger and Andreas Stahel and Josef Goette and Marcel Jacomet", editor = "{IEEE}", booktitle = "{2010 Fifth IEEE International Symposium on Electronic Design, Test \& Applications: 13--15 January 2010 Ho Chi Minh City, Vietnam}", title = "An Efficient Hardware Implementation for a Reciprocal Unit", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "183--187", year = "2010", DOI = "https://doi.org/10.1109/delta.2010.65", bibdate = "Thu Apr 10 13:16:33 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, } @InProceedings{Hsiao:2010:LCD, author = "Shen-Fu Hsiao and Chia-Sheng Wen and Ming-Yu Tsai", editor = "????", booktitle = "{2010 International Symposium on Next Generation Electronics: 18--19 November 2010}", title = "Low-cost design of reciprocal function units using shared multipliers and adders for polynomial approximation and {Newton--Raphson} interpolation", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "40--43", month = nov, year = "2010", DOI = "https://doi.org/10.1109/isne.2010.5669204", bibdate = "Thu Apr 10 13:30:38 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, } @Book{ISO:2010:IIIa, author = "{ISO}", title = "{ISO\slash IEC 29124:2010}: Information technology --- Programming languages, their environments and system software interfaces --- Extensions to the {C++ Library} to support mathematical special functions", publisher = pub-ISO, address = pub-ISO:adr, year = "2010", LCCN = "????", bibdate = "Thu Nov 25 08:56:44 2010", bibsource = "http://www.iso.org/iso/search.htm; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/isostd.bib", series = "Technical report", URL = "http://www.iso.org/iso/iso_catalogue/catalogue_tc/catalogue_detail.htm?csnumber=50511", acknowledgement = ack-nhfb, subject = "programming languages (electronic computers)", } @Article{Li:2010:NRB, author = "Rong Li and Pooi Yuen Kam and Hua Fu", title = "New representations and bounds for the generalized {Marcum} {$Q$}-function via a geometric approach, and an application", journal = j-IEEE-TRANS-COMM, volume = "58", number = "1", pages = "157--169", month = jan, year = "2010", CODEN = "IECMBT", DOI = "https://doi.org/10.1109/tcomm.2010.01.070426", ISSN = "0090-6778 (print), 1558-0857 (electronic)", ISSN-L = "0090-6778", bibdate = "Sat Dec 16 16:52:49 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "https://ieeexplore.ieee.org/document/5397910/", acknowledgement = ack-nhfb, fjournal = "IEEE Transactions on Communications", } @Book{Muller:2010:HFP, author = "Jean-Michel Muller and Nicolas Brisebarre and Florent de Dinechin and Claude-Pierre Jeannerod and Vincent Lef{\`e}vre and Guillaume Melquiond and Nathalie Revol and Damien Stehl{\'e} and Serge Torres", title = "Handbook of Floating-Point Arithmetic", publisher = pub-BIRKHAUSER-BOSTON, address = pub-BIRKHAUSER-BOSTON:adr, pages = "xxiii + 572", year = "2010", DOI = "https://doi.org/10.1007/978-0-8176-4705-6", ISBN = "0-8176-4704-X (hardcover), 0-8176-4705-8 (e-book)", ISBN-13 = "978-0-8176-4704-9 (hardcover), 978-0-8176-4705-6 (e-book)", LCCN = "QA76.9.C62 H36 2010", MRnumber = "MR2568265", bibdate = "Thu Jan 27 16:18:58 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/master.bib; https://www.math.utah.edu/pub/tex/bib/mathcw.bib", price = "US\$90 (est.)", acknowledgement = ack-nhfb, tableofcontents = "Preface \\ List of Figures \\ List of Tables \\ I Introduction, Basic Definitions, and Standards \\ 1 Introduction \\ 2 Definitions and Basic Notions \\ 3 Floating-Point Formats and Environment \\ II Cleverly Using Floating-Point Arithmetic \\ 4 Basic Properties and Algorithms \\ 5 The Fused Multiply-Add Instruction \\ 6 Enhanced Floating-Point Sums, Dot Products, and Polynomial Values \\ 7 Languages and Compilers \\ III Implementing Floating-Point Operators \\ 8 Algorithms for the Five Basic Operations \\ 9 Hardware Implementation of Floating-Point Arithmetic \\ 10 Software Implementation of Floating-Point Arithmetic", } @Article{Nandagopal:2010:NEF, author = "Mohankumar Nandagopal and Soubhadra Sen and Ajay Rawat", title = "A Note on the Error Function", journal = j-COMPUT-SCI-ENG, volume = "12", number = "4", pages = "84--88", month = jul # "\slash " # aug, year = "2010", CODEN = "CSENFA", DOI = "https://doi.org/10.1109/MCSE.2010.79", ISSN = "1521-9615 (print), 1558-366X (electronic)", ISSN-L = "1521-9615", bibdate = "Tue Jul 27 16:37:11 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "See improvements \cite{Iacono:2021:BEF}.", acknowledgement = ack-nhfb, fjournal = "Computing in Science and Engineering", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=5992", } @Article{Paszkowski:2010:UMC, author = "Stefan Paszkowski", title = "Untypical methods of convergence acceleration", journal = j-NUMER-ALGORITHMS, volume = "54", number = "??", pages = "??--??", month = "????", year = "2010", CODEN = "NUALEG", DOI = "https://doi.org/10.1007/s11075-010-9381-1", ISSN = "1017-1398 (print), 1572-9265 (electronic)", ISSN-L = "1017-1398", bibdate = "Mon May 17 14:24:01 MDT 2010", bibsource = "http://springerlink.metapress.com/openurl.asp?genre=issue&issn=1017-1398&volume=0&issue=0; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.springerlink.com/openurl.asp?genre=article&issn=1017-1398&volume=0&issue=0&spage=??", acknowledgement = ack-nhfb, fjournal = "Numerical Algorithms", journal-URL = "http://link.springer.com/journal/11075", keywords = "convergence acceleration", remark = "Article in press, not yet assigned to an issue.", } @Article{Prevost:2010:RVZ, author = "Marc Pr{\'e}vost", title = "Recurrence for values of the zeta function", journal = j-APPL-NUM-MATH, volume = "60", number = "12", pages = "1382--1394", month = dec, year = "2010", CODEN = "ANMAEL", DOI = "https://doi.org/10.1016/j.apnum.2010.05.011", ISSN = "0168-9274 (print), 1873-5460 (electronic)", ISSN-L = "0168-9274", bibdate = "Sat Oct 16 16:17:49 MDT 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/applnummath.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", abstract = "Using the Pad{\'e} approximation of the exponential function, we obtain a general recurrence relation for values of the zeta function which contains, as particular cases, many relations already proved. Applications to Bernoulli polynomials are given. Finally, we derive some new recurrence relations with gap of length 4 for zeta numbers.", acknowledgement = ack-nhfb, fjournal = "Applied Numerical Mathematics: Transactions of IMACS", journal-URL = "http://www.sciencedirect.com/science/journal/01689274/", keywords = "Pad{\'e} approximants; zeta function", } @Article{Qi:2010:CMS, author = "Feng Qi and Senlin Guo and Bai-Ni Guo", title = "Complete monotonicity of some functions involving polygamma functions", journal = j-J-COMPUT-APPL-MATH, volume = "233", number = "9", pages = "2149--2160", day = "1", month = mar, year = "2010", CODEN = "JCAMDI", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Sat Feb 25 13:24:22 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2010.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0377042709006682", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @Article{Safouhi:2010:BSC, author = "Hassan Safouhi", title = "{Bessel}, sine and cosine functions and extrapolation methods for computing molecular multi-center integrals", journal = j-NUMER-ALGORITHMS, volume = "54", number = "1", pages = "141--167", month = may, year = "2010", CODEN = "NUALEG", ISSN = "1017-1398 (print), 1572-9265 (electronic)", ISSN-L = "1017-1398", bibdate = "Mon May 17 14:08:57 MDT 2010", bibsource = "http://springerlink.metapress.com/openurl.asp?genre=issue&issn=1017-1398&volume=54&issue=1; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.springerlink.com/openurl.asp?genre=article&issn=1017-1398&volume=54&issue=1&spage=141", acknowledgement = ack-nhfb, fjournal = "Numerical Algorithms", journal-URL = "http://link.springer.com/journal/11075", } @Article{Slevinsky:2010:RAT, author = "Richard M. Slevinsky and Hassan Safouhi", title = "A recursive algorithm for the {$G$} transformation and accurate computation of incomplete {Bessel} functions", journal = j-APPL-NUM-MATH, volume = "60", number = "12", pages = "1411--1417", month = dec, year = "2010", CODEN = "ANMAEL", DOI = "https://doi.org/10.1016/j.apnum.2010.04.005", ISSN = "0168-9274 (print), 1873-5460 (electronic)", ISSN-L = "0168-9274", bibdate = "Sat Oct 16 16:17:49 MDT 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/applnummath.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", abstract = "In the present contribution, we develop an efficient algorithm for the recursive computation of the $ G_n^{(1)} $ source transformation for the approximation of infinite-range integrals. Previous to this contribution, the theoretically powerful $ G_n^{(1)} $ transformation was handicapped by the lack of an algorithmic implementation. Our proposed algorithm removes this handicap by introducing a recursive computation of the successive $ G_n^{(1)} $ transformations with respect to the order $n$. This recursion, however, introduces the $ (x^2 d / d x) $ source operator applied to the integrand. Consequently, we employ the Slevinsky--Safouhi formula I for the analytical and numerical developments of these required successive derivatives.\par Incomplete Bessel functions, which pose as a numerical challenge, are computed to high pre-determined accuracies using the developed algorithm. The numerical results obtained show the high efficiency of the new method, which does not resort to any numerical integration in the computation.", acknowledgement = ack-nhfb, fjournal = "Applied Numerical Mathematics: Transactions of IMACS", journal-URL = "http://www.sciencedirect.com/science/journal/01689274/", keywords = "Extrapolation methods; Incomplete Bessel functions; Nonlinear transformations; Slevinsky--Safouhi formulae", } @InProceedings{Sofotasios:2010:NEM, author = "Paschalis C. Sofotasios and Steven Freear", booktitle = "2010 7th International Symposium on Wireless Communication Systems", title = "Novel expressions for the {Marcum} and one dimensional {$Q$}-functions", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "", month = sep, year = "2010", DOI = "https://doi.org/10.1109/iswcs.2010.5624374", ISBN = "1-4244-6315-7 (print), 1-4244-6317-3 (e-book), 1-4244-6316-5 (CD-ROM)", ISBN-13 = "978-1-4244-6315-2 (print), 978-1-4244-6317-6 (e-book), 978-1-4244-6316-9 (CD-ROM)", ISSN = "2154-0217 (print), 2154-0225 (electronic)", ISSN-L = "2154-0225", bibdate = "Sat Dec 16 17:36:08 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "https://ieeexplore.ieee.org/document/5624374/", acknowledgement = ack-nhfb, } @Book{Vallee:2010:AFA, author = "Olivier Vall{\'e}e and Manuel Soares", title = "{Airy} Functions and Applications to Physics", publisher = "Imperial College Press", address = "London WC26 9HE, UK", edition = "Second", pages = "x + 202", year = "2010", ISBN = "1-84816-548-X, 1-84816-550-1", ISBN-13 = "978-1-84816-548-9, 978-1-84816-550-2", LCCN = "QA351", bibdate = "Tue Dec 5 10:05:10 MST 2023", bibsource = "fsz3950.oclc.org:210/WorldCat; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", abstract = "Addressed mainly to physicist and chemical physicist, this textbook is the result of a broad compilation of current knowledge on analytical properties of Airy functions. In particular, the calculus implying the Airy functions is developed with care. In the latter chapters, examples are given to succinctly illustrate the use of Airy functions in classical and quantum physics. The physicist, for instance in fluid mechanics, can find what he is looking for, in the references for works of molecular physics or in physics of surfaces, and vice versa. The knowledge on Airy functions is frequently reviewed. The reason may be found in the need to express a physical phenomenon in terms of an effective and comprehensive analytical form for the whole scientific community.", acknowledgement = ack-nhfb, remark = "See also first edition \cite{Vallee:2004:AFA}", subject = "Airy functions; Airy-Funktion; Mathematische Physik", tableofcontents = "Preface / x \\ 1: A Historical Introduction: Sir George Biddell Airy / 1 \\ 2: Definitions and Properties / 5 \\ 2.1: Homogeneous Airy functions \\ 2.1.1: The Airy equation \\ 2.1.2: Elementary properties \\ 2.1.3: Integral representations \\ 2.1.4: Ascending and asymptotic series \\ 2.2: Properties of Airy functions \\ 2.2.1: Zeros of Airy functions \\ 2.2.2: The spectral zeta function \\ 2.2.3: Inequalities \\ 2.2.4: Connection with Bessel functions \\ 2.2.5: Modulus and phase of Airy functions \\ 2.3: Inhomogeneous Airy functions \\ 2.3.1: Definitions \\ 2.3.2: Properties of inhomogeneous Airy functions \\ 2.3.3: Ascending series and asymptotic expansion \\ 2.3.4: Zeros of the Scorer functions \\ 2.4: Squares and products of Airy functions \\ 2.4.1: Differential equation and integral representation \\ 2.4.2: A remarkable identity \\ 2.4.3: The product $\Ai(x) \Ai(-x)$: Airy wavelets \\ 3: Primitives and Integrals of Airy Functions / 37 \\ 3.1: Primitives containing one Airy function \\ 3.1.1: In terms of Airy functions \\ 3.1.2: Ascending series \\ 3.1.3: Asymptotic expansions \\ 3.1.4: Primitives of Scorer functions \\ 3.1.5: Repeated primitives \\ 3.2: Product of Airy functions \\ 3.2.1: The method of Albright \\ 3.2.2: Some primitives \\ 3.3: Other primitives \\ 3.4: Miscellaneous \\ 3.5: Elementary integrals \\ 3.5.1: Particular integrals \\ 3.5.2: Integrals containing a single Airy function \\ 3.5.3: Integrals of products of two Airy functions \\ 3.6: Other integrals \\ 3.6.1: Integrals involving the Volterra $\mu$-function \\ 3.6.2: Canonisation of cubic forms \\ 3.6.3: Integrals with three Airy functions \\ 3.6.4: Integrals with four Airy functions \\ 3.6.5: Double integrals \\ 4: Transformations of Airy functions / 69 \\ 4.1: Causal properties of Airy functions \\ 4.1.1: Causal relations \\ 4.1.2: Green's function of the Airy equation \\ 4.1.3: Fractional derivatives of Airy functions \\ 4.2: The Airy transform \\ 4.2.1: Definitions and elementary properties \\ 4.2.2: Some examples \\ 4.2.3: Airy polynomials \\ 4.2.4: A particular case: correlation Airy transform \\ 4.3: Other kinds of transformations \\ 4.3.1: Laplace transform of Airy functions \\ 4.3.2: Mellin transform of Airy functions \\ 4.3.3: Fourier transform of Airy functions \\ 4.3.4: Hankel transform and the Airy kernel \\ 4.4: Expansion into Fourier--Airy series \\ 5: The Uniform Approximation / 101 \\ 5.1: Oscillating integrals \\ 5.1.1: The method of stationary phase \\ 5.1.2: The uniform approximation of oscillating integrals \\ 5.1.3: The Airy uniform approximation \\ 5.2: Differential equations of the second order \\ 5.2.1: The JWKB method \\ 5.2.2: The Langer generalisation \\ 5.3: Inhomogeneous differential equations \\ 6: Generalisation of Airy Functions / 111 \\ 6.1: Generalisation of the Airy integral \\ 6.1.1: The generalisation of Watson \\ 6.1.2: Oscillating integrals and catastrophes \\ 6.2: Third order differential equations \\ 6.2.1: The linear third order differential equation \\ 6.2.2: Asymptotic solutions \\ 6.2.3: The comparison equation \\ 6.3: A differential equation of the fourth order \\ 7: Applications to Classical Physics / 127 \\ 7.1: Optics and electromagnetism \\ 7.2: Fluid mechanics \\ 7.2.1: The Tricomi equation \\ 7.2.2: The Orr--Sommerfeld equation \\ 7.3: Elasticity \\ 7.4: The heat equation \\ 7.5: Nonlinear physics \\ 7.5.1: Korteweg--de Vries equation \\ 7.5.2: The Second Painlev{\'e} equation \\ 8: Applications to Quantum Physics / 147 \\ 8.1: The Schr{\"o}dinger equation \\ 8.1.1: Particle in a Uniform field \\ 8.1.2: The 8.1.3: Uniform approximation of the Schr{\"o}dinger equation \\ 8.2: Evaluation of the Franck--Condon factors \\ 8.2.1: The Franck--Condon principle \\ 8.2.2: The JWKB approximation \\ 8.2.3: The uniform approximation \\ 8.3: The semiclassical Wigner distribution \\ 8.3.1: The Weyl--Wigner formalism \\ 8.3.2: The one-dimensional Wigner distribution \\ 8.3.3: The two-dimensional Wigner distribution \\ 8.3.4: Configuration of the Wigner distribution \\ 8.4: Airy transform of the Schr{\"o}dinger equation \\ Appendix A: Numerical Computation of the Airy Functions / 185 \\ A.1: Homogeneous functions \\ A.2: Inhomogeneous functions \\ Bibliography / 191 \\ Index / 201", } @Article{Weniger:2010:SDP, author = "E. J. Weniger", title = "Summation of divergent power series by means of factorial series", journal = j-APPL-NUM-MATH, volume = "60", number = "12", pages = "1429--1441", month = "????", year = "2010", CODEN = "ANMAEL", ISSN = "0168-9274 (print), 1873-5460 (electronic)", ISSN-L = "0168-9274", bibdate = "Thu Dec 01 10:37:55 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "Presented at Approximation and Extrapolation of Convergent and Divergent Sequences and Series (CIRM, Luminy --- France, 2009).", acknowledgement = ack-nhfb, fjournal = "Applied Numerical Mathematics: Transactions of IMACS", journal-URL = "http://www.sciencedirect.com/science/journal/01689274/", } @Article{Wozny:2010:EAS, author = "Pawe{\l} Wo{\'z}ny", title = "Efficient algorithm for summation of some slowly convergent series", journal = j-APPL-NUM-MATH, volume = "60", number = "12", pages = "1442--1453", month = "????", year = "2010", CODEN = "ANMAEL", ISSN = "0168-9274 (print), 1873-5460 (electronic)", ISSN-L = "0168-9274", bibdate = "Thu Dec 01 09:26:24 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "Presented at Approximation and Extrapolation of Convergent and Divergent Sequences and Series (CIRM, Luminy - France, 2009).", acknowledgement = ack-nhfb, fjournal = "Applied Numerical Mathematics: Transactions of IMACS", journal-URL = "http://www.sciencedirect.com/science/journal/01689274/", keywords = "convergence acceleration", } @InProceedings{Ye:2010:FIC, author = "Min Ye and Taijun Liu and Yan Ye and Gaoming Xu and Tiefeng Xu", booktitle = "{2010 6th International Conference on Wireless Communications Networking and Mobile Computing (WiCOM)}", title = "{FPGA} Implementation of {CORDIC}-based Square Root Operation for Parameter Extraction of Digital Pre-Distortion for Power Amplifiers", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "1--4", year = "2010", DOI = "https://doi.org/10.1109/WICOM.2010.5600929", bibdate = "Mon Oct 27 10:32:44 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, keywords = "Arrays; Field programmable gate arrays; Hardware; Parameter extraction; Power amplifiers; Software; Software algorithms", } @Article{Zhu:2010:JTI, author = "Ling Zhu", title = "{Jordan} type inequalities involving the {Bessel} and modified {Bessel} functions", journal = j-COMPUT-MATH-APPL, volume = "59", number = "2", pages = "724--736", month = jan, year = "2010", CODEN = "CMAPDK", ISSN = "0898-1221 (print), 1873-7668 (electronic)", ISSN-L = "0898-1221", bibdate = "Wed Mar 1 21:50:34 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/computmathappl2010.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0898122109007196", acknowledgement = ack-nhfb, fjournal = "Computers and Mathematics with Applications", journal-URL = "http://www.sciencedirect.com/science/journal/08981221", } @Article{Ali:2011:NGJ, author = "Ahmad T. Ali", title = "New generalized {Jacobi} elliptic function rational expansion method", journal = j-J-COMPUT-APPL-MATH, volume = "235", number = "14", pages = "4117--4127", day = "15", month = may, year = "2011", CODEN = "JCAMDI", DOI = "https://doi.org/10.1016/j.cam.2011.03.002", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Sat Feb 25 13:24:28 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2010.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0377042711001257", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @InProceedings{Borwein:2011:SVG, author = "Jonathan M. Borwein and Armin Straub", title = "Special values of generalized log-sine integrals", crossref = "Schost:2011:IPI", pages = "43--50", year = "2011", DOI = "https://doi.org/10.1145/1993886.1993899", bibdate = "Fri Mar 14 12:20:08 MDT 2014", bibsource = "http://portal.acm.org/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/issac.bib; https://www.math.utah.edu/pub/tex/bib/mathematica.bib", abstract = "We study generalized log-sine integrals at special values. At $ \pi $ and multiples thereof explicit evaluations are obtained in terms of Nielsen polylogarithms at $ \pm 1 $. For general arguments we present algorithmic evaluations involving Nielsen polylogarithms at related arguments. In particular, we consider log-sine integrals at $ \pi / 3 $ which evaluate in terms of polylogarithms at the sixth root of unity. An implementation of our results for the computer algebra systems Mathematica and SAGE is provided.", acknowledgement = ack-nhfb, } @InProceedings{Brisebarre:2011:APS, author = "Nicolas Brisebarre and Mioara Joldes and Peter Kornerup and {\'E}rik Martin-Dorel and Jean-Michel Muller", title = "Augmented Precision Square Roots and {$2$-D} Norms, and Discussion on Correctly Rounding $ \sqrt {x^2 + y^2} $", crossref = "Schwarz:2011:PIS", pages = "23--30", year = "2011", DOI = "https://doi.org/10.1109/ARITH.2011.13", bibdate = "Sat Aug 20 09:00:00 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5992105", acknowledgement = ack-nhfb, keywords = "ARITH-20; hypotenuse", } @InProceedings{Butts:2011:RDR, author = "J. Adam Butts and Ping Tak Peter Tang and Ron O. Dror and David E. Shaw", title = "Radix-8 Digit-by-Rounding: Achieving High-Performance Reciprocals, Square Roots, and Reciprocal Square Roots", crossref = "Schwarz:2011:PIS", pages = "149--158", year = "2011", DOI = "https://doi.org/10.1109/ARITH.2011.28", bibdate = "Sat Aug 20 09:00:00 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5992120", acknowledgement = ack-nhfb, keywords = "ARITH-20", } @Article{Cai:2011:CSB, author = "Liang-Wu Cai", title = "On the computation of spherical {Bessel} functions of complex arguments", journal = j-COMP-PHYS-COMM, volume = "182", number = "3", pages = "663--668", month = mar, year = "2011", CODEN = "CPHCBZ", DOI = "https://doi.org/10.1016/j.cpc.2010.11.019", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", bibdate = "Sat Feb 11 10:10:56 MST 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/compphyscomm2010.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0010465510004650", acknowledgement = ack-nhfb, fjournal = "Computer Physics Communications", journal-URL = "http://www.sciencedirect.com/science/journal/00104655", } @Article{Cardoso:2011:IFP, author = "Jo{\~a}o Ribeiro Cardoso and Ana F. Loureiro", title = "Iteration functions for $p$ th roots of complex numbers", journal = j-NUMER-ALGORITHMS, volume = "57", number = "3", pages = "329--356", month = jul, year = "2011", CODEN = "NUALEG", ISSN = "1017-1398 (print), 1572-9265 (electronic)", ISSN-L = "1017-1398", bibdate = "Fri Jul 22 09:48:58 MDT 2011", bibsource = "http://springerlink.metapress.com/openurl.asp?genre=issue&issn=1017-1398&volume=57&issue=3; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.springerlink.com/openurl.asp?genre=article&issn=1017-1398&volume=57&issue=3&spage=329", abstract = "A novel way of generating higher-order iteration functions for the computation of pth roots of complex numbers is the main contribution of the present work. The behavior of some of these iteration functions will be analyzed and the conditions on the starting values that guarantee the convergence will be stated. The illustration of the basins of attractions of the pth roots will be carried out by some computer generated plots. In order to compare the performance of the iterations some numerical examples will be considered.", acknowledgement = ack-nhfb, fjournal = "Numerical Algorithms", journal-URL = "http://link.springer.com/journal/11075", } @Article{Chang:2011:CTB, author = "Seok-Ho Chang and Pamela C. Cosman and Laurence B. Milstein", title = "{Chernoff}-Type Bounds for the {Gaussian} Error Function", journal = j-IEEE-TRANS-COMM, volume = "59", number = "11", pages = "2939--2944", month = nov, year = "2011", CODEN = "IECMBT", DOI = "https://doi.org/10.1109/tcomm.2011.072011.100049", ISSN = "0090-6778 (print), 1558-0857 (electronic)", ISSN-L = "0090-6778", bibdate = "Fri Jul 22 09:48:58 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "IEEE Transactions on Communications", } @Article{Chen:2011:ECV, author = "Rui-Lin Chen and Chichyang Chen", title = "Efficient computation of very high effective-precision exponential function with additive normalization method", journal = j-J-CHINESE-INST-ENG, volume = "34", number = "7", pages = "935--946", month = oct, year = "2011", CODEN = "CKCKDZ", DOI = "https://doi.org/10.1080/02533839.2011.591963", ISSN = "2158-7299", fjournal = "Journal of the Chinese Institute of Engineers = Chung-kuo kung ch'eng hsueh kan", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", bibdate = "Tue Nov 11 14:57:36 2025", acknowledgement = ack-nhfb, } @Article{Chen:2011:SPF, author = "Chao-Ping Chen", title = "Some properties of functions related to the gamma, psi and tetragamma functions", journal = j-COMPUT-MATH-APPL, volume = "62", number = "9", pages = "3389--3395", month = nov, year = "2011", CODEN = "CMAPDK", ISSN = "0898-1221 (print), 1873-7668 (electronic)", ISSN-L = "0898-1221", bibdate = "Wed Mar 1 21:51:02 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/computmathappl2010.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0898122111007267", acknowledgement = ack-nhfb, fjournal = "Computers and Mathematics with Applications", journal-URL = "http://www.sciencedirect.com/science/journal/08981221", } @InProceedings{Chen:2011:TSA, author = "Jianxun Chen and Yongzhong Huang and Shaozhong Guo and Shimiao Chen and Wei Wang", booktitle = "{2011 Third International Conference on Measuring Technology and Mechatronics Automation (ICMTMA)}", title = "Test Standardization and Analyse Model of Mathematical Functions for Precision", volume = "3", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "652--655", year = "2011", DOI = "https://doi.org/10.1109/ICMTMA.2011.734", ISBN = "0-7695-4296-4", ISBN-13 = "978-0-7695-4296-6", bibdate = "Tue Sep 27 08:11:02 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5721571", abstract = "This article describes problems of meet the requirements to implementations of mathematical functions working with floating-point numbers, and so facilitate the comprehensive testing of mathematical functions. Inconsistency and incompleteness of available standards in the domain is demonstrated. Correct rounding requirement is suggested to guarantee preservation of all important properties of functions and to support high level of interoperability between different mathematical libraries and software using them. The article also concerns precision analyse of mathematical functions. Conformance test construction method is proposed based on different sources of test data.", acknowledgement = ack-nhfb, book-URL = "http://ieeexplore.ieee.org/xpl/mostRecentIssue.jsp?punumber=5720445", remark = "This paper contains unattributed plagiaristic copying of material from \url{https://www.math.utah.edu/~beebe/software/ieee/index.html}.", } @Article{Chlebus:2011:RSI, author = "Edward Chlebus", title = "A Recursive Scheme for Improving the Original Rate of Convergence to the {Euler--Mascheroni} Constant", journal = j-AMER-MATH-MONTHLY, volume = "118", number = "3", pages = "268--274", month = mar, year = "2011", CODEN = "AMMYAE", DOI = "https://doi.org/10.4169/amer.math.monthly.118.03.268", ISSN = "0002-9890 (print), 1930-0972 (electronic)", ISSN-L = "0002-9890", bibdate = "Mon Jan 30 08:58:19 MST 2012", bibsource = "http://www.jstor.org/journals/00029890.html; http://www.jstor.org/stable/10.4169/amermathmont.118.issue-3; https://www.math.utah.edu/pub/tex/bib/amermathmonthly2010.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.jstor.org/stable/pdfplus/10.4169/amer.math.monthly.118.03.268.pdf", abstract = "We have used Euler--Maclaurin summation to develop a recursive scheme for modifying the original approximation for the Euler--Mascheroni constant $ \gamma $. Convergence to $ \gamma $ resulting from successively employing the proposed scheme has been significantly accelerated while the form of the approximation originally introduced by Euler is still preserved.", acknowledgement = ack-nhfb, fjournal = "American Mathematical Monthly", journal-URL = "https://www.jstor.org/journals/00029890.htm", remark = "The author derives relations between $ \gamma $ and the $n$-th partial sum of the harmonic series that have an error $ O(n^{-2 k}) $ for increasing $k$. He also references prior work from 2009 that computes $ \gamma $ to 29,844,489,545 decimal digits.", } @Article{Choi:2011:AFT, author = "Junesang Choi and H. M. Srivastava", title = "Asymptotic formulas for the triple {Gamma} function {$ \Gamma_3 $} by means of its integral representation", journal = j-APPL-MATH-COMP, volume = "218", number = "6", pages = "2631--2640", day = "15", month = nov, year = "2011", CODEN = "AMHCBQ", DOI = "https://doi.org/10.1016/j.amc.2011.08.002", ISSN = "0096-3003 (print), 1873-5649 (electronic)", ISSN-L = "0096-3003", bibdate = "Tue Oct 25 09:03:08 MDT 2011", bibsource = "http://www.sciencedirect.com/science/journal/00963003; https://www.math.utah.edu/pub/tex/bib/applmathcomput2010.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0096300311010289", acknowledgement = ack-nhfb, fjournal = "Applied Mathematics and Computation", journal-URL = "http://www.sciencedirect.com/science/journal/00963003", } @Article{Colman:2011:VCC, author = "Michel Colman and Annie Cuyt and Joris {Van Deun}", title = "Validated computation of certain hypergeometric functions", journal = j-TOMS, volume = "38", number = "2", pages = "11:1--11:20", month = dec, year = "2011", CODEN = "ACMSCU", DOI = "https://doi.org/10.1145/2049673.2049675", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Fri Dec 30 17:43:07 MST 2011", bibsource = "http://www.acm.org/pubs/contents/journals/toms/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib", abstract = "We present an efficient algorithm for the validated high-precision computation of real continued fractions, accurate to the last digit. The algorithm proceeds in two stages. In the first stage, computations are done in double precision. A forward error analysis and some heuristics are used to obtain an a priori error estimate. This estimate is used in the second stage to compute the fraction to the requested accuracy in high precision (adaptively incrementing the precision for reasons of efficiency). A running error analysis and techniques from interval arithmetic are used to validate the result.", acknowledgement = ack-nhfb, articleno = "11", fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", } @Article{deDinechin:2011:CFP, author = "Florent de Dinechin and Christoph Lauter and Guillaume Melquiond", title = "Certifying the Floating-Point Implementation of an Elementary Function Using {Gappa}", journal = j-IEEE-TRANS-COMPUT, volume = "60", number = "2", pages = "242--253", month = feb, year = "2011", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/TC.2010.128", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Sun Feb 20 19:15:33 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", abstract = "High confidence in floating-point programs requires proving numerical properties of final and intermediate values. One may need to guarantee that a value stays within some range, or that the error relative to some ideal value is well bounded. This certification may require a time-consuming proof for each line of code, and it is usually broken by the smallest change to the code, e.g., for maintenance or optimization purpose. Certifying floating-point programs by hand is, therefore, very tedious and error-prone. The Gappa proof assistant is designed to make this task both easier and more secure, due to the following novel features: It automates the evaluation and propagation of rounding errors using interval arithmetic. Its input format is very close to the actual code to validate. It can be used incrementally to prove complex mathematical properties pertaining to the code. It generates a formal proof of the results, which can be checked independently by a lower level proof assistant like Coq. Yet it does not require any specific knowledge about automatic theorem proving, and thus, is accessible to a wide community. This paper demonstrates the practical use of this tool for a widely used class of floating-point programs: implementations of elementary functions in a mathematical library.", acknowledgement = ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", } @InProceedings{Fu:2011:ETB, author = "Hua Fu and Pooi-Yuen Kam", booktitle = "2011 {IEEE} Global Telecommunications Conference --- {GLOBECOM 2011}", title = "Exponential-Type Bounds on the First-Order {Marcum} Q-Function", publisher = pub-IEEE, address = pub-IEEE:adr, month = dec, year = "2011", DOI = "https://doi.org/10.1109/glocom.2011.6133801", bibdate = "Sat Dec 16 16:28:28 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "https://ieeexplore.ieee.org/document/6133801/", acknowledgement = ack-nhfb, } @Article{Fukushima:2011:PFC, author = "Toshio Fukushima", title = "Precise and fast computation of the general complete elliptic integral of the second kind", journal = j-MATH-COMPUT, volume = "80", number = "275", pages = "1725--1743", month = jul, year = "2011", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Mon Apr 18 06:32:30 MDT 2011", bibsource = "http://www.ams.org/mcom/2011-80-275; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.ams.org/journals/mcom/2011-80-275/S0025-5718-2011-02455-5/home.html; http://www.ams.org/journals/mcom/2011-80-275/S0025-5718-2011-02455-5/S0025-5718-2011-02455-5.pdf", abstract = "We developed an efficient procedure to evaluate two auxiliary complete elliptic integrals of the second kind $ B(m) $ and $ D(m) $ by using their Taylor series expansions, the definition of Jacobi's nome, and Legendre's relation. The developed procedure is more precise than the existing ones in the sense that the maximum relative errors are 1--3 machine epsilons, and it runs drastically faster; around 5 times faster than Bulirsch's cel2 and 16 times faster than Carlson's $ R_F $ and $ R_D $.", acknowledgement = ack-nhfb, fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Gautschi:2011:LWF, author = "Walter Gautschi", title = "The {Lambert} {$W$}-functions and some of their integrals: a case study of high-precision computation", journal = j-NUMER-ALGORITHMS, volume = "57", number = "1", pages = "27--34", month = may, year = "2011", CODEN = "NUALEG", DOI = "https://doi.org/10.1007/s11075-010-9409-6", ISSN = "1017-1398 (print), 1572-9265 (electronic)", ISSN-L = "1017-1398", bibdate = "Wed Apr 27 08:44:14 MDT 2011", bibsource = "http://springerlink.metapress.com/openurl.asp?genre=issue&issn=1017-1398&volume=57&issue=1; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.springerlink.com/openurl.asp?genre=article&issn=1017-1398&volume=57&issue=1&spage=27", abstract = "The real-valued Lambert W-functions considered here are $ w_0 (y) $ and $ w_{-1}(y) $, solutions of $ w e^w = y $, $ - 1 / e < y < 0 $, with values respectively in $ ( - 1, 0) $ and $ ( - \infty, - 1) $. A study is made of the numerical evaluation to high precision of these functions and of the integrals, $ \alpha > 0 $, $ \beta \in \mathbb {R} $, and $ \alpha > - 1 $, $ \beta < 1 $. For the latter we use known integral representations and their evaluation by nonstandard Gaussian quadrature, if $ \alpha \neq \beta $, and explicit formulae involving the trigamma function, if $ \alpha = \beta $.", acknowledgement = ack-nhfb, fjournal = "Numerical Algorithms", journal-URL = "http://link.springer.com/journal/11075", keywords = "Integrals of Lambert W-functions; Lambert W-functions; Nonstandard Gaussian quadrature; Variable-precision computation", } @Article{Gil:2011:APC, author = "Amparo Gil and Javier Segura and Nico M. Temme", title = "{Algorithm 914}: {Parabolic} cylinder function {$ W(a, x) $} and its derivative", journal = j-TOMS, volume = "38", number = "1", pages = "6:1--6:5", month = nov, year = "2011", CODEN = "ACMSCU", DOI = "https://doi.org/10.1145/2049662.2049668", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Thu Dec 15 08:59:34 MST 2011", bibsource = "http://www.acm.org/pubs/contents/journals/toms/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fortran3.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib", abstract = "A Fortran 90 program for the computation of the real parabolic cylinder functions $ W(a, \pm x) $, $ x \geq 0 $ and their derivatives is presented. The code also computes scaled functions for $ a > 50 $. The functions $ W(a, \pm x) $ are a numerically satisfactory pair of solutions of the parabolic cylinder equation $ y^\prime + (x^2 / 4 - a)y = 0 $, $ x \geq 0 $. Using Wronskian tests, we claim a relative accuracy better than $ 5 \times 10^{-13} $ in the computable range of unscaled functions, while for scaled functions the aimed relative accuracy is better than $ 5 \times 10^{-14} $. This code, together with the algorithm and related software described in Gil et al.", acknowledgement = ack-nhfb, articleno = "6", fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", } @Article{Gil:2011:FAC, author = "Amparo Gil and Javier Segura and Nico M. Temme", title = "Fast and accurate computation of the {Weber} parabolic cylinder function {$ W(a, x) $}", journal = j-IMA-J-NUMER-ANAL, volume = "31", number = "3", pages = "1194--1216", month = jul, year = "2011", CODEN = "IJNADH", DOI = "https://doi.org/10.1093/imanum/drq012", ISSN = "0272-4979 (print), 1464-3642 (electronic)", ISSN-L = "0272-4979", bibdate = "Fri Jul 15 12:37:42 MDT 2011", bibsource = "http://imanum.oxfordjournals.org/content/31/3.toc; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://imajna.oxfordjournals.org/content/31/3/1194.full.pdf+html", acknowledgement = ack-nhfb, fjournal = "IMA Journal of Numerical Analysis", journal-URL = "http://imajna.oxfordjournals.org/content/by/year", onlinedate = "July 7, 2010", } @Article{Jaime:2011:HSA, author = "F. J. Jaime and M. A. S{\'a}nchez and J. Hormigo and J. Villalba and E. L. Zapata", title = "High-Speed Algorithms and Architectures for Range Reduction Computation", journal = j-IEEE-TRANS-VLSI-SYST, volume = "19", number = "3", pages = "512--516", month = "????", year = "2011", CODEN = "IEVSE9", DOI = "https://doi.org/10.1109/TVLSI.2009.2033932", ISSN = "1063-8210 (print), 1557-9999 (electronic)", ISSN-L = "1063-8210", bibdate = "Tue Sep 27 08:11:02 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5308221", abstract = "Range reduction is a crucial step for accuracy in trigonometric functions evaluation. This paper shows and compares a set of algorithms for additive range reduction computation and their corresponding application-specific integrated circuit implementations (ensuring an accuracy of one unit in the last place). A word-serial architecture implementation has been used as a reference for clearer comparisons. Besides, a new table-based pipelined architecture for range reduction has also been proposed.", acknowledgement = ack-nhfb, book-URL = "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=92", fjournal = "IEEE Transactions on Very Large Scale Integration (VLSI) Systems", } @Article{Jang:2011:CTS, author = "Won Mee Jang", title = "Corrections to {``A Simple Upper Bound of the Gaussian $Q$-Function with Closed-form Error Bound''}", journal = j-IEEE-COMMUN-LET, volume = "15", number = "12", pages = "1274--1274", month = dec, year = "2011", CODEN = "ICLEF6", DOI = "https://doi.org/10.1109/lcomm.2011.101911.111996", ISSN = "1089-7798 (print), 1558-2558 (electronic)", ISSN-L = "1089-7798", bibdate = "Sat Dec 16 16:46:05 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "https://ieeexplore.ieee.org/document/6065242/", acknowledgement = ack-nhfb, fjournal = "IEEE Communications Letters", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=4234", } @Article{Jang:2011:SUB, author = "Won Mee Jang", title = "A Simple Upper Bound of the {Gaussian} {$Q$}-Function with Closed-Form Error Bound", journal = j-IEEE-COMMUN-LET, volume = "15", number = "2", pages = "157--159", month = feb, year = "2011", CODEN = "ICLEF6", DOI = "https://doi.org/10.1109/lcomm.2011.011011.102207", ISSN = "1089-7798 (print), 1558-2558 (electronic)", ISSN-L = "1089-7798", bibdate = "Sat Dec 16 16:47:28 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "https://ieeexplore.ieee.org/document/5692888/", acknowledgement = ack-nhfb, fjournal = "IEEE Communications Letters", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=4234", } @Article{Jeannerod:2011:CFP, author = "Claude-Pierre Jeannerod and Herv{\'e} Knochel and Christophe Monat and Guillaume Revy", title = "Computing Floating-Point Square Roots via Bivariate Polynomial Evaluation", journal = j-IEEE-TRANS-COMPUT, volume = "60", number = "2", pages = "214--227", month = feb, year = "2011", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/TC.2010.152", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Sun Feb 20 19:15:33 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput2010.bib", abstract = "In this paper, we show how to reduce the computation of correctly rounded square roots of binary floating-point data to the fixed-point evaluation of some particular integer polynomials in two variables. By designing parallel and accurate evaluation schemes for such bivariate polynomials, we show further that this approach allows for high instruction-level parallelism (ILP) exposure, and thus, potentially low-latency implementations. Then, as an illustration, we detail a C implementation of our method in the case of IEEE 754-2008 binary32 floating-point data (formerly called single precision in the 1985 version of the IEEE 754 standard). This software implementation, which assumes 32-bit unsigned integer arithmetic only, is almost complete in the sense that it supports special operands, subnormal numbers, and all rounding-direction attributes, but not exception handling (that is, status flags are not set). Finally, we have carried out experiments with this implementation on the ST231, an integer processor from the STMicroelectronics' ST200 family, using the ST200 family VLIW compiler. The results obtained demonstrate the practical interest of our approach in that context: for all rounding-direction attributes, the generated assembly code is optimally scheduled and has indeed low latency (23 cycles).", acknowledgement = ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", } @Article{Johansson:2011:CRE, author = "Bo G{\"o}ran Johansson", title = "Cube root extraction in medieval mathematics", journal = j-HIST-MATH, volume = "38", number = "3", pages = "338--367", month = aug, year = "2011", CODEN = "HIMADS", ISSN = "0315-0860 (print), 1090-249X (electronic)", ISSN-L = "0315-0860", bibdate = "Wed Jun 26 06:21:13 MDT 2013", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/histmath.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0315086010000546", acknowledgement = ack-nhfb, fjournal = "Historia Mathematica", journal-URL = "http://www.sciencedirect.com/science/journal/03150860", } @Article{Knessl:2011:EAF, author = "Charles Knessl and Mark W. Coffey", title = "An effective asymptotic formula for the {Stieltjes} constants", journal = j-MATH-COMPUT, volume = "80", number = "273", pages = "379--386", month = jan, year = "2011", CODEN = "MCMPAF", DOI = "https://doi.org/10.1090/S0025-5718-2010-02390-7", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Wed Oct 13 16:46:42 MDT 2010", bibsource = "http://www.ams.org/mcom/2011-80-273; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.ams.org/journals/mcom/2011-80-273/S0025-5718-2010-02390-7/home.html; http://www.ams.org/journals/mcom/2011-80-273/S0025-5718-2010-02390-7/S0025-5718-2010-02390-7.pdf", abstract = "The Stieltjes constants $ \gamma_k $ appear in the coefficients in the regular part of the Laurent expansion of the Riemann zeta function $ \zeta (s) $ about its only pole at $ s = 1 $. We present an asymptotic expression for $ \gamma_k $ for $ k \gg 1 $. This form encapsulates both the leading rate of growth and the oscillations with $k$. Furthermore, our result is effective for computation, consistently in close agreement (for both magnitude and sign) for even moderate values of $k$. Comparison to some earlier work is made.", acknowledgement = ack-nhfb, fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Kodama:2011:AMC, author = "Masao Kodama", title = "{Algorithm 912}: a Module for Calculating Cylindrical Functions of Complex Order and Complex Argument", journal = j-TOMS, volume = "37", number = "4", pages = "47:1--47:25", month = feb, year = "2011", CODEN = "ACMSCU", DOI = "https://doi.org/10.1145/1916461.1916471", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Tue Mar 1 16:05:18 MST 2011", bibsource = "http://www.acm.org/pubs/contents/journals/toms/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib", abstract = "The present algorithm provides a module for calculating the cylindrical functions $ J_\nu (z) $, $ Y_\nu (z) $, $ H_{\nu (1)}(z) $, and $ H_{\nu (2)}(z) $, where the order $ \nu $ is complex and the complex argument $z$ satisfies $ - \pi < \arg z \leq \pi $. The algorithm is written in Fortran 90 and calculates the functions using real and complex numbers of any intrinsic data type whose kind type parameter the user's Fortran system accepts. The methods of calculating the functions are based on two kinds of series expansions and numerical integration. Wronskian tests examine the functional values computed by this algorithm with double precision at 4,100,625 pseudorandom test points in the region $ | \Re \nu | \leq 60 $, $ | \Im \nu | \leq 60 $, $ | \Re z| \leq 300 $, $ | \Im z| \leq 300 $.", acknowledgement = ack-nhfb, articleno = "47", fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", } @Article{Kormanyos:2011:APC, author = "Christopher Kormanyos", title = "{Algorithm 910}: a Portable {C++} Multiple-Precision System for Special-Function Calculations", journal = j-TOMS, volume = "37", number = "4", pages = "45:1--45:27", month = feb, year = "2011", CODEN = "ACMSCU", DOI = "https://doi.org/10.1145/1916461.1916469", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Tue Mar 1 16:05:18 MST 2011", bibsource = "http://www.acm.org/pubs/contents/journals/toms/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib", abstract = "This article presents a portable C++ system for multiple precision calculations of special functions called {\tt e\_float}. It has an extendable architecture with a uniform C++ layer which can be used with any suitably prepared MP type. The system implements many high-precision special functions and extends some of these to very large parameter ranges. It supports calculations with 30 \ldots{} 300 decimal digits of precision. Interoperabilities with Microsoft's CLR, Python, and Mathematica{\reg} are supported. The {\tt e\_float} system and its usage are described in detail. Implementation notes, testing results, and performance measurements are provided.", acknowledgement = ack-nhfb, articleno = "45", fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", } @Article{Lopez-Benitez:2011:VAA, author = "Miguel L{\"o}pez-Benitez and Fernando Casadevall", title = "Versatile, Accurate, and Analytically Tractable Approximation for the {Gaussian} {$Q$}-Function", journal = j-IEEE-TRANS-COMM, volume = "59", number = "4", pages = "917--922", month = apr, year = "2011", CODEN = "IECMBT", DOI = "https://doi.org/10.1109/tcomm.2011.012711.100105", ISSN = "0090-6778 (print), 1558-0857 (electronic)", ISSN-L = "0090-6778", bibdate = "Sat Dec 16 17:01:00 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "https://ieeexplore.ieee.org/document/5706433/", acknowledgement = ack-nhfb, fjournal = "IEEE Transactions on Communications", } @InProceedings{Matula:2011:PLP, author = "David W. Matula and Mihai T. Panu", title = "A Prescale-Lookup-Postscale Additive Procedure for Obtaining a Single Precision Ulp Accurate Reciprocal", crossref = "Schwarz:2011:PIS", pages = "177--183", year = "2011", DOI = "https://doi.org/10.1109/ARITH.2011.31", bibdate = "Sat Aug 20 09:00:00 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5992123", acknowledgement = ack-nhfb, keywords = "ARITH-20", } @Article{Mortici:2011:IAF, author = "Cristinel Mortici", title = "Improved asymptotic formulas for the gamma function", journal = j-COMPUT-MATH-APPL, volume = "61", number = "11", pages = "3364--3369", month = jun, year = "2011", CODEN = "CMAPDK", DOI = "https://doi.org/10.1016/j.camwa.2011.04.036", ISSN = "0898-1221 (print), 1873-7668 (electronic)", ISSN-L = "0898-1221", bibdate = "Wed Mar 1 21:50:47 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/computmathappl2010.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0898122111003373", acknowledgement = ack-nhfb, fjournal = "Computers and Mathematics with Applications", journal-URL = "http://www.sciencedirect.com/science/journal/08981221", } @InProceedings{Nannarelli:2011:RCD, author = "Alberto Nannarelli", title = "Radix-16 Combined Division and Square Root Unit", crossref = "Schwarz:2011:PIS", pages = "169--176", year = "2011", DOI = "https://doi.org/10.1109/ARITH.2011.30", bibdate = "Sat Aug 20 09:00:00 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5992122", acknowledgement = ack-nhfb, keywords = "ARITH-20; sqrt(x); square root", } @Article{Paszkowski:2011:UMC, author = "Stefan Paszkowski", title = "Untypical methods of convergence acceleration", journal = j-NUMER-ALGORITHMS, volume = "56", number = "2", pages = "185--209", month = "????", year = "2011", CODEN = "NUALEG", ISSN = "1017-1398 (print), 1572-9265 (electronic)", ISSN-L = "1017-1398", MRclass = "65B10 (33C20 33E05 41Axx)", MRnumber = "MR2755669", bibdate = "Thu Dec 01 09:27:45 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Numerical Algorithms", journal-URL = "http://link.springer.com/journal/11075", keywords = "convergence acceleration", } @Article{Pawellek:2011:GJE, author = "Michael Pawellek", title = "On a generalization of {Jacobi}'s elliptic functions and the double {sine--Gordon} kink chain", journal = j-J-MATH-PHYS, volume = "52", number = "11", pages = "113701", month = nov, year = "2011", CODEN = "JMAPAQ", DOI = "https://doi.org/10.1063/1.3656873", ISSN = "0022-2488 (print), 1089-7658 (electronic), 1527-2427", ISSN-L = "0022-2488", bibdate = "Wed Jan 4 08:04:23 MST 2012", bibsource = "http://www.aip.org/ojs/jmp.html; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jmathphys2010.bib", URL = "http://jmp.aip.org/resource/1/jmapaq/v52/i11/p113701_s1", acknowledgement = ack-nhfb, fjournal = "Journal of Mathematical Physics", journal-URL = "http://jmp.aip.org/", onlinedate = "4 November 2011", pagecount = "18", } @Article{Pegoraro:2011:ECV, author = "Vincent Pegoraro and Philipp Slusallek", title = "On the Evaluation of the Complex-Valued Exponential Integral", journal = j-J-GRAPHICS-GPU-GAME-TOOLS, volume = "15", number = "3", pages = "183--198", year = "2011", CODEN = "????", DOI = "https://doi.org/10.1080/2151237X.2011.617177", ISSN = "2151-237X", bibdate = "Wed Dec 14 10:31:39 MST 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jgraphtools.bib", abstract = "Although its applications span a broad scope of scientific fields ranging from applied physics to computer graphics, the exponential integral is a nonelementary special function available in specialized software packages but not in standard libraries, consequently requiring custom implementations on most platforms. In this paper, we provide a concise and comprehensive description of how to evaluate the complex-valued exponential integral. We first introduce some theoretical background on the main characteristics of the function, and outline available third-party proprietary implementations. We then provide an analysis of the various known representations of the function and present an effective algorithm allowing the computation of results within a desired accuracy, together with the corresponding pseudocode in order to facilitate portability onto various systems. An application to the calculation of the closed-form solution to single light scattering in homogeneous participating media illustrates the practical benefits of the provided implementation with the hope that, in the long term, the latter will contribute to standardizing the availability of the complex-valued exponential integral on graphics platforms.", acknowledgement = ack-nhfb, journal-URL = "http://www.tandfonline.com/loi/ujgt20", onlinedate = "21 Oct 2011", } @InProceedings{Pouyan:2011:VIL, author = "Peyman Pouyan and Erik Hertz and Peter Nilsson", booktitle = "{2011 20th European Conference on Circuit Theory and Design (ECCTD)}", title = "A {VLSI} implementation of logarithmic and exponential functions using a novel parabolic synthesis methodology compared to the {CORDIC} algorithm", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "709--712", year = "2011", DOI = "https://doi.org/10.1109/ECCTD.2011.6043642", bibdate = "Mon Oct 27 10:32:44 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, keywords = "Accuracy; Algorithm design and analysis; Application specific integrated circuits; Approximation methods; Computer architecture; CORDIC; Exponential; Field programmable gate arrays; FPGA; Hardware; Logarithmic; Parabolic Synthesis; VLSI", } @TechReport{Roegel:2011:RTB, author = "Denis Roegel", title = "A reconstruction of the tables of {Briggs}' \booktitle{Arithmetica logarithmica} (1624)", type = "Report", number = "????", institution = inst-LORIA-INRIA-LORRAINE, address = inst-LORIA-INRIA-LORRAINE:adr, pages = "334", day = "11", month = jan, year = "2011", bibdate = "Mon Nov 10 08:51:59 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "Revised 30 November 2014. See \cite{Briggs:1624:AL}.", URL = "https://locomat.loria.fr/briggs1624/briggs1624doc.pdf", acknowledgement = ack-nhfb, } @Article{Shi:2011:AEA, author = "Qinghua Shi and Y. Karasawa", title = "An Accurate and Efficient Approximation to the {Gaussian} {$Q$}-Function and its Applications in Performance Analysis in {Nakagami}-$m$ Fading", journal = j-IEEE-COMMUN-LET, volume = "15", number = "5", pages = "479--481", month = may, year = "2011", CODEN = "ICLEF6", DOI = "https://doi.org/10.1109/lcomm.2011.032111.102440", ISSN = "1089-7798 (print), 1558-2558 (electronic)", ISSN-L = "1089-7798", bibdate = "Sat Dec 16 17:32:51 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "https://ieeexplore.ieee.org/document/5740503/", acknowledgement = ack-nhfb, fjournal = "IEEE Communications Letters", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=4234", } @Article{Smith:2011:AMP, author = "David M. Smith", title = "{Algorithm 911}: Multiple-Precision Exponential Integral and Related Functions", journal = j-TOMS, volume = "37", number = "4", pages = "46:1--46:16", month = feb, year = "2011", CODEN = "ACMSCU", DOI = "https://doi.org/10.1145/1916461.1916470", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Tue Mar 1 16:05:18 MST 2011", bibsource = "http://www.acm.org/pubs/contents/journals/toms/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib", abstract = "This article describes a collection of Fortran-95 routines for evaluating the exponential integral function, error function, sine and cosine integrals, Fresnel integrals, Bessel functions, and related mathematical special functions using the FM multiple-precision arithmetic package.", acknowledgement = ack-nhfb, articleno = "46", fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", } @Article{Srivastava:2011:ADZ, author = "H. M. Srivastava and Jian-Rong Zhou and Zhi-Gang Wang", title = "Asymptotic distributions of the zeros of certain classes of hypergeometric functions and polynomials", journal = j-MATH-COMPUT, volume = "80", number = "275", pages = "1769--1784", month = jul, year = "2011", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Mon Apr 18 06:32:30 MDT 2011", bibsource = "http://www.ams.org/mcom/2011-80-275; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp2010.bib", URL = "http://www.ams.org/journals/mcom/2011-80-275/S0025-5718-2011-02409-9/home.html; http://www.ams.org/journals/mcom/2011-80-275/S0025-5718-2011-02409-9/S0025-5718-2011-02409-9.pdf", acknowledgement = ack-nhfb, fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Strollo:2011:EFH, author = "Antonio Giuseppe Maria Strollo and Davide {De Caro} and Nicola Petra", title = "Elementary Functions Hardware Implementation Using Constrained Piecewise-Polynomial Approximations", journal = j-IEEE-TRANS-COMPUT, volume = "60", pages = "418--432", year = "2011", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/TC.2010.127", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Sun Feb 20 19:10:07 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", abstract = "A novel technique for designing piecewise-polynomial interpolators for hardware implementation of elementary functions is investigated in this paper. In the proposed approach, the interval where the function is approximated is subdivided in equal length segments and two adjacent segments are grouped in a segment pair. Suitable constraints are then imposed between the coefficients of the two interpolating polynomials in each segment pair. This allows reducing the total number of stored coefficients. It is found that the increase in the approximation error due to constraints between polynomial coefficients can easily be overcome by increasing the fractional bits of the coefficients. Overall, compared with standard unconstrained piecewise-polynomial approximation having the same accuracy, the proposed method results in a considerable advantage in terms of the size of the lookup table needed to store polynomial coefficients. The calculus of the coefficients of constrained polynomials and the optimization of coefficients bit width is also investigated in this paper. Results for several elementary functions and target precision ranging from 12 to 42 bits are presented. The paper also presents VLSI implementation results, targeting a 90 nm CMOS technology, and using both direct and Horner architectures for constrained degree-1, degree-2, and degree-3 approximations.", acknowledgement = ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", keywords = "computer arithmetic; elementary functions; min-max approximation; polynomial approximation; VLSI systems.", } @InProceedings{Tang:2011:TCT, author = "Ping Tak Peter Tang and J. Adam Butts and Ron O. Dror and David E. Shaw", title = "Tight Certification Techniques for Digit-by-Rounding Algorithms with Application to a New $ 1 / \sqrt {x} $ Design", crossref = "Schwarz:2011:PIS", pages = "159--168", year = "2011", DOI = "https://doi.org/10.1109/ARITH.2011.29", bibdate = "Sat Aug 20 09:00:00 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5992121", acknowledgement = ack-nhfb, keywords = "ARITH-20; reciprocal square root; rsqrt(x)", } @Article{Trudgian:2011:ITM, author = "Timothy Trudgian", title = "Improvements to {Turing}'s method", journal = j-MATH-COMPUT, volume = "80", number = "276", pages = "2259--2279", month = oct, year = "2011", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Mon Oct 24 10:33:34 MDT 2011", bibsource = "http://www.ams.org/mcom/2011-80-276; https://www.math.utah.edu/pub/bibnet/authors/t/turing-alan-mathison.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp2010.bib", note = "See \cite{Turing:1953:SCR,Lehman:1970:DZR}.", URL = "http://www.ams.org/journals/mcom/2011-80-276/S0025-5718-2011-02470-1/home.html; http://www.ams.org/journals/mcom/2011-80-276/S0025-5718-2011-02470-1/S0025-5718-2011-02470-1.pdf; http://www.ams.org/mathscinet-getitem?mr=2813359", abstract = "This article improves the estimate of the size of the definite integral of {$ S(t) $}, the argument of the Riemann zeta-function. The primary application of this improvement is Turing's Method for the Riemann zeta-function. Analogous improvements are given for the arguments of Dirichlet {$L$}-functions and of Dedekind zeta-functions.", acknowledgement = ack-nhfb, fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{VanDeun:2011:RIC, author = "Joris {Van Deun} and Lloyd N. Trefethen", title = "A robust implementation of the {Carath{\'e}odory-Fej{\'e}r} method for rational approximation", journal = j-BIT-NUM-MATH, volume = "51", number = "??", pages = "??--??", month = "????", year = "2011", CODEN = "BITTEL, NBITAB", DOI = "https://doi.org/10.1007/s10543-011-0331-7", ISSN = "0006-3835 (print), 1572-9125 (electronic)", ISSN-L = "0006-3835", bibdate = "Thu Sep 29 07:17:26 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.springerlink.com/content/ag2514840142707r/", abstract = "Best rational approximations are notoriously difficult to compute. However, the difference between the best rational approximation to a function and its Carath{\'e}odory-Fej{\'e}r (CF) approximation is often so small as to be negligible in practice, while CF approximations are far easier to compute. We present a robust and fast implementation of this method in the Chebfun software system and illustrate its use with several examples. Our implementation handles both polynomial and rational approximation and substantially improves upon earlier published software.", acknowledgement = ack-nhfb, journal-URL = "http://link.springer.com/journal/10543", keywords = "Carath{\'e}odory-Fej{\'e}r approximation; Chebfun; Near-best rational approximation", onlinedate = "04 May 2011", } @Article{Veling:2011:GIG, author = "E. J. M. Veling", title = "The Generalized Incomplete Gamma Function as sum over Modified {Bessel} Functions of the First Kind", journal = j-J-COMPUT-APPL-MATH, volume = "235", number = "14", pages = "4107--4116", day = "15", month = may, year = "2011", CODEN = "JCAMDI", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Sat Feb 25 13:24:28 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2010.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0377042711001245", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @InProceedings{Wu:2011:NEL, author = "Mingwei Wu and Xuzheng Lin and Pooi-Yuen Kam", booktitle = "{2011 IEEE 73rd Vehicular Technology Conference (VTC Spring)}", title = "New Exponential Lower Bounds on the {Gaussian} {$Q$}-Function via {Jensen}'s Inequality", publisher = pub-IEEE, address = pub-IEEE:adr, month = may, year = "2011", DOI = "https://doi.org/10.1109/vetecs.2011.5956392", bibdate = "Sat Dec 16 16:53:49 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "https://ieeexplore.ieee.org/document/5956392/", acknowledgement = ack-nhfb, } @Article{Yerukala:2011:ACS, author = "R. Yerukala and N. K. Boiroju and M. K. Reddy", title = "An Approximation to the {CDF} of Standard Normal Distribution", journal = "International Journal of Mathematical Archive", volume = "2", number = "7", pages = "1077--1079", month = "????", year = "2011", ISSN = "2229-5046", ISSN-L = "2229-5046", bibdate = "Sat Dec 16 18:03:12 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://ijma.info/index.php/ijma/article/view/393", acknowledgement = ack-nhfb, ajournal = "Int. J. Math. Arch.", } @Article{Zaghloul:2011:ACF, author = "Mofreh R. Zaghloul and Ahmed N. Ali", title = "{Algorithm 916}: Computing the {Faddeyeva} and {Voigt} functions", journal = j-TOMS, volume = "38", number = "2", pages = "15:1--15:22", month = dec, year = "2011", CODEN = "ACMSCU", DOI = "https://doi.org/10.1145/2049673.2049679", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Fri Dec 30 17:43:07 MST 2011", bibsource = "http://www.acm.org/pubs/contents/journals/toms/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/matlab.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib", note = "See remark \cite{Zaghloul:2016:RAC}.", abstract = "We present a MATLAB function for the numerical evaluation of the Faddeyeva function $ w(z) $. The function is based on a newly developed accurate algorithm. In addition to its higher accuracy, the software provides a flexible accuracy vs efficiency trade-off through a controlling parameter that may be used to reduce accuracy and computational time and vice versa. Verification of the flexibility, reliability, and superior accuracy of the algorithm is provided through comparison with standard algorithms available in other libraries and software packages.", acknowledgement = ack-nhfb, articleno = "15", fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", } @Article{Zhou:2011:ADZ, author = "Jian-Rong Zhou and Yu-Qiu Zhao", title = "Asymptotic distributions of the zeros of certain classes of {Gauss} hypergeometric polynomials", journal = j-APPL-MATH-COMP, volume = "218", number = "3", pages = "1153--1159", day = "1", month = oct, year = "2011", CODEN = "AMHCBQ", DOI = "https://doi.org/10.1016/j.amc.2011.05.106", ISSN = "0096-3003 (print), 1873-5649 (electronic)", ISSN-L = "0096-3003", bibdate = "Tue Oct 25 09:02:50 MDT 2011", bibsource = "http://www.sciencedirect.com/science/journal/00963003; https://www.math.utah.edu/pub/tex/bib/applmathcomput2010.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "Special Issue in Honour of Hari M. Srivastava on his 70th birth anniversary.", URL = "http://www.sciencedirect.com/science/article/pii/S0096300311007892", acknowledgement = ack-nhfb, fjournal = "Applied Mathematics and Computation", journal-URL = "http://www.sciencedirect.com/science/journal/00963003", } @Article{Adlaj:2012:EFP, author = "Semjon Adlaj", title = "An Eloquent Formula for the Perimeter of an Ellipse", journal = j-NAMS, volume = "59", number = "8", pages = "1094--1099", month = sep, year = "2012", CODEN = "AMNOAN", ISSN = "0002-9920 (print), 1088-9477 (electronic)", ISSN-L = "0002-9920", bibdate = "Wed Sep 05 09:12:25 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.ams.org/notices/201208/rtx120801094p.pdf", acknowledgement = ack-nhfb, fjournal = "Notices of the American Mathematical Society", journal-URL = "http://www.ams.org/notices/", keywords = "complete elliptic integral; pendulum; perimeter of ellipse", remark = "This paper introduces several arithmetic-geometric mean (AGM) algorithms for fast and practical computation of complete elliptic integrals.", } @Article{Al-Mohy:2012:MAB, author = "Awad H. Al-Mohy", title = "A more accurate {Briggs} method for the logarithm", journal = j-NUMER-ALGORITHMS, volume = "59", number = "3", pages = "393--402", month = mar, year = "2012", CODEN = "NUALEG", DOI = "https://doi.org/10.1007/s11075-011-9496-z", ISSN = "1017-1398 (print), 1572-9265 (electronic)", ISSN-L = "1017-1398", bibdate = "Fri Oct 26 08:07:24 MDT 2012", bibsource = "http://springerlink.metapress.com/openurl.asp?genre=issue&issn=1017-1398&volume=59&issue=3; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/numeralgorithms.bib", URL = "http://www.springerlink.com/content/4110609h521kg66m/; http://www.springerlink.com/openurl.asp?genre=article&issn=1017-1398&volume=59&issue=3&spage=393", abstract = "A new approach for computing an expression of the form $ a^{1 / 2^k} - 1 $ is presented that avoids the danger of subtractive cancellation in floating point arithmetic, where $a$ is a complex number not belonging to the closed negative real axis and $k$ is a nonnegative integer. We also derive a condition number for the problem. The algorithm therefore allows highly accurate numerical calculation of $ \log (a) $ using Briggs' method.", acknowledgement = ack-nhfb, fjournal = "Numerical Algorithms", journal-URL = "http://link.springer.com/journal/11075", keywords = "Briggs' method; Briggs' tables; Inverse scaling and squaring method; Logarithm function", } @Misc{Anonymous:2012:FIS, author = "Anonymous", title = "Fast inverse square root", howpublished = "Wikipedia article.", day = "20", month = mar, year = "2012", bibdate = "Mon Apr 02 17:03:18 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", note = "This article describes an algorithm for the inverse square root. The only novel feature is use of two IEEE-754 specific magic constants for 32-bit and 64-bit binary arithmetic that allow obtaining fast starting estimates for Newton--Raphson iterations by manipulating the floating-point representations as integers. The code fails to handle signed zero, Infinity, and NaN arguments, uses too few iterations, and does not adjust for rounding errors to obtain correctly-rounded results. See \cite{Blinn:1997:JBC}.", URL = "http://en.wikipedia.org/wiki/Fast_inverse_square_root", acknowledgement = ack-nhfb, } @Article{Bailey:2012:AIS, author = "David H. Bailey and Jonathan M. Borwein", title = "Ancient {Indian} Square Roots: An Exercise in Forensic Paleo-Mathematics", journal = j-AMER-MATH-MONTHLY, volume = "119", number = "8", pages = "646--657", month = oct, year = "2012", CODEN = "AMMYAE", DOI = "https://doi.org/10.4169/amer.math.monthly.119.08.646", ISSN = "0002-9890 (print), 1930-0972 (electronic)", ISSN-L = "0002-9890", bibdate = "Thu Nov 8 07:34:21 MST 2012", bibsource = "http://www.jstor.org/journals/00029890.html; http://www.jstor.org/stable/10.4169/amermathmont.119.issue-8; https://www.math.utah.edu/pub/tex/bib/amermathmonthly2010.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.jstor.org/stable/pdfplus/10.4169/amer.math.monthly.119.08.646.pdf", acknowledgement = ack-nhfb, fjournal = "American Mathematical Monthly", journal-URL = "https://www.jstor.org/journals/00029890.htm", } @InProceedings{Brisebarre:2012:MPK, author = "Nicolas Brisebarre and Milo D. Ercegovac and Jean-Michel Muller", editor = "{IEEE}", booktitle = "{2012 IEEE 23rd International Conference on Application-Specific Systems, Architectures and Processors, 9--11 July 2012. Delft, The Netherlands}", title = "{$ (M, p, k) $}-Friendly Points: a Table-Based Method for Trigonometric Function Evaluation", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "46--52", year = "2012", DOI = "https://doi.org/10.1109/ASAP.2012.17", ISBN = "0-7695-4768-0", ISBN-13 = "978-0-7695-4768-8", ISSN = "1063-6862", ISSN-L = "1063-6862", bibdate = "Fri Sep 29 10:49:22 2023", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, } @Article{Chen:2012:SIM, author = "Chao-Ping Chen and Necdet Batir", title = "Some inequalities and monotonicity properties associated with the gamma and psi functions", journal = j-APPL-MATH-COMP, volume = "218", number = "17", pages = "8217--8225", day = "1", month = may, year = "2012", CODEN = "AMHCBQ", DOI = "https://doi.org/10.1016/j.amc.2012.02.007", ISSN = "0096-3003 (print), 1873-5649 (electronic)", ISSN-L = "0096-3003", bibdate = "Thu Apr 5 06:00:26 MDT 2012", bibsource = "http://www.sciencedirect.com/science/journal/00963003; https://www.math.utah.edu/pub/tex/bib/applmathcomput2010.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0096300312001257", acknowledgement = ack-nhfb, fjournal = "Applied Mathematics and Computation", journal-URL = "http://www.sciencedirect.com/science/journal/00963003", remark = "New bounds on the gamma function in terms of the psi function, and a new estimate for the error in Stirling's formula, $ \Gamma (x + 1) \approx x^x e^{-x} \sqrt {2 \pi x} $.", } @Article{Cohl:2012:TEF, author = "Howard S. Cohl", title = "Table Errata to {``Formulas and theorems for the special functions of mathematical physics'' by W. Magnus, F. Oberhettinger \& R. P. Soni (1966)}", journal = j-MATH-COMPUT, volume = "81", number = "280", pages = "2251--2251", month = oct, year = "2012", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Nov 6 09:52:53 MST 2012", bibsource = "http://www.ams.org/mcom/2012-81-280; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp2010.bib", note = "See \cite{Magnus:1966:FTS}.", URL = "http://www.ams.org/journals/mcom/2012-81-280/S0025-5718-2012-02612-3; http://www.ams.org/journals/mcom/2012-81-280/S0025-5718-2012-02612-3/S0025-5718-2012-02612-3.pdf", acknowledgement = ack-nhfb, fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", remark = "Report of sign error in a sum of two Gauss hypergeometric functions for Ferrers function of the second kind.", } @Article{Cote:2012:CTL, author = "F. D. C{\^o}t{\'e} and I. N. Psaromiligkos and W. J. Gross", title = "A {Chernoff}-type lower bound for the {Gaussian} {$Q$}-function", journal = "arxiv.org", volume = "??", number = "??", pages = "??--??", year = "2012", bibdate = "Sat Dec 16 16:04:00 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "https://arxiv.org/abs/1202.6483", acknowledgement = ack-nhfb, pagecount = "3", } @Book{Crandall:2012:ARS, author = "Richard E. Crandall", title = "Algorithmic Reflections: Selected Works", publisher = "PSI Press", address = "Portland, OR, USA", edition = "First Perfectly Scientific Press paperback", pages = "410", year = "2012", ISBN = "1-935638-19-X", ISBN-13 = "978-1-935638-19-3", LCCN = "QA958 .C736 2012", bibdate = "Fri Jun 30 11:14:26 MDT 2023", bibsource = "fsz3950.oclc.org:210/WorldCat; https://www.math.utah.edu/pub/bibnet/authors/c/crandall-richard-e.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, author-dates = "1947--2012", subject = "Algorithms; Algorithmes", tableofcontents = "Part I: Number theory \\ On the $ 3 x + 1$ problem \\ On a conjecture of Crandall concerning the $ q n + 1$ problem \\ A search for Wieferich and Wilson primes \\ Parallelization of Pollard-rho factorization \\ Three new factors of Fermat numbers \\ Random generators and normal numbers \\ The googol-th bit of the Erd{\H{o}}s--Borwein constant \\ Part II: Analytical algorithms \\ Fast evaluation of multiple zeta sums \\ On the Khintchine Constant \\ On the dynamics of certain recurrence relations \\ Effective Laguerre asymptotics \\ Theory of ROOF walks \\ Unified algorithms for polylogarithm, $L$-series, and zeta variants \\ Part III: Physics, biology, epidemics, and physiology \\ The potential within a crystal lattice \\ The fractal character of space-time epidemics \\ Mathematical signatures as tools for visual dysfunction \\ NLA system for medical-data classification \\ On the fractal distribution of brain synapses", } @Misc{Crandall:2012:UAP, author = "R. E. Crandall", title = "Unified algorithms for polylogarithm, {$L$}-series, and zeta variants", type = "Preprint", pages = "53", year = "2012", bibdate = "Tue Sep 09 11:50:04 2014", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/c/crandall-richard-e.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "Published in \cite{Crandall:2012:ARS}.", URL = "http://www.perfscipress.com/papers/universalTOC25.pdf; https://web.archive.org/web/20130430193005/http://www.perfscipress.com/papers/universalTOC25.pdf", abstract = "We describe a general computational scheme for evaluation of a wide class of number-theoretical functions. We avoid asymptotic expansions in favor of manifestly convergent series that lend themselves naturally to rigorous error bounds. By employing three fundamental series algorithms we achieve a unified strategy to compute the various functions via parameter selection. This work amounts to a compendium of methods to establish extreme-precision results as typify modern experimental mathematics. A fortuitous byproduct of this unified approach is automatic analytic continuation over complex parameters. Another byproduct is a host of converging series for various fundamental constants.", acknowledgement = ack-nhfb, remark-1 = "In memory of gentle colleague Jerry Keiper (1953--1995).", remark-2 = "Host in URL field no longer exists; cited in \cite{Coffey:2014:SRR}.", } @Article{DeSchrijver:2012:DPRa, author = "Steven K. {De Schrijver} and El-Houssaine Aghezzaf and Hendrik Vanmaele", title = "Double precision rational approximation algorithm for the inverse standard normal first order loss function", journal = j-APPL-MATH-COMP, volume = "219", number = "3", pages = "1375--1382", day = "15", month = oct, year = "2012", CODEN = "AMHCBQ", DOI = "https://doi.org/10.1016/j.amc.2012.07.011", ISSN = "0096-3003 (print), 1873-5649 (electronic)", ISSN-L = "0096-3003", bibdate = "Thu Oct 25 09:05:16 MDT 2012", bibsource = "http://www.sciencedirect.com/science/journal/00963003; https://www.math.utah.edu/pub/tex/bib/applmathcomput2010.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0096300312007114", acknowledgement = ack-nhfb, fjournal = "Applied Mathematics and Computation", journal-URL = "http://www.sciencedirect.com/science/journal/00963003", } @Article{DeSchrijver:2012:DPRb, author = "Steven K. {De Schrijver} and El-Houssaine Aghezzaf and Hendrik Vanmaele", title = "Double precision rational approximation algorithms for the standard normal first and second order loss functions", journal = j-APPL-MATH-COMP, volume = "219", number = "4", pages = "2320--2330", day = "1", month = nov, year = "2012", CODEN = "AMHCBQ", DOI = "https://doi.org/10.1016/j.amc.2012.08.012", ISSN = "0096-3003 (print), 1873-5649 (electronic)", ISSN-L = "0096-3003", bibdate = "Thu Oct 25 09:05:21 MDT 2012", bibsource = "http://www.sciencedirect.com/science/journal/00963003; https://www.math.utah.edu/pub/tex/bib/applmathcomput2010.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0096300312008041", acknowledgement = ack-nhfb, fjournal = "Applied Mathematics and Computation", journal-URL = "http://www.sciencedirect.com/science/journal/00963003", } @Article{Develi:2012:NAB, author = "I. Develi", title = "A new approximation based on the differential evolution algorithm for the {Gaussian} {$Q$}-function", journal = "Int. J. Innov. Comput. Inf. Control", volume = "8", number = "10(B)", pages = "7095--7102", month = "????", year = "2012", bibdate = "Sat Dec 16 16:14:27 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://link.springer.com/content/pdf/10.1007/s10957-012-0217-0.pdf; http://www.ijicic.org/ijicic-11-08039.pdf", acknowledgement = ack-nhfb, } @InProceedings{Dong:2012:ISP, author = "Chen Dong and Chen He and Sun Xing and Pang Long", booktitle = "{2012 International Conference on Control Engineering and Communication Technology}", title = "Implementation of Single-Precision Floating-Point Trigonometric Functions with Small Area", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "589--592", year = "2012", DOI = "https://doi.org/10.1109/ICCECT.2012.186", bibdate = "Mon Oct 27 10:32:44 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, keywords = "Algorithm design and analysis; Computer architecture; CORDIC; Equations; Field programmable gate arrays; floating-point; FPGA; Hardware; Mathematical model; Transforms; trigonometric functions", } @Article{Fukushima:2012:SES, author = "Toshio Fukushima", title = "Series expansions of symmetric elliptic integrals", journal = j-MATH-COMPUT, volume = "81", number = "278", pages = "957--990", month = "", year = "2012", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Sat Feb 4 09:28:39 MST 2012", bibsource = "http://www.ams.org/mcom/2012-81-278; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp2010.bib", URL = "http://www.ams.org/journals/mcom/2012-81-278/S0025-5718-2011-02531-7; http://www.ams.org/journals/mcom/2012-81-278/S0025-5718-2011-02531-7/S0025-5718-2011-02531-7.pdf", abstract = "Based on general discussion of series expansions of Carlson's symmetric elliptic integrals, we developed fifteen kinds of them including eleven new ones by utilizing the symmetric nature of the integrals. Thanks to the special addition formulas of the integrals, we also obtained their complementary series expansions. By considering the balance between the speed of convergence and the amount of computational labor, we chose four of them as the best series expansions. Practical evaluation of the integrals is conducted by the most suitable one among these four series expansions. Its selection rule was analytically specified in terms of the numerical values of given parameters. As a by-product, we obtained an efficient asymptotic expansion of the integrals around their logarithmic singularities. Numerical experiments confirmed the effectiveness of these new series expansions.", acknowledgement = ack-nhfb, fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Gaudreau:2012:CTP, author = "Philippe Gaudreau and Richard M. Slevinsky and Hassan Safouhi", title = "Computation of Tail Probabilities via Extrapolation Methods and Connection with Rational and {Pad{\'e}} Approximants", journal = j-SIAM-J-SCI-COMP, volume = "34", number = "1", pages = "B65--B85", month = jan, year = "2012", CODEN = "SJOCE3", DOI = "https://doi.org/10.1137/100803778", ISSN = "1064-8275 (print), 1095-7197 (electronic)", ISSN-L = "1064-8275", bibdate = "Sat Dec 16 16:33:00 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/siamjscicomput.bib", URL = "http://epubs.siam.org/doi/abs/10.1137/100803778", acknowledgement = ack-nhfb, fjournal = "SIAM Journal on Scientific Computing", journal-URL = "http://epubs.siam.org/sisc", } @Article{Gil:2012:CRZ, author = "Amparo Gil and Javier Segura", title = "Computing the real zeros of cylinder functions and the roots of the equation {$ x C^\prime_\nu (x) + \gamma C_\nu (x) = 0 $}", journal = j-COMPUT-MATH-APPL, volume = "64", number = "1", pages = "11--21", month = jul, year = "2012", CODEN = "CMAPDK", DOI = "https://doi.org/10.1016/j.camwa.2012.02.032", ISSN = "0898-1221 (print), 1873-7668 (electronic)", ISSN-L = "0898-1221", bibdate = "Wed Mar 1 21:51:09 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/computmathappl2010.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0898122112001460", acknowledgement = ack-nhfb, fjournal = "Computers and Mathematics with Applications", journal-URL = "http://www.sciencedirect.com/science/journal/08981221", remark = "From the abstract: ``Fast methods to compute the zeros of general cylinder functions $ C_\nu (x) = \cos \alpha J_\nu (x) - \sin \alpha Y_\nu (x) C_\nu (x) = \cos \alpha J_\nu (x) - \sin \alpha Y_\nu (x) $ in real intervals can be obtained from an approximate integration of the second order ODE satisfied by these functions, leading to fourth order methods with global convergence.''", } @Article{Gil:2012:EAA, author = "Amparo Gil and Javier Segura and Nico M. Temme", title = "Efficient and Accurate Algorithms for the Computation and Inversion of the Incomplete Gamma Function Ratios", journal = j-SIAM-J-SCI-COMP, volume = "34", number = "6", pages = "A2965--A2981", month = "????", year = "2012", CODEN = "SJOCE3", DOI = "https://doi.org/10.1137/120872553", ISSN = "1064-8275 (print), 1095-7197 (electronic)", ISSN-L = "1064-8275", bibdate = "Fri Jul 19 07:43:33 MDT 2013", bibsource = "http://epubs.siam.org/sam-bin/dbq/toc/SISC/34/6; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/siamjscicomput.bib", acknowledgement = ack-nhfb, fjournal = "SIAM Journal on Scientific Computing", journal-URL = "http://epubs.siam.org/sisc", onlinedate = "January 2012", } @Article{Gil:2012:IAF, author = "Amparo Gil and Javier Segura and Nico M. Temme", title = "An improved algorithm and a {Fortran 90} module for computing the conical function $ p^m_{1 / 2 + i \tau }(x) $", journal = j-COMP-PHYS-COMM, volume = "183", number = "3", pages = "794--799", month = mar, year = "2012", CODEN = "CPHCBZ", DOI = "https://doi.org/10.1016/j.cpc.2011.11.025", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", bibdate = "Sat Feb 11 10:11:02 MST 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/compphyscomm2010.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0010465511003936", acknowledgement = ack-nhfb, fjournal = "Computer Physics Communications", journal-URL = "http://www.sciencedirect.com/science/journal/00104655", } @Article{Jablonski:2012:EAC, author = "A. Jablonski", title = "An effective algorithm for calculating the {Chandrasekhar} function", journal = j-COMP-PHYS-COMM, volume = "183", number = "8", pages = "1773--1782", month = aug, year = "2012", CODEN = "CPHCBZ", DOI = "https://doi.org/10.1016/j.cpc.2012.02.022", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", bibdate = "Tue Apr 24 06:33:31 MDT 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/compphyscomm2010.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0010465512000847", acknowledgement = ack-nhfb, fjournal = "Computer Physics Communications", journal-URL = "http://www.sciencedirect.com/science/journal/00104655", } @InCollection{Jargstorff:2012:AEF, author = "Frank Jargstorff", title = "Approximating the {{\tt erfinv}} Function", crossref = "Hwu:2012:GCG", chapter = "10", pages = "??--??", year = "2012", bibdate = "Sat Feb 08 19:05:23 2014", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, } @Article{Jentschura:2012:NCB, author = "U. D. Jentschura and E. L{\"o}tstedt", title = "Numerical calculation of {Bessel}, {Hankel} and {Airy} functions", journal = j-COMP-PHYS-COMM, volume = "183", number = "3", pages = "506--519", month = mar, year = "2012", CODEN = "CPHCBZ", DOI = "https://doi.org/10.1016/j.cpc.2011.11.010", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", bibdate = "Sat Feb 11 10:11:02 MST 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/compphyscomm2010.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0010465511003729", acknowledgement = ack-nhfb, fjournal = "Computer Physics Communications", journal-URL = "http://www.sciencedirect.com/science/journal/00104655", } @Misc{Johnson:2012:FPF, author = "Steven G. Johnson", title = "{Faddeeva} package, a free\slash open-source {C++} Software to compute the various error functions of arbitrary complex arguments", howpublished = "Web site", year = "2012", bibdate = "Sat Feb 17 14:11:45 2024", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://ab-initio.mit.edu/wiki/index.php/Faddeeva_Package", acknowledgement = ack-nhfb, keywords = "$\erf(x)$ (the error function); $\erfc()$ (complementary error function = $1 - \erf(z)$); $\erfcx()$ (scaled complementary error function $e^{z^2} \erfc(z) = w(i z)$); $\erfi(z)$ (imaginary error function = $-i \erf(i z)$); $w$ (Faddeeva function $w(z) = e^{-z^2} \erfc(-i z)$); Dawson ($((\sqrt \pi)/2) e^{-z^2} \erfi(z)$)", } @Article{Li:2012:DIF, author = "L. S. Li", title = "Design and Implementation of Floating-Point Exponential Function", journal = "Microelectronics", volume = "42", number = "??", pages = "703--709", month = "????", year = "2012", DOI = "", bibdate = "Tue Nov 11 20:07:11 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "", acknowledgement = ack-nhfb, } @InProceedings{Mahapatra:2012:FIS, author = "Chinmaya Mahapatra and Saad Mahboob and Victor C. M. Leung and Thanos Stouraitis", booktitle = "{2012 International Conference on Control Engineering and Communication Technology}", title = "Fast Inverse Square Root Based Matrix Inverse for {MIMO-LTE} Systems", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "321--324", year = "2012", DOI = "https://doi.org/10.1109/ICCECT.2012.253", bibdate = "Mon Oct 27 10:32:44 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, keywords = "Arrays; CORDIC; Fast inverse square root; Field programmable gate arrays; Hardware; Matrix decomposition; MIMO; MIMO LTE; Pipeline processing; Pipelining; QR decomposition; Systolic array; Xilinx virtex6 FPGA", } @InBook{Markovi:2012:CDS, author = "Dejan Markovi and Robert W. Brodersen", booktitle = "{DSP} Architecture Design Essentials", title = "{CORDIC}, Divider, Square Root", publisher = "Springer US", pages = "91--110", year = "2012", DOI = "https://doi.org/10.1007/978-1-4419-9660-2_6", ISBN = "1-4419-9660-5", ISBN-13 = "978-1-4419-9660-2", bibdate = "Tue Oct 28 07:04:09 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, } @InProceedings{Mukunoki:2012:IEQ, author = "Daichi Mukunoki and Daisuke Takahashi", title = "Implementation and Evaluation of Quadruple Precision {BLAS} Functions on {GPUs}", crossref = "Jonasson:2012:APS", pages = "249--259", year = "2012", DOI = "https://doi.org/10.1007/978-3-642-28151-8_25q", bibdate = "Fri Apr 25 15:01:14 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, } @InProceedings{Phong:2012:EAG, author = "Dao Ngoc Phong and Nguyen Quang Uy and Nguyen Xuan Hoai and R. I. (Bob) McKay", editor = "????", booktitle = "Proceedings of the 2012 {IEEE} Congress on Evolutionary Computation, June 10--15, 2012 --- Brisbane, {QLD}, Australia", title = "Evolving approximations for the {Gaussian} {$Q$}-function by genetic programming with semantic based crossover", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "1--6", year = "2012", DOI = "https://doi.org/10.1109/CEC.2012.6256588", bibdate = "Sat Dec 16 16:08:39 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "https://ieeexplore.ieee.org/document/6256588/", acknowledgement = ack-nhfb, } @Article{Poelke:2012:DCC, author = "Konstantin Poelke and Konrad Polthier", title = "Domain Coloring of Complex Functions: An Implementation-Oriented Introduction", journal = j-IEEE-CGA, volume = "32", number = "5", pages = "90--97", month = sep # "\slash " # oct, year = "2012", CODEN = "ICGADZ", DOI = "https://doi.org/10.1109/MCG.2012.100", ISSN = "0272-1716 (print), 1558-1756 (electronic)", ISSN-L = "0272-1716", bibdate = "Mon Oct 22 06:56:23 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/ieeecga.bib", acknowledgement = ack-nhfb, fjournal = "IEEE Computer Graphics and Applications", journal-URL = "http://www.computer.org/portal/web/csdl/magazines/cga", } @Article{Rzadkowski:2012:SEE, author = "Grzegorz Rz{\k{a}}dkowski", title = "On some expansions for the {Euler} Gamma function and the {Riemann} Zeta function", journal = j-J-COMPUT-APPL-MATH, volume = "236", number = "15", pages = "3710--3719", month = sep, year = "2012", CODEN = "JCAMDI", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Sat Feb 25 13:24:35 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2010.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0377042711004663", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @Book{Srivastava:2012:ZZF, author = "H. M. Srivastava and Choi Junesang", title = "Zeta and $q$-Zeta functions and associated series and integrals", publisher = pub-ELSEVIER, address = pub-ELSEVIER:adr, pages = "xvi + 657", year = "2012", ISBN = "0-12-385218-8", ISBN-13 = "978-0-12-385218-2", LCCN = "QA351 .S745 2012", bibdate = "Wed Jun 10 16:19:46 MDT 2015", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; z3950.loc.gov:7090/Voyager", series = "Elsevier Insights", URL = "http://www.sciencedirect.com/science/book/9780123852182", acknowledgement = ack-nhfb, remark = "Revised, enlarged, and updated version of \cite{Srivastava:2001:SAZ}.", subject = "Functions, Zeta", tableofcontents = "1. Introduction and preliminaries \\ 2. The zeta and related functions \\ 3. Series involving zeta functions \\ 4. Evaluations and series representations \\ 5. Determinants of the laplacians \\ 6. q-Extensions of some special functions and polynomials \\ 7. Miscellaneous results", } @Article{Sudha:2012:NMC, author = "J. Sudha and M. C. Hanumantharaju and V. Venkateswarulu and Jayalaxmi H", title = "A Novel Method for Computing Exponential Function Using {CORDIC} Algorithm", journal = "Procedia Engineering", volume = "30", pages = "519--528", year = "2012", DOI = "https://doi.org/10.1016/j.proeng.2012.01.893", ISSN = "1877-7058", ISSN-L = "1877-7058", bibdate = "Wed Oct 29 14:21:46 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", note = "International Conference on Communication Technology and System Design 2011", URL = "https://www.sciencedirect.com/science/article/pii/S1877705812009034", acknowledgement = ack-nhfb, keywords = "2D Gaussian Function; Exponential Function; FPGA; Hyperbolic {CORDIC} Algorithm", } @Article{Vazquez-Leal:2012:HAS, author = "Hector Vazquez-Leal and Roberto Castaneda-Sheissa and Uriel Filobello-Nino and Arturo Sarmiento-Reyes and Jesus Sanchez Orea", title = "High Accurate Simple Approximation of Normal Distribution Integral", journal = "Mathematical Problems in Engineering", volume = "2012", pages = "1--22", year = "2012", DOI = "https://doi.org/10.1155/2012/124029", ISSN = "1024-123X (print), 1563-5147 (electronic)", bibdate = "Sat Dec 16 17:54:09 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "https://www.hindawi.com/journals/mpe/2012/124029/", acknowledgement = ack-nhfb, } @Article{Veberic:2012:LFA, author = "Darko Veberi{\v{c}}", title = "{Lambert} {$W$} function for applications in physics", journal = j-COMP-PHYS-COMM, volume = "183", number = "12", pages = "2622--2628", month = dec, year = "2012", CODEN = "CPHCBZ", DOI = "https://doi.org/10.1016/j.cpc.2012.07.008", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", bibdate = "Tue Aug 28 17:36:53 MDT 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/compphyscomm2010.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0010465512002366", acknowledgement = ack-nhfb, fjournal = "Computer Physics Communications", journal-URL = "http://www.sciencedirect.com/science/journal/00104655", } @Article{Willis:2012:AGH, author = "Joshua L. Willis", title = "Acceleration of generalized hypergeometric functions through precise remainder asymptotics", journal = j-NUMER-ALGORITHMS, volume = "59", number = "??", pages = "??--??", month = "????", year = "2012", CODEN = "NUALEG", DOI = "https://doi.org/10.1007/s11075-011-9499-9", ISSN = "1017-1398 (print), 1572-9265 (electronic)", ISSN-L = "1017-1398", bibdate = "Wed Nov 30 06:42:07 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://arxiv.org/abs/1102.3003; http://www.springerlink.com/content/k413064448600815/", abstract = "We express the asymptotics of the remainders of the partial sums $ \{ s_n \} $ of the generalized hypergeometric function through an inverse power series $ z^n n^\lambda \sum c_k / n_k $, where the exponent $ \lambda $ and the asymptotic coefficients $ \{ c_k \} $ may be recursively computed to any desired order from the hypergeometric parameters and argument. From this we derive a new series acceleration technique that can be applied to any such function, even with complex parameters and at the branch point $ z = 1 $. For moderate parameters (up to approximately ten) a C implementation at fixed precision is very effective at computing these functions; for larger parameters an implementation in higher than machine precision would be needed. Even for larger parameters, however, our C implementation is able to correctly determine whether or not it has converged; and when it converges, its estimate of its error is accurate.", acknowledgement = ack-nhfb, fjournal = "Numerical Algorithms", journal-URL = "http://link.springer.com/journal/11075", keywords = "Generalized hypergeometric functions; Recurrence asymptotics; Series acceleration", } @InProceedings{Yang:2012:CDS, author = "Bohan Yang and Dong Wang and Leibo Liu", booktitle = "{2012 2nd International Conference on Consumer Electronics, Communications and Networks (CECNet)}", title = "Complex division and square-root using {CORDIC}", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "2464--2468", year = "2012", DOI = "https://doi.org/10.1109/CECNet.2012.6201840", bibdate = "Mon Oct 27 10:32:44 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, keywords = "Accuracy; Adders; Algorithm design and analysis; Complex; Computer architecture; CORDIC; Division; Field programmable gate arrays; FPGA; Hardware; Registers; Square-root", } @InCollection{Arfken:2013:BF, author = "George B. (George Brown) Arfken and Hans-J{\"u}rgen Weber and Frank E. Harris", title = "{Bessel} Functions", crossref = "Arfken:2013:MMP", chapter = "14", pages = "643--713", year = "2013", bibdate = "Thu Dec 5 05:54:14 MST 2013", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/h/harris-frank-e.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/B9780123846549000141", acknowledgement = ack-nhfb, } @InCollection{Arfken:2013:GFb, author = "George B. (George Brown) Arfken and Hans-J{\"u}rgen Weber and Frank E. Harris", title = "{Gamma} Function", crossref = "Arfken:2013:MMP", chapter = "13", pages = "599--641", year = "2013", bibdate = "Thu Dec 5 05:54:14 MST 2013", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/h/harris-frank-e.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/B978012384654900013X", acknowledgement = ack-nhfb, } @Article{Babusci:2013:SME, author = "D. Babusci and G. Dattoli and K. G{\'o}rska and K. A. Penson", title = "Symbolic methods for the evaluation of sum rules of {Bessel} functions", journal = j-J-MATH-PHYS, volume = "54", number = "7", pages = "073501", month = jul, year = "2013", CODEN = "JMAPAQ", DOI = "https://doi.org/10.1063/1.4812325", ISSN = "0022-2488 (print), 1089-7658 (electronic), 1527-2427", ISSN-L = "0022-2488", bibdate = "Wed Feb 12 12:24:18 MST 2014", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jmathphys2010.bib", acknowledgement = ack-nhfb, fjournal = "Journal of Mathematical Physics", journal-URL = "http://jmp.aip.org/", } @InProceedings{Bangqiang:2013:BLI, author = "Liu Bangqiang and He Ling and Yan Xiao", booktitle = "{2013 IEEE 9th International Colloquium on Signal Processing and its Applications}", title = "Base-{$N$} logarithm implementation on {FPGA} for the data with random decimal point positions", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "17--20", year = "2013", DOI = "https://doi.org/10.1109/CSPA.2013.6530006", bibdate = "Mon Oct 27 10:32:44 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, keywords = "Algorithm design and analysis; CORDIC algorithm; decimal point positions; Educational institutions; Electronic mail; Field programmable gate arrays; IP networks; Logarithm; Signal processing; Signal processing algorithms", } @Article{Beals:2013:MFG, author = "Richard Beals and Jacek Szmigielski", title = "{Meijer} {$G$}-functions: a gentle introduction", journal = j-NAMS, volume = "60", number = "7", pages = "866--872", month = aug, year = "2013", CODEN = "AMNOAN", DOI = "https://doi.org/10.1090/noti1016", ISSN = "0002-9920 (print), 1088-9477 (electronic)", ISSN-L = "0002-9920", MRclass = "33C60", MRnumber = "3086637", MRreviewer = "Gianni Pagnini", bibdate = "Thu Aug 15 07:17:02 2024", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/nams2010.bib", acknowledgement = ack-nhfb, ajournal = "Notices Amer. Math. Soc.", fjournal = "Notices of the American Mathematical Society", journal-URL = "http://www.ams.org/notices/", } @Article{Booker:2013:BAB, author = "Andrew R. Booker and Andreas Str{\"o}mbergsson and Holger Then", title = "Bounds and algorithms for the {$K$-Bessel} function of imaginary order", journal = j-LMS-J-COMPUT-MATH, volume = "16", pages = "78--108", year = "2013", CODEN = "????", DOI = "https://doi.org/10.1112/S1461157013000028", ISSN = "1461-1570", ISSN-L = "1461-1570", MRclass = "26D07; 33C10; 33F05; 34D05; 41A58 (primary); 41A80; 65D05; 40H05; 26B99 (secondary)", bibdate = "Sat Jun 22 11:29:28 MDT 2013", bibsource = "http://journals.cambridge.org/action/displayJournal?jid=JCM; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/lms-j-comput-math.bib", acknowledgement = ack-nhfb, ajournal = "LMS J. Comput. Math.", fjournal = "LMS Journal of Computation and Mathematics", journal-URL = "http://journals.cambridge.org/action/displayJournal?jid=JCM", onlinedate = "10 April 2013", } @InProceedings{Brent:2013:FCB, author = "Richard P. Brent and David Harvey", editor = "D. H. Bailey and H. H. Bauschke and P. Borwein and F. Garvan and M. Th{\'e}ra and J. D. Vanderwerff and H. Wolkowicz", booktitle = "Computational and Analytical Mathematics", title = "Fast Computation of {Bernoulli}, Tangent and Secant Numbers", volume = "50", publisher = pub-SV, address = pub-SV:adr, bookpages = "xv + 701", pages = "127--142", year = "2013", DOI = "https://doi.org/10.1007/978-1-4614-7621-4_8", ISBN = "1-4614-7620-8, 1-4614-7621-6 (e-book)", ISBN-13 = "978-1-4614-7620-7, 978-1-4614-7621-4 (e-book)", ISSN = "2194-1009", bibdate = "Sat Jun 8 08:38:45 2024", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, } @Article{Chen:2013:CFE, author = "Chao-Ping Chen", title = "Continued fraction estimates for the psi function", journal = j-APPL-MATH-COMP, volume = "219", number = "19", pages = "9865--9871", day = "1", month = jun, year = "2013", CODEN = "AMHCBQ", DOI = "https://doi.org/10.1016/j.amc.2013.03.134", ISSN = "0096-3003 (print), 1873-5649 (electronic)", ISSN-L = "0096-3003", bibdate = "Mon May 20 19:05:31 MDT 2013", bibsource = "http://www.sciencedirect.com/science/journal/00963003; https://www.math.utah.edu/pub/tex/bib/applmathcomput2010.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcw.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0096300313003962", acknowledgement = ack-nhfb, fjournal = "Applied Mathematics and Computation", journal-URL = "http://www.sciencedirect.com/science/journal/00963003", } @Article{Chen:2013:LIA, author = "Chao-Ping Chen and Cristinel Mortici", title = "Limits and inequalities associated with the {Euler--Mascheroni} constant", journal = j-APPL-MATH-COMP, volume = "219", number = "18", pages = "9755--9761", day = "15", month = may, year = "2013", CODEN = "AMHCBQ", DOI = "https://doi.org/10.1016/j.amc.2013.03.089", ISSN = "0096-3003 (print), 1873-5649 (electronic)", ISSN-L = "0096-3003", bibdate = "Mon May 20 19:05:27 MDT 2013", bibsource = "http://www.sciencedirect.com/science/journal/00963003; https://www.math.utah.edu/pub/tex/bib/applmathcomput2010.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0096300313003500", acknowledgement = ack-nhfb, fjournal = "Applied Mathematics and Computation", journal-URL = "http://www.sciencedirect.com/science/journal/00963003", keywords = "asymptotic expansion; Euler-Mascheroni constant; harmonic numbers; inequality; polygamma functions; psi function", } @Article{Chen:2013:UTS, author = "Chao-Ping Chen", title = "Unified treatment of several asymptotic formulas for the gamma function", journal = j-NUMER-ALGORITHMS, volume = "64", number = "2", pages = "311--319", month = oct, year = "2013", CODEN = "NUALEG", DOI = "https://doi.org/10.1007/s11075-012-9667-6", ISSN = "1017-1398 (print), 1572-9265 (electronic)", ISSN-L = "1017-1398", bibdate = "Mon Dec 2 18:18:08 MST 2013", bibsource = "http://springerlink.metapress.com/openurl.asp?genre=issue&issn=1017-1398&volume=64&issue=2; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/numeralgorithms.bib", URL = "http://link.springer.com/article/10.1007/s11075-012-9667-6", acknowledgement = ack-nhfb, fjournal = "Numerical Algorithms", journal-URL = "http://link.springer.com/journal/11075", } @InProceedings{Chevillard:2013:MPE, author = "Sylvain Chevillard and Marc Mezzarobba", title = "Multiple-Precision Evaluation of the {Airy} {Ai} Function with Reduced Cancellation", crossref = "IEEE:2013:PIS", pages = "175--182", year = "2013", DOI = "https://doi.org/10.1109/ARITH.2013.33", ISSN = "1063-6889", bibdate = "Sat Aug 1 09:38:32 2015", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, keywords = "Accuracy; Airy Ai function; algorithm; Algorithm design and analysis; Approximation algorithms; Approximation methods; arbitrary precision; ARITH-21; asymptotics; cancellation reduction; classical Miller algorithm; correct rounding; differential equations; Equations; error bounds; ill-conditioned three-term recurrence; linear ordinary differential equation; Miller method; multiple-precision evaluation; nonnegative Taylor expansions; numerical evaluation; series (mathematics); series expansion; Shape; Special functions; Taylor coefficients; Taylor series", } @Article{deDinechin:2013:FPT, author = "Florent de Dinechin and Matei Istoan and Guillaume Sergent", title = "Fixed-point trigonometric functions on {FPGAs}", journal = j-COMP-ARCH-NEWS, volume = "41", number = "5", pages = "83--88", month = dec, year = "2013", CODEN = "CANED2", DOI = "https://doi.org/10.1145/2641361.2641375", ISSN = "0163-5964 (print), 1943-5851 (electronic)", ISSN-L = "0163-5964", bibdate = "Mon Aug 18 17:12:43 MDT 2014", bibsource = "http://portal.acm.org/; https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/sigarch.bib", abstract = "Three approaches for computing sines and cosines on FPGAs are studied in this paper, with a focus of high-throughput pipelined architecture, and state-of-the-art implementation techniques. The first approach is the classical CORDIC iteration, for which we suggest a reduced iteration technique and fine optimizations in datapath width and latency. The second is an ad-hoc architecture specifically designed around trigonometric identities. The third uses a generic table- and DSP-based polynomial approximator. These three architectures are implemented and compared in the FloPoCo framework.", acknowledgement = ack-nhfb, fjournal = "ACM SIGARCH Computer Architecture News", journal-URL = "https://dl.acm.org/loi/sigarch", keywords = "CORDIC; cosine; sine", } @Article{Develi:2013:HAA, author = "I. Develi and A. Basturk", title = "Highly Accurate Analytic Approximation to the {Gaussian} {$Q$}-function Based on the Use of Nonlinear Least Squares Optimization Algorithm", journal = j-J-OPT-THEORY-APPL, volume = "159", number = "1", pages = "183--191", day = "01", month = oct, year = "2013", CODEN = "JOTABN", DOI = "https://doi.org/10.1007/s10957-012-0217-0", ISSN = "1573-2878", ISSN-L = "0022-3239", bibdate = "Sat Dec 16 16:18:18 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "https://link.springer.com/article/10.1007/s10957-012-0217-0", acknowledgement = ack-nhfb, ajournal = "J. Optim. Theory Appl.", fjournal = "Journal of Optimization Theory and Applications", journal-URL = "http://link.springer.com/journal/volumesAndIssues/10957", } @Article{Erricolo:2013:AFS, author = "Danilo Erricolo and Giuseppe Carluccio", title = "{Algorithm 934}: {Fortran 90} subroutines to compute {Mathieu} functions for complex values of the parameter", journal = j-TOMS, volume = "40", number = "1", pages = "8:1--8:19", month = sep, year = "2013", CODEN = "ACMSCU", DOI = "https://doi.org/10.1145/2513109.2513117", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Mon Sep 30 16:05:58 MDT 2013", bibsource = "http://www.acm.org/pubs/contents/journals/toms/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fortran3.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib", abstract = "Software to compute angular and radial Mathieu functions is provided in the case that the parameter q is a complex variable and the independent variable x is real. After an introduction on the notation and the definitions of Mathieu functions and their related properties, Fortran 90 subroutines to compute them are described and validated with some comparisons. A sample application is also provided.", acknowledgement = ack-nhfb, articleno = "8", fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", } @Article{Feng:2013:TFA, author = "Lei Feng and Weiping Wang", title = "Two families of approximations for the gamma function", journal = j-NUMER-ALGORITHMS, volume = "64", number = "3", pages = "403--416", month = nov, year = "2013", CODEN = "NUALEG", DOI = "https://doi.org/10.1007/s11075-012-9671-x", ISSN = "1017-1398 (print), 1572-9265 (electronic)", ISSN-L = "1017-1398", bibdate = "Mon Dec 2 18:18:12 MST 2013", bibsource = "http://springerlink.metapress.com/openurl.asp?genre=issue&issn=1017-1398&volume=64&issue=3; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/numeralgorithms.bib", URL = "http://link.springer.com/article/10.1007/s11075-012-9671-x", acknowledgement = ack-nhfb, fjournal = "Numerical Algorithms", journal-URL = "http://link.springer.com/journal/11075", } @Article{Fukushima:2013:PFC, author = "Toshio Fukushima", title = "Precise and fast computation of {Jacobian} elliptic functions by conditional duplication", journal = j-NUM-MATH, volume = "123", number = "4", pages = "585--605", month = apr, year = "2013", CODEN = "NUMMA7", DOI = "https://doi.org/10.1007/s00211-012-0498-0", ISSN = "0029-599X (print), 0945-3245 (electronic)", ISSN-L = "0029-599X", bibdate = "Sat Apr 27 13:30:29 MDT 2013", bibsource = "http://springerlink.metapress.com/openurl.asp?genre=issue&issn=0029-599X&volume=123&issue=4; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/nummath2010.bib", URL = "http://link.springer.com/article/10.1007/s00211-012-0498-0", acknowledgement = ack-nhfb, fjournal = "Numerische Mathematik", journal-URL = "http://link.springer.com/journal/211", } @Article{Fukushima:2013:RCD, author = "Toshio Fukushima", title = "Recursive computation of derivatives of elliptic functions and of incomplete elliptic integrals", journal = j-APPL-MATH-COMP, volume = "221", number = "??", pages = "21--31", day = "15", month = sep, year = "2013", CODEN = "AMHCBQ", ISSN = "0096-3003 (print), 1873-5649 (electronic)", ISSN-L = "0096-3003", bibdate = "Mon Dec 2 12:34:28 MST 2013", bibsource = "http://www.sciencedirect.com/science/journal/00963003; https://www.math.utah.edu/pub/tex/bib/applmathcomput2010.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0096300313006152", acknowledgement = ack-nhfb, fjournal = "Applied Mathematics and Computation", journal-URL = "http://www.sciencedirect.com/science/journal/00963003", } @Article{Gonzalez-Morales:2013:NII, author = "M. J. Gonz{\'a}lez-Morales and R. Mahillo-Isla and C. Dehesa-Mart{\'\i}nez", title = "A new integral identity involving the elliptic integral {$ E(m) $}", journal = j-APPL-MATH-COMP, volume = "221", number = "??", pages = "568--570", day = "15", month = sep, year = "2013", CODEN = "AMHCBQ", ISSN = "0096-3003 (print), 1873-5649 (electronic)", ISSN-L = "0096-3003", bibdate = "Mon Dec 2 12:34:28 MST 2013", bibsource = "https://www.math.utah.edu/pub/tex/bib/applmathcomput2010.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0096300313007303", acknowledgement = ack-nhfb, fjournal = "Applied Mathematics and Computation", journal-URL = "http://www.sciencedirect.com/science/journal/00963003/", } @Article{Hale:2013:FAC, author = "Nicholas Hale and Alex Townsend", title = "Fast and Accurate Computation of {Gauss--Legendre} and {Gauss--Jacobi} Quadrature Nodes and Weights", journal = j-SIAM-J-SCI-COMP, volume = "35", number = "2", pages = "A652--A674", month = "????", year = "2013", CODEN = "SJOCE3", DOI = "https://doi.org/10.1137/120889873", ISSN = "1064-8275 (print), 1095-7197 (electronic)", ISSN-L = "1064-8275", bibdate = "Fri Jul 19 07:43:46 MDT 2013", bibsource = "http://epubs.siam.org/sam-bin/dbq/toc/SISC/35/2; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/siamjscicomput.bib", acknowledgement = ack-nhfb, fjournal = "SIAM Journal on Scientific Computing", journal-URL = "http://epubs.siam.org/sisc", onlinedate = "January 2013", } @Article{Huang:2013:NNE, author = "Zhi-Wei Huang and Jueping Liu", title = "{NumExp}: Numerical epsilon expansion of hypergeometric functions", journal = j-COMP-PHYS-COMM, volume = "184", number = "8", pages = "1973--1980", month = aug, year = "2013", CODEN = "CPHCBZ", DOI = "https://doi.org/10.1016/j.cpc.2013.03.016", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", bibdate = "Wed May 15 07:02:08 MDT 2013", bibsource = "https://www.math.utah.edu/pub/tex/bib/compphyscomm2010.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0010465513001136", acknowledgement = ack-nhfb, fjournal = "Computer Physics Communications", journal-URL = "http://www.sciencedirect.com/science/journal/00104655", } @Article{Jablonski:2013:IAC, author = "A. Jablonski", title = "Improved algorithm for calculating the {Chandrasekhar} function", journal = j-COMP-PHYS-COMM, volume = "184", number = "2", pages = "440--442", month = feb, year = "2013", CODEN = "CPHCBZ", DOI = "https://doi.org/10.1016/j.cpc.2012.08.020", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", bibdate = "Fri Nov 2 11:55:56 MDT 2012", bibsource = "https://www.math.utah.edu/pub/tex/bib/compphyscomm2010.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/S001046551200286X", acknowledgement = ack-nhfb, fjournal = "Computer Physics Communications", journal-URL = "http://www.sciencedirect.com/science/journal/00104655", } @InProceedings{Jiang:2013:AFE, author = "Hao Jiang and Stef Graillat and Roberto Barrio", title = "Accurate and Fast Evaluation of Elementary Symmetric Functions", crossref = "IEEE:2013:PIS", pages = "183--190", year = "2013", DOI = "https://doi.org/10.1109/ARITH.2013.18", ISSN = "1063-6889", bibdate = "Sat Aug 1 09:38:32 2015", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, keywords = "Accuracy; accurate algorithm; Algorithm design and analysis; ARITH-21; compensated algorithm; double-double library; elementary symmetric functions; error-free transformation; error-free transformations; floating point arithmetic; floating-point arithmetic; forward roundoff error bound; Libraries; mathematics computing; MATLAB poly function; Polynomials; psychological measurement; Rasch model; roundoff error; Roundoff errors; running error bound; shaper bound; summation algorithm; Vectors", } @Article{Lopez:2013:NSE, author = "Jos{\'e} L. L{\'o}pez and Nico M. Temme", title = "New series expansions of the {Gauss} hypergeometric function", journal = j-ADV-COMPUT-MATH, volume = "39", number = "2", pages = "349--365", month = aug, year = "2013", CODEN = "ACMHEX", DOI = "https://doi.org/10.1007/s10444-012-9283-y", ISSN = "1019-7168 (print), 1572-9044 (electronic)", ISSN-L = "1019-7168", MRclass = "33C05 (33F05 41A58 65D20)", MRnumber = "3082518", MRreviewer = "Jochen Denzler", bibdate = "Sat Feb 3 18:23:06 MST 2018", bibsource = "https://www.math.utah.edu/pub/tex/bib/advcomputmath.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://link.springer.com/article/10.1007/s10444-012-9283-y", acknowledgement = ack-nhfb, fjournal = "Advances in Computational Mathematics", journal-URL = "http://link.springer.com/journal/10444", keywords = "Gauss hypergeometric function $_2F_1(a,b,c; z)$", remark = "Improvement on \cite{Buhring:1987:ACH,Buhring:1987:BUA} and \cite[\S 2.3]{Gil:2007:NMS} by removal of points excluded from the domain of convergence.", } @Article{Low:2013:MET, author = "Joshua Yung Lih Low and Ching Chuen Jong", title = "A Memory-Efficient Tables-and-Additions Method for Accurate Computation of Elementary Functions", journal = j-IEEE-TRANS-COMPUT, volume = "62", number = "5", pages = "858--872", month = may, year = "2013", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/TC.2012.43", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Tue Apr 30 12:26:22 2013", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput2010.bib", acknowledgement = ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", } @Article{Mastin:2013:LQB, author = "Andrew Mastin and Patrick Jaillet", title = "Log-quadratic bounds for the {Gaussian} {$Q$}-function", journal = "arxiv.org", volume = "??", number = "??", pages = "??--??", day = "9", month = apr, year = "2013", bibdate = "Sat Dec 16 17:09:03 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://arxiv.org/abs/1304.2488", acknowledgement = ack-nhfb, } @Article{Nemes:2013:EBE, author = "Gerg{\H{o}} Nemes", title = "Error bounds and exponential improvement for {Hermite}'s asymptotic expansion for the gamma function", journal = "Applicable Analysis and Discrete Mathematics", volume = "7", number = "1", pages = "161--179", month = apr, year = "2013", DOI = "https://doi.org/10.2298/AADM130124002N", ISSN = "1452-8630 (print), 2406-100X (electronic)", ISSN-L = "1452-8630", MRclass = "41A60 (33B15 41A80)", MRnumber = "3086174", MRreviewer = "Junesang Choi", bibdate = "Fri Oct 18 16:34:50 2024", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, ajournal = "Appl. Anal. Discrete Math.", fjournal = "Applicable Analysis and Discrete Mathematics", journal-URL = "https://pefmath.etf.bg.ac.rs/", } @Article{Neta:2013:FHL, author = "Beny Neta and Melvin Scott", title = "On a family of {Halley}-like methods to find simple roots of nonlinear equations", journal = j-APPL-MATH-COMP, volume = "219", number = "15", pages = "7940--7944", day = "1", month = apr, year = "2013", CODEN = "AMHCBQ", DOI = "https://doi.org/10.1016/j.amc.2013.02.035", ISSN = "0096-3003 (print), 1873-5649 (electronic)", ISSN-L = "0096-3003", bibdate = "Mon May 6 18:04:12 MDT 2013", bibsource = "https://www.math.utah.edu/pub/tex/bib/applmathcomput2010.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0096300313001574", acknowledgement = ack-nhfb, fjournal = "Applied Mathematics and Computation", journal-URL = "http://www.sciencedirect.com/science/journal/00963003/", keywords = "basin of attraction; Euler--Chebyshev method; Halley method; nonlinear equations; simple roots", } @Book{Osipov:2013:PSW, author = "Andrei Osipov", title = "Prolate Spheroidal Wave Functions of Order Zero: Mathematical Tools for Bandlimited Approximation", publisher = pub-SV, address = pub-SV:adr, pages = "????", year = "2013", ISBN = "1-4614-8258-5", ISBN-13 = "978-1-4614-8258-1", LCCN = "????", bibdate = "Sat Apr 1 14:32:29 MDT 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; z3950.loc.gov:7090/Voyager", URL = "http://www.loc.gov/catdir/enhancements/fy1315/2013945079-b.html; http://www.loc.gov/catdir/enhancements/fy1315/2013945079-d.html; http://www.loc.gov/catdir/enhancements/fy1315/2013945079-t.html", acknowledgement = ack-nhfb, tableofcontents = "Introduction \\ Mathematical and Numerical Preliminaries \\ Overview \\ Analysis of the Differential Operator \\ Analysis of the Integral Operator \\ Rational Approximations of PSWFs \\ Miscellaneous Properties of PSWFs \\ Asymptotic Analysis of PSWFs \\ Quadrature Rules and Interpolation via PSWFs \\ Numerical Algorithms", } @Article{Ray:2013:CBV, author = "Kailash Chandra Ray and Anindya Sundar Dhar", title = "{CORDIC}-Based {VLSI} Architecture for Implementing {Kaiser--Bessel} Window in Real Time Spectral Analysis", journal = "Journal of Signal Processing Systems", volume = "74", number = "2", pages = "235--244", month = jun, year = "2013", DOI = "https://doi.org/10.1007/s11265-013-0781-z", ISSN = "1939-8115", bibdate = "Tue Oct 28 07:04:09 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, } @InProceedings{Rodriguez-Garcia:2013:FFP, author = "A. Rodriguez-Garcia and L. Pizano-Escalante and R. Parra-Michel and O. Longoria-Gandara and J. Cortez", editor = "Ren{\'e} Cumplido and Eduardo de la Torre and Mike Wirthlin", booktitle = "{2013 International Conference on Reconfigurable Computing and FPGAs (ReConFig): Cancun, Mexico, December 9--11, 2013}", title = "Fast fixed-point divider based on {Newton--Raphson} method and piecewise polynomial approximation", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "1--6", month = dec, year = "2013", DOI = "https://doi.org/10.1109/reconfig.2013.6732291", ISBN = "1-4799-2079-7", ISBN-13 = "978-1-4799-2079-2", bibdate = "Thu Apr 10 15:12:10 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, } @Article{Russinoff:2013:CFV, author = "David M. Russinoff", title = "Computation and Formal Verification of {SRT} Quotient and Square Root Digit Selection Tables", journal = j-IEEE-TRANS-COMPUT, volume = "62", number = "5", pages = "900--913", month = may, year = "2013", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/TC.2012.40", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Tue Apr 30 12:26:22 2013", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput2010.bib", acknowledgement = ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", } @InProceedings{Saravanan:2013:SCG, author = "P. Saravanan and S. Ramasamy", booktitle = "{2013 Fourth International Conference on Computing, Communications and Networking Technologies (ICCCNT)}", title = "Sine\slash cos generator for direct digital frequency synthesizer using pipelined {CORDIC} processor", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "1--6", year = "2013", DOI = "https://doi.org/10.1109/ICCCNT.2013.6726502", bibdate = "Mon Oct 27 10:32:44 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, keywords = "Algorithm design and analysis; ASIC; Computer architecture; CORDIC; DDFS; Equations; FPGA; Generators; Mathematical model; Pipelined processor; Signal processing algorithms; sine and cosine generator; Vectors", } @Article{Swetz:2013:MTA, author = "Frank J. Swetz", title = "Mathematical Treasure: {{\booktitle{Arithmetica Logarithmica}}} of {Henry Briggs}", journal = "Loci", volume = "??", number = "??", month = mar, year = "2013", DOI = "https://doi.org/10.4169/loci003959", bibdate = "Mon Nov 10 09:14:45 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, remark = "DOI does not resolve on 10 November 2025. Publisher and journal uncertain: journal name is too common to be readily found in library catalog searches.", xxISSN = "2829-4262 (print), 2829-3827 (electronic)", xxjournal = "Locus", xxpublisher = "The MAA Mathematical Sciences Digital Library", } @Article{Szmytkowski:2013:EBT, author = "Rados{\l}aw Szmytkowski", title = "Erratum to {{\booktitle{Formulas and Theorems for the Special Functions of Mathematical Physics}} by W. Magnus, F. Oberhettinger, R. P. Soni}", journal = j-MATH-COMPUT, volume = "82", number = "283", pages = "1709--1710", month = "????", year = "2013", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Apr 30 16:18:02 MDT 2013", bibsource = "http://www.ams.org/mcom/2013-82-283; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp2010.bib", URL = "http://www.ams.org/journals/mcom/2013-82-283/S0025-5718-2013-02671-3; http://www.ams.org/journals/mcom/2013-82-283/S0025-5718-2013-02671-3/S0025-5718-2013-02671-3.pdf", acknowledgement = ack-nhfb, fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Thompson:2013:AIG, author = "Ian Thompson", title = "{Algorithm 926}: Incomplete {Gamma} Functions with Negative Arguments", journal = j-TOMS, volume = "39", number = "2", pages = "14:1--14:9", month = feb, year = "2013", CODEN = "ACMSCU", DOI = "https://doi.org/10.1145/2427023.2427031", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Wed Feb 20 16:46:13 MST 2013", bibsource = "http://www.acm.org/pubs/contents/journals/toms/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib", abstract = "An algorithm for accurately computing the lower incomplete gamma function $ \gamma (a, t) $ in the case where $ a = n + 1 / 2 $, $ n \in Z $ and $ t < 0 $ is described. Series expansions and analytic continuation are employed to compute the function for certain critical values of $n$, and these results are used to initiate stable recurrence. The algorithm has been implemented in Fortran 2003, with precomputations carried out in Maple.", acknowledgement = ack-nhfb, articleno = "14", fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", } @Article{Yukcu:2013:SEB, author = "Niyazi Y{\"u}k{\c{c}}{\"u} and Emin {\"O}ztekin", title = "Strategies on the evaluation of binomial coefficients for all integers", journal = j-COMPUT-MATH-MATH-PHYS, volume = "53", number = "1", pages = "1--7", month = jan, year = "2013", DOI = "https://doi.org/10.1134/s0965542513010119", ISSN = "0965-5425 (print), 1555-6662 (electronic)", ISSN-L = "0965-5425", bibdate = "Tue Aug 5 06:59:58 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Computational Mathematics and Mathematical Physics", remark = "The mishandling of binomials of negative arguments reported in this paper for Mathematica 7 was repaired in later Mathematica versions.", } @Article{Zhong:2013:AKF, author = "Min Zhong and R. J. Loy and R. S. Anderssen", title = "Approximating the {Kohlrausch} function by sums of exponentials", journal = j-ANZIAM-J, volume = "54", number = "4", pages = "306--323", month = apr, year = "2013", CODEN = "AJNOA2", DOI = "https://doi.org/10.1017/S1446181113000229", ISSN = "1446-1811 (print), 1446-8735 (electronic)", ISSN-L = "1446-1811", bibdate = "Fri Apr 26 16:14:05 MDT 2019", bibsource = "https://www.math.utah.edu/pub/tex/bib/anziamj.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "https://www.cambridge.org/core/journals/anziam-journal/article/approximating-the-kohlrausch-function-by-sums-of-exponentials/1F2BD299466198D202D9D2355E34116F", acknowledgement = ack-nhfb, ajournal = "ANZIAM J.", fjournal = "The ANZIAM Journal. The Australian \& New Zealand Industrial and Applied Mathematics Journal", journal-URL = "http://journals.cambridge.org/action/displayJournal?jid=ANZ", keywords = "Kohlrausch function $\exp(-t^\beta)$, with $\beta \in (0,1)$", onlinedate = "04 September 2013", } @Article{Adj:2014:SRC, author = "G. Adj and F. Rodriguez-Henriquez", title = "Square Root Computation over Even Extension Fields", journal = j-IEEE-TRANS-COMPUT, volume = "63", number = "11", pages = "2829--2841", month = nov, year = "2014", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/TC.2013.145", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Thu Nov 06 07:39:04 2014", bibsource = "https://www.math.utah.edu/pub/tex/bib/cryptography2010.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput2010.bib", acknowledgement = ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", keywords = "Algorithm design and analysis; Complexity theory; Computational efficiency; Computer science; Elliptic curve cryptography; Elliptic curves; even extension fields; finite extension fields; finite field arithmetic; Modular square root; number theoretical problem; number theory; square root computation; Taxonomy", } @Misc{Anonymous:2014:CLL, author = "Anonymous", title = "{CR-Libm} --- a library of correctly rounded elementary functions in double-precision", howpublished = "Web site", year = "2014", bibdate = "Sat Oct 31 07:21:21 2015", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "http://lipforge.ens-lyon.fr/www/crlibm/", abstract = "CRlibm is a free mathematical library (libm) that provides: (1) implementations of the double-precision C99 standard elementary functions; (2) correctly rounded in the four IEEE-754 rounding modes; (3) with a comprehensive proof of both the algorithms used and their implementation; (4) sufficiently efficient in average time, worst-case time, and memory consumption to replace existing libms transparently.", acknowledgement = ack-nhfb, keywords = "CR-Libm; scslib (software carry save library)", } @Article{Babusci:2014:SBS, author = "D. Babusci and G. Dattoli and K. G{\'o}rska and K. A. Penson", title = "The spherical {Bessel} and {Struve} functions and operational methods", journal = j-APPL-MATH-COMP, volume = "238", number = "??", pages = "1--6", day = "1", month = jul, year = "2014", CODEN = "AMHCBQ", ISSN = "0096-3003 (print), 1873-5649 (electronic)", ISSN-L = "0096-3003", bibdate = "Fri May 23 10:53:19 MDT 2014", bibsource = "https://www.math.utah.edu/pub/tex/bib/applmathcomput2010.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0096300314005086", acknowledgement = ack-nhfb, fjournal = "Applied Mathematics and Computation", journal-URL = "http://www.sciencedirect.com/science/journal/00963003/", } @Article{Backeljauw:2014:VES, author = "Franky Backeljauw and Stefan Becuwe and Annie Cuyt and Joris {Van Deun} and Daniel W. Lozier", title = "Validated evaluation of special mathematical functions", journal = j-SCI-COMPUT-PROGRAM, volume = "90 (part A)", number = "??", pages = "2--20", day = "15", month = sep, year = "2014", CODEN = "SCPGD4", DOI = "https://doi.org/10.1016/j.scico.2013.05.006", ISSN = "0167-6423 (print), 1872-7964 (electronic)", ISSN-L = "0167-6423", bibdate = "Thu May 22 07:49:47 MDT 2014", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/scicomputprogram.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0167642313001263", acknowledgement = ack-nhfb, fjournal = "Science of Computer Programming", journal-URL = "http://www.sciencedirect.com/science/journal/01676423/", } @Book{Bartsch:2014:TMF, author = "Hans-Jochen Bartsch", title = "{Taschenbuch mathematischer Formeln f{\"u}r Ingenieure und Naturwissenschaftler: [F{\"u}r Studium und Beruf]}. ({German}) [{Pocketbook} of mathematical formulas for engineers and natural sciences: [For study and job]]", publisher = "Fachbuchverlag Leipzig im Hanser-Verlag", address = "M{\"u}nchen, Germany", edition = "Twenty-third", pages = "832", year = "2014", ISBN = "3-446-43800-9", ISBN-13 = "978-3-446-43800-2", LCCN = "????", bibdate = "Wed Mar 1 17:30:07 MST 2017", bibsource = "fsz3950.oclc.org:210/WorldCat; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://d-nb.info/1045240478/04", acknowledgement = ack-nhfb, language = "German", tableofcontents = "1 Logik, Mengen, Zahlensysteme / 21 \\ 2 Arithmetik / 46 \\ 3 Gleichungen und Ungleichungen / 91 \\ 4 Elementare Geometrie / 124 \\ 5 Lineare Algebra / 168 \\ 6 Vektoren, Analytische Geometrie / 244 \\ 7 Funktionen und Kurven / 335 \\ 8 Differenzialrechnung / 421 \\ 9 Integralrechnung / 467 \\ 10 Vektoranalysis / 512 \\ 11 Differenzialgleichungen / 536 \\ 12 Reihen, F- und L-/ Transformation \\ 13 Statistik, Stochastik / 643 \\ 14 Integraltabellen / 719", } @Book{Boyd:2014:STE, author = "John P. (John Philip) Boyd", title = "Solving transcendental equations: the {Chebyshev} polynomial proxy and other numerical rootfinders, perturbation series, and oracles", publisher = pub-SIAM, address = pub-SIAM:adr, pages = "xviii + 460", year = "2014", ISBN = "1-61197-351-1 (paperback)", ISBN-13 = "978-1-61197-351-8 (paperback)", LCCN = "QA353.T7 B69 2014", bibdate = "Wed Sep 23 17:10:53 MDT 2015", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/numana2010.bib; z3950.loc.gov:7090/Voyager", URL = "http://www.loc.gov/catdir/enhancements/fy1503/2014017078-b.html; http://www.loc.gov/catdir/enhancements/fy1503/2014017078-d.html; http://www.loc.gov/catdir/enhancements/fy1503/2014017078-t.html", acknowledgement = ack-nhfb, author-dates = "1951--", subject = "Transcendental functions; Chebyshev polynomials; Transcendental numbers", tableofcontents = "I: Introduction and overview \\ Introduction: Key themes in rootfinding \\ II: the Chebyshev-Proxy rootfinder and its generalizations \\ The Chebyshev-Proxy/Companion matrix rootfinder \\ Adaptive Chebyshev interpolation \\ Adaptive Fourier interpolation and rootfinding \\ Complex zeros: Interpolation on a disk, the Delves--Lyness algorithm, and contour integrals \\ III: Fundamentals: Iterations, bifurcation, and continuation \\ Newton iteration and its kin \\ Bifurcation theory \\ Continuation in a parameter \\ IV: Polynomials \\ Polynomial equations and the irony of Galois Theory \\ The Quadratic Equation \\ Roots of a cubic polynomial \\ Roots of a quartic polynomial \\ V: Analytical methods \\ Methods for explicit solutions \\ Regular perturbation methods for roots \\ Singular perturbation methods: fractional powers, logarithms, and exponential asymptotics \\ VI: Classics, special functions, inverses, and oracles \\ Classical methods for solving one equation in one unknown \\ Special algorithms for special functions \\ Inverse functions of one unknown \\ Oracles: Theorems and algorithms for determining the existence, nonexistence, and number of zeros \\ VII: Bivariate systems \\ Two equations in two unknowns \\ VIII: Challenges \\ Past and future \\ A: Companion matrices \\ B: Chebyshev interpolation and quadrature \\ Marching triangles \\ D: Imbricate-Fourier series and the Poisson Summation Theorem", } @Article{Buehler:2014:CCH, author = "Stephan Buehler and Claude Duhr", title = "{CHAPLIN-Complex Harmonic Polylogarithms} in {Fortran}", journal = j-COMP-PHYS-COMM, volume = "185", number = "10", pages = "2703--2713", month = oct, year = "2014", CODEN = "CPHCBZ", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", bibdate = "Sat Aug 16 08:37:41 MDT 2014", bibsource = "https://www.math.utah.edu/pub/tex/bib/compphyscomm2010.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fortran3.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0010465514001969", acknowledgement = ack-nhfb, fjournal = "Computer Physics Communications", journal-URL = "http://www.sciencedirect.com/science/journal/00104655/", } @Article{Choudhury:2014:SAA, author = "Amit Choudhury", title = "A simple approximation to the area under standard normal curve", journal = "Mathematics and Statistics", volume = "2", number = "3", pages = "147--149", month = "????", year = "2014", DOI = "https://doi.org/10.13189/ms.2014.020307", ISSN = "2332-2071 (print), 2332-2144 (electronic)", ISSN-L = "2332-2071", bibdate = "Sat Dec 16 15:57:04 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.hrpub.org/journals/article_info.php?aid=1470; https://www.hrpub.org/download/20140305/MS7-13401470.pdf", acknowledgement = ack-nhfb, } @Book{Dunkl:2014:OPS, author = "Charles F. Dunkl and Yuan Xu", title = "Orthogonal Polynomials of Several Variables", volume = "155", publisher = pub-CAMBRIDGE, address = pub-CAMBRIDGE:adr, edition = "Second", pages = "xvii + 420", year = "2014", ISBN = "1-107-07189-5, 1-316-05717-8 (e-book)", ISBN-13 = "978-1-107-07189-6, 978-1-316-05717-9 (e-book)", LCCN = "QA404.5 .D86 2014", bibdate = "Sat Nov 11 06:43:34 MST 2023", bibsource = "fsz3950.oclc.org:210/WorldCat; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", series = "Encyclopedia of mathematics and its applications", URL = "http://catdir.loc.gov/catdir/enhancements/fy1408/2014001846-b.html; http://catdir.loc.gov/catdir/enhancements/fy1408/2014001846-d.html; http://catdir.loc.gov/catdir/enhancements/fy1408/2014001846-t.html; http://digitale-objekte.hbz-nrw.de/storage2/2014/11/17/file_11/5877666.pdf", acknowledgement = ack-nhfb, remark = "See also first edition \cite{Dunkl:2001:OPS}", shorttableofcontents = "Preface to the Second Edition / xiii \\ Preface to the First Edition / xv \\ 1 Background / 1 \\ 2 Orthogonal Polynomials in Two Variables / 28 \\ 3 General Properties of Orthogonal Polynomials in Several Variables / 57 \\ 4 Orthogonal Polynomials on the Unit Sphere / 114 \\ 5 Examples of Orthogonal Polynomials in Several Variables / 137 \\ 6 Root Systems and Coxeter Groups / 174 \\ 7 Spherical Harmonics Associated with Reflection Groups / 208 \\ 8 Generalized Classical Orthogonal Polynomials / 258 \\ 9 Summability of Orthogonal Expansions / 289 \\ 10 Orthogonal Polynomials Associated with Symmetric Groups / 318 \\ 11 Orthogonal Polynomials Associated with Octahedral Groups and Applications / 364 \\ References / 396 \\ Author Index / 413 \\ Symbol Index / 416 \\ Subject Index / 418", subject = "Orthogonal polynomials; Functions of several real variables; Polyn{\^o}mes orthogonaux; Fonctions de plusieurs variables r{\'e}elles; Functions of several real variables; Orthogonal polynomials; Orthogonale reeksen; Ortogonalpolynom", tableofcontents = "Preface to the Second Edition / xiii \\ Preface to the First Edition / xv \\ \\ 1 Background / 1 \\ 1.1 The Gamma and Beta Functions / 1 \\ 1.2 Hypergeometric Series / 3 \\ 1.2.1 Lauricella series / 5 \\ 1.3 Orthogonal Polynomials of One Variable / 6 \\ 1.3.1 General properties / 6 \\ 1.3.2 Three-term recurrence / 9 \\ 1.4 Classical Orthogonal Polynomials / 13 \\ 1.4.1 Hermite polynomials / 13 \\ 1.4.2 Laguerre polynomials / 14 \\ 1.4.3 Gegenbauer polynomials / 16 \\ 1.4.4 Jacobi polynomials / 20 \\ 1.5 Modified Classical Polynomials / 22 \\ 1.5.1 Generalized Hermite polynomials / 24 \\ 1.5.2 Generalized Gegenbauer polynomials / 25 \\ 1.5.3 A limiting relation / 27 \\ 1.6 Notes / 27 \\ \\ 2 Orthogonal Polynomials in Two Variables / 28 \\ 2.1 Introduction / 28 \\ 2.2 Product Orthogonal Polynomials / 29 \\ 2.3 Orthogonal Polynomials on the Unit Disk / 30 \\ 2.4 Orthogonal Polynomials on the Triangle / 35 \\ 2.5 Orthogonal Polynomials and Differential Equations / 37 \\ 2.6 Generating Orthogonal Polynomials of Two Variables / 38 \\ 2.6.1 A method for generating orthogonal polynomials / 38 \\ 2.6.2 Orthogonal polynomials for a radial weight / 40 \\ 2.6.3 Orthogonal polynomials in complex variables / 41 \\ 2.7 First Family of Koornwinder Polynomials / 45 \\ 2.8 A Related Family of Orthogonal Polynomials / 43 \\ 2.9 Second Family of Koornwinder Polynomials / 50 \\ 2.10 Notes / 54 \\ \\ 3 General Properties of Orthogonal Polynomials in Several Variables / 57 \\ 3.1 Notation and Preliminaries / 58 \\ 3.2 Moment Functionals and Orthogonal Polynomials in Several Variables / 60 \\ 3.2.1 Definition of orthogonal polynomials / 60 \\ 3.2.2 Orthogonal polynomials and moment matrices / 64 \\ 3.2.3 The moment problem / 67 \\ 3.3 The Three-Term Relation / 70 \\ 3.3.1 Definition and basic properties / 70 \\ 3.3.2 Favard's theorem / 73 \\ 3.3.3 Centrally symmetric integrals / 76 \\ 3.3.4 Examples / 79 \\ 3.4 Jacobi Matrices and Commuting Operators / 82 \\ 3.5 Further Properties of the Three-Term Relation / 87 \\ 3.5.1 Recurrence formula / 87 \\ 3.5.2 General solutions of the three-term relation / 94 \\ 3.6 Reproducing Kernels and Fourier Orthogonal Series / 96 \\ 3.6.1 Reproducing kernels / 97 \\ 3.6.2 Fourier orthogonal series / 101 \\ 3.7 Common Zeros of Orthogonal Polynomials in Several Variables / 103 \\ 3.8 Gaussian Cubature Formulae / 107 \\ 3.9 Notes / 112 \\ \\ 4 Orthogonal Polynomials on the Unit Sphere / 114 \\ 4.1 Spherical Harmonics / 114 \\ 4.2 Orthogonal Structures on $S^d$ and on $B^d$ / 119 \\ 4.3 Orthogonal Structures on $B^d$ and on $S^{d + m - 1}$ / 125 \\ 4.4 Orthogonal Structures on the Simplex / 129 \\ 4.5 Van der Corput--Schaake Inequality / 133 \\ 4.6 Notes / 136 \\ \\ 5 Examples of Orthogonal Polynomials in Several Variables / 137 \\ 5.1 Orthogonal Polynomials for Simple Weight Functions / 137 \\ 5.1.1 Product weight functions / 138 \\ 5.1.2 Rotation-invariant weight functions / 138 \\ 5.1.3 Multiple Hermite polynomials on $\mathbb{R}^d$ / 139 \\ 5.1.4 Multiple Laguerre polynomials on $\mathbb{R}^d__$ / 141 \\ 5.2 Classical Orthogonal Polynomials on the Unit Ball / 141 \\ 5.2.1 Orthonormal bases / 142 \\ 5.2.2 Appell's monic orthogonal and biorthogonal polynomials / 143 \\ 5.2.3 Reproducing kernel with respect to $W_\mu^B$ on $B^d$ / 148 \\ 53.3 Classical Orthogonal Polynomials on the Simplex / 150 \\ 5.4 Orthogonal Polynomials via Symmetric Functions / 154 \\ 5.4.1 Two general families of orthogonal polynomials / 154 \\ 5.4.2 Common zeros and Gaussian cubature formulae / 156 \\ 5.5 Chebyshev Polynomials of Type ${\cal A}_d$ / 165 \\ 5.6 Sobolev Orthogonal Polynomials on the Unit Ball / 165 \\ 5.6.1 Sobolev orthogonal polynomials defined via the gradient operator / 165 \\ 5.6.2 Sobolev orthogonal polynomials defined via the Laplacian operator / 168 \\ 5.7 Notes / 171 \\ \\ 6 Root Systems and Coxeter Groups / 174 \\ 6.1 Introduction and Overview / 174 \\ 6.2 Root Systems / 176 \\ 6.2.1 Type $A_{d - 1}$ / 179 \\ 6.2.2 Type $B_d$ / 179 \\ 6.2.3 Type $I_2(m)$ / 180 \\ 6.2.4 Type $D_d$ / 181 \\ 6.2.5 Type $H_3$ / 181 \\ 6.2.6 Type $F_4$ / 182 \\ 6.2.7 Other types / 182 \\ 6.2.8 Miscellaneous results / 182 \\ 6.3 Invariant Polynomials / 183 \\ 6.3.1 Type $A_{d - 1}$ invariants / 183 \\ 6.3.2 Type $B_d$ invariants / 186 \\ 6.3.3 Type $D_d$ invariants / 186 \\ 6.3.4 Type $I_2(m)$ invariants / 186 \\ 6.3.5 Type $H_3$ invariants / 186 \\ 6.3.6 Type $F_4$ invariants / 187 \\ 6.4 Differential--Difference Operators / 187 \\ 6.5 The Intertwining Operator / 192 \\ 6.6 The $\kappa$-Analogue of the Exponential / 200 \\ 6.7 Invariant Differential Operators / 202 \\ 6.8 Notes / 207 \\ \\ 7 Spherical Harmonics Associated with Reflection Groups / 208 \\ 7.1 $h$-Harmonic Polynomials / 208 \\ 7.2 Inner Products on Polynomials / 217 \\ 7.3 Reproducing Kernels and the Poisson Kernel / 221 \\ 7.4 Integration of the Intertwining Operator / 224 \\ 7.5 Example: Abelian Group ${\cal Z}_2^d$ / 228 \\ 7.5.1 Orthogonal basis for $h$-harmonics / 228 \\ 7.5.2 Intertwining and projection operators / 232 \\ 7.5.3 Monic orthogonal basis / 235 \\ 7.6 Example: Dihedral Groups / 240 \\ 7.6.1 An orthonormal basis of ${\cal H}_(h^2_{\alpha, \beta})$ / 241 \\ 7.6.2 Cauchy and Poisson kernels / 248 \\ 7.7 The Dunkl Transform / 250 \\ 7.8 Notes / 256 \\ \\ 8 Generalized Classical Orthogonal Polynomials / 258 \\ 8.1 Generalized Classical Orthogonal Polynomials on the Ball / 258 \\ 8.1.1 Definition and differential-difference equations / 258 \\ 8.1.2 Orthogonal basis and reproducing kernel / 263 \\ 8.1.3 Orthogonal polynomials for ${\cal Z}_2^d$-invariant weight functions / 266 \\ 8.1.4 Reproducing kernel for ${\cal Z}_2^d$-invariant weight functions / 268 \\ 8.2 Generalized Classical Orthogonal Polynomials on the Simplex / 271 \\ 8.2.1 Weight function and differential-difference equation / 271 \\ 8.2.2 Orthogonal basis and reproducing kernel / 273 \\ 8.2.3 Monic orthogonal polynomials / 287 \\ 8.3 Generalized Hermite Polynomials / 278 \\ 8.4 Generalized Laguerre Polynomials / 283 \\ 8.5 Notes / 287 \\ \\ 9 Summability of Orthogonal Expansions / 289 \\ 9.1 General Results on Orthogonal Expansions / 289 \\ 9.1.1 Uniform convergence of partial sums / 289 \\ 9.1.2 Ces{\`a}ro means of the orthogonal expansion / 293 \\ 9.2 Orthogonal Expansion on the Sphere / 296 \\ 9.3 Orthogonal Expansion on the Ball / 299 \\ 9.4 Orthogonal Expansion on the Simplex / 304 \\ 9.5 Orthogonal Expansion of Laguerre and Hermite Polynomials / 306 \\ 9.6 Multiple Jacobi Expansion / 311 \\ 9.7 Notes / 315 \\ \\ 10 Orthogonal Polynomials Associated with Symmetric Groups / 318 \\ 10.1 Partitions, Compositions and Orderings / 318 \\ 10.2 Commuting Self-Adjoint Operators / 320 \\ 10.3 The Dual Polynomial Basis / 322 \\ 10.4 $S_d$-Invariant Subspaces / 329 \\ 10.5 Degree-Changing Recurrences / 334 \\ 10.6 Norm Formulae / 337 \\ 10.6.1 Hook-length products and the pairing norm / 337 \\ 10.6.2 The biorthogonal-type norm / 341 \\ 10.6.3 The torus inner product / 343 \\ 10.6.4 Monic polynomials / 346 \\ 10.6.5 Normalizing constants / 346 \\ 10.7 Symmetric Functions and Jack Polynomials / 350 \\ 10.8 Miscellaneous Topics / 357 \\ 10.9 Notes / 362 \\ \\ 11 Orthogonal Polynomials Associated with Octahedral Groups and Applications / 364 \\ 11.1 Introduction / 364 \\ 11.2 Operators of Type $B$ / 365 \\ 11.3 Polynomial Eigenfunctions of Type $B$ / 368 \\ 11.4 Generalized Binomial Coefficients / 376 \\ 11.5 Hermite Polynomials of Type $B$ / 373 \\ 11.6 Calogero--Sutherland Systems / 385 \\ 11.6.1 The simple harmonic oscillator / 386 \\ 11.6.2 Root systems and the Laplacian / 387 \\ 11.6.3 Type $A$ models on the line / 387 \\ 11.6.4 Type $A$ models on the circle / 389 \\ 11.6.5 Type $B$ models on the line / 392 \\ 11.7 Notes / 394 \\ \\ References / 396 \\ Author Index / 413 \\ Symbol Index / 416 \\ Subject Index / 418", } @Article{Fukushima:2014:ACG, author = "Toshio Fukushima", title = "Analytical computation of generalized {Fermi--Dirac} integrals by truncated {Sommerfeld} expansions", journal = j-APPL-MATH-COMP, volume = "234", number = "??", pages = "417--433", day = "15", month = may, year = "2014", CODEN = "AMHCBQ", ISSN = "0096-3003 (print), 1873-5649 (electronic)", ISSN-L = "0096-3003", bibdate = "Mon Apr 21 18:04:13 MDT 2014", bibsource = "https://www.math.utah.edu/pub/tex/bib/applmathcomput2010.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0096300314002926", acknowledgement = ack-nhfb, fjournal = "Applied Mathematics and Computation", journal-URL = "http://www.sciencedirect.com/science/journal/00963003/", } @Article{Fukushima:2014:CGI, author = "Toshio Fukushima", title = "Computation of a general integral of {Fermi--Dirac} distribution by {McDougall--Stoner} method", journal = j-APPL-MATH-COMP, volume = "238", number = "??", pages = "485--510", day = "1", month = jul, year = "2014", CODEN = "AMHCBQ", ISSN = "0096-3003 (print), 1873-5649 (electronic)", ISSN-L = "0096-3003", bibdate = "Fri May 23 10:53:19 MDT 2014", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/d/dirac-p-a-m.bib; https://www.math.utah.edu/pub/bibnet/authors/f/fermi-enrico.bib; https://www.math.utah.edu/pub/tex/bib/applmathcomput2010.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/S009630031400561X", acknowledgement = ack-nhfb, fjournal = "Applied Mathematics and Computation", journal-URL = "http://www.sciencedirect.com/science/journal/00963003/", } @Article{Gil:2014:ACM, author = "Amparo Gil and Javier Segura and Nico M. Temme", title = "{Algorithm 939}: Computation of the {Marcum} {$Q$}-Function", journal = j-TOMS, volume = "40", number = "3", pages = "20:1--20:21", month = apr, year = "2014", CODEN = "ACMSCU", DOI = "https://doi.org/10.1145/2591004", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Mon Apr 21 17:42:14 MDT 2014", bibsource = "http://www.acm.org/pubs/contents/journals/toms/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib", abstract = "Methods and an algorithm for computing the generalized Marcum $Q$-function $ (Q_\mu (x, y))$ and the complementary function $ (P_\mu (x, y))$ are described. These functions appear in problems of different technical and scientific areas such as, for example, radar detection and communications, statistics, and probability theory, where they are called the noncentral chi-square or the noncentral gamma cumulative distribution functions. The algorithm for computing the Marcum functions combines different methods of evaluation in different regions: series expansions, integral representations, asymptotic expansions, and use of three-term homogeneous recurrence relations. A relative accuracy close to $ 10^{-12}$ can be obtained in the parameter region $ (x, y, \mu) \in [0, A] \times [0, A] \times [1, A]$, $ A = 200$, while for larger parameters the accuracy decreases (close to $ 10^{-11}$ for $ A = 1000$ and close to $ 5 \times 10^{-11}$ for $ A = 10000$).", acknowledgement = ack-nhfb, articleno = "20", fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", } @Article{Gil:2014:CZA, author = "Amparo Gil and Javier Segura", title = "On the complex zeros of {Airy} and {Bessel} functions and those of their derivatives", journal = j-ANAL-APPL, volume = "12", number = "5", pages = "537--561", month = aug, year = "2014", DOI = "https://doi.org/10.1142/s0219530514500341", ISSN = "0219-5305 (print), 1793-6861 (electronic)", ISSN-L = "0219-5305", bibdate = "Thu Nov 16 07:32:34 2023", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/o/olver-frank-w-j.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "Special Issue: Dedicated to the Memory of Frank Olver (Part II).", acknowledgement = ack-nhfb, ajournal = "Anal. Appl. (Singapore)", fjournal = "Analysis and Applications (Singapore)", journal-URL = "https://www.worldscientific.com/worldscinet/aa", subject-dates = "Frank William John Olver (15 December 1924--23 April 2013)", } @Article{Gil:2014:RSD, author = "Amparo Gil and Javier Segura and Nico M. Temme", title = "Recent software developments for special functions in the {Santander--Amsterdam} project", journal = j-SCI-COMPUT-PROGRAM, volume = "90 (part A)", number = "??", pages = "42--54", day = "15", month = sep, year = "2014", CODEN = "SCPGD4", DOI = "https://doi.org/10.1016/j.scico.2013.11.004", ISSN = "0167-6423 (print), 1872-7964 (electronic)", ISSN-L = "0167-6423", bibdate = "Thu May 22 07:49:47 MDT 2014", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/scicomputprogram.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0167642313002888", acknowledgement = ack-nhfb, fjournal = "Science of Computer Programming", journal-URL = "http://www.sciencedirect.com/science/journal/01676423/", } @Article{Goerg:2014:ULW, author = "Georg M. Goerg", title = "Usage of the {Lambert} {$W$} function in statistics", journal = j-ANN-APPL-STAT, volume = "8", number = "4", pages = "2567--2567", month = dec, year = "2014", CODEN = "????", ISSN = "1932-6157 (print), 1941-7330 (electronic)", ISSN-L = "1932-6157", bibdate = "Wed Feb 11 19:26:08 MST 2015", bibsource = "https://www.math.utah.edu/pub/tex/bib/annapplstat.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://projecteuclid.org/euclid.aoas/1419001755", acknowledgement = ack-nhfb, fjournal = "Annals of Applied Statistics", journal-URL = "http://projecteuclid.org/all/euclid.aoas/; http://www.jstor.org/journals/19326157.html", } @InProceedings{Greuel:2014:SIS, author = "Gert-Martin Greuel and Wolfram Sperber", title = "{swMATH} --- an Information Service for Mathematical Software", crossref = "Hong:2014:MSI", pages = "691--701", year = "2014", DOI = "https://doi.org/10.1007/978-3-662-44199-2_103", bibdate = "Tue Sep 26 10:21:48 2023", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, } @Article{Greynat:2014:NAE, author = "David Greynat and Javier Sesma", title = "A new approach to the epsilon expansion of generalized hypergeometric functions", journal = j-COMP-PHYS-COMM, volume = "185", number = "2", pages = "472--478", month = feb, year = "2014", CODEN = "CPHCBZ", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", bibdate = "Mon Dec 2 12:05:01 MST 2013", bibsource = "https://www.math.utah.edu/pub/tex/bib/compphyscomm2010.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/S001046551300324X", acknowledgement = ack-nhfb, fjournal = "Computer Physics Communications", journal-URL = "http://www.sciencedirect.com/science/journal/00104655", } @Article{Harvey:2014:SAC, author = "David Harvey", title = "A subquadratic algorithm for computing the $n$-th {Bernoulli} number", journal = j-MATH-COMPUT, volume = "83", number = "289", pages = "2471--2477", year = "2014", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Sep 9 11:37:57 MDT 2014", bibsource = "http://www.ams.org/mcom/2014-83-289; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp2010.bib", URL = "http://www.ams.org/journals/mcom/2014-83-289/S0025-5718-2014-02832-9; http://www.ams.org/journals/mcom/2014-83-289/S0025-5718-2014-02832-9/S0025-5718-2014-02832-9.pdf", acknowledgement = ack-nhfb, fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Johansson:2014:EIE, author = "Fredrik Johansson", title = "Efficient implementation of elementary functions in the medium-precision range", journal = "arxiv.org", volume = "??", number = "??", pages = "??--??", day = "27", month = oct, year = "2014", bibdate = "Mon Jun 12 16:12:02 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "https://arxiv.org/abs/1410.7176", abstract = "We describe a new implementation of the elementary transcendental functions exp, sin, cos, log and atan for variable precision up to approximately 4096 bits. Compared to the MPFR library, we achieve a maximum speedup ranging from a factor 3 for cos to 30 for atan. Our implementation uses table-based argument reduction together with rectangular splitting to evaluate Taylor series. We collect denominators to reduce the number of divisions in the Taylor series, and avoid overhead by doing all multiprecision arithmetic using the mpn layer of the GMP library. Our implementation provides rigorous error bounds.", acknowledgement = ack-nhfb, } @PhdThesis{Johansson:2014:FRC, author = "Fredrik Johansson", title = "Fast and Rigorous Computation of Special Functions to High Precision", type = "{Ph.D.} thesis", school = "Johannes Kepler University", address = "Linz, Austria", pages = "ix + 109", day = "24", month = mar, year = "2014", bibdate = "Sat Aug 09 09:01:13 2014", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://permalink.obvsg.at/AC10776210", abstract = "The problem of efficiently evaluating special functions to high precision has been considered by numerous authors. Important tools used for this purpose include algorithms for evaluation of linearly recurrent sequences, and algorithms for power series arithmetic.\par In this work, we give new baby-step, giant-step algorithms for evaluation of linearly recurrent sequences involving an expensive parameter (such as a high-precision real number) and for computing compositional inverses of power series. Our algorithms do not have the best asymptotic complexity, but they are faster than previous algorithms in practice over a large input range.\par Using a combination of techniques, we also obtain efficient new algorithms for numerically evaluating the gamma function $ \Gamma (z) $ and the Hurwitz zeta function $ \zeta (s, a) $, or Taylor series expansions of those functions, with rigorous error bounds. Our methods achieve softly optimal complexity when computing a large number of derivatives to proportionally high precision.\par Finally, we show that isolated values of the integer partition function $ p(n) $ can be computed rigorously with softly optimal complexity by means of the Hardy--Ramanujan--Rademacher formula and careful numerical evaluation. We provide open source implementations which run significantly faster than previously published software. The implementations are used for record computations of the partition function, including the tabulation of several billion Ramanujan-type congruences, and of Taylor series associated with the Riemann zeta function.", acknowledgement = ack-nhfb, remark = "Reviewed in \booktitle{ACM Communications in Computer Algebra}, {\bf 48}(2) 28--28 (2014).", } @Article{Krasikov:2014:ABA, author = "Ilia Krasikov", title = "Approximations for the {Bessel} and {Airy} functions with an explicit error term", journal = j-LMS-J-COMPUT-MATH, volume = "17", number = "1", pages = "209--225", year = "2014", CODEN = "????", DOI = "https://doi.org/10.1112/S1461157013000351", ISSN = "1461-1570", bibdate = "Tue Sep 9 12:34:08 MDT 2014", bibsource = "http://journals.cambridge.org/action/displayJournal?jid=JCM; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/lms-j-comput-math.bib", acknowledgement = ack-nhfb, ajournal = "LMS J. Comput. Math.", fjournal = "LMS Journal of Computation and Mathematics", journal-URL = "http://journals.cambridge.org/action/displayJournal?jid=JCM", onlinedate = "19 May 2014", } @InProceedings{Kupriianova:2014:MMF, author = "Olga Kupriianova and Christoph Lauter", title = "{Metalibm}: A Mathematical Functions Code Generator", crossref = "Hong:2014:MSI", pages = "713--717", year = "2014", DOI = "https://doi.org/10.1007/978-3-662-44199-2_106", bibdate = "Tue Sep 26 10:21:48 2023", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, } @Article{Li:2014:ICH, author = "Dingfang Li and Ping Liu and Jisheng Kou", title = "An improvement of {Chebyshev--Halley} methods free from second derivative", journal = j-APPL-MATH-COMP, volume = "235", number = "??", pages = "221--225", day = "25", month = may, year = "2014", CODEN = "AMHCBQ", ISSN = "0096-3003 (print), 1873-5649 (electronic)", ISSN-L = "0096-3003", bibdate = "Mon Apr 21 18:04:20 MDT 2014", bibsource = "https://www.math.utah.edu/pub/tex/bib/applmathcomput2010.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0096300314003312", acknowledgement = ack-nhfb, fjournal = "Applied Mathematics and Computation", journal-URL = "http://www.sciencedirect.com/science/journal/00963003/", } @Article{Lu:2014:GAF, author = "Dawei Lu and Jinghai Feng and Congxu Ma", title = "A general asymptotic formula of the gamma function based on the {Burnside}'s formula", journal = j-J-NUMBER-THEORY, volume = "145", number = "??", pages = "317--328", month = dec, year = "2014", CODEN = "JNUTA9", DOI = "https://doi.org/10.1016/j.jnt.2014.06.016", ISSN = "0022-314X (print), 1096-1658 (electronic)", ISSN-L = "0022-314X", bibdate = "Wed Jul 15 08:49:09 MDT 2020", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jnumbertheory2010.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0022314X14002224", acknowledgement = ack-nhfb, ajournal = "J. Number Theory", fjournal = "Journal of Number Theory", journal-URL = "http://www.sciencedirect.com/science/journal/0022314X", } @Article{Lu:2014:GAG, author = "Dawei Lu and Lixin Song and Congxu Ma", title = "A generated approximation of the gamma function related to {Windschitl}'s formula", journal = j-J-NUMBER-THEORY, volume = "140", number = "??", pages = "215--225", month = jul, year = "2014", CODEN = "JNUTA9", DOI = "https://doi.org/10.1016/j.jnt.2014.01.023", ISSN = "0022-314X (print), 1096-1658 (electronic)", ISSN-L = "0022-314X", bibdate = "Wed Jul 15 08:49:07 MDT 2020", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jnumbertheory2010.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0022314X14000687", acknowledgement = ack-nhfb, ajournal = "J. Number Theory", fjournal = "Journal of Number Theory", journal-URL = "http://www.sciencedirect.com/science/journal/0022314X", } @Article{Lu:2014:NAE, author = "Dawei Lu and Xiaoguang Wang", title = "A new asymptotic expansion and some inequalities for the gamma function", journal = j-J-NUMBER-THEORY, volume = "140", number = "??", pages = "314--323", month = jul, year = "2014", CODEN = "JNUTA9", DOI = "https://doi.org/10.1016/j.jnt.2014.01.025", ISSN = "0022-314X (print), 1096-1658 (electronic)", ISSN-L = "0022-314X", bibdate = "Wed Jul 15 08:49:07 MDT 2020", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jnumbertheory2010.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0022314X14000705", acknowledgement = ack-nhfb, ajournal = "J. Number Theory", fjournal = "Journal of Number Theory", journal-URL = "http://www.sciencedirect.com/science/journal/0022314X", } @Article{Lu:2014:SNI, author = "Dawei Lu", title = "Some new improved classes of convergence towards {Euler}'s constant", journal = j-APPL-MATH-COMP, volume = "243", number = "??", pages = "24--32", day = "15", month = sep, year = "2014", CODEN = "AMHCBQ", DOI = "https://doi.org/10.1016/j.amc.2014.05.098", ISSN = "0096-3003 (print), 1873-5649 (electronic)", ISSN-L = "0096-3003", bibdate = "Sat Aug 16 10:10:22 MDT 2014", bibsource = "https://www.math.utah.edu/pub/tex/bib/applmathcomput2010.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/S009630031400798X", acknowledgement = ack-nhfb, fjournal = "Applied Mathematics and Computation", journal-URL = "http://www.sciencedirect.com/science/journal/00963003/", keywords = "Continued fraction; Euler's constant; Inequalities; Rate of convergence", } @Article{Mortici:2014:SBG, author = "Cristinel Mortici", title = "Sharp bounds for gamma function in terms of $ x^{x - 1} $", journal = j-APPL-MATH-COMP, volume = "249", number = "??", pages = "278--285", day = "15", month = dec, year = "2014", CODEN = "AMHCBQ", ISSN = "0096-3003 (print), 1873-5649 (electronic)", ISSN-L = "0096-3003", bibdate = "Wed Nov 26 10:49:00 MST 2014", bibsource = "https://www.math.utah.edu/pub/tex/bib/applmathcomput2010.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0096300314013939", acknowledgement = ack-nhfb, fjournal = "Applied Mathematics and Computation", journal-URL = "http://www.sciencedirect.com/science/journal/00963003/", } @Article{Ogburn:2014:FDC, author = "Daniel X. Ogburn and Colin L. Waters and Murray D. Sciffer and Jeff A. Hogan and Paul C. Abbott", title = "A finite difference construction of the spheroidal wave functions", journal = j-COMP-PHYS-COMM, volume = "185", number = "1", pages = "244--253", month = jan, year = "2014", CODEN = "CPHCBZ", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", bibdate = "Mon Dec 2 12:04:56 MST 2013", bibsource = "https://www.math.utah.edu/pub/tex/bib/compphyscomm2010.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0010465513002610", acknowledgement = ack-nhfb, fjournal = "Computer Physics Communications", journal-URL = "http://www.sciencedirect.com/science/journal/00104655", } @Article{Piparo:2014:SHE, author = "Danilo Piparo and Vincenzo Innocente and Thomas Hauth", title = "Speeding up {HEP} experiment software with a library of fast and auto-vectorisable mathematical functions", journal = j-J-PHYS-CONF-SER, volume = "513", number = "5", pages = "052027", month = jun, year = "2014", CODEN = "JPCSDZ", DOI = "https://doi.org/10.1088/1742-6596/513/5/052027", ISSN = "1742-6588 (print), 1742-6596 (electronic)", ISSN-L = "1742-6588", bibdate = "Tue Sep 24 14:55:02 2024", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Journal of Physics: Conference Series", journal-URL = "http://www.iop.org/EJ/journal/conf", } @Book{Potter:2014:APC, author = "Ronald W. Potter", title = "Arbitrary Precision Calculation of Selected Higher Functions", publisher = "Lulu", address = "????", pages = "????", year = "2014", ISBN = "1-312-59943-X", ISBN-13 = "978-1-312-59943-7", LCCN = "QA76.9.A43 P56 2014", bibdate = "Sat Dec 10 15:39:37 MST 2022", bibsource = "fsz3950.oclc.org:210/WorldCat; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, subject = "Computer algorithms; Computational complexity; Functional programming languages; Mathematics; Algorithms; Algorithmes; Complexit{\'e} de calcul (Informatique); Langages de programmation fonctionnels; Math{\'e}matiques; algorithms; mathematics; applied mathematics; Algorithms; Computational complexity; Computer algorithms; Functional programming languages; Mathematics", tableofcontents = "Preface / ii \\ Introduction / x \\ 1: Basic Arithmetic / 1-1 \\ 2: High Precision Computational Techniques / 2-1 \\ 3: Elementary Functions / 3-1 \\ 4: Euler's Constant / 4-1 \\ 5: Gamma and Polygamma Functions / 5-1 \\ 6: Elliptic Integrals and $ \pi $ / 6-1 \\ 7: Jacobian Elliptic Functions / 7-1 \\ 8: Theta Functions / 8-1 \\ 9: Incomplete Gamma Functions, Chi$^2$ and Inverse Chi$^2$ Distribution / 9-1 \\ 10: Beta and Incomplete Beta Functions, Student's $t$ and $F$-Distributions and Their Inverses / 10-1 \\ 11: Error Functions, Gaussian Distribution and Inverse / 11-1 \\ 12: Modified Bessel Functions / 12-1 \\ 13: Ordinary Bessel Functions / 13-1 \\ 14: Zeros of Ordinary Bessel Functions / 14-1 \\ 15: Spherical Bessel Functions / 15-1 \\ 16: Airy Functions and Zeros / 16-1 \\ 17: Kelvin Functions / 17-1 \\ 18: Struve Functions / 18-1 \\ 19: Fresnel Integrals / 19-1 \\ 20: Exponential Integrals / 20-1 \\ 21: Sine\slash Cosine and Sinh\slash Cosh Integrals / 21-1 \\ 22: Orthogonal Polynomials / 22-1 \\ 23: Polynomial Roots / 23-1 \\ 24: Matrix Operations / 24-1 \\ 25: Geometric Operations / 25-1 \\ Appendix A: Fast Fourier Transform (FFT) / A-1 \\ Appendix B: The AGM Algorithm / B-1 \\ Appendix C: Contours of Bessel Function Zeros / C-1 \\ Appendix D: A Few Numbers (6071 digits per number) / D-1 \\ Appendix E: 315061 Decimal Digits of Euler's Constant / E-1 \\ Index / I-1 \\ About the Author", } @Article{Qi:2014:IRC, author = "Feng Qi", title = "Integral representations and complete monotonicity related to the remainder of {Burnside}'s formula for the gamma function", journal = j-J-COMPUT-APPL-MATH, volume = "268", number = "??", pages = "155--167", day = "1", month = oct, year = "2014", CODEN = "JCAMDI", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Sat Feb 25 13:34:45 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2010.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0377042714001356", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @InProceedings{Rappoport:2014:MSM, author = "Juri Rappoport", title = "Mathematical Software for Modified {Bessel} Functions", crossref = "Hong:2014:MSI", pages = "325--332", year = "2014", DOI = "https://doi.org/10.1007/978-3-662-44199-2_51", bibdate = "Tue Sep 26 10:17:51 2023", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, } @Article{Ratnanather:2014:ATI, author = "J. Tilak Ratnanather and Jung H. Kim and Sirong Zhang and Anthony M. J. Davis and Stephen K. Lucas", title = "{Algorithm 935}: {{\tt IIPBF}}, a {{\tt MATLAB}} toolbox for infinite integral of products of two {Bessel} functions", journal = j-TOMS, volume = "40", number = "2", pages = "14:1--14:12", month = feb, year = "2014", CODEN = "ACMSCU", DOI = "https://doi.org/10.1145/2508435", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Fri Mar 14 06:30:41 MDT 2014", bibsource = "http://www.acm.org/pubs/contents/journals/toms/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/matlab.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib", abstract = "A {\tt MATLAB} toolbox, {\tt IIPBF}, for calculating infinite integrals involving a product of two Bessel functions $ J_a(\rho x) J_b(\tau x) $, $ J_a(\rho x) Y_b(\tau x) $, and $ Y_a(\rho x) Y_b(\tau x) $, for non-negative integers $a$, $b$, and a well-behaved function $ f(x) $, is described. Based on the Lucas algorithm previously developed for $ J_a(\rho x) J_b(\tau x) $ only, {\tt IIPBF} recasts each product as the sum of two functions whose oscillatory behavior is exploited in the three-step procedure of adaptive integration, summation, and extrapolation. The toolbox uses customised {\tt QUADPACK} and {\tt IMSL} functions from a {\tt MATLAB} conversion of the {\tt SLATEC} library. In addition, {\tt MATLAB}'s own {\tt quadgk} function for adaptive Gauss--Kronrod quadrature results in a significant speed up compared with the original algorithm. Usage of {\tt IIPBF} is described and eighteen test cases illustrate the robustness of the toolbox; five additional ones are used to compare {\tt IIPBF} with the {\tt BESSELINT} code for rational and exponential forms of $ f(x) $ with $ J_a(\rho x) J_b(\tau x) $. Reliability for a broad range of values of $ \rho $ and $ \tau $ for the three different product types as well as different orders in one case is demonstrated. An electronic appendix provides a novel derivation of formulae for five cases.", acknowledgement = ack-nhfb, articleno = "14", fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", } @Article{Shukla:2014:LLH, author = "R. Shukla and K. C. Ray", title = "Low Latency Hybrid {CORDIC} Algorithm", journal = j-IEEE-TRANS-COMPUT, volume = "63", number = "12", pages = "3066--3078", month = dec, year = "2014", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/TC.2013.173", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Thu Dec 4 10:36:57 2014", bibsource = "https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput2010.bib", acknowledgement = ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", keywords = "adders; Approximation algorithms; communication systems; Computer architecture; coordinate rotational digital computer; CORDIC algorithm; Delays; digital arithmetic; Digital computers; digital computers; double step branching; fast adders; first order hardware architecture; hardware complexity; hybrid CORDIC algorithm; image processing; low latency; low latency hybrid CORDIC algorithm; Mathematical model; radix-4; redundant arithmetic; scale factor calculation; signal processing; Signal processing algorithms", } @Article{Soranzo:2014:VSE, author = "Alessandro Soranzo and Emanuela Epure", title = "Very simply explicitly invertible approximations of normal cumulative and normal quantile function", journal = j-APPL-MATH-SCI-RUSE, volume = "8", pages = "4323--4341", year = "2014", DOI = "https://doi.org/10.12988/ams.2014.45338", ISSN = "1312-885X (print), 1314-7552 (electronic)", ISSN-L = "1312-885X", bibdate = "Sat Dec 16 17:41:14 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://m-hikari.com/ams/ams-2014/ams-85-88-2014/epureAMS85-88-2014.pdf", acknowledgement = ack-nhfb, fjournal = "Applied Mathematical Sciences (Ruse)", journal-URL = "http://www.m-hikari.com/ams/", } @Article{Wang:2014:CFA, author = "Dong Wang and Milo{\v{s}} D. Ercegovac and Yang Xiao", title = "Complex Function Approximation Using Two-Dimensional Interpolation", journal = j-IEEE-TRANS-COMPUT, volume = "63", number = "12", pages = "2948--2960", month = dec, year = "2014", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/TC.2013.181", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Thu Dec 4 10:36:57 2014", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput2010.bib", acknowledgement = ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", keywords = "2D convolution algorithm; 2D interpolation; Approximation error; ASIC; bipartite schemes; bivariate functions; coefficient table; complex exponential; complex function approximation; complex function evaluation; complex reciprocal; Complex reciprocal; Computational complexity; cubic interpolation; exponential functions; field programmable gate arrays; FPGA; Function approximation; generic hardware architecture; interpolation; interpolation degree; interpolation kernels; Lagrange interpolation; Lagrangian functions; linear interpolation; lookup tables; memory requirements; multipartite schemes; quadratic interpolation; Quadratic programming; table lookup; tabulated function; two-dimensional interpolation", } @Article{Wang:2014:FPT, author = "Dong Wang and Jean-Michel Muller and Nicolas Brisebarre and Milo D. Ercegovac", title = "{$ (M, p, k)$-Friendly} Points: a Table-Based Method to Evaluate Trigonometric Function", journal = j-IEEE-TRANS-CIRCUITS-SYST-II-EXPRESS-BRIEFS, volume = "61", number = "9", pages = "711--715", year = "2014", DOI = "https://doi.org/10.1109/TCSII.2014.2331094", ISSN = "1549-7747 (print), 1558-3791 (electronic)", ISSN-L = "1549-7747", bibdate = "Fri Sep 29 10:46:18 2023", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "IEEE Transactions on Circuits and Systems II: Express Briefs", journal-URL = "https://ieeexplore.ieee.org/xpl/issues?punumber=8920", } @Article{Xu:2014:SII, author = "Ai-Min Xu and Zhong-Di Cen", title = "Some identities involving exponential functions and {Stirling} numbers and applications", journal = j-J-COMPUT-APPL-MATH, volume = "260", number = "??", pages = "201--207", month = apr, year = "2014", CODEN = "JCAMDI", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Sat Feb 25 13:34:42 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2010.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0377042713005323", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @Article{Yun:2014:AHA, author = "Beong In Yun", title = "An ad hoc approximation to the {Gauss} error function and a correction method", journal = j-APPL-MATH-SCI-RUSE, volume = "8", pages = "4261--4273", year = "2014", DOI = "https://doi.org/10.12988/ams.2014.45345", ISSN = "1312-885X (print), 1314-7552 (electronic)", bibdate = "Sat Dec 16 18:06:18 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.m-hikari.com/ams/ams-2014/ams-85-88-2014/yunAMS85-88-2014.pdf", acknowledgement = ack-nhfb, fjournal = "Applied Mathematical Sciences (Ruse)", journal-URL = "http://www.m-hikari.com/ams/", } @InProceedings{Zafar:2014:HAD, author = "Saad Zafar and Raviteja Adapa", booktitle = "2014 International Conference on Advances in Electrical Engineering {(ICAEE)}", title = "Hardware architecture design and mapping of ``{Fast Inverse Square Root}'' algorithm", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "1--4", month = jan, year = "2014", DOI = "https://doi.org/10.1109/icaee.2014.6838433", bibdate = "Wed Dec 20 07:29:37 2023", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", abstract = "The Fast Inverse Square Root algorithm has been used in 3D games of past for lighting and reflection calculations, because it offers up to four times performance gains. This paper presents a hardware implementation of the algorithm on an FPGA board by designing the complete architecture and successfully mapping it on Xilinx Spartan 3E after thorough functional verification. The results show that this implementation provides a very efficient single-precision floating point inverse square root calculator with practically accurate results being made available after just 12 short clock cycles. This performance measure is far superior to the software counterpart of the algorithm, and is not processor dependent like rsqrtss of x86 SSE instruction set. Results of this work can aid FPGA based vector processors or graphic processing units with 3D rendering. The hardware design can also form part of a larger floating point arithmetic unit for dedicated reciprocal square root calculations.", acknowledgement = ack-nhfb, } @Misc{Anonymous:2015:L, author = "Anonymous", title = "libcerf", howpublished = "Web site", year = "2015", bibdate = "Mon Jun 12 16:08:24 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://apps.jcns.fz-juelich.de/doku/sc/libcerf", abstract = "This is the home page of libcerf, a self-contained numeric library that provides an efficient and accurate implementation of complex error functions, along with Dawson, Faddeeva, and Voigt functions.", acknowledgement = ack-nhfb, } @Article{Bailey:2015:CCI, author = "D. H. Bailey and J. M. Borwein", title = "{Crandall}'s computation of the incomplete Gamma function and the {Hurwitz} zeta function, with applications to {Dirichlet} {$L$}-series", journal = j-APPL-MATH-COMP, volume = "268", number = "??", pages = "462--477", day = "1", month = oct, year = "2015", CODEN = "AMHCBQ", ISSN = "0096-3003 (print), 1873-5649 (electronic)", ISSN-L = "0096-3003", bibdate = "Wed Sep 16 06:56:32 MDT 2015", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/c/crandall-richard-e.bib; https://www.math.utah.edu/pub/tex/bib/applmathcomput2015.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0096300315008292", acknowledgement = ack-nhfb, fjournal = "Applied Mathematics and Computation", journal-URL = "http://www.sciencedirect.com/science/journal/00963003/", } @Article{Beliakov:2015:ARZ, author = "Gleb Beliakov and Yuri Matiyasevich", title = "Approximation of {Riemann}'s Zeta Function by Finite {Dirichlet} Series: A Multiprecision Numerical Approach", journal = j-EXP-MATH, volume = "24", number = "2", pages = "150--161", year = "2015", CODEN = "????", DOI = "https://doi.org/10.1080/10586458.2014.976801", ISSN = "1058-6458 (print), 1944-950X (electronic)", ISSN-L = "1058-6458", bibdate = "Mon Jun 8 17:49:44 MDT 2015", bibsource = "http://www.tandfonline.com/toc/uexm20/24/2; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/expmath.bib", acknowledgement = ack-nhfb, fjournal = "Experimental Mathematics", journal-URL = "http://www.tandfonline.com/loi/uexm20", } @Article{Boyd:2015:FWC, author = "John P. Boyd", title = "Four ways to compute the inverse of the complete elliptic integral of the first kind", journal = j-COMP-PHYS-COMM, volume = "196", number = "??", pages = "13--18", month = nov, year = "2015", CODEN = "CPHCBZ", DOI = "https://doi.org/10.1016/j.cpc.2015.05.006", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", bibdate = "Tue Sep 22 13:45:19 MDT 2015", bibsource = "https://www.math.utah.edu/pub/tex/bib/compphyscomm2010.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/maple-extract.bib; https://www.math.utah.edu/pub/tex/bib/matlab.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0010465515001733", acknowledgement = ack-nhfb, fjournal = "Computer Physics Communications", journal-URL = "http://www.sciencedirect.com/science/journal/00104655/", } @Article{Brent:2015:BET, author = "Richard P. Brent and Fredrik Johansson", title = "A bound for the error term in the {Brent--McMillan} algorithm", journal = j-MATH-COMPUT, volume = "84", number = "295", pages = "2351--2359", month = "", year = "2015", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Aug 4 08:33:55 MDT 2015", bibsource = "http://www.ams.org/mcom/2015-84-295; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp2010.bib", URL = "http://www.ams.org/journals/mcom/2015-84-295/S0025-5718-2015-02931-7; http://www.ams.org/journals/mcom/2015-84-295/S0025-5718-2015-02931-7/S0025-5718-2015-02931-7.pdf", abstract = "The Brent--McMillan algorithm B3 (1980), when implemented with binary splitting, is the fastest known algorithm for high-precision computation of Euler's constant. However, no rigorous error bound for the algorithm has ever been published. We provide such a bound and justify the empirical observations of Brent and McMillan. We also give bounds on the error in the asymptotic expansions of functions related to the Bessel functions $ I_0 (x) $ and $ K_0 (x) $ for positive real $x$.", acknowledgement = ack-nhfb, fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "Euler's constant; multiple-precision arithmetic", } @InProceedings{Brunie:2015:CGM, author = "Nicolas Brunie and Florent de Dinechin and Olga Kupriianova and Christoph Lauter", title = "Code Generators for Mathematical Functions", crossref = "Muller:2015:ISC", pages = "66--73", year = "2015", DOI = "https://doi.org/10.1109/ARITH.2015.22", bibdate = "Sat Aug 01 08:05:52 2015", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, keywords = "ARITH-22", } @Article{Chen:2015:AEC, author = "Chao-Ping Chen and Neven Elezovi{\'c}", title = "Asymptotic expansions and completely monotonic functions associated with the gamma, psi and polygamma functions", journal = j-APPL-MATH-COMP, volume = "269", number = "??", pages = "232--241", day = "15", month = oct, year = "2015", CODEN = "AMHCBQ", ISSN = "0096-3003 (print), 1873-5649 (electronic)", ISSN-L = "0096-3003", bibdate = "Wed Sep 16 06:56:33 MDT 2015", bibsource = "https://www.math.utah.edu/pub/tex/bib/applmathcomput2015.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0096300315009637", acknowledgement = ack-nhfb, fjournal = "Applied Mathematics and Computation", journal-URL = "http://www.sciencedirect.com/science/journal/00963003/", } @Article{Chen:2015:IAEa, author = "Chao-Ping Chen and Richard B. Paris", title = "Inequalities, asymptotic expansions and completely monotonic functions related to the gamma function", journal = j-APPL-MATH-COMP, volume = "250", number = "??", pages = "514--529", day = "1", month = jan, year = "2015", CODEN = "AMHCBQ", ISSN = "0096-3003 (print), 1873-5649 (electronic)", ISSN-L = "0096-3003", bibdate = "Wed Jan 7 16:27:08 MST 2015", bibsource = "https://www.math.utah.edu/pub/tex/bib/applmathcomput2015.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/S009630031401515X", acknowledgement = ack-nhfb, fjournal = "Applied Mathematics and Computation", journal-URL = "http://www.sciencedirect.com/science/journal/00963003/", } @Article{Chen:2015:IAEb, author = "Chao-Ping Chen", title = "Inequalities and asymptotic expansions associated with the {Ramanujan} and {Nemes} formulas for the gamma function", journal = j-APPL-MATH-COMP, volume = "261", number = "??", pages = "337--350", day = "15", month = jun, year = "2015", CODEN = "AMHCBQ", ISSN = "0096-3003 (print), 1873-5649 (electronic)", ISSN-L = "0096-3003", bibdate = "Wed May 13 09:01:41 MDT 2015", bibsource = "https://www.math.utah.edu/pub/tex/bib/applmathcomput2015.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0096300315004610", acknowledgement = ack-nhfb, fjournal = "Applied Mathematics and Computation", journal-URL = "http://www.sciencedirect.com/science/journal/00963003/", } @Article{Chen:2015:ICM, author = "Chao-Ping Chen", title = "Inequalities and completely monotonic functions associated with the ratios of functions resulting from the gamma function", journal = j-APPL-MATH-COMP, volume = "259", number = "??", pages = "790--799", day = "15", month = may, year = "2015", CODEN = "AMHCBQ", ISSN = "0096-3003 (print), 1873-5649 (electronic)", ISSN-L = "0096-3003", bibdate = "Fri Apr 24 18:27:24 MDT 2015", bibsource = "https://www.math.utah.edu/pub/tex/bib/applmathcomput2015.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0096300315003148", acknowledgement = ack-nhfb, fjournal = "Applied Mathematics and Computation", journal-URL = "http://www.sciencedirect.com/science/journal/00963003/", } @InProceedings{deDinechin:2015:HIF, author = "Florent de Dinechin and Matei Istoan", title = "Hardware Implementations of Fixed-Point {Atan2}", crossref = "Muller:2015:ISC", pages = "34--41", year = "2015", DOI = "https://doi.org/10.1109/ARITH.2015.23", bibdate = "Sat Aug 01 08:05:52 2015", bibsource = "https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, keywords = "ARITH-22; atan; atan2; atan2pi; CORDIC; hypot; hypotenuse", remark-1 = "From page 35: ``This work essentially focuses on FPGAs. An unexpected result is that, even on modern FPGAs enhanced with DSP blocks and memories, CORDIC is a clear winner.''", remark-2 = "From page 37, on $w$-bit computation: ``We therefore need $g_\alpha = 1 + \lceil \log_2 ((w - 1) \times 0.5) \rceil$ guard bits to absorb all these errors.'' For the four IEEE 754 binary formats, that is 5, 6, 6, and 7 extra bits, respectively.", remark-3 = "From page 40: ``On the other hand, the latency of CORDIC does not seem quadratic, it seems linear in w. This is explained by the fact that the carry propagation delay is about 30 times faster than the standard routing used between two iterations. It justifies a posteriori the choice of ignoring redundant versions of CORDIC''.", remark-4 = "From page 41: ``To make things even better for CORDIC, it should be noted that it may also compute the module $\sqrt{x^2 + y^2}$ along with the angle [1]. This costs only one additional constant multiplication by $1 / K$.", } @Article{Elezovic:2015:EPF, author = "Neven Elezovi{\'c}", title = "Estimations of psi function and harmonic numbers", journal = j-APPL-MATH-COMP, volume = "258", number = "??", pages = "192--205", day = "1", month = may, year = "2015", CODEN = "AMHCBQ", DOI = "https://doi.org/10.1016/j.amc.2015.02.008", ISSN = "0096-3003 (print), 1873-5649 (electronic)", ISSN-L = "0096-3003", bibdate = "Thu Mar 19 09:03:22 MDT 2015", bibsource = "https://www.math.utah.edu/pub/tex/bib/applmathcomput2015.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0096300315001617", acknowledgement = ack-nhfb, fjournal = "Applied Mathematics and Computation", journal-URL = "http://www.sciencedirect.com/science/journal/00963003/", } @Article{Flocke:2015:AAE, author = "N. Flocke", title = "{Algorithm 954}: an Accurate and Efficient Cubic and Quartic Equation Solver for Physical Applications", journal = j-TOMS, volume = "41", number = "4", pages = "30:1--30:24", month = oct, year = "2015", CODEN = "ACMSCU", DOI = "https://doi.org/10.1145/2699468", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Mon Oct 26 17:31:15 MDT 2015", bibsource = "http://www.acm.org/pubs/contents/journals/toms/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib", abstract = "We report on an accurate and efficient algorithm for obtaining all roots of general real cubic and quartic polynomials. Both the cubic and quartic solvers give highly accurate roots and place no restrictions on the magnitude of the polynomial coefficients. The key to the algorithm is a proper rescaling of both polynomials. This puts upper bounds on the magnitude of the roots and is very useful in stabilizing the root finding process. The cubic solver is based on dividing the cubic polynomial into six classes. By analyzing the root surface for each class, a fast convergent Newton--Raphson starting point for a real root is obtained at a cost no higher than three additions and four multiplications. The quartic solver uses the cubic solver in getting information about stationary points and, when the quartic has real roots, stable Newton--Raphson iterations give one of the extreme real roots. The remaining roots follow by composite deflation to a cubic. If the quartic has only complex roots, the present article shows that a stable Newton--Raphson iteration on a derived symmetric sixth degree polynomial can be formulated for the real parts of the complex roots. The imaginary parts follow by solving suitable quadratics.", acknowledgement = ack-nhfb, articleno = "30", fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", } @Article{Fukushima:2015:PFCa, author = "Toshio Fukushima", title = "Precise and fast computation of inverse {Fermi--Dirac} integral of order $ 1 / 2 $ by minimax rational function approximation", journal = j-APPL-MATH-COMP, volume = "259", number = "??", pages = "698--707", day = "15", month = may, year = "2015", CODEN = "AMHCBQ", ISSN = "0096-3003 (print), 1873-5649 (electronic)", ISSN-L = "0096-3003", bibdate = "Fri Apr 24 18:27:24 MDT 2015", bibsource = "https://www.math.utah.edu/pub/tex/bib/applmathcomput2015.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0096300315003094", acknowledgement = ack-nhfb, fjournal = "Applied Mathematics and Computation", journal-URL = "http://www.sciencedirect.com/science/journal/00963003/", } @Article{Fukushima:2015:PFCb, author = "Toshio Fukushima", title = "Precise and fast computation of {Fermi--Dirac} integral of integer and half integer order by piecewise minimax rational approximation", journal = j-APPL-MATH-COMP, volume = "259", number = "??", pages = "708--729", day = "15", month = may, year = "2015", CODEN = "AMHCBQ", ISSN = "0096-3003 (print), 1873-5649 (electronic)", ISSN-L = "0096-3003", bibdate = "Fri Apr 24 18:27:24 MDT 2015", bibsource = "https://www.math.utah.edu/pub/tex/bib/applmathcomput2015.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0096300315003033", acknowledgement = ack-nhfb, fjournal = "Applied Mathematics and Computation", journal-URL = "http://www.sciencedirect.com/science/journal/00963003/", } @InProceedings{Fukushima:2015:PFCc, author = "Toshio Fukushima", title = "Precise and Fast Computation of Elliptic Integrals and Functions", crossref = "Muller:2015:ISC", pages = "50--57", year = "2015", DOI = "https://doi.org/10.1109/ARITH.2015.15", bibdate = "Sat Aug 01 08:05:52 2015", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, keywords = "ARITH-22", } @Article{Fukushima:2015:PFCd, author = "Toshio Fukushima", title = "Precise and fast computation of generalized {Fermi--Dirac} integral by parameter polynomial approximation", journal = j-APPL-MATH-COMP, volume = "270", number = "??", pages = "802--807", day = "1", month = nov, year = "2015", CODEN = "AMHCBQ", ISSN = "0096-3003 (print), 1873-5649 (electronic)", ISSN-L = "0096-3003", bibdate = "Thu Nov 5 06:24:28 MST 2015", bibsource = "https://www.math.utah.edu/pub/tex/bib/applmathcomput2015.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0096300315011509", acknowledgement = ack-nhfb, fjournal = "Applied Mathematics and Computation", journal-URL = "http://www.sciencedirect.com/science/journal/00963003/", } @Article{Gil:2015:CKF, author = "Amparo Gil and Javier Segura and Nico M. Temme", title = "Computing the {Kummer} function {$ U(a, b, z) $} for small values of the arguments", journal = j-APPL-MATH-COMP, volume = "271", number = "??", pages = "532--539", day = "15", month = nov, year = "2015", CODEN = "AMHCBQ", ISSN = "0096-3003 (print), 1873-5649 (electronic)", ISSN-L = "0096-3003", bibdate = "Fri Nov 13 08:52:33 MST 2015", bibsource = "https://www.math.utah.edu/pub/tex/bib/applmathcomput2015.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0096300315012837", acknowledgement = ack-nhfb, fjournal = "Applied Mathematics and Computation", journal-URL = "http://www.sciencedirect.com/science/journal/00963003/", } @Article{Gil:2015:GPI, author = "Amparo Gil and Javier Segura and Nico M. Temme", title = "{GammaCHI}: a package for the inversion and computation of the gamma and chi-square cumulative distribution functions (central and noncentral)", journal = j-COMP-PHYS-COMM, volume = "191", number = "??", pages = "132--139", month = jun, year = "2015", CODEN = "CPHCBZ", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", bibdate = "Fri Apr 24 18:44:55 MDT 2015", bibsource = "https://www.math.utah.edu/pub/tex/bib/compphyscomm2010.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0010465515000107", acknowledgement = ack-nhfb, fjournal = "Computer Physics Communications", journal-URL = "http://www.sciencedirect.com/science/journal/00104655/", } @Article{Graillat:2015:ECF, author = "Stef Graillat and Christoph Lauter and Ping Tak Peter Tang and Naoya Yamanaka and Shin'ichi Oishi", title = "Efficient Calculations of Faithfully Rounded $ l_2$-Norms of $n$-Vectors", journal = j-TOMS, volume = "41", number = "4", pages = "24:1--24:20", month = oct, year = "2015", CODEN = "ACMSCU", DOI = "https://doi.org/10.1145/2699469", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Mon Oct 26 17:31:15 MDT 2015", bibsource = "http://www.acm.org/pubs/contents/journals/toms/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib", abstract = "In this article, we present an efficient algorithm to compute the faithful rounding of the $ l_2 $-norm of a floating-point vector. This means that the result is accurate to within 1 bit of the underlying floating-point type. This algorithm does not generate overflows or underflows spuriously, but does so when the final result calls for such a numerical exception to be raised. Moreover, the algorithm is well suited for parallel implementation and vectorization. The implementation runs up to 3 times faster than the netlib version on current processors.", acknowledgement = ack-nhfb, articleno = "24", fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", } @Article{Graillat:2015:MRE, author = "Stef Graillat and Vincent Lef{\`e}vre and Jean-Michel Muller", title = "On the maximum relative error when computing integer powers by iterated multiplications in floating-point arithmetic", journal = j-NUMER-ALGORITHMS, volume = "70", number = "3", pages = "653--667", month = nov, year = "2015", CODEN = "NUALEG", DOI = "https://doi.org/10.1007/s11075-015-9967-8", ISSN = "1017-1398 (print), 1572-9265 (electronic)", ISSN-L = "1017-1398", bibdate = "Sun Oct 25 07:27:50 MDT 2015", bibsource = "http://link.springer.com/journal/11075/70/3; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/numeralgorithms.bib", URL = "http://link.springer.com/article/10.1007/s11075-015-9967-8", acknowledgement = ack-nhfb, fjournal = "Numerical Algorithms", journal-URL = "http://link.springer.com/journal/11075", remark = "The authors show via a complex multipage proof that the iterated product for $ x^n $ in p-bit binary arithmetic with default IEEE 754 rounding (to nearest with ties to even) produces a worst-case relative error in the product that is no larger than $ (n - 1) u $, where $ u = 2^{-p} $ is the rounding unit.", } @InProceedings{Johansson:2015:EIE, author = "Fredrik Johansson", title = "Efficient Implementation of Elementary Functions in the Medium-Precision Range", crossref = "Muller:2015:ISC", pages = "83--89", year = "2015", DOI = "https://doi.org/10.1109/ARITH.2015.16", bibdate = "Sat Aug 01 08:05:52 2015", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, keywords = "ARITH-22", } @Article{Johansson:2015:RHP, author = "Fredrik Johansson", title = "Rigorous high-precision computation of the {Hurwitz} zeta function and its derivatives", journal = j-NUMER-ALGORITHMS, volume = "69", number = "2", pages = "253--270", month = jun, year = "2015", CODEN = "NUALEG", DOI = "https://doi.org/10.1007/s11075-014-9893-1", ISSN = "1017-1398 (print), 1572-9265 (electronic)", ISSN-L = "1017-1398", bibdate = "Thu May 28 15:00:06 MDT 2015", bibsource = "http://link.springer.com/journal/11075/69/2; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/numeralgorithms.bib", URL = "http://link.springer.com/article/10.1007/s11075-014-9893-1", acknowledgement = ack-nhfb, fjournal = "Numerical Algorithms", journal-URL = "http://link.springer.com/journal/11075", } @Article{Kuznetsov:2015:CTT, author = "A. Kuznetsov", title = "Computing the truncated theta function via {Mordell} integral", journal = j-MATH-COMPUT, volume = "84", number = "296", pages = "2911--2926", month = "", year = "2015", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 20 16:30:35 MDT 2015", bibsource = "http://www.ams.org/mcom/2015-84-296; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp2010.bib", URL = "http://www.ams.org/journals/mcom/2015-84-296/S0025-5718-2015-02953-6; http://www.ams.org/journals/mcom/2015-84-296/S0025-5718-2015-02953-6/S0025-5718-2015-02953-6.pdf", acknowledgement = ack-nhfb, fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @InProceedings{Lauter:2015:SAF, author = "Christoph Lauter and Marc Mezzarobba", title = "Semi-Automatic Floating-Point Implementation of Special Functions", crossref = "Muller:2015:ISC", pages = "58--65", year = "2015", DOI = "https://doi.org/10.1109/ARITH.2015.12", bibdate = "Sat Aug 01 08:05:52 2015", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, keywords = "ARITH-22", } @Article{Lu:2015:NSC, author = "Dawei Lu and Lixin Song and Yang Yu", title = "New sequences with continued fraction towards {Euler}'s constant", journal = j-APPL-MATH-COMP, volume = "259", number = "??", pages = "12--20", day = "15", month = may, year = "2015", CODEN = "AMHCBQ", ISSN = "0096-3003 (print), 1873-5649 (electronic)", ISSN-L = "0096-3003", bibdate = "Fri Apr 24 18:27:24 MDT 2015", bibsource = "https://www.math.utah.edu/pub/tex/bib/applmathcomput2015.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0096300315001745", acknowledgement = ack-nhfb, fjournal = "Applied Mathematics and Computation", journal-URL = "http://www.sciencedirect.com/science/journal/00963003/", } @Article{Lu:2015:SNA, author = "Dawei Lu and Lixin Song and Congxu Ma", title = "Some new asymptotic approximations of the gamma function based on {Nemes}' formula, {Ramanujan}'s formula and {Burnside}'s formula", journal = j-APPL-MATH-COMP, volume = "253", number = "??", pages = "1--7", day = "15", month = feb, year = "2015", CODEN = "AMHCBQ", DOI = "https://doi.org/10.1016/j.amc.2014.12.077", ISSN = "0096-3003 (print), 1873-5649 (electronic)", ISSN-L = "0096-3003", bibdate = "Wed Feb 18 09:36:23 MST 2015", bibsource = "https://www.math.utah.edu/pub/tex/bib/applmathcomput2015.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0096300314017317", acknowledgement = ack-nhfb, fjournal = "Applied Mathematics and Computation", journal-URL = "http://www.sciencedirect.com/science/journal/00963003/", } @Article{Lu:2015:SNQ, author = "Dawei Lu and Congxu Ma", title = "Some new quicker continued fraction approximations for the gamma function related to the {Nemes}' formula", journal = j-NUMER-ALGORITHMS, volume = "70", number = "4", pages = "825--833", month = dec, year = "2015", CODEN = "NUALEG", DOI = "https://doi.org/10.1007/s11075-015-9975-8", ISSN = "1017-1398 (print), 1572-9265 (electronic)", ISSN-L = "1017-1398", bibdate = "Mon Jan 25 08:55:03 MST 2016", bibsource = "http://link.springer.com/journal/11075/70/4; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/numeralgorithms.bib", URL = "http://link.springer.com/article/10.1007/s11075-015-9975-8", acknowledgement = ack-nhfb, fjournal = "Numerical Algorithms", journal-URL = "http://link.springer.com/journal/11075", } @Article{Mortici:2015:PAG, author = "Cristinel Mortici and Hari M. Srivastava", title = "A product approximation of the gamma function", journal = j-NUMER-ALGORITHMS, volume = "69", number = "3", pages = "595--610", month = jul, year = "2015", CODEN = "NUALEG", DOI = "https://doi.org/10.1007/s11075-014-9915-z", ISSN = "1017-1398 (print), 1572-9265 (electronic)", ISSN-L = "1017-1398", bibdate = "Sat Aug 8 13:58:48 MDT 2015", bibsource = "http://link.springer.com/journal/11075/69/3; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/numeralgorithms.bib", URL = "http://link.springer.com/article/10.1007/s11075-014-9915-z", acknowledgement = ack-nhfb, fjournal = "Numerical Algorithms", journal-URL = "http://link.springer.com/journal/11075", } @Article{Nadarajah:2015:CGH, author = "Saralees Nadarajah", title = "On the Computation of {Gauss} Hypergeometric Functions", journal = j-AMER-STAT, volume = "69", number = "2", pages = "146--148", year = "2015", CODEN = "ASTAAJ", DOI = "https://doi.org/10.1080/00031305.2015.1028595", ISSN = "0003-1305 (print), 1537-2731 (electronic)", ISSN-L = "0003-1305", bibdate = "Sun Aug 9 16:54:48 MDT 2015", bibsource = "http://www.tandfonline.com/toc/utas20/69/2; https://www.math.utah.edu/pub/tex/bib/amstat2010.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "The American Statistician", journal-URL = "http://amstat.tandfonline.com/loi/utas20", onlinedate = "24 Mar 2015", } @Article{Natalini:2015:BPM, author = "Pierpaolo Natalini and Paolo Emilio Ricci", title = "{Bell} polynomials and modified {Bessel} functions of half-integral order", journal = j-APPL-MATH-COMP, volume = "268", number = "??", pages = "270--274", day = "1", month = oct, year = "2015", CODEN = "AMHCBQ", DOI = "https://doi.org/10.1016/j.amc.2015.06.069", ISSN = "0096-3003 (print), 1873-5649 (electronic)", ISSN-L = "0096-3003", bibdate = "Wed Sep 16 06:56:32 MDT 2015", bibsource = "https://www.math.utah.edu/pub/tex/bib/applmathcomput2015.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0096300315008504", acknowledgement = ack-nhfb, fjournal = "Applied Mathematics and Computation", journal-URL = "http://www.sciencedirect.com/science/journal/00963003/", keywords = "Bell polynomials; Bessel functions; Combinatorial analysis; Hankel functions", } @Article{Qi:2015:SIT, author = "Feng Qi and Cristinel Mortici", title = "Some inequalities for the trigamma function in terms of the digamma function", journal = j-APPL-MATH-COMP, volume = "271", number = "??", pages = "502--511", day = "15", month = nov, year = "2015", CODEN = "AMHCBQ", ISSN = "0096-3003 (print), 1873-5649 (electronic)", ISSN-L = "0096-3003", bibdate = "Fri Nov 13 08:52:33 MST 2015", bibsource = "https://www.math.utah.edu/pub/tex/bib/applmathcomput2015.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0096300315012758", acknowledgement = ack-nhfb, fjournal = "Applied Mathematics and Computation", journal-URL = "http://www.sciencedirect.com/science/journal/00963003/", } @Book{Schwalm:2015:EFE, author = "William A. Schwalm", title = "Elliptic Functions and Elliptic Integrals", publisher = "Morgan and Claypool Publishers and IOP Publishing", address = "San Rafael, CA, USA and Bristol, UK", pages = "67", year = "2015", DOI = "https://doi.org/10.1088/978-1-6817-4230-4", ISBN = "1-68174-166-0 (print), 1-68174-230-6 (e-book), 1-68174-102-4 (mobi)", ISBN-13 = "978-1-68174-166-6 (print), 978-1-68174-230-4 (e-book), 978-1-68174-102-4 (mobi)", ISSN = "2054-7307", LCCN = "QA343 .S355 2015", bibdate = "Tue Mar 14 07:38:46 MDT 2023", bibsource = "fsz3950.oclc.org:210/WorldCat; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", series = "IOP concise physics", URL = "http://iopscience.iop.org/book/978-1-6817-4230-4", abstract = "This volume is a basic introduction to certain aspects of elliptic functions and elliptic integrals. Primarily, the elliptic functions stand out as closed solutions to a class of physical and geometrical problems giving rise to nonlinear differential equations. While these nonlinear equations may not be the types of greatest interest currently, the fact that they are solvable exactly in terms of functions about which much is known makes up for this. The elliptic functions of Jacobi, or equivalently the Weierstrass elliptic functions, inhabit the literature on current problems in condensed matter and statistical physics, on solitons and conformal representations, and all sorts of famous problems in classical mechanics. The lectures on elliptic functions have evolved as part of the first semester of a course on theoretical and mathematical methods given to first- and second-year graduate students in physics and chemistry at the University of North Dakota. They are for graduate students or for researchers who want an elementary introduction to the subject that nevertheless leaves them with enough of the details to address real problems. The style is supposed to be informal. The intention is to introduce the subject as a moderate extension of ordinary trigonometry in which the reference circle is replaced by an ellipse. This entre depends upon fewer tools and has seemed less intimidating that other typical introductions to the subject that depend on some knowledge of complex variables. The first three lectures assume only calculus, including the chain rule and elementary knowledge of differential equations. In the later lectures, the complex analytic properties are introduced naturally so that a more complete study becomes possible", acknowledgement = ack-nhfb, tableofcontents = "Preface \\ 1. Elliptic functions as trigonometry \\ 1.1. Definition of Jacobian elliptic functions and trigonometric identities \\ 1.2. Differential equations \\ 1.3. Anharmonic oscillator \\ 2. Differential equations satisfied by the Jacobi elliptic functions: pendula \\ 2.1. Oscillatory motion of a pendulum at large amplitude \\ 2.2. Motion traversing the whole circle \\ 2.3. The sine-Gordon equation: a series of pendula \\ 2.4. Series of pendula: 'super luminal' case \\ 3. General reduction of the DE in terms of Jacobi functions \\ 3.1. Linear fractional transformation and cross ratio \\ 3.2. Reduction of general quartic case \\ 3.3. Finding the coefficients of the linear fractional transformation \\ 4. Elliptic integrals \\ 4.1. Review of complex variables up through residues \\ 4.2. Branching and multi-valued functions in complex planes \\ 4.3. Elliptic integrals and elliptic functions in complex planes \\ 4.4. Example \\ 4.5. Reduction of the most general elliptic integral in terms of the three Legendre forms", } @Article{Sun:2015:LEG, author = "Qiming Sun", title = "{Libcint}: an efficient general integral library for {Gaussian} basis functions", journal = j-J-COMPUT-CHEM, volume = "36", number = "22", pages = "1664--1671", day = "15", month = aug, year = "2015", CODEN = "JCCHDD", DOI = "https://doi.org/10.1002/jcc.23981", ISSN = "0192-8651 (print), 1096-987X (electronic)", ISSN-L = "0192-8651", bibdate = "Sat Jul 25 20:32:36 MDT 2015", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputchem2010.bib", acknowledgement = ack-nhfb, fjournal = "Journal of Computational Chemistry", journal-URL = "http://www.interscience.wiley.com/jpages/0192-8651", onlinedate = "30 Jun 2015", } @InProceedings{T:2015:IHC, author = "Yamunadevi T. and Parmasivam C.", booktitle = "{2015 International Conference on Innovations in Information, Embedded and Communication Systems (ICIIECS)}", title = "Implementation of Hyperbolic {CORDIC}-based {VLSI} architecture for {Kaiser--Bessel} Window techniques in spectral analysis", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "1--5", year = "2015", DOI = "https://doi.org/10.1109/ICIIECS.2015.7193234", bibdate = "Mon Oct 27 10:32:44 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, keywords = "Bessel argument generator; Bessel Function; DH-HEMTs; Generators; Hyperbolic CORDIC; Indexes; Integrated circuits; Kaiser Bessel Window Co-efficient; Silicon", } @Book{Temme:2015:AMI, author = "Nico M. Temme", title = "Asymptotic Methods for Integrals", volume = "6", publisher = pub-WORLD-SCI, address = pub-WORLD-SCI:adr, pages = "xxii + 605", year = "2015", ISBN = "981-4612-15-4 (hardcover), 981-4612-16-2 (e-book)", ISBN-13 = "978-981-4612-15-9 (hardcover), 978-981-4612-16-6 (e-book)", MRclass = "41-02 (33Cxx 33E20 65D30)", MRnumber = "3328507", MRreviewer = "Jos{\'e} Luis L{\'o}pez", bibdate = "Tue Feb 06 11:44:21 2018", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/numana2010.bib", series = "Series in Analysis", abstract = "This book gives introductory chapters on the classical basic and standard methods for asymptotic analysis, such as Watson's lemma, Laplace's method, the saddle point and steepest descent methods, stationary phase and Darboux's method. The methods, explained in great detail, will obtain asymptotic approximations of the well-known special functions of mathematical physics and probability theory. After these introductory chapters, the methods of uniform asymptotic analysis are described in which several parameters have influence on typical phenomena: turning points and transition points, coinciding saddle and singularities. In all these examples, the special functions are indicated that describe the peculiar behavior of the integrals. The text extensively covers the classical methods with an emphasis on how to obtain expansions, and how to use the results for numerical methods, in particular for approximating special functions. In this way, we work with a computational mind: how can we use certain expansions in numerical analysis and in computer programs, how can we compute coefficients, and so on.", acknowledgement = ack-nhfb, shorttableofcontents = "Basic methods for integrals \\ Basic methods: examples for special functions \\ Other methods for integrals \\ Uniform methods for integrals \\ Uniform methods for Laplace-type integrals \\ Uniform examples for special functions \\ A class of cumulative distribution factors", tableofcontents = "Preface / vii \\ Acknowledgments / ix \\ Part 1: Basic Methods for Integrals / 1 \\ 1. Introduction / 3 \\ 1.1 Symbols used in asymptotic estimates / 3 \\ 1.2 Asymptotic expansions / 4 \\ 1.3 A first example: Exponential integral / 5 \\ 1.4 Generalized asymptotic expansions / 7 \\ 1.5 Properties of asymptotic power series / 8 \\ 1.6 Optimal truncation of asymptotic expansions / 10 \\ 2. Expansions of Laplace-type integrals: Watson's lemma / 13 \\ 2.1 Watson's lemma / 13 \\ 2.1.1 Watson's lemma for extended sectors / 14 \\ 2.1.2 More general forms of Watson's lemma / 16 \\ 2.2 Watson's lemma for loop integrals / 16 \\ 2.3 More general forms of Laplace-type integrals / 19 \\ 2.3.1 Transformation to the standard form / 19 \\ 2.4 How to compute the coefficients / 20 \\ 2.4.1 Inversion method for computing the coefficients / 20 \\ 2.4.2 Integrating by parts / 22 \\ 2.4.3 Manipulating power series / 23 \\ 2.4.4 Explicit forms of the coefficients in the expansion / 25 \\ 2.5 Other kernels / 26 \\ 2.6 Exponentially improved asymptotic expansions / 27 \\ 2.7 Singularities of the integrand / 29 \\ 2.7.1 A pole near the endpoint / 29 \\ 2.7.2 More general cases / 32 \\ 3. The method of Laplace / 33 \\ 3.1 A theorem for the general case / 33 \\ 3.2 Constructing the expansion / 35 \\ 3.2.1 Inversion method for computing the coefficients / 36 \\ 3.3 Explicit forms of the coefficients in the expansion / 37 \\ 3.4 The complementary error function / 38 \\ 4. The saddle point method and paths of steepest descent / 41 \\ 4.1 The axis of the valley at the saddle point / 43 \\ 4.2 Examples with simple exponentials / 43 \\ 4.2.1 A first example / 43 \\ 4.2.2 A cosine transform / 44 \\ 4.3 Steepest descent paths not through a saddle point / 44 \\ 4.3.1 A gamma function example / 45 \\ 4.3.2 An integral related to the error function / 46 \\ 4.4 An example with strong oscillations: A 100-digit challenge / 48 \\ 4.5 A Laplace inversion formula for $ \erfc z $ / 49 \\ 4.6 A non-oscillatory integral for $ \erfc z $, $ z \in \mathbb{C} $ / 50 \\ 4.7 The complex Airy function / 50 \\ 4.8 A parabolic cylinder function / 53 \\ 5. The Stokes phenomenon / 57 \\ 5.1 The Airy function / 57 \\ 5.2 The recent interest in the Stokes phenomenon / 58 \\ 5.3 Exponentially small terms in the Airy expansions / 59 \\ 5.4 Expansions in connection with the Stokes phenomenon / 60 \\ 5.4.1 Applications to a Kummer function / 61 \\ Part 2: Basic Methods: Examples for Special Functions / 63 \\ 6. The gamma function / 65 \\ 6.1 $ \Gamma(z) $ by Laplace's method / 66 \\ 6.1.1 Calculating the coefficients / 67 \\ 6.1.2 Details on the transformation / 68 \\ 6.2 $ 1 / \Gamma(z) $ by the saddle point method / 71 \\ 6.2.1 Another integral representation of $ 1 / \Gamma(z) $ / 72 \\ 6.3 The logarithm of the gamma function / 72 \\ 6.3.1 Estimations of the remainder / 73 \\ 6.4 Expansions of $ \Gamma(z + a) $ and $ 1 / \Gamma(z + a) $ / 75 \\ 6.5 The ratio of two gamma functions / 76 \\ 6.5.1 A simple expansion / 77 \\ 6.5.2 A more efficient expansion / 78 \\ 6.6 A binomial coefficient / 80 \\ 6.6.1 A uniform expansion of the binomial coefficient / 83 \\ 6.7 Asymptotic expansion of a product of gamma functions / 85 \\ 6.8 Expansions of ratios of three gamma functions / 88 \\ 7. Incomplete gamma functions / 91 \\ 7.1 Integral representations / 91 \\ 7.2 $ \Gamma(a, x) $ : Asymptotic expansion for $ x \gg a $ / 92 \\ 7.3 $ \gamma(a, x) $ : Asymptotic expansion for $ a > x $ / 93 \\ 7.3.1 Singularity of the integrand / 94 \\ 7.3.2 More details on the transformation $ u = \phi(t) $ / 96 \\ 7.4 $ \Gamma(a, x) $ : Asymptotic expansion for $ x > a $ / 97 \\ 8. The Airy functions / 101 \\ 8.1 Expansions of $ \Ai(z) $, $ \Bi(z) $ / 102 \\ 8.1.1 Transforming the saddle point contour / 102 \\ 8.2 Expansions of $ \Ai(-z) $, $ \Bi(-z) $ / 105 \\ 8.3 Two simple ways to obtain the coefficients / 106 \\ 8.4 A generalized form of the Airy function / 107 \\ 9. Bessel functions: Large argument / 109 \\ 9.1 The modified Bessel function $ K_\nu(z) $ / 109 \\ 9.2 The ordinary Bessel functions / 110 \\ 9.3 The modified Bessel function $ I_\nu(z) $ / 111 9.3.1 A compound expansion of $ I_\nu(z) $ / 111 9.4 Saddle point method for $ K_\nu(z) $, $ z \in \mathbb{C} $ / 113 \\ 9.4.1 Integral representations from saddle point analysis / 115 \\ 9.4.2 Saddle point method for $ J_\nu(x) $, $ x < \nu $ / 116 \\ 9.5 Debye-type expansions of the modified Bessel functions / 117 \\ 9.6 Modified Bessel functions of purely imaginary order / 119 \\ 9.6.1 The monotonic case: $ x > \nu > 0 $ / 120 \\ 9.6.2 The oscillatory case: $ \nu > x > 0 $ / 123 \\ 9.7 A $ J $ -Bessel integral / 126 \\ 10. Kummer functions / 129 \\ 10.1 General properties / 129 \\ 10.2 Asymptotic expansions for large $ z $ / 131 \\ 10.3 Expansions for large $ a $ / 132 \\ 10.3.1 Tricomi's function $ E_\nu(z) $ / 132 \\ 10.3.2 Expansion of $ U(a, c, z) $, $ a \to +\infty $ / 133 \\ 10.3.3 Expansion of $ _1F_1(a; c; z) $, $ a \to +\infty $ / 135 \\ 10.3.4 Expansion of $ _1F_1(a; c; z) $, $ a \to -\infty $ / 137 \\ 10.3.5 Expansion of $ U(a, c, z) $, $ a \to -\infty $ / 138 \\ 10.3.6 Slater's results for large $ a $ / 140 \\ 10.4 Expansions for large $ c $ / 142 \\ 10.4.1 Expansion of $ _1F_1(a; c; z) $, $ c \to +\infty $ / 142 \\ 10.4.2 Expansion of $ U(a, c, z) $, $ c \to +\infty $, $ z < c $ / 143 \\ 10.4.3 Expansion of $ U(a, c, z) $, $ c \to +\infty $, $ z > e $ / 144 \\ 10.4.4 Expansion of $ U(a, c, z) $, $ c \to -\infty $ / 145 \\ 10.4.5 Expansion of $ _1F_1(a; c; z) $, $ c \to -\infty $ / 147 \\ 10.5 Uniform expansions of the Kummer functions / 147 \\ 11. Parabolic cylinder functions: Large argument / 149 \\ 11.1 A few properties of the parabolic cylinder functions / 149 \\ 11.2 The function $ U(a, z) $ / 150 \\ 11.3 The function $ U(a, -z) $ / 152 \\ 11.4 The function $ V(a, z) $ / 153 \\ 11.5 Expansions of the derivatives / 154 \\ 12. The Gauss hypergeometric function / 155 \\ 12.1 Large values of $ c $ / 156 \\ 12.1.1 Large positive $ c $; $ |z| < z_0 $ / 156 \\ 12.1.2 Large negative $ c $; $ |z| < z_0 $ / 157 \\ 12.1.3 Large positive $ c $; $ |z| > z_0 $ / 158 \\ 12.1.4 Large negative $ c $; $ |z| > z_0 $ / 158 \\ 12.2 Large values of $ b $ / 158 \\ 12.2.1 Large negative $ b $; $ |z| > z_0 $ / 159 \\ 12.2.2 Large $ b $, $ |z| < z_0 $ / 159 \\ 12.3 Other large parameter cases / 160 \\ 12.3.1 Jacobi polynomials of large degree / 161 \\ 12.3.2 An example of the case $ _2F_1(a, b - \lambda; c + \lambda; z) $ / 163 \\ 13. Examples of $ _3F_2 $ -polynomials / 167 \\ 13.1 A $ _3F_2 $ associated with the Catalan--Larcombe--French sequence / 167 \\ 13.1.1 Transformations / 169 \\ 13.1.2 Asymptotic analysis / 170 \\ 13.1.3 Asymptotic expansion / 172 \\ 13.1.4 An alternative method / 173 \\ 13.2 An integral of Laguerre polynomials / 175 \\ 13.2.1 A first approach / 176 \\ 13.2.2 A generating function approach / 178 \\ Part 3: Other Methods for Integrals / 181 \\ 14. The method of stationary phase / 183 \\ 14.1 Critical points / 183 \\ 14.2 Integrating by parts: No stationary points / 184 \\ 14.3 Three critical points: A formal approach / 185 \\ 14.4 On the use of neutralizes / 186 \\ 14.5 How to avoid neutralizes? / 188 \\ 14.5.1 A few details about the Fresnel integral / 190 \\ 14.6 Algebraic singularities at both endpoints: Erdelyi's example / 191 \\ 14.6.1 Application: A conical function / 192 \\ 14.6.2 Avoiding neutralizes in Erdelyi's example / 193 \\ 14.7 Transformation to standard form / 194 \\ 14.8 General order stationary points / 196 \\ 14.8.1 Integrating by parts / 196 \\ 14.9 The method fails: Examples / 197 \\ 14.9.1 The Airy function / 198 \\ 14.9.2 A more complicated example / 198 \\ 15. Coefficients of a power series. Darboux's method / 203 \\ 15.1 A first example: A binomial coefficient / 204 \\ 15.2 Legendre polynomials of large degree / 205 \\ 15.2.1 A paradox in asymptotics / 207 \\ 15.3 Gegenbauer polynomials of large degree / 208 \\ 15.4 Jacobi polynomials of large degree / 209 \\ 15.5 Laguerre polynomials of large degree / 209 \\ 15.6 Generalized Bernoulli polynomials $ B_n^{(\mu)}(z) $ / 210 \\ 15.6.1 Asymptotic expansions for large degree / 211 \\ 15.6.2 An alternative expansion / 213 \\ 15.7 Generalized Euler polynomials $ E_n^{(\mu)}(z) $ / 215 \\ 15.7.1 Asymptotic expansions for large degree / 215 \\ 15.7.2 An alternative expansion / 216 \\ 15.8 Coefficients of expansions of the $ _1F_1 $ -function / 218 \\ 15.8.1 Coefficients of Tricomi's expansion / 218 \\ 15.8.2 Coefficients of Buchholz's expansion / 221 \\ 16. Mellin--Barnes integrals and Mellin convolution integrals / 225 \\ 16.1 Mellin--Barnes integrals / 226 \\ 16.2 Mellin convolution integrals / 228 \\ 16.3 Error bounds / 230 \\ 17. Alternative expansions of Laplace-type integrals / 231 \\ 17.1 Hadamard-type expansions / 231 \\ 17.2 An expansion in terms of Kummer functions / 233 \\ 17.3 An expansion in terms of factorial series / 234 \\ 17.4 The Franklin--Friedman expansion / 237 \\ 18. Two-point Taylor expansions / 241 \\ 18.1 The expansions / 242 \\ 18.2 An alternative form of the expansion / 243 \\ 18.3 Explicit forms of the coefficients / 244 \\ 18.4 Manipulations with two-point Taylor expansions / 245 \\ 19. Hermite polynomials as limits of other classical orthogonal polynomials / 249 \\ 19.1 Limits between orthogonal polynomials / 249 \\ 19.2 The Askey scheme of orthogonal polynomials / 251 \\ 19.3 Asymptotic representations / 251 \\ 19.4 Gegenbauer polynomials / 253 \\ 19.5 Laguerre polynomials / 254 \\ 19.6 Generalized Bessel polynomials / 255 \\ 19.7 Meixner--Pollaczek polynomials into Laguerre polynomials / 257 \\ Part 4: Uniform Methods for Integrals / 259 \\ 20. An overview of standard forms / 261 \\ 20.1 Comments on the table / 263 \\ 21. A saddle point near a pole / 267 \\ 21.1 A saddle point near a pole: Van der Waerden's method / 267 \\ 21.2 An alternative expansion / 269 \\ 21.3 An example from De Bruijn / 270 \\ 21.4 A pole near a double saddle point / 271 \\ 21.5 A singular perturbation problem and $ K $ -Bessel integrals / 272 \\ 21.5.1 A Bessel $ K_0 $ integral / 272 \\ 21.5.2 A similar Bessel $ K_1 $ integral / 274 \\ 21.5.3 A singular perturbation problem / 275 \\ 21.6 A double integral with poles near saddle points / 277 \\ 21.6.1 Application to a singular perturbation problem / 278 \\ 21.7 The Fermi--Dirac integral / 281 \\ 22. Saddle point near algebraic singularity / 285 \\ 22.1 A saddle point near an endpoint of the interval / 285 \\ 22.2 The Bleistein expansion / 286 \\ 22.3 Extending the role of the parameter /3 / 289 \\ 22.4 Contour integrals / 291 \\ 22.5 Kummer functions in terms of parabolic cylinder functions / 292 \\ 22.5.1 Uniform expansion of $ U(a, c, z) $, $ c \to +\infty $ / 293 \\ 22.5.2 Uniform expansion of $ _1F_1(a; c; z) $, $ c \to +\infty $ / 296 \\ 23. Two coalescing saddle points: Airy-type expansions / 299 \\ 23.1 The standard form / 299 \\ 23.2 An integration by parts method / 300 \\ 23.3 How to compute the coefficients / 302 \\ 23.4 An Airy-type expansion of the Hermite polynomial / 305 \\ 23.4.1 The cubic transformation / 306 \\ 23.4.2 Details on the coefficients / 308 \\ 23.5 An Airy-type expansion of the Bessel function $ J_\nu(z) $ / 309 \\ 23.6 A semi-infinite interval: Incomplete Scorer function / 313 \\ 23.6.1 A singular perturbation problem inside a circle / 315 \\ 24. Hermite-type expansions of integrals / 319 \\ 24.1 An expansion in terms of Hermite polynomials / 320 \\ 24.1.1 Cauchy-type integrals for the coefficients / 321 \\ 24.2 Gegenbauer polynomials / 323 \\ 24.2.1 Preliminary steps / 324 \\ 24.2.2 A first approximation / 325 \\ 24.2.3 Transformation to the standard form / 326 \\ 24.2.4 Special cases of the expansion / 331 \\ 24.2.5 Approximating the zeros / 332 \\ 24.2.6 The relativistic Hermite polynomials / 333 \\ 24.3 Tricomi--Carlitz polynomials / 333 \\ 24.3.1 Contour integral and saddle points / 335 \\ 24.3.2 A first approximation / 337 \\ 24.3.3 Transformation to the standard form / 337 \\ 24.3.4 Approximating the zeros / 339 \\ 24.4 More examples / 340 \\ Part 5: Uniform Methods for Laplace-Type Integrals / 341 \\ 25. The vanishing saddle point / 343 \\ 25.1 Expanding at the saddle point / 344 \\ 25.2 An integration by parts method / 346 \\ 25.2.1 Representing coefficients as a Cauchy-type integral / 347 \\ 25.3 Expansions for loop integrals / 348 \\ 25.4 Rummer functions / 350 \\ 25.5 Generalized zeta function / 350 \\ 25.6 Transforming to the standard form / 351 \\ 25.6.1 The ratio of two gamma functions / 352 \\ 25.6.2 Parabolic cylinder functions / 354 \\ 26. A moving endpoint: Incomplete Laplace integrals / 355 \\ 26.1 The standard form / 355 \\ 26.2 Constructing the expansion / 356 \\ 26.2.1 The complementary function / 357 \\ 26.3 Application to the incomplete beta function / 358 \\ 26.3.1 Expansions of the coefficients / 361 \\ 26.4 A corresponding loop integral / 362 \\ 26.4.1 Application to the incomplete beta function / 363 \\ 27. An essential singularity: Bessel-type expansions / 365 \\ 27.1 Expansions in terms of modified Bessel functions / 365 \\ 27.2 A corresponding loop integral / 368 \\ 27.3 Expansion at the internal saddle point / 368 \\ 27.4 Application to Kummer functions / 369 \\ 27.4.1 Expansion of $ U(a, c, z) $, $ a \to +\infty $, $ z > 0 $ / 369 \\ 27.4.2 Auxiliary expansions and further details / 372 \\ 27.4.3 Expansion of $ _1F_1(a: c; z) $, $ a \to +\infty $, $ z > 0 $ / 374 \\ 27.4.4 Expansion of $ _1F_1(a; c: z) $, $ a \to -\infty $, $ 0 < z < -4a $ / 375 \\ 27.4.5 Expansion of $ U(a, c, z) $, $ a \to -\infty $, $ 0 < z < -4a $ / 377 \\ 27.5 A second uniformity parameter / 378 \\ 27.5.1 Expansion of $ U(a, c, z) $, $ a \to \infty $, $ z > 0 $, $ c < 1 $ / 380 \\ 27.5.2 Expansion of $ _1F_1(a; c; z), $ a \to \infty $, $ z > 0 $, $ c > 1 $ / 381 \\ 28. Expansions in terms of Kummer functions / 383 \\ 28.1 Approximation in terms of the Kummer J7-function / 383 \\ 28.1.1 Constructing the expansions / 384 \\ 28.1.2 Expansion for the loop integral / 387 \\ 28.2 The $ _2F_1 $ function, large $ c $, in terms of $ U $ / 387 \\ 28.2.1 Legendre polynomials: Uniform expansions / 388 \\ 28.3 The $ _2F_1 $ -function, large $ b $ : in terms of $ _1F_1 $ / 389 \\ 28.3.1 Using a real integral / 390 \\ 28.3.2 Using a loop integral / 394 \\ 28.4 Jacobi polynomials of large degree: Laguerre-type expansion / 394 \\ 28.4.1 Laguerre-type expansion for large values of /3 / 398 \\ 28.5 Expansion of a Dirichlet-type integral / 401 \\ Part 6: Uniform Examples for Special Functions / 403 \\ 29. Legendre functions / 405 \\ 29.1 Expansions of $ P_\nu^\mu(z) $, $ Q_\nu^\mu(z) $; $ \nu \to \infty $, $ z \geq 1 $ / 406 \\ 29.1.1 Expansions for $ z > z_0 > 1 $ / 400 \\ 29.1.2 Expansion in terms of modified Bessel functions / 407 \\ 29.1.3 Expansions of $ P_\nu^\mu(z) $ and $ Q_\nu^\mu(z) $ in terms of Bessel functions / 411 \\ 29.2 Expansions of $ P_\nu^\mu(z) $, $ Q_\nu^\mu(z) $; $ p \to \infty $, $ z > 1 $ / 412 \\ 29.2.1 Expansions for bounded $ z $ / 412 \\ 29.2.2 Expansions in terms of modified Bessel functions / 412 \\ 29.2.3 Expansions of $ P_\nu^\mu(z) $ and $ Q_\nu^\mu(z) $ / 413 \\ 29.3 Integrals with nearly coincident branch points / 414 \\ 29.3.1 Ursell's expansions of Legendre functions / 415 \\ 29.3.2 Coefficients of the expansion / 416 \\ 29.3.3 An alternative expansion of $ P_n^m(\cosh z) $ / 417 \\ 29.3.4 A related integral with nearly coincident branch points / 418 \\ 29.4 Toroidal harmonics and conical functions / 418 \\ 30. Parabolic cylinder functions: Large parameter / 419 \\ 30.1 Notation for uniform asymptotic expansions / 419 \\ 30.2 The case $ a < 0 $ / 421 \\ 30.2.1 The case $ z > 2\sqrt{-a} $ : $ -a + z \to \infty $ / 421 \\ 30.2.2 The case $ z < -2\sqrt{-a} $ : $ -a - z \to \infty $ / 422 \\ 30.2.3 The case -2\sqrt{-a} < z < 2\sqrt{-a} / 423 \\ 30.3 The case $ a > 0 $ / 424 \\ 30.3.1 The case $ z > 0 $, $ a + z \to \infty $ / 425 \\ 30.3.2 The case $ z < 0 $, $ a - z \to \infty $ / 425 \\ 30.4 Expansions from integral representations / 426 \\ 30.4.1 The case $ a > 0 $, $ z > 0 $; $ a + z \to \infty $ / 426 \\ 30.4.2 The case $ a > 0 $, $ z < 0 $; $ a - z \to \infty $ / 428 \\ 30.4.3 The case $ a < 0 $, $ |z| > 2\sqrt{-a} $; $ -a + |z| \to \infty $ / 429 \\ 30.5 Airy-type expansions / 430 \\ 31. Coulomb wave functions / 433 \\ 31.1 Contour integrals for Coulomb functions / 434 \\ 31.2 Expansions for $ \rho \to \infty $ and bounded $ \eta $ / / 435 \\ 31.3 Expansions for $ \eta \to \infty $ and bounded $ \rho $ / 437 \\ 31.4 Expansions for $ \eta \to -\infty $ and bounded $ \rho $ / 439 \\ 31.5 Expansions for $ \eta \to -\infty and $ \rho \geq \rho_0 > 0 $ / 440 \\ 31.6 Expansions for $ \eta \to -\infty $ and $ \rho \geq 0 $ / 442 \\ 31.7 Expansions for $ \eta $, $ \rho \to \infty $; Airy-type expansions / 444 \\ 32. Laguerre polynomials: Uniform expansions / 449 \\ 32.1 An expansion for bounded $ z $ and $ a $ / 449 \\ 32.2 An expansion for bounded $ z $; $ a $ depends on $ n $ / 451 \\ 32.3 Expansions for bounded $ a $; $ z $ depends on $ n $ / 454 \\ 32.3.1 An expansion in terms of Airy functions / 455 \\ 32.3.2 An expansion in terms of Bessel functions / 456 \\ 32.4 An expansion in terms of Hermite polynomials; large $ a $ / 458 \\ 32.4.1 A first approximation / 459 \\ 32.4.2 Transformation to the standard form / 460 \\ 32.4.3 Approximating the zeros / 462 \\ 33. Generalized Bessel polynomials / 465 \\ 33.1 Relations to Bessel and Kummer functions / 466 \\ 33.2 An expansion in terms of Laguerre polynomials / 467 \\ 33.3 Expansions in terms of elementary functions / 470 \\ 33.3.1 The case $ |\ph z| < \pi/2 $ / 470 \\ 33.3.2 The case $ |\ph(-z)| < \pi/2 $ / 471 \\ 33.3.3 Integral representations / 472 \\ 33.3.4 Construction of the expansions / 472 \\ 33.4 Expansions in terms of modified Bessel functions / 476 \\ 33.4.1 Construction of the expansion / 476 \\ 34. Stirling numbers / 479 \\ 34.1 Definitions and integral representations / 479 \\ 34.2 Stirling number of the second kind / 481 \\ 34.2.1 Higher-order approximations / 483 \\ 34.2.2 About the positive saddle point / 486 \\ 34.2.3 About the quantity $ A $ / 487 \\ 34.3 Stirling numbers of the first kind / 488 \\ 35. Asymptotics of the integral $ \int_0^1 \cos(b x + a / x) \, dx $ / 491 \\ 35.1 The case $ b < a $ / 491 \\ 35.2 The case $ a = b $ / 493 \\ 35.3 The case $ b > a $ / 494 \\ 35.3.1 The contribution from $ \mathcal{P}_1 $ / 495 \\ 35.3.2 The contribution from $ \mathcal{P}_2 $ / 496 \\ 35.4 A Fresnel-type expansion / 497 \\ Part 7: A Class of Cumulative Distribution Functions / 499 \\ 36. Expansions of a class of cumulative distribution functions / 501 \\ 36.1 Cumulative distribution functions: A standard form / 501 \\ 36.2 An incomplete normal distribution function / 505 \\ 36.3 The Sievert integral / 506 \\ 36.4 The Pearson type IV distribution / 507 \\ 36.5 The Von Mises distribution / 509 \\ 36.5.1 An expansion near the lower endpoint of integration / 511 \\ 37. Incomplete gamma functions: Uniform expansions / 513 \\ 37.1 Using the standard integral representations / 513 \\ 37.2 Representations by contour integrals / 514 \\ 37.2.1 Constructing the expansions / 516 \\ 37.2.2 Details on the coefficients / 518 \\ 37.2.3 Relations to the coefficients of earlier expansions / 520 \\ 37.3 Incomplete gamma functions, negative parameters / 520 \\ 37.3.1 Expansions near the transition point / 522 \\ 37.3.2 A real expansion of 7*(-a, -z) / 524 \\ 38. Incomplete beta function / 525 \\ 38.1 A power series expansion for large p / 526 \\ 38.2 A uniform expansion for large p / 526 \\ 38.3 The nearly symmetric case / 527 \\ 38.4 The general error function case / 529 \\ 39. Non-central chi-square, Marcum functions / 531 \\ 39.1 Properties of the Marcum functions / 532 \\ 39.2 More integral representations / 533 \\ 39.3 Asymptotic expansion; $ \mu $ fixed, $ \xi $ large / 535 \\ 39.4 Asymptotic expansion; $ \xi + \mu $ large / 537 \\ 39.5 An expansion in terms of the incomplete gamma function / 540 \\ 39.6 Comparison of the expansions numerically / 543 \\ 40. A weighted sum of exponentials / 545 \\ 40.1 An integral representation / 546 \\ 40.2 Saddle point analysis / 547 \\ 40.3 Details on the coefficients / 548 \\ 40.4 Auxiliary expansions / 550 \\ 40.5 Numerical verification / 551 \\ 41. A generalized incomplete gamma function / 553 \\ 41.1 An expansion in terms of incomplete gamma functions / 554 \\ 41.2 An expansion in terms of Laguerre polynomials / 554 \\ 41.3 An expansion in terms of Kummer functions / 555 \\ 41.4 An expansion in terms of the error function / 555 \\ 42. Asymptotic inversion of cumulative distribution functions / 559 \\ 42.1 The asymptotic inversion method / 559 \\ 42.2 Asymptotic inversion of the gamma distribution / 561 \\ 42.2.1 Numerical verification / 563 \\ 42.2.2 Other asymptotic inversion methods / 564 \\ 42.2.3 Asymptotics of the zeros of $ \Gamma(a, z) $ / 565 \\ 42.3 Asymptotic inversion of the incomplete beta function / 567 \\ 42.3.1 Inverting by using the error function / 568 \\ 42.3.2 Inverting by using the incomplete gamma function / 569 \\ 42.3.3 Numerical verification / 572 \\ 42.4 The hyperbolic cumulative distribution / 573 \\ 42.4.1 Numerical verification / 574 \\ 42.5 The Marcum functions / 575 \\ 42.5.1 Asymptotic inversion / 576 \\ 42.5.2 Asymptotic inversion with respect to $ x $ / 576 \\ 42.5.3 Asymptotic inversion with respect to $ y $ / 579 \\ Bibliography / 583 \\ Index / 597", } @Article{Weiss:2015:ROS, author = "Alexander K. H. Weiss and Christian Ochsenfeld", title = "A rigorous and optimized strategy for the evaluation of the {Boys} function kernel in molecular electronic structure theory", journal = j-J-COMPUT-CHEM, volume = "36", number = "18", pages = "1390--1398", day = "5", month = jul, year = "2015", CODEN = "JCCHDD", DOI = "https://doi.org/10.1002/jcc.23935", ISSN = "0192-8651 (print), 1096-987X (electronic)", ISSN-L = "0192-8651", bibdate = "Sat Jul 25 20:32:35 MDT 2015", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputchem2010.bib", acknowledgement = ack-nhfb, fjournal = "Journal of Computational Chemistry", journal-URL = "http://www.interscience.wiley.com/jpages/0192-8651", onlinedate = "13 May 2015", } @Article{Xu:2015:CFC, author = "Ai-Min Xu and Zhong-Di Cen", title = "Closed formulas for computing higher-order derivatives of functions involving exponential functions", journal = j-APPL-MATH-COMP, volume = "270", number = "??", pages = "136--141", day = "1", month = nov, year = "2015", CODEN = "AMHCBQ", DOI = "https://doi.org/10.1016/j.amc.2015.08.051", ISSN = "0096-3003 (print), 1873-5649 (electronic)", ISSN-L = "0096-3003", bibdate = "Thu Nov 5 06:24:28 MST 2015", bibsource = "https://www.math.utah.edu/pub/tex/bib/applmathcomput2015.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0096300315011066", acknowledgement = ack-nhfb, fjournal = "Applied Mathematics and Computation", journal-URL = "http://www.sciencedirect.com/science/journal/00963003/", keywords = "closed formula; derivatives of exponential functions; derivatives of trigonometric functions; higher-order derivative; hyperbolic function; tangent number; trigonometric function", remark = "The authors derive closed-form $n$-term sums for the $n$-th order derivatives of exponential and trigonometric functions. The sums involve factorials, powers, and Stirling numbers of the first and second kinds. At the end of their paper, they derive a new computationally-stable formula for the tangent numbers, $ T_{2 n + 1} = \sum_{k = 1}^n \binom {2 n}{2 k - 1} T_{2 k - 1} T_{2(n - k) + 1}$, a sum that involves only positive terms. There is a stable recurrence relation discussed in the MathCW book that is likely faster, because it requires only 2 multiplies and 1 add in each term of the recurrence.", } @Article{Yang:2015:AFG, author = "Zhen-Hang Yang and Yu-Ming Chu", title = "Asymptotic formulas for gamma function with applications", journal = j-APPL-MATH-COMP, volume = "270", number = "??", pages = "665--680", day = "1", month = nov, year = "2015", CODEN = "AMHCBQ", DOI = "https://doi.org/10.1016/j.amc.2015.08.087", ISSN = "0096-3003 (print), 1873-5649 (electronic)", ISSN-L = "0096-3003", bibdate = "Thu Nov 5 06:24:28 MST 2015", bibsource = "https://www.math.utah.edu/pub/tex/bib/applmathcomput2015.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0096300315011431", acknowledgement = ack-nhfb, fjournal = "Applied Mathematics and Computation", journal-URL = "http://www.sciencedirect.com/science/journal/00963003/", } @Article{Yang:2015:SBP, author = "Zhen-Hang Yang and Yu-Ming Chu and Xiao-Hui Zhang", title = "Sharp bounds for psi function", journal = j-APPL-MATH-COMP, volume = "268", number = "??", pages = "1055--1063", day = "1", month = oct, year = "2015", CODEN = "AMHCBQ", DOI = "https://doi.org/10.1016/j.amc.2015.07.012", ISSN = "0096-3003 (print), 1873-5649 (electronic)", ISSN-L = "0096-3003", bibdate = "Wed Sep 16 06:56:32 MDT 2015", bibsource = "https://www.math.utah.edu/pub/tex/bib/applmathcomput2015.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0096300315009248", acknowledgement = ack-nhfb, fjournal = "Applied Mathematics and Computation", journal-URL = "http://www.sciencedirect.com/science/journal/00963003/", keywords = "Gamma function; Monotonicity; Psi function", } @Article{Zhang:2015:EAR, author = "Jianfeng Zhang and Paul Chow and Hengzhu Liu", title = "An Enhanced Adaptive Recoding Rotation {CORDIC}", journal = j-TRETS, volume = "9", number = "1", pages = "4:1--4:??", month = nov, year = "2015", CODEN = "????", DOI = "https://doi.org/10.1145/2812813", ISSN = "1936-7406 (print), 1936-7414 (electronic)", ISSN-L = "1936-7406", bibdate = "Tue Dec 22 16:19:56 MST 2015", bibsource = "http://portal.acm.org/; https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/trets.bib", abstract = "The Conventional Coordinate Rotation Digital Computer (CORDIC) algorithm has been widely used in many applications, particularly in Direct Digital Frequency Synthesizers (DDS) and Fast Fourier Transforms (FFT). However, CORDIC is constrained by the excessive number of iterations, angle data path, and scaling factor compensation. In this article, an enhanced adaptive recoding CORDIC (EARC) is proposed. It uses the enhanced adaptive recoding method to reduce the required iterations and adopts the trigonometric transformation scheme to scale up the rotation angles. Computing sine and cosine is used first to compare the core functionality of EARC with basic CORDIC; then a 16-bit DDS and a 1,024-point FFT based on EARC are evaluated to demonstrate the benefits of EARC in larger applications. All the proposed architectures are validated on a Virtex 5 FPGA development platform. Compared with a commercial implementation of CORDIC, EARC requires 33.3\% less hardware resources, provides a twofold speedup, dissipates 70.4\% less power, and improves accuracy in terms of the Bit Error Position (BEP). Compared to the state-of-the-art Hybrid CORDIC, EARC reduces latency by 11.1\% and consumes 17\% less power. Compared with a commercial implementation of DDS, the dissipated power of the proposed DDS is reduced by 27.2\%. The proposed DDS improves Spurious-Free Dynamic Range (SFDR) by nearly 7 dBc and dissipates 21.8\% less power when compared with a recently published DDS circuit. The FFT based on EARC dissipates a factor of 2.05 less power than the commercial FFT even when choosing the 100\% toggle rate for the FFT based on EARC and the 12.5\% toggle rate for the commercial FFT. Compared with a recently published FFT, the FFT based on EARC improves Signal-to-Noise Ratio (SNR) by 8.9 dB and consumes 7.78\% less power.", acknowledgement = ack-nhfb, articleno = "4", fjournal = "ACM Transactions on Reconfigurable Technology and Systems (TRETS)", journal-URL = "http://portal.acm.org/toc.cfm?id=J1151", } @Article{Abel:2016:HOA, author = "Ulrich Abel", title = "High order algorithms for calculating roots", journal = j-MATH-GAZ, volume = "100", number = "549", pages = "420--428", month = nov, year = "2016", CODEN = "MAGAAS", DOI = "https://doi.org/10.1017/mag.2016.106", ISSN = "0025-5572 (print), 2056-6328 (electronic)", ISSN-L = "0025-5572", bibdate = "Thu Nov 17 10:32:54 MST 2016", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathgaz2010.bib", URL = "https://www.cambridge.org/core/product/2FACD442DB78364B04DA2E64BA06F269", acknowledgement = ack-nhfb, ajournal = "Math. Gaz.", fjournal = "The Mathematical Gazette", journal-URL = "http://journals.cambridge.org/action/displayIssue?jid=MAG; http://www.m-a.org.uk/jsp/index.jsp?lnk=620", onlinedate = "17 October 2016", } @Article{Aprahamian:2016:MIT, author = "Mary Aprahamian and Nicholas J. Higham", title = "Matrix Inverse Trigonometric and Inverse Hyperbolic Functions: Theory and Algorithms", journal = j-SIAM-J-MAT-ANA-APPL, volume = "37", number = "4", pages = "1453--1477", month = "????", year = "2016", CODEN = "SJMAEL", DOI = "https://doi.org/10.1137/16M1057577", ISSN = "0895-4798 (print), 1095-7162 (electronic)", ISSN-L = "0895-4798", bibdate = "Fri Aug 25 09:01:43 MDT 2017", bibsource = "http://epubs.siam.org/sam-bin/dbq/toc/SIMAX/37/4; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/siamjmatanaappl.bib", acknowledgement = ack-nhfb, fjournal = "SIAM Journal on Matrix Analysis and Applications", journal-URL = "http://epubs.siam.org/simax", onlinedate = "January 2016", } @Article{Bailey:2016:CSC, author = "D. H. Bailey and J. M. Borwein", title = "Computation and structure of character polylogarithms with applications to character {Mordell--Tornheim--Witten} sums", journal = j-MATH-COMPUT, volume = "85", number = "297", pages = "295--324", month = "", year = "2016", CODEN = "MCMPAF", DOI = "https://doi.org/10.1090/mcom/2974", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Mon Feb 8 17:02:07 MST 2016", bibsource = "http://www.ams.org/mcom/2016-85-297; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp2010.bib", URL = "http://www.ams.org/journals/mcom/2016-85-297/S0025-5718-2015-02974-3; http://www.ams.org/journals/mcom/2016-85-297/S0025-5718-2015-02974-3/S0025-5718-2015-02974-3.pdf; http://www.ams.org/mathscinet/search/author.html?authorName=Borwein%2C%20J.%20M; http://www.ams.org/mathscinet/search/author.html?mrauthid=29355", acknowledgement = ack-nhfb, fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Bao:2016:SAO, author = "Vo Nguyen Quoc Bao and Luu Pham Tuyen and Huynh Huu Tue", title = "A Survey on Approximations of One-Dimensional {Gaussian} {$Q$}-Function", journal = "{REV} Journal on Electronics and Communications", volume = "5", number = "1--2", month = feb, year = "2016", DOI = "https://doi.org/10.21553/rev-jec.92", bibdate = "Sat Dec 16 15:18:22 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.rev-jec.org/index.php/rev-jec/article/view/92", acknowledgement = ack-nhfb, journal-URL = "http://www.rev-jec.org/index.php/rev-jec/", } @Article{Bytev:2016:HHF, author = "Vladimir V. Bytev and Bernd A. Kniehl", title = "{HYPERDIRE} --- {HYPERgeometric functions DIfferential REduction}: {Mathematica}-based packages for the differential reduction of generalized hypergeometric functions: {Lauricella} function {$ F_c $} of three variables", journal = j-COMP-PHYS-COMM, volume = "206", number = "??", pages = "78--83", month = sep, year = "2016", CODEN = "CPHCBZ", DOI = "https://doi.org/10.1016/j.cpc.2016.04.016", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", bibdate = "Fri Jun 10 18:27:25 MDT 2016", bibsource = "https://www.math.utah.edu/pub/tex/bib/compphyscomm2010.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathematica.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0010465516301059", acknowledgement = ack-nhfb, fjournal = "Computer Physics Communications", journal-URL = "http://www.sciencedirect.com/science/journal/00104655/", } @Article{Chen:2016:AEG, author = "Chao-Ping Chen", title = "On the asymptotic expansions of the gamma function related to the {Nemes}, {Gosper} and {Burnside} formulas", journal = j-APPL-MATH-COMP, volume = "276", number = "??", pages = "417--431", day = "5", month = mar, year = "2016", CODEN = "AMHCBQ", ISSN = "0096-3003 (print), 1873-5649 (electronic)", ISSN-L = "0096-3003", bibdate = "Tue Jan 26 17:22:21 MST 2016", bibsource = "https://www.math.utah.edu/pub/tex/bib/applmathcomput2015.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0096300315016057", acknowledgement = ack-nhfb, fjournal = "Applied Mathematics and Computation", journal-URL = "http://www.sciencedirect.com/science/journal/00963003/", } @Article{Chen:2016:IAEa, author = "Chao-Ping Chen and Long Lin", title = "Inequalities and asymptotic expansions for the gamma function related to {Mortici}'s formula", journal = j-J-NUMBER-THEORY, volume = "162", number = "??", pages = "578--588", month = may, year = "2016", CODEN = "JNUTA9", DOI = "https://doi.org/10.1016/j.jnt.2015.09.014", ISSN = "0022-314X (print), 1096-1658 (electronic)", ISSN-L = "0022-314X", bibdate = "Wed Jul 15 08:49:20 MDT 2020", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jnumbertheory2010.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0022314X15003133", acknowledgement = ack-nhfb, ajournal = "J. Number Theory", fjournal = "Journal of Number Theory", journal-URL = "http://www.sciencedirect.com/science/journal/0022314X", } @Article{Chen:2016:IAEb, author = "Chao-Ping Chen", title = "Inequalities and asymptotic expansions for the psi function and the {Euler--Mascheroni} constant", journal = j-J-NUMBER-THEORY, volume = "163", number = "??", pages = "596--607", month = jun, year = "2016", CODEN = "JNUTA9", DOI = "https://doi.org/10.1016/j.jnt.2015.10.013", ISSN = "0022-314X (print), 1096-1658 (electronic)", ISSN-L = "0022-314X", bibdate = "Wed Jul 15 08:49:20 MDT 2020", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jnumbertheory2010.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0022314X15003558", acknowledgement = ack-nhfb, ajournal = "J. Number Theory", fjournal = "Journal of Number Theory", journal-URL = "http://www.sciencedirect.com/science/journal/0022314X", } @Article{Chen:2016:IAEc, author = "Chao-Ping Chen", title = "Inequalities and asymptotics for the {Euler--Mascheroni} constant based on {DeTemple's} result", journal = j-NUMER-ALGORITHMS, volume = "73", number = "3", pages = "761--774", month = nov, year = "2016", CODEN = "NUALEG", DOI = "https://doi.org/10.1007/s11075-016-0116-9", ISSN = "1017-1398 (print), 1572-9265 (electronic)", ISSN-L = "1017-1398", bibdate = "Wed Mar 1 09:12:13 MST 2017", bibsource = "http://link.springer.com/journal/11075/73/3; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/numeralgorithms.bib", URL = "http://link.springer.com/article/10.1007/s11075-016-0116-9", acknowledgement = ack-nhfb, fjournal = "Numerical Algorithms", journal-URL = "http://link.springer.com/journal/11075", } @Article{Chen:2016:MAA, author = "Chao-Ping Chen", title = "A more accurate approximation for the gamma function", journal = j-J-NUMBER-THEORY, volume = "164", number = "??", pages = "417--428", month = jul, year = "2016", CODEN = "JNUTA9", DOI = "https://doi.org/10.1016/j.jnt.2015.11.007", ISSN = "0022-314X (print), 1096-1658 (electronic)", ISSN-L = "0022-314X", bibdate = "Wed Jul 15 08:49:21 MDT 2020", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jnumbertheory2010.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0022314X16000068", acknowledgement = ack-nhfb, ajournal = "J. Number Theory", fjournal = "Journal of Number Theory", journal-URL = "http://www.sciencedirect.com/science/journal/0022314X", } @Article{Chen:2016:MPI, author = "Chao-Ping Chen", title = "Monotonicity properties, inequalities and asymptotic expansions associated with the gamma function", journal = j-APPL-MATH-COMP, volume = "283", number = "??", pages = "385--396", day = "20", month = jun, year = "2016", CODEN = "AMHCBQ", ISSN = "0096-3003 (print), 1873-5649 (electronic)", ISSN-L = "0096-3003", bibdate = "Tue Apr 5 07:51:07 MDT 2016", bibsource = "https://www.math.utah.edu/pub/tex/bib/applmathcomput2015.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0096300316301515", acknowledgement = ack-nhfb, fjournal = "Applied Mathematics and Computation", journal-URL = "http://www.sciencedirect.com/science/journal/00963003/", } @Article{Chen:2016:SIAa, author = "Chao-Ping Chen and Wei-Wei Tong", title = "Sharp inequalities and asymptotic expansions for the gamma function", journal = j-J-NUMBER-THEORY, volume = "160", number = "??", pages = "418--431", month = mar, year = "2016", CODEN = "JNUTA9", DOI = "https://doi.org/10.1016/j.jnt.2015.09.021", ISSN = "0022-314X (print), 1096-1658 (electronic)", ISSN-L = "0022-314X", bibdate = "Wed Jul 15 08:49:18 MDT 2020", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jnumbertheory2010.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0022314X15003200", acknowledgement = ack-nhfb, ajournal = "J. Number Theory", fjournal = "Journal of Number Theory", journal-URL = "http://www.sciencedirect.com/science/journal/0022314X", } @Article{Erascu:2016:RQE, author = "Madalina Erascu and Hoon Hong", title = "Real quantifier elimination for the synthesis of optimal numerical algorithms (Case study: Square root computation)", journal = j-J-SYMBOLIC-COMP, volume = "75", number = "??", pages = "110--126", month = jul # "\slash " # aug, year = "2016", CODEN = "JSYCEH", ISSN = "0747-7171 (print), 1095-855X (electronic)", ISSN-L = "0747-7171", bibdate = "Mon Jan 25 06:25:01 MST 2016", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jsymcomp.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0747717115001091", acknowledgement = ack-nhfb, fjournal = "Journal of Symbolic Computation", journal-URL = "http://www.sciencedirect.com/science/journal/07477171/", keywords = "interval arithmetic; interval square root", } @TechReport{Fateman:2016:CUA, author = "Richard J. Fateman", title = "Comments on Unrestricted Algorithms for {Bessel} Functions in Computer Algebra: Arbitrary Precision, The Backwards Recurrence, {Taylor} Series, {Hermite} Interpolation", type = "Report", institution = "University of California, Berkeley", address = "Berkeley, CA 947220-1776, USA", day = "4", month = jun, year = "2016", bibdate = "Fri Feb 24 09:55:02 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "https://people.eecs.berkeley.edu/~fateman/papers/hermite.pdf", abstract = "We explore various ways of implementing ``unrestricted algorithms'' [3] for approximating Bessel ($J$) functions. An unrestricted algorithm for a function $ f(x)$ provides a result to any requested precision in the answer. The emphasis is on higher-than-normal precision with the precision specified as an extra argument to the function. That is, the precision is specified at run-time. We require that the algorithm provide at least the requested number of correct digits, contrary to some existing codes which provide only ``absolute error'' near critical points. We use $ J_0 $ of real non-negative argument as an example, although much of the reasoning generalizes to other Bessel functions or related functions.\par Since it is plausible that there will be requests for additional values of $ J_0 $ at the same (high) precision at a collection of nearby arguments, we consider implementations that cache certain re-usable key constants (namely zeros of $ J_0 $ near the argument values).", acknowledgement = ack-nhfb, } @Article{Gautschi:2016:AER, author = "Walter Gautschi", title = "Algorithm 957: Evaluation of the Repeated Integral of the Coerror Function by Half-Range {Gauss--Hermite} Quadrature", journal = j-TOMS, volume = "42", number = "1", pages = "9:1--9:10", month = feb, year = "2016", CODEN = "ACMSCU", DOI = "https://doi.org/10.1145/2735626", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Tue Mar 1 17:07:56 MST 2016", bibsource = "http://www.acm.org/pubs/contents/journals/toms/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/matlab.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib", abstract = "Nonstandard Gaussian quadrature is applied to evaluate the repeated integral $ i^n \erfc x $ of the coerror function for $ n \in N_0 $, $ x \in R $ in an appropriate domain of the $ (n, x)$-plane. Relevant software in MATLAB is provided: in particular, two routines evaluating the function to an accuracy of 12 respective 30-decimal digits.", acknowledgement = ack-nhfb, articleno = "9", fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", } @Article{Gil:2016:ACI, author = "Amparo Gil and Diego Ruiz-Antol{\'\i}n and Javier Segura and Nico M. Temme", title = "{Algorithm 969}: Computation of the Incomplete Gamma Function for Negative Values of the Argument", journal = j-TOMS, volume = "43", number = "3", pages = "26:1--26:9", month = nov, year = "2016", CODEN = "ACMSCU", DOI = "https://doi.org/10.1145/2972951", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Tue Nov 22 17:45:25 MST 2016", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib", URL = "http://dl.acm.org/citation.cfm?id=2972951", abstract = "An algorithm for computing the incomplete gamma function $ \gamma^(a, z) $ for real values of the parameter $a$ and negative real values of the argument $z$ is presented. The algorithm combines the use of series expansions, Poincar{\'e}-type expansions, uniform asymptotic expansions, and recurrence relations, depending on the parameter region. A relative accuracy $ \approx 10^{-13}$ in the parameter region $ (a, z) \in [500, 500] \times [500, 0)$ can be obtained when computing the function $ \gamma^\ast (a, z)$ with the Fortran 90 module IncgamNEG implementing the algorithm.", acknowledgement = ack-nhfb, articleno = "26", fjournal = "ACM Transactions on Mathematical Software", journal-URL = "https://dl.acm.org/loi/toms", } @Article{Giles:2016:AAI, author = "Michael B. Giles", title = "Algorithm 955: Approximation of the Inverse {Poisson} Cumulative Distribution Function", journal = j-TOMS, volume = "42", number = "1", pages = "7:1--7:22", month = feb, year = "2016", CODEN = "ACMSCU", DOI = "https://doi.org/10.1145/2699466", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Tue Mar 1 17:07:56 MST 2016", bibsource = "http://www.acm.org/pubs/contents/journals/toms/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib", abstract = "New approximations for the inverse of the incomplete gamma function are derived, which are used to develop efficient evaluations of the inverse Poisson cumulative distribution function. An asymptotic approximation based on the standard Normal approximation is particularly good for CPUs with MIMD cores, while for GPUs and other hardware with vector units, a second asymptotic approximation based on Temme's approximation of the incomplete gamma function is more efficient due to conditional branching within each vector. The accuracy and efficiency of the software implementations is assessed on both CPUs and GPUs.", acknowledgement = ack-nhfb, articleno = "7", fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", } @Article{Jameson:2016:IGF, author = "G. J. O. Jameson", title = "The incomplete gamma functions", journal = j-MATH-GAZ, volume = "100", number = "548", pages = "298--306", month = jul, year = "2016", CODEN = "MAGAAS", DOI = "https://doi.org/10.1017/mag.2016.67", ISSN = "0025-5572 (print), 2056-6328 (electronic)", ISSN-L = "0025-5572", bibdate = "Tue Sep 27 10:11:13 MDT 2016", bibsource = "http://journals.cambridge.org/action/displayIssue?jid=MAG; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathgaz2010.bib", URL = "https://www.cambridge.org/core/product/9373A31AD28D793AB5431E35EA5C2AF6", acknowledgement = ack-nhfb, ajournal = "Math. Gaz.", fjournal = "The Mathematical Gazette", journal-URL = "http://journals.cambridge.org/action/displayIssue?jid=MAG; http://www.m-a.org.uk/jsp/index.jsp?lnk=620", onlinedate = "14 June 2016", remark = "This paper exhibits and proves several useful identities for the incomplete gamma functions, but does not discuss their stable numerical computation.", } @Article{Johansson:2016:CHF, author = "Fredrik Johansson", title = "Computing hypergeometric functions rigorously", journal = "arxiv.org", volume = "??", number = "??", pages = "2--29", day = "22", month = jun, year = "2016", bibdate = "Thu Jun 23 07:39:32 2016", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://arxiv.org/abs/1606.06977", abstract = "We present an efficient implementation of hypergeometric functions in arbitrary-precision interval arithmetic. The functions 0F1, 1F1, 2F1 and 2F0 (or the Kummer U-function) are supported for unrestricted complex parameters and argument, and by extension, we cover exponential and trigonometric integrals, error functions, Fresnel integrals, incomplete gamma and beta functions, Bessel functions, Airy functions, Legendre functions, Jacobi polynomials, complete elliptic integrals, and other special functions. The output can be used directly for interval computations or to generate provably correct floating-point approximations in any format. Performance is competitive with earlier arbitrary-precision software, and sometimes orders of magnitude faster. We also partially cover the generalized hypergeometric function pFq and computation of high-order parameter derivatives.", acknowledgement = ack-nhfb, } @Article{Johansson:2016:FAE, author = "H. T. Johansson and C. Forss{\'e}n", title = "Fast and Accurate Evaluation of {Wigner} 3$j$, 6$j$, and 9$j$ Symbols Using Prime Factorization and Multiword Integer Arithmetic", journal = j-SIAM-J-SCI-COMP, volume = "38", number = "1", pages = "A376--A384", month = "????", year = "2016", CODEN = "SJOCE3", DOI = "https://doi.org/10.1137/15M1021908", ISSN = "1064-8275 (print), 1095-7197 (electronic)", ISSN-L = "1064-8275", bibdate = "Tue Jun 21 08:11:55 MDT 2016", bibsource = "http://epubs.siam.org/toc/sjoce3/38/1; https://www.math.utah.edu/pub/bibnet/authors/w/wigner-eugene.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/siamjscicomput.bib", acknowledgement = ack-nhfb, fjournal = "SIAM Journal on Scientific Computing", journal-URL = "http://epubs.siam.org/sisc", onlinedate = "January 2016", } @Article{Koelink:2016:AST, author = "Erik Koelink", title = "Applications of spectral theory to special functions", journal = "ArXiv e-prints", volume = "??", pages = "1--63", month = dec, year = "2016", bibdate = "Sat Feb 18 09:23:20 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "https://arxiv.org/abs/1612.07035", abstract = "Many special functions are eigenfunctions to explicit operators, such as difference and differential operators, which is in particular true for the special functions occurring in the Askey-scheme, its $q$-analogue and extensions. The study of the spectral properties of such operators leads to explicit information for the corresponding special functions. We discuss several instances of this application, involving orthogonal polynomials and their matrix-valued analogues.", acknowledgement = ack-nhfb, eprint = "1612.07035", keywords = "Mathematics --- Classical Analysis and ODEs; Mathematics --- Functional Analysis", primaryclass = "math.CA", } @Article{Kutsuna:2016:ARM, author = "Takuro Kutsuna and Yoshinao Ishii", title = "Abstraction and refinement of mathematical functions toward {SMT}-based test-case generation", journal = j-INT-J-SOFTW-TOOLS-TECHNOL-TRANSFER, volume = "18", number = "1", pages = "109--120", month = feb, year = "2016", CODEN = "????", DOI = "https://doi.org/10.1007/s10009-015-0389-7", ISSN = "1433-2779 (print), 1433-2787 (electronic)", ISSN-L = "1433-2779", bibdate = "Mon Jan 25 08:12:53 MST 2016", bibsource = "http://link.springer.com/journal/10009/18/1; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/multithreading.bib; https://www.math.utah.edu/pub/tex/bib/sttt.bib", URL = "http://link.springer.com/article/10.1007/s10009-015-0389-7", acknowledgement = ack-nhfb, fjournal = "International Journal on Software Tools for Technology Transfer (STTT)", journal-URL = "http://link.springer.com/journal/10009", } @InProceedings{Langhammer:2016:SPN, author = "Martin Langhammer and Bogdan Pasca", title = "Single Precision Natural Logarithm Architecture for Hard Floating-Point and {DSP}-Enabled {FPGAs}", crossref = "Montuschi:2016:ISC", pages = "164--171", year = "2016", DOI = "https://doi.org/10.1109/ARITH.2016.20", bibdate = "Fri Dec 16 15:17:20 MST 2016", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, keywords = "ARITH-23", } @InProceedings{LeMaire:2016:CFP, author = "Julien {Le Maire} and Nicolas Brunie and Florent de Dinechin and Jean-Michel Muller", title = "Computing floating-point logarithms with fixed-point operations", crossref = "Montuschi:2016:ISC", pages = "156--163", year = "2016", DOI = "https://doi.org/10.1109/ARITH.2016.24", bibdate = "Fri Dec 16 15:17:20 MST 2016", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, keywords = "ARITH-23", } @Article{Lu:2016:QCF, author = "Dawei Lu and Lixin Song and Congxu Ma", title = "A quicker continued fraction approximation of the gamma function related to the {Windschitl}'s formula", journal = j-NUMER-ALGORITHMS, volume = "72", number = "4", pages = "865--874", month = aug, year = "2016", CODEN = "NUALEG", DOI = "https://doi.org/10.1007/s11075-015-0070-y", ISSN = "1017-1398 (print), 1572-9265 (electronic)", ISSN-L = "1017-1398", bibdate = "Tue Sep 20 10:57:47 MDT 2016", bibsource = "http://link.springer.com/journal/11075/72/4; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/numeralgorithms.bib", URL = "http://link.springer.com/article/10.1007/s11075-015-0070-y", acknowledgement = ack-nhfb, fjournal = "Numerical Algorithms", journal-URL = "http://link.springer.com/journal/11075", remark = "The tables at the end of this paper compare six algorithms for approximating $ n! $ for $ n = 50, 100, 500, 2500 $. The Burnside, Nemes, and Windschitl formulas are slightly less accurate than the traditional Stirling approximation. The new formula, and the Mortici formula, are slightly better than Stirling's.", } @Article{Maignan:2016:FGL, author = "Aude Maignan and Tony C. Scott", title = "Fleshing out the generalized {Lambert} {$W$} function", journal = j-ACM-COMM-COMP-ALGEBRA, volume = "50", number = "2", pages = "45--60", month = jun, year = "2016", CODEN = "????", DOI = "https://doi.org/10.1145/2992274.2992275", ISSN = "1932-2232 (print), 1932-2240 (electronic)", ISSN-L = "1932-2232", bibdate = "Thu Aug 25 17:57:39 MDT 2016", bibsource = "http://portal.acm.org/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/sigsam.bib", abstract = "Herein, we use Hardy's notion of the ``false derivative'' to obtain exact multiple roots in closed form of the transcendental--algebraic equations representing the generalized Lambert $W$ function. In this fashion, we flesh out the generalized Lambert $W$ function by complementing previous developments to produce a more complete and integrated body of work. Finally, we demonstrate the usefulness of this special function with some applications.", acknowledgement = ack-nhfb, fjournal = "ACM Communications in Computer Algebra", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J1000", } @Article{Martin-Dorel:2016:PTB, author = "{\'E}rik Martin-Dorel and Guillaume Melquiond", title = "Proving Tight Bounds on Univariate Expressions with Elementary Functions in {Coq}", journal = j-J-AUTOM-REASON, volume = "57", number = "3", pages = "187--217", month = oct, year = "2016", CODEN = "JAREEW", DOI = "https://doi.org/10.1007/s10817-015-9350-4", ISSN = "0168-7433 (print), 1573-0670 (electronic)", ISSN-L = "0168-7433", bibdate = "Fri Sep 2 06:39:36 MDT 2016", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/jautomreason.bib", URL = "http://link.springer.com/accesspage/article/10.1007/s10817-015-9350-4", acknowledgement = ack-nhfb, ajournal = "J. Autom. Reason.", fjournal = "Journal of Automated Reasoning", journal-URL = "http://link.springer.com/journal/10817", } @Article{Mohankumar:2016:VAN, author = "N. Mohankumar and A. Natarajan", title = "On the very accurate numerical evaluation of the {Generalized Fermi--Dirac Integrals}", journal = j-COMP-PHYS-COMM, volume = "207", number = "??", pages = "193--201", month = oct, year = "2016", CODEN = "CPHCBZ", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", bibdate = "Tue Aug 30 18:08:51 MDT 2016", bibsource = "https://www.math.utah.edu/pub/tex/bib/compphyscomm2010.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0010465516301667", acknowledgement = ack-nhfb, fjournal = "Computer Physics Communications", journal-URL = "http://www.sciencedirect.com/science/journal/00104655/", } @Article{Moroz:2016:FCI, author = "Leonid V. Moroz and Cezary J. Walczyk and Andriy Hrynchyshyn and Vijay Holimath and Jan L. Cie{\'s}li{\'n}ski", title = "Fast calculation of inverse square root with the use of magic constant --- analytical approach", journal = "arXiv.org", volume = "??", number = "??", pages = "1--23", day = "14", month = mar, year = "2016", DOI = "https://doi.org/10.48550/arXiv.1603.04483", bibdate = "Wed Dec 20 07:34:12 2023", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "https://arxiv.org/pdf/1603.04483.pdf", abstract = "We present a mathematical analysis of transformations used in fast calculation of inverse square root for single-precision floating-point numbers. Optimal values of the so called magic constants are derived in a systematic way, minimizing either absolute or relative errors at subsequent stages of the discussed algorithm.", acknowledgement = ack-nhfb, } @Book{Muller:2016:EFA, author = "Jean-Michel Muller", title = "Elementary Functions: Algorithms and Implementation", publisher = pub-BIRKHAUSER-BOSTON, address = pub-BIRKHAUSER-BOSTON:adr, edition = "Third", pages = "xxv + 283", year = "2016", DOI = "https://doi.org/10.1007/978-1-4899-7983-4", ISBN = "1-4899-7981-6 (print), 1-4899-7983-2 (e-book)", ISBN-13 = "978-1-4899-7981-0 (print), 978-1-4899-7983-4 (e-book)", LCCN = "QA331 .M866 2016", bibdate = "Sun Dec 04 15:12:36 2016", bibsource = "https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/maple-extract.bib; https://www.math.utah.edu/pub/tex/bib/master.bib; https://www.math.utah.edu/pub/tex/bib/mathcw.bib; https://www.math.utah.edu/pub/tex/bib/numana2010.bib; z3950.loc.gov:7090/Voyager", abstract = "This textbook presents the concepts and tools necessary to understand, build, and implement algorithms for computing elementary functions (e.g., logarithms, exponentials, and the trigonometric functions). Both hardware- and software-oriented algorithms are included, along with issues related to accurate floating-point implementation. This third edition has been updated and expanded to incorporate the most recent advances in the field, new elementary function algorithms, and function software. After a preliminary chapter that briefly introduces some fundamental concepts of computer arithmetic, such as floating-point arithmetic and redundant number systems, the text is divided into three main parts. Part I considers the computation of elementary functions using algorithms based on polynomial or rational approximations and using table-based methods; the final chapter in this section deals with basic principles of multiple-precision arithmetic. Part II is devoted to a presentation of shift-and-add algorithms (hardware-oriented algorithms that use additions and shifts only). Issues related to accuracy, including range reduction, preservation of monotonicity, and correct rounding, as well as some examples of implementation are explored in Part III. Numerous examples of command lines and full programs are provided throughout for various software packages, including Maple, Sollya, and Gappa. New to this edition are an in-depth overview of the IEEE-754-2008 standard for floating-point arithmetic; a section on using double- and triple-word numbers; a presentation of new tools for designing accurate function software; and a section on the Toom--Cook family of multiplication algorithms. The techniques presented in this book will be of interest to implementors of elementary function libraries or circuits and programmers of numerical applications. Additionally, graduate and advanced undergraduate students, professionals, and researchers in scientific computing, numerical analysis, software engineering, and computer engineering will find this a useful reference and resource.", acknowledgement = ack-nhfb, subject = "Functions; Data processing; Algorithms", tableofcontents = "Introduction \\ Introduction to Computer Arithmetic \\ Part I: Algorithms Based on Polynomial Approximation and/or Table Lookup, Multiple-Precision Evaluation of Functions \\ The Classical Theory of Polynomial or Rational Approximations \\ Polynomial Approximations with Special Constraints \\ Polynomial Evaluation \\ Table-Based Methods \\ Multiple-Precision Evaluation of Functions \\ Part II: Shift-and-Add Algorithms \\ Introduction to Shift-and-Add Algorithms \\ The CORDIC Algorithm \\ Some Other Shift-and-Add Algorithms \\ Part III: Range Reduction, Final Rounding, and Exceptions \\ Range Reduction \\ Final Rounding \\ Miscellaneous \\ Examples of Implementation \\ References \\ Index", } @Misc{Munshi:2016:OCS, author = "Aaftab Munshi and Lee Howes and Bartosz Sochacki and {Khronos OpenCL Working Group}", title = "The {OpenCL} {C} Specification Version: 2.0 Document Revision: 33", howpublished = "Web document.", pages = "205", day = "13", month = apr, year = "2016", bibdate = "Mon Apr 16 14:05:49 2018", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/pvm.bib", URL = "https://www.khronos.org/registry/OpenCL/specs/opencl-2.0-openclc.pdf", acknowledgement = ack-nhfb, remark = "Section 6.1.3.2 Math Functions, pages 74ff, defines a function repertoire extended beyond that of ISO C, including {\tt acospi}, {\tt asinpi}, {\tt atanpi}, {\tt atan2pi}, {\tt cospi}, {\tt sinpi}, {\tt tanpi}, {\tt cospi}, {\tt fract}, {\tt lgamma\_r}, {\tt mad} (approximation to {\tt a * b + c}), {\tt minmag}, {\tt pown}, {\tt rootn}, {\tt sincos}, {\tt sinpi}, and {\tt tanpi}.", } @InProceedings{Navas-Palencia:2016:CCH, author = "Guillermo Navas-Palencia and Argimiro Arratia", title = "On the Computation of Confluent Hypergeometric Functions for Large Imaginary Part of Parameters $b$ and $z$", crossref = "Greuel:2016:MSI", pages = "241--248", year = "2016", DOI = "https://doi.org/10.1007/978-3-319-42432-3_30", bibdate = "Mon Feb 5 08:27:34 2018", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, } @Article{OSullivan:2016:ZD, author = "Cormac O'Sullivan", title = "Zeros of the dilogarithm", journal = j-MATH-COMPUT, volume = "85", number = "302", pages = "2967--2993", month = nov, year = "2016", CODEN = "MCMPAF", DOI = "https://doi.org/10.1090/mcom/3065", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Sat Nov 5 12:22:19 MDT 2016", bibsource = "http://www.ams.org/mcom/2016-85-302; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp2010.bib", URL = "http://www.ams.org/journals/mcom/2016-85-302/S0025-5718-2016-03065-3; http://www.ams.org/journals/mcom/2016-85-302/S0025-5718-2016-03065-3/S0025-5718-2016-03065-3.pdf; http://www.ams.org/mathscinet/search/author.html?mrauthid=658848", acknowledgement = ack-nhfb, fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Ozcag:2016:RPI, author = "Emin {\"O}zc{\d{a}}{\u{g}} and {\.I}nci Ege", title = "Remarks on polygamma and incomplete gamma type functions", journal = j-J-NUMBER-THEORY, volume = "169", number = "??", pages = "369--387", month = dec, year = "2016", CODEN = "JNUTA9", DOI = "https://doi.org/10.1016/j.jnt.2016.05.021", ISSN = "0022-314X (print), 1096-1658 (electronic)", ISSN-L = "0022-314X", bibdate = "Wed Jul 15 08:49:24 MDT 2020", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jnumbertheory2010.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0022314X16301378", acknowledgement = ack-nhfb, ajournal = "J. Number Theory", fjournal = "Journal of Number Theory", journal-URL = "http://www.sciencedirect.com/science/journal/0022314X", } @Article{Paris:2016:UAE, author = "R. B. Paris", title = "A uniform asymptotic expansion for the incomplete gamma functions revisited", journal = "arxiv.org", volume = "??", number = "??", pages = "1--9", day = "2", month = nov, year = "2016", bibdate = "Sat Feb 18 09:13:43 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "https://arxiv.org/abs/1611.00548", abstract = "A new uniform asymptotic expansion for the incomplete gamma function $ \Gamma (a, z) $ valid for large values of $z$ was given by the author in \cite{Paris:2002:UAE}. This expansion contains a complementary error function of an argument measuring transition across the point $ z = a$, with easily computable coefficients that do not involve a removable singularity in the neighbourhood of this point. In this note we correct a misprint in the listing of certain coefficients in this expansion and discuss in more detail the situation corresponding to $ \Gamma (a, a)$.", acknowledgement = ack-nhfb, remark = "Page 9 gives corrections to \cite[Eq. 8.12.18--8.12.20]{Olver:2010:NHM}.", } @Article{Piparo:2016:CPT, author = "D. Piparo and V. Innocente", title = "The {CptnHook Profiler} --- a tool to investigate usage patterns of mathematical functions", journal = j-J-PHYS-CONF-SER, volume = "762", pages = "012038:1--012038:", month = oct, year = "2016", CODEN = "JPCSDZ", DOI = "https://doi.org/10.1088/1742-6596/762/1/012038", ISSN = "1742-6588 (print), 1742-6596 (electronic)", ISSN-L = "1742-6588", bibdate = "Thu Sep 19 14:53:02 2024", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Journal of Physics: Conference Series", journal-URL = "http://www.iop.org/EJ/journal/conf", } @InProceedings{Revy:2016:ADF, author = "Guillaume Revy", title = "Automated Design of Floating-Point Logarithm Functions on Integer Processors", crossref = "Montuschi:2016:ISC", pages = "172--180", year = "2016", DOI = "https://doi.org/10.1109/ARITH.2016.28", bibdate = "Fri Dec 16 15:17:20 MST 2016", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, keywords = "ARITH-23", } @Article{Sayed:2016:WCR, author = "Wafaa S. Sayed and Hossam A. H. Fahmy", title = "What are the Correct Results for the Special Values of the Operands of the Power Operation?", journal = j-TOMS, volume = "42", number = "2", pages = "14:1--14:17", month = jun, year = "2016", CODEN = "ACMSCU", DOI = "https://doi.org/10.1145/2809783", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Fri Jun 3 18:52:21 MDT 2016", bibsource = "http://www.acm.org/pubs/contents/journals/toms/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib", abstract = "Language standards such as C99 and C11, as well as the IEEE Standard for Floating-Point Arithmetic 754 (IEEE Std 754-2008) specify the expected behavior of binary and decimal floating-point arithmetic in computer-programming environments and the handling of special values and exception conditions. Many researchers focus on verifying the compliance of implementations for binary and decimal floating-point operations with these standards. In this article, we are concerned with the special values of the operands of the power function Z = X$^Y$. We study how the standards define the correct results for this operation, propose a mathematically justified definition for the correct results of the power function on the occurrence of these special values as its operands, test how different software implementations for the power function deal with these special values, and classify the behavior of different programming languages from the viewpoint of how much they conform to the standards and our proposed mathematical definition. We present inconsistencies between the implementations and the standards, and discuss incompatibilities between different versions of the same software.", acknowledgement = ack-nhfb, articleno = "14", fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", } @Article{Schmidt:2016:ZSG, author = "Maxie D. Schmidt", title = "Zeta Series Generating Function Transformations Related to Generalized {Stirling} Numbers and Partial Sums of the {Hurwitz} Zeta Function", journal = "arxiv.org", month = nov, year = "2016", bibdate = "Sat Feb 18 09:26:39 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://adsabs.harvard.edu/abs/2016arXiv161100957S", abstract = "We define a generalized class of modified zeta series transformations generating the partial sums of the Hurwitz zeta function and series expansions of the Lerch transcendent function. The new transformation coefficients we define within the article satisfy expansions by generalized harmonic number sequences, or the partial sums of the Hurwitz zeta function, which are analogous to known properties for the Stirling numbers of the first kind and for the known transformation coefficients employed to enumerate variants of the polylogarithm function series. Applications of the new results we prove in the article include new series expansions of the Dirichlet beta function, the Legendre chi function, BBP-type series identities for special constants, alternating and exotic Euler sum variants, alternating zeta functions with powers of quadratic denominators, and particular series defining special cases of the Riemann zeta function constants at the positive integers $ s \geq 3 $.", acknowledgement = ack-nhfb, eprint = "1611.00957", keywords = "Mathematics - Combinatorics, Mathematics - Number Theory", primaryclass = "math.CO", } @Article{Stange:2016:CAM, author = "J. Stange and N. Loginova and T. Dickhaus", title = "Computing and approximating multivariate chi-square probabilities", journal = j-J-STAT-COMPUT-SIMUL, volume = "86", number = "6", pages = "1233--1247", year = "2016", CODEN = "JSCSAJ", DOI = "https://doi.org/10.1080/00949655.2015.1058798", ISSN = "0094-9655 (print), 1026-7778 (electronic), 1563-5163", ISSN-L = "0094-9655", bibdate = "Thu Feb 4 07:57:25 MST 2016", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jstatcomputsimul.bib", URL = "http://www.tandfonline.com/doi/abs/10.1080/00949655.2015.1058798", acknowledgement = ack-nhfb, fjournal = "Journal of Statistical Computation and Simulation", journal-URL = "http://www.tandfonline.com/loi/gscs20", } @Article{Stefanica:2016:SAA, author = "Dan Stefanica and Rado{\v{s}} Radoi{\v{c}}i{\'c}", title = "A sharp approximation for {ATM}-forward option prices and implied volatilities", journal = "International Journal of Financial Engineering", volume = "3", number = "1", pages = "1650002", month = mar, year = "2016", DOI = "https://doi.org/10.1142/s242478631650002x", ISSN = "2424-7863", bibdate = "Sat Dec 16 17:46:33 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, } @Article{Wang:2016:AFG, author = "Miao-Kun Wang and Yu-Ming Chu and Ying-Qing Song", title = "Asymptotical formulas for {Gaussian} and generalized hypergeometric functions", journal = j-APPL-MATH-COMP, volume = "276", number = "??", pages = "44--60", day = "5", month = mar, year = "2016", CODEN = "AMHCBQ", ISSN = "0096-3003 (print), 1873-5649 (electronic)", ISSN-L = "0096-3003", bibdate = "Tue Jan 26 17:22:21 MST 2016", bibsource = "https://www.math.utah.edu/pub/tex/bib/applmathcomput2015.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0096300315015908", acknowledgement = ack-nhfb, fjournal = "Applied Mathematics and Computation", journal-URL = "http://www.sciencedirect.com/science/journal/00963003/", } @Article{Wang:2016:UAA, author = "Weiping Wang", title = "Unified approaches to the approximations of the gamma function", journal = j-J-NUMBER-THEORY, volume = "163", number = "??", pages = "570--595", month = jun, year = "2016", CODEN = "JNUTA9", DOI = "https://doi.org/10.1016/j.jnt.2015.12.016", ISSN = "0022-314X (print), 1096-1658 (electronic)", ISSN-L = "0022-314X", bibdate = "Wed Jul 15 08:49:20 MDT 2020", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jnumbertheory2010.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0022314X16000470", acknowledgement = ack-nhfb, ajournal = "J. Number Theory", fjournal = "Journal of Number Theory", journal-URL = "http://www.sciencedirect.com/science/journal/0022314X", } @Article{Xu:2016:AEG, author = "Aimin Xu and Yongcai Hu and Peipei Tang", title = "Asymptotic expansions for the gamma function", journal = j-J-NUMBER-THEORY, volume = "169", number = "??", pages = "134--143", month = dec, year = "2016", CODEN = "JNUTA9", DOI = "https://doi.org/10.1016/j.jnt.2016.05.020", ISSN = "0022-314X (print), 1096-1658 (electronic)", ISSN-L = "0022-314X", bibdate = "Wed Jul 15 08:49:24 MDT 2020", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jnumbertheory2010.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0022314X16301366", acknowledgement = ack-nhfb, ajournal = "J. Number Theory", fjournal = "Journal of Number Theory", journal-URL = "http://www.sciencedirect.com/science/journal/0022314X", } @Article{Zaghloul:2016:RAC, author = "Mofreh R. Zaghloul", title = "Remark on {``Algorithm 916: Computing the Faddeyeva and Voigt Functions''}: Efficiency Improvements and {Fortran} Translation", journal = j-TOMS, volume = "42", number = "3", pages = "26:1--26:9", month = may, year = "2016", CODEN = "ACMSCU", DOI = "https://doi.org/10.1145/2806884", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Mon May 23 16:40:02 MDT 2016", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fortran3.bib; https://www.math.utah.edu/pub/tex/bib/matlab.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib", note = "See \cite{Zaghloul:2011:ACF}.", abstract = "This remark describes efficiency improvements to Algorithm 916 [Zaghloul and Ali 2011]. It is shown that the execution time required by the algorithm, when run at its highest accuracy, may be improved by more than a factor of 2. A better accuracy vs efficiency tradeoff scheme is also implemented; this requires the user to supply the number of significant figures desired in the computed values as an extra input argument to the function. Using this tradeoff, it is shown that the efficiency of the algorithm may be further improved significantly while maintaining reasonably accurate and safe results that are free of the pitfalls and complete loss of accuracy seen in other competitive techniques. The current version of the code is provided in Matlab and Scilab in addition to a Fortran translation prepared to meet the needs of real-world problems where very large numbers of function evaluations would require the use of a compiled language. To fulfill this last requirement, a recently proposed reformed version of Huml{\'\i}cek's w4 routine, shown to maintain the claimed accuracy of the algorithm over a wide and fine grid, is implemented in the present Fortran translation for the case of four significant figures. This latter modification assures the reliability of the code in the solution of practical problems requiring numerous evaluation of the function for applications requiring low-accuracy computations ($ < 10^{-4}$).", acknowledgement = ack-nhfb, articleno = "26", fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", } @Article{Alonso:2017:EAA, author = "Pedro Alonso and Javier Ib{\'a}{\~n}ez and Jorge Sastre and Jes{\'u}s Peinado and Emilio Defez", title = "Efficient and accurate algorithms for computing matrix trigonometric functions", journal = j-J-COMPUT-APPL-MATH, volume = "309", number = "??", pages = "325--332", day = "1", month = jan, year = "2017", CODEN = "JCAMDI", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Sat Feb 25 13:35:53 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2015.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0377042716302321", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @Article{Baikov:2017:AID, author = "Nikita Baikov", title = "Algorithm and Implementation Details for Complementary Error Function", journal = j-IEEE-TRANS-COMPUT, volume = "66", number = "7", pages = "1106--1118", month = jul, year = "2017", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/TC.2016.2641960", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Thu Jun 8 10:22:00 MDT 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput2010.bib", URL = "https://www.computer.org/csdl/trans/tc/2017/07/07792222-abs.html", acknowledgement = ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", } @Book{Beebe:2017:MFC, author = "Nelson H. F. Beebe", title = "The Mathematical-Function Computation Handbook: Programming Using the {MathCW} Portable Software Library", publisher = pub-SV, address = pub-SV:adr, pages = "xxxvi + 1114", year = "2017", DOI = "https://doi.org/10.1007/978-3-319-64110-2", ISBN = "3-319-64109-3 (hardcover), 3-319-64110-7 (e-book)", ISBN-13 = "978-3-319-64109-6 (hardcover), 978-3-319-64110-2 (e-book)", LCCN = "QA75.5-76.95", bibdate = "Sat Jul 15 19:34:43 MDT 2017", bibsource = "fsz3950.oclc.org:210/WorldCat; https://www.math.utah.edu/pub/bibnet/authors/b/beebe-nelson-h-f.bib; https://www.math.utah.edu/pub/tex/bib/axiom.bib; https://www.math.utah.edu/pub/tex/bib/cryptography2010.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/maple-extract.bib; https://www.math.utah.edu/pub/tex/bib/master.bib; https://www.math.utah.edu/pub/tex/bib/mathematica.bib; https://www.math.utah.edu/pub/tex/bib/matlab.bib; https://www.math.utah.edu/pub/tex/bib/mupad.bib; https://www.math.utah.edu/pub/tex/bib/numana2010.bib; https://www.math.utah.edu/pub/tex/bib/prng.bib; https://www.math.utah.edu/pub/tex/bib/redbooks.bib; https://www.math.utah.edu/pub/tex/bib/utah-math-dept-books.bib", URL = "http://www.springer.com/us/book/9783319641096", acknowledgement = ack-nhfb, ORCID-numbers = "Beebe, Nelson H. F./0000-0001-7281-4263", tableofcontents = "List of figures / xxv \\ List of tables / xxxi \\ Quick start / xxxv \\ 1: Introduction / 1 \\ 1.1: Programming conventions / 2 \\ 1.2: Naming conventions / 4 \\ 1.3: Library contributions and coverage / 5 \\ 1.4: Summary / 6 \\ 2: Iterative solutions and other tools / 7 \\ 2.1: Polynomials and Taylor series / 7 \\ 2.2: First-order Taylor series approximation / 8 \\ 2.3: Second-order Taylor series approximation / 9 \\ 2.4: Another second-order Taylor series approximation / 9 \\ 2.5: Convergence of second-order methods / 10 \\ 2.6: Taylor series for elementary functions / 10 \\ 2.7: Continued fractions / 12 \\ 2.8: Summation of continued fractions / 17 \\ 2.9: Asymptotic expansions / 19 \\ 2.10: Series inversion / 20 \\ 2.11: Summary / 22 \\ 3: Polynomial approximations / 23 \\ 3.1: Computation of odd series / 23 \\ 3.2: Computation of even series / 25 \\ 3.3: Computation of general series / 25 \\ 3.4: Limitations of Cody\slash Waite polynomials / 28 \\ 3.5: Polynomial fits with Maple / 32 \\ 3.6: Polynomial fits with Mathematica / 33 \\ 3.7: Exact polynomial coefficients / 42 \\ 3.8: Cody\slash Waite rational polynomials / 43 \\ 3.9: Chebyshev polynomial economization / 43 \\ 3.10: Evaluating Chebyshev polynomials / 48 \\ 3.11: Error compensation in Chebyshev fits / 50 \\ 3.12: Improving Chebyshev fits / 51 \\ 3.13: Chebyshev fits in rational form / 52 \\ 3.14: Chebyshev fits with Mathematica / 56 \\ 3.15: Chebyshev fits for function representation / 57 \\ 3.16: Extending the library / 57 \\ 3.17: Summary and further reading / 58 \\ 4: Implementation issues / 61 \\ 4.1: Error magnification / 61 \\ 4.2: Machine representation and machine epsilon / 62 \\ 4.3: IEEE 754 arithmetic / 63 \\ 4.4: Evaluation order in C / 64 \\ 4.5: The {\tt volatile} type qualifier / 65 \\ 4.6: Rounding in floating-point arithmetic / 66 \\ 4.7: Signed zero / 69 \\ 4.8: Floating-point zero divide / 70 \\ 4.9: Floating-point overflow / 71 \\ 4.10: Integer overflow / 72 \\ 4.11: Floating-point underflow / 77 \\ 4.12: Subnormal numbers / 78 \\ 4.13: Floating-point inexact operation / 79 \\ 4.14: Floating-point invalid operation / 79 \\ 4.15: Remarks on NaN tests / 80 \\ 4.16: Ulps --- units in the last place / 81 \\ 4.17: Fused multiply-add / 85 \\ 4.18: Fused multiply-add and polynomials / 88 \\ 4.19: Significance loss / 89 \\ 4.20: Error handling and reporting / 89 \\ 4.21: Interpreting error codes / 93 \\ 4.22: C99 changes to error reporting / 94 \\ 4.23: Error reporting with threads / 95 \\ 4.24: Comments on error reporting / 95 \\ 4.25: Testing function implementations / 96 \\ 4.26: Extended data types on Hewlett--Packard HP-UX IA-64 / 100 \\ 4.27: Extensions for decimal arithmetic / 101 \\ 4.28: Further reading / 103 \\ 4.29: Summary / 104 \\ 5: The floating-point environment / 105 \\ 5.1: IEEE 754 and programming languages / 105 \\ 5.2: IEEE 754 and the mathcw library / 106 \\ 5.3: Exceptions and traps / 106 \\ 5.4: Access to exception flags and rounding control / 107 \\ 5.5: The environment access pragma / 110 \\ 5.6: Implementation of exception-flag and rounding-control access / 110 \\ 5.7: Using exception flags: simple cases / 112 \\ 5.8: Using rounding control / 115 \\ 5.9: Additional exception flag access / 116 \\ 5.10: Using exception flags: complex case / 120 \\ 5.11: Access to precision control / 123 \\ 5.12: Using precision control / 126 \\ 5.13: Summary / 127 \\ 6: Converting floating-point values to integers / 129 \\ 6.1: Integer conversion in programming languages / 129 \\ 6.2: Programming issues for conversions to integers / 130 \\ 6.3: Hardware out-of-range conversions / 131 \\ 6.4: Rounding modes and integer conversions / 132 \\ 6.5: Extracting integral and fractional parts / 132 \\ 6.6: Truncation functions / 135 \\ 6.7: Ceiling and floor functions / 136 \\ 6.8: Floating-point rounding functions with fixed rounding / 137 \\ 6.9: Floating-point rounding functions: current rounding / 138 \\ 6.10: Floating-point rounding functions without {\em inexact\/} exception / 139 \\ 6.11: Integer rounding functions with fixed rounding / 140 \\ 6.12: Integer rounding functions with current rounding / 142 \\ 6.13: Remainder / 143 \\ 6.14: Why the remainder functions are hard / 144 \\ 6.15: Computing {\tt fmod} / 146 \\ 6.16: Computing {\tt remainder} / 148 \\ 6.17: Computing {\tt remquo} / 150 \\ 6.18: Computing one remainder from the other / 152 \\ 6.19: Computing the remainder in nonbinary bases / 155 \\ 6.20: Summary / 156 \\ 7: Random numbers / 157 \\ 7.1: Guidelines for random-number software / 157 \\ 7.2: Creating generator seeds / 158 \\ 7.3: Random floating-point values / 160 \\ 7.4: Random integers from floating-point generator / 165 \\ 7.5: Random integers from an integer generator / 166 \\ 7.6: Random integers in ascending order / 168 \\ 7.7: How random numbers are generated / 169 \\ 7.8: Removing generator bias / 178 \\ 7.9: Improving a poor random number generator / 178 \\ 7.10: Why long periods matter / 179 \\ 7.11: Inversive congruential generators / 180 \\ 7.12: Inversive congruential generators, revisited / 189 \\ 7.13: Distributions of random numbers / 189 \\ 7.14: Other distributions / 195 \\ 7.15: Testing random-number generators / 196 \\ 7.16: Applications of random numbers / 202 \\ 7.17: The \textsf {mathcw} random number routines / 208 \\ 7.18: Summary, advice, and further reading / 214 \\ 8: Roots / 215 \\ 8.1: Square root / 215 \\ 8.2: Hypotenuse and vector norms / 222 \\ 8.3: Hypotenuse by iteration / 227 \\ 8.4: Reciprocal square root / 233 \\ 8.5: Cube root / 237 \\ 8.6: Roots in hardware / 240 \\ 8.7: Summary / 242 \\ 9: Argument reduction / 243 \\ 9.1: Simple argument reduction / 243 \\ 9.2: Exact argument reduction / 250 \\ 9.3: Implementing exact argument reduction / 253 \\ 9.4: Testing argument reduction / 265 \\ 9.5: Retrospective on argument reduction / 265 \\ 10: Exponential and logarithm / 267 \\ 10.1: Exponential functions / 267 \\ 10.2: Exponential near zero / 273 \\ 10.3: Logarithm functions / 282 \\ 10.4: Logarithm near one / 290 \\ 10.5: Exponential and logarithm in hardware / 292 \\ 10.6: Compound interest and annuities / 294 \\ 10.7: Summary / 298 \\ 11: Trigonometric functions / 299 \\ 11.1: Sine and cosine properties / 299 \\ 11.2: Tangent properties / 302 \\ 11.3: Argument conventions and units / 304 \\ 11.4: Computing the cosine and sine / 306 \\ 11.5: Computing the tangent / 310 \\ 11.6: Trigonometric functions in degrees / 313 \\ 11.7: Trigonometric functions in units of $ \pi $ / 315 \\ 11.8: Computing the cosine and sine together / 320 \\ 11.9: Inverse sine and cosine / 323 \\ 11.10: Inverse tangent / 331 \\ 11.11: Inverse tangent, take two / 336 \\ 11.12: Trigonometric functions in hardware / 338 \\ 11.13: Testing trigonometric functions / 339 \\ 11.14: Retrospective on trigonometric functions / 340 \\ 12: Hyperbolic functions / 341 \\ 12.1: Hyperbolic functions / 341 \\ 12.2: Improving the hyperbolic functions / 345 \\ 12.3: Computing the hyperbolic functions together / 348 \\ 12.4: Inverse hyperbolic functions / 348 \\ 12.5: Hyperbolic functions in hardware / 350 \\ 12.6: Summary / 352 \\ 13: Pair-precision arithmetic / 353 \\ 13.1: Limitations of pair-precision arithmetic / 354 \\ 13.2: Design of the pair-precision software interface / 355 \\ 13.3: Pair-precision initialization / 356 \\ 13.4: Pair-precision evaluation / 357 \\ 13.5: Pair-precision high part / 357 \\ 13.6: Pair-precision low part / 357 \\ 13.7: Pair-precision copy / 357 \\ 13.8: Pair-precision negation / 358 \\ 13.9: Pair-precision absolute value / 358 \\ 13.10: Pair-precision sum / 358 \\ 13.11: Splitting numbers into pair sums / 359 \\ 13.12: Premature overflow in splitting / 362 \\ 13.13: Pair-precision addition / 365 \\ 13.14: Pair-precision subtraction / 367 \\ 13.15: Pair-precision comparison / 368 \\ 13.16: Pair-precision multiplication / 368 \\ 13.17: Pair-precision division / 371 \\ 13.18: Pair-precision square root / 373 \\ 13.19: Pair-precision cube root / 377 \\ 13.20: Accuracy of pair-precision arithmetic / 379 \\ 13.21: Pair-precision vector sum / 384 \\ 13.22: Exact vector sums / 385 \\ 13.23: Pair-precision dot product / 385 \\ 13.24: Pair-precision product sum / 386 \\ 13.25: Pair-precision decimal arithmetic / 387 \\ 13.26: Fused multiply-add with pair precision / 388 \\ 13.27: Higher intermediate precision and the FMA / 393 \\ 13.28: Fused multiply-add without pair precision / 395 \\ 13.29: Fused multiply-add with multiple precision / 401 \\ 13.30: Fused multiply-add, Boldo/\penalty \exhyphenpenalty Melquiond style / 403 \\ 13.31: Error correction in fused multiply-add / 406 \\ 13.32: Retrospective on pair-precision arithmetic / 407 \\ 14: Power function / 411 \\ 14.1: Why the power function is hard to compute / 411 \\ 14.2: Special cases for the power function / 412 \\ 14.3: Integer powers / 414 \\ 14.4: Integer powers, revisited / 420 \\ 14.5: Outline of the power-function algorithm / 421 \\ 14.6: Finding $a$ and $p$ / 423 \\ 14.7: Table searching / 424 \\ 14.8: Computing $\log_n(g/a)$ / 426 \\ 14.9: Accuracy required for $\log_n(g/a)$ / 429 \\ 14.10: Exact products / 430 \\ 14.11: Computing $w$, $w_1$ and $w_2$ / 433 \\ 14.12: Computing $n^{w_2}$ / 437 \\ 14.13: The choice of $q$ / 438 \\ 14.14: Testing the power function / 438 \\ 14.15: Retrospective on the power function / 440 \\ 15: Complex arithmetic primitives / 441 \\ 15.1: Support macros and type definitions / 442 \\ 15.2: Complex absolute value / 443 \\ 15.3: Complex addition / 445 \\ 15.4: Complex argument / 445 \\ 15.5: Complex conjugate / 446 \\ 15.6: Complex conjugation symmetry / 446 \\ 15.7: Complex conversion / 448 \\ 15.8: Complex copy / 448 \\ 15.9: Complex division: C99 style / 449 \\ 15.10: Complex division: Smith style / 451 \\ 15.11: Complex division: Stewart style / 452 \\ 15.12: Complex division: Priest style / 453 \\ 15.13: Complex division: avoiding subtraction loss / 455 \\ 15.14: Complex imaginary part / 456 \\ 15.15: Complex multiplication / 456 \\ 15.16: Complex multiplication: error analysis / 458 \\ 15.17: Complex negation / 459 \\ 15.18: Complex projection / 460 \\ 15.19: Complex real part / 460 \\ 15.20: Complex subtraction / 461 \\ 15.21: Complex infinity test / 462 \\ 15.22: Complex NaN test / 462 \\ 15.23: Summary / 463 \\ 16: Quadratic equations / 465 \\ 16.1: Solving quadratic equations / 465 \\ 16.2: Root sensitivity / 471 \\ 16.3: Testing a quadratic-equation solver / 472 \\ 16.4: Summary / 474 \\ 17: Elementary functions in complex arithmetic / 475 \\ 17.1: Research on complex elementary functions / 475 \\ 17.2: Principal values / 476 \\ 17.3: Branch cuts / 476 \\ 17.4: Software problems with negative zeros / 478 \\ 17.5: Complex elementary function tree / 479 \\ 17.6: Series for complex functions / 479 \\ 17.7: Complex square root / 480 \\ 17.8: Complex cube root / 485 \\ 17.9: Complex exponential / 487 \\ 17.10: Complex exponential near zero / 492 \\ 17.11: Complex logarithm / 495 \\ 17.12: Complex logarithm near one / 497 \\ 17.13: Complex power / 500 \\ 17.14: Complex trigonometric functions / 502 \\ 17.15: Complex inverse trigonometric functions / 504 \\ 17.16: Complex hyperbolic functions / 509 \\ 17.17: Complex inverse hyperbolic functions / 514 \\ 17.18: Summary / 520 \\ 18: The Greek functions: gamma, psi, and zeta / 521 \\ 18.1: Gamma and log-gamma functions / 521 \\ 18.2: The {\tt psi} and {\tt psiln} functions / 536 \\ 18.3: Polygamma functions / 547 \\ 18.4: Incomplete gamma functions / 560 \\ 18.5: A Swiss diversion: Bernoulli and Euler / 568 \\ 18.6: An Italian excursion: Fibonacci numbers / 575 \\ 18.7: A German gem: the Riemann zeta function / 579 \\ 18.8: Further reading / 590 \\ 18.9: Summary / 591 \\ 19: Error and probability functions / 593 \\ 19.1: Error functions / 593 \\ 19.2: Scaled complementary error function / 598 \\ 19.3: Inverse error functions / 600 \\ 19.4: Normal distribution functions and inverses / 610 \\ 19.5: Summary / 617 \\ 20: Elliptic integral functions / 619 \\ 20.1: The arithmetic-geometric mean / 619 \\ 20.2: Elliptic integral functions of the first kind / 624 \\ 20.3: Elliptic integral functions of the second kind / 627 \\ 20.4: Elliptic integral functions of the third kind / 630 \\ 20.5: Computing $K(m)$ and $K'(m)$ / 631 \\ 20.6: Computing $E(m)$ and $E'(m)$ / 637 \\ 20.7: Historical algorithms for elliptic integrals / 643 \\ 20.8: Auxiliary functions for elliptic integrals / 645 \\ 20.9: Computing the elliptic auxiliary functions / 648 \\ 20.10: Historical elliptic functions / 650 \\ 20.11: Elliptic functions in software / 652 \\ 20.12: Applications of elliptic auxiliary functions / 653 \\ 20.13: Elementary functions from elliptic auxiliary functions / 654 \\ 20.14: Computing elementary functions via $R_C(x,y)$ / 655 \\ 20.15: Jacobian elliptic functions / 657 \\ 20.16: Inverses of Jacobian elliptic functions / 664 \\ 20.17: The modulus and the nome / 668 \\ 20.18: Jacobian theta functions / 673 \\ 20.19: Logarithmic derivatives of the Jacobian theta functions / 675 \\ 20.20: Neville theta functions / 678 \\ 20.21: Jacobian Eta, Theta, and Zeta functions / 679 \\ 20.22: Weierstrass elliptic functions / 682 \\ 20.23: Weierstrass functions by duplication / 689 \\ 20.24: Complete elliptic functions, revisited / 690 \\ 20.25: Summary / 691 \\ 21: Bessel functions / 693 \\ 21.1: Cylindrical Bessel functions / 694 \\ 21.2: Behavior of $J_n(x)$ and $Y_n(x)$ / 695 \\ 21.3: Properties of $J_n(z)$ and $Y_n(z)$ / 697 \\ 21.4: Experiments with recurrences for $J_0(x)$ / 705 \\ 21.5: Computing $J_0(x)$ and $J_1(x)$ / 707 \\ 21.6: Computing $J_n(x)$ / 710 \\ 21.7: Computing $Y_0(x)$ and $Y_1(x)$ / 713 \\ 21.8: Computing $Y_n(x)$ / 715 \\ 21.9: Improving Bessel code near zeros / 716 \\ 21.10: Properties of $I_n(z)$ and $K_n(z)$ / 718 \\ 21.11: Computing $I_0(x)$ and $I_1(x)$ / 724 \\ 21.12: Computing $K_0(x)$ and $K_1(x)$ / 726 \\ 21.13: Computing $I_n(x)$ and $K_n(x)$ / 728 \\ 21.14: Properties of spherical Bessel functions / 731 \\ 21.15: Computing $j_n(x)$ and $y_n(x)$ / 735 \\ 21.16: Improving $j_1(x)$ and $y_1(x)$ / 740 \\ 21.17: Modified spherical Bessel functions / 743 \\ 21.18: Software for Bessel-function sequences / 755 \\ 21.19: Retrospective on Bessel functions / 761 \\ 22: Testing the library / 763 \\ 22.1: Testing {\tt tgamma} and {\tt lgamma} / 765 \\ 22.2: Testing {\tt psi} and {\tt psiln} / 768 \\ 22.3: Testing {\tt erf} and {\tt erfc} / 768 \\ 22.4: Testing cylindrical Bessel functions / 769 \\ 22.5: Testing exponent/\penalty \exhyphenpenalty significand manipulation / 769 \\ 22.6: Testing inline assembly code / 769 \\ 22.7: Testing with Maple / 770 \\ 22.8: Testing floating-point arithmetic / 773 \\ 22.9: The Berkeley Elementary Functions Test Suite / 774 \\ 22.10: The AT\&T floating-point test package / 775 \\ 22.11: The Antwerp test suite / 776 \\ 22.12: Summary / 776 \\ 23: Pair-precision elementary functions / 777 \\ 23.1: Pair-precision integer power / 777 \\ 23.2: Pair-precision machine epsilon / 779 \\ 23.3: Pair-precision exponential / 780 \\ 23.4: Pair-precision logarithm / 787 \\ 23.5: Pair-precision logarithm near one / 793 \\ 23.6: Pair-precision exponential near zero / 793 \\ 23.7: Pair-precision base-$n$ exponentials / 795 \\ 23.8: Pair-precision trigonometric functions / 796 \\ 23.9: Pair-precision inverse trigonometric functions / 801 \\ 23.10: Pair-precision hyperbolic functions / 804 \\ 23.11: Pair-precision inverse hyperbolic functions / 808 \\ 23.12: Summary / 808 \\ 24: Accuracy of the Cody\slash Waite algorithms / 811 \\ 25: Improving upon the Cody\slash Waite algorithms / 823 \\ 25.1: The Bell Labs libraries / 823 \\ 25.2: The {Cephes} library / 823 \\ 25.3: The {Sun} libraries / 824 \\ 25.4: Mathematical functions on EPIC / 824 \\ 25.5: The GNU libraries / 825 \\ 25.6: The French libraries / 825 \\ 25.7: The NIST effort / 826 \\ 25.8: Commercial mathematical libraries / 826 \\ 25.9: Mathematical libraries for decimal arithmetic / 826 \\ 25.10: Mathematical library research publications / 826 \\ 25.11: Books on computing mathematical functions / 827 \\ 25.12: Summary / 828 \\ 26: Floating-point output / 829 \\ 26.1: Output character string design issues / 830 \\ 26.2: Exact output conversion / 831 \\ 26.3: Hexadecimal floating-point output / 832 \\ 26.4: Octal floating-point output / 850 \\ 26.5: Binary floating-point output / 851 \\ 26.6: Decimal floating-point output / 851 \\ 26.7: Accuracy of output conversion / 865 \\ 26.8: Output conversion to a general base / 865 \\ 26.9: Output conversion of Infinity / 866 \\ 26.10: Output conversion of NaN / 866 \\ 26.11: Number-to-string conversion / 867 \\ 26.12: The {\tt printf} family / 867 \\ 26.13: Summary / 878 \\ 27: Floating-point input / 879 \\ 27.1: Binary floating-point input / 879 \\ 27.2: Octal floating-point input / 894 \\ 27.3: Hexadecimal floating-point input / 895 \\ 27.4: Decimal floating-point input / 895 \\ 27.5: Based-number input / 899 \\ 27.6: General floating-point input / 900 \\ 27.7: The {\tt scanf} family / 901 \\ 27.8: Summary / 910 \\ A: Ada interface / 911 \\ A.1: Building the Ada interface / 911 \\ A.2: Programming the Ada interface / 912 \\ A.3: Using the Ada interface / 915 \\ B: C\# interface / 917 \\ B.1: C\# on the CLI virtual machine / 917 \\ B.2: Building the C\# interface / 918 \\ B.3: Programming the C\# interface / 920 \\ B.4: Using the C\# interface / 922 \\ C: C++ interface / 923 \\ C.1: Building the C++ interface / 923 \\ C.2: Programming the C++ interface / 924 \\ C.3: Using the C++ interface / 925 \\ D: Decimal arithmetic / 927 \\ D.1: Why we need decimal floating-point arithmetic / 927 \\ D.2: Decimal floating-point arithmetic design issues / 928 \\ D.3: How decimal and binary arithmetic differ / 931 \\ D.4: Initialization of decimal floating-point storage / 935 \\ D.5: The {\tt } header file / 936 \\ D.6: Rounding in decimal arithmetic / 936 \\ D.7: Exact scaling in decimal arithmetic / 937 \\ E: Errata in the Cody\slash Waite book / 939 \\ F: Fortran interface / 941 \\ F.1: Building the Fortran interface / 943 \\ F.2: Programming the Fortran interface / 944 \\ F.3: Using the Fortran interface / 945 \\ H: Historical floating-point architectures / 947 \\ H.1: CDC family / 949 \\ H.2: Cray family / 952 \\ H.3: DEC PDP-10 / 953 \\ H.4: DEC PDP-11 and VAX / 956 \\ H.5: General Electric 600 series / 958 \\ H.6: IBM family / 959 \\ H.7: Lawrence Livermore S-1 Mark IIA / 965 \\ H.8: Unusual floating-point systems / 966 \\ H.9: Historical retrospective / 967 \\ I: Integer arithmetic / 969 \\ I.1: Memory addressing and integers / 971 \\ I.2: Representations of signed integers / 971 \\ I.3: Parity testing / 975 \\ I.4: Sign testing / 975 \\ I.5: Arithmetic exceptions / 975 \\ I.6: Notations for binary numbers / 977 \\ I.7: Summary / 978 \\ J: Java interface / 979 \\ J.1: Building the Java interface / 979 \\ J.2: Programming the Java MathCW class / 980 \\ J.3: Programming the Java C interface / 982 \\ J.4: Using the Java interface / 985 \\ L: Letter notation / 987 \\ P: Pascal interface / 989 \\ P.1: Building the Pascal interface / 989 \\ P.2: Programming the Pascal MathCW module / 990 \\ P.3: Using the Pascal module interface / 993 \\ P.4: Pascal and numeric programming / 994 \\ Bibliography / 995 \\ Author/editor index / 1039 \\ Function and macro index / 1049 \\ Subject index / 1065 \\ Colophon / 1115", } @TechReport{Brent:2017:JBP, author = "Richard P. Brent", title = "{Jonathan Borwein}, Pi and the {AGM}", type = "Talk slides", institution = "Australian National University and CARMA, University of Newcastle", address = "Canberra, ACT and Newcastle, NSW, Australia", pages = "76", day = "26", month = sep, year = "2017", bibdate = "Fri Sep 04 17:08:54 2020", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib; https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "https://carma.newcastle.edu.au/meetings/jbcc/abstracts/pdf/JBCC-Richard_Brent.pdf", abstract = "We consider some of Jon Borwein's contributions to the high-precision computation of $ \pi $ and the elementary functions, with particular reference to the fascinating book \booktitle{Pi and the AGM}(Wiley, 1987) by Jon and his brother Peter Borwein. Here ``AGM'' is the arithmetic-geometric mean, first studied by Euler, Gauss and Legendre. Because the AGM has second-order convergence, it can be combined with fast multiplication algorithms to give fast algorithms for the $n$-bit computation of $ \pi $, and more generally the elementary functions. These algorithms run in ``almost linear' time $ O(M(n) \log n)$, where $ M(n)$ is the time for $n$-bit multiplication. The talk will survey some of the results and algorithms, from the time of Archimedes to the present day, that were of interest to Jon. In several cases they were discovered or improved by him", acknowledgement = ack-nhfb, ORCID-numbers = "Borwein, Jonathan/0000-0002-1263-0646", subject-dates = "Jonathan Michael Borwein (20 May 1951--2 August 2016)", } @Article{Chen:2017:UTS, author = "Chao-Ping Chen and Junesang Choi", title = "Unified treatment of several asymptotic expansions concerning some mathematical constants", journal = j-APPL-MATH-COMP, volume = "305", number = "??", pages = "348--363", day = "15", month = jul, year = "2017", CODEN = "AMHCBQ", DOI = "https://doi.org/10.1016/j.amc.2017.02.001", ISSN = "0096-3003 (print), 1873-5649 (electronic)", ISSN-L = "0096-3003", bibdate = "Sun Mar 12 13:31:57 MDT 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/applmathcomput2015.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0096300317300978", acknowledgement = ack-nhfb, fjournal = "Applied Mathematics and Computation", journal-URL = "http://www.sciencedirect.com/science/journal/00963003", keywords = "Asymptotic expansion; Choi--Srivastava constants; constants of Landau and Lebesgue; Euler--Mascheroni constant; Glaisher--Kinkelin constant; psi function (logarithmic derivative of gamma function)", } @Article{Gil:2017:ECL, author = "Amparo Gil and Javier Segura and Nico M. Temme", title = "Efficient computation of {Laguerre} polynomials", journal = j-COMP-PHYS-COMM, volume = "210", number = "??", pages = "124--131", month = jan, year = "2017", CODEN = "CPHCBZ", DOI = "https://doi.org/10.17632/3jkk659cn8.1", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", bibdate = "Thu Dec 1 14:31:09 MST 2016", bibsource = "https://www.math.utah.edu/pub/tex/bib/compphyscomm2010.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0010465516302727", acknowledgement = ack-nhfb, fjournal = "Computer Physics Communications", journal-URL = "http://www.sciencedirect.com/science/journal/00104655/", } @Article{Horsley:2017:BPF, author = "David E. Horsley", title = "{Bessel} phase functions: calculation and application", journal = j-NUM-MATH, volume = "136", number = "3", pages = "679--702", month = jul, year = "2017", CODEN = "NUMMA7", ISSN = "0029-599X (print), 0945-3245 (electronic)", ISSN-L = "0029-599X", bibdate = "Wed Jun 7 17:52:44 MDT 2017", bibsource = "http://link.springer.com/journal/211/136/3; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/nummath2010.bib", acknowledgement = ack-nhfb, fjournal = "Numerische Mathematik", journal-URL = "http://link.springer.com/journal/211", } @InProceedings{Istoan:2017:FFP, author = "M. Istoan and B. Pasca", title = "Flexible Fixed-Point Function Generation for {FPGAs}", crossref = "Burgess:2017:ISC", pages = "123--130", month = jul, year = "2017", DOI = "https://doi.org/10.1109/ARITH.2017.31", ISSN = "1063-6889", bibdate = "Fri Nov 17 09:10:14 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", abstract = "Efficient fixed-point function implementation is critical in many FPGA application domains including convolutional neural networks, computer vision, and communication systems. In this work we focus on functions of the form $ x^p $, with $ p \in \{ - 1, - 1 / 2, 1 / 2 \} $ as part of a function generator targeting FPGAs. The generator implements architectures based on new but also existing algorithms. In this work we present three distinct methods implemented in this generator that outperform state-of-the-art implementations for certain configurations. Traditionally, fixed-point function implementation requires a normalization stage, compute and denormalization (reconstruction) of the result. The first proposed method implements the function holistically, thus saving the logic and latency required during the normalize and reconstruct stages. The second proposed method is based on a novel second order Taylor implementation. The third method is based on the cubic convergence of Halley's method, which is novel in this context. The proposed methods are compared and contrasted against state-of-the art implementations in the context of FPGA targets.", acknowledgement = ack-nhfb, keywords = "arithmetic; communication systems; computer vision; convolutional neural networks; cubic convergence; Digital signal processing; Field programmable gate arrays; field programmable gate arrays; fixed point arithmetic; fixed-point; flexible fixed-point function generation; FPGA; FPGAs; generator; Generators; Halley method; Kernel; Memory management; reciprocal; reciprocal sqrt; second order Taylor implementation; Signal generators; sqrt", } @InProceedings{Jeannerod:2017:REC, author = "Claude-Pierre Jeannerod and Jean-Michel Muller", editor = "Michael B. Matthews", booktitle = "{2017 51st Asilomar Conference on Signals, Systems, and Computers. October 29--November 1, 2017. Pacific Grove, California}", title = "On the relative error of computing complex square roots in floating-point arithmetic", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "737--740", year = "2017", DOI = "https://doi.org/10.1109/ACSSC.2017.8335442", ISBN = "1-5386-1823-0", ISBN-13 = "978-1-5386-1823-3", bibdate = "Fri Sep 29 10:59:32 2023", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", abstract = "We study the accuracy of a classical approach to computing complex square-roots in floating-point arithmetic. Our analyses are done in binary floating-point arithmetic in precision p, and we assume that the (real) arithmetic operations $+$, $-$, $ \times $, $ \div $, $ \sqrt {} $ are rounded to nearest, so the unit roundoff is $ u = 2^{-p} $. We show that in the absence of underflow and overflow, the componentwise and normwise relative errors of this approach are at most $ 7 / 2 u $ and $ \sqrt {37} / 2 u $, respectively, and this without having to neglect terms of higher order in $u$. We then provide some input examples showing that these bounds are reasonably sharp for the three basic binary interchange formats (binary32, binary64, and binary128) of the IEEE 754 standard for floating-point arithmetic.", acknowledgement = ack-nhfb, } @Article{Jeffrey:2017:BSI, author = "David J. Jeffrey", title = "Branch Structure and Implementation of {Lambert} {$W$}", journal = j-MATH-COMPUT-SCI, volume = "11", number = "3--4", pages = "341--350", month = dec, year = "2017", CODEN = "????", DOI = "https://doi.org/10.1007/s11786-017-0320-6", ISSN = "1661-8270 (print), 1661-8289 (electronic)", ISSN-L = "1661-8270", bibdate = "Mon Oct 2 10:24:36 MDT 2017", bibsource = "http://link.springer.com/journal/11786/11/3; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/math-comput-sci.bib", acknowledgement = ack-nhfb, fjournal = "Mathematics in Computer Science", journal-URL = "http://link.springer.com/journal/11786", } @InProceedings{Langhammer:2017:FPT, author = "M. Langhammer and B. Pasca", title = "Floating Point Tangent Implementation for {FPGAs}", crossref = "Burgess:2017:ISC", pages = "64--65", month = jul, year = "2017", DOI = "https://doi.org/10.1109/ARITH.2017.25", ISSN = "1063-6889", bibdate = "Fri Nov 17 09:10:14 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", abstract = "This paper presents an implementation of the floating-point (FP) tangent function, optimized for an FPGA containing hard floating point (HFP) DSP Blocks. This function inputs values in the interval [- /2, /2], uses the IEEE-754 single-precision (SP) format, and has an accuracy conforming to OpenCL requirements. The presented architecture is based on a combination of mathematical identities and properties of the tangent function in FP. The resultant design outperforms generic polynomial approximation methods targeting the same resource utilization spectrum, and provides better resource trade-offs than classical CORDIC-based implementations. The presented work is widely available as part of the Intel DSP Builder Advanced Blockset.", acknowledgement = ack-nhfb, keywords = "Approximation methods; classical CORDIC-based implementations; Digital arithmetic; Digital signal processing; digital signal processing chips; field programmable gate arrays; Field programmable gate arrays; fixed point arithmetic; floating point arithmetic; floating point tangent function; FPGAs; generic polynomial approximation methods; hard floating point DSP blocks; HFP DSP; IEEE-754 single-precision format; Intel DSP Builder Advanced Blockset; OpenCL; reconfigurable architectures; Resource management; resource utilization spectrum; Table lookup", } @Article{Langhammer:2017:SPL, author = "Martin Langhammer and Bogdan Pasca", title = "Single Precision Logarithm and Exponential Architectures for Hard Floating-Point Enabled {FPGAs}", journal = j-IEEE-TRANS-COMPUT, volume = "66", number = "12", pages = "2031--2043", month = "????", year = "2017", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/TC.2017.2703923", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Fri Nov 10 08:32:25 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput2010.bib", URL = "https://ieeexplore.ieee.org/document/7927449/", acknowledgement = ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", } @Article{Matic:2017:PBA, author = "Ivan Mati{\'c} and Rado{\v{s}} Radoi{\v{c}}i{\'c} and Dan Stefanica", title = "{P{\'o}lya}-based approximation for the {ATM}-forward implied volatility", journal = "International Journal of Financial Engineering", volume = "4", number = "2--3", pages = "1--15", month = jun # "\slash " # sep, year = "2017", DOI = "https://doi.org/10.1142/S2424786317500323", ISSN = "2424-7863", bibdate = "Sat Dec 16 17:12:10 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.worldscientific.com/doi/abs/10.1142/S2424786317500323", acknowledgement = ack-nhfb, ajournal = "Int. J. Finan. Eng.", journal-URL = "http://www.worldscientific.com/worldscinet/ijfe", } @Article{Mopuri:2017:LCM, author = "Suresh Mopuri and Amit Acharyya", title = "Low-Complexity Methodology for Complex Square-Root Computation", journal = j-IEEE-TRANS-VLSI-SYST, volume = "25", number = "11", pages = "3255--3259", year = "2017", CODEN = "IEVSE9", DOI = "https://doi.org/10.1109/TVLSI.2017.2740343", ISSN = "1063-8210 (print), 1557-9999 (electronic)", ISSN-L = "1063-8210", bibdate = "Mon Oct 27 10:32:44 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "IEEE Transactions on Very Large Scale Integration (VLSI) Systems", journal-URL = "https://ieeexplore.ieee.org/xpl/issues?punumber=92", keywords = "Complex square root; Complexity theory; Computer architecture; coordinate rotation digital computer (CORDIC); Field programmable gate arrays; Hardware; Logic gates; square root; Transistors; Very large scale integration", } @Article{Pearson:2017:NMC, author = "John W. Pearson and Sheehan Olver and Mason A. Porter", title = "Numerical methods for the computation of the confluent and {Gauss} hypergeometric functions", journal = j-NUMER-ALGORITHMS, volume = "74", number = "3", pages = "821--866", month = mar, year = "2017", CODEN = "NUALEG", DOI = "https://doi.org/10.1007/s11075-016-0173-0", ISSN = "1017-1398 (print), 1572-9265 (electronic)", ISSN-L = "1017-1398", bibdate = "Wed Mar 1 09:12:15 MST 2017", bibsource = "http://link.springer.com/journal/11075/74/3; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/numeralgorithms.bib", URL = "http://link.springer.com/article/10.1007/s11075-016-0173-0", acknowledgement = ack-nhfb, fjournal = "Numerical Algorithms", journal-URL = "http://link.springer.com/journal/11075", } @InProceedings{Saha:2017:AEA, author = "Anurup Saha and K. Gaurav Kumar and Archisman Ghosh and Mrinal Kanti Naskar", booktitle = "{2017 International Conference on Circuits, Controls, and Communications (CCUBE)}", title = "Area efficient architecture of Hyperbolic functions for high frequency applications", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "139--142", year = "2017", DOI = "https://doi.org/10.1109/CCUBE.2017.8394139", bibdate = "Mon Oct 27 10:32:44 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, keywords = "CORDIC; Field programmable gate arrays; FPGA; Hardware; Hardware design languages; High-frequency architecture; hyperbolic functions; Signal processing algorithms; Table lookup; Unified Architecture", } @Article{Saint-Genies:2017:ELT, author = "Hugues de Lassus Saint-Geni{\`e}s and David Defour and Guillaume Revy", title = "Exact Lookup Tables for the Evaluation of Trigonometric and Hyperbolic Functions", journal = j-IEEE-TRANS-COMPUT, volume = "66", number = "12", pages = "2058--2071", month = "????", year = "2017", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/TC.2017.2703870", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Fri Nov 10 08:32:25 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput2010.bib", URL = "https://ieeexplore.ieee.org/document/7927421/", acknowledgement = ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", } @Article{Shi:2017:RIF, author = "X. W. Shi and C. Wang and C. L. Zhang", title = "Research and Implementation of Floating-Point Exponential Function Algorithm Based on {FPGA}", journal = "Journal of Computer Measurement and Control", volume = "10", number = "??", pages = "226--228", month = "????", year = "2017", DOI = "", bibdate = "Tue Nov 11 20:15:53 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "", acknowledgement = ack-nhfb, } @Article{Staunton:2017:PP, author = "Mike Staunton", title = "Power to {P{\'o}lya}", journal = "Wilmott Magazine", volume = "90", pages = "36--37", month = jul, year = "2017", DOI = "https://doi.org/10.1002/wilm.10605", bibdate = "Sat Dec 16 17:41:48 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://onlinelibrary.wiley.com/doi/10.1002/wilm.10605/full", acknowledgement = ack-nhfb, journal-URL = "http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)1541-8286; https://www.wilmott.com/category/magazine/", remark = "No issues online at Wiley before year 2011, or at Wilmott before 2006.", } @Article{Tihanyi:2017:CEL, author = "Norbert Tihanyi and Attila Kov{\'a}cs and J{\'o}zsef Kov{\'a}cs", title = "Computing Extremely Large Values of the {Riemann} Zeta Function", journal = j-J-GRID-COMP, volume = "15", number = "4", pages = "527--534", month = dec, year = "2017", CODEN = "????", DOI = "https://doi.org/10.1007/s10723-017-9416-0", ISSN = "1570-7873 (print), 1572-9184 (electronic)", ISSN-L = "1570-7873", bibdate = "Sat Jan 6 08:41:37 MST 2018", bibsource = "http://link.springer.com/journal/10723/15/4; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jgridcomp.bib", URL = "https://link.springer.com/article/10.1007/s10723-017-9416-0", acknowledgement = ack-nhfb, fjournal = "Journal of Grid Computing", journal-URL = "http://link.springer.com/journal/10723", } @Article{Xu:2017:AEP, author = "Aimin Xu and Zhongdi Cen", title = "Asymptotic expansions for the psi function and the {Euler--Mascheroni} constant", journal = j-J-NUMBER-THEORY, volume = "180", number = "??", pages = "360--372", month = nov, year = "2017", CODEN = "JNUTA9", DOI = "https://doi.org/10.1016/j.jnt.2017.04.014", ISSN = "0022-314X (print), 1096-1658 (electronic)", ISSN-L = "0022-314X", bibdate = "Wed Jul 15 08:49:31 MDT 2020", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jnumbertheory2010.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0022314X17302007", acknowledgement = ack-nhfb, ajournal = "J. Number Theory", fjournal = "Journal of Number Theory", journal-URL = "http://www.sciencedirect.com/science/journal/0022314X", } @Article{Ye:2017:SDP, author = "Liangjie Ye", title = "A symbolic decision procedure for relations arising among {Taylor} coefficients of classical {Jacobi} theta functions", journal = j-J-SYMBOLIC-COMP, volume = "82", number = "??", pages = "134--163", month = sep # "\slash " # oct, year = "2017", CODEN = "JSYCEH", DOI = "https://doi.org/10.1016/j.jsc.2017.01.005", ISSN = "0747-7171 (print), 1095-855X (electronic)", ISSN-L = "0747-7171", bibdate = "Fri Feb 17 12:14:20 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jsymcomp.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0747717117300135", acknowledgement = ack-nhfb, fjournal = "Journal of Symbolic Computation", journal-URL = "http://www.sciencedirect.com/science/journal/07477171/", } @Article{Zaghloul:2017:ASE, author = "Mofreh R. Zaghloul", title = "Algorithm 985: Simple, Efficient, and Relatively Accurate Approximation for the Evaluation of the {Faddeyeva} Function", journal = j-TOMS, volume = "44", number = "2", pages = "22:1--22:9", month = sep, year = "2017", CODEN = "ACMSCU", DOI = "https://doi.org/10.1145/3119904", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Tue Sep 19 17:19:59 MDT 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib", URL = "http://dl.acm.org/citation.cfm?id=3119904", abstract = "We present a new simple algorithm for efficient, and relatively accurate computation of the Faddeyeva function $ w(z) $. The algorithm carefully exploits previous approximations by Hui et al. (1978) and Huml{\'\i}cek (1982) along with asymptotic expressions from Laplace continued fractions. Over a wide and fine grid of the complex argument, $ z = x + i y $, numerical results from the present approximation show a maximum relative error less than $ 4.0 \times 10^{-5} $ for both real and imaginary parts of $w$ while running in a relatively shorter execution time than other competitive techniques. In addition to the calculation of the Faddeyeva function, $w$, partial derivatives of the real and imaginary parts of the function can easily be calculated and returned as optional output.", acknowledgement = ack-nhfb, articleno = "22", fjournal = "ACM Transactions on Mathematical Software", journal-URL = "https://dl.acm.org/loi/toms", } @Article{Abrarov:2018:RAD, author = "Sanjar M. Abrarov and Brendan M. Quine", title = "A rational approximation of the {Dawson}'s integral for efficient computation of the complex error function", journal = j-APPL-MATH-COMP, volume = "321", number = "??", pages = "526--543", day = "15", month = mar, year = "2018", CODEN = "AMHCBQ", ISSN = "0096-3003 (print), 1873-5649 (electronic)", ISSN-L = "0096-3003", bibdate = "Sat Dec 9 07:21:49 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/applmathcomput2015.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0096300317307312", acknowledgement = ack-nhfb, fjournal = "Applied Mathematics and Computation", journal-URL = "http://www.sciencedirect.com/science/journal/00963003", } @Article{Bober:2018:NCR, author = "Jonathan W. Bober and Ghaith A. Hiary", title = "New Computations of the {Riemann} Zeta Function on the Critical Line", journal = j-EXP-MATH, volume = "27", number = "2", pages = "125--137", year = "2018", CODEN = "????", DOI = "https://doi.org/10.1080/10586458.2016.1233083", ISSN = "1058-6458 (print), 1944-950X (electronic)", ISSN-L = "1058-6458", bibdate = "Thu Sep 27 18:22:33 MDT 2018", bibsource = "http://www.tandfonline.com/toc/uexm20/27/2; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/expmath.bib", URL = "http://www.tandfonline.com/doi/full/10.1080/10586458.2016.1233083", acknowledgement = ack-nhfb, fjournal = "Experimental Mathematics", journal-URL = "http://www.tandfonline.com/loi/uexm20", onlinedate = "14 Oct 2016", } @Article{Borwein:2018:GFM, author = "Jonathan M. Borwein and Robert M. Corless", title = "Gamma and Factorial in the {{\booktitle{Monthly}}}", journal = j-AMER-MATH-MONTHLY, volume = "125", number = "5", pages = "400--424", month = may, year = "2018", CODEN = "AMMYAE", DOI = "https://doi.org/10.1080/00029890.2018.1420983", ISSN = "0002-9890 (print), 1930-0972 (electronic)", ISSN-L = "0002-9890", MRclass = "33B15", MRnumber = "3785875", bibdate = "Tue Apr 17 09:02:26 2018", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", abstract = "Since its inception in 1894, the Monthly has printed 50 articles on the $ \Gamma $ function or Stirling's asymptotic formula, including the magisterial 1959 paper by Phillip J. Davis, which won the 1963 Chauvenet prize, and the eye-opening 2000 paper by the Fields medalist Manjul Bhargava. In this article, we look back and comment on what has been said, and why, and try to guess what will be said about the $ \Gamma $ function in future Monthly issues.1 We also identify some gaps, which surprised us: phase plots, Riemann surfaces, and the functional inverse of $ \Gamma $ make their first appearance in the Monthly here. We also give a new elementary treatment of the asymptotics of $ n! $ and the first few terms of a new asymptotic formula for inv$ \Gamma $.", acknowledgement = ack-nhfb, author-dates = "Jonathan Michael Borwein (20 May 1951--2 August 2016)", fjournal = "American Mathematical Monthly", journal-URL = "http://www.jstor.org/journals/00029890.html", ORCID-numbers = "Borwein, Jonathan/0000-0002-1263-0646", } @Article{Braumann:2018:RGF, author = "C. A. Braumann and J.-C. Cort{\'e}s and L. J{\'o}dar and L. Villafuerte", title = "On the random gamma function: Theory and computing", journal = j-J-COMPUT-APPL-MATH, volume = "335", number = "??", pages = "142--155", month = jun, year = "2018", CODEN = "JCAMDI", DOI = "https://doi.org/10.1016/j.cam.2017.11.045", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Tue Mar 6 07:50:18 MST 2018", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2015.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0377042717306064", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @Article{Bremer:2018:ANE, author = "James Bremer", title = "An algorithm for the numerical evaluation of the associated {Legendre} functions that runs in time independent of degree and order", journal = j-J-COMPUT-PHYS, volume = "360", number = "??", pages = "15--38", day = "1", month = may, year = "2018", CODEN = "JCTPAH", DOI = "https://doi.org/10.1016/j.jcp.2018.01.014", ISSN = "0021-9991 (print), 1090-2716 (electronic)", ISSN-L = "0021-9991", bibdate = "Thu Mar 15 15:42:48 MDT 2018", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputphys2015.bib", URL = "http://www.sciencedirect.com/science/article/pii/S002199911830024X", acknowledgement = ack-nhfb, fjournal = "Journal of Computational Physics", journal-URL = "http://www.sciencedirect.com/science/journal/00219991", } @Article{Ceretani:2018:AME, author = "Andrea N. Ceretani and Natalia N. Salva and Domingo A. Tarzia", title = "Approximation of the modified error function", journal = j-APPL-MATH-COMP, volume = "337", number = "??", pages = "591--606", day = "15", month = nov, year = "2018", CODEN = "AMHCBQ", DOI = "https://doi.org/10.1016/j.amc.2018.05.054", ISSN = "0096-3003 (print), 1873-5649 (electronic)", ISSN-L = "0096-3003", bibdate = "Fri Sep 14 08:14:13 MDT 2018", bibsource = "https://www.math.utah.edu/pub/tex/bib/applmathcomput2015.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0096300318304715", acknowledgement = ack-nhfb, fjournal = "Applied Mathematics and Computation", journal-URL = "http://www.sciencedirect.com/science/journal/00963003", } @Article{Chen:2018:NEH, author = "Ruyun Chen and Gang Yang", title = "Numerical evaluation of highly oscillatory {Bessel} transforms", journal = j-J-COMPUT-APPL-MATH, volume = "342", number = "??", pages = "16--24", month = nov, year = "2018", CODEN = "JCAMDI", DOI = "https://doi.org/10.1016/j.cam.2018.03.026", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Fri Aug 10 18:10:42 MDT 2018", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2015.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0377042718301894", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @InProceedings{De:2018:MLH, author = "Debaprasad De and Archisman Ghosh and K. Gaurav Kumar and Anurup Saha and Mrinal Kanti Naskar", booktitle = "{2018 IEEE Applied Signal Processing Conference (ASPCON)}", title = "Multiplier-less Hardware Realization of Trigonometric Functions for High Speed Applications", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "149--152", year = "2018", DOI = "https://doi.org/10.1109/ASPCON.2018.8748709", bibdate = "Mon Oct 27 10:32:44 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, keywords = "Conferences; CORDIC; Field programmable gate arrays; FPGA; Hardware; Signal processing; Signal processing algorithms; Table lookup; Trigonometric functions; Unified architecture", } @Article{DelPunta:2018:LRC, author = "Jessica A. {Del Punta} and Gustavo Gasaneo and Lorenzo U. Ancarani", title = "On the {Laguerre} Representation of {Coulomb} Functions and the Relation to Orthogonal Polynomials", chapter = "4", journal = j-ADV-QUANTUM-CHEM, volume = "76", pages = "79--101", year = "2018", CODEN = "AQCHA9", DOI = "https://doi.org/10.1016/bs.aiq.2017.06.005", ISSN = "0065-3276", ISSN-L = "0065-3276", bibdate = "Thu Feb 1 07:08:30 2018", bibsource = "https://www.math.utah.edu/pub/tex/bib/advquantumchem.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "https://www.sciencedirect.com/science/article/pii/S0065327617300643", acknowledgement = ack-nhfb, ajournal = "Adv. Quantum Chem.", fjournal = "Advances in Quantum Chemistry", journal-URL = "http://www.sciencedirect.com/science/bookseries/00653276/", keywords = "Coulomb functions; Laguerre basis; Orthogonal polynomials", } @Article{Dunster:2018:UAE, author = "T. M. Dunster and A. Gil and J. Segura", title = "Uniform asymptotic expansions for {Laguerre} polynomials and related confluent hypergeometric functions", journal = j-ADV-COMPUT-MATH, volume = "44", number = "5", pages = "1441--1474", month = oct, year = "2018", CODEN = "ACMHEX", DOI = "https://doi.org/10.1007/s10444-018-9589-5", ISSN = "1019-7168 (print), 1572-9044 (electronic)", ISSN-L = "1019-7168", bibdate = "Thu May 30 08:11:44 MDT 2024", bibsource = "https://www.math.utah.edu/pub/tex/bib/advcomputmath.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "https://link.springer.com/article/10.1007/s10444-018-9589-5", acknowledgement = ack-nhfb, ajournal = "Adv. Comput. Math.", fjournal = "Advances in Computational Mathematics", journal-URL = "http://link.springer.com/journal/10444", } @Article{Hanson:2018:RAM, author = "Richard J. Hanson and Tim Hopkins", title = "Remark on {Algorithm 539: A Modern Fortran Reference Implementation for Carefully Computing the Euclidean Norm}", journal = j-TOMS, volume = "44", number = "3", pages = "24:1--24:23", month = apr, year = "2018", CODEN = "ACMSCU", DOI = "https://doi.org/10.1145/3134441", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Mon Jan 22 17:49:32 MST 2018", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fortran3.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib", URL = "https://dl.acm.org/citation.cfm?id=3134441", abstract = "We propose a set of new Fortran reference implementations, based on an algorithm proposed by Kahan, for the Level 1 BLAS routines *NRM2 that compute the Euclidean norm of a real or complex input vector. The principal advantage of these routines over the current offerings is that, rather than losing accuracy as the length of the vector increases, they generate results that are accurate to almost machine precision for vectors of length $ N < N_{\rm max} $ where $ N_{\rm max} $ depends upon the precision of the floating point arithmetic being used. In addition, we make use of intrinsic modules, introduced in the latest Fortran standards, to detect occurrences of non-finite numbers in the input data and return suitable values as well as setting IEEE floating point status flags as appropriate. A set of C interface routines is also provided to allow simple, portable access to the new routines. To improve execution speed, we advocate a hybrid algorithm; a simple loop is used first and, only if IEEE floating point exception flags signal, do we fall back on Kahan's algorithm. Since most input vectors are ``easy,'' i.e., they do not require the sophistication of Kahan's algorithm, the simple loop improves performance while the use of compensated summation ensures high accuracy. We also report on a comprehensive suite of test problems that has been developed to test both our new implementation and existing codes for both accuracy and the appropriate settings of the IEEE arithmetic status flags.", acknowledgement = ack-nhfb, articleno = "24", fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", xxnote = "See \cite{Lawson:1979:ABL}.", } @Article{Higham:2018:UN, author = "Nicholas J. Higham", title = "The Unwinding Number", journal = j-SIAM-NEWS, volume = "51", number = "8", pages = "??--??", month = oct, year = "2018", ISSN = "0036-1437", ISSN-L = "0036-1437", bibdate = "Sat Oct 06 08:46:15 2018", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/h/higham-nicholas-john.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "https://sinews.siam.org/Details-Page/the-unwinding-number", abstract = "While Fortran 66 had a complex data type, this was not true of most other early programming languages, such as Algol 60. As a result, programmers had to write their own procedures to implement complex arithmetic and transcendental functions in terms of separately stored real and imaginary parts. They quickly realized that this is not a trivial task; in the early 1960s, it took five published attempts over three years to obtain a correct implementation of the complex logarithm in Algol 60.", acknowledgement = ack-nhfb, } @InProceedings{Hsiao:2018:AEF, author = "Shen-Fu Hsiao and Yu-Chang Chen and Hsiang-Hao Liang", editor = "Martin Novotn{\'y} and Nikos Konofaos and Amund Skavhaug", booktitle = "{21st Euromicro Conference on Digital System Design: DSD 2018: 29--31 August 2018, Prague, Czech Republic}", title = "Architectural Exploration of Function Computation Based on Cubic Polynomial Interpolation with Application in Deep Neural Networks", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "22--29", month = aug, year = "2018", DOI = "https://doi.org/10.1109/dsd.2018.00020", ISBN = "1-5386-7376-2", ISBN-13 = "978-1-5386-7376-8", bibdate = "Thu Apr 10 13:24:27 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, } @Article{Johansson:2018:FRA, author = "Fredrik Johansson and Marc Mezzarobba", title = "Fast and Rigorous Arbitrary-Precision Computation of {Gauss--Legendre} Quadrature Nodes and Weights", journal = j-SIAM-J-SCI-COMP, volume = "40", number = "6", pages = "C726--C747", month = "????", year = "2018", CODEN = "SJOCE3", DOI = "https://doi.org/10.1137/18M1170133", ISSN = "1064-8275 (print), 1095-7197 (electronic)", ISSN-L = "1064-8275", bibdate = "Fri Jan 25 18:37:30 MST 2019", bibsource = "http://epubs.siam.org/toc/sjoce3/40/6; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/siamjscicomput.bib", acknowledgement = ack-nhfb, ajournal = "SIAM J. Sci. Comput.", fjournal = "SIAM Journal on Scientific Computing", journal-URL = "http://epubs.siam.org/sisc", onlinedate = "January 2018", } @Misc{Kahan:2018:TD, author = "William Kahan", title = "The tanpi Dilemma", howpublished = "Web document.", day = "16", month = apr, year = "2018", bibdate = "Tue Apr 17 06:52:47 2018", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/k/kahan-william-m.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://754r.ucbtest.org/background/tanpi.txt; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", abstract = "The function tanpi(x) satisfies two familiar identities, tanpi(-x) = -tanpi(x), and tanpi(x + integer) = tanpi(x), that cannot both be satisfied {\em everywhere\/} by IEEE 754's arithmetic; the obvious failures occur when tanpi is infinite: does tanpi(-2.5) = -tanpi(2.5) or does tanpi(-2.5) = tanpi(-2.5 + 4) = +tanpi(2.5)? Whoever puts a tanpi subprogram into the Math library has no choice but to disappoint somebody.", acknowledgement = ack-nhfb, } @InBook{Kumari:2018:RLS, author = "Aishwarya Kumari and D. P. Acharya", booktitle = "Recent Findings in Intelligent Computing Techniques", title = "Reduced Latency Square-Root Calculation for Signal Processing Using Radix-4 Hyperbolic {CORDIC}", publisher = "Springer Singapore", pages = "219--225", year = "2018", DOI = "https://doi.org/10.1007/978-981-10-8636-6_23", ISBN = "981-10-8636-2", ISBN-13 = "978-981-10-8636-6", ISSN = "2194-5365", bibdate = "Tue Oct 28 07:04:09 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, } @Article{Lopez:2018:CEB, author = "Jos{\'e} L. L{\'o}pez", title = "Convergent expansions of the {Bessel} functions in terms of elementary functions", journal = j-ADV-COMPUT-MATH, volume = "44", number = "1", pages = "277--294", month = feb, year = "2018", CODEN = "ACMHEX", DOI = "https://doi.org/10.1007/s10444-017-9543-y", ISSN = "1019-7168 (print), 1572-9044 (electronic)", ISSN-L = "1019-7168", MRclass = "33C10 (41A58)", MRnumber = "3755750", bibdate = "Sat Feb 3 18:23:33 MST 2018", bibsource = "https://www.math.utah.edu/pub/tex/bib/advcomputmath.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://link.springer.com/article/10.1007/s10444-017-9543-y", acknowledgement = ack-nhfb, fjournal = "Advances in Computational Mathematics", journal-URL = "http://link.springer.com/journal/10444", } @Article{Matic:2018:SPB, author = "Ivan Mati{\'c} and Rado{\v{s}} Radoi{\v{c}}i{\'c} and Dan Stefanica", title = "A sharp {P{\'o}lya}-based approximation to the normal cumulative distribution function", journal = j-APPL-MATH-COMP, volume = "322", number = "??", pages = "111--122", day = "1", month = apr, year = "2018", CODEN = "AMHCBQ", DOI = "https://doi.org/10.1016/j.amc.2017.10.019", ISSN = "0096-3003 (print), 1873-5649 (electronic)", ISSN-L = "0096-3003", bibdate = "Fri Dec 15 10:03:09 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/applmathcomput2015.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/S009630031730718X", acknowledgement = ack-nhfb, fjournal = "Applied Mathematics and Computation", journal-URL = "http://www.sciencedirect.com/science/journal/00963003", remark = "Although the accuracy of the approximations developed is low (6 to 10 digits), the article shows how it can be increased by taking more series terms. The article is an excellent overview of prior work on computing the normal and inverse normal cumulative distribution function, almost all of which is low accuracy (2 to 4 digits). The authors supply 89 references to prior work, all of which are now in this bibliography as of 16 December 2017.", } @InProceedings{Mikaitis:2018:AFP, author = "Mantas Mikaitis and David R. Lester and Delong Shang and Steve Furber and Gengting Liu and Jim Garside and Stefan Scholze and Sebastian H{\"o}ppner and Andreas Dixius", title = "Approximate Fixed-Point Elementary Function Accelerator for the {SpiNNaker-2} Neuromorphic Chip", crossref = "Tenca:2018:PIS", pages = "37--44", year = "2018", DOI = "https://doi.org/10.1109/ARITH.2018.8464785", bibdate = "Fri Jan 31 08:05:31 2020", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", abstract = "Neuromorphic chips are used to model biologically inspired Spiking-Neural-Networks (SNNs) where most models are based on differential equations. Equations for most SNN algorithms usually contain variables with one or more ex components. SpiNNaker is a digital neuromorphic chip that has so far been using pre-calculated look-up tables for exponential function. However this approach is limited because the memory requirements grow as more complex neural models are developed. To save already limited memory resources in the next generation SpiNNaker chip, we are including a fast exponential function in the silicon. In this paper we analyse iterative algorithms for elementary functions and show how to build a single hardware accelerator for exp and natural log, for a neuromorphic chip prototype, to be manufactured in a 22 nm FDSOI process. We present the accelerator that has algorithmic level approximation control, allowing it to trade precision for latency and energy efficiency. As an addition to neuromorphic chip application, we provide analysis of a parameterized elementary function unit that can be tailored for other systems with different power, area, accuracy and latency constraints.", acknowledgement = ack-nhfb, keywords = "Adders; algorithmic level approximation control; approximate arithmetic; approximate fixed-point elementary function accelerator; ARITH-25; Biological system modeling; biologically inspired spiking-neural-networks; complex neural models; Computational modeling; Convergence; differential equations; digital neuromorphic chip; energy efficiency; exponential function; fast exponential function; FDSOI process; fixed-point arithmetic; hardware accelerators; iterative algorithms; iterative methods; logarithm function; Mathematical model; memory requirements; memory resources; MPSoC; neural chips; neuromorphic chip prototype; neuromorphic computing; Neuromorphics; next generation SpiNNaker chip; parameterized elementary function unit; pre-calculated look-up tables; single hardware accelerator; size 22.0 nm; SNN algorithms; SpiNNaker-2 neuromorphic chip; SpiNNaker2; table lookup; Table lookup", } @Article{Moroz:2018:FCI, author = "Leonid V. Moroz and Cezary J. Walczyk and Andriy Hrynchyshyn and Vijay Holimath and Jan L. Cie{\'s}li{\'n}ski", title = "Fast calculation of inverse square root with the use of magic constant --- analytical approach", journal = j-APPL-MATH-COMP, volume = "316", number = "??", pages = "245--255", day = "1", month = jan, year = "2018", CODEN = "AMHCBQ", DOI = "https://doi.org/10.1016/j.amc.2017.08.025", ISSN = "0096-3003 (print), 1873-5649 (electronic)", ISSN-L = "0096-3003", bibdate = "Tue Oct 10 15:56:03 MDT 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/applmathcomput2015.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0096300317305763", acknowledgement = ack-nhfb, fjournal = "Applied Mathematics and Computation", journal-URL = "http://www.sciencedirect.com/science/journal/00963003", keywords = "single-precision 32-bit IEEE 754 binary arithmetic", } @Book{Muller:2018:HFP, author = "Jean-Michel Muller and Nicolas Brunie and Florent de Dinechin and Claude-Pierre Jeannerod and Mioara Joldes and Vincent Lef{\`e}vre and Guillaume Melquiond and Nathalie Revol and Serge Torres", title = "Handbook of Floating-Point Arithmetic", publisher = pub-BIRKHAUSER, address = pub-BIRKHAUSER:adr, edition = "Second", pages = "xxv + 627", year = "2018", DOI = "https://doi.org/10.1007/978-3-319-76526-6", ISBN = "3-319-76525-6, 3-319-76526-4 (e-book)", ISBN-13 = "978-3-319-76525-9, 978-3-319-76526-6 (e-book)", LCCN = "QA76.9.C62", bibdate = "Fri Jun 1 06:59:01 MDT 2018", bibsource = "fsz3950.oclc.org:210/WorldCat; https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/master.bib", abstract = "This handbook aims to provide a complete overview of modern floating-point arithmetic. This includes a detailed treatment of the current (IEEE-754) and next (preliminarily called IEEE-754R) standards for floating-point arithmetic.", acknowledgement = ack-nhfb, shorttableofcontents = "Introduction / 3--14 \\ Definitions and Basic Notions / 15--45 \\ Floating-Point Formats and Environment / 47--93 \\ Basic Properties and Algorithms / 97--162 \\ Enhanced Floating-Point Sums, Dot Products, and Polynomial Values / 163--192 \\ Languages and Compilers / 193--230 \\ Algorithms for the Basic Operations / 233--266 \\ Hardware Implementation of Floating-Point Arithmetic / 267--320 \\ Software Implementation of Floating-Point Arithmetic / 321--374 \\ Evaluating Floating-Point Elementary Functions / 375--433 \\ Complex Numbers / 437--452 \\ Interval Arithmetic / 453--477 \\ Verifying Floating-Point Algorithms / 479--511 \\ Extending the Precision / 513--552", subject = "Floating-point arithmetic; Handbooks, manuals, etc; Computer arithmetic; COMPUTERS / Computer Literacy; COMPUTERS / Computer Science; COMPUTERS / Data Processing; COMPUTERS / Hardware / General; COMPUTERS / Information Technology; COMPUTERS / Machine Theory; COMPUTERS / Reference.", tableofcontents = "Intro \\ Contents \\ List of Figures \\ List of Tables \\ Preface \\ I Introduction, Basic Definitions, and Standards \\ 1 Introduction \\ 1.1 Some History \\ 1.2 Desirable Properties \\ 1.3 Some Strange Behaviors \\ 1.3.1 Some famous bugs \\ 1.3.2 Difficult problems \\ 1.3.2.1 A sequence that seems to converge to a wrong limit \\ 1.3.2.2 The Chaotic Bank Society \\ 1.3.2.3 Rump's example \\ 2 Definitions and Basic Notions \\ 2.1 Floating-Point Numbers \\ 2.1.1 Main definitions \\ 2.1.2 Normalized representations, normal and subnormal numbers \\ 2.1.3 A note on underflow \\ 2.1.4 Special floating-point data \\ 2.2 Rounding \\ 2.2.1 Rounding functions \\ 2.2.2 Useful properties \\ 2.3 Tools for Manipulating Floating-Point Errors \\ 2.3.1 Relative error due to rounding \\ 2.3.2 The ulp function \\ 2.3.3 Link between errors in ulps and relative errors \\ 2.3.3.1 Converting from errors in ulps to relative errors \\ 2.3.3.2 Converting from relative errors to errors in ulps \\ 2.3.3.3 Loss of information during these conversions \\ 2.3.4 An example: iterated products \\ 2.4 The Fused Multiply-Add (FMA) Instruction \\ 2.5 Exceptions \\ 2.6 Lost and Preserved Properties of Real Arithmetic \\ 2.7 Note on the Choice of the Radix \\ 2.7.1 Representation errors \\ 2.7.2 A case for radix 10 \\ 2.8 Reproducibility \\ 3 Floating-Point Formats and Environment \\ 3.1 The IEEE 754-2008 Standard \\ 3.1.1 Formats \\ 3.1.1.1 Binary interchange format encodings \\ 3.1.1.2 Decimal interchange format encodings \\ 3.1.1.3 Larger formats \\ 3.1.1.4 Extended and extendable precisions \\ 3.1.1.5 Little-endian, big-endian \\ 3.1.2 Attributes and rounding \\ 3.1.2.1 Rounding direction attributes \\ 3.1.2.2 Alternate exception-handling attributes \\ 3.1.2.3 Preferred width attributes \\ 3.1.2.4 Value-changing optimization attributes \\ 3.1.2.5 Reproducibility attributes \\ 3.1.3 Operations specified by the standard \\ 3.1.3.1 Arithmetic operations and square root \\ 3.1.3.2 Remainders \\ 3.1.3.3 Preferred exponent for arithmetic operations in the decimal format \\ 3.1.3.4 scaleB and logB \\ 3.1.3.5 Miscellaneous \\ 3.1.4 Comparisons \\ 3.1.5 Conversions to/from string representations \\ 3.1.6 Default exception handling \\ 3.1.6.1 Invalid operation \\ 3.1.6.2 Division by zero \\ 3.1.6.3 Overflow \\ 3.1.6.4 Underflow \\ 3.1.6.5 Inexact \\ 3.1.7 Special values \\ 3.1.7.1 NaN: Not a Number \\ 3.1.7.2 Arithmetic of infinities and zeros \\ 3.1.8 Recommended functions \\ 3.2 On the Possible Hidden Use of a Higher Internal Precision \\ 3.3 Revision of the IEEE 754-2008 Standard \\ 3.4 Floating-Point Hardware in Current Processors \\ 3.4.1 The common hardware denominator \\ 3.4.2 Fused multiply-add \\ 3.4.3 Extended precision and 128-bit formats \\ 3.4.4 Rounding and precision control \\ 3.4.5 SIMD instructions \\ 3.4.6 Binary16 (half-precision) support \\ 3.4.7 Decimal arithmetic \\ 3.4.8 The legacy x87 processor \\ 3.5 Floating-Point Hardware in Recent Graphics Processing Units \\ 3.6 IEEE Support in Programming Languages \\ 3.7 Checking the Environment \\ 3.7.1 MACHAR \\ 3.7.2 Paranoia \\ \ldots{} \\ Basic Properties and Algorithms \\ Enhanced Floating-Point Sums, Dot Products, and Polynomial Values \\ Languages and Compilers \\ Algorithms for the Basic Operations \\ Hardware Implementation of Floating-Point Arithmetic \\ Software Implementation of Floating-Point Arithmetic \\ Evaluating Floating-Point Elementary Functions \\ Complex Numbers \\ Interval Arithmetic \\ Verifying Floating-Point Algorithms \\ Extending the Precision", } @Article{Munoz-Coreas:2018:CQO, author = "Edgard Mu{\~n}oz-Coreas and Himanshu Thapliyal", title = "{T}-count and Qubit Optimized Quantum Circuit Design of the Non-Restoring Square Root Algorithm", journal = j-JETC, volume = "14", number = "3", pages = "36:1--36:15", month = oct, year = "2018", CODEN = "????", DOI = "https://doi.org/10.1145/3264816", ISSN = "1550-4832", bibdate = "Thu Nov 1 16:44:41 MDT 2018", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/jetc.bib", abstract = "Quantum circuits for basic mathematical functions such as the square root are required to implement scientific computing algorithms on quantum computers. Quantum circuits that are based on Clifford+T gates can easily be made fault tolerant, but the T gate is very costly to implement. As a result, reducing T-count has become an important optimization goal. Further, quantum circuits with many qubits are difficult to realize, making designs that save qubits and produce no garbage outputs desirable. In this work, we present a T-count optimized quantum square root circuit with only $ 2 s n + 1 $ qubits and no garbage output. To make a fair comparison against existing work, the Bennett's garbage removal scheme is used to remove garbage output from existing works. We determined that our proposed design achieves an average T-count savings of 43.44\%, 98.95\%, 41.06\%, and 20.28\% as well as qubit savings of 85.46\%, 95.16\%, 90.59\%, and 86.77\% compared to existing works.", acknowledgement = ack-nhfb, articleno = "36", fjournal = "ACM Journal on Emerging Technologies in Computing Systems (JETC)", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J967", } @Article{Myland:2018:JEF, author = "Jan C. Myland and Keith B. Oldham", title = "{Jacobian} elliptic functions describe the properties of the diffuse charge region in narrow electrochemical cells", journal = j-J-MATH-CHEM, volume = "56", number = "4", pages = "1184--1205", month = apr, year = "2018", CODEN = "JMCHEG", DOI = "https://doi.org/10.1007/s10910-017-0847-4", ISSN = "0259-9791 (print), 1572-8897 (electronic)", ISSN-L = "0259-9791", bibdate = "Tue Mar 6 07:08:26 MST 2018", bibsource = "http://link.springer.com/journal/10910/56/4; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jmathchem.bib", URL = "https://link.springer.com/article/10.1007/s10910-017-0847-4", acknowledgement = ack-nhfb, ajournal = "J. Math. Chem.", fjournal = "Journal of Mathematical Chemistry", journal-URL = "http://link.springer.com/journal/10910", journalabr = "J. Math. Chem.", } @Article{Navas-Palencia:2018:FAA, author = "Guillermo Navas-Palencia", title = "Fast and accurate algorithm for the generalized exponential integral {$ E_\nu (x) $} for positive real order", journal = j-NUMER-ALGORITHMS, volume = "77", number = "2", pages = "603--630", month = feb, year = "2018", CODEN = "NUALEG", DOI = "https://doi.org/10.1007/s11075-017-0331-z", ISSN = "1017-1398 (print), 1572-9265 (electronic)", ISSN-L = "1017-1398", bibdate = "Thu Jan 25 09:50:15 MST 2018", bibsource = "http://link.springer.com/journal/11075/77/2; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/numeralgorithms.bib", URL = "https://link.springer.com/article/10.1007/s11075-017-0331-z", acknowledgement = ack-nhfb, fjournal = "Numerical Algorithms", journal-URL = "http://link.springer.com/journal/11075", } @Article{Navas-Palencia:2018:HPC, author = "Guillermo Navas-Palencia", title = "High-precision computation of the confluent hypergeometric functions via {Franklin--Friedman} expansion", journal = j-ADV-COMPUT-MATH, volume = "44", number = "3", pages = "841--859", month = jun, year = "2018", CODEN = "ACMHEX", DOI = "https://doi.org/10.1007/s10444-017-9565-5", ISSN = "1019-7168 (print), 1572-9044 (electronic)", ISSN-L = "1019-7168", bibdate = "Thu May 30 08:11:42 MDT 2024", bibsource = "https://www.math.utah.edu/pub/tex/bib/advcomputmath.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "https://link.springer.com/article/10.1007/s10444-017-9565-5", acknowledgement = ack-nhfb, ajournal = "Adv. Comput. Math.", fjournal = "Advances in Computational Mathematics", journal-URL = "http://link.springer.com/journal/10444", keywords = "Arbitrary-precision arithmetic; confluent hypergeometric functions; Franklin--Friedman expansion; generalized exponential integer $E_\nu(z) = z^{\nu - 1} U(\nu, \nu, z)$; Kummer function $U(a,b,z)$; modified Bessel function $K_\nu(z)$; uniform series expansion", } @Article{Pakes:2018:LFN, author = "Anthony G. Pakes", title = "The {Lambert} {$W$} function, {Nuttall}'s integral, and the {Lambert} law", journal = j-STAT-PROB-LETT, volume = "139", number = "??", pages = "53--60", month = aug, year = "2018", CODEN = "SPLTDC", DOI = "https://doi.org/10.1016/j.spl.2018.03.015", ISSN = "0167-7152 (print), 1879-2103 (electronic)", ISSN-L = "0167-7152", bibdate = "Thu Nov 8 12:34:02 MST 2018", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/statproblett2010.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0167715218301354", acknowledgement = ack-nhfb, fjournal = "Statistics \& Probability Letters", journal-URL = "http://www.sciencedirect.com/science/journal/01677152", } @Article{Patterson:2018:SCS, author = "T. N. L. Patterson", title = "Sines, Cosines, Square Roots, and Binary Bits", journal = j-AMER-MATH-MONTHLY, volume = "125", number = "8", pages = "750--754", year = "2018", CODEN = "AMMYAE", DOI = "https://doi.org/10.1080/00029890.2018.1498695", ISSN = "0002-9890 (print), 1930-0972 (electronic)", ISSN-L = "0002-9890", bibdate = "Mon Dec 13 17:59:05 MST 2021", bibsource = "https://www.math.utah.edu/pub/tex/bib/amermathmonthly2010.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "American Mathematical Monthly", journal-URL = "http://www.jstor.org/journals/00029890.html; https://www.tandfonline.com/loi/uamm20", onlinedate = "28 Sep 2018", } @Article{Punta:2018:CFL, author = "Jessica A. Del Punta and Gustavo Gasaneo and Lorenzo U. Ancarani", title = "On the {Laguerre} Representation of {Coulomb} Functions and the Relation to Orthogonal Polynomials", chapter = "4", journal = j-ADV-QUANTUM-CHEM, volume = "76", pages = "79--101", year = "2018", CODEN = "AQCHA9", DOI = "https://doi.org/10.1016/bs.aiq.2017.06.005", ISSN = "0065-3276", ISSN-L = "0065-3276", bibdate = "Thu Feb 1 07:08:30 2018", bibsource = "https://www.math.utah.edu/pub/tex/bib/advquantumchem.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "https://www.sciencedirect.com/science/article/pii/S0065327617300643", acknowledgement = ack-nhfb, fjournal = "Advances in Quantum Chemistry", journal-URL = "http://www.sciencedirect.com/science/bookseries/00653276", keywords = "Coulomb functions; Laguerre basis; Orthogonal polynomials", } @Article{Qi:2018:DME, author = "Hongyuan Qi and Jinchen Xu and Shaozhong Guo", title = "Detection of the maximum error of mathematical functions", journal = j-J-SUPERCOMPUTING, volume = "74", number = "11", pages = "6275--6290", month = nov, year = "2018", CODEN = "JOSUED", DOI = "https://doi.org/10.1007/s11227-018-2552-x", ISSN = "0920-8542 (print), 1573-0484 (electronic)", ISSN-L = "0920-8542", bibdate = "Thu Oct 10 15:31:09 MDT 2019", bibsource = "http://link.springer.com/journal/11227/74/11; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jsuper.bib", acknowledgement = ack-nhfb, fjournal = "The Journal of Supercomputing", journal-URL = "http://link.springer.com/journal/11227", } @Article{Quan:2018:ACA, author = "Le Phuong Quan and Th{\'a}i Anh Nhan", title = "Applying Computer Algebra Systems in Approximating the Trigonometric Functions", journal = j-MATH-COMPUT-APPL, volume = "23", number = "3", pages = "??--??", month = sep, year = "2018", CODEN = "????", DOI = "https://doi.org/10.3390/mca23030037", ISSN = "2297-8747", ISSN-L = "2297-8747", bibdate = "Sun Feb 18 06:28:34 MST 2024", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/math-comput-appl.bib", URL = "https://www.mdpi.com/2297-8747/23/3/37", acknowledgement = ack-nhfb, ajournal = "Math. Comput. Appl.", articleno = "37", fjournal = "Mathematical and Computational Applications", journal-URL = "https://www.mdpi.com/journal/mca", } @Article{Quan:2018:CMM, author = "Le Phuong Quan", title = "A Computational Method with {MAPLE} for a Piecewise Polynomial Approximation to the Trigonometric Functions", journal = j-MATH-COMPUT-APPL, volume = "23", number = "4", pages = "??--??", month = dec, year = "2018", CODEN = "????", DOI = "https://doi.org/10.3390/mca23040063", ISSN = "2297-8747", ISSN-L = "2297-8747", bibdate = "Sun Feb 18 06:28:34 MST 2024", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/maple-extract.bib; https://www.math.utah.edu/pub/tex/bib/math-comput-appl.bib", URL = "https://www.mdpi.com/2297-8747/23/4/63", acknowledgement = ack-nhfb, ajournal = "Math. Comput. Appl.", articleno = "63", fjournal = "Mathematical and Computational Applications", journal-URL = "https://www.mdpi.com/journal/mca", } @InProceedings{Rekha:2018:FIE, author = "R. Rekha and Karunakara P. Menon", booktitle = "{2018 3rd IEEE International Conference on Recent Trends in Electronics, Information \& Communication Technology (RTEICT)}", title = "{FPGA} implementation of exponential function using {CORDIC} {IP} core for extended input range", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "597--600", year = "2018", DOI = "https://doi.org/10.1109/RTEICT42901.2018.9012611", bibdate = "Mon Oct 27 10:32:44 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/matlab.bib", acknowledgement = ack-nhfb, keywords = "Computer architecture; CORDIC IP Core; Exponential function; Field programmable gate arrays; FPGA; Hardware; hardware implementation; IP networks; Matlab; Signal processing algorithms; Simulation", } @PhdThesis{Saint-Genies:2018:EFT, author = "Hugues de Lassus Saint-Geni{\`e}s", title = "Elementary functions: towards automatically generated, efficient, and vectorizable implementations", type = "{Ph.D.} thesis", school = "Universit{\'e} de Perpignan", address = "Perpignan, France", pages = "xxxviii + 128", day = "17", month = jul, year = "2018", bibdate = "Tue Mar 01 06:09:03 2022", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "https://tel.archives-ouvertes.fr/tel-01841424/document", acknowledgement = ack-nhfb, } @Article{Schneider:2018:NFP, author = "Barry I. Schneider and Javier Segura and Amparo Gil and Xiaoxu Guan and Klaus Bartschat", title = "A new {Fortran 90} program to compute regular and irregular associated {Legendre} functions (new version announcement)", journal = j-COMP-PHYS-COMM, volume = "225", number = "??", pages = "192--193", month = apr, year = "2018", CODEN = "CPHCBZ", DOI = "https://doi.org/10.1016/j.cpc.2017.12.013", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", bibdate = "Wed Feb 28 14:39:27 MST 2018", bibsource = "https://www.math.utah.edu/pub/tex/bib/compphyscomm2010.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fortran3.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0010465517304186", acknowledgement = ack-nhfb, fjournal = "Computer Physics Communications", journal-URL = "http://www.sciencedirect.com/science/journal/00104655", } @Article{Walczyk:2018:IAF, author = "Cezary J. Walczyk and Leonid V. Moroz and Jan L. Cie{\'s}li{\'n}ski", title = "Improving the accuracy of the fast inverse square root algorithm", journal = "arXiv.org", volume = "??", number = "??", pages = "1--21", day = "17", month = feb, year = "2018", DOI = "https://doi.org/10.48550/arXiv.1802.06302", bibdate = "Wed Dec 20 07:55:45 2023", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "https://arxiv.org/abs/1802.06302", abstract = "We present improved algorithms for fast calculation of the inverse square root for single-precision floating-point numbers. The algorithms are much more accurate than the famous fast inverse square root algorithm and have the same or similar computational cost. The main idea of our work consists in modifying the Newton-Raphson method and demanding that the maximal error is as small as possible. Such modification is possible when the distribution of Newton-Raphson corrections is not symmetric (e.g., if they are non-positive functions).", acknowledgement = ack-nhfb, } @Article{Xue:2018:RCL, author = "Changfeng Xue and Shaozhong Deng", title = "Recursive Computation of Logarithmic Derivatives, Ratios, and Products of Spheroidal Harmonics and Modified {Bessel} Functions and Applications", journal = j-J-SCI-COMPUT, volume = "75", number = "1", pages = "128--156", month = oct, year = "2018", CODEN = "JSCOEB", DOI = "https://doi.org/10.1007/s10915-017-0527-3", ISSN = "0885-7474 (print), 1573-7691 (electronic)", ISSN-L = "0885-7474", bibdate = "Fri Mar 29 16:29:33 MDT 2019", bibsource = "http://link.springer.com/journal/10915/75/1; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jscicomput.bib", URL = "https://link.springer.com/article/10.1007/s10915-017-0527-3; https://link.springer.com/content/pdf/10.1007/s10915-017-0527-3.pdf", acknowledgement = ack-nhfb, fjournal = "Journal of Scientific Computing", journal-URL = "http://link.springer.com/journal/10915", } @Article{Alonso:2019:CMT, author = "Pedro Alonso and Jes{\'u}s Peinado and Javier Ib{\'a}{\~n}ez and Jorge Sastre and Emilio Defez", title = "Computing matrix trigonometric functions with {GPUs} through {Matlab}", journal = j-J-SUPERCOMPUTING, volume = "75", number = "3", pages = "1227--1240", month = mar, year = "2019", CODEN = "JOSUED", DOI = "https://doi.org/10.1007/s11227-018-2354-1", ISSN = "0920-8542 (print), 1573-0484 (electronic)", ISSN-L = "0920-8542", bibdate = "Thu Oct 10 15:31:18 MDT 2019", bibsource = "http://link.springer.com/journal/11227/75/3; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jsuper.bib; https://www.math.utah.edu/pub/tex/bib/matlab.bib", acknowledgement = ack-nhfb, fjournal = "The Journal of Supercomputing", journal-URL = "http://link.springer.com/journal/11227", } @InProceedings{Arzelier:2019:EAE, author = "Denis Arzelier and Florent Br{\'e}hard and Mioara Joldes", title = "Exchange Algorithm for Evaluation and Approximation Error-Optimized Polynomials", crossref = "Takagi:2019:ISC", pages = "30--37", month = jun, year = "2019", DOI = "https://doi.org/10.1109/ARITH.2019.00014", bibdate = "Fri Jan 31 08:18:07 2020", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", abstract = "Machine implementation of mathematical functions often relies on polynomial approximations. The particularity is that rounding errors occur both when representing the polynomial coefficients on a finite number of bits, and when evaluating it in finite precision. Hence, for finding the best polynomial (for a given fixed degree, norm and interval), one has to consider both types of errors: approximation and evaluation. While efficient algorithms were already developed for taking into account the approximation error, the evaluation part is usually a posteriori handled, in an ad-hoc manner. Here, we formulate a semi-infinite linear optimization problem whose solution is the best polynomial with respect to the supremum norm of the sum of both errors. This problem is then solved with an iterative exchange algorithm, which can be seen as an extension of the well-known Remez algorithm. A discussion and comparison of the obtained results on different examples are finally presented.", acknowledgement = ack-nhfb, keywords = "Approximation algorithms; Approximation error; approximation error; approximation error-optimized polynomials; ARITH-26; Digital arithmetic; evaluation error; exchange algorithm; function approximation; Indexes; Input variables; iterative exchange algorithm; iterative methods; learning (artificial intelligence); libm; linear programming; machine implementation; mathematical functions; mathematics computing; optimisation; Optimization; polynomial approximation; polynomial approximations; polynomial coefficients; Programming; remez algorithm; Remez algorithm; semi-infinite programming; semiinfinite linear optimization problem", } @Article{Batista:2019:ECM, author = "Milan Batista", title = "\pkg{Elfun18} --- a collection of {MATLAB} functions for the computation of elliptic integrals and {Jacobian} elliptic functions of real arguments", journal = j-SOFTWAREX, volume = "10", number = "??", pages = "Article 100245", month = jul # "\slash " # dec, year = "2019", CODEN = "????", DOI = "https://doi.org/10.1016/j.softx.2019.100245", ISSN = "2352-7110", ISSN-L = "2352-7110", bibdate = "Fri Apr 9 16:04:36 MDT 2021", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/matlab.bib; https://www.math.utah.edu/pub/tex/bib/softwarex.bib", URL = "http://www.sciencedirect.com/science/article/pii/S2352711018302796", acknowledgement = ack-nhfb, fjournal = "SoftwareX", journal-URL = "https://www.sciencedirect.com/journal/softwarex/issues", } @TechReport{Bernstein:2019:FCT, author = "Daniel J. Bernstein and Bo-Yin Yang", title = "Fast constant-time gcd computation and modular inversion", institution = "International Association for Cryptologic Research", address = "????", pages = "1--59", day = "13", month = apr, year = "2019", bibdate = "Tue May 24 07:23:13 2022", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "https://eprint.iacr.org/2019/266.pdf", abstract = "This paper introduces streamlined constant-time variants of Euclid's algorithm, both for polynomial inputs and for integer inputs. As concrete applications, this paper saves time in (1) modular inversion for Curve25519, which was previously believed to be handled much more efficiently by Fermat's method, and (2) key generation for the ntruhrss701 and sntrup4591761 lattice-based cryptosystems", acknowledgement = ack-nhfb, keywords = "algorithm design; branchless algorithms; constant-time computations; Curve25519; Euclid's algorithm; gcd; greatest common divisor; modular inversion; modular reciprocal; NTRU", } @Article{Borges:2019:IAH, author = "Carlos F. Borges", title = "An Improved Algorithm for {\tt hypot(a,b)}", journal = "arXiv.org", volume = "??", number = "??", pages = "1--15", day = "14", month = jun, year = "2019", bibdate = "Fri Apr 19 05:40:55 2024", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "https://arxiv.org/abs/1904.09481", abstract = "We develop a fast and accurate algorithm for evaluating $ \sqrt {x^2 + y^2} $ for two floating point numbers $a$ and $b$. Library functions that perform this computation are generally named {\tt hypot(a,b)}. We will compare four approaches that we will develop in this paper to the current resident library function that is delivered with Julia 1.1 and to the code that has been distributed with the C math library for decades. We will demonstrate the performance of our algorithms by simulation.", acknowledgement = ack-nhfb, } @Article{Bremer:2019:ARN, author = "James Bremer", title = "An algorithm for the rapid numerical evaluation of {Bessel} functions of real orders and arguments", journal = j-ADV-COMPUT-MATH, volume = "45", number = "1", pages = "173--211", month = feb, year = "2019", CODEN = "ACMHEX", DOI = "https://doi.org/10.1007/s10444-018-9613-9", ISSN = "1019-7168 (print), 1572-9044 (electronic)", ISSN-L = "1019-7168", bibdate = "Thu May 30 08:11:46 MDT 2024", bibsource = "https://www.math.utah.edu/pub/tex/bib/advcomputmath.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "https://link.springer.com/article/10.1007/s10444-018-9613-9", acknowledgement = ack-nhfb, ajournal = "Adv. Comput. Math.", fjournal = "Advances in Computational Mathematics", journal-URL = "http://link.springer.com/journal/10444", } @Article{Bujanda:2019:CEC, author = "Blanca Bujanda and Jos{\'e} and L. L{\'o}pez and Pedro J. Pagola", title = "Convergent expansions of the confluent hypergeometric functions in terms of elementary functions", journal = j-MATH-COMPUT, volume = "88", number = "318", pages = "1773--1789", month = apr, year = "2019", CODEN = "MCMPAF", DOI = "https://doi.org/10.1090/mcom/3389", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Jul 14 06:45:40 MDT 2020", bibsource = "http://www.ams.org/mcom/2019-88-318; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp2010.bib", URL = "https://www.ams.org/journals/mcom/2019-88-318/S0025-5718-2018-03389-0; https://www.ams.org/journals/mcom/2019-88-318/S0025-5718-2018-03389-0/S0025-5718-2018-03389-0.pdf; https://www.ams.org/mathscinet/search/authors.html?authorName=Lopez%2C%20Jose%20L.; https://www.ams.org/mathscinet/search/authors.html?mrauthid=636519; https://www.ams.org/mathscinet/search/authors.html?mrauthid=806866", acknowledgement = ack-nhfb, fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Campos-Pinto:2019:APP, author = "Martin Campos-Pinto and Fr{\'e}d{\'e}rique Charles and Bruno Despr{\'e}s", title = "Algorithms For Positive Polynomial Approximation", journal = j-SIAM-J-NUMER-ANAL, volume = "57", number = "1", pages = "148--172", month = "????", year = "2019", CODEN = "SJNAAM", DOI = "https://doi.org/10.1137/17M1131891", ISSN = "0036-1429 (print), 1095-7170 (electronic)", ISSN-L = "0036-1429", bibdate = "Mon Mar 18 13:37:59 MDT 2019", bibsource = "http://epubs.siam.org/http://epubs.siam.org/toc/sjnaam/57/1; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/siamjnumeranal2010.bib", acknowledgement = ack-nhfb, fjournal = "SIAM Journal on Numerical Analysis", journal-URL = "http://epubs.siam.org/sinum", onlinedate = "January 2019", } @Article{Cardoso:2019:CMG, author = "Jo{\~a}o R. Cardoso and Amir Sadeghi", title = "Computation of matrix gamma function", journal = j-BIT-NUM-MATH, volume = "59", number = "2", pages = "343--370", month = jun, year = "2019", CODEN = "BITTEL, NBITAB", DOI = "https://doi.org/10.1007/s10543-018-00744-1", ISSN = "0006-3835 (print), 1572-9125 (electronic)", ISSN-L = "0006-3835", bibdate = "Fri Sep 6 09:16:11 MDT 2019", bibsource = "https://www.math.utah.edu/pub/tex/bib/bit.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://link.springer.com/article/10.1007/s10543-018-00744-1", acknowledgement = ack-nhfb, fjournal = "BIT Numerical Mathematics", journal-URL = "http://link.springer.com/journal/10543", } @Article{Fedotov:2019:CWM, author = "Alexander Fedotov and Fr{\'e}d{\'e}ric Klopp", title = "The Complex {WKB} Method for Difference Equations and {Airy} Functions", journal = j-SIAM-J-MATH-ANA, volume = "51", number = "6", pages = "4413--4447", month = "????", year = "2019", CODEN = "SJMAAH", DOI = "https://doi.org/10.1137/18M1228694", ISSN = "0036-1410 (print), 1095-7154 (electronic)", ISSN-L = "0036-1410", bibdate = "Fri Apr 24 15:47:49 MDT 2020", bibsource = "http://epubs.siam.org/sam-bin/dbq/toc/SIMA/51/6; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/siamjmathana2010.bib", acknowledgement = ack-nhfb, fjournal = "SIAM Journal on Mathematical Analysis", journal-URL = "http://epubs.siam.org/sima", onlinedate = "January 2019", } @Article{Green:2019:DFE, author = "Kevin R. Green and Tanner A. Bohn and Raymond J. Spiteri", title = "Direct Function Evaluation versus Lookup Tables: When to Use Which?", journal = j-SIAM-J-SCI-COMP, volume = "41", number = "3", pages = "C194--C218", month = "????", year = "2019", CODEN = "SJOCE3", DOI = "https://doi.org/10.1137/18M1201421", ISSN = "1064-8275 (print), 1095-7197 (electronic)", ISSN-L = "1064-8275", bibdate = "Thu Oct 10 06:58:05 MDT 2019", bibsource = "http://epubs.siam.org/toc/sjoce3/41/3; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/siamjscicomput.bib", acknowledgement = ack-nhfb, fjournal = "SIAM Journal on Scientific Computing", journal-URL = "http://epubs.siam.org/sisc", onlinedate = "January 2019", } @Article{Greengard:2019:AEI, author = "Philip Greengard and Vladimir Rokhlin", title = "An algorithm for the evaluation of the incomplete gamma function", journal = j-ADV-COMPUT-MATH, volume = "45", number = "1", pages = "23--49", month = feb, year = "2019", CODEN = "ACMHEX", DOI = "https://doi.org/10.1007/s10444-018-9604-x", ISSN = "1019-7168 (print), 1572-9044 (electronic)", ISSN-L = "1019-7168", bibdate = "Thu May 30 08:11:46 MDT 2024", bibsource = "https://www.math.utah.edu/pub/tex/bib/advcomputmath.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "https://link.springer.com/article/10.1007/s10444-018-9604-x", acknowledgement = ack-nhfb, ajournal = "Adv. Comput. Math.", fjournal = "Advances in Computational Mathematics", journal-URL = "http://link.springer.com/journal/10444", } @Article{Horyachyy:2019:SEF, author = "Oleh Horyachyy and Leonid Moroz and Viktor Otenko", title = "Simple Effective Fast Inverse Square Root Algorithm with Two Magic Constants", journal = "International Journal of Computing", volume = "18", number = "4", pages = "461--470", month = dec, year = "2019", ISSN = "1727-6209 (print), 2312-5381 (electronic)", ISSN-L = "1727-6209", bibdate = "Thu Feb 11 11:01:47 2021", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "https://www.computingonline.net/computing/article/view/1616; https://www.researchgate.net/publication/349173096_SIMPLE_EFFECTIVE_FAST_INVERSE_SQUARE_ROOT_ALGORITHM_WITH_TWO_MAGIC_CONSTANTS", acknowledgement = ack-nhfb, keywords = "FISR algorithm; floating-point arithmetic; FMA function; Householder.; IEEE 754 standard; initial approximation; inverse square root; magic constant; maximum relative error; Newton-Raphson", } @Article{Johansson:2019:CHF, author = "Fredrik Johansson", title = "Computing Hypergeometric Functions Rigorously", journal = j-TOMS, volume = "45", number = "3", pages = "30:1--30:26", month = aug, year = "2019", CODEN = "ACMSCU", DOI = "https://doi.org/10.1145/3328732", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Tue Sep 3 17:49:22 MDT 2019", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib", URL = "https://dl.acm.org/citation.cfm?id=3328732", abstract = "We present an efficient implementation of hypergeometric functions in arbitrary-precision interval arithmetic. The functions $_0 F_1$, $_1 F_1$, $_2 F_1$, and $_2 F_0$ (or the Kummer $U$-function) are supported for unrestricted complex parameters and argument, and, by extension, we cover exponential and trigonometric integrals, error functions, Fresnel integrals, incomplete gamma and beta functions, Bessel functions, Airy functions, Legendre functions, Jacobi polynomials, complete elliptic integrals, and other special functions. The output can be used directly for interval computations or to generate provably correct floating-point approximations in any format. Performance is competitive with earlier arbitrary-precision software and sometimes orders of magnitude faster. We also partially cover the generalized hypergeometric function $_p F_q$ and computation of high-order parameter derivatives.", acknowledgement = ack-nhfb, articleno = "30", fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", } @InProceedings{Kumar:2019:FIT, author = "Puli Anil Kumar", booktitle = "{2019 5th International Conference on Advanced Computing \& Communication Systems (ICACCS)}", title = "{FPGA} Implementation of the Trigonometric Functions Using the {CORDIC} Algorithm", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "894--900", year = "2019", DOI = "https://doi.org/10.1109/ICACCS.2019.8728315", bibdate = "Mon Oct 27 10:32:44 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, keywords = "ArcTangent; Communication systems; Computational modeling; CORDIC; Exponential; Field programmable gate arrays; Hardware; Logarithm; Polar to Rectangular conversion; Signal processing algorithms; Table lookup; Trigonometric function", } @Article{Lemire:2019:FRD, author = "Daniel Lemire and Owen Kaser and Nathan Kurz", title = "Faster remainder by direct computation: Applications to compilers and software libraries", journal = j-SPE, volume = "49", number = "6", pages = "953--970", month = jun, year = "2019", CODEN = "SPEXBL", DOI = "https://doi.org/10.1002/spe.2689", ISSN = "0038-0644 (print), 1097-024X (electronic)", ISSN-L = "0038-0644", bibdate = "Sat Oct 12 09:43:47 MDT 2019", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/spe.bib", acknowledgement = ack-nhfb, fjournal = "Software --- Practice and Experience", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)1097-024X", keywords = "integer division; integer remainder", onlinedate = "27 February 2019", } @InProceedings{Melquiond:2019:FVS, author = "Guillaume Melquiond and Raphael Rieu-Helft", title = "Formal Verification of a State-of-the-Art Integer Square Root", crossref = "Takagi:2019:ISC", pages = "183--186", month = jun, year = "2019", DOI = "https://doi.org/10.1109/ARITH.2019.00041", bibdate = "Fri Jan 31 08:18:07 2020", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", abstract = "We present the automatic formal verification of a state-of-the-art algorithm from the GMP library that computes the square root of a 64-bit integer. Although it uses only integer operations, the best way to understand the program is to view it as a fixed-point arithmetic algorithm that implements Newton's method. The C code is short but intricate, involving magic constants and intentional arithmetic overflows. We have verified the algorithm using the Why3 tool and automated solvers such as Gappa.", acknowledgement = ack-nhfb, keywords = "64-bit integer; Approximation algorithms; ARITH-26; automatic formal verification; C code; C language; Convergence; Digital arithmetic; electronic engineering computing; fixed point arithmetic; Fixed-point arithmetic; fixed-point arithmetic algorithm; floating point arithmetic; Floors; Formal verification; GMP library; integer operations; integer square root; intentional arithmetic overflows; Libraries; Newton method; program verification; programming; Tools; Why3 tool", } @Article{Miyajima:2019:VCM, author = "Shinya Miyajima", title = "Verified computation for the matrix {Lambert} {$W$} function", journal = j-APPL-MATH-COMP, volume = "362", number = "??", pages = "Article 124555", day = "1", month = dec, year = "2019", CODEN = "AMHCBQ", ISSN = "0096-3003 (print), 1873-5649 (electronic)", ISSN-L = "0096-3003", bibdate = "Fri Sep 6 09:21:26 MDT 2019", bibsource = "https://www.math.utah.edu/pub/tex/bib/applmathcomput2015.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "https://www.sciencedirect.com/science/article/pii/S0096300319305387", acknowledgement = ack-nhfb, fjournal = "Applied Mathematics and Computation", journal-URL = "http://www.sciencedirect.com/science/journal/00963003", } @Article{Mopuri:2019:CRB, author = "Suresh Mopuri and Swati Bhardwaj and Amit Acharyya", title = "Coordinate Rotation-Based Design Methodology for Square Root and Division Computation", journal = j-IEEE-TRANS-CIRCUITS-SYST-II-EXPRESS-BRIEFS, volume = "66", number = "7", pages = "1227--1231", year = "2019", DOI = "https://doi.org/10.1109/TCSII.2018.2878599", ISSN = "1549-7747 (print), 1558-3791 (electronic)", ISSN-L = "1549-7747", bibdate = "Mon Oct 27 10:32:44 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "IEEE Transactions on Circuits and Systems II: Express Briefs", journal-URL = "https://ieeexplore.ieee.org/xpl/issues?punumber=8920", keywords = "Circuits and systems; Computer architecture; CORDIC; Covariance matrices; Design methodology; division; Hardware; Matrix decomposition; Monitoring; Square root", } @Article{Mopuri:2019:CRM, author = "Suresh Mopuri and Amit Acharyya", title = "Configurable Rotation Matrix of Hyperbolic {CORDIC} for Any Logarithm and Its Inverse computation", journal = j-CSSP, volume = "39", number = "5", pages = "2551--2573", month = sep, year = "2019", CODEN = "CSSPEH", DOI = "https://doi.org/10.1007/s00034-019-01277-w", ISSN = "1531-5878", ISSN-L = "0278-081X", bibdate = "Tue Oct 28 07:04:09 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Circuits, systems, and signal processing: {CSSP}", journal-URL = "http://link.springer.com/journal/34", } @Article{Moroz:2019:EFP, author = "Leonid Moroz and Volodymyr Samotyy", title = "Efficient Floating-Point Division for Digital Signal Processing Application [Tips \& Tricks]", journal = j-IEEE-SIGNAL-PROCESS-MAG, volume = "36", number = "1", pages = "159--163", month = jan, year = "2019", CODEN = "ISPRE6", DOI = "https://doi.org/10.1109/msp.2018.2875977", ISSN = "1053-5888 (print), 1558-0792 (electronic)", ISSN-L = "1053-5888", bibdate = "Thu Apr 10 15:04:29 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, fjournal = "IEEE Signal Processing Magazine", } @Article{Nemes:2019:AEI, author = "Gerg{\H{o}} Nemes and Adri B. Olde Daalhuis", title = "Asymptotic expansions for the incomplete gamma function in the transition regions", journal = j-MATH-COMPUT, volume = "88", number = "318", pages = "1805--1827", month = apr, year = "2019", CODEN = "MCMPAF", DOI = "https://doi.org/10.1090/mcom/3391", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Jul 14 06:45:40 MDT 2020", bibsource = "http://www.ams.org/mcom/2019-88-318; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp2010.bib", URL = "https://www.ams.org/journals/mcom/2019-88-318/S0025-5718-2018-03391-9; https://www.ams.org/journals/mcom/2019-88-318/S0025-5718-2018-03391-9/S0025-5718-2018-03391-9.pdf; https://www.ams.org/mathscinet/search/authors.html?authorName=Nemes%2C%20Gergo; https://www.ams.org/mathscinet/search/authors.html?mrauthid=293428", acknowledgement = ack-nhfb, fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Parhi:2019:CAF, author = "Keshab K. Parhi and Yin Liu", title = "Computing Arithmetic Functions Using Stochastic Logic by Series Expansion", journal = j-IEEE-TRANS-EMERG-TOP-COMPUT, volume = "7", number = "1", pages = "44--59", month = jan # "\slash " # mar, year = "2019", DOI = "https://doi.org/10.1109/TETC.2016.2618750", ISSN = "2168-6750 (print), 2376-4562 (electronic)", bibdate = "Thu Sep 21 14:02:06 2023", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetransemergtopcomput.bib", abstract = "Stochastic logic implementations of complex arithmetic functions, such as trigonometric, exponential, and sigmoid, are derived based on truncated versions of their Maclaurin series expansions. This paper makes three contributions. First, it is shown that a polynomial can be implemented using multiple levels of NAND gates based on Horner's rule, if the coefficients are alternately positive and negative and their magnitudes are monotonically decreasing. Truncated Maclaurin series expansions of arithmetic functions are used to generate polynomials which satisfy these constraints. The input and output in these functions are represented by unipolar representation. Functions including sine, cosine, tangent hyperbolic, logarithm and exponential can be implemented using this method. Second, for a polynomial that does not satisfy these constraints, it still can be implemented based on Horner's rule if each factor of the polynomial satisfies these constraints. It is shown that functions such as $ \sin \pi x / \pi $, $ e^{-a x} $, $ \tanh a x $ and $ \sigmoid (a x^3) $ (for values of $ a > 1$) can be implemented using stochastic logic using factorization in combination with Horner's rule. Third, format conversion is proposed for arithmetic functions with input and output represented in different formats, such as $ \cos \pi x$ given $ x \in [0, 1]$ and $ \sigmoid (x)$ given $ x \in [ - 1, 1]$. Polynomials are transformed to equivalent forms that naturally exploit format conversions. The proposed stochastic logic circuits outperform the well-known Bernstein polynomial based and finite-state-machine (FSM) based implementations. Furthermore, the hardware complexity and the critical path of the proposed implementations are less than the well-known Bernstein polynomial based and FSM based implementations for most cases", acknowledgement = ack-nhfb, fjournal = "IEEE Transactions on Emerging Topics in Computing", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=6245516", } @Article{Qi:2019:CMD, author = "Feng Qi and Ai-Qi Liu", title = "Completely monotonic degrees for a difference between the logarithmic and psi functions", journal = j-J-COMPUT-APPL-MATH, volume = "361", number = "??", pages = "366--371", day = "1", month = dec, year = "2019", CODEN = "JCAMDI", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Fri Sep 6 08:23:29 MDT 2019", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2015.bib", URL = "https://www.sciencedirect.com/science/article/pii/S0377042719302298", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @Article{Shterenlikht:2019:QIF, author = "A. Shterenlikht", title = "On Quality of Implementation of {Fortran 2008} Complex Intrinsic Functions on Branch Cuts", journal = j-TOMS, volume = "45", number = "1", pages = "11:1--11:9", month = mar, year = "2019", CODEN = "ACMSCU", DOI = "https://doi.org/10.1145/3301318", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Mon May 6 18:23:42 MDT 2019", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fortran3.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib", URL = "https://dl.acm.org/citation.cfm?id=3301318", abstract = "Branch cuts in complex functions have important uses in fracture mechanics, jet flow, and aerofoil analysis. This article introduces tests for validating Fortran 2008 complex functions-LOG, SQRT, ASIN, ACOS, ATAN, ASINH, ACOSH, and ATANH-on branch cuts with arguments of all 3 IEEE floating-point binary formats: binary32, binary64, and binary128, including signed zero and signed infinity. Multiple test failures were revealed, such as wrong signs of results or unexpected overflow, underflow, or NaN. We conclude that the quality of implementation of these Fortran 2008 intrinsics in many compilers is not yet sufficient to remove the need for special code for branch cuts. The electronic appendix contains the full test results with 8 Fortran 2008 compilers: GCC, Flang, Cray, Oracle, PGI, Intel, NAG, and IBM, detailed derivations of the values of these functions on branch cuts and conformal maps of the branch cuts, to be used as a reference. The tests and the results are freely available from https://cmplx.sourceforge.io. This work will be of interest to engineers who use complex functions, as well as to compiler and math library developers.", acknowledgement = ack-nhfb, articleno = "11", fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", } @InProceedings{Volkova:2019:SAI, author = "Anastasia Volkova and Jean-Michel Muller", title = "Semi-Automatic Implementation of the Complementary Error Function", crossref = "Takagi:2019:ISC", pages = "167--174", month = jun, year = "2019", DOI = "https://doi.org/10.1109/ARITH.2019.00039", bibdate = "Fri Jan 31 08:18:07 2020", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", abstract = "The normal and complementary error functions are ubiquitous special functions for any mathematical library. They have a wide range of applications. Practical applications call for customized implementations that have strict accuracy requirements. Accurate numerical implementation of these functions is, however, non-trivial. In particular, the complementary error function erfc for large positive arguments heavily suffers from cancellation, which is largely due to its asymptotic behavior. We provide a semi-automatic code generator for the erfc function which is parameterized by the user-given bound on the relative error. Our solution exploits the asymptotic expression of erfc and leverages the automatic code generator Metalibm that provides accurate polynomial approximations. A fine-grained a priori error analysis provides a libm developer with the required accuracy for each step of the evaluation. In critical parts, we exploit double-word arithmetic to achieve implementations that are fast, yet accurate up to 50 bits, even for large input arguments. We demonstrate that for high required accuracies the automatically generated code has performance comparable to that of the standard libm and for lower ones our code demonstrated roughly 25\% speedup.", acknowledgement = ack-nhfb, keywords = "a priori error analysis; ARITH-26; asymptotic behavior; asymptotic expression; complementary error functions; Digital arithmetic; Error analysis; error analysis; error function; floating-point arithmetic; Generators; Libraries; Lips; mathematical library; Metalibm; normal error functions; polynomial approximation; polynomial approximations; program compilers; semi-automated code generation; semiautomatic code generator; semiautomatic implementation; Standards; Tools; ubiquitous special functions", } @Article{Walczyk:2019:MFI, author = "Cezary J. Walczyk and Leonid V. Moroz and Jan L. Cie{\'s}li{\'n}ski", title = "A Modification of the Fast Inverse Square Root Algorithm", journal = "Computation", volume = "7", number = "3", pages = "41", month = aug, year = "2019", DOI = "https://doi.org/10.3390/computation7030041", ISSN = "2079-3197", ISSN-L = "2079-3197", bibdate = "Thu Apr 10 15:20:23 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, } @Article{Zaghloul:2019:RO, author = "Mofreh R. Zaghloul", title = "Remark on {`Algorithm 680: Evaluation of the Complex Error Function': Cause and Remedy for the Loss of Accuracy Near the Real Axis}", journal = j-TOMS, volume = "45", number = "2", pages = "24:1--24:3", month = apr, year = "2019", CODEN = "ACMSCU", DOI = "https://doi.org/10.1145/3309681", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Mon May 6 18:23:42 MDT 2019", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib", URL = "https://dl.acm.org/citation.cfm?id=3309681", abstract = "In this remark, we identify the cause of the loss of accuracy in the computation of the Faddeyeva function, $ w(z) $, near the real axis when using Algorithm 680. We provide a simple correction to this problem that allows us to restore this code as one of the important reference routines for accuracy comparisons.", acknowledgement = ack-nhfb, articleno = "24", fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", } @Article{Zou:2019:ARH, author = "Xiafeng Zou and Mingjiang Wang", title = "Algorithm Research and Hardware Implementation of High Precision Floating Point Exponential Function", journal = j-J-PHYS-CONF-SER, volume = "1345", number = "4", pages = "042085", month = nov, year = "2019", CODEN = "JPCSDZ", DOI = "https://doi.org/10.1088/1742-6596/1345/4/042085", ISSN = "1742-6588 (print), 1742-6596 (electronic)", ISSN-L = "1742-6588", bibdate = "Tue Nov 11 13:46:40 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Journal of Physics: Conference Series", journal-URL = "http://www.iop.org/EJ/journal/conf", keywords = "CORD vector mode; exp(); IEEE 754 64-bit floating-point", } @Article{Abergel:2020:AFA, author = "R{\'e}my Abergel and Lionel Moisan", title = "{Algorithm 1006}: Fast and Accurate Evaluation of a Generalized Incomplete Gamma Function", journal = j-TOMS, volume = "46", number = "1", pages = "10:1--10:24", month = mar, year = "2020", CODEN = "ACMSCU", DOI = "https://doi.org/10.1145/3365983", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Tue Apr 7 10:39:23 MDT 2020", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib", URL = "https://dl.acm.org/doi/abs/10.1145/3365983", abstract = "We present a computational procedure to evaluate the integral $ \int^y_x s^{p - 1} e^{- \mu s} \, d s $ for $ 0 \leq x < y \leq + \infty $, $ \mu = \pm 1 $, $ p > 0 $, which generalizes the lower $ (x = 0) $ and upper $ (y = + \infty) $ incomplete gamma functions. To allow for large values of $x$, $y$, and $p$ while avoiding under\slash overflow issues in the standard double precision floating point arithmetic, we use an explicit normalization that is much more efficient than the classical ratio with the complete gamma function. The generalized incomplete gamma function is estimated with continued fractions, with integrations by parts, or, when $ x \approx y$, with the Romberg numerical integration algorithm. We show that the accuracy reached by our algorithm improves a recent state-of-the-art method by two orders of magnitude, and it is essentially optimal considering the limitations imposed by floating point arithmetic. Moreover, the admissible parameter range of our algorithm $ (0 \leq p, x, y \leq 10^{15})$ is much larger than competing algorithms, and its robustness is assessed through massive usage in an image processing application.", acknowledgement = ack-nhfb, articleno = "10", fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", } @Article{Adell:2020:RAE, author = "Jos{\'e} A. Adell and Alberto Lekuona", title = "Rational approximation to {Euler}'s constant at a geometric rate of convergence", journal = j-MATH-COMPUT, volume = "89", number = "325", pages = "2553--2561", month = jan, year = "2020", CODEN = "MCMPAF", DOI = "https://doi.org/10.1090/mcom/3528", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Jul 14 07:56:12 MDT 2020", bibsource = "http://www.ams.org/mcom/2020-89-325; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp2020.bib", URL = "https://www.ams.org/AMSMathViewer; https://www.ams.org/journals/mcom/2020-89-325/S0025-5718-2020-03528-5; https://www.ams.org/journals/mcom/2020-89-325/S0025-5718-2020-03528-5/S0025-5718-2020-03528-5.pdf; https://www.ams.org/mathscinet/search/authors.html?mrauthid=340766; https://www.ams.org/mathscinet/search/authors.html?mrauthid=663604", acknowledgement = ack-nhfb, fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Chen:2020:HCB, author = "Hui Chen and Kaifeng Cheng and Zhonghai Lu and Yuxiang Fu and Li Li", title = "Hyperbolic {CORDIC}-based Architecture for Computing Logarithm and Its Implementation", journal = j-IEEE-TRANS-CIRCUITS-SYST-II-EXPRESS-BRIEFS, volume = "67", number = "11", pages = "2652--2656", year = "2020", DOI = "https://doi.org/10.1109/TCSII.2020.2971974", ISSN = "1549-7747 (print), 1558-3791 (electronic)", ISSN-L = "1549-7747", bibdate = "Mon Oct 27 10:32:44 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "IEEE Transactions on Circuits and Systems II: Express Briefs", journal-URL = "https://ieeexplore.ieee.org/xpl/issues?punumber=8920", keywords = "CMOS technology; Computational modeling; Computer architecture; Convergence; Hardware; Hyperbolic CORDIC; logarithm; pipelined structure; shift-add operations; Software", } @Article{Chin:2020:PPW, author = "Wooyoung Chin", title = "A Probabilistic Proof of a {Wallis}-type Formula for the Gamma Function", journal = j-AMER-MATH-MONTHLY, volume = "127", number = "1", pages = "75--79", year = "2020", CODEN = "AMMYAE", DOI = "https://doi.org/10.1080/00029890.2020.1668708", ISSN = "0002-9890 (print), 1930-0972 (electronic)", ISSN-L = "0002-9890", bibdate = "Mon Dec 13 15:45:45 MST 2021", bibsource = "https://www.math.utah.edu/pub/tex/bib/amermathmonthly2020.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "American Mathematical Monthly", journal-URL = "http://www.jstor.org/journals/00029890.html; https://www.tandfonline.com/loi/uamm20", onlinedate = "19 Dec 2019", } @InProceedings{Dutt:2020:HSL, author = "Rashi Dutt and Amit Acharyya", booktitle = "{2020 European Conference on Circuit Theory and Design (ECCTD)}", title = "A High Speed and Low Complexity Architecture Design Methodology for Square Root Unscented {Kalman} Filter based {SLAM}", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "1--4", year = "2020", DOI = "https://doi.org/10.1109/ECCTD49232.2020.9218287", bibdate = "Mon Oct 27 10:32:44 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, keywords = "Complexity theory; Computational modeling; Computer architecture; Covariance matrices; Householder CORDIC; Kalman filters; Low Complexity Architecture; Simultaneous localization and mapping; Simultaneous Localization and Mapping; Square Root Unscented Kalman Filter", } @Article{Ewart:2020:PES, author = "Timoth{\'e}e Ewart and Francesco Cremonesi and Felix Sch{\"u}rmann and Fabien Delalondre", title = "Polynomial Evaluation on Superscalar Architecture, Applied to the Elementary Function $ e^x $", journal = j-TOMS, volume = "46", number = "3", pages = "28:1--28:22", month = sep, year = "2020", CODEN = "ACMSCU", DOI = "https://doi.org/10.1145/3408893", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Sat Sep 26 07:28:19 MDT 2020", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib", URL = "https://dl.acm.org/doi/10.1145/3408893", abstract = "The evaluation of small degree polynomials is critical for the computation of elementary functions. It has been extensively studied and is well documented. In this article, we evaluate existing methods for polynomial evaluation on superscalar architecture. In addition, we have completed this work with a factorization method, which is surprisingly neglected in the literature. This work focuses on out-of-order Intel processors, amongst others, of which computational units are available. Moreover, we applied our work on the elementary function $ e^x $ that requires, in the current implementation, an evaluation of a polynomial of degree 10 for a satisfying precision and performance. Our results show that the factorization scheme is the fastest in benchmarks, and that latency and throughput are intrinsically dependent on each other on superscalar architecture.", acknowledgement = ack-nhfb, articleno = "28", fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", } @Article{Gil:2020:NEA, author = "A. Gil and J. Segura and N. M. Temme", title = "Numerical evaluation of {Airy}-type integrals arising in uniform asymptotic analysis", journal = j-J-COMPUT-APPL-MATH, volume = "371", number = "??", pages = "Article 112717", month = jun, year = "2020", CODEN = "JCAMDI", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", bibdate = "Wed May 13 06:58:32 MDT 2020", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2020.bib", URL = "http://www.sciencedirect.com/science/article/pii/S037704272030008X", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @Article{Gimbutas:2020:EAF, author = "Zydrunas Gimbutas and Shidong Jiang and Li-Shi Luo", title = "Evaluation of {Abramowitz} functions in the right half of the complex plane", journal = j-J-COMPUT-PHYS, volume = "405", number = "??", pages = "Article 109169", day = "15", month = mar, year = "2020", CODEN = "JCTPAH", DOI = "https://doi.org/10.1016/j.jcp.2019.109169", ISSN = "0021-9991 (print), 1090-2716 (electronic)", ISSN-L = "0021-9991", bibdate = "Mon Mar 9 18:28:24 MDT 2020", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputphys2020.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0021999119308745", acknowledgement = ack-nhfb, fjournal = "Journal of Computational Physics", journal-URL = "http://www.sciencedirect.com/science/journal/00219991", keywords = "Abramowitz functions; Laurent series; Least squares method", remark = "The Abramowitz functions of order n, defined by $ J_n(z) = \int_0^\infty t^n \exp ( - t^2 - z / t) \, d t $, for $ n \in \mathbb {Z} $.", } @Article{Godunov:2020:ACC, author = "A. Godunov", title = "Algorithms for Calculating Correctly Rounded Exponential Function in Double-Precision Arithmetic", journal = j-IEEE-TRANS-COMPUT, volume = "69", number = "9", pages = "1388--1400", month = sep, year = "2020", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/TC.2020.2972901", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Wed Aug 12 14:58:16 2020", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput2020.bib", abstract = "Correct rounding provides the best approximation of the exponential function by double-precision numbers. To obtain the correctly rounded exponential of some arguments, the exponential should be calculated with high accuracy. For small arguments, even higher accuracy is required. This article presents simple and very fast algorithms for small arguments. Yet another algorithm presented here demonstrates a good maximum execution time, which may be important for critical applications. This algorithm can be combined with some other already existing algorithms to achieve the best maximum and average execution times. All proposed algorithms calculate the correctly rounded exponential function for all rounding modes and use only double-precision arithmetic for computation. In the argument reduction step, precalculated tables are used. Test implementations of these algorithms were developed in C language and are portable. Full proofs are presented either in this article itself or in its appendices.", acknowledgement = ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", } @Article{Harper:2020:AGH, author = "J. F. Harper", title = "Asymptotics of a {Gauss} hypergeometric function with two large parameters: a new case", journal = j-ANZIAM-J, volume = "62", number = "4", pages = "446--452", month = oct, year = "2020", CODEN = "AJNOA2", DOI = "https://doi.org/10.1017/S1446181119000166", ISSN = "1446-1811 (print), 1446-8735 (electronic)", ISSN-L = "1446-1811", bibdate = "Fri May 14 17:04:43 MDT 2021", bibsource = "https://www.math.utah.edu/pub/tex/bib/anziamj.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "https://www.cambridge.org/core/journals/anziam-journal/article/asymptotics-of-a-gauss-hypergeometric-function-with-two-large-parameters-a-new-case/32B40986E7DB85F500FC9024F846E527", acknowledgement = ack-nhfb, ajournal = "ANZIAM J.", fjournal = "The ANZIAM Journal. The Australian \& New Zealand Industrial and Applied Mathematics Journal", journal-URL = "http://journals.cambridge.org/action/displayJournal?jid=ANZ", onlinedate = "10 December 2019", } @Article{Hrycak:2020:ELP, author = "Tomasz Hrycak and Sebastian Schmutzhard", title = "Evaluation of {Legendre} polynomials by a three-term recurrence in floating-point arithmetic", journal = j-IMA-J-NUMER-ANAL, volume = "40", number = "1", pages = "587--605", month = jan, year = "2020", CODEN = "IJNADH", DOI = "https://doi.org/10.1093/imanum/dry079", ISSN = "0272-4979 (print), 1464-3642 (electronic)", ISSN-L = "0272-4979", bibdate = "Sat Feb 29 14:22:43 MST 2020", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/imajnumeranal.bib", URL = "http://academic.oup.com/imajna/article/40/1/587/5162990", acknowledgement = ack-nhfb, fjournal = "IMA Journal of Numerical Analysis", journal-URL = "http://imajna.oxfordjournals.org/content/by/year", } @Article{Jablonski:2020:IAC, author = "A. Jablonski", title = "Improved algorithm for calculating high accuracy values of the {Chandrasekhar} function", journal = j-COMP-PHYS-COMM, volume = "251", number = "??", pages = "Article 107237", month = jun, year = "2020", CODEN = "CPHCBZ", DOI = "https://doi.org/10.1016/j.cpc.2020.107237", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", bibdate = "Fri May 29 07:03:02 MDT 2020", bibsource = "https://www.math.utah.edu/pub/tex/bib/compphyscomm2020.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0010465520300709", acknowledgement = ack-nhfb, fjournal = "Computer Physics Communications", journal-URL = "http://www.sciencedirect.com/science/journal/00104655", } @Article{Johansson:2020:CLW, author = "Fredrik Johansson", title = "Computing the {Lambert $W$} function in arbitrary-precision complex interval arithmetic", journal = j-NUMER-ALGORITHMS, volume = "83", number = "1", pages = "221--242", month = jan, year = "2020", CODEN = "NUALEG", DOI = "https://doi.org/10.1007/s11075-019-00678-x", ISSN = "1017-1398 (print), 1572-9265 (electronic)", ISSN-L = "1017-1398", bibdate = "Wed Jan 22 08:40:22 MST 2020", bibsource = "http://link.springer.com/journal/11075/83/1; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/numeralgorithms.bib", acknowledgement = ack-nhfb, fjournal = "Numerical Algorithms", journal-URL = "http://link.springer.com/journal/11075", } @InCollection{Johansson:2020:FSL, author = "Fredrik Johansson", title = "{FunGrim}: A Symbolic Library for Special Functions", crossref = "Bigatti:2020:MSI", pages = "315--323", year = "2020", DOI = "https://doi.org/10.1007/978-3-030-52200-1_31", bibdate = "Sat Sep 23 06:47:37 2023", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, } @Article{Johnson:2020:EAHa, author = "Jeff Johnson", title = "Efficient, arbitrarily high precision hardware logarithmic arithmetic for linear algebra", journal = "arxiv.org", volume = "??", number = "??", pages = "1--8", day = "14", month = may, year = "2020", bibdate = "Tue Jul 06 18:17:13 2021", bibsource = "https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "https://arxiv.org/pdf/2004.09313.pdf", abstract = "The logarithmic number system (LNS) is arguably not broadly used due to exponential circuit overheads for summation tables relative to arithmetic precision. Methods to reduce this overhead have been proposed, yet still yield designs with high chip area and power requirements. Use remains limited to lower precision or high multiply/add ratio cases, while much of linear algebra (near 1:1 multiply/add ratio) does not qualify.\par We present a dual-base approximate logarithmic arithmetic comparable to floating point in use, yet unlike LNS it is easily fully pipelined, extendable to arbitrary precision with $ O(n^2) $ overhead, and energy efficient at a 1:1 multiply/add ratio.Compared to float32 or float64 vector inner product with FMA, our design is respectively $ 2.3 \times $ and $ 4.6 \times $ more energy efficient in 7 nm CMOS. It depends on exp and log evaluation $ 5.4 \times $ and $ 3.2 \times $ more energy efficient, at $ 0.23 \times $ and $ 0.37 \times $ the chip area for equivalent accuracy versus standard hyperbolic CORDIC using shift-and-add and approximated ODE integration in the style of Revol and Yakoubsohn. This technique is a novel alternative for low power, high precision hardened linear algebra in computer vision, graphics and machine learning applications.", acknowledgement = ack-nhfb, keywords = "approximate arithmetic; elementary function evaluation; hardware linear algebra; logarithmic arithmetic", remark = "Published in \cite{Johnson:2020:EAHb}.", } @InProceedings{Johnson:2020:EAHb, author = "Jeff Johnson", title = "Efficient, arbitrarily high precision hardware logarithmic arithmetic for linear algebra", crossref = "Cornea:2020:ISC", pages = "25--32", month = jun, year = "2020", DOI = "https://doi.org/10.1109/ARITH48897.2020.00013", ISSN = "2576-2265", bibdate = "Wed Jul 7 06:24:52 2021", bibsource = "https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", abstract = "The logarithmic number system (LNS) is arguably not broadly used due to exponential circuit overheads for summation tables relative to arithmetic precision. Methods to reduce this overhead have been proposed, yet still yield designs with high chip area and power requirements. Use remains limited to lower precision or high multiply/add ratio cases, while much of linear algebra (near 1:1 multiply/add ratio) does not qualify. We present a dual-base approximate logarithmic arithmetic comparable to floating point in use, yet unlike LNS it is easily fully pipelined, extendable to arbitrary precision with $ O(n^2) $ overhead, and energy efficient at a 1:1 multiply/add ratio. Compared to float32 or float64 vector inner product with FMA, our design is respectively $ 2.3 \times $ and $ 4.6 \times $ more energy efficient in 7 nm CMOS. It depends on exp and log evaluation 5.4 and $ 3.2 \times $ more energy efficient, at $ 0.23 \times $ and $ 0.37 \times $ the chip area for equivalent accuracy versus standard hyperbolic CORDIC using shift-and-add and approximated ODE integration in the style of Revol and Yakoubsohn. This technique is a novel alternative for low power, high precision hardened linear algebra in computer vision, graphics and machine learning applications.", acknowledgement = ack-nhfb, keywords = "Adders; approximate arithmetic; Clocks; elementary function evaluation; Hardware; hardware linear algebra; Linear algebra; logarithmic arithmetic; Pipeline processing; Read only memory; Switches", } @Book{Korotkov:2020:IRE, author = "N. E. (Nikola{\'y}i Efimovich) Korotkov and Alexander N. Korotkov", title = "Integrals Related to the Error Function", publisher = "CRC Press, Taylor and Francis Group", address = "Boca Raton, FL, USA", pages = "xx + 227", year = "2020", ISBN = "0-367-40820-1 (hardcover), 0-367-80923-0 (e-book), 1-000-03307-4 (e-book), 1-000-03308-2 (Mobipocket e-book), 1-000-03309-0 (e-Pub)", ISBN-13 = "978-0-367-40820-6 (hardcover), 978-0-367-80923-2 (e-book), 978-1-000-03307-6 (e-book), 978-1-000-03308-3 (Mobipocket e-book), 978-1-000-03309-0 (e-Pub)", LCCN = "QA308 .K67 2020", bibdate = "Fri Feb 5 17:54:22 MST 2021", bibsource = "fsz3950.oclc.org:210/WorldCat; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", abstract = "\booktitle{Integrals Related to the Error Function} presents a table of integrals related to the error function, including indefinite and improper definite integrals. Most of the formulas in this book have not been presented in other tables of integrals or have been presented only for some special cases of parameters or for integration only along the real axis of the complex plane. Many of the integrals presented here cannot be obtained using a computer (except via an approximate numerical integration). Additionally, for improper integrals, this book emphasizes the necessary and sufficient conditions for the validity of the presented formulas, including trajectory for going to infinity on the complex plane; such conditions are usually not given in computer-assisted analytical integration and often not presented in the previously published tables of integrals. Features The first book in English language to present a comprehensive collection of integrals related to the error function Useful for researchers whose work involves the error function (e.g., via probability integrals in communication theory). Additionally, it can also be used by broader audience.", acknowledgement = ack-nhfb, subject = "Integrals; Tables; Error functions", tableofcontents = "Indefinite integrals \\ Definite integrals \\ Appendix: Some useful formulas for obtaining other integrals.", } @Article{Muller:2020:EFA, author = "Jean-Michel Muller", title = "Elementary Functions and Approximate Computing", journal = j-PROC-IEEE, volume = "108", number = "12", pages = "2136--2149", month = dec, year = "2020", CODEN = "IEEPAD", DOI = "https://doi.org/10.1109/jproc.2020.2991885", ISSN = "0018-9219 (print), 1558-2256 (electronic)", ISSN-L = "0018-9219", bibdate = "Tue Mar 1 06:07:02 2022", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", abstract = "In this article, we review some of the classical methods used for quickly obtaining low-precision approximations to the elementary functions. Then, for each of the three main classes of elementary function algorithms (shift-and-add algorithms, polynomial or rational approximations, and table-based methods) and for the additional, specific to approximate computing, ``bit-manipulation'' techniques, we examine what can be done for obtaining very fast estimates of a function, at the cost of a (controlled) loss in terms of accuracy.", acknowledgement = ack-nhfb, fjournal = "Proceedings of the IEEE", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=5", } @Article{Naterop:2020:HRN, author = "L. Naterop and A. Signer and Y. Ulrich", title = "{handyG} --- Rapid numerical evaluation of generalised polylogarithms in {Fortran}", journal = j-COMP-PHYS-COMM, volume = "253", number = "??", pages = "Article 107165", month = aug, year = "2020", CODEN = "CPHCBZ", DOI = "https://doi.org/10.1016/j.cpc.2020.107165", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", bibdate = "Fri Jun 19 07:19:48 MDT 2020", bibsource = "https://www.math.utah.edu/pub/tex/bib/compphyscomm2020.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fortran3.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0010465520300230", acknowledgement = ack-nhfb, fjournal = "Computer Physics Communications", journal-URL = "http://www.sciencedirect.com/science/journal/00104655", } @TechReport{Pornin:2020:OBG, author = "Thomas Pornin", title = "Optimized Binary {GCD} for Modular Inversion", type = "Report", number = "??", institution = "International Association for Cryptologic Research", address = "????", pages = "16", day = "23", month = aug, year = "2020", bibdate = "Mon May 30 07:10:10 2022", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "https://eprint.iacr.org/2020/972.pdf", abstract = "In this short note, we describe a practical optimization of the well-known extended binary GCD algorithm, for the purpose of computing modular inverses. The method is conceptually simple and is applicable to all odd moduli (including non-prime moduli). When implemented for inversion in the field of integers modulo the prime $ 2^{255} - 19 $, on a recent x86 CPU (Coffee Lake core), we compute the inverse in 6253 cycles, with a fully constant-time implementation.", acknowledgement = ack-nhfb, } @InProceedings{Raveendran:2020:NPF, author = "Aneesh Raveendran and Sandra Jean and J. Mervin and D. Vivian and David Selvakumar", editor = "{IEEE}", booktitle = "{2020 33rd International Conference on VLSI Design and 2020 19th International Conference on Embedded Systems (VLSID), Bengaluru, India, 4--8 January 2020}", title = "A Novel Parametrized Fused Division and Square-Root {POSIT} Arithmetic Architecture", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "207--212", month = jan, year = "2020", DOI = "https://doi.org/10.1109/vlsid49098.2020.00053", ISBN = "1-72815-701-3", ISBN-13 = "978-1-72815-701-6", ISSN = "1063-9667 (print), 2380-6923 (electronic)", ISSN-L = "1063-9667", bibdate = "Fri Dec 15 07:29:26 2023", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, } @Article{Shibata:2020:SPV, author = "Naoki Shibata and Francesco Petrogalli", title = "{SLEEF}: a Portable Vectorized Library of {C} Standard Mathematical Functions", journal = j-IEEE-TRANS-PAR-DIST-SYS, volume = "31", number = "6", pages = "1316--1327", month = jun, year = "2020", CODEN = "ITDSEO", DOI = "https://doi.org/10.1109/TPDS.2019.2960333", ISSN = "1045-9219 (print), 1558-2183 (electronic)", ISSN-L = "1045-9219", bibdate = "Thu Feb 20 10:08:58 2020", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranspardistsys2020.bib", acknowledgement = ack-nhfb, ajournal = "IEEE Trans. Parallel Distrib. Syst.", fjournal = "IEEE Transactions on Parallel and Distributed Systems", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=71", keywords = "elementary functions; floating-point arithmetic; Parallel and vector implementations; SIMD processors", } @Article{Wakhare:2020:TCJ, author = "Tanay Wakhare and Christophe Vignat", title = "{Taylor} coefficients of the {Jacobi} $ \theta_3 (q) $ function", journal = j-J-NUMBER-THEORY, volume = "216", number = "??", pages = "280--306", month = nov, year = "2020", CODEN = "JNUTA9", DOI = "https://doi.org/10.1016/j.jnt.2020.03.002", ISSN = "0022-314X (print), 1096-1658 (electronic)", ISSN-L = "0022-314X", bibdate = "Sat Aug 8 09:41:52 MDT 2020", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jnumbertheory2020.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0022314X20300858", acknowledgement = ack-nhfb, ajournal = "J. Number Theory", fjournal = "Journal of Number Theory", journal-URL = "http://www.sciencedirect.com/science/journal/0022314X", } @Article{Xiao:2020:PAH, author = "Feibao Xiao and Feng Liang and Bin Wu and Junzhe Liang and Shuting Cheng and Guohe Zhang", title = "Posit Arithmetic Hardware Implementations with The Minimum Cost Divider and Square Root", journal = j-ELECTRONICS, volume = "9", number = "10", pages = "1622:1--1622:16", month = oct, year = "2020", DOI = "https://doi.org/10.3390/electronics9101622", ISSN = "2079-9292", ISSN-L = "2079-9292", bibdate = "Fri Dec 15 07:25:40 2023", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, fjournal = "Electronics", journal-URL = "https://www.mdpi.com/journal/electronics", } @Article{Akram:2021:XFA, author = "Riad Akram and Shantanu Mandal and Abdullah Muzahid", title = "{XMeter}: Finding Approximable Functions and Predicting Their Accuracy", journal = j-IEEE-TRANS-COMPUT, volume = "70", number = "7", pages = "1081--1093", month = jul, year = "2021", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/TC.2020.3005083", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Thu Jun 10 15:51:57 2021", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput2020.bib", acknowledgement = ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", } @Misc{Bailey:2021:PMN, author = "David H. Bailey", title = "\pkg{MPFUN2020}: a new thread-safe arbitrary precision package", howpublished = "Web document", pages = "54", day = "18", month = may, year = "2021", bibdate = "Mon Dec 05 07:32:16 2022", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "https://www.davidhbailey.com/dhbpapers/mpfun2020.pdf", abstract = "Numerous research studies have arisen, particularly in mathematical physics and experimental mathematics, that require extremely high numeric precision. Such precision greatly magnifies computer run times, so software packages to support high-precision computing must be designed for thread-based parallel processing. This paper describes a new arbitrary precision software package (``MPFUN2020'') that features several significant improvements over an earlier package. It comes in two versions: a self-contained all-Fortran version, and a version based on the MPFR package, which is even faster. Both versions feature: (a) a completely thread-safe design, so user codes can be converted for parallel execution at the application level; (b) a full-featured high-level Fortran interface, so that most applications can be converted to multiprecision with relatively minor changes to source code; (c) full support for both real and complex datatypes; (d) a wide variety of transcendental functions and special functions; (e) run-time checking and other facilities to overcome problems with converting double precision constants and data; (f) a medium precision datatype, which improves performance and reduces memory cost on large variable precision applications; and (g) interoperability --- with a simple restriction, application codes written for one version can be run with the other without change.", acknowledgement = ack-nhfb, } @Article{Borges:2021:AIA, author = "Carlos F. Borges", title = "{Algorithm 1014}: an Improved Algorithm for {\tt hypot(x,y)}", journal = j-TOMS, volume = "47", number = "1", pages = "9:1--9:12", month = jan, year = "2021", CODEN = "ACMSCU", DOI = "https://doi.org/10.1145/3428446", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Thu Jan 7 10:31:04 MST 2021", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/julia.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib", URL = "https://dl.acm.org/doi/10.1145/3428446", abstract = "We develop fast and accurate algorithms for evaluating $ \sqrt {x^2 + y^2} $ for two floating-point numbers $x$ and $y$. Library functions that perform this computation are generally named {\tt hypot(x,y)}. We compare five approaches that we will develop in this article to the current resident library function that is delivered with Julia 1.1 and to the code that has been distributed with the C math library for decades. We will investigate the accuracy of our algorithms by simulation.", acknowledgement = ack-nhfb, articleno = "9", fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", } @Article{Borges:2021:CRN, author = "Carlos F. Borges", title = "A Correctly Rounded {Newton} Step for the Reciprocal Square Root", journal = "arXiv.org", volume = "??", number = "??", pages = "1--8", day = "28", month = dec, year = "2021", bibdate = "Fri Sep 22 16:08:53 2023", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "https://arxiv.org/abs/2112.14321", abstract = "The reciprocal square root is an important computation for which many sophisticated algorithms exist (see for example \ldots{} and the references therein). A common theme is the use of Newton's method to refine the estimates. In this paper we develop a correctly rounded Newton step that can be used to improve the accuracy of a naive calculation (using methods similar to those developed in \ldots{}) The approach relies on the use of the fused multiply-add (FMA) which is widely available in hardware on a variety of modern computer architectures. We then introduce the notion of {\em weak rounding} and prove that our proposed algorithm meets this standard. We then show how to leverage the exact Newton step to get a Halley's method compensation which requires one additional FMA and one additional multiplication. This method appears to give correctly rounded results experimentally and we show that it can be combined with a square root free method for estimating the reciprocal square root to get a method that is both very fast (in computing environments with a slow square root) and, experimentally, highly accurate.", acknowledgement = ack-nhfb, } @Article{Borges:2021:FCA, author = "Carlos F. Borges", title = "Fast compensated algorithms for the reciprocal square root, the reciprocal hypotenuse, and {Givens} rotations", journal = "arXiv.org", volume = "??", number = "??", pages = "1--11", day = "23", month = feb, year = "2021", bibdate = "Fri Sep 22 16:05:47 2023", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "https://arxiv.org/abs/2103.08694", abstract = "The reciprocal square root is an important computation for which many very sophisticated algorithms exist (see for example \ldots{} and the references therein). In this paper we develop a simple differential compensation (much like those developed in \ldots{}) that can be used to improve the accuracy of a naive calculation. The approach relies on the use of the fused multiply-add (FMA) which is widely available in hardware on a variety of modern computer architectures. We then demonstrate how to combine this approach with a somewhat inaccurate but fast square root free method for estimating the reciprocal square root to get a method that is both fast (in computing environments with a slow square root) and, experimentally, highly accurate. Finally, we show how this same approach can be extended to the reciprocal hypotenuse calculation and, most importantly, to the construction of Givens rotations.", acknowledgement = ack-nhfb, } @Article{Chen:2021:LCH, author = "Hui Chen and Zongguang Yu and Yonggang Zhang and Zhonghai Lu and Yuxiang Fu and Li Li", title = "Low-Complexity High-Precision Method and Architecture for Computing the Logarithm of Complex Numbers", journal = j-IEEE-TRANS-CIRCUITS-SYST-1, volume = "68", number = "8", pages = "3293--3304", year = "2021", DOI = "https://doi.org/10.1109/TCSI.2021.3081517", ISSN = "1549-8328 (print), 1558-0806 (electronic)", ISSN-L = "1549-8328", bibdate = "Mon Oct 27 10:32:44 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "IEEE Transactions on Circuits and Systems I: Regular Papers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=8919", keywords = "complex logarithm; Complex number; Complexity theory; Computer architecture; Convergence; CORDIC; Hardware; high hardware efficiency; low design complexity; Software; Synthetic aperture radar; Table lookup", } @Article{Iacono:2021:BEF, author = "Roberto Iacono", title = "Bounding the Error Function", journal = j-COMPUT-SCI-ENG, volume = "23", number = "4", pages = "65--68", year = "2021", CODEN = "CSENFA", DOI = "https://doi.org/10.1109/MCSE.2021.3083778", ISSN = "1521-9615 (print), 1558-366X (electronic)", ISSN-L = "1521-9615", bibdate = "Thu Jul 29 07:00:57 2021", bibsource = "https://www.math.utah.edu/pub/tex/bib/computscieng.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", abstract = "Prompted by previous work published in this magazine, in this article we focus on the derivation of global analytical bounds for the error function of a real argument. Using an integral representation of this function, we obtain two simple and accurate lower bounds, which complement a well-known upper bound given long ago by P{\'o}lya.", acknowledgement = ack-nhfb, fjournal = "Computing in Science and Engineering", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=5992", } @Article{Johansson:2021:APC, author = "Fredrik Johansson", title = "Arbitrary-Precision Computation of the Gamma Function", journal = "arXiv.org", pages = "1--51", day = "17", month = sep, year = "2021", DOI = "https://doi.org/10.48550/arXiv.2109.0839", bibdate = "Sun Dec 04 11:01:23 2022", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "https://arxiv.org/abs/2109.08392", abstract = "We discuss the best methods available for computing the gamma function $ \Gamma (z) $ in arbitrary-precision arithmetic with rigorous error bounds. We address different cases: rational, algebraic, real or complex arguments; large or small arguments; low or high precision; with or without precomputation. The methods also cover the log-gamma function $ \log \Gamma (z) $, the digamma function $ \psi (z) $, and derivatives $ \Gamma^{(n)}(z) $ and $ \psi^{(n)}(z) $. Besides attempting to summarize the existing state of the art, we present some new formulas, estimates, bounds and algorithmic improvements and discuss implementation results.", acknowledgement = ack-nhfb, } @Article{Kang:2021:NEE, author = "Hongchao Kang and Hong Wang", title = "Numerical evaluation and error analysis of many different oscillatory {Bessel} transforms via confluent hypergeometric function", journal = j-APPL-NUM-MATH, volume = "160", number = "??", pages = "23--41", month = feb, year = "2021", CODEN = "ANMAEL", ISSN = "0168-9274 (print), 1873-5460 (electronic)", ISSN-L = "0168-9274", bibdate = "Tue Dec 29 07:52:55 MST 2020", bibsource = "https://www.math.utah.edu/pub/tex/bib/applnummath.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0168927420302932", acknowledgement = ack-nhfb, fjournal = "Applied Numerical Mathematics: Transactions of IMACS", journal-URL = "http://www.sciencedirect.com/science/journal/01689274", } @Article{Langdon:2021:GID, author = "William B. Langdon and Oliver Krauss", title = "Genetic Improvement of Data for Maths Functions", journal = j-TELO, volume = "1", number = "2", pages = "7:1--7:30", month = jun, year = "2021", CODEN = "????", DOI = "https://doi.org/10.1145/3461016", ISSN = "2688-299X (print), 2688-3007 (electronic)", ISSN-L = "2688-299X", bibdate = "Sat Aug 21 15:11:10 MDT 2021", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/telo.bib", URL = "https://dl.acm.org/doi/10.1145/3461016", abstract = "We use continuous optimisation and manual code changes to evolve up to 1024 Newton--Raphson numerical values embedded in an open source GNU C library glibc square root sqrt to implement a double precision cube root routine cbrt, binary logarithm log2 and reciprocal square root function for C in seconds. The GI inverted square root $ x{-1 / 2} $ is far more accurate than Quake's InvSqrt, Quare root. GI shows potential for automatically creating mobile or low resource mote smart dust bespoke custom mathematical libraries with new functionality.", acknowledgement = ack-nhfb, articleno = "7", fjournal = "ACM Transactions on Evolutionary Learning and Optimization", journal-URL = "https://dl.acm.org/loi/telo", } @InProceedings{Lim:2021:HPC, author = "Jay P. Lim and Santosh Nagarakatte", booktitle = "Proceedings of the {42nd ACM SIGPLAN International Conference on Programming Language Design and Implementation}", title = "High performance correctly rounded math libraries for 32-bit floating point representations", publisher = pub-ACM, address = pub-ACM:adr, month = jun, year = "2021", DOI = "https://doi.org/10.1145/3453483.3454049", bibdate = "Tue Sep 24 15:06:44 2024", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "https://github.com/rutgers-apl/rlibm-32", abstract = "This paper proposes a set of techniques to develop correctly rounded math libraries for 32-bit float and posit types. It enhances our RLIBM approach that frames the problem of generating correctly rounded libraries as a linear programming problem in the context of 16-bit types to scale to 32-bit types. Specifically, this paper proposes new algorithms to (1) generate polynomials that produce correctly rounded outputs for all inputs using counterexample guided polynomial generation, (2) generate efficient piecewise polynomials with bit-pattern based domain splitting, and (3) deduce the amount of freedom available to produce correct results when range reduction involves multiple elementary functions. The resultant math library for the 32-bit float type is faster than state-of-the-art math libraries while producing the correct output for all inputs. We have also developed a set of correctly rounded elementary functions for 32-bit posits.", acknowledgement = ack-nhfb, description = "RLIBM-32 is both a math library that provides correctly rounded result for all inputs and tools used to generate the correct polynomials. The techniques behind the tools will be appearing at PLDI 2021. Currently, RLIBM-32 supports a number of elementary functions for float and posit32 representations.\par List of float functions supported by RLIBM-32:\par log(x), log2(x), log10(x) \\ exp(x), exp2(x), exp10(x) \\ sinh(x), cosh(x) \\ sinpi(x), cospi(x)\par List of posit32 functions supported by RLIBM-32:\par log(x), log2(x), log10(x) \\ exp(x), exp2(x), exp10(x) \\ sinh(x), cosh(x)", keywords = "binary32; correctly rounded functions; posit32", remark-1 = "From page 360: ``Our RLibm approach [31, 32] generates polynomials that approximate the correctly rounded result rather than the real value of the elementary function. \ldots{} Using the RLibm approach, we have been successful in generating correctly rounded libraries with 16-bit types such as bfloat16 and posit16.''", remark-2 = "From page 360: ``A naive use of the RLibm approach with 32-bit types will generate more than a billion constraints, which is beyond the capabilities of current LP solvers.''", remark-3 = "From page 361: ``Our elementary functions for floats are faster than existing libraries: Intel's libm, Glibc's libm, CR-LIBM [13], and Metalibm [25]. Unlike existing libraries, our functions produce correctly rounded results for all inputs. We have developed the first correctly rounded implementations of functions for 32-bit posits.''", remark-4 = "From page 363: ``All internal computation such as range reduction, polynomial evaluation, and output compensation is performed in representation H where H has higher precision than T. To attain good performance, H is a representation that is supported in hardware (e.g., double).''", remark-5 = "From page 368: ``CR-LIBM has correctly rounded functions for double precision. However, CR-LIBM does not produce correctly rounded results for 32-bit floats due to double rounding. There are no math libraries available for posit32. All posit32 values can be exactly represented in double.''", remark-6 = "From page 370: ``All three re-purposed math libraries produce wrong results for some inputs. RLibm-32 provides the first correctly rounded functions for the posit32 type.''", remark-7 = "From page 372: ``This paper extends our prior work on RLibm [31, 32] and John Gustafson's Minefield method [20], which advocate approximating the correctly rounded value rather than real value of an elementary function.''", remark-8 = "From page 372: ``We have used RLibm to create correctly rounded functions for 16-bit types: bfloat16 and posit16.''", } @Article{Lipovetsky:2021:BRI, author = "Stan Lipovetsky", title = "Book Review: {{\booktitle{Integrals Related to the Error Function}}, by Nikolai E. Korotkov and Alexander N. Korotkov. Boca Raton, FL: Chapman and Hall\slash CRC Press, Taylor \& Francis Group, 2020, 228 pp., \$140.00 (hardback), \$46.36 (eBook), ISBN: 978-0-367-40820-6 (hardback)}", journal = j-TECHNOMETRICS, volume = "62", number = "4", pages = "560--560", year = "2021", CODEN = "TCMTA2", DOI = "https://doi.org/10.1080/00401706.2020.1825632", ISSN = "0040-1706 (print), 1537-2723 (electronic)", ISSN-L = "0040-1706", bibdate = "Fri Feb 5 17:42:52 MST 2021", bibsource = "http://www.tandf.co.uk/journals/titles/00401706.html; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/technometrics2020.bib", acknowledgement = ack-nhfb, fjournal = "Technometrics", journal-URL = "http://www.tandfonline.com/loi/utch20", onlinedate = "23 Oct 2020", } @TechReport{MPFR:2021:MLA, author = "{The MPFR Team}", title = "The {MPFR} Library: Algorithms and Proofs", type = "Report", institution = "????", address = "????", pages = "69", day = "5", month = nov, year = "2021", bibdate = "Tue Mar 14 13:13:13 2023", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "https://www.mpfr.org/algorithms.pdf", acknowledgement = ack-nhfb, } @Article{Patankar:2021:RBC, author = "Udayan S. Patankar and Ants Koel", title = "Review of Basic Classes of Dividers Based on Division Algorithm", journal = j-IEEE-ACCESS, volume = "9", pages = "23035--23069", year = "2021", DOI = "https://doi.org/10.1109/access.2021.3055735", ISSN = "2169-3536", ISSN-L = "2169-3536", bibdate = "Thu Apr 10 15:08:12 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, fjournal = "IEEE Access", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=6287639", } @Article{Snyder:2021:CRA, author = "W. Van Snyder", title = "Corrigendum: {Remark on Algorithm 723: Fresnel Integrals}", journal = j-TOMS, volume = "47", number = "4", pages = "37:1--37:1", month = dec, year = "2021", CODEN = "ACMSCU", DOI = "https://doi.org/10.1145/3452336", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Wed Sep 29 06:58:41 MDT 2021", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib", note = "See \cite{Snyder:1993:AFI}.", URL = "https://dl.acm.org/doi/10.1145/3452336", abstract = "There are mistakes and typographical errors in Remark on Algorithm 723: Fresnel Integrals, which appeared in ACM Transactions on Mathematical Software 22, 4 (December 1996). This remark corrects those errors. The software provided to Collected Algorithms of the ACM was correct.", acknowledgement = ack-nhfb, articleno = "37", fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", } @Article{Walczyk:2021:IAF, author = "Cezary J. Walczyk and Leonid V. Moroz and Jan L. Cie{\'s}li{\'n}ski", title = "Improving the Accuracy of the Fast Inverse Square Root by Modifying {Newton--Raphson} Corrections", journal = j-ENTROPY, volume = "23", number = "1", pages = "86:1--86:20", month = jan, year = "2021", CODEN = "ENTRFG", DOI = "https://doi.org/10.3390/e23010086", ISSN = "1099-4300", ISSN-L = "1099-4300", bibdate = "Wed Dec 20 07:52:39 2023", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, fjournal = "Entropy", journal-URL = "https://www.mdpi.com/journal/entropy/", } @InProceedings{Xu:2021:LCA, author = "Jin Xu and Lin Jiang and Hui Chen and Yuxiang Fu and Li Li", booktitle = "{2021 18th International SoC Design Conference (ISOCC)}", title = "A Low-Complexity Architecture for Implementing Square to Tenth Root of Complex Numbers", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "15--16", year = "2021", DOI = "https://doi.org/10.1109/ISOCC53507.2021.9613873", bibdate = "Mon Oct 27 10:32:44 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, keywords = "complex number Nth root; Complexity theory; Computer architecture; Convergence; CORDIC; Digital computers; Hardware; high efficiency; high precision; low hardware complexity; Power demand; Quantization (signal)", } @Misc{Anonymous:2022:DLM, author = "Anonymous", title = "Digital Library of Mathematical Functions: Date: 2010", howpublished = "NIST Web site", day = "14", month = mar, year = "2022", bibdate = "Wed Oct 25 18:20:12 2023", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "https://dlmf.nist.gov/; https://www.nist.gov/mathematics-statistics/digital-library-mathematical-functions", abstract = "In 2010, NIST released the Digital Library of Mathematical Functions (DLMF), an online successor to the classic Abramowitz and Stegun \booktitle{Handbook of Mathematical Functions}. [\cite{Abramowitz:1964:HMF}]", acknowledgement = ack-nhfb, remark = "From the site: ``By the late 1990s it [the 1964 Handbook] was still the most widely distributed and cited publication of all time, regularly seeing more than 2000 citations per year.''", } @InProceedings{Borges:2022:HLA, author = "Carlos F. Borges and Claude-Pierre Jeannerod and Jean-Michel Muller", title = "High-level algorithms for correctly-rounded reciprocal square roots", crossref = "IEEE:2022:ISC", pages = "18--25", year = "2022", DOI = "https://doi.org/10.1109/ARITH54963.2022.00013", bibdate = "Thu Sep 21 10:14:25 2023", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, keywords = "ARITH-29", } @InProceedings{Bruguera:2022:LLH, author = "Javier D. Bruguera", title = "Low-Latency and High-Bandwidth Pipelined Radix-64 Division and Square Root Unit", crossref = "IEEE:2022:ISC", pages = "10--17", year = "2022", DOI = "https://doi.org/10.1109/ARITH54963.2022.00012", bibdate = "Thu Sep 21 10:14:25 2023", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, keywords = "ARITH-29", } @Article{Causley:2022:GFI, author = "Matthew F. Causley", title = "The gamma function via interpolation", journal = j-NUMER-ALGORITHMS, volume = "90", number = "2", pages = "687--707", month = jun, year = "2022", CODEN = "NUALEG", DOI = "https://doi.org/10.1007/s11075-021-01204-8", ISSN = "1017-1398 (print), 1572-9265 (electronic)", ISSN-L = "1017-1398", bibdate = "Sun May 8 06:36:19 MDT 2022", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/numeralgorithms.bib", URL = "https://link.springer.com/article/10.1007/s11075-021-01204-8", acknowledgement = ack-nhfb, ajournal = "Numer. Algorithms", fjournal = "Numerical Algorithms", journal-URL = "http://link.springer.com/journal/11075", } @Article{Cojean:2022:GML, author = "Terry Cojean and Yu-Hsiang Mike Tsai and Hartwig Anzt", title = "{Ginkgo} --- a math library designed for platform portability", journal = j-PARALLEL-COMPUTING, volume = "111", number = "??", pages = "??--??", month = jul, year = "2022", CODEN = "PACOEJ", DOI = "https://doi.org/10.1016/j.parco.2022.102902", ISSN = "0167-8191 (print), 1872-7336 (electronic)", ISSN-L = "0167-8191", bibdate = "Mon May 9 07:06:37 MDT 2022", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/parallelcomputing.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0167819122000096", abstract = "In an era of increasing computer system diversity, the portability of software from one system to another plays a central role. Software portability is important for the software developers as many software projects have a lifetime longer than a specific system, e.g., a supercomputer, and it is important for the domain scientists that realize their scientific application in a software framework and want to be able to run on one or another system. On a high level, there exist two approaches for realizing platform portability: (1) implementing software using a portability layer leveraging any technique which always generates specific kernels from another language or through an interface for running on different architectures; and (2) providing backends for different hardware architectures, with the backends typically differing in how and in which programming language functionality is realized due to using the language of choice for each hardware (e.g., CUDA kernels for NVIDIA GPUs, SYCL (DPC++) kernels to targeting Intel GPUs and other supported hardware, \ldots). In practice, these two approaches can be combined in applications to leverage their respective strengths. In this paper, we present how we realize portability across different hardware architectures for the Ginkgo library by following the second strategy and the goal to not only port to new hardware architectures but also achieve good performance. We present the Ginkgo library design, separating algorithms from hardware-specific kernels forming the distinct hardware executors, and report our experience when adding execution backends for NVIDIA, AMD, and Intel GPUs. We also present the performance we achieve with this approach for distinct hardware backends.", acknowledgement = ack-nhfb, articleno = "102902", fjournal = "Parallel Computing", journal-URL = "http://www.sciencedirect.com/science/journal/01678191", } @InProceedings{Gao:2022:TFI, author = "Zhanyuan Gao and Laiping Zhao and Haonan Chen", editor = "{IEEE}", booktitle = "{2022 IEEE\slash ACIS 22nd International Conference on Computer and Information Science (ICIS)}", title = "A Trigonometric Function Instruction Set Extension Method Based on {RISC-V}", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "119--126", year = "2022", DOI = "https://doi.org/10.1109/ICIS54925.2022.9882453", bibdate = "Sat Dec 16 15:51:40 2023", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/risc-v.bib", acknowledgement = ack-nhfb, } @Article{Hao:2022:DVP, author = "Jiangwei Hao and Jinchen Xu and YuanYuan Xia", title = "Design of variable precision transcendental function automatic generator", journal = j-J-SUPERCOMPUTING, volume = "78", number = "2", pages = "2196--2218", month = feb, year = "2022", CODEN = "JOSUED", DOI = "https://doi.org/10.1007/s11227-021-03937-8", ISSN = "0920-8542 (print), 1573-0484 (electronic)", ISSN-L = "0920-8542", bibdate = "Mon Feb 28 16:44:34 MST 2022", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jsuper.bib", URL = "https://link.springer.com/article/10.1007/s11227-021-03937-8", acknowledgement = ack-nhfb, ajournal = "J. Supercomputing", fjournal = "The Journal of Supercomputing", journal-URL = "http://link.springer.com/journal/11227", } @Article{Howard:2022:AAA, author = "Roy M. Howard", title = "Arbitrarily Accurate Analytical Approximations for the Error Function", journal = j-MATH-COMPUT-APPL, volume = "27", number = "1", pages = "14:1--14:44", month = feb, year = "2022", CODEN = "????", DOI = "https://doi.org/10.3390/mca27010014", ISSN = "2297-8747", ISSN-L = "2297-8747", bibdate = "Sun Feb 18 06:28:42 MST 2024", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/math-comput-appl.bib", URL = "https://www.mdpi.com/2297-8747/27/1/14", acknowledgement = ack-nhfb, ajournal = "Math. Comput. Appl.", articleno = "14", fjournal = "Mathematical and Computational Applications", journal-URL = "https://www.mdpi.com/journal/mca", } @Article{Lim:2022:OPA, author = "Jay P. Lim and Santosh Nagarakatte", title = "One polynomial approximation to produce correctly rounded results of an elementary function for multiple representations and rounding modes", journal = j-PACMPL, volume = "6", number = "POPL", pages = "3:1--3:28", month = jan, year = "2022", CODEN = "????", DOI = "https://doi.org/10.1145/3498664", ISSN = "2475-1421 (electronic)", ISSN-L = "2475-1421", bibdate = "Thu May 26 06:32:48 MDT 2022", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/pacmpl.bib", URL = "https://dl.acm.org/doi/10.1145/3498664", abstract = "Mainstream math libraries for floating point (FP) do not produce correctly rounded results for all inputs. In contrast, CR-LIBM and RLIBM provide correctly rounded implementations for a specific FP representation with one rounding mode. Using such libraries for a representation with a new rounding mode or with different precision will result in wrong results due to double rounding. This paper proposes a novel method to generate a single polynomial approximation that produces correctly rounded results for all inputs for multiple rounding modes and multiple precision configurations. To generate a correctly rounded library for n-bits, our key idea is to generate a polynomial approximation for a representation with n+2-bits using the round-to-odd mode. We prove that the resulting polynomial approximation will produce correctly rounded results for all five rounding modes in the standard and for multiple representations with k-bits such that $ |E| + 1 < k \leq n $, where $ |E| $ is the number of exponent bits in the representation. Similar to our prior work in the RLIBM project, we approximate the correctly rounded result when we generate the library with n+2-bits using the round-to-odd mode. We also generate polynomial approximations by structuring it as a linear programming problem but propose enhancements to polynomial generation to handle the round-to-odd mode. Our prototype is the first 32-bit float library that produces correctly rounded results with all rounding modes in the IEEE standard for all inputs with a single polynomial approximation. It also produces correctly rounded results for any FP configuration ranging from 10-bits to 32-bits while also being faster than mainstream libraries.", acknowledgement = ack-nhfb, articleno = "3", fjournal = "Proceedings of the ACM on Programming Languages (PACMPL)", journal-URL = "https://dl.acm.org/loi/pacmpl", keywords = "correct rounding; elementary functions", } @InProceedings{Oh:2022:EPA, author = "Hyun Woo Oh and Won Sik Jeong and Seung Eun Lee", editor = "{IEEE}", booktitle = "{2022 19th International SoC Design Conference (ISOCC)}", title = "Evaluation of Posit Arithmetic on Machine Learning based on Approximate Exponential Functions", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "358--359", year = "2022", DOI = "https://doi.org/10.1109/ISOCC56007.2022.10031524", bibdate = "Fri Dec 15 09:21:55 2023", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, } @InProceedings{Patel:2022:LPG, author = "Pragnesh Patel and Aman Arora and Earl Swartzlander and Lizy John", booktitle = "{2022 23rd International Symposium on Quality Electronic Design (ISQED)}", title = "{LogGen}: a Parameterized Generator for Designing Floating-Point Logarithm Units for Deep Learning", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "1--7", year = "2022", DOI = "https://doi.org/10.1109/ISQED54688.2022.9806139", bibdate = "Wed Oct 1 06:40:18 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, keywords = "ASIC; Deep learning; Deep Learning Hardware; Delays; Digital signal processing; Floating-point Arithmetic; FPGA; Generator; Generators; Logarithm; Measurement; Memory management; Pipelines; Tools", } @InProceedings{Puntsri:2022:RAG, author = "Kidsananapong Puntsri and Bussakorn Bunsri and Yaowarat Pittayang and Tanatip Bubpawan and Wuttichai Partipralam and Watid Phakphisut", booktitle = "{2022 37th International Technical Conference on Circuits/Systems, Computers and Communications (ITC-CSCC)}", title = "Reconfigurable {AWGN} Generator Using {Box--Muller} Method with {CORDIC}-based Square Root Calculation", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "1--4", year = "2022", DOI = "https://doi.org/10.1109/ITC-CSCC55581.2022.9894924", bibdate = "Mon Oct 27 10:32:44 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, keywords = "Box--Muller Method; CODIC method; Gaussian Noise Generator; Generators; Noise generators; Parity check codes; Polar codes; Real-time systems; Registers; Table lookup", } @InBook{Ramani:2022:DDI, author = "Narnindi Ramani and Saroj Mondal", booktitle = "{VLSI} Design and Test", title = "A Deep Dive into {CORDIC} Architectures to Implement Trigonometric Functions", publisher = "Springer Nature Switzerland", address = "", pages = "551--561", year = "2022", DOI = "https://doi.org/10.1007/978-3-031-21514-8_45", ISBN = "3-031-21514-1", ISBN-13 = "978-3-031-21514-8", ISSN = "1865-0937", bibdate = "Tue Oct 28 07:04:09 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, } @Article{Takekawa:2022:FPC, author = "Takashi Takekawa", title = "Fast parallel calculation of modified {Bessel} function of the second kind and its derivatives", journal = j-SOFTWAREX, volume = "17", number = "??", pages = "??--??", month = jan, year = "2022", CODEN = "????", DOI = "https://doi.org/10.1016/j.softx.2021.100923", ISSN = "2352-7110", ISSN-L = "2352-7110", bibdate = "Mon Feb 28 10:41:25 MST 2022", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/softwarex.bib", URL = "http://www.sciencedirect.com/science/article/pii/S2352711021001655", acknowledgement = ack-nhfb, articleno = "100923", fjournal = "SoftwareX", journal-URL = "https://www.sciencedirect.com/journal/softwarex/issues", } @InProceedings{Vakil:2022:TCB, author = "Ardavan Vakil and Miad Faezipour", booktitle = "{2022 International Conference on Computational Science and Computational Intelligence (CSCI)}", title = "Toward {CORDIC}-based Hyperbolic Function Implementation for Neural Engineering Hardware", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "416--418", year = "2022", DOI = "https://doi.org/10.1109/CSCI58124.2022.00080", bibdate = "Mon Oct 27 10:32:44 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, keywords = "Biological system modeling; Computational modeling; CORDIC; Hardware; hyperbolic functions; Mathematical models; Neural engineering; neuronal spiking patterns; Real-time systems; Scientific computing; trigonometric functions", } @InProceedings{Vinh:2022:FIT, author = "Truong Quang Vinh and Tran Ba Thanh and Dang Hoang Viet", booktitle = "{2022 9th NAFOSTED Conference on Information and Computer Science (NICS)}", title = "{FPGA} Implementation of Trigonometric Function Using Loop-Optimized Radix-4 {CORDIC}", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "217--222", year = "2022", DOI = "https://doi.org/10.1109/NICS56915.2022.10013467", bibdate = "Mon Oct 27 10:32:44 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, keywords = "Computer science; Convolution; Convolutional neural networks; CORDIC; FPGA; Hardware; hardware implementation; Performance evaluation; Quaternions; Radix-4 CORDIC; Signal processing algorithms; trigonometric function", } @Article{Akdemir:2023:ABC, author = "S. Akdemir and S. {\"O}zay and E. {\"O}ztekin", title = "Asymptotic behavior of {Clebsch--Gordan} coefficients", journal = j-J-MATH-CHEM, volume = "62", number = "10", pages = "2761--2775", month = nov, year = "2023", CODEN = "JMCHEG", DOI = "https://doi.org/10.1007/s10910-023-01544-x", ISSN = "0259-9791 (print), 1572-8897 (electronic)", ISSN-L = "0259-9791", bibdate = "Tue Aug 5 06:18:24 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Journal of Mathematical Chemistry", journal-URL = "http://link.springer.com/journal/10910", } @Article{Ananthanarayan:2023:EAD, author = "B. Ananthanarayan and Souvik Bera and S. Friot and O. Marichev and Tanay Pathak", title = "On the evaluation of the {Appell} {$ F_2 $} double hypergeometric function", journal = j-COMP-PHYS-COMM, volume = "284", number = "??", pages = "Article 108589", month = mar, year = "2023", CODEN = "CPHCBZ", DOI = "https://doi.org/10.1016/j.cpc.2022.108589", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", bibdate = "Sat Feb 25 06:01:54 MST 2023", bibsource = "https://www.math.utah.edu/pub/tex/bib/compphyscomm2020.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0010465522003083", acknowledgement = ack-nhfb, fjournal = "Computer Physics Communications", journal-URL = "http://www.sciencedirect.com/science/journal/00104655", } @InProceedings{Bavier:2023:VNF, author = "Eric Bavier and Nicholas Knight and Hugues de Lassus Saint-Geni{\`e}s and Eric Love", title = "Vectorized Nonlinear Functions with the {RISC-V} Vector Extension", crossref = "IEEE:2023:PIS", pages = "127--130", year = "2023", DOI = "https://doi.org/10.1109/ARITH58626.2023.00032", bibdate = "Wed May 8 09:17:31 2024", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/risc-v.bib", acknowledgement = ack-nhfb, keywords = "ARITH-30; floating point; Instruction sets; Libraries; Pipelines; RISC-V vectors; scalable vectors; Software; Software algorithms; vector mathematical functions; Vectors; Writing", } @Article{Blanchard:2023:NMD, author = "Jeffrey D. Blanchard and Marc Chamberland", title = "{Newton}'s Method Without Division", journal = j-AMER-MATH-MONTHLY, volume = "130", number = "7", pages = "606--617", year = "2023", CODEN = "AMMYAE", DOI = "https://doi.org/10.1080/00029890.2022.2093573", ISSN = "0002-9890 (print), 1930-0972 (electronic)", ISSN-L = "0002-9890", bibdate = "Fri Aug 25 08:24:37 MDT 2023", bibsource = "https://www.math.utah.edu/pub/tex/bib/amermathmonthly2020.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, fjournal = "American Mathematical Monthly", journal-URL = "http://www.jstor.org/journals/00029890.html; https://www.tandfonline.com/loi/uamm20", onlinedate = "04 Aug 2023", } @InProceedings{Brisebarre:2023:TME, author = "Nicolas Brisebarre and Silviu-Ioan Filip", title = "Towards Machine-Efficient Rational {$ L^\infty $}-Approximations of Mathematical Functions", crossref = "IEEE:2023:PIS", pages = "119--126", year = "2023", DOI = "https://doi.org/10.1109/ARITH58626.2023.00029", bibdate = "Wed May 8 09:17:31 2024", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, keywords = "Approximation algorithms; ARITH-30; Behavioral sciences; Digital arithmetic; Software", } @InProceedings{deLamarliere:2023:SFP, author = "Paul Geneau de Lamarli{\`e}re and Guillaume Melquiond and Florian Faissole", title = "Slimmer Formal Proofs for Mathematical Libraries", crossref = "IEEE:2023:PIS", pages = "32--35", year = "2023", DOI = "https://doi.org/10.1109/ARITH58626.2023.00026", bibdate = "Wed May 8 09:17:31 2024", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, keywords = "ARITH-30; Behavioral sciences; Codes; Coq proof assistant; Costs; Digital arithmetic; Floating-point arithmetic; Formal methods; Libraries; Mathematical libraries; Writing", } @Article{Gil:2023:EGF, author = "Amparo Gil and Andrzej Odrzywo{\l}ek and Javier Segura and Nico M. Temme", title = "Evaluation of the generalized {Fermi--Dirac} integral and its derivatives for moderate\slash large values of the parameters", journal = j-COMP-PHYS-COMM, volume = "283", number = "??", pages = "Article 108563", month = feb, year = "2023", CODEN = "CPHCBZ", DOI = "https://doi.org/10.1016/j.cpc.2022.108563", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", bibdate = "Mon Dec 5 09:16:39 MST 2022", bibsource = "https://www.math.utah.edu/pub/tex/bib/compphyscomm2020.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/S001046552200282X", acknowledgement = ack-nhfb, fjournal = "Computer Physics Communications", journal-URL = "http://www.sciencedirect.com/science/journal/00104655", } @InProceedings{Graillat:2023:PCH, author = "Stef Graillat and Youness Ibrahimy and Clothilde Jeangoudoux and Christoph Lauter", title = "A parallel compensated {Horner} scheme for {SIMD} architecture", crossref = "IEEE:2023:PIS", pages = "131--138", year = "2023", DOI = "https://doi.org/10.1109/ARITH58626.2023.00010", bibdate = "Wed May 8 09:17:31 2024", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, keywords = "ARITH-30; AVX; compensated algorithms; Computer architecture; Computers; Costs; Digital arithmetic; error-free transformations; Horner scheme; Limiting; parallel algorithms; polynomial evaluation; Registers; rounding errors; Scalability; SIMD", } @TechReport{Hubrecht:2023:TCRa, author = "Tom Hubrecht and Claude-Pierre Jeannerod and Paul Zimmermann", title = "Towards a correctly-rounded and fast power function in binary64 arithmetic", type = "Report", institution = "DI-ENS --- D{\'e}partement d'informatique --- ENS Paris", address = "Paris, France", day = "12", month = jul, year = "2023", bibdate = "Mon May 13 12:00:21 2024", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "https://inria.hal.science/hal-04159652v1/", abstract = "We design algorithms for the correct rounding of the power function $ x^y $ in the binary64 IEEE 754 format, for all rounding modes, modulo the knowledge of hardest-to-round cases. Our implementation of these algorithms largely outperforms previous correctly-rounded implementations and is not far from the efficiency of current mathematical libraries, which are not correctly-rounded. Still, we expect our algorithms can be further improved for speed. The proofs of correctness are fully detailed in an extended version of this paper, with the goal to enable a formal proof of these algorithms. We hope this work will motivate the next IEEE 754 revision committee to require correct rounding for mathematical functions.", acknowledgement = ack-nhfb, remark = "This is a longer version of \cite{Hubrecht:2023:TCRb} with proofs.", } @InProceedings{Hubrecht:2023:TCRb, author = "Tom Hubrecht and Claude-Pierre Jeannerod and Paul Zimmermann", title = "Towards a correctly-rounded and fast power function in binary64 arithmetic", crossref = "IEEE:2023:PIS", pages = "111--118", year = "2023", DOI = "https://doi.org/10.1109/ARITH58626.2023.00028", bibdate = "Fri Dec 08 15:03:08 2023", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "https://arith2023.arithsymposium.org/slides/S6_PaulZimmermannS6P1.pdf; https://inria.hal.science/hal-04159652v1/file/pow.pdf; https://inria.hal.science/hal-04326201", abstract = "We design algorithms for the correct rounding of the power function $ x^y $ in the binary64 IEEE 754 format, for all rounding modes, modulo the knowledge of hardest-to-round cases. Our implementation of these algorithms largely outperforms previous correctly-rounded implementations and is not far from the efficiency of current mathematical libraries, which are not correctly-rounded. Still, we expect our algorithms can be further improved for speed. The proofs of correctness are fully detailed, with the goal to enable a formal proof of these algorithms. We hope this work will motivate the next IEEE 754 revision committee to require correct rounding for mathematical functions.", acknowledgement = ack-nhfb, keywords = "ARITH-30; binary64 format; correct rounding; Digital arithmetic; double precision; efficiency; Error analysis; IEEE 754; Libraries; power function; Prediction algorithms; Software; Software algorithms; Switches", remark = "See also longer versions \cite{Hubrecht:2023:TCRa,Hubrecht:2024:TCR}.", } @Article{Mansfield:2023:MSR, author = "Daniel F. Mansfield", title = "{Mesopotamian} square root approximation by a sequence of rectangles", journal = j-BRITISH-J-HIST-MATH, volume = "38", number = "3", pages = "175--188", year = "2023", CODEN = "????", DOI = "https://doi.org/10.1080/26375451.2023.2215652", ISSN = "1749-8430 (print), 1749-8341 (electronic)", ISSN-L = "1749-8341", bibdate = "Thu Apr 25 11:06:30 MDT 2024", bibsource = "https://www.math.utah.edu/pub/tex/bib/bshm.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "http://www.tandfonline.com/doi/full/10.1080/26375451.2023.2215652", acknowledgement = ack-nhfb, ajournal = "BSHM Bull.", fjournal = "BSHM Bulletin: Journal of the British Society for the History of Mathematics", journal-URL = "http://www.tandfonline.com/loi/tbsh20", onlinedate = "09 Jun 2023", } @Article{Pradhan:2023:ETB, author = "Chetana Pradhan and Martin Letras and J{\"u}rgen Teich", title = "Efficient Table-based Function Approximation on {FPGAs} Using Interval Splitting and {BRAM} Instantiation", journal = j-TECS, volume = "22", number = "4", pages = "73:1--73:??", month = jul, year = "2023", CODEN = "????", DOI = "https://doi.org/10.1145/3580737", ISSN = "1539-9087 (print), 1558-3465 (electronic)", ISSN-L = "1539-9087", bibdate = "Thu Aug 10 07:21:24 MDT 2023", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/tecs.bib", URL = "https://dl.acm.org/doi/10.1145/3580737", abstract = "This article proposes a novel approach for the generation of memory-efficient table-based function approximation circuits for edge devices in general and FPGAs in particular. Given a function $ f(x) $ to be approximated in a given interval $ [x_0, x_0 + a) $ and a maximum approximation error $ E_a $, the goal is to determine a function table implementation with a minimized memory footprint, i.e., number of entries that need to be stored. Rather than state-of-the-art work performing an equidistant sampling of the given interval by so-called breakpoints and using linear interpolation between two adjacent breakpoints to determine $ f(x) $ at the maximum error bound, we propose and compare three algorithms for splitting the given interval into sub-intervals to reduce the required memory footprint drastically based on the observation that in sub-intervals of low gradient, a coarser sampling grid may be assumed while guaranteeing the maximum interpolation error bound $ E_a $. Experiments on elementary mathematical functions show that a large fraction in memory footprint may be saved. Second, a hardware architecture implementing the sub-interval selection, breakpoint lookup, and interpolation at a latency of just 9 clock cycles is introduced. Third, for each generated circuit design, BRAMs are automatically instantiated rather than synthesizing the reduced footprint function table using LUT primitives, providing an additional degree of resource efficiency. The approach presented here for FPGAs can equally be applied to other circuit technologies for fast and, at the same time, memory-optimized function approximation at the edge.", acknowledgement = ack-nhfb, ajournal = "ACM Trans. Embed. Comput. Syst.", articleno = "73", fjournal = "ACM Transactions on Embedded Computing Systems", journal-URL = "https://dl.acm.org/loi/tecs", } @TechReport{Scott:2023:HBH, author = "Jennifer Scott", title = "{HSL@60}: a brief history of the {HSL} mathematical software library", type = "Technical Repo", number = "STFC-TR-2023-002", institution = "RAL Library STFC Rutherford Appleton Laboratory", address = "Harwell Oxford, Didcot OX11 0Q, UK", month = may, year = "2023", DOI = "https://doi.org/10.5286/stfctr.2023002", ISSN = "2753-5797", bibdate = "Wed Dec 17 10:54:12 2025", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/p/powell-m-j-d.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", abstract = "The HSL Mathematical Software Library (https://www.hsl.rl.ac.uk/) started life in 1963 as the Harwell Subroutine Library. It is the world s oldest mathematical software library to have been in continual use. Over the years, the HSL Library has continually evolved and has been developed and maintained by a small research group of numerical analysts that has been at the Rutherford Appleton Laboratory since 1990. The group s interests lie in the development of numerical algorithms and their underlying theory and then implementing these algorithms in state-of-the-art software. The focus for many years has been on sparse problems. The HSL Library has been extensively used on a wide range of computing platforms, from supercomputers to modern PCs and laptops. The current version of the HSL Library is still widely used and it remains a highly respected source of software for solving sparse problems. This report celebrates the 60th anniversary by presenting a short historic overview, summarising the key milestones of the HSL Library 1963--2023.", acknowledgement = ack-nhfb, ORCID-numbers = "Scott, Jennifer/0000-0003-2130-1091", remark-1 = "From page 6: ``\ldots{} it was not until HSL 2000 that agreement was reached that individual packages could be made available without cost to academic users for research purposes. \ldots{} Initially, only UK academics were offered free access; this was extended to academics worldwide in 2010. The only requirement became having a valid academic email address in a country not on any UK government list of banned countries. HSL Archive has always been freely-available for non-commercial use.''", remark-2 = "From page 7: ``HSL software continues to be written in Fortran, with the version of modern Fortran used being dependent on the widespread availability of reliable compilers. Matlab and C interfaces for some key routines were included for the first time in HSL 2011.''", remark-3 = "From page 8: ``The Group's Sparse Parallel Robust Algorithms Library (SPRAL) is a small open-source library that began in 2015 for sparse linear algebra and associated algorithms.'' See https://www.numerical.rl.ac.uk/spral/.", remark-4 = "From page 8: ``Details of how to access the software are available at https://www.hsl.rl.ac.uk/ (or contact hsl@stfc.ac.uk).''", } @Article{Slevinsky:2023:RAE, author = "Richard M. Slevinsky and Hassan Safouhi", title = "A recursive algorithm for an efficient and accurate computation of incomplete {Bessel} functions", journal = j-NUMER-ALGORITHMS, volume = "92", number = "1", pages = "973--983", month = jan, year = "2023", CODEN = "NUALEG", DOI = "https://doi.org/10.1007/s11075-022-01438-0", ISSN = "1017-1398 (print), 1572-9265 (electronic)", ISSN-L = "1017-1398", bibdate = "Mon Jan 30 12:22:09 MST 2023", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/numeralgorithms.bib", URL = "https://link.springer.com/article/10.1007/s11075-022-01438-0", acknowledgement = ack-nhfb, ajournal = "Numer. Algorithms", fjournal = "Numerical Algorithms", journal-URL = "http://link.springer.com/journal/11075", } @Article{Ananthanarayan:2024:OWR, author = "B. Ananthanarayan and Souvik Bera and S. Friot and Tanay Pathak", title = "\pkg{Olsson.wl} and \pkg{ROC2.wl}: {Mathematica} packages for transformations of multivariable hypergeometric functions and regions of convergence for their series representations in the two variables case", journal = j-COMP-PHYS-COMM, volume = "300", number = "??", pages = "??--??", month = jul, year = "2024", CODEN = "CPHCBZ", DOI = "https://doi.org/10.1016/j.cpc.2024.109162", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", bibdate = "Mon May 6 07:51:16 MDT 2024", bibsource = "https://www.math.utah.edu/pub/tex/bib/compphyscomm2020.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathematica.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0010465524000857", acknowledgement = ack-nhfb, articleno = "109162", fjournal = "Computer Physics Communications", journal-URL = "http://www.sciencedirect.com/science/journal/00104655", } @Article{Briggs:2024:ISM, author = "Ian Briggs and Yash Lad and Pavel Panchekha", title = "Implementation and Synthesis of Math Library Functions", journal = j-PACMPL, volume = "8", number = "POPL", pages = "32:1--32:??", month = jan, year = "2024", CODEN = "????", DOI = "https://doi.org/10.1145/3632874", ISSN = "2475-1421 (electronic)", ISSN-L = "2475-1421", bibdate = "Fri May 10 10:23:39 MDT 2024", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/pacmpl.bib", URL = "https://dl.acm.org/doi/10.1145/3632874", abstract = "Achieving speed and accuracy for math library functions like exp, sin, and log is difficult. This is because low-level implementation languages like C do not help math library developers catch mathematical errors, build implementations incrementally, or separate high-level and low-level decision making. This ultimately puts development of such functions out of reach for all but the most experienced experts. To address this, we introduce MegaLibm, a domain-specific language for implementing, testing, and tuning math library implementations. MegaLibm is safe, modular, and tunable. Implementations in MegaLibm can automatically detect mathematical mistakes like sign flips via semantic wellformedness checks, and components like range reductions can be implemented in a modular, composable way, simplifying implementations. Once the high-level algorithm is done, tuning parameters like working precisions and evaluation schemes can be adjusted through orthogonal tuning parameters to achieve the desired speed and accuracy. MegaLibm also enables math library developers to work interactively, compiling, testing, and tuning their implementations and invoking tools like Sollya and type-directed synthesis to complete components and synthesize entire implementations. MegaLibm can express 8 state-of-the-art math library implementations with comparable speed and accuracy to the original C code, and can synthesize 5 variations and 3 from-scratch implementations with minimal guidance.", acknowledgement = ack-nhfb, ajournal = "Proc. ACM Program. Lang.", articleno = "32", fjournal = "Proceedings of the ACM on Programming Languages (PACMPL)", journal-URL = "https://dl.acm.org/loi/pacmpl", } @TechReport{Brisebarre:2024:CRE, author = "Nicolas Brisebarre and Guillaume Hanrot and Jean-Michel Muller and Paul Zimmermann", title = "Correctly-rounded evaluation of a function: why, how, and at what cost?", type = "Report", number = "hal-04474530", institution = "CNRS --- Centre National de la Recherche Scientifique and others", address = "Paris, France", pages = "29", day = "23", month = feb, year = "2024", bibdate = "Fri Feb 23 16:11:08 2024", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "https://hal.science/hal-04474530", abstract = "The goal of this paper is to convince the reader that a future standard for floating-point arithmetic should require the availability of a correctly-rounded version of a well-chosen core set of elementary functions. We discuss the interest and feasibility of this requirement. We also give answers to common objections we have received over the last 10 years.", acknowledgement = ack-nhfb, keywords = "algorithmic number theory; approximation theory; Computer arithmetic; elementary functions; floating-point arithmetic; lattice basis reduction; LLL algorithm; standardization", } @Article{Cameron:2024:AHM, author = "Thomas R. Cameron and Stef Graillat", title = "Accurate {Horner} methods in real and complex floating-point arithmetic", journal = j-BIT-NUM-MATH, volume = "64", number = "2", pages = "??--??", month = jun, year = "2024", CODEN = "BITTEL, NBITAB", DOI = "https://doi.org/10.1007/s10543-024-01017-w", ISSN = "0006-3835 (print), 1572-9125 (electronic)", ISSN-L = "0006-3835", bibdate = "Tue May 28 15:02:24 MDT 2024", bibsource = "https://www.math.utah.edu/pub/tex/bib/bit.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "https://link.springer.com/article/10.1007/s10543-024-01017-w", acknowledgement = ack-nhfb, ajournal = "Bit Num. Math.", articleno = "17", fjournal = "BIT Numerical Mathematics", journal-URL = "http://link.springer.com/journal/10543", } @Article{Dalloo:2024:LPL, author = "Ayad M. Dalloo and Amjad Jaleel Humaidi and Ammar K. {Al Mhdawi} and Hamed Al-Raweshidy", title = "Low-Power and Low-Latency Hardware Implementation of Approximate Hyperbolic and Exponential Functions for Embedded System Applications", journal = j-IEEE-ACCESS, volume = "12", pages = "24151--24163", year = "2024", DOI = "https://doi.org/10.1109/ACCESS.2024.3364361", ISSN = "2169-3536", ISSN-L = "2169-3536", bibdate = "Mon Oct 27 10:32:44 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "IEEE Access", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=6287639", keywords = "approximate computing; Approximation algorithms; Approximation methods; Computer architecture; CORDIC; elementary functions; Exponential distribution; exponential function; Field programmable gate arrays; Hyperbolic functions; machine learning; Machine learning; Signal processing algorithms; Source coding; Table lookup; table-driven algorithm; Taylor series", } @Book{deDinechin:2024:ASA, author = "Florent de Dinechin and Martin Kumm", title = "Application-specific Arithmetic: Computing Just Right for the Reconfigurable Computer and the Dark Silicon Era", publisher = pub-SV-CHAM, address = pub-SV-CHAM:adr, pages = "xxiii + 804", year = "2024", DOI = "https://doi.org/10.1007/978-3-031-42808-1", ISBN = "3-031-42807-2, 3-031-42808-0 (e-book), 3-031-42809-9, 3-031-42810-2", ISBN-13 = "978-3-031-42807-4, 978-3-031-42808-1 (e-book), 978-3-031-42809-8, 978-3-031-42810-4", LCCN = "QA76.9.C62 D56 2024", bibdate = "Fri Dec 8 13:09:29 MST 2023", bibsource = "fsz3950.oclc.org:210/WorldCat; https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "http://perso.citi-lab.fr/fdedinec/ASA-book/errata.pdf; https://link.springer.com/book/10.1007/978-3-031-42808-1", abstract = "Written by two experts of the domain, this book presents the most recent advances in computer arithmetic hardware, with a focus on application-specific arithmetic beyond the classic operators and the standard precisions. It targets silicon designers who have to do better with less in the post-Moore era, and FPGA developers who want to exploit the full possibilities of reconfigurable computing platforms. Presents a unique focus on application-specific computer arithmetic; Helps developers gain a deep understanding of the arithmetic in their projects, and tailor it to their application; Illustrates concepts and architectures by actual implementations, using the FloPoCo open-source hardware generator.", acknowledgement = ack-nhfb, tableofcontents = "1: Introduction \\ 2: Number Formats \\ 3: Computing Just Right: Accuracy Specification and Error Analysis \\ 4: Field Programmable Gate Arrays \\ \\ Part 1 Revisiting Classic Arithmetic \\ 5: Fixed-Point Addition \\ 6: Fixed-Point Comparison \\ 7: Sums of Weighted Bits \\ 8: Fixed-Point Multiplication \\ 9: Fixed-Point Division \\ 10: Shifters and Leading Bit Counters \\ 11: Basic Floating-Point Operators \\ \\ Part 2 Operator Specialization \\ 12: Multiplication by Constants \\ 13: Division by Constants \\ 14: Fixed-Point Squares, Cubes, and Other Integer Powers \\ 15: Specialization and Fusion of Floating-Point Operators \\ \\ Part 3 Generic Methods for Fixed-Point Function Approximation \\ 16: Generalities on Fixed-Point Function Approximation \\ 17: Function Evaluation Using Tables and Additions \\ 18: Polynomial-Based Architectures for Function Evaluation \\ 19: Digit Recurrence for Algebraic Functions \\ \\ Part 4 Example Composite Operators \\ 20: Fixed-Point Sine and Cosine \\ 21: Floating-Point Accumulation and Sum-of-Products \\ 22: Floating-Point Exponential \\ \\ Part 5 Application Domains \\ 23: Arithmetic in The Design of Linear Time-Invariant Filters \\ 24: Arithmetic for Deep Learning \\ \\ Part 6 Appendix \\ 25: Appendix A: Custom Arithmetic Datapath Design with FloPoCo \\ 26: Appendix B: Optimization Using Integer Linear Programming", } @Article{Dempsey:2024:PBI, author = "Kevin M. Dempsey", title = "Principal Branches of Inverse Trigonometric and Inverse Hyperbolic Functions", journal = j-ACM-COMM-COMP-ALGEBRA, volume = "58", number = "3", pages = "45--56", month = sep, year = "2024", CODEN = "????", DOI = "https://doi.org/10.1145/3717582.3717583", ISSN = "1932-2232 (print), 1932-2240 (electronic)", ISSN-L = "1932-2232", bibdate = "Sat Apr 19 05:48:11 MDT 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/sigsam.bib", abstract = "We discuss principal branches for five square root functions and for the inverse trigonometric and inverse hyperbolic functions. We take the standard reference in this area to be the NIST Digital Library of Mathematical Functions (DLMF). We adopt the notation for and the definitions of the principal branches of the inverse functions in the DLMF. Similarly, the branch cuts for the inverse functions are defined as per the DLMF. Our goal is to use complex analysis to turn the definitions of the principal branches in the DLMF into concrete expressions that hold on the entirety of their respective cut planes. The square root principal branch expressions are new breakthrough discoveries that lead smoothly to four of the concrete expressions. We expand the number of concrete expressions in Sections 4.23 and 4.37 in the DLMF from two to eight. Three of these eight concrete expressions were in print in 1924. One of the latter is still awaiting inclusion in the DLMF. Taken altogether, we provide a computationally efficient resource for computer algebra in programming languages, specifically for the principal branches of the inverse trigonometric and inverse hyperbolic functions.", acknowledgement = ack-nhfb, ajournal = "ACM Commun. Computer Algebr.", fjournal = "ACM Communications in Computer Algebra", journal-URL = "https://dl.acm.org/loi/sigsam-cca", } @Article{Dunster:2024:CPC, author = "T. M. Dunster and A. Gil and J. Segura", title = "Computation of parabolic cylinder functions having complex argument", journal = j-APPL-NUM-MATH, volume = "197", number = "??", pages = "230--242", month = mar, year = "2024", CODEN = "ANMAEL", DOI = "https://doi.org/10.1016/j.apnum.2023.11.017", ISSN = "0168-9274 (print), 1873-5460 (electronic)", ISSN-L = "0168-9274", bibdate = "Mon Dec 18 15:47:31 MST 2023", bibsource = "https://www.math.utah.edu/pub/tex/bib/applnummath.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0168927423002969", acknowledgement = ack-nhfb, fjournal = "Applied Numerical Mathematics: Transactions of IMACS", journal-URL = "http://www.sciencedirect.com/science/journal/01689274", } @Article{Harris:2024:UDS, author = "David Harris and James Stine and Milo Ercegovac and Alberto Nannarelli and Katherine Parry and Cedar Turek", title = "Unified Digit Selection for Radix-4 Recurrence Division and Square Root", journal = j-IEEE-TRANS-COMPUT, volume = "73", number = "1", pages = "292--300", month = jan, year = "2024", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/TC.2023.3305760", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Wed Dec 27 15:37:27 2023", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput2020.bib; https://www.math.utah.edu/pub/tex/bib/risc-v.bib", acknowledgement = ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", keywords = "division; minimally-redundant radix-4; RISC-V; square root; SRT", } @TechReport{Hubrecht:2024:TCR, author = "Tom Hubrecht and Claude-Pierre Jeannerod and Paul Zimmermann and Laurence Rideau and Laurent Th{\'e}ry", title = "Towards a correctly-rounded and fast power function in binary64 arithmetic", type = "Report", institution = "DI-ENS --- D{\'e}partement d'informatique --- ENS Paris", address = "Paris, France", day = "8", month = feb, year = "2024", bibdate = "Mon May 13 12:00:21 2024", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "https://inria.hal.science/hal-04159652v2/", abstract = "We design algorithms for the correct rounding of the power function $ x^y $ in the binary64 IEEE 754 format, for all rounding modes, modulo the knowledge of hardest-to-round cases. Our implementation of these algorithms largely outperforms previous correctly-rounded implementations and is not far from the efficiency of current mathematical libraries, which are not correctly-rounded. Still, we expect our algorithms can be further improved for speed. The proofs of correctness are fully detailed and have been formally verified. We hope this work will motivate the next IEEE 754 revision committee to require correct rounding for mathematical functions.", acknowledgement = ack-nhfb, remark = "This is a longer version of \cite{Hubrecht:2023:TCRb} with proofs and remarks by the final two authors on the formal verification.", } @Article{Kowalenko:2024:AVT, author = "Victor Kowalenko", title = "Algorithms for Various Trigonometric Power Sums", journal = j-ALGORITHMS-BASEL, volume = "17", number = "8", year = "2024", CODEN = "ALGOCH", DOI = "https://doi.org/10.3390/a17080373", ISSN = "1999-4893 (electronic)", ISSN-L = "1999-4893", bibdate = "Fri Aug 30 05:57:31 2024", bibsource = "https://www.math.utah.edu/pub/tex/bib/algorithms.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "https://www.mdpi.com/1999-4893/17/8/373", abstract = "In this paper, algorithms for different types of trigonometric power sums are developed and presented. Although interesting in their own right, these trigonometric power sums arise during the creation of an algorithm for the four types of twisted trigonometric power sums defined in the introduction. The primary aim in evaluating these sums is to obtain exact results in a rational form, as opposed to standard or direct evaluation, which often results in machine-dependent decimal values that can be affected by round-off errors. Moreover, since the variable, m, appearing in the denominators of the arguments of the trigonometric functions in these sums, can remain algebraic in the algorithms/codes, one can also obtain polynomial solutions in powers of m and the variable r that appears in the cosine factor accompanying the trigonometric power. The degrees of these polynomials are found to be dependent upon v, the value of the trigonometric power in the sum, which must always be specified.", acknowledgement = ack-nhfb, articleno = "373", fjournal = "Algorithms (Basel)", journal-URL = "https://www.mdpi.com/journal/algorithms", remark = "See \cite{Rosenberg:1963:FTP,Dronyuk:2025:ACG}.", } @Misc{Muller:2024:SNC, author = "Jean-Michel Muller", title = "Some notes on correct rounding of functions", howpublished = "Attachment to STDS-754 mailing list", pages = "31", day = "18", month = sep, year = "2024", bibdate = "Thu Sep 19 15:27:07 2024", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, } @InProceedings{Murillo:2024:SRU, author = "Raul Murillo and Alberto A. {Del Barrio} and Guillermo Botella", title = "Square Root Unit with Minimum Iterations for Posit Arithmetic", crossref = "IEEE:2024:PIS", pages = "132--138", year = "2024", DOI = "https://doi.org/10.1109/arith61463.2024.00030", bibdate = "Thu Nov 13 11:37:34 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, keywords = "ARITH 2024; ARITH-31; posit arithmetic", } @InProceedings{vanderHoeven:2024:FMP, author = "Joris van der Hoeven and Fredrik Johansson", title = "Fast multiple precision $ \exp (x) $ with precomputations", crossref = "IEEE:2024:PIS", pages = "80--87", year = "2024", DOI = "https://doi.org/10.1109/arith61463.2024.00023", bibdate = "Thu Nov 13 11:37:34 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, keywords = "ARITH 2024; ARITH-31", } @InProceedings{Wong:2024:HGT, author = "Paul Wong and Dania Susanne Mosuli and Xuechen Zhang and Xiaokun Yang", booktitle = "{2024 IEEE International Conference on Big Data (BigData)}", title = "Hardware Generation on Trigonometric Functions", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "7571--7576", year = "2024", DOI = "https://doi.org/10.1109/BigData62323.2024.10825243", bibdate = "Mon Oct 27 10:32:44 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, keywords = "Chisel; Codes; Computer architecture; CORDIC; floating-point; FPGA; Generators; Hardware; Hardware acceleration; Hardware design languages; Pipelines; Registers; Resource management; Scientific computing; trigonometry", } @Article{Yoshida:2024:CIB, author = "Toshio Yoshida and Yoshinori Adachi", title = "Computation of incomplete beta function ratios {$ I_x(a, b) $} with {Deuflhard}'s algorithm", journal = j-NUMER-ALGORITHMS, volume = "97", number = "1", pages = "373--390", month = sep, year = "2024", CODEN = "NUALEG", DOI = "https://doi.org/10.1007/s11075-023-01707-6", ISSN = "1017-1398 (print), 1572-9265 (electronic)", ISSN-L = "1017-1398", bibdate = "Wed Aug 7 06:17:56 MDT 2024", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/numeralgorithms.bib", URL = "https://link.springer.com/article/10.1007/s11075-023-01707-6", acknowledgement = ack-nhfb, ajournal = "Numer. Algorithms", fjournal = "Numerical Algorithms", journal-URL = "http://link.springer.com/journal/11075", } @Article{Zaghloul:2024:CFI, author = "Mofreh R. Zaghloul and Leen Alrawas", title = "Calculation of {Fresnel} integrals of real and complex arguments up to 28 significant digits", journal = j-NUMER-ALGORITHMS, volume = "96", number = "2", pages = "489--506", month = jun, year = "2024", CODEN = "NUALEG", DOI = "https://doi.org/10.1007/s11075-023-01654-2", ISSN = "1017-1398 (print), 1572-9265 (electronic)", ISSN-L = "1017-1398", bibdate = "Tue May 21 07:35:40 MDT 2024", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/numeralgorithms.bib", URL = "https://link.springer.com/article/10.1007/s11075-023-01654-2", acknowledgement = ack-nhfb, ajournal = "Numer. Algorithms", fjournal = "Numerical Algorithms", journal-URL = "http://link.springer.com/journal/11075", keywords = "complex cosine integral C(z); complex sine integral S(z); Fresnel functions; real cosine integral C(x); real sine integral S(x)", } @Article{Zaghloul:2024:EMP, author = "Mofreh R. Zaghloul", title = "Efficient multiple-precision computation of the scaled complementary error function and the {Dawson} integral", journal = j-NUMER-ALGORITHMS, volume = "95", number = "3", pages = "1291--1308", month = mar, year = "2024", CODEN = "NUALEG", DOI = "https://doi.org/10.1007/s11075-023-01608-8", ISSN = "1017-1398 (print), 1572-9265 (electronic)", ISSN-L = "1017-1398", bibdate = "Wed Feb 14 08:54:32 MST 2024", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/numeralgorithms.bib", note = "See correction \cite{Zaghloul:2026:CEM}.", URL = "https://link.springer.com/article/10.1007/s11075-023-01608-8", acknowledgement = ack-nhfb, ajournal = "Numer. Algorithms", fjournal = "Numerical Algorithms", journal-URL = "http://link.springer.com/journal/11075", keywords = "Dawson(x); erf(x); erfc(x)", } @Article{Zaghloul:2024:ENA, author = "Mofreh R. Zaghloul", title = "Efficient numerical algorithms for multi-precision and multi-accuracy calculation of the error functions and {Dawson} integral with complex arguments", journal = j-NUMER-ALGORITHMS, volume = "95", number = "??", pages = "1--19", month = "????", year = "2024", CODEN = "NUALEG", DOI = "https://doi.org/10.1007/s11075-023-01727-2", ISSN = "1017-1398 (print), 1572-9265 (electronic)", ISSN-L = "1017-1398", bibdate = "Tue Feb 20 15:50:26 2024", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "Numerical Algorithms", journal-URL = "http://link.springer.com/journal/11075", keywords = "Dawson(x); erf(x); erfc(x)", remark = "[20-Feb-2024] Available at journal Web site, but not yet assigned to a journal volume.", } @Article{Anonymous:2025:Cd, author = "Anonymous", title = "Correction", journal = j-J-STAT-COMPUT-SIMUL, volume = "95", number = "15", pages = "3397--3399", year = "2025", CODEN = "JSCSAJ", DOI = "https://doi.org/10.1080/00949655.2025.2525617", ISSN = "0094-9655 (print), 1026-7778 (electronic), 1563-5163", ISSN-L = "0094-9655", bibdate = "Thu Oct 2 07:55:18 MDT 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jstatcomputsimul2020.bib", note = "See \cite{Das:2025:NMC}.", acknowledgement = ack-nhfb, fjournal = "Journal of Statistical Computation and Simulation", journal-URL = "http://www.tandfonline.com/loi/gscs20", onlinedate = "16 Jul 2025", } @Article{Anton:2025:FER, author = "Ramona Anton and Nicolae Mihalache and Fran{\c{c}}ois Vigneron", title = "Fast evaluation of real and complex polynomials", journal = j-NUM-MATH, volume = "157", number = "1", pages = "355--408", month = feb, year = "2025", CODEN = "NUMMA7", DOI = "https://doi.org/10.1007/s00211-025-01454-x", ISSN = "0029-599X (print), 0945-3245 (electronic)", ISSN-L = "0029-599X", bibdate = "Sat Feb 1 09:47:51 MST 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/nummath2020.bib", URL = "https://link.springer.com/article/10.1007/s00211-025-01454-x", acknowledgement = ack-nhfb, ajournal = "Num. Math.", fjournal = "Numerische Mathematik", journal-URL = "http://link.springer.com/journal/211", keywords = "Horner's nested form; number of multiplications to evaluate a polynomial", } @Article{Arzelier:2025:EAO, author = "Denis Arzelier and Florent Brehard and Tom Hubrecht and Mioara Joldes", title = "An Exchange Algorithm for Optimizing Both Approximation and Finite-Precision Evaluation Errors in Polynomial Approximations", journal = j-TOMS, volume = "51", number = "4", pages = "25:1--25:32", month = dec, year = "2025", CODEN = "ACMSCU", DOI = "https://doi.org/10.1145/3770066", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Tue Dec 23 05:39:42 MST 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib", abstract = "The finite precision implementation of mathematical functions frequently depends on polynomial approximations. A key characteristic of this approach is that rounding errors occur both when representing the coefficients of the polynomial on a finite number of bits, and when evaluating it in finite precision arithmetic. Hence, to find a best polynomial, for a given fixed degree, norm, and interval, it is necessary to account for both the approximation error and the floating-point evaluation error. While efficient algorithms were already developed for taking into account the approximation error, the evaluation part is usually a posteriori handled, in an ad hoc manner. Here, we formulate a semi-infinite linear optimization problem whose solution is a best polynomial with respect to the supremum norm of the sum of both errors. This problem is then solved with an iterative exchange algorithm, which can be seen as an extension of the well-known Remez exchange algorithm. An open source C implementation using the Sollya library is presented and tested on several examples, which are then analyzed and compared against state-of-the-art Sollya routines.", acknowledgement = ack-nhfb, ajournal = "ACM Trans. Math. Softw.", articleno = "25", fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", } @Article{Das:2025:NMC, author = "Abhranil Das", title = "New methods to compute the generalized chi-square distribution", journal = j-J-STAT-COMPUT-SIMUL, volume = "95", number = "12", pages = "2608--2642", year = "2025", CODEN = "JSCSAJ", DOI = "https://doi.org/10.1080/00949655.2025.2501401", ISSN = "0094-9655 (print), 1026-7778 (electronic), 1563-5163", ISSN-L = "0094-9655", bibdate = "Thu Aug 7 05:47:44 MDT 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jstatcomputsimul2020.bib", note = "See correction \cite{Anonymous:2025:Cd}.", acknowledgement = ack-nhfb, fjournal = "Journal of Statistical Computation and Simulation", journal-URL = "http://www.tandfonline.com/loi/gscs20", onlinedate = "19 May 2025", } @Article{Dronyuk:2025:ACG, author = "Ivanna Dronyuk", title = "Algorithms for Calculating Generalized Trigonometric Functions", journal = j-ALGORITHMS-BASEL, volume = "18", number = "2", month = feb, year = "2025", CODEN = "ALGOCH", DOI = "https://doi.org/10.3390/a18020060", ISSN = "1999-4893 (electronic)", ISSN-L = "1999-4893", bibdate = "Fri Feb 28 06:02:21 MST 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/algorithms.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "https://www.mdpi.com/1999-4893/18/2/60", acknowledgement = ack-nhfb, articleno = "60", fjournal = "Algorithms (Basel)", journal-URL = "https://www.mdpi.com/journal/algorithms", pagecount = "13", remark = "See \cite{Rosenberg:1963:FTP,Kowalenko:2024:AVT}.", } @Article{Dunster:2025:NAC, author = "T. M. Dunster and A. Gil and J. Segura", title = "A numerical algorithm for computing the zeros of parabolic cylinder functions in the complex plane", journal = j-BIT-NUM-MATH, volume = "65", number = "2", pages = "??--??", month = jun, year = "2025", CODEN = "BITTEL, NBITAB", DOI = "https://doi.org/10.1007/s10543-025-01065-w", ISSN = "0006-3835 (print), 1572-9125 (electronic)", ISSN-L = "0006-3835", bibdate = "Wed Jul 16 14:17:01 MDT 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/bit.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "https://link.springer.com/article/10.1007/s10543-025-01065-w", acknowledgement = ack-nhfb, ajournal = "Bit Num. Math.", articleno = "21", fjournal = "BIT Numerical Mathematics", journal-URL = "http://link.springer.com/journal/10543", } @Article{Guseinov:2025:CEI, author = "I. I. Guseinov and B. A. Mamedov", title = "Correction to: {Evaluation} of incomplete gamma functions using downward recursion and analytical relations", journal = j-J-MATH-CHEM, volume = "63", number = "2", pages = "650--650", month = feb, year = "2025", CODEN = "JMCHEG", DOI = "https://doi.org/10.1007/s10910-024-01686-6", ISSN = "0259-9791 (print), 1572-8897 (electronic)", ISSN-L = "0259-9791", bibdate = "Tue Mar 4 11:35:40 MST 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jmathchem.bib", note = "See \cite{Guseinov:2004:EIG}.", URL = "https://link.springer.com/article/10.1007/s10910-024-01686-6", acknowledgement = ack-nhfb, ajournal = "J. Math. Chem.", fjournal = "Journal of Mathematical Chemistry", journal-URL = "http://link.springer.com/journal/10910", } @Article{Kokosinski:2025:FAA, author = "Zbigniew Kokosi{\'n}ski and Pawe{\l} Gepner and Nataliia Gavkalova", title = "Fast and accurate approximation algorithms for computing floating point square root", journal = j-NUMER-ALGORITHMS, volume = "99", number = "4", pages = "1791--1804", month = aug, year = "2025", CODEN = "NUALEG", DOI = "https://doi.org/10.1007/s11075-024-01932-7", ISSN = "1017-1398 (print), 1572-9265 (electronic)", ISSN-L = "1017-1398", bibdate = "Wed Jul 16 11:11:45 MDT 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/numeralgorithms.bib", URL = "https://link.springer.com/article/10.1007/s11075-024-01932-7", acknowledgement = ack-nhfb, ajournal = "Numer. Algorithms", fjournal = "Numerical Algorithms", journal-URL = "http://link.springer.com/journal/11075", } @Article{Marcheva:2025:QCD, author = "Plamena I. Marcheva and Ivan K. Ivanov and Stoil I. Ivanov", title = "On the {Q}-Convergence and Dynamics of a Modified {Weierstrass} Method for the Simultaneous Extraction of Polynomial Zeros", journal = j-ALGORITHMS-BASEL, volume = "18", number = "4", month = apr, year = "2025", CODEN = "ALGOCH", DOI = "https://doi.org/10.3390/a18040205", ISSN = "1999-4893 (electronic)", ISSN-L = "1999-4893", bibdate = "Mon Apr 28 09:25:31 MDT 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/algorithms.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "https://www.mdpi.com/1999-4893/18/4/205", acknowledgement = ack-nhfb, articleno = "205", fjournal = "Algorithms (Basel)", journal-URL = "https://www.mdpi.com/journal/algorithms", pagecount = "??", } @Article{Ozay:2025:CCS, author = "S. {\"O}zay and S. Akdemir and E. {\"O}ztekin", title = "The coupling coefficients with six parameters and the generalized hypergeometric functions", journal = j-COMP-PHYS-COMM, volume = "315", number = "??", pages = "??--??", month = oct, year = "2025", CODEN = "CPHCBZ", DOI = "https://doi.org/10.1016/j.cpc.2025.109656", ISSN = "0010-4655 (print), 1879-2944 (electronic)", ISSN-L = "0010-4655", bibdate = "Mon Aug 4 11:17:58 MDT 2025", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/w/wigner-eugene.bib; https://www.math.utah.edu/pub/tex/bib/compphyscomm2020.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathematica.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0010465525001584", acknowledgement = ack-nhfb, ajournal = "Comput. Phys. Commun.", articleno = "109656", fjournal = "Computer Physics Communications", journal-URL = "http://www.sciencedirect.com/science/journal/00104655", keywords = "angular momentum; binomial coefficients; Clebsch-Gordon coefficients; Gaunt coefficients; generalized hypergeometric functions; Mathematica; spherical harmonics; Wigner 3-j symbols; Wigner 6-j symbols; Wigner 9-j symbols", } @InProceedings{Swords:2025:REE, author = "Sol Swords and Cuong Chau", title = "Robust, End-to-end Correctness Proofs of Industrial Divide and Square Root {RTL} Designs", crossref = "IEEE:2025:PIS", pages = "149--156", year = "2025", DOI = "https://doi.org/10.1109/arith64983.2025.00031", bibdate = "Thu Nov 13 12:36:40 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, keywords = "ARITH 2025; ARITH-32", } @Article{Trim:2025:AEF, author = "Sean J. Trim and Raymond J. Spiteri", title = "Algorithm 1054: \pkg{ellipFor}, a {Fortran} Software Library for {Legendre} Elliptic Integrals and {Jacobi} Elliptic Functions with Generalized Input Arguments", journal = j-TOMS, volume = "51", number = "1", pages = "6:1--6:??", month = mar, year = "2025", CODEN = "ACMSCU", DOI = "https://doi.org/10.1145/3709136", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Thu Apr 10 08:03:40 MDT 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib", abstract = "Legendre elliptic integrals and Jacobi elliptic functions arise in multiple applications within the physical sciences, including oscillations, celestial mechanics, and geodynamics. In this study, we describe the Fortran library ellipFor capable of evaluating the following for generalized input values: (1) the complete Legendre elliptic integrals of the first and second kinds, (2) the incomplete Legendre elliptic integrals of the first and second kinds, and (3) the principal Jacobi elliptic functions. Our software builds upon previously developed Fortran routines, which were designed with restrictions on input parameters that may be limiting in applications. Our routines apply multiple transformations to allow for more general input values, such as elliptic moduli greater than unity for points 1--3, arbitrary real Jacobi amplitudes for points 1--2, and complex first arguments for point 3. In addition, our routines are thread-safe, allowing for parallel computations. Our routines were compared with values from the computer algebra system SageMath over a wide range of input parameters. Values from ellipFor and SageMath agreed to within tolerances commensurate with the limitations of floating-point arithmetic used for the elliptic integrals and Jacobi elliptic functions listed in points 1, 2, and 3 above for generalized input arguments.", acknowledgement = ack-nhfb, ajournal = "ACM Trans. Math. Softw.", articleno = "6", fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", } @Article{Walczyk:2025:OAX, author = "Cezary J. Walczyk and Leonid V. Moroz and Volodymyr Samotyy and Jan L. Cie{\'s}li{\'n}ski", title = "Optimal Approximation of the $ 1 / x $ Function using {Chebyshev} Polynomials and Magic Constants", journal = j-TOMS, volume = "51", number = "1", pages = "2:1--2:??", month = mar, year = "2025", CODEN = "ACMSCU", DOI = "https://doi.org/10.1145/3708472", ISSN = "0098-3500 (print), 1557-7295 (electronic)", ISSN-L = "0098-3500", bibdate = "Thu Apr 10 08:03:40 MDT 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib", abstract = "In this article we analyze low-cost accurate approximation of the function $ 1 / x $ using Chebyshev polynomials of the first kind and minimizing number of elementary operations in computer codes (in particular, by using the so-called magic constants). It is shown that Newton-Raphson iterative method is not optimal and a new approach is proposed. We prove that optimal Chebyshev polynomials can be factorized in terms of Chebyshev polynomials of lower order which leads to new optimal iteration schemes. We also construct a family of new algorithms by dividing the considered interval into sub-intervals where different magic constants and multiplicative factors are used (in order to increase the accuracy). Theoretical considerations and proofs are completed with numerical tests on three types of microcontroller processors.", accepted = "4 December 2024", acknowledgement = ack-nhfb, ajournal = "ACM Trans. Math. Softw.", articleno = "2", fjournal = "ACM Transactions on Mathematical Software (TOMS)", journal-URL = "https://dl.acm.org/loi/toms", received = "3 September 2022", revised = "25 September 2024", } @Article{Zhang:2025:RRH, author = "Yangyi Zhang and Xianglong Wang and Lei Chen and Fengwei An", title = "{ReHIT}: Reconfigurable High-Radix Iterative-Taylor Architecture for Ultraprecise Logarithm\slash Exponential Functions in {FPGA}-Based {Softmax} Accelerators", journal = j-IEEE-TRANS-VLSI-SYST, volume = "33", number = "10", pages = "2690--2701", year = "2025", CODEN = "IEVSE9", DOI = "https://doi.org/10.1109/TVLSI.2025.3566433", ISSN = "1063-8210 (print), 1557-9999 (electronic)", ISSN-L = "1063-8210", bibdate = "Wed Oct 1 06:40:18 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, fjournal = "IEEE Transactions on Very Large Scale Integration (VLSI) Systems", journal-URL = "https://ieeexplore.ieee.org/xpl/issues?punumber=92", keywords = "Accuracy; Approximation algorithms; Architecture; Computer architecture; exponential function; Field programmable gate arrays; field-programmable gate array (FPGA); Hardware; logarithm function; Logic; Polynomials; softmax function; Table lookup; Taylor formula; Taylor series; Technological innovation", } @InProceedings{Zhao:2025:RVP, author = "Ruixiao Zhao and Min Xie and Xinchen Li and Yujie Zhang", booktitle = "{2025 IEEE 14th International Conference on Communications, Circuits and Systems (ICCCAS)}", title = "A {RISC-V} Processor with Optimized {CORDIC}-based Trigonometric Function Accelerator", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "155--159", year = "2025", DOI = "https://doi.org/10.1109/ICCCAS65806.2025.11102696", bibdate = "Mon Oct 27 10:32:44 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/risc-v.bib", acknowledgement = ack-nhfb, keywords = "Accuracy; Circuits and systems; Computational efficiency; Computer architecture; CORDIC; Hardware; instruction set extension; Instruction sets; RISC-V; Table lookup; trigonometric function", } @Article{Zullo:2025:LFP, author = "Federico Zullo", title = "{Lommel} functions, {Pad{\'e}} approximants and hypergeometric functions", journal = j-APPL-NUM-MATH, volume = "209", number = "??", pages = "275--284", month = mar, year = "2025", CODEN = "ANMAEL", DOI = "https://doi.org/10.1016/j.apnum.2024.11.003", ISSN = "0168-9274 (print), 1873-5460 (electronic)", ISSN-L = "0168-9274", bibdate = "Sat Dec 28 08:12:54 MST 2024", bibsource = "https://www.math.utah.edu/pub/tex/bib/applnummath.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/S016892742400299X", acknowledgement = ack-nhfb, fjournal = "Applied Numerical Mathematics: Transactions of IMACS", journal-URL = "http://www.sciencedirect.com/science/journal/01689274", } @Article{Zaghloul:2026:CEM, author = "Mofreh R. Zaghloul", title = "Correction to: {Efficient} multiple-precision computation of the scaled complementary error function and the {Dawson} integral", journal = j-NUMER-ALGORITHMS, volume = "101", number = "3", pages = "2151--2151", month = mar, year = "2026", CODEN = "NUALEG", DOI = "https://doi.org/10.1007/s11075-025-02057-1", ISSN = "1017-1398 (print), 1572-9265 (electronic)", ISSN-L = "1017-1398", bibdate = "Fri Mar 6 07:14:16 MST 2026", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/numeralgorithms.bib", URL = "https://link.springer.com/article/10.1007/s11075-025-02057-1", acknowledgement = ack-nhfb, ajournal = "Numer. Algorithms", fjournal = "Numerical Algorithms", journal-URL = "http://link.springer.com/journal/11075", } @Misc{Hyland:20xx:FIS, author = "Adam Hyland", title = "The Fast Inverse Square Root", howpublished = "Web site.", year = "20xx", bibdate = "Fri Dec 12 15:05:49 2025", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/h/hartree-douglas-r.bib; https://www.math.utah.edu/pub/bibnet/authors/k/kahan-william-m.bib; https://www.math.utah.edu/pub/bibnet/authors/m/moler-cleve-b.bib; https://www.math.utah.edu/pub/bibnet/authors/t/turing-alan-mathison.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "https://0x5f37642f.com/", acknowledgement = ack-nhfb, keywords = "inverse square root; reciprocal square root", remark = "The Internet domain name in the URL contains a magic hexadecimal constant that is used in computing a rough approximation to $ 1 / \sqrt {x} $ in IEEE 754 binary32 arithmetic. The Web site supplies a considerable number of references to earlier work on fast approximations for this function, dating back to 1949 on the Manchester Mark I. Tests of the Web site code, which requires replacement of `long' by `int32_t' on modern platforms, show maximum errors of about 14700 ulps for $x$ on [1/2,1] and [1,1024], corresponding to perturbations in the fourth decimal digit of the computed reciprocal square root.", } %%% ==================================================================== %%% Cross-referenced entries must come last: @Book{Knuth:1998:SA, author = "Donald E. Knuth", title = "Seminumerical Algorithms", volume = "2", publisher = pub-AW, address = pub-AW:adr, edition = "Third", pages = "xiii + 762", year = "1998", ISBN = "0-201-89684-2", ISBN-13 = "978-0-201-89684-8", LCCN = "QA76.6 .K64 1997", bibdate = "Fri Jul 11 15:41:22 1997", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/v/von-neumann-john.bib; https://www.math.utah.edu/pub/bibnet/subjects/acc-stab-num-alg-2ed.bib; https://www.math.utah.edu/pub/tex/bib/benfords-law.bib; https://www.math.utah.edu/pub/tex/bib/css.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/master.bib; https://www.math.utah.edu/pub/tex/bib/prng.bib; https://www.math.utah.edu/pub/tex/bib/texbook2.bib", price = "US\$52.75", series = "The Art of Computer Programming", acknowledgement = ack-nhfb, tableofcontents = "3: Random Numbers / 1 \\ 3.1. Introduction / 1 \\ 3.2. Generating Uniform Random Numbers / 10 \\ 3.2.1. The Linear Congruential Method / 10 \\ 3.2 1.1. Choice of modulus / 12 \\ 3.2.1.2 Choice of multiplier / 16 \\ 3.2.1.3. Potency / 23 \\ 3.2.2. Other Methods / 26 \\ 3.3. Statistical Tests / 41 \\ 3.3.1. General Test Procedures for Studying Random Data / 41 \\ 3.3.2. Empirical Tests / 61 \\ *3.3.3. Theoretical Tests / 80 \\ 3.3.4. The Spectral Test / 93 \\ 3.4. Other Types of Random Quantities / 119 \\ 3.4 1. Numerical Distributions / 119 \\ 3.4.2. Random Sampling and Shuffling / 142 \\ *3.5. What Is a Random Sequence? / 149 \\ 3.6. Summary / 184 \\ 4: Arithmetic / 194 \\ 4.1. Positional Number Systems / 195 \\ 4.2. Floating Point Arithmetic / 214 \\ 4.2.1. Single-Precision Calculations / 214 \\ 4.2 2. Accuracy of Floating Point Arithmetic / 229 \\ *4.2.3. Double-Precision Calculations / 246 \\ 4.2.4. Distribution of Floating Point Numbers / 253 \\ 4.3 Multiple Precision Arithmetic / 265 \\ 4.3.1. The Classical Algorithms / 265 \\ *4.3.2. Modular Arithmetic / 284 \\ *4.3.3. How Fast Can We Multiply? / 294 \\ 4.4. Radix Conversion / 319 \\ 4.5. Rational Arithmetic / 330 \\ 4.5.1. Fractions / 330 \\ 4.5.2. The Greatest Common Divisor / 333 \\ *4.5.3. Analysis of Euclid's Algorithm / 356 \\ 4.5.4. Factoring into Primes / 379 \\ 4.6. Polynomial Arithmetic / 418 \\ 4.6.1. Division of Polynomials / 420 \\ *4.6.2. Factorization of Polynomials / 439 \\ 4.6.3. Evaluation of Powers / 461 \\ 4.6.4. Evaluation of Polynomials / 485 \\ *4.7. Manipulation of Power Series / 525 \\ Answers to Exercises / 538 \\ Appendix A: Tables of Numerical Quantities / 726 \\ 1. Fundamental Constants (decimal) / 726 \\ 2; Fundamental Constants ( octal) / 727 \\ 3. Harmonic Numbers, Bernoulli Numbers, Fibonacci Numbers / 728 \\ Appendix B: Index to Notations / 730 \\ Index and Glossary / 735", } @Book{Borwein:2007:RHR, editor = "Peter Borwein and Stephen Choi and Brendan Rooney and Andrea Weirathmueller and others", title = "The {Riemann Hypothesis}: a resource for the afficionado and virtuoso alike", volume = "27", publisher = "Springer Science+Business Media, LLC", address = "New York, NY, USA", pages = "xiv + 533", year = "2007", DOI = "https://doi.org/10.1007/978-0-387-72126-2", ISBN = "0-387-72125-8 (hardcover), 0-387-72126-6 (e-book)", ISBN-13 = "978-0-387-72125-5 (hardcover), 978-0-387-72126-2 (e-book)", LCCN = "QA246 .R53 2008", bibdate = "Thu Sep 1 07:07:49 MDT 2022", bibsource = "fsz3950.oclc.org:210/WorldCat; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", series = "CMS books in mathematics", URL = "http://libanswers.liverpool.ac.uk/faq/182315", abstract = "This book is an introduction to the theory surrounding the Riemann Hypothesis. Part I serves as a compendium of known results and as a primer for the material presented in the 20 original papers contained in Part II. The original papers place the material into historical context and illustrate the motivations for research on and around the Riemann Hypothesis. Several of these papers focus on computation of the zeta function, while others give proofs of the Prime Number Theorem, since the Prime Number Theorem is so closely connected to the Riemann Hypothesis. The text is suitable for a graduate course or seminar or simply as a reference for anyone interested in this extraordinary conjecture.", acknowledgement = ack-nhfb, shorttableofcontents = "To the Riemann Hypothesis \\ Why This Book \\ Analytic Preliminaries \\ Algorithms for Calculating ?(s) \\ Empirical Evidence \\ Equivalent Statements \\ Extensions of the Riemann Hypothesis \\ Assuming the Riemann Hypothesis and Its Extensions \ldots{} \\ Failed Attempts at Proof \\ Formulas \\ Timeline \\ Original Papers \\ Expert Witnesses \\ The Experts Speak for Themselves", subject = "Mathematics; History; Number theory; Math{\'e}matiques; Histoire; Th{\'e}orie des nombres; Mathematics.; Number theory.", tableofcontents = "Part 1: Introduction to the Riemann hypothesis \\ 1: Why this book \\ 1.1: The Holy Grail \\ 1.2: Riemann's zeta and Liouville's lambda \\ 1.3: The prime number theorem \\ 2: Analytic preliminaries \\ 2.1: The Riemann zeta function \\ 2.2: Zero-free region \\ 2.3: Counting the zeros of [cedilla](s) \\ 2.4: Hardy's theorem \\ 3: Algorithms for calculating [cedilla](s) \\ 3.1: Euler--MacLaurin summation \\ 3.2: Backlund \\ 3.3: Hardy's function \\ 3.4: The Riemann--Siegel formula \\ 3.5: Gram's law \\ 3.6: Turing \\ 3.7: The Odlyzko--Sch{\"o}nhage algorithm \\ 3.8: A simple algorithm for the zeta function \\ 3.9: Further reading \\ 4: Empirical evidence \\ 4.1: Verification in an interval \\ 4.2: A brief history of computational evidence \\ 4.3: The Riemann hypothesis and random matrices \\ 4.4: The Skewes number \\ 5: Equivalent statements \\ 5.1: Number-theoretic equivalences \\ 5.2: Analytic equivalences \\ 5.3: Other equivalences \\ 6: Extensions of the Riemann hypothesis \\ 6.1: The Riemann hypothesis \\ 6.2: The generalized Riemann hypothesis \\ 6.3: The extended Riemann hypothesis \\ 6.4: An equivalent extended Riemann hypothesis \\ 6.5: Another extended Riemann hypothesis \\ 6.6: The Grand Riemann hypothesis \\ 7: Assuming the Riemann hypothesis and its extensions \\ 7.1: Another proof of the prime number theorem \\ 7.2: Goldbach's conjecture \\ 7.3: More Goldbach \\ 7.4: Primes in a given interval \\ 7.5: The least prime in arithmetic progressions \\ 7.6: Primality testing \\ 7.7: Artin's primitive root conjecture \\ 7.8: Bounds on Dirichlet $L$-series \\ 7.9: The Lindel{\"o}f hypothesis \\ 7.10: Titchmarsh's [delta](T) function \\ 7.11: Mean values of [cedilla](s)8: Failed attempts at proof \\ 8.1: Stieltjes and Mertens' conjecture \\ 8.2: Hans Rademacher and false hopes \\ 8.3: Tur{\'a}n's condition \\ 8.4: Louis de Branges's approach \\ 8.5: No really good idea \\ 9: Formulas \\ 10: Timeline \\ pt. 2: Original papers \\ 11: Expert witnesses \\ 11. 1: E. Bombieri (2000--2001) \\ 11.2: P. Sarnak (2004) \\ 11.3: J.B. Conrey (2003) \\ 11.4: A. Ivi{\'c} (2003) \\ 12: The experts speak for themselves \\ 12.1: P.L. Chebyshev (1852) \\ 12.2: B. Riemann (1859) \\ 12.3: J. Hadamard (1896) \\ 12.4: C. de la Vall{\'e}e Poussin (1899) \\ 12.5: G.H. Hardy (1914) \\ 12.6: G.H. Hardy (1915) \\ 12.7: G.H. Hardy and J.E. Littlewood (1915) \\ 12.8: A. Weil (1941) \\ 12.9: P. Tur{\'a}n (1948) \\ 12.10: A. Selberg (1949) \\ 12.11: P. Erdo$\cdot$s (1949) \\ 12.12: S. Skewes (1955) \\ 12.13: C.B. Haselgrove (1958) \\ 12.14: H. Montgomery (1973) \\ 12.15: D.J. Newman (1980) \\ 12.16: J. Korevaar (1982) \\ 12.17: H. Daboussi (1984) \\ 12.18: A. Hildebrand (1986) \\ 12.19: D. Goldston and H. Montgomery (1987) \\ 12.20: M. Agrawal, N. Kayal, and N. Saxena (2004) \\ References \\ References \\ Index", } @Proceedings{Bowden:1953:FTT, editor = "{Baron} Bertram Vivian Bowden", booktitle = "Faster Than Thought: a Symposium on Digital Computing Machines", title = "Faster Than Thought: a Symposium on Digital Computing Machines", publisher = "Sir Isaac Pitman and Sons, Ltd.", address = "London, UK", pages = "xix + 416 + 21", year = "1953", LCCN = "QA76.5 .B66", bibdate = "Sun May 15 10:03:12 MDT 2011", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/b/babbage-charles.bib; https://www.math.utah.edu/pub/bibnet/authors/l/lovelace-ada-augusta.bib; https://www.math.utah.edu/pub/bibnet/authors/t/turing-alan-mathison.bib; https://www.math.utah.edu/pub/tex/bib/adabooks.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; z3950.loc.gov:7090/Voyager", note = "With a foreword by the Right Honourable the Earl of Halsbury.", URL = "https://archive.org/details/FasterThanThought", acknowledgement = ack-nhfb, listofcontributors = "Miss M. Audrey Bates, Ferranti Ltd., Moston, Manchester (Chapter 25) \\ Dr. J. M. Bennett, Ferranti Ltd., Moston, Manchester (Chapters 5, 17, 20) \\ Dr. A. D. Booth, Director of the Electronic Computation Laboratory, Birkbeck College, London (Chapter 13) \\ Dr. B. V. Bowden, Ferranti Ltd., Moston, Manchester (Chapters 1--4, 14, 22, 25, 26) \\ Mr. R. H. A. Carter, Telecommunications Research Establishment, Malvern (M.O.S.) (Chapter 10) \\ Mr. E. H. Cooke-Yarborough, Atomic Energy Research Establishment, Harwell (M.O.S.) (Chapter 9) \\ Mr. A. E. Glennie, Research Establishment, Fort Halstead (M.O.S.) (Chapters 5, 19) \\ Dr. S. H. Hollingdale, Head of the Mathematical Services Department, Royal Aircraft Establishment, Farnborough (M.O.S.) \\ (Chapter 12) \\ Dr. T. Kilburn, Senior Lecturer, Electrical Engineering Dept., Manchester University (Chapter 6) \\ Mr. S. Michaelson, Imperial College of Science and Technology, London (Chapter 11) \\ Dr. G. Morton, Lecturer In Economics, London School of Economics And Political Science (Chapter 23) \\ Mr. B. W. Pollard, Ferranti Ltd., Moston, Manchester (Chapter 2) \\ Miss Cicely M. Popplewell, Staff Member of the Royal Society Computing Laboratory, Manchester University (Chapter 24) \\ Dr. D. G. Prinz, Ferranti Ltd., Moston, Manchester (Chapter 15) \\ Dr. R. S. Scorer, Lecturer, Department of Meteorology, Imperial College of Science and Technology, London (Chapter 18) \\ Mr. J. B. Smith, Ferranti Ltd., Crewe Toll, Edinburgh (Chapter 15) \\ Mr. R. Stuart-Williams, Sometime of Ferranti Ltd., Moston, Manchester, now at the R.C.A. Research Laboratories, Princeton, New Jersey, U.S.A. (Chapter 16) \\ Mr. B. B. Swann, Ferranti Ltd., Moston, Manchester (Chapter 21) \\ Mr. C. Strachey, National Research Development Corporation (Chapter 25) \\ Dr. K. D. Tocher, Imperial College of Science and Technology, London (Chapter 11) \\ Dr. A. M. Turing, F.R.S., Assistant Director of the Royal Society Computing Laboratory, Manchester University (Chapter 25) \\ Dr. A. M. Uttley, Telecommunications Research Establishment, Malvern (M.O.S.) (Chapter 10) \\ Dr. M. V. Wilkes, Director of the University Mathematical Laboratory, Cambridge (Chapter 17) \\ Professor F. C. Williams, O.B.E., F.R.S. (Professor of Electrical Engineering) Director of the Royal Society Computing Laboratory, Manchester University (Chapter 6) \\ Chapter 8 is reprinted from \booktitle{Engineering} by kind permission of the Publishers", listofplates = "Ada Augusta, Countess of Lovelace / Frontispiece \\ I. Charles Babbage / 12 \\ II. Part of Babbage's Difference Engine / 28 \\ III. Two Hollerith Punch Cards of the Type Used in the A.C.E. / 29 \\ IV. The Magnetic Drum of the Manchester Machine / 60 \\ V. The Photo-Electric Tape-Reader of the Manchester Machine / 112 \\ VI. A Typical Stored Pattern on a Cathode-Ray-Tube Screen / 120 \\ VII. The First Manchester University Computer / 121 \\ VIII. A General View of the Manchester University Computer Without Covers / 124 \\ IX. A General View of the Manchester University Computer and Control Desk / 126 \\ X. The Control Desk of the Manchester University Computer, Showing the Console / 127 \\ XI. A General View of the E.D.S.A.C. / 132 \\ XII. One Unit of the A.C.E. / 136 \\ XIII. A View of the A.C.E. Showing Delay Units / 138 \\ XIV. A View of the A.C.E. Showing the Hollerith Equipment Used for Input and Output / 139 \\ XV. A Cathode-Ray-Tube Store Pattern / 148 \\ XVI. The Ferranti (Edinburgh) Logical Computer and Feedback Computer / 188 \\ XVII. ``Nimrod'' at the Science Exhibition, South Kensington / 200 \\ XVIII. The $b$ Patterson Projection of Whale Myoglobin Printed in Contour Form / 204", remark-01 = "Portrait of Ada Augusta, Countess of Lovelace, faces title page.", remark-02 = "Chapter authors are credited only in the List of Contributors on page xv; their names, and order, fail to appear on chapter papers. No author is credited for Chapters 7 and 8", remark-03 = "From page ix: ``The principles on which all modern computing machines are based were enunciated more than a hundred years ago by a Cambridge mathematician named Charles Babbage, who devoted his life and fortune to an unsuccessful attempt to construct one. Modern developments in electronics have made his dream come true in the last decade, and there are now a dozen or more machines in the world which do all and more than he expected.", remark-04 = "From page ix: ``A rough count showed that about 150 digital computers are being built at this moment, most of them in universities and other research establishments. It will be interesting to see if these machines play in the next decade the part of the cyclotrons and high voltage generators in the `thirties'.''", remark-05 = "From page x: ``It seems probable that we shall have a second Industrial Revolution on our hands before long. The first one replaced men's muscles by machines, and eve1y worker in England now has an average of more than 3 horse power to help him. In the next revolution machines may replace men's brains and relieve them of much of the drudgery and boredom which is now the lot of so many white collar workers.''", remark-06 = "From page x: ``Nowadays many of these dedicated men spend their time in computing prime numbers. The search for the largest known prime is a hobby which is at least as useful and interesting as playing bridge, and computing machines have helped enormously. The reader will not be surprised to hear that nowadays the biggest primes are found in America. The largest which has been discovered so far (January, 1953) consists of 2281 consecutive `ones,' when it is expressed in the binary scale (see page 33).''", remark-07 = "From page xi: ``The early history of these machines and the story of poor Babbage's struggles is very interesting. We owe our best account of Babbage's `Engines' to the Countess of Lovelace, who was a mathematician of great competence and one of the very few people who understood what Babbage was trying to do. Her ideas are so modern that they have become of great topical interest once again, and since her paper has long been out of print (it appeared more than a hundred years ago) it has been reproduced as an appendix to this book. Lady Lovelace's grand-daughter, the Right Hon. Lady Wentworth, has very kindly allowed me to read many of Lady Lovelace's most interesting papers; I was so surprised by the connexion that I found between digital computers and thoroughbred horses that I have given a brief account of the story, for further details of which the reader is referred to Lady Wentworth's own books.''", remark-08 = "From page xi: ``After I had finished the book, I saw a microfilm of a life of Babbage which had been written by his executor, the late Mr. L. H. D. Buxton. Mr. Whitwell of the Powers Samas Company found the manuscript in the Museum of the History of Science in Oxford. It contains a more detailed account of the construction of Babbage's Engines than any I have seen elsewhere, and it is to be hoped that the material will some day be published.''", remark-09 = "From page xiii: ``Much of this book derives from the work of those prolific authors `Anon' and `Ibid' who have done so much to put our English platitudes on a sound literary basis.''", remark-10 = "From page xiii: ``I must express my thanks to all my collaborators; to Lord Halsbury for writing the foreword; to Lady Wentworth who gave me so much information about Lady Lovelace, and who allowed me to reproduce the portrait which has been used as a frontispiece. I am also indebted to Miss Draper who read all the Lovelace paper; and gave me a great deal of help. I must thank Miss Dyke for preparing the flow sheets which I used in Chapter 22. Dr. Gilles and Mr. Whitewell told me the story of Dr. Comrie; Dr. Bullard found some of Babbage's writing in the archives of the National Physical Laboratory; and Professor Aitken, Mr. W. Klein, Dr. van Wijngaarden, Dr. Stokvis, Mr. Seeber, Mr. Ferris and Dr. Gabor gave me much of the information on which Chapter 26 is based. The Portrait of Babbage is included by courtesy of the Director of the Science Museum, South Kensington.''", remark-11 = "From glossary entry on page 411: ``{\em Computor}. `Bad spelling of Computer' --- Oxford English Dictionary.''", remark-12 = "From glossary entry on page 411: ``{\em Cybernetics}. A word invented by Professor Wiener to describe the field of control and communication theory, whether in the machine or in the animal. None of the authors quite understands what the word means, so it has not been used in this book.", remark-13 = "From glossary entry on page 412: ``{\em Hartree Constant}. The time which is expected to elapse before a particular electronic computing machine is finished and working. It was Professor Hartree who first pointed out that this estimated time usually remains constant at about six months for a period of several years during the development of a machine. This phenomenon was well known to Babbage. Few engineers are worried unless the `constant' shows signs of increasing monotonically as the years go by.''", remark-14 = "From glossary entry on page 413: ``{\em Mill}. Babbage's name for the arithmetic unit of his machine.''", remark-15 = "From glossary entry on page 414: ``{\em Programmer}. One who prepares programmes for a machine, `a harmless drudge'.''", remark-16 = "From glossary entry on page 414: ``{\em T{\"u}ring Machine}. In 1936 Dr. Turing wrote a paper on the design and the limitations of computing machines. For this reason they are sometimes known by his name. The umlaut is an unearned and undesirable addition, due, presumably, to an impression that anything so incomprehensible must be Teutonic.''", subject = "Electronic digital computers", tableofcontents = "Foreword / v \\ Preface / vii \\ List of Contributors / xv \\ Part One: The History and Theory of Computing Machines \\ 1. A Brief History of Computation / B. V. Bowden / 3 \\ 2. The Circuit Components of Digital Computers / B. V. Bowden and B. W. Pollard / 32 \\ 3. The Organization of a Typical Machine / B. V. Bowden / 67 \\ 4. The Construction, Performance and Maintenance of Digital Computers / B. V. Bowden / 78 \\ 5. Programming For High-Speed Digital Calculating Machines / J. M. Bennett and A. E. Glennie / 101 \\ Part Two: Electronic Computing Machines in Britain and America / \\ 6. The University of Manchester Computing Machine / T. Kilburn and F. C. Williams / 117 \\ 7. Calculating Machine Development at Cambridge / 130 \\ 8. Automatic Computation at the National Physical Laboratory / 135 \\ 9. The Harwell Electronic Digital Computer / E. H. Cooke-Yarborough / 140 \\ 10. The Telecommunications Research Establishment Parallel Electronic Digital Computer / R. H. A. Carter and A. M. Uttley / 144 \\ 11. The Imperial College Computing Engine / S. Michaelson and K. D. Tocher / 161 \\ 12. The Royal Aircraft Establishment Sequence-Controlled Calculator / S. H. Hollingdale / 165 \\ 13. Calculating Machines at the Birkbeck College Computation Laboratory / A. D. Booth / 170 \\ 14. Computers in America / B. V. Bowden / 173 \\ Part Three: Applications of Electronic Computing Machines \\ 15. Machines for the Solution of Logical Problems / D. G. Prinz and J. B. Smith / 181 \\ 16. Special-Purpose Automatic Computers / R. Stuart-Williams / 199 \\ 17. Digital Computation and the Crystallographer / J. M. Bennett and M. V. Wilkes / 203 \\ 18. The Use of High-Speed Computing Machines in Meteorology / R. S. Scorer / 210 \\ 19. An Application to Ballistics / A. E. Glennie / 216 \\ 20. Digital Computers and the Engineer / J. M. Bennett / 223 \\ 21. Machines in Government Calculations / B. B. Swann / 234 \\ 22. The Application of Digital Computers to Business and Commerce / B. V. Bowden / 246 \\ 23. Electronic Machines and Economics / G. Morton / 272 \\ 24. Problems of Dynamical Astronomy / Cicely M. Popplewell / 282 \\ 25. Digital Computers Applied to Games / M. Audrey Bates, B. V. Bowden, C. Strachey, and A. M. Turing / 286 \\ 26. Thought and Machine Processes / B. V. Bowden / 311 \\ Appendix 1: Extracts From \booktitle{Taylor's Scientific Memoirs}, Vol. III / 341 \\ Appendix 2: Extracts From the \booktitle{Lovelace Papers} / 409 \\ Glossary / 411 \\ Index / 415 \\ Insets \\ Flow Sheet For P.A.Y.E. Calculation / 254 \\ Computation of Bernoulli Numbers / 404", } @Book{Birkhoff:1954:SMM, editor = "Garrett Birkhoff", booktitle = "Studies in Mathematics and Mechanics Presented to {Richard von Mises} by friends, colleagues, and pupils", title = "Studies in Mathematics and Mechanics Presented to {Richard von Mises} by friends, colleagues, and pupils", publisher = pub-ACADEMIC, address = pub-ACADEMIC:adr, pages = "ix + 353", year = "1954", ISBN = "1-4832-6356-8", ISBN-13 = "978-1-4832-6356-4", LCCN = "QA3 .S853", bibdate = "Fri Oct 20 10:13:10 2023", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", abstract = "Studies in Mathematics and Mechanics is a collection of studies presented to Professor Richard von Mises as a token of reverence and appreciation on the occasion of his seventieth birthday which occurred on April 19, 1953. von Mises' thought has been a stimulus in many seemingly unconnected fields of mathematics, science, and philosophy, to which he has contributed decisive results and new formulations of fundamental concepts. The book contains 42 chapters organized into five parts. Part I contains papers on algebra, number theory and geometry. These include a study of Poincar{\'e}'s representation of a hyperbolic space on an Euclidean half-space and elementary estimates for the least primitive root. Part II on analysis includes papers on a generalization of Green's Formula and its application to the Cauchy problem for a hyperbolic equation, and the fundamental solutions of a singular Beltrami operator. Part III deals with theoretical mechanics and covers topics such as turbulent flow, axially symmetric flow, and oscillating wakes. The papers in Part IV focus on applied mechanics. These include studies on plastic flow under high stresses and the problem of inelastic thermal stresses. Part V presents studies on probability and statistics, including a finite frequency theory of probability and the problem of expansion of clusters of galaxies.", acknowledgement = ack-nhfb, subject-dates = "Richard von Mises (1883--1953)", } @Book{Langer:1959:NAP, editor = "R. E. Langer", booktitle = "On numerical approximation. {Proceedings of a Symposium, Madison, April 21--23, 1958}", title = "On numerical approximation. {Proceedings of a Symposium, Madison, April 21--23, 1958}", publisher = "The University of Wisconsin Press", address = "Madison, WI, USA", pages = "x + 462", year = "1959", LCCN = "QA3 .U45 no. 1", bibdate = "Tue Jun 19 06:45:47 2018", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/b/bauer-friedrich-ludwig.bib; https://www.math.utah.edu/pub/bibnet/authors/s/stiefel-eduard.bib; https://www.math.utah.edu/pub/bibnet/authors/t/tukey-john-w.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "Publication no. 1 of the Mathematics Research Center, U.S. Army, the University of Wisconsin.", acknowledgement = ack-nhfb, subjects = "statistics", tableofcontents = "1. On trends and problems in numerical approximation / Ostrowski \\ 2. Linear spaces and approximation theory / Buck \\ 3. Operational methods in numerical analysis based on rational approximations / Kopal \\ 4. On the numerical integration of periodic analytic functions / Davis \\ S. Some new divided difference algorithms in two variables / Salzer \\ 6. Numerical evaluation of multiple integrals / Hammer \\ 7. Optimal approximation and error bounds / Golomb and Weinberger \\ 8. The rationale of approximation / Sard \\ 9. On extremal approximations / Walsh \\ 10. Numerical methods of Tchebycheff approximation / Stiefel \\ 11. Minimax methods in table construction / Fox \\ 12. Existence of essentially nonlinear families suitable for oscillatory approximation / Motzkin \\ 13. On variation diminishing approximation methods / Schoenberg \\ 14. Approximation by functions of fewer variables / Golomb \\ 15. Extremal approximations --- a summary / Miller \\ 16. Survey of recent Russian literature on approximation / Buck \\ 17. The quotient--difference and epsilon algorithms / Bauer \\ 18. Some sufficient conditions for the existence of an asymptotic formula or an asymptotic expansion / Rosser \\ 19. The estimation of (power) spectra and related quantities / Tukey \\ 20. Approximation in partial differential equations / Collatz \\ 21. Special polynomials in numerical analysis / Todd", } @Book{Ralston:1960:MMD, editor = "Anthony Ralston and Herbert S. Wilf", booktitle = "Mathematical methods for digital computers", title = "Mathematical methods for digital computers", publisher = pub-WILEY, address = pub-WILEY:adr, pages = "various", year = "1960--1977", ISBN = "0-471-70690-6", ISBN-13 = "978-0-471-70690-8", LCCN = "QA39 .R26", bibdate = "Mon Jan 13 10:36:06 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "Three volumes.", acknowledgement = ack-nhfb, } @Book{Abramowitz:1964:HMF, editor = "Milton Abramowitz and Irene A. Stegun", key = "NBS", booktitle = "Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables", title = "Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables", volume = "55", publisher = "U. S. Department of Commerce", address = "Washington, DC, USA", pages = "xiv + 1046", year = "1964", LCCN = "QA47.A161 1972; QA 55 A16h 1972", bibdate = "Thu Jan 27 07:58:12 2000", bibsource = "https://www.math.utah.edu/pub/bibnet/subjects/acc-stab-num-alg.bib; https://www.math.utah.edu/pub/bibnet/subjects/han-wri-mat-sci-2ed.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/master.bib", note = "Tenth printing, with corrections (December 1972). This book is also available online at \path=http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP= in bitmap image format.", series = "Applied mathematics series", abstract = "This book is a compendium of mathematical formulas, tables, and graphs. It contains a table of analytical integrals, differential equations, and numerical series; and includes tables of trigonometric and hyperbolic functions, tables for numerical integration, rules for differentiation and integration, and techniques for point interpolation and function approximation. Additionally, it devotes a entire section to mathematical and physical constants as fractions and powers of Pi, e, and prime numbers; and discusses statistics by presenting combinatorial analysis and probability functions.", acknowledgement = ack-nhfb, tableofcontents = "Mathematical constants / David S. Liepman \\ Physical constants and conversion factors / A. G. McNish \\ Elementary analytical methods / Milton Abramowitz \\ Elementary transcendental functions: logarithmic, exponential, circular and hyperbolic functions / Ruth Zucker \\ Exponential integral and related functions / Walter Gautschi and William F. Cahill \\ Gamma function and related functions / Philip J. Davis \\ Error function and Fresnel integrals / Walter Gautschi \\ Legendre functions / Irene A. Stegun \\ Bessel functions of integer order / F. W. J. Olver \\ Bessell functions of fractional order / H. A. Antosiewicz \\ Integrals of Bessel functions / Yudell L. Luke \\ Struve functions and related functions / Milton Abramowitz \\ Confluent hypergeometric functions / Lucy Joan Slater \\ Coulomb wave functions / Milton Abramowitz \\ Hypergeometric functions / Fritz Oberhettinger \\ Jacobian elliptic functions and theta functions; Elliptic integrals / L. M. Milne-Thomson \\ Weierstrass elliptic and related functions / Thomas H. Southard \\ Parabolic cylinder functions / J. C. P. Miller\ldots{} Mathieu functions / Gertrude Blanch \\ Spheroidal wave functions / Arnold N. Lowan \\ Orthogonal polynomials / Urs W. Hochstrasser \\ Bernoulli and Euler polynomials, Riemann zeta function / Emilie V. Haynesworth and Karl Goldberg \\ Combinatorial analysis / K. Goldberg, M. Newman and E. Haynesworth \\ Numerical interpolation, differentiation and integration / Philip J. Davis and Ivan Polonsky \\ Probability functions / Marvin Zelen and Norman C. Severo \\ Miscellaneous functions / Irene A. Stegun \\ Scales of notation / S. Peavy and A. Schopf \\ Laplace transforms", } @Book{Magnus:1966:FTS, author = "Wilhelm Magnus and Fritz Oberhettinger and Raj Pal Soni", booktitle = "Formulas and theorems for the special functions of mathematical physics", title = "Formulas and theorems for the special functions of mathematical physics", publisher = pub-SV, address = pub-SV:adr, edition = "Third", pages = "viii + 508", year = "1966", LCCN = "QA1 G88 v. 52, 1966", bibdate = "Sat Oct 30 18:23:25 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp2010.bib", note = "See errata \cite{Cohl:2012:TEF,Szmytkowski:2013:EBT}.", acknowledgement = ack-nhfb, } @Book{Klerer:1967:DCU, editor = "Melvin Klerer and Granino A. Korn", title = "Digital Computer User's Handbook", booktitle = "Digital Computer User's Handbook", publisher = pub-MCGRAW-HILL, address = pub-MCGRAW-HILL:adr, year = "1967", LCCN = "QA76.5 .K524", bibdate = "Wed Dec 15 17:52:19 1993", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/h/hamming-richard-w.bib; https://www.math.utah.edu/pub/bibnet/authors/w/wilkinson-james-hardy.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/master.bib; https://www.math.utah.edu/pub/tex/bib/mathcw.bib; https://www.math.utah.edu/pub/tex/bib/prng.bib; https://www.math.utah.edu/pub/tex/bib/technometrics1960.bib", acknowledgement = ack-nhfb, tableofcontents = "Contributors / v \\ Preface / vii \\ \\ Part I Topics in Programming \\ \\ 1.1. Elements of Programming / Melvin Klerer / 1-3 \\ 1.2. Computer Number Systems and Arithmetic / Melvin Klerer / 1-34 \\ 1.3. Errors, Loss of Significance, and Data Presentation / Melvin Klerer / 1-67 \\ 1.4. Computer Characteristics Table / Charles W. Adams Associates / 1-81 \\ 1.5. Algorithmic Compiler Design / A. A. Grau / 1-141 \\ 1.6. Structure and Use of ALGOL 60 / H. Bottenbruch / 1-181 \\ 1.7. List-processing Languages / Paul W. Abrahams / 1-239 \\ 1.8. Computer Languages for System Simulation / Howard S. Krasnow / 1-258 \\ 1.9. PERT/CPM / William C. Geoghan / 1-278 \\ 1.10. Sorting and Merging / Martin A. Goetz / 1-292 \\ \\ Part II Numerical Techniques \\ \\ 2.1. A Survey of Function-approximation Techniques / Granino A. Korn / 2-3 \\ 2.2. Solution of Linear Algebraic Equations and Matrix Problems by Direct Method / James H. Wilkinson / 2-18 \\ 2.3. Solution of Nonlinear Equations / Royce E. Beckett / 2-56 \\ 2.4. Interpolation, Curve Fitting, and Differentiation / Kaiser S. Kunz / 2-82 \\ 2.5. Numerical Integration / A. H. Stroud / 2-117 \\ 2.6. Numerical Solution of Ordinary Differential Equations / R. W. Hamming / 2-144 \\ 2.7. Numerical Solution of Partial Differential Equations / Walter J. Karplus and Venkateswararao Vemuri / 2-163 \\ \\ Part III Statistical Methods \\ \\ 3.1. Introduction to Statistical Methods / Granino A. Korn / 3-3 \\ 3.2. Statistical Techniques and Computations / Henry Tucker / 3-18 \\ 3.3. Computation of Power Spectra / Melvin Klerer / 3-53 \\ 3.4. Random-number Generation and Monte-Carlo Methods / T. E. Hull / 3-63 \\ \\ Part IV Computer Applications \\ \\ 4.1. Symbolic Logic and Practical Applications / J. V. Wait / 4-3 \\ 4.2. Information Theory and Codes / Harvey L. Garner / 4-29 \\ 4.3. Linear Programming / Lloyd Rosenberg / 4-63 \\ 4.4. Nonlinear Programming / E. M. L. Beale / 4-117 \\ 4.5. Commercial Data Processing / Robert V. Head / 4-153 \\ 4.6. Digital Computers for Logical Design / Richard E. Merwin and Jere L. Sanborn / 4-167 \\ 4.7. Information Retrieval / Jack Belzer and Orrin E. Taulbee / 4-193 \\ 4.8. Some Parameter-optimization Techniques / Robert B. McGhee / 4-234 \\ 4.9. Scheduling and Inventory Control / Jerry L. Sanders / 4-256 \\ 4.10. Real-time Operations with Small General-purpose Computers / Barbera W. Stephenson / 4-263 \\ \\ Index", xxISBN = "none", } @Proceedings{AFIPS:1969:ACPb, key = "AFIPS FJCC '69", booktitle = "1969 Fall Joint Computer Conference, November 18--20, 1969, Las Vegas, Nevada", title = "1969 Fall Joint Computer Conference, November 18--20, 1969, Las Vegas, Nevada", volume = "35", publisher = pub-AFIPS, address = pub-AFIPS:adr, pages = "807", year = "1969", LCCN = "TK7885.A1 J6 1969", bibdate = "Sat Sep 24 01:06:00 1994", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/c/cody-william-j.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", series = "AFIPS conference proceedings", acknowledgement = ack-nhfb, } @Proceedings{AFIPS:1971:ACP, key = "AFIPS SJCC '71", booktitle = "1971 Spring Joint Computer Conference, May 18--20, 1971, Atlantic City, New Jersey", title = "1971 Spring Joint Computer Conference, May 18--20, 1971, Atlantic City, New Jersey", volume = "38", publisher = pub-AFIPS, address = pub-AFIPS:adr, pages = "631", year = "1971", LCCN = "????", bibdate = "Fri Sep 16 10:47:01 1994", bibsource = "https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", series = "AFIPS conference proceedings", acknowledgement = ack-nj # " and " # ack-nhfb, } @Proceedings{Rice:1971:MS, editor = "John R. Rice", booktitle = "Mathematical Software", title = "Mathematical Software", publisher = pub-ACADEMIC, address = pub-ACADEMIC:adr, pages = "xvii + 515", year = "1971", ISBN = "0-12-587250-X", ISBN-13 = "978-0-12-587250-8", LCCN = "QA1 .M26", bibdate = "Thu Sep 15 18:56:52 1994", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/c/cody-william-j.bib; https://www.math.utah.edu/pub/bibnet/authors/r/rice-john-r.bib; https://www.math.utah.edu/pub/mirrors/ftp.ira.uka.de/bibliography/Math/elefunt.bib; https://www.math.utah.edu/pub/mirrors/ftp.ira.uka.de/bibliography/Math/fparith.bib; https://www.math.utah.edu/pub/mirrors/ftp.ira.uka.de/bibliography/Misc/Bibnet/authors/c/cody-william-j.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", note = "Based on the proceedings of the Mathematical Software Symposium held at Purdue University, Lafayette, Indiana, USA, April 1--3, 1970.", acknowledgement = ack-nhfb, tableofcontents = "Preface \\ Acknowledgments \\ Part One: Prologue \\ Chapter 1. Historical Notes \\ I. Introduction \\ II. Chronological Record \\ References \\ Chapter 2. The Distribution and Sources of Mathematical Software \\ I. Introduction \\ II. Local Distribution Methods \\ III. Assessment of General Sources \\ IV. Summary \\ Chapter 3. The Challenge for Mathematical Software \\ I. Introduction \\ II. Algorithm Construction \\ III. Evaluation --- Charting the Unknown \\ IV. Dissemination --- Some Alternatives \\ V. Two Recommendation \\ References \\ Chapter 4. Discussion of the Papers \\ I. The User's Voice \\ II. Arithmetic \\ III. Libraries \\ IV. The Automation of Numerical Analysis \\ V. Comparative Evaluation \\ VI. Systems for Mathematical Software \\ VII. Nonnumerical Software \\ VIII. Mathematical Procedures \\ Part Two: Proceedings of the Symposium \\ Chapter 5. The Papers \\ 5.1 A user's experience with sophisticated least-squares software in the discovery of the lunar mass concentrations (MASCONS) \\ I. Nature of the Data Reduction \\ II. Implication for Program Development and Distribution \\ III. Summary of Conclusions \\ Reference \\ 5.2 User Modifiable Software \\ I. The Argument for Easy-to-Modify Software \\ II. Writing Easy-to-Modify Software \\ 5.3 Number Representation and Significance Monitoring \\ I. Number Representation \\ II. Error Classification \\ III. Significance Analysis \\ IV. Significance Monitoring \\ V. Mathematical Software \\ References \\ 5.4 The Estimation of Significance \\ I. Introduction \\ II. Discussion of Rules \\ III. Implementation \\ IV. Elementary Functions \\ V. Numerical Experiments \\ References \\ 5.5 Nonstandard Arithmetic \\ I. Reliability \\ II. Subroutine Library \\ III. Efficiency in Execution \\ IV. Ease of Use \\ V. Implementation of Nonstandard Arithmetic \\ VI. Use of Precompiler \\ VII. Type Other \\ VIII. Conclusion \\ References \\ 5.6 Making Special Arithmetics Available \\ References \\ 5.7 The Production of Mathematical Software for a Mass Audience \\ I. Introduction \\ II. Discussion Assumptions \\ III. Problems in Mathematical Software Production \\ IV. Environmental Conditions Affecting Mathematical Software Production \\ V. Production of Mathematical Software \\ VI. User Attitudes \\ VII. Summary \\ 5.8 High Quality Portable Numerical Mathematics Software \\ I. Introduction \\ II. The Bell Laboratories Numerical Mathematics Program Library One \\ III. Status of Library One \\ IV. ZERBND \\ V. Portability \\ VI. Testing \\ References \\ 5.9 The Development and Maintenance of a Technical Subprogram Library \\ I. Introduction \\ II. Coding Standards \\ III. Documentation Format \\ IV. Review Procedures \\ V. Maintenance Procedures \\ VI. Multiple Precision in Fortran \\ VII. Support and Maintenance Requirements \\ VIII. Access Procedures \\ IX. Summary and Conclusions \\ X. Current Category Index \\ XI. Sample Documentation \\ 5.10 The Boeing library and Handbook of Mathematical Routines \\ Reference", } @Proceedings{Macon:1971:SJC, editor = "Nathaniel Macon", key = "AFIPS SJCC '71", booktitle = "1971 Spring Joint Computer Conference, May 18--20, 1971, Atlantic City, New Jersey", title = "1971 Spring Joint Computer Conference, May 18--20, 1971, Atlantic City, New Jersey", volume = "38", publisher = pub-AFIPS, address = pub-AFIPS:adr, pages = "631", year = "1971", DOI = "https://doi.org/10.1145/1478786", ISBN = "1-4503-7907-9", ISBN-13 = "978-1-4503-7907-6", LCCN = "????", bibdate = "Fri Sep 16 10:47:01 1994", bibsource = "https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fortran1.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", series = "AFIPS conference proceedings", acknowledgement = ack-nj # " and " # ack-nhfb, tableofcontents = "Computing Machines --- Menace or Messiah? --- Panel Session (No papers in this volume) \\ \\ Image of the Industry --- Panel Session (No papers in this volume) \\ \\ The New Technology --- Hardware Design and Evaluation \\ \\ The DINKIAC I --- A pseudo-virtual-memoried mini --- For stand-alone interactive use / R. W. Conn / 1 \\ A multi-channel CRC register / A. M. Patel / 11 \\ Features of an advanced front-end CPU / R. B. Hibbs / 15 \\ Interpreting the results of a hardware systems monitor / J. S. Cockrum and E. D. Crockett / 23 \\ \\ Law Enforcement and Judicial Administration --- Panel Session (No papers in this volume) \\ \\ Applications Requiring Multiprocessors \\ \\ 4-way parallel processor partition of an atmospheric primitive-equation prediction model / E. Morenoff, W. Beckett, P. G. Kesel, F. J. Winninghoff, and P. M. Wolff / 39 \ An associative processor for air traffic control / K. J. Thurber / 49 \\ \\ Computer Aided management Of Earth Resources --- Panel Session (No papers in this volume) \\ \\ Responsive Government --- Panel Session (No papers in this volume) \\ \\ Computers in Transport --- For Management Needs or Suppliers' Delight? \\ \\ A computer-aided traffic forecasting technique --- The trans Hudson model / E. J. Lessieu / 61 \\ Computer graphics for transportation problems / D. Cohen and J. M. McQuillan / 77 \\ Real time considerations for an airline / J. Loo, B. T. O'Donald, and I. R. Whiteman / 83 \\ A computer simulation model of train operations in CTC territory D. Borch / 93 \\ \\ Present and Future Data Networks --- Panel Session (No papers in this volume) \\ \\ Terminal Oriented Displays \\ \\ A general display terminal system J. H. Botterill and G. F. Heyne / 103 \\ AIDS --- Advanced interactive display system / T. R. Stack and S. T. Walker / 113 \\ CRT display system for industrial process / T. Konishe, N. Hamada and I. Yasuda / 123 \\ Computer generated closed circuit TV displays with remote terminal control / S. Winkler, G. W. Price / 131 \\ \\ Competitive Evaluation of Interactive Systems --- Panel Session (No papers in this volume) \\ \\ Computers in the Electoral Process \\ \\ The theory and practice of bipartisan constitutional computer-aided redistricting / S. S. Nagel / 137 \\ ``Second-generation'' computer vote count systems --- Assuming a professional responsibility / C. H. Springer and M. R. Alkus / 143 \\ \\ Microprogramming and Emulation \\ \\ Evaluation of hardware--firmware--software trade-offs with mathematical modeling / H. Barsamiam and A. DeCegama / 151 \\ System/370 integrated emulation under OS and DOS / G. R. Allred / 163 \\ A high-level microprogramming language (MPL) / R. H. Eckhouse, Jr. / 169 \\ A firmware APL time-sharing system / R. Zaks, D. Steingart, and J. Moore / 179 \\ \\ Interactive Applications and Systems \\ \\ Designing a large scale on-line real-time system / S. Ishizaki / 191 \\ PERT --- A computer-aided game / J. Richter-Nielsen / 199 \\ Interactive problem-solving --- An experimental study of ``lockout'' effects / B. W. Boehm, M. J. Seven, and R. A. Watson / 205 \\ TYMNET --- A terminal-oriented communication network / L. R. Tymes / 211 \\ Implementation of an interactive conference system / T. W. Hall / 217 \\ \\ Computational Complexity --- Panel Session (No papers in this volume) \\ \\ The Evolution of Computer Animation --- Panel Session (No papers in this volume) \\ \\ Serving Users in Higher Education \\ \\ Who are the users? --- An analysis of computer use in a university computer center / E. Hunt, G. Diehr, and D. Garnatz / 231 \\ \\ Information and Data Management \\ \\ An initial operational problem oriented medical record system --- For storage, manipulation and retrieval of medical data / J. R. Schultz, S. V. Cantrill, and K. G. Morgan / 239 \\ Laboratory verification of patient identity / S. Raymond, L. Chalmers, and W. Steuber / 265 \\ The data system environment simulator (DASYS) / L. E. DeCuir and R. W. Garrett / 271 \\ Management information systems --- What happens after implementation? / D. E. Thomas, Jr. / 277 \\ A methodology for the design and optimization of information processing systems / J. F. Nunamaker, Jr. / 283 \\ \\ Computer Assisted Instruction \\ \\ Computer generated repeatable tests / F. Prosser and D. D. Jensen / 295 \\ R2 --- A natural language question-answering system / K. Biss, R. Chien, and F. Stahl / 303 \\ \\ The New Technology --- Storage \\ \\ Performance evaluation of direct access storage devices with a fixed head per track / T. Manocha, W. L. Martin, and K. W. Stevens / 309 \\ Drum queueing model / G. P. Jain and S. R. Arora / 319 \\ Storage hierarchy systems / H. Katzan, Jr. / 325 \\ Optimal sizing, loading and re-loading in a multi-level memory hierarchy system / S. R. Arora and A. Gallo / 337 \\ The TABLON mass storage network / R. B. Gentile and J. R. Lucas, Jr. / 345 \\ \\ Topics in Computer Arithmetic and in Artificial Intelligence \\ \\ A structure for systems that plan abstractly / W. W. Jacobs / 357 \\ Unconventional superspeed computer systems / T. C. Chen / 365 \\ High speed division for binary computers / H. Ling / 373 \\ A unified algorithm for elementary functions / S. Walther / 379 \\ A software system for tracing numerical significance during computer program execution / H. S. Bright, B. A. Colhoun, and F. B. Mallory / 387 \\ Software Liability and Responsibility --- Panel Session (No papers in this volume) \\ \\ Venture Capital --- Financing Young Companies --- Panel Session (No papers in this volume) \\ \\ From the User's Viewpoint --- Panel Session (No papers in this volume) \\ \\ Peripheral Processing --- Panel Session (No papers in this volume) \\ \\ Computer Pictorics \\ \\ Automated interpretation and editing of fuzzy line drawings / S. K. Chang / 393 \\ Computer graphics study of array response / G. W. Byram, G. V. Olds, and L. P. LaLumiere / 401 \\ Computer manipulation of digitized pictures / N. Macon and M. E. Kiefer / 407 \\ An International View --- Panel Session (No papers in this volume) \\ \\ Simulation of Computer Systems \\ \\ The design of a meta-system / A. S. Noetzel / 415 \\ An interactive simulator generating system for small computers / J. L. Brame and C. V. Ramamoorthy / 425 \\ Application of Computers to Training --- Panel Session (No papers in this volume) \\ \\ The New Technology --- Diagnostics and Recovery \\ \\ Multiband automatic test equipment --- A computer controlled check-out system / T. Kuroda and T. C. Bush / 451 \\ Coding techniques for failure recovery in a distributive modular memory organization / S. A. Szygenda and M. J. Flynn / 459 \\ Recovery through programming system/370 / D. L. Droulette / 467 \\ On automatic testing of one-line, real-time systems / J. S. Gould / 477 \\ \\ The New Technology --- Systems Software \\ \\ PORTS --- A method for dynamic interprogram communication and job control / R. M. Balzer / 485 \\ Automatic program segmentation based on boolean connectivity / E. W. Ver Hoef / 491 \\ Partial recompilation / R. B. Ayres and R. L. Derrenbacher / 497 \\ PL/C --- The design of a high-performance compiler for PL/I / H. L. Morgan and R. A. Wagner / 503 \\ GPL/I --- A PL/I extension for computer graphics / D. N. Smith / 511 \\ ETC --- An extendible macro-based compiler / B. N. Dickman / 529 \\ \\ The Computer Professional and the Changing Job Market --- Panel Session (No papers in this volume) \\ \\ The New Technology --- File Organization \\ \\ A file organization method using multiple keys / M. L. O'Connell / 539 \\ Arranging frequency dependent data on sequential memories / C. V. Ramamoorthy and P. R. Blevins / 545 \\ Associative processing of line drawings / N. J. Stillman, C. R. Defiore, and P. B. Berra / 557 \\ \\ The New Technology --- Computer Architecture \\ \\ The hardware-implemented high-level machine language for symbol / G. D. Chesley and W. R. Smith / 563 \\ SYMBOL --- A major departure from classic software dominated von Neumann computing systems / R. Rice and W. R. Smith / 575 \\ The physical attributes and testing aspects of the symbol system / B. E. Cowart, R. Rice, and S. F. Lundstrom / 589 \\ SYMBOL --- A large experimental system exploring major hardware replacement of software / W. R. Smith, R. Rice, G. D. Chesley, T. A. Laliotis, S. F. Lundstrom, M. A. Calhoun, L. D. Gerould, and T. G. Cook / 601 \\ \\ Educational Requirements for Systems Analysts \\ \\ A semi-automatic relevancy generation technique for data processing education and career development / J. D. Benenati / 617 \\ An architectural framework for systems analysis and evaluation / P. Freeman / 629 \\ \\ Computer Acquisition --- Purchase or Lease --- Panel Session (No papers in this volume) \\ \\ Computation, Decision Making, and the Environment --- Panel Session (No papers in this volume)", } @Proceedings{Askey:1975:TAS, editor = "Richard Askey", booktitle = "{Theory and application of special functions: proceedings of an advanced seminar sponsored by the Mathematics Research Center, the University of Wisconsin-Madison, March 31--April 2, 1975}", title = "{Theory and application of special functions: proceedings of an advanced seminar sponsored by the Mathematics Research Center, the University of Wisconsin-Madison, March 31--April 2, 1975}", number = "35", publisher = pub-ACADEMIC, address = pub-ACADEMIC:adr, pages = "xi + 560", year = "1975", ISBN = "0-12-064850-4", ISBN-13 = "978-0-12-064850-4", LCCN = "QA3 .U45 no. 35 QA351", bibdate = "Sat Oct 30 07:41:32 MDT 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; z3950.loc.gov:7090/Voyager", series = "Publication of the Mathematics Research Center, the University of Wisconsin", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", meetingname = "Advanced Seminar on Special Functions, Madison, Wis., 1975.", subject = "Functions, Special; Congresses", } @Proceedings{Miller:1975:TNA, editor = "John J. H. Miller", booktitle = "Topics in numerical analysis: proceedings of the Royal Irish Academy Conference on Numerical Analysis, 1972, 1974, 1976", title = "Topics in numerical analysis: proceedings of the Royal Irish Academy Conference on Numerical Analysis, 1972, 1974, 1976", publisher = pub-ACADEMIC, address = pub-ACADEMIC:adr, pages = "various", year = "1975", ISBN = "0-12-496950-X", ISBN-13 = "978-0-12-496950-6", LCCN = "QA297 .R69 1973", bibdate = "Mon Jan 13 10:41:13 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, } @Proceedings{Traub:1976:ACC, editor = "J. F. (Joseph Frederick) Traub", booktitle = "{Analytic computational complexity: Proceedings of the Symposium on Analytic Computational Complexity, held by the Computer Science Department, Carnegie-Mellon University, Pittsburgh, Pennsylvania, on April 7--8, 1975}", title = "{Analytic computational complexity: Proceedings of the Symposium on Analytic Computational Complexity, held by the Computer Science Department, Carnegie-Mellon University, Pittsburgh, Pennsylvania, on April 7--8, 1975}", publisher = pub-ACADEMIC, address = pub-ACADEMIC:adr, pages = "ix + 239", year = "1976", ISBN = "0-12-697560-4", ISBN-13 = "978-0-12-697560-4", LCCN = "QA297.S9151 1975", bibdate = "Mon Jan 13 10:18:33 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, } @Proceedings{Cowell:1977:PNS, editor = "Wayne Cowell", booktitle = "Portability of Numerical Software, Workshop, {Oak Brook, Illinois, June 21--23, 1976}", title = "Portability of Numerical Software, Workshop, {Oak Brook, Illinois, June 21--23, 1976}", volume = "57", publisher = pub-SV, address = pub-SV:adr, pages = "viii + 539", year = "1977", DOI = "https://doi.org/10.1007/3-540-08446-0", ISBN = "0-387-08446-0, 3-540-08446-0, 3-540-37071-4", ISBN-13 = "978-0-387-08446-6, 978-3-540-08446-4, 978-3-540-37071-0", ISSN = "0302-9743", LCCN = "QA297 .W65 1976", bibdate = "Thu Dec 11 14:25:53 MST 2025", bibsource = "fsz3950.oclc.org:210/WorldCat; https://www.math.utah.edu/pub/bibnet/authors/c/cody-william-j.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fortran1.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", series = ser-LNCS, URL = "http://link.springer.com/10.1007/3-540-08446-0", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", tableofcontents = "What is portability? / 1--2 \\ Some side effects of striving for portability / Christian Reinsch / 3--19 \\ Machine characteristics and portability / 20--21 \\ Machine requirements for reliable, portable software / T. J. Dekker / 22--36 \\ Semantics of floating point arithmetic and elementary functions / T. E. Hull / 37--48 \\ Machine parameters for numerical analysis / W. J. Cody / 49--67 \\ Preparing conventions for parameters for transportable numerical software / B. Ford / 68--91 \\ Programming languages and portability / 92--94 \\ Algol 68 as a language for numerical software / L. M. Delves / 95--126 \\ Writing the elementary function procedures for the ALGOL68C compiler / P. Kemp / 127--144 \\ Criteria for transportable Algol libraries / Pieter W. Hemker / 145--157 \\ Fortran portability via models and tools / W. S. Brown, A. D. Hall / 158--164 \\ Port --- A portable mathematical subroutine library / P. A. Fox / 165--177 \\ Fortran poisoning and antidotes / Brian T. Smith / 178--256 \\ Two numerical analysts' views on the Draft Proposed ANS Fortran / C. L. Lawson, J. K. Reid / 257--268 \\ Intermediate languages: Current status / W. M. Waite / 269--303 \\ Computer-assisted portability / 304--304 \\ Mathematical software transportability systems --- have the variations a theme? / James M. Boyle / 305--360 \\ Features for Fortran portability / Fred T. Krogh / 361--367 \\ The IMSL Fortran converter: an approach to solving portability problems / T. J. Aird / 368--388 \\ Aids to portability within the NAG project / J. J. Du Croz, S. J. Hague, J. L. Siemieniuch / 389--404 \\ Multiple program realizations using the TAMPR system / Kenneth W. Dritz / 405--423 \\ Software design to facilitate portability / 424--424 \\ The production and testing of special function software in the NAG library / J. L. Schonfelder / 425--451 \\ Portable special function routines / L. Wayne Fullerton / 452--483 \\ The importance of standardized interfaces for portable statistical software / N. Victor, M. Sund / 484--503 \\ Exploring the impact of portability / 504--504 \\ On the enhancement of portability within the NAG project --- a statistical survey / J. Bentley, B. Ford / 505--528 \\ A study of portability in technical and scientific computing / Ingemar Dahlstrand / 529--539", } @Proceedings{IEEE:1978:PSC, editor = "{IEEE}", booktitle = "Proceedings of the Symposium on Computer Arithmetic {(4th: 1978: Santa Monica, CA)}", title = "Proceedings of the Symposium on Computer Arithmetic {(4th: 1978: Santa Monica, CA)}", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "xi + 274", year = "1978", ISSN = "1063-6889", LCCN = "QA76.6 .S919a", bibdate = "Mon May 19 15:22:15 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "IEEE catalog no. 78 CH1412-6C.", acknowledgement = ack-nhfb, keywords = "Computer arithmetic --- Congresses.; Electronic digital computers --- Programming --- Congresses.; Floating-point arithmetic --- Congresses.", } @Proceedings{Alefeld:1980:PSE, editor = "G. Alefeld and R. D. Grigorieff and R. Albrecht and U. Kulisch and F. Stummel", booktitle = "{Fundamentals of numerical computation (computer-oriented numerical analysis). Proceedings of a conference held June 5--8, 1979, on the occasion of the centennial of the Technical University of Berlin}", title = "{Fundamentals of numerical computation (computer-oriented numerical analysis). Proceedings of a conference held June 5--8, 1979, on the occasion of the centennial of the Technical University of Berlin}", volume = "2", publisher = pub-SV, address = pub-SV:adr, pages = "229", year = "1980", ISBN = "0-387-81566-X", ISBN-13 = "978-0-387-81566-4", ISSN = "0344-8029", LCCN = "QA297 .F84", bibdate = "Mon Jan 13 10:20:47 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", series = "Computing. Supplementum", acknowledgement = ack-nhfb, } @Proceedings{Dieudonne:1980:SFL, editor = "Jean Dieudonn{\'e}", booktitle = "{Special functions and linear representations of Lie groups}", title = "{Special functions and linear representations of Lie groups}", volume = "42", publisher = "Published for the Conference Board of the Mathematical Sciences by the American Mathematical Society", address = "Providence, RI, USA", pages = "iii + 59", year = "1980", ISBN = "0-8218-1692-6", ISBN-13 = "978-0-8218-1692-9", LCCN = "????", bibdate = "Sat Oct 30 17:14:47 MDT 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; z3950.bibsys.no:2100/BIBSYS", series = "Regional conference series in mathematics", acknowledgement = ack-nhfb, remark = "Expository lectures from the CBMS regional conference held at East Carolina University, March 5--9, 1979.", } @Proceedings{Lavington:1980:IPP, editor = "Simon Hugh Lavington", booktitle = "Information Processing 80: Proceedings of {IFIP} Congress 80, Tokyo, Japan, October 6--9, 1980, Melbourne, Australia, October 14--17, 1980", title = "Information Processing 80: Proceedings of {IFIP} Congress 80, Tokyo, Japan, October 6--9, 1980, Melbourne, Australia, October 14--17, 1980", publisher = pub-ENH, address = pub-ENH:adr, pages = "xiii + 1070", year = "1980", ISBN = "0-444-86034-7", ISBN-13 = "978-0-444-86034-7", LCCN = "QA 75.5 I57 1980", bibdate = "Thu Sep 01 23:09:20 1994", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, } @Proceedings{IEEE:1981:PIS, editor = "{IEEE}", booktitle = "Proceedings: 5th Symposium on Computer Arithmetic, May 18-19, 1981, University of Michigan, Ann Arbor, Michigan", title = "Proceedings: 5th Symposium on Computer Arithmetic, May 18-19, 1981, University of Michigan, Ann Arbor, Michigan", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "vii + 278", year = "1981", LCCN = "QA76.9.C62 S95 1981", bibdate = "Mon May 19 13:17:13 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "IEEE Catalog No. 81CH1630-3. Computer Society Order No. 347.", acknowledgement = ack-nhfb, xxISBN = "none", } @Proceedings{IEEE:1981:PSC, key = "IEEE CA5 '81", booktitle = "Proceedings: 5th Symposium on Computer Arithmetic, May 18--19, 1981, University of Michigan, Ann Arbor, Michigan", title = "Proceedings: 5th Symposium on Computer Arithmetic: May 18--19, 1981, University of Michigan, Ann Arbor, Michigan", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "vii + 278", year = "1981", LCCN = "QA 76.6 S985t 1981", bibdate = "Sat Feb 24 15:01:45 MST 1996", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", note = "IEEE catalog number 81CH1630-C.", acknowledgement = ack-nhfb, keywords = "ARITH-5; Computer arithmetic and logic units --- Congresses.; Electronic digital computers --- Programming --- Congresses.; Floating-point arithmetic Congresses.", xxISBN = "(none)", } @Proceedings{Mulvey:1982:EMP, editor = "J. M. Mulvey", booktitle = "{Evaluating Mathematical Programming Techniques: Proceedings of a Conference Held at the National Bureau of Standards, Boulder, Colorado, January 5--6, 1981}", title = "{Evaluating Mathematical Programming Techniques: Proceedings of a Conference Held at the National Bureau of Standards, Boulder, Colorado, January 5--6, 1981}", volume = "199", publisher = pub-SV, address = pub-SV:adr, pages = "xi + 379", year = "1982", ISBN = "0-387-11495-5", ISBN-13 = "978-0-387-11495-8", LCCN = "QA402.5 .E94 1982", bibdate = "Thu Nov 17 06:36:49 1994", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/c/cody-william-j.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", series = "Lecture Notes in Economics and Mathematical Systems", acknowledgement = ack-nhfb, } @Book{Jones:1984:CFA, author = "William B. (William Branham) Jones and Wolfgang J. Thron", booktitle = "Continued Fractions: Analytic Theory and Applications", title = "Continued Fractions: Analytic Theory and Applications", volume = "11", publisher = pub-CAMBRIDGE, address = pub-CAMBRIDGE:adr, pages = "xxviii + 428", year = "1984", DOI = "https://doi.org/10.1017/CBO9780511759550", ISBN = "0-521-30231-5", ISBN-13 = "978-0-521-30231-9", LCCN = "QA295 .J64 1984", bibdate = "Sat Jul 10 09:47:15 MDT 2010", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/h/henrici-peter.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcw.bib; z3950.bibsys.no:2100/BIBSYS", series = "Encyclopedia of mathematics and its applications", acknowledgement = ack-nhfb, tableofcontents = "Editor's Statement \\ Section Editor's Foreword \\ Introduction by Peter Henrici \\ Preface \\ Symbols \\ 1: Introduction \\ 1.1 History \\ 1.1.1 Beginnings \\ 1.1.2 Number-Theoretic Results \\ 1.1.3 Analytic Theory \\ 1.2 Overview of Contents \\ \\ 2: Elementary Properties of Continued Fractions \\ 2.1 Preliminaries \\ 2.1.1 Basic Definitions and Theorems \\ 2.1.2 Regular Continued Fractions \\ 2.1.3 Other Continued-Fraction Expansions \\ 2.1.4 Algorithms for Computing Approximants \ldots{} \\ 2.2 Sequences Generated by Linear Fractional Transformations \\ 2.3 Equivalence Transformations \\ 2.3.1 Equivalent Continued Fractions \\ 2.3.2 Euler's [1748] Connection between Continued Fractions and Infinite Series \\ 2.4 Contractions and Extensions \\ 2.4.1 Contraction of Continued Fractions \\ 2.4.2 Even Part of a Continued Fraction \\ 2.4.3 Odd Part of a Continued Fraction \\ 2.4.4 Extension of a Continued Fraction \\ \\ 3: Periodic Continued Fractions \\ 3.1 Introduction \\ 3.2 Convergence of Periodic Continued Fractions \ldots{} \\ 3.3 Dual Periodic Continued Fractions \\ \\ 4: Convergence of Continued Fractions \\ 4.1 Introduction \\ 4.2 Element Regions, Value Regions, and Sequences of Nested Circular Disks \\ 4.3 Necessary Conditions for Convergence \\ 4.3.1 Stern--Stolz Theorem \\ 4.3.2 Necessary Conditions for Best Value Regions and Convergence Regions \\ 4.4 Sufficient Conditions for Convergence: Constant Elements \\ 4.4.1 Classical Results and Generalizations \\ 4.4.2 Parabolic Convergence Regions \\ 4.4.3 Convergence Neighborhoods for K(an/1) \ldots{} \\ 4.4.4 Twin Convergence Regions \\ 4.4.5 Miscellaneous Convergence Criteria \\ 4.5 Sufficient Conditions for Convergence \\ Variable Elements \\ 4.5.1 Introduction \\ Classification of Continued Fractions \\ 4.5.2 Regular C-fractions \\ 4.5.3 Positive Definite/-fractions \\ 4.5.4 General T-fractions \\ \\ 5: Methods for Representing Analytic Functions by Continued Fractions \\ 5.1 Correspondence \\ 5.2 Three-Term Recurrence Relations \\ 5.3 Minimal Solutions of Three-Term Recurrence Relations \\ 5.4 Uniform Convergence \\ 5.5 Pad{\'e} Table \\ 5.5.1 Pad{\'e} Approximants \\ 5.5.2 Multiple-Point Pad{\'e} Tables \\ \\ 6: Representations of Analytic Functions by Continued Fractions \\ 6.1 Continued Fractions of Gauss \\ 6.1.1 Hypergeometric Functions $F(a, b; c; z)$ \ldots{} \\ 6.1.2 Confluent Hypergeometric Functions $\Phi(b; c; z)$ \\ 6.1.3 Confluent Hypergeometric Functions $\Psi(c; z)$ \\ 6.1.4 Confluent Hypergeometric Functions $\Omega(a, b; z)$ \\ 6.2 Representations from Minimal Solutions \\ \\ 7: Types of Corresponding Continued \\ Fractions and Related Algorithms \\ 7.1 Regular $C$-Fractions \\ 7.1.1 Correspondence of Regular $C$-Fractions \\ 7.1.2 Quotient-Difference Algorithm \\ 7.1.3 $g$-Fractions \\ 7.2 Associated Continued Fractions and $J$-Fractions \\ 7.2.1 Correspondence of Associated Continued Fractions \\ 7.2.2 $J$-Fractions and Orthogonal Polynomials \\ 7.3 General $T$-Fractions \\ 7.3.1 Correspondence of General $T$-Fractions \\ 7.3.2 FG Algorithms \\ 7.3.3 Representation of Analytic Functions \\ 7.4 Stable Polynomials \\ \\ 8: Truncation-Error Analysis \\ 8.1 Introduction \\ 8.2 General Theory of Inclusion Regions and Truncation Errors \\ 8.3 Explicit Results on Inclusion Regions and Truncation-Error Bounds \\ 8.4 Accelerating Convergence \\ \\ 9: Asymptotic Expansions and Moment Problems \\ 9.1 Introduction \\ 9.2 Moment Problems \\ 9.3 Integral Representations of Continued Fractions \\ 9.4 Asymptotic Expansions for Continued Fractions \\ 9.5 Solutions of the Moment Problems \\ 9.6 Representations of Analytic Functions \\ \\ 10: Numerical Stability in Evaluating Continued Fractions \\ 10.1 General Estimates of Relative Roundoff Error \\ 10.2 Methods for Estimating $g_^{(n)}$ \\ 10.3 Applications \\ \\ 11: Application of Continued Fractions to \\ Birth--Death Processes \\ 11.1 Birth--Death Processes \\ 11.2 Computational Procedures \\ \\ 12: Miscellaneous Results \\ 12.1 $T$-Fraction Expansions for Families of Bounded Functions \\ 12.2 $T$-Fractions Corresponding to Rational Functions \\ 12.3 Location of Singular Points of Analytic Functions Represented by Continued Fractions \\ 12.4 Univalence of Functions Represented by Continued Fractions \\ \\ Appendix A. Classification of Special Types of Continued Fractions \\ Appendix B. Additional Results on Minimal Solutions of Three-Term Recurrence Relations \\ Bibliography \\ Author Index \\ Subject Index", } @Proceedings{Hwang:1985:PSC, editor = "Kai Hwang", booktitle = "Proceedings: 7th Symposium on Computer Arithmetic, June 4--6, 1985, University of Illinois, Urbana, Illinois", title = "Proceedings: 7th Symposium on Computer Arithmetic, June 4--6, 1985, University of Illinois, Urbana, Illinois", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "xi + 343", year = "1985", ISBN = "0-8186-0632-0 (paperback), 0-8186-8632-4 (hard), 0-8186-4632-2 (microfiche)", ISBN-13 = "978-0-8186-0632-8 (paperback), 978-0-8186-8632-0 (hard), 978-0-8186-4632-4 (microfiche)", LCCN = "QA76.9.C62 S95 1985", bibdate = "Thu Sep 08 00:11:41 1994", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", note = "IEEE catalog number 85CH2146-9. IEEE Computer Society order number 632.", acknowledgement = ack-nj, keywords = "ARITH-7", } @Proceedings{IEEE:1985:ERC, key = "IEEE Region 5 '85", booktitle = "1985 {IEEE} Region 5 Conference, March 13--15, 1985, Holiday Inn Civic Center, Lubbock, Texas", title = "1985 {IEEE} Region 5 Conference, March 13--15, 1985, Holiday Inn Civic Center, Lubbock, Texas", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "vi + 71", year = "1985", LCCN = "TK 7801 N56 1985", bibdate = "Thu Sep 15 18:50:54 1994", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, xxISBN = "(none)", } @Proceedings{Marron:1985:FEP, editor = "J. S. Marron", booktitle = "{Function estimates: proceedings of a conference held July 28--August 3, 1985}", title = "{Function estimates: proceedings of a conference held July 28--August 3, 1985}", volume = "59", publisher = pub-AMS, address = pub-AMS:adr, pages = "ix + 178", year = "1985", ISBN = "0-8218-5062-8", ISBN-13 = "978-0-8218-5062-6", ISSN = "0271-4132 (print), 1098-3627 (electronic)", LCCN = "QA276.8 .C651 1985", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "Held at Humboldt State University, Arcata, California.", series = "Contemporary mathematics (American Mathematical Society)", acknowledgement = ack-nhfb, bibno = "18241", catcode = "G.1.2", CRclass = "G.1.2 Approximation; G.1.2 Elementary function approximation", descriptor = "Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation", genterm = "algorithms; theory", guideno = "1986-12215", procdate = "July 28--Aug. 3, 1985", procloc = "Arcata, CA", subject = "G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS", } @Proceedings{Miranker:1985:ASC, editor = "Willard L. Miranker and Richard A. Toupin", booktitle = "Accurate Scientific Computations: Symposium, Bad Neuenahr, {FRG}, March 12--14, 1985: Proceedings", title = "Accurate Scientific Computations: Symposium, Bad Neuenahr, {FRG}, March 12--14, 1985: Proceedings", volume = "235", publisher = pub-SV, address = pub-SV:adr, pages = "x + 205", year = "1985", DOI = "https://doi.org/10.1007/3-540-16798-6", ISBN = "0-387-16798-6", ISBN-13 = "978-0-387-16798-5", LCCN = "QA76.95 .A231 1986", bibdate = "Sat Sep 03 12:24:08 1994", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", series = ser-LNCS, acknowledgement = ack-nhfb, } @Proceedings{Miranker:1986:ASC, editor = "Willard L. Miranker and Richard A. Toupin", booktitle = "Accurate scientific computations: symposium, Bad Neuenahr, {FRG}, March 12--14, 1985: proceedings", title = "Accurate scientific computations: symposium, Bad Neuenahr, {FRG}, March 12--14, 1985: proceedings", volume = "235", publisher = pub-SV, address = pub-SV:adr, pages = "x + 205", year = "1986", CODEN = "LNCSD9", ISBN = "0-387-16798-6 (USA: paperback)", ISBN-13 = "978-0-387-16798-5 (USA: paperback)", ISSN = "0302-9743 (print), 1611-3349 (electronic)", LCCN = "QA76.95 .A231 1986", bibdate = "Fri Apr 12 07:14:49 1996", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "Symposium sponsored by IBM Deutschland.", series = ser-LNCS, acknowledgement = ack-nhfb, keywords = "mathematics --- data processing --- congresses; numerical calculations --- congresses", } @Proceedings{ACM:1987:UAA, editor = "{ACM}", booktitle = "Using Ada: {ACM} {SIGAda} international conference, Boston, Massachusetts, December 8--11, 1987", title = "Using Ada: {ACM} {SIGAda} international conference, Boston, Massachusetts, December 8--11, 1987", publisher = pub-ACM, address = pub-ACM:adr, pages = "viii + 240", year = "1987", ISBN = "0-89791-243-8", ISBN-13 = "978-0-89791-243-3", LCCN = "QA 76.73 A35 U85 1987", bibdate = "Mon May 19 13:18:54 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, } @Proceedings{Iserles:1987:SAN, editor = "A. Iserles and M. J. D. Powell", booktitle = "The State of the Art in Numerical Analysis: Proceedings of the Joint {IMA}\slash {SIAM} Conference on the State of the Art in Numerical Analysis held at the University of Birmingham, 14--18 April 1986", title = "The State of the Art in Numerical Analysis: Proceedings of the Joint {IMA}\slash {SIAM} Conference on the State of the Art in Numerical Analysis held at the University of Birmingham, 14--18 April 1986", publisher = pub-OXFORD, address = pub-OXFORD:adr, pages = "xiv + 719", year = "1987", ISBN = "0-19-853614-3", ISBN-13 = "978-0-19-853614-7", LCCN = "QA297 .S781 1987", bibdate = "Thu Sep 08 00:41:24 1994", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", price = "UK\pounds55.00, US\$77.50", acknowledgement = ack-nj # " and " # ack-nhfb, } @Proceedings{Mason:1987:AAB, editor = "J. C. Mason and M. G. Cox", booktitle = "{Algorithms for approximation: based on the proceedings of the IMA Conference on Algorithms for the Approximation of Functions and Data, held at the Royal Military College of Science, Shrivenham, July 1985}", title = "{Algorithms for approximation: based on the proceedings of the IMA Conference on Algorithms for the Approximation of Functions and Data, held at the Royal Military College of Science, Shrivenham, July 1985}", volume = "10", publisher = pub-CLARENDON, address = pub-CLARENDON:adr, pages = "xvi + 694 + 8", year = "1987", ISBN = "0-19-853612-7", ISBN-13 = "978-0-19-853612-3", LCCN = "QA221 .A5361 1987; QA221 .I47 1985", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/bibnet/authors/p/powell-m-j-d.bib; https://www.math.utah.edu/pub/bibnet/authors/r/rice-john-r.bib; https://www.math.utah.edu/pub/bibnet/authors/r/ruhe-axel.bib; https://www.math.utah.edu/pub/bibnet/authors/t/trefethen-lloyd-n.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; z3950.loc.gov:7090/Voyager", price = "US\$90", series = "The Institute of Mathematics and Its Applications conference series, new series", acknowledgement = ack-nhfb, bibno = "39820", catcode = "G.1.2; G.1.2", CRclass = "G.1.2 Approximation; G.1.2 Approximation; G.1.2 Elementary function approximation", descriptor = "Mathematics of Computing, NUMERICAL ANALYSIS, Approximation; Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation", genterm = "theory; algorithms", guideno = "1987-16080", meetingname = "IMA Conference on Algorithms for the Approximation of Functions and Data (1985: Royal Military College of Science, Shrivenham)", procdate = "The Institute of mathematics and its applications conference series; 10 July 1985", procloc = "Shrivenham, UK", sub = "Proceedings of the IMA Conference on Algorithms for the approximation of functions", subject = "Approximation theory; Data processing; Congresses; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS", tableofcontents = "Preface / v \\ Contributors / xiii \\ \\ I Development of Algorithms \\ \\ 1. Spline Approximation and Smoothing \\ \\ G. T. Anthony and M. G. Cox / The fitting of extremely large data sets by bivariate splines / 5 \\ W. Dahmen / Subdivision algorithms --- recent results, some extensions and further developments / 21 \\ P. Dierckx / Fast algorithms for smoothing data over a disc or a sphere using tensor product splines / 51 \\ T. Lyche and K. M{\o}rken / A discrete approach to knot removal and degree reduction algorithms for splines / 67 \\ R. H. J. Gmelig Meyling / On algorithms and applications for bivariate B-splines / 83 \\ \\ 2. Spline Interpolation and Shape Preservation \\ \\ R. E. Carlson / Shape preserving interpolation / 97 \\ M. G. Cox and H. M. Jones / Shape preserving spline approximation in the $\ell_1$-norm / 115 \\ J. A. Gregory / A review of curve interpolation with shape control / 131 \\ \\ 3. Multivariate Interpolation \\ \\ M. J. D. Powell / Radial basis functions for multivariable interpolation: a review / 143 \\ R. A. Lorentz / On the determinant of a bivariate Birkhoff interpolation problem / 169 \\ A. Le Mehaute / Interpolation with piecewise polynomials in more than one variable / 181 \\ \\ 4. Least Square Methods \\ \\ R. Farwig / Multivariate interpolation of scattered data by moving least squares methods / 193 \\ F. Yoshimoto / Least squares approximation by one-pass methods with piecewise polynomials / 213 \\ \\ 5. Rational Approximation \\ \\ L. N. Trefethen and M. H. Gutknecht / Pad{\'e}, stable Pad{\'e}, and Chebyshev--Pad{\'e} approximation / 227 \\ P. T. Breuer / A new method for real rational uniform approximation / 265 \\ C. B. Dunham / Rationals with repeated poles / 285 \\ A. Iserles and S. P. N{\o}rsett / Error control of rational approximation with a matrix argument / 293 \\ \\ 6. Complex and Nonlinear Approximation \\ \\ K. Madsen / General algorithms for discrete non-linear parameter estimation / 309 \\ G. Opfer / Complex rational approximation with numerical experiments / 327 \\ G. A. Watson / Data fitting by positive sums of exponentials / 337 \\ J. C. Mason and P. Owen / Some simple algorithms for constrained complex and rational approximation / 357 \\ \\ 7. Computer-Aided Design and Blending \\ \\ L. L. Schumaker / Numerical aspects of spaces of piecewise polynomials on triangulations / 373 \\ M. V. Golitschek / The $H$-sets of the blending functions / 407 \\ D. Levin / Multidimensional reconstruction by set-valued approximations/ 421 \\ \\ II Applications \\ \\ 8. Applications in Numerical Analysis \\ \\ H. P. Blatt, A, Iserles and E. B. Saff / Remarks on the behaviour of zeros of best approximating polynomials and rational functions / 437 \\ J. Gilbert and W. A. Light / Multigrid methods and the alternating algorithm / 447 \\ K. Jetter and J. St{\"o}ckler / On the computation of Gauss--Birkhoff quadrature formulas / 459 \\ E. Schock / Error bounds for the solution of integral equations by Galerkin-like methods / 471 \\ N. M. Temme / On the computation of the incomplete gamma functions for large values of the parameters / 479 \\ \\ 9. Applications in Partial Differential Equations \\ \\ J. R. Rice / Adaptive tensor product grids for singular problems / 493 \\ W. Freeden / Harmonic splines for solving boundary value problems of potential theory / 507 \\ D. C. Hanscomb / Recovery of fluid flow fields / 531 \\ L. Reichel / The selection of subspace and collocation points in the boundary collocation method for some plane elliptic boundary problems / 541 \\ \\ 10. Applications in Other Disciplines \\ \\ L. Andersson, K. Holmstr{\"o}m and A. Ruhe / Complex formation constants --- a problem from solution chemistry / 557 \\ D. E. Roberts and P. R. Graves-Morris / The application of generalised inverse rational interpolants in the model analysis of vibrating structures I / 573 \\ A. Daman and J. C. Mason / A generalised cross-validation method for meteorological data with gaps / 595 \\ K. P. Jackson and J. C. Mason / The approximation by complex functions of stresses in cracked domains / 611 \\ J. H. McDonnell / Equally spaced cubic splines for representing time histories / 623 \\ B. L. Rahimi and S. W. Ellacott / Dynamic phase analysis of heart anomalies / 641 \\ \\ III Software \\ \\ J. G. Hayes / NAG algorithms for the approximation of functions and data / 653 \\ G. T. Anthony and M. G. Cox / The National Physical Laboratory's Data Approximation Subroutine Library / 669 \\ \\ M. G. Cox (editor) / Panel Discussion / 689", } @Proceedings{USENIX:1988:UPC, editor = "{USENIX Association}", booktitle = "{USENIX} proceedings: {C++} Conference, Denver, {CO}, October 17--21, 1988", title = "{USENIX} proceedings: {C++} Conference, Denver, {CO}, October 17--21, 1988", publisher = pub-USENIX, address = pub-USENIX:adr, pages = "362", year = "1988", bibdate = "Sun Feb 18 07:46:09 MST 1996", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, keywords = "C++ (Computer program language) --- Congresses.", } @Proceedings{ACM:1989:PAI, editor = "{ACM}", booktitle = "Proceedings of the {ACM-SIGSAM 1989} International Symposium on Symbolic and Algebraic Computation, {ISSAC '89}", title = "{Proceedings of the ACM--SIGSAM 1989 International Symposium on Symbolic and Algebraic Computation, ISSAC '89}", publisher = pub-ACM, address = pub-ACM:adr, pages = "399", year = "1989", ISBN = "0-89791-325-6", ISBN-13 = "978-0-89791-325-6", LCCN = "QA76.95.I59 1989", bibdate = "Tue Sep 17 06:46:18 MDT 1996", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, confdate = "17--19 July 1989", conflocation = "Portland, OR, USA", confsponsor = "ACM", pubcountry = "USA", } @Book{Campbell-Kelly:1989:WCB-3, editor = "Martin Campbell-Kelly", booktitle = "The works of {Charles Babbage}, Vol. 3, The analytical engine and mechanical notation", title = "The works of {Charles Babbage}, Vol. 3, The analytical engine and mechanical notation", publisher = "William Pickering", address = "London, UK", pages = "253", year = "1989", ISBN = "1-85196-503-3, 1-85196-005-8 (set)", ISBN-13 = "978-1-85196-503-8, 978-1-85196-005-7 (set)", LCCN = "????", MRclass = "01A75 (68-03)", MRnumber = "998151 (90g:01064)", MRreviewer = "A. D. Booth", bibdate = "Sat Jan 12 22:42:35 MST 2013", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/l/lovelace-ada-augusta.bib; https://www.math.utah.edu/pub/tex/bib/adabooks.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; z3950.libris.kb.se:210/libr", acknowledgement = ack-nhfb, subject = "Mathematics; Science; 1961; mathematics", } @Proceedings{Ercegovac:1989:PSC, editor = "Milo{\v{s}} D. Ercegovac and Earl E. {Swartzlander, Jr.}", booktitle = "Proceedings: 9th Symposium on Computer Arithmetic: September 6--8, 1989, Santa Monica, California, {USA}", title = "Proceedings: 9th Symposium on Computer Arithmetic: September 6--8, 1989, Santa Monica, California, {USA}", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "xv + 247", year = "1989", ISBN = "0-8186-8963-3 (case), 0-8186-5963-7 (microfiche)", ISBN-13 = "978-0-8186-8963-5 (case), 978-0-8186-5963-8 (microfiche)", LCCN = "QA 76.9 C62 S95 1989", bibdate = "Thu Sep 01 22:36:52 1994", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/c/clenshaw-charles-w.bib; https://www.math.utah.edu/pub/bibnet/authors/o/olver-frank-w-j.bib; https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", note = "IEEE catalog no. 89CH2757-3.", acknowledgement = ack-nhfb, confdate = "6-8 Sept. 1989", conflocation = "Santa Monica, CA, USA", confsponsor = "IEEE; IFIP; University of California", keywords = "ARITH-9", } @Proceedings{IEE:1989:EEC, editor = "{IEE}", booktitle = "{ECCTD 89}: European Conference on Circuit Theory and Design, 5--8 September 1989: venue, University of Sussex, Brighton, United Kingdom", title = "{ECCTD} 89: European Conference on Circuit Theory and Design, 5--8 September 1989: venue, University of Sussex, Brighton, United Kingdom", publisher = pub-IEE, address = pub-IEE:adr, bookpages = "xviii + 680", year = "1989", ISBN = "0-85296-383-1", ISBN-13 = "978-0-85296-383-8", LCCN = "????", bibdate = "Sat Nov 29 08:19:35 2003", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", note = "Conference publication no. 308.", acknowledgement = ack-nhfb, confdate = "5-8 Sept. 1989", conflocation = "Brighton, UK", confsponsor = "IEE", pubcountry = "UK", } @Proceedings{IEEE:1989:ASF, editor = "{IEEE}", booktitle = "30th annual Symposium on Foundations of Computer Science, October 30--November 1, 1989, Research Triangle Park, North Carolina", title = "30th annual Symposium on Foundations of Computer Science, October 30--November 1, 1989, Research Triangle Park, North Carolina", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "xvii + 632", year = "1989", CODEN = "ASFPDV", ISBN = "0-8186-1982-1 (casebound), 0-8186-5982-3 (microfiche)", ISBN-13 = "978-0-8186-1982-3 (casebound), 978-0-8186-5982-9 (microfiche)", ISSN = "0272-5428", LCCN = "QA 76 S979 1989; TK7885.A1 S92 1989", bibdate = "Thu Dec 3 07:11:18 MST 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "Formerly called the Annual Symposium on Switching and Automata Theory. IEEE catalog no. 89CH2808-4. Computer Society order no. 1982.", acknowledgement = ack-nhfb, keywords = "computational complexity --- congresses; electronic data processing --- congresses; machine theory --- congresses", } @Proceedings{IEEE:1989:PII, key = "IEEE ICCD '89", booktitle = "Proceedings: 1989 {IEEE} International Conference on Computer Design: {VLSI} in Computer and Processors, {ICCD} '89, Hyatt Regency Cambridge, Cambridge, Massachusetts, October 2--4, 1989", title = "Proceedings: 1989 {IEEE} International Conference on Computer Design: {VLSI} in Computer and Processors, {ICCD} '89, Hyatt Regency Cambridge, Cambridge, Massachusetts, October 2--4, 1989", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "xvii + 587", year = "1989", ISBN = "0-8186-1971-6 (paper), 0-8186-5971-8 (microfiche), 0-8186-8971-4 (case)", ISBN-13 = "978-0-8186-1971-7 (paper), 978-0-8186-5971-3 (microfiche), 978-0-8186-8971-0 (case)", LCCN = "TK 7888.4 I23 1989", bibdate = "Wed Dec 13 18:26:58 1995", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", note = "IEEE catalog number 89CH2794-6.", acknowledgement = ack-nj, confdate = "2-4 Oct. 1989", conflocation = "Cambridge, MA, USA", confsponsor = "IEEE", } @Proceedings{MacNair:1989:WSC, editor = "Edward A. MacNair and Kenneth J. Musselman and Philip Heidelberger", booktitle = "{1989 Winter Simulation Conference proceedings: December 4--6, 1989, the Capital Hilton Hotel, Washington, DC}", title = "{1989 Winter Simulation Conference proceedings: December 4--6, 1989, the Capital Hilton Hotel, Washington, DC}", publisher = pub-IEEE, address = pub-IEEE:adr, bookpages = "xx + 1139", pages = "xx + 1139", year = "1989", ISBN = "0-911801-58-8", ISBN-13 = "978-0-911801-58-3", LCCN = "QA76.9.C65 W56 1989", bibdate = "Fri Nov 8 18:01:57 MST 2002", bibsource = "ACM Computing Archive CD-ROM database (1991); https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/prng.bib", note = "IEEE order number 89CH2778-9.", URL = "https://ieeexplore.ieee.org/servlet/opac?punumber=5823", acknowledgement = ack-nhfb, bibno = "76750", catcode = "I.6.3; G.1.6; G.3; G.1.2", CRclass = "I.6.3 Applications; G.1.6 Optimization; G.1.2 Approximation; G.1.2 Elementary function approximation", descriptor = "Computing Methodologies, SIMULATION AND MODELING, Applications; Mathematics of Computing, NUMERICAL ANALYSIS, Optimization; Mathematics of Computing, PROBABILITY AND STATISTICS; Mathematics of Computing, NUMERICAL ANALYSIS, Approximation, Elementary function approximation", genterm = "algorithms; design; performance", guideno = "1989-12012", procdate = "December 4-6, 1989", procloc = "Washington, D. C.", subject = "I. Computing Methodologies; I.6 SIMULATION AND MODELING; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS; G. Mathematics of Computing; G.3 PROBABILITY AND STATISTICS; G. Mathematics of Computing; G.1 NUMERICAL ANALYSIS", } @Proceedings{Megiddo:1989:PMP, editor = "N. Megiddo", booktitle = "Progress in Mathematical Programming: Interior-Point and Related Methods", title = "Progress in Mathematical Programming: Interior-Point and Related Methods", publisher = pub-SV, address = pub-SV:adr, pages = "x + 158", year = "1989", ISBN = "0-387-96847-4", ISBN-13 = "978-0-387-96847-6", LCCN = "QA402.5 .P7851 1989", bibdate = "Sat Nov 09 07:07:37 2002", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "Proceedings of the conference held at the Asilomar conference center in Pacific Grove, California, March 1--4, 1987.", acknowledgement = ack-nhfb, } @Book{Srivastava:1989:UFF, editor = "H. M. Srivastava and Shigeyoshi Owa", booktitle = "Univalent functions, fractional calculus, and their applications (K{\=o}riyama, 1988)", title = "Univalent functions, fractional calculus, and their applications ({K}{\=o}riyama, 1988)", publisher = pub-ELLIS-HORWOOD, address = pub-ELLIS-HORWOOD:adr, pages = "404", year = "1989", ISBN = "0-470-21630-1, 0-7458-0701-1", ISBN-13 = "978-0-470-21630-9, 978-0-7458-0701-0", LCCN = "QA331 .U55 1989", bibdate = "Mon Jan 13 09:52:29 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", price = "UK\pounds 39.95", acknowledgement = ack-nhfb, } @Proceedings{Cray:1990:PCU, editor = "????", key = "Cray UG '90", booktitle = "Proceedings Cray User Group", title = "Proceedings Cray User Group", publisher = "????", address = "????", pages = "????", month = "Spring", year = "1990", ISBN = "????", ISBN-13 = "????", LCCN = "????", bibdate = "Thu Sep 08 08:56:01 1994", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nj # " and " # ack-nhfb, } @Proceedings{Mason:1990:AAI, editor = "J. C. Mason and M. G. Cox", booktitle = "{Algorithms for approximation II: based on the proceedings of the Second International Conference on Algorithms for Approximation, held at Royal Military College of Science, Shrivenham, July 1988}", title = "{Algorithms for approximation II: based on the proceedings of the Second International Conference on Algorithms for Approximation, held at Royal Military College of Science, Shrivenham, July 1988}", publisher = pub-CHAPMAN-HALL, address = pub-CHAPMAN-HALL:adr, pages = "514", year = "1990", ISBN = "0-412-34580-3", ISBN-13 = "978-0-412-34580-7", LCCN = "QA221 .I54 1988", bibdate = "Thu Sep 01 23:55:44 1994", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/g/grosse-eric.bib; https://www.math.utah.edu/pub/bibnet/authors/p/powell-m-j-d.bib; https://www.math.utah.edu/pub/bibnet/authors/t/trefethen-lloyd-n.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; z3950.loc.gov:7090/Voyager", acknowledgement = ack-nhfb, meetingname = "International Conference on Algorithms for Approximation (2nd: 1988: Royal Military College of Science, Shrivenham, England)", subject = "Approximation theory; Data processing; Congresses", tableofcontents = "Part One: Development of Algorithms / 1 \\ 1. Spline Approximation / 3 \\ E. Arge, M. Dcehlen, T. Lyche and K. Morken / Constrained spline approximation of functions and data based on constrained knot removal / 4 \\ G. T. Anthony and M. G. Cox / Near real-time spline fitting of long sequences of uniformly-spaced data / 21 \\ M. Bozzini and F. de Tisi / An algorithm for knot location in bivariate least squares spline approximation / 30 \\ M. G. Cox, P. M. Harris and H. M. Jones / A knot placement strategy for least squares spline fitting based on the use of local polynomial approximations / 37 \\ G. Opfer / An algorithm for nonlinear splines with non-negativity constraints / 46 \\ C. Potier and C. Vercken / Spline curve fitting of digitized contours / 54 \\ C. Rabut / A B-spline approximation algorithm for quasi-interpolation or filtering / 62 \\ P. W. Smith / On knots and nodes for spline interpolation / 72 \\ 2. Polynomial and Piecewise Polynomial Approximation / 79 \\ W. Dahmen / A basis for certain spaces of multivariate polynomials and exponentials / 80 \\ F. N. Fritschi / Monotone piecewise cubic data fitting / 99 \\ M. Heilmann and M. W. M{\"u}ller / Direct and converse results on simultaneous approximation by the method of Bernstein--Durrmeyer operators / 107 \\ A. Iserles, P. E. Koch, S. P. N{\o}rsett and J. M. Sanz-Serna / Orthogonality and approximation in a Sobolev space / 117 \\ M. A. Lachance / Piecewise polynomial approximation of polynomial curves / 125 \\ E. Quak and L. L. Schumaker / Calculation of the energy of a piecewise polynomial surface / 134 \\ 3. Interpolation / 145 \\ M. D. Buhmann and M. J. D. Powell / Radial basis function interpolation on an infinite regular grid / 146 \\ L. Brutman / The Fourier operator of even order and its application to an extremum problem in interpolation / 170 \\ N. Dyn and A. Ron / On multivariate polynomial interpolation / 177 \\ N. Dyn, D. Levin and S. Rippen / Algorithms for the construction of data dependent triangulations / 185 \\ C. Rademacher and K. Scherer / Algorithms for computing best parametric cubic interpolation / 193 \\ 4. Smoothing and Constraint Methods / 209 \\ M. Von Golitschek and L. L. Schumaker / Data fitting by penalized least squares / 210 \\ K. W. Bosworth / A semiinfinite programming algorithm for constrained best approximation / 228 \\ M. Bozzini and L. Lenarduzzi / Inference region for a method of local approximation by using the residuals / 236 \\ 5. Complex Approximation / 245 \\ G. A. Watson / Numerical methods for Chebyshev approximation of complex-valued functions / 246 \\ P. T. P. Tang / A fast algorithm for linear complex Chebyshev approximation / 265 \\ Part Two: Applications / 275 \\ 6. Computer Aided Design and Geometric Modelling / 277 \\ N. Dyn, J. A. Gregory and D. Levin / Uniform subdivision algorithms for curves and surfaces / 278 \\ T. B. Boffey, M. G. Cox, L. M. Delves and C. J. Pursglove / Approximation by spheres / 296 \\ T. A. Foley / Interpolation of scattered data on a spherical domain / 303 \\ A. B. Forbes / Least squares best fit geometric elements / 311 \\ W. Freeden and J. C. Mason / Uniform piecewise approximation on the sphere / 320 \\ 7. Applications in Numerical Analysis / 335 \\ L. N. Trefethen / Approximation theory and numerical linear algebra / 336 \\ M. Frontini, G. Rodriguez and S. Seatzu / An algorithm for computing minimum norm solutions of the finite moment problem / 361 \\ R. H. J. Gmelig Meyling / Numerical solution of the biharmonic equation using different types of bivariate spline functions / 369 \\ G. O. Olaofe / Quadrature solution of integral equations: a uniform treatment of Fredholm and Volterra equations / 377 \\ G. Walz / Increasing the convergence modulus of an asymptotic expansion: an algorithm for numerical differentiation / 387 \\ J. Williams / Approximation and parameter estimation in ordinary differential equations / 395 \\ 8. Applications in Other Disciplines / 405 \\ C. Zala and I. Barrodale / Applications of discrete $L_1$ methods in science and engineering / 406 \\ J. C. Mason, A. E. Trefethen and S. J. Wilde / Constrained complex approximation algorithms in communication engineering / 424 \\ R. W. Allen and J. G. Metcalfe / Integration of absolute amplitude from a decibel B-spline fit / 449 \\ M. G. Cox and H. M. Jones / A nonlinear least squares data fitting problem arising in microwave measurement / 458 \\ J. C. Mason and S. J. Wilde / A complex minimax algorithm for phase-only adaptation in antenna arrays / 466 \\ Part Three: Catalogue of Algorithms / 477 \\ E. Grosse / A catalogue of algorithms for approximation / 479", } @Proceedings{Ullrich:1990:CCA, editor = "Christian Ullrich", booktitle = "Contributions to Computer Arithmetic and Self-Validating Numerical Methods. (Proceedings of {SCAN 89}, held in Basel, Oct. 2--6, 1989)", title = "Contributions to Computer Arithmetic and Self-Validating Numerical Methods. (Proceedings of {SCAN} 89, held in Basel, Oct. 2--6, 1989)", volume = "7", publisher = pub-BALTZER, address = pub-BALTZER:adr, pages = "526", year = "1990", ISBN = "????", ISBN-13 = "????", LCCN = "QA76.9.C62 C664 1990", bibdate = "Sat Nov 29 08:36:57 2003", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/numana1990.bib", series = "IMACS annals on computing and applied mathematics", acknowledgement = ack-nhfb, keywords = "computer arithmetic --- congresses; numerical analysis --- congresses", xxbooktitle = "SCAN-89, International Symposium on Scientific Computing, Computer Arithmetic, and Numeric Validation [October 1989, Basel, Switzerland]", } @Proceedings{Anonymous:1991:PIS, editor = "Anonymous", booktitle = "Proceedings of the International Symposium on Supercomputing: Fukuoka, Japan, November 6--8, 1991", title = "Proceedings of the International Symposium on Supercomputing: Fukuoka, Japan, November 6--8, 1991", publisher = "Kyushu University Press", address = "Fukuoka, Japan", pages = "iv + 261", year = "1991", ISBN = "4-87378-284-8", ISBN-13 = "978-4-87378-284-3", LCCN = "QA76.88.I 1991", bibdate = "Sat Jan 11 10:14:06 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, searchkey = "ti:elementary function", } @Proceedings{Bowers:1991:CCI, editor = "K. L. (Kenneth L.) Bowers and J. (John) Lund", booktitle = "{Computation and control II: proceedings of the second Bozeman conference, Bozeman, Montana, August 1--7, 1990}", title = "{Computation and control II: proceedings of the second Bozeman conference, Bozeman, Montana, August 1--7, 1990}", volume = "11", publisher = pub-BIRKHAUSER, address = pub-BIRKHAUSER:adr, pages = "369", year = "1991", ISBN = "0-8176-3611-0", ISBN-13 = "978-0-8176-3611-1", LCCN = "TA329 .C644 1991", bibdate = "Wed May 9 08:56:08 MDT 2007", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/s/stenger-frank.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; z3950.loc.gov:7090/Voyager", price = "US\$65.00", series = "Progress in systems and control theory", acknowledgement = ack-nhfb, keywords = "convergence acceleration", subject = "Engineering mathematics; Congresses; Feedback control systems", } @Proceedings{IEEE:1991:PSA, editor = "{IEEE}", booktitle = "Proceedings, Supercomputing '91: Albuquerque, New Mexico, November 18--22, 1991", title = "Proceedings, Supercomputing '91: Albuquerque, New Mexico, November 18--22, 1991", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "xxiii + 917", year = "1991", ISBN = "0-8186-9158-1 (IEEE case), 0-8186-2158-3 (IEEE paper), 0-8186-6158-5 (IEEE microfiche), 0-89791-459-7 (ACM)", ISBN-13 = "978-0-8186-9158-4 (IEEE case), 978-0-8186-2158-1 (IEEE paper), 978-0-8186-6158-7 (IEEE microfiche), 978-0-89791-459-8 (ACM)", LCCN = "QA76.5 .S894 1991", bibdate = "Fri Aug 30 08:01:51 MDT 1996", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; University of California MELVYL catalog.", note = "ACM order number 415913. IEEE Computer Society Press order number 2158. IEEE catalog number 91CH3058-5.", acknowledgement = ack-nhfb, classification = "C5440 (Multiprocessor systems and techniques); C5470 (Performance evaluation and testing); C6110P (Parallel programming)", keywords = "combinatorial algorithms; data dependence; distributed memory code generation; high school environment; latency tolerance; memory access; numerical algorithms; parallel processing; parallel programming; performance evaluation; performance tools; processor design; program analysis; storage hierarchy optimization; supercomputer benchmarks; supercomputer congresses; supercomputing; system issues", } @Proceedings{Koopman:1991:PST, editor = "Philip J. {Koopman, Jr.}", booktitle = "{The proceedings of the second and third annual workshops for the ACM Special Interest Group on Forth: SIGForth '90, February 16--18, 1990, Dallas, Texas \ldots{} SIGForth '91, March 7--9, 1991, San Antonio, Texas}", title = "{The proceedings of the second and third annual workshops for the ACM Special Interest Group on Forth: SIGForth '90, February 16--18, 1990, Dallas, Texas \ldots{} SIGForth '91, March 7--9, 1991, San Antonio, Texas}", publisher = pub-ACM, address = pub-ACM:adr, pages = "ii + 134", year = "1991", ISBN = "0-89791-462-7", ISBN-13 = "978-0-89791-462-8", LCCN = "QA 76.73 F24 S53 1991", bibdate = "Tue May 04 07:39:28 1999", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", note = "ACM order number 817911.", acknowledgement = ack-nhfb, } @Proceedings{Kornerup:1991:PIS, editor = "Peter Kornerup and David W. Matula", booktitle = "Proceedings: 10th {IEEE} Symposium on Computer Arithmetic: June 26--28, 1991, Grenoble, France", title = "Proceedings: 10th {IEEE} Symposium on Computer Arithmetic: June 26--28, 1991, Grenoble, France", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "xiii + 282", year = "1991", ISBN = "0-8186-9151-4 (case), 0-8186-6151-8 (microfiche), 0-7803-0187-0 (library binding)", ISBN-13 = "978-0-8186-9151-5 (case), 978-0-8186-6151-8 (microfiche), 978-0-7803-0187-0 (library binding)", LCCN = "QA76.9.C62 S95 1991", bibdate = "Thu Sep 01 23:18:52 1994", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, } @Book{Lewin:1991:SPP, editor = "Leonard Lewin", booktitle = "Structural Properties of Polylogarithms", title = "Structural Properties of Polylogarithms", volume = "37", publisher = pub-AMS, address = pub-AMS:adr, pages = "xviii + 412", year = "1991", ISBN = "0-8218-1634-9, 1-4704-1264-0 (e-book)", ISBN-13 = "978-0-8218-1634-9, 978-1-4704-1264-7 (e-book)", ISSN = "0076-5376", MRclass = "33E20, 00B15, 11-02, 11F67, 11R70, 11R42, 19F27, 33-02, 33-06, 33B99", bibdate = "Fri Jun 16 14:03:50 MDT 2023", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; z3950.loc.gov:7090/Voyager", series = "Mathematical surveys and monographs", acknowledgement = ack-nhfb, editor-dates = "22-Jul-1919--13-Aug-2007", editor-url = "https://en.wikipedia.org/wiki/Leonard_Lewin_(telecommunications_engineer)", subject = "Logarithmic functions; Fonctions logarithmes; Mathematics; Algebra; Intermediate; Logarithmic functions; Fonctions logarithmes", tableofcontents = "Preface / xiii \\ Acknowledgments / xv \\ List of Contributors / xvii \\ \\ 1: The Evolution of the Ladder Concept / L. Lewin / 1 \\ 1.1 Early History / 1 \\ 1.2 Functional Equations / 2 \\ 1.3 More Recent Numerical Results / 4 \\ 1.4 Current Developments / 6 \\ 1.5 Base on the Unit Circle and Clausen Function Ladders / 8 \\ References / 9 \\ \\ 2: Dilogarithmic Ladders / L. Lewin / 11 \\ 2.1 Derivation from Kummer's Functional Equation / 11 \\ 2.2 Relation to Clausen's Function / 15 \\ 2.3 A Three-Variable Dilogarithmic Functional Equation / 17 \\ 2.4 Functional Equations in the Complex Plane / 18 \\ 2.5 Cyclotomic Equations and Rogers' Function / 20 \\ 2.6 Accessible and Analytic Ladders / 21 \\ 2.7 Inaccessible Ladders / 23 \\ References / 25 \\ \\ 3: Polylogarithmic Ladders / M. Abouzahra and L. Lewin / 27 \\ 3.1 Kummer's Function and its Relation to the Polylogarithm / 27 \\ 3.2 Functional Equations for the Polylogarithm / 28 \\ 3.3 A Generalization of Rogers' Function to the $n$th Order / 31 \\ 3.4 Ladder Order-Independence on Reduction of Order / 33 \\ 3.5 Generic Ladders for the Base Equation $u^p + u^q = 1$ / 34 \\ 3.6 Examples of Ladders for $n < 3$ / 40 \\ 3.7 Examples of Ladders for $n < 4$ / 44 \\ 3.8 Examples of Ladders for $n < 5$ / 45 \\ 3.9 Polynomial Relations for Ladders / 46 \\ References / 47 \\ \\ 4: Ladders in the Trans-Kummer Region / M. Abouzahra and L. Lewin / 49 \\ 4.1 Ladder Results to $n = 9$ for the Base p / 49 \\ 4.2 Ladder Results to $n = 9$ for the Base co / 53 \\ 4.3 Ladder Results to $n = 6$ for the Base 6 / 62 \\ 4.4 The Nonexistence of Functional Equations at $n = 6$ with Arguments Limited to $\pm z^m (1 - z)^r (1 + z)^s$ / 65 \\ References / 67 \\ \\ 5: Supemumary Ladders / M. Abouzahra and L. Lewin / 69 \\ 5.1 The Concept of Supemumary Results / 69 \\ 5.2 Supemumary Results for $p = 4$ / 71 \\ 5.3 Supemumary Results for $p = 5$ / 76 \\ 5.4 Supemumary Results for $p = 6$ / 78 \\ 5.5 Supemumary Results for the Equation-family $u^{6m + 1} u^{6r - 1}$ = 1 / 80 \\ 5.6 Supemumary Results for an Irreducible Quintic / 82 \\ 5.7 Supemumary Ladders from a 15-Term Functional Equation / 84 \\ 5.8 Supemumary Ladders on the Unit Circle / 90 \\ References \\ 6: Functional Equations and Ladders / L. Lewin / 97 \\ 6.1 New Categories of Functional Equations / 97 \\ 6.2 The $\rho$-family of Equations / 100 \\ 6.3 The $\omega$-family of Equations / 109 \\ 6.4 The $\theta$-family of Equations / 115 \\ Acknowledgements / 121 \\ References / 121 \\ \\ 7: Multivariable Polylogarithm Identities / G. A. Ray / 123 \\ 7.0 Introduction / 123 \\ 7.1 A General Identity for the Dilogarithm / 123 \\ 7.2 A General Identity for the Bloch-Wigner Function / 135 \\ 7.3 A General Identity for the Trilogarithm and $D_3(z)$ / 141 \\ 7.4 Linear Power Relations among Dilogarithms / 147 \\ 7.5 Cyclotomic Equations and Bases for Polylogarithm Relations / 154 \\ 7.6 Mahler's Measure and Salem/Pisot Numbers / 160 \\ 7.7 Recent Results for Supemumary Ladders / 165 \\ References / 168 \\ \\ 8: Functional Equations of Hyperlogarithms / G. Wechsung / 171 \\ 8.1 Hyperlogarithms / 171 \\ 8.2 Logarithmic Singularities / 172 \\ 8.3 The Linear Spaces LI$_n$ and PLI$_n$ / 176 \\ 8.4 Functional Equations of Hyperlogarithms / 177 \\ 8.5 A Reduction Problem / 181 \\ References / 184 \\ \\ 9: Kummer-Type Functional Equations of Polylogarithms / G. Wechsung / 185 \\ 9.1 Automorphic Functions / 185 \\ 9.2 Kummer-Type Functional Equations / 186 \\ 9.3 A Method to Construct Functional Equations / 191 \\ 9.4 The Nonexistence of a Kummer-Type Functional Equation for $\Li_6$ / 197 \\ References / 203 \\ \\ 10: The Basic Structure of Polylogarithmic Equations / Z. Wojtkowiak / 205 \\ 10.1 Introduction / 205 \\ 10.2 Canonical Unipotent Connection on $P^1(\mathbb{C})\{a_1, \ldots{}, a_{n+1}\}$ / 211 \\ 10.3 Horizontal Sections / 213 \\ 10.4 Easy Lemmas about Monodromy / 215 \\ 10.5 Functional Equations / 216 \\ 10.6 Functional Equations of Polylogarithms / 218 \\ 10.7 Functional Equations of Lower Degree Polylogarithms / 223 \\ 10.8 Generalized Bloch Groups / 228 \\ Acknowledgements / 231 \\ References / 231 \\ \\ 11: $K$-Theory, Cyclotomic Equations and Clausen's Function / J. Browkin / 233 \\ 11.1 Algebraic Background / 233 \\ 11.2 Analytic Background / 238 \\ 11.3 $K$-theoretic Background / 248 \\ 11.4 Examples / 251 \\ 11.5 Problems and Conjectures / 270 \\ References / 272 \\ \\ 12: Function Theory of Polylogarithms / S. Bloch / 275 \\ \\ 13: Partition Identities and the Dilogarithm / J. H. Loxton / 287 \\ 13.1 Introduction / 287 \\ 13.2 Cyclotomic Equations / 290 \\ 13.3 Accessible Relations / 291 \\ 13.4 Partition Identities / 292 \\ 13.5 Generalisations and Extensions / 297 \\ References / 299 \\ \\ 14: The Dilogarithm and Volumes of Hyperbolic Polytopes / R. Kellerhals / 301 \\ 14.0 Introduction / 301 \\ 14.1 A Particular Class of Hyperbolic Polytopes / 303 \\ 14.2 The Volume of a rf-Truncated Orthoscheme / 309 \\ 14.3 Applications / 321 \\ 14.4 Further Aspects / 328 \\ References / 335 \\ \\ 15: Introduction to Higher Logarithms / R. M. Hain and R. MacPherson / 337 \\ 15.1 The Problem of Generalizing the Logarithm and the Dilogarithm / 337 \\ 15.2 The Quest for Higher Logarithms / 340 \\ 15.3 Higher Logarithms / 341 \\ 15.4 The Higher Logarithm Bicomplex / 343 \\ 15.5 Multivalued Deligne Cohomology / 346 \\ 15.6 Higher Logarithms as Deligne Cohomology Classes / 350 \\ Acknowledgements 3 / 51 \\ References / 352 \\ \\ 16: Some Miscellaneous Results / L. Lewin / 355 \\ 16.1 Clausen's Function and the Di-Gamma Function for Rational Arguments / 355 \\ 16.2 An Infinite Integral of a Product of Two Polylogarithms / 359 \\ 16.3 Cyclotomic and Polylogarithmic Equations for a Salem Number / 364 \\ 16.4 New Functional Equations / 373 \\ References / 374 \\ \\ Appendix A. Special Values and Functional Equations of Polylogarithms / D. Zagier / 377 \\ 0. Introduction / 377 \\ 1. The Basic Algebraic Relation and the Definition of $\mathcal{A}_m(F)$ / 378 \\ 2. Examples of Dilogarithm Relations / 383 \\ 3. Examples for Higher Order Polylogarithms / 385 \\ 4. Examples: Ladders / 387 \\ 5. Existence of Relations among Polylogarithm Values of Arbitrarily High Order / 390 \\ 6. A Conjecture on Linear Independence / 391 \\ 7. Functional Equations / 392 \\ References / 399 \\ \\ Appendix B. Summary of the Informal Polylogarithm Workshop, November 17--18, 1990, MIT, Cambridge, Massachusetts / 401 \\ R. MacPherson and H. Sah / List of Participants / 401 \\ Abbreviated Summary / 402 \\ Bibliography / 405 \\ Index / 409", } @Proceedings{EC2:1992:DJN, key = "AEF'92", booktitle = "{Deuxi{\`e}mes journ{\'e}es nationales: Les applications des ensembles flous, en l'honneur du Professeur A. Kaufmann, Nimes, 2--3 novembre 1992, conference scientifique (English: Second national conference: Application of Fuzzy Sets, in honor of Professor A. Kaufman, Nimes, 2--3 November 1992, scientific conference)}", title = "{Deuxi{\`e}mes journ{\'e}es nationales: Les applications des ensembles flous, en l'honneur du Professeur A. Kaufmann, Nimes, 2--3 novembre 1992, conference scientifique (English: Second national conference: Application of Fuzzy Sets, in honor of Professor A. Kaufman, Nimes, 2--3 November 1992, scientific conference)}", publisher = "EC2", address = "Nanterre Cedex, France", pages = "384", year = "1992", ISBN = "2-906899-78-X", ISBN-13 = "978-2-906899-78-0", LCCN = "????", bibdate = "Wed Jan 10 07:40:53 1996", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, } @Proceedings{Richards:1992:HFD, editor = "Donald St P. Richards", booktitle = "{Hypergeometric functions on domains of positivity, Jack polynomials, and applications: proceedings of an AMS Special Session held March 22--23, 1991 in Tampa, Florida}", title = "{Hypergeometric functions on domains of positivity, Jack polynomials, and applications: proceedings of an AMS Special Session held March 22--23, 1991 in Tampa, Florida}", volume = "138", publisher = pub-AMS, address = pub-AMS:adr, pages = "x + 259", year = "1992", ISBN = "0-8218-5159-4", ISBN-13 = "978-0-8218-5159-3", ISSN = "0271-4132 (print), 1098-3627 (electronic)", LCCN = "QA353.H9 H97 1992", bibdate = "Sat Oct 30 21:12:24 MDT 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; z3950.loc.gov:7090/Voyager", series = "Contemporary mathematics", acknowledgement = ack-nhfb, subject = "Hypergeometric functions; Congresses", } @Book{Adams:1993:ACA, editor = "E. Adams and U. Kulisch", booktitle = "Scientific Computing with Automatic Result Verification", title = "Scientific Computing with Automatic Result Verification", volume = "189", publisher = pub-ACADEMIC, address = pub-ACADEMIC:adr, pages = "x + 612", year = "1993", ISBN = "0-12-044210-8", ISBN-13 = "978-0-12-044210-2", LCCN = "QA76.S368 1993", bibdate = "Mon Jan 13 09:58:58 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", series = "Mathematics in science and engineering", acknowledgement = ack-nhfb, tableofcontents = "Contributors \\ Preface \\ Acknowledgements \\ Introduction \\ Part I. Language and Programming Support for Verified Scientific Computation \\ 1. PASCAL-XSC, New Concepts for Scientific Computation and Numerical Data Processing \\ 2. ACRITH-XSC, A Fortran-like Language for Verified Scientific Computing \\ 3. C-XSC, A Programming Environment for Verified Scientific Computing and Numerical Data Processing \\ 4. Proposal for Accurate Floating-Point Vector Arithmetic", } @Proceedings{Albrecht:1993:VNT, editor = "R. Albrecht and G. Alefeld and H. J. Stetter", booktitle = "Validation numerics: theory and applications", title = "Validation numerics: theory and applications", volume = "9", publisher = pub-SPRINGER-WIEN, address = pub-SPRINGER-WIEN:adr, pages = "291", year = "1993", CODEN = "COSPDM", ISBN = "0-387-82451-0 (New York), 3-211-82451-0 (Vienna)", ISBN-13 = "978-0-387-82451-2 (New York), 978-3-211-82451-1 (Vienna)", ISSN = "0344-8029", LCCN = "QA297 .V27 1993", bibdate = "Wed Oct 13 18:45:11 1999", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "Dedicated to Ulrich Kulisch on the occasion of his 60th birthday.", series = j-COMPUTING-SUPPLEMENTUM, acknowledgement = ack-nhfb, keywords = "convergence acceleration", } @Proceedings{Allasia:1993:PIJ, editor = "G. Allasia and Luighi Gatteshi and Francesco Lerda", booktitle = "Proceedings of the International Joint Symposium on Special Functions and Artificial Intelligence, (1993: Turin, Italy)", title = "Proceedings of the International Joint Symposium on Special Functions and Artificial Intelligence, (1993: Turin, Italy)", volume = "2(1/4)", publisher = "Baltzer Science Publishers", address = "Amsterdam, The Netherlands", pages = "474", year = "1993", ISSN = "1021-2655", LCCN = "QA297 A614 v. 2, no. 1/4", bibdate = "Sat Oct 30 18:57:57 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", series = "Annals of numerical mathematics", acknowledgement = ack-nhfb, } @Proceedings{Sincovec:1993:PSS, editor = "Richard F. Sincovec and David E. Keyes and Michael R. Leuze", booktitle = "{Proceedings of the Sixth SIAM Conference on Parallel Processing for Scientific Computing, held March 22--24, 1993, in Norfolk, VA, USA}", title = "{Proceedings of the Sixth SIAM Conference on Parallel Processing for Scientific Computing, held March 22--24, 1993, in Norfolk, VA, USA}", publisher = pub-SIAM, address = pub-SIAM:adr, pages = "xix + 1041 + iv", year = "1993", ISBN = "0-89871-315-3", ISBN-13 = "978-0-89871-315-2", LCCN = "QA76.58 .S55 1993 v.1-2", bibdate = "Tue Oct 11 12:21:40 1994", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/b/berger-marsha-j.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "Two volumes.", acknowledgement = ack-nhfb, } @Proceedings{Swartzlander:1993:SCA, editor = "Earl {Swartzlander, Jr.} and Mary Jane Irwin and Graham Jullien", booktitle = "Proceedings: 11th Symposium on Computer Arithmetic, June 29--July 2, 1993, Windsor, Ontario", title = "Proceedings: 11th Symposium on Computer Arithmetic, June 29--July 2, 1993, Windsor, Ontario", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "xii + 284", year = "1993", ISBN = "0-7803-1401-8 (softbound), 0-8186-3862-1 (casebound), 0-8186-3861-3 (microfiche)", ISBN-13 = "978-0-7803-1401-6 (softbound), 978-0-8186-3862-6 (casebound), 978-0-8186-3861-9 (microfiche)", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", LCCN = "QA 76.9 C62 S95 1993", bibdate = "Thu Sep 01 22:58:49 1994", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "IEEE Transactions on Computers {\bf 43(8)}, 1994", acknowledgement = ack-nhfb, } @Proceedings{Brown:1994:PCL, editor = "J. David Brown and Moody T. Chu and Donald C. Ellison and Robert J. Plemmons", booktitle = "{Proceedings of the Cornelius Lanczos International Centenary Conference, Raleigh, North Carolina, December 12--17, 1993}", title = "{Proceedings of the Cornelius Lanczos International Centenary Conference, Raleigh, North Carolina, December 12--17, 1993}", volume = "73", publisher = pub-SIAM, address = pub-SIAM:adr, pages = "lxv + 644", year = "1994", ISBN = "0-89871-339-0", ISBN-13 = "978-0-89871-339-8", LCCN = "QC19.2 .C67 1993", bibdate = "Wed Jun 8 14:42:43 MDT 2011", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/d/dirac-p-a-m.bib; https://www.math.utah.edu/pub/bibnet/authors/g/golub-gene-h.bib; https://www.math.utah.edu/pub/bibnet/authors/h/heisenberg-werner.bib; https://www.math.utah.edu/pub/bibnet/authors/l/lanczos-cornelius.bib; https://www.math.utah.edu/pub/bibnet/authors/p/parlett-beresford-n.bib; https://www.math.utah.edu/pub/bibnet/authors/s/saad-yousef.bib; https://www.math.utah.edu/pub/bibnet/authors/s/stewart-gilbert-w.bib; https://www.math.utah.edu/pub/bibnet/authors/t/tukey-john-w.bib; https://www.math.utah.edu/pub/bibnet/authors/v/vandervorst-henk-a.bib; https://www.math.utah.edu/pub/bibnet/authors/y/young-david-m.bib; https://www.math.utah.edu/pub/bibnet/subjects/acc-stab-num-alg.bib; https://www.math.utah.edu/pub/tex/bib/einstein.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; z3950.loc.gov:7090/Voyager", series = "Proceedings in Applied Mathematics", acknowledgement = ack-nhfb, meetingname = "Cornelius Lanczos International Centenary Conference (1993:Raleigh, NC)", subject = "Mathematical physics; Congresses; Astrophysics; Mathematics; Lanczos, Cornelius; Physicists; Hungary; Biography; Mathematicians", subject-dates = "1893--1974", tableofcontents = "The Life and Works of Cornelius Lanczos \\ \\ A Photographic Essay / / xvii \\ Cornelius Lanczos: A Biographical Essay / Barbara Gellai / xxi \\ Cornelius Lanczos (1893-1974), and the Hungarian Phenomenon in Science and Mathematics / Peter D. Lax / xlix \\ The Roots of Cornelius Lanczos / George Marx / liii \\ Reminiscences of Cornelius Lanczos / Jon Todd / lviii \\ Published Papers and Books of Cornelius Lanczos / / lx \\ \\ Plenary Presentations: Computational Mathematics \\ \\ Lanczos and the FFT: A Discovery Before its Time / James W. Cooley / 3 \\ Lanczos Algorithms for Large Scale Symmetric and Nonsymmetric Matrix Eigenvalue Problems / Jane K. Cullum / 11 \\ The Look-Ahead Lanczos Process for Nonsymmetric Matrices and its Applications / Roland W Freund / 33 \\ The Lanczos and Conjugate Gradient Algorithms in Finite Precision Arithmetic / Anne Greenbaum / 49 \\ The Lanczos Process and Pade Approximation / Martin H. Gutknecht / 61 \\ The Tau Method and the Numerical Solution of Differential Equations: Past Research and Recent Research / Eduardo L. Ortiz / 77 \\ Krylov Subspace Processes, Krylov Subspace Methods, and Iteration Polynomials / C. C. Paige / 83 \\ Do We Fully Understand the Symmetric Lanczos Algorithm Yet? / Beresford N. Parlett / 93 \\ On Generalized Band Matrices and Their Inverses / P{\'a}l R{\'o}sa, Francesco Romani, and Roberto Bevilacqua / 109 \\ Theoretical Error Bounds and General Analysis of a Few Lanczos-Type Algorithms / Youcef Saad / 123 \\ Lanczos and Linear Systems / G. W. Stewart / 135 \\ \\ Plenary Presentations: Theoretical Physics and Astrophysics \\ \\ Integration on the Space of Connections Modulo Gauge Transformations / Abbay Ashtekar, Donald Marolf, and Jose Mourdo / 143 \\ Quasiclassical Domains in a Quantum Universe / James B. Hartle / 161 \\ Gauge Invariant Energy-Momentum Tensor in Spinar Electrodynamics / D. Petiot and Y. Takahashi / 173 \\ $\gamma$-Ray Bursts and Neutron Star Mergers / Tsvi Piran / 187 \\ Lanczos's Early Contributions to Relativity and His Relationship with Einstein / John Stachel / 201 \\ Topological Roots of Black Hole Entropy / Claudio Teitelboim / 223 \\ Variational Principles, Local Symmetries, and Black Hole Entropy / Robert M. Wald / 231 \\ \\ Mathematics Minisymposia \\ \\ Eigenvalue Computations: Theory and Algorithms / / 241 \\ Eigenvalue Computations: Applications / / 249 \\ Moments in Numerical Analysis / / 265 \\ Iterative Methods for Linear Systems / / 277 \\ Least Squares / / 301 \\ Software for Lanczos-based Algorithms / / 311 \\ Tau Method / / 335 \\ Chebyshev Polynomials / / 357 \\ Lanczos Methods in Control and Signal Processing / / 375 \\ Development of the FFT / / 393 \\ The FFT in Signal Processing / / 399 \\ Wavelets / / 411 \\ \\ Physics Minisymposia \\ \\ Computational Magnetohydrodynamics in Astrophysics / / 431 \\ Numerical Simulations of Collisionless Space Plasmas / / 453 \\ Detection of Gravitational Radiation from Astrophysical Sources / / 477 \\ Lanczos $H$-tensor / / 489 \\ Cosmic Censorship / / 513 \\ Cauchy Problem of General Relativity / / 527 \\ Black Hole Evaporation and Thermodynamics / / 543 \\ The Problem of Time in Quantum Gravity / / 555 \\ New Variables and Loop Quantization / / 571 \\ Decoherence and the Foundations of Quantum Mechanics / / 589 \\ Open Questions in Particle Theory / / 603 \\ Supercollider Physics / / 621 \\ Symplectic Methods in Physics / / 633", } @Proceedings{Cuyt:1994:NNM, editor = "Annie Cuyt", booktitle = "Nonlinear numerical methods and rational approximation {II}", title = "Nonlinear numerical methods and rational approximation {II}", volume = "296", publisher = pub-KLUWER, address = pub-KLUWER:adr, pages = "xviii + 446", year = "1994", ISBN = "0-7923-2967-8", ISBN-13 = "978-0-7923-2967-1", LCCN = "QA297 .N642 1994", bibdate = "Wed Nov 3 09:30:14 MST 1999", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "Proceedings of an international conference held at the University of Antwerp, Belgium, Sept. 5--11, 1993.", series = "Mathematics and its applications", acknowledgement = ack-nhfb, keywords = "approximation theory -- congresses; numerical analysis -- congresses", } @Proceedings{Gautschi:1994:MCH, editor = "Walter Gautschi", booktitle = "{Mathematics of computation, 1943--1993: a half-century of computational mathematics: Mathematics of Computation 50th Anniversary Symposium, August 9--13, 1993, Vancouver, British Columbia}", title = "{Mathematics of computation, 1943--1993: a half-century of computational mathematics: Mathematics of Computation 50th Anniversary Symposium, August 9--13, 1993, Vancouver, British Columbia}", volume = "48", publisher = pub-AMS, address = pub-AMS:adr, pages = "xix + 643", year = "1994", ISBN = "0-8218-0291-7, 0-8218-0353-0 (pt. 1), 0-8218-0354-9 (pt. 2)", ISBN-13 = "978-0-8218-0291-5, 978-0-8218-0353-0 (pt. 1), 978-0-8218-0354-7 (pt. 2)", ISSN = "0160-7634", LCCN = "QA1 .A56 v.48 1994; QA297.M385 1993", MRclass = "00B25 (11-06 65-06)", MRnumber = "95j:00014", bibdate = "Mon Oct 24 11:37:20 2011", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/b/berger-marsha-j.bib; https://www.math.utah.edu/pub/bibnet/authors/g/gautschi-walter.bib; https://www.math.utah.edu/pub/bibnet/authors/h/higham-nicholas-john.bib; https://www.math.utah.edu/pub/bibnet/authors/l/lehmer-derrick-henry.bib; https://www.math.utah.edu/pub/bibnet/authors/o/olver-frank-w-j.bib; https://www.math.utah.edu/pub/bibnet/authors/v/varga-richard-steven.bib; https://www.math.utah.edu/pub/bibnet/authors/w/wigner-eugene.bib; https://www.math.utah.edu/pub/bibnet/subjects/acc-stab-num-alg.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp1940.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp1950.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp1960.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp1970.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp1980.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp1990.bib; https://www.math.utah.edu/pub/tex/bib/mathcw.bib", note = "See also SIAM Review, September 1995, {\bf 37}(3), p. 483.", series = "Proceedings of Symposia in Applied Mathematics", acknowledgement = ack-nhfb, author-dates = "Frank William John Olver (15 December 1924--23 April 2013)", tableofcontents = "Preface / xi \\ Mathematics of Computation: A brief history / Eugene Isaacson / xvii \\ \\ Part I. Symposium on Numerical Analysis \\ \\ Invited Papers \\ \\ On the development of multigrid methods and their analysis / James H. Bramble / 5 \\ An introduction to inverse problems / Margaret Cheney / 21 \\ Algorithms for unconstrained optimization: A review of recent developments / Donald Goldfarb / 33 \\ A survey of componentwise perturbation theory in numerical linear algebra / Nicholas J. Higham / 49 \\ Numerical evaluation of special functions / D. W. Lozier and F. W. J. Olver / 79 \\ A survey of numerical cubature over triangles / J. N. Lyness and Ronald Cools / 127 \\ New trends in the use and analysis of integral equations / J. C. Nedelec / 151 \\ Applications of multivariate splines / Larry L. Schumaker / 177 \\ Initial value problems for ordinary differential equations: Development of ideas, techniques, and implementation / Hans J. Stetter / 205 \\ Multiresolution methods for partial differential equations / Roger Temam / 225 \\ \\ Contributed Papers \\ \\ A comparison of techniques for solving ill-conditioned problems arising from the immersed boundary method / Loyce Adams and Zhiyun Yang / 243 \\ A mixed spectral-collocation and operator splitting method for the Wigner-Poisson equation / Anton Arnold / 249 \\ Finite volume methods for irregular one-dimensional grids / M. J. Berger, R. J. Leveque, and L. G. Stern / 255 \\ Linear rational interpolation of continuous functions over an interval / Jean-Paul Berrut / 261 \\ A von Neumann reflection for the 2-D Burgers equation / M. Brio and J. K. Hunter / 265 \\ Slow evolution from the boundary: A new stabilizing constraint in ill-posed continuation problems / Alfred S. Carasso / 269 \\ A finite element method for the 2D drift-diffusion semiconductor model / Zhangxin Chen / 275 \\ Splitting functions and numerical analysis of WR-type methods for evolutionary and stationary problems / S. De Marchi, M. Vianello, and R. Zanovello / 281 \\ Error estimates for a quadrature rule for Cauchy principal value integrals / Kai Diethelm / 287 \\ A numerical radius approach to stable difference schemes for parabolic systems / Moshe Goldberg / 293 \\ An extension of the Olver--Sookne method for the solution of second-order linear difference equations / Takemitsu Hasegawa and Tatsuo Torii / 297 \\ The Faber polynomials for circular arcs / Matthew He / 301 \\ Finite element approximation for optimal control of electrically conducting fluid flows / L. S. Hou and S. S. Ravindran / 305 \\ ADI methods for heat equations with discontinuities along an arbitrary interface / Zhilin Li and Anita Mayo / 311 \\ Eigenvalue approximation of Fredholm integral operators / E. B. Lin / 317 \\ Spectral methods for singular perturbation problems / Wenbin Liu and Tao Tang / 323 \\ A quaternion-Jacobi method for symmetric matrices / Niloufer Mackey / 327 \\ On constructing Chebyshev series solutions of differential equations / Allan J. MacLeod / 333 \\ Multiquadric collocation methods in the numerical solution of Volterra integral and integro-differential equations / Athena Makroglou / 337 \\ Methods for solving large eigenvalue problems associated with configuration interaction electronic structure calculations / Kristyn J. Maschhoff / 343 \\ Computing limiting normals to real surfaces / Donal O'Shea and Les Wilson / 349 \\ Orthogonal spline collocation solution of nonlinear Schr{\"o}dinger equations / Mark P. Robinson / 355 \\ Who invented the computer? The debate from the viewpoint of computer architecture / Ra{\'u}l Rojas / 361 \\ Locking and boundary layer effects in the finite element approximation of the Reissner--Mindlin plate model / Christoph Schwab and Manil Suri / 367 \\ Efficient spectral Galerkin methods for some elliptic problems / Jie Shen / 373 \\ Periodic solutions of higher-order difference equations in two independent variables / Qin Sheng and Ravi P. Agarwal / 377 \\ Front tracking based on high-resolution wave propagation methods / Keh-Ming Shyue / 383 \\ Time-splitting methods for nonhomogeneous conservation laws / Tao Tang and Zhen-Huan Teng / 389 \\ Numerical aspects of uniform Airy-type asymptotic expansions / N. M. Temme / 395 \\ Local dynamics and bifurcation consistencies of continuous-time dynamical systems and their numerical discretizations / Xin Wang, Edward K. Blum, and Qingnan Li / 399 \\ Computing integrals of the complex error function / J. A. C. Weideman / 403 \\ Quadratures for improper integrals and their applications in integral equations / Yuesheng Xu and Yunhe Zhao / 409 \\ Spline harmonic analysis and wavelet bases / Valery A. Zheludev / 415 \\ \\ Part II. Minisymposium on Computational Number Theory Dedicated to the memory of Derrick Henry Lehmer \\ \\ Invited Papers \\ \\ Algorithms for quadratic orders / Ingrid Biehl and Johannes Buchmann / 425 \\ Analytic computations in number theory / Andrew M. Odlyzko / 451 \\ The number field sieve / Carl Pomerance / 465 \\ Factoring integers before computers / H. C. Williams and J. O. Shallit / 481 \\ \\ Contributed Papers \\ \\ Explicit bounds for primes in residue classes / Eric Bach and Jonathan Sorenson / 535 \\ Ramanujan and Euler's constant / Richard P. Brent / 541 \\ Congruential sieves on FPGA computers / Nathan D. Bronson and Duncan A. Buell / 547 \\ Lehmer pairs of zeros and the Riemann $\xi$-function / George Csordas, Wayne Smith, and Richard S. Varga / 553 \\ A record Aliquot sequence / Andrew W. P. Guy and Richard K. Guy / 557 \\ Implications of computational mathematics for the philosophy of mathematics / Andrew J. Lazarus / 561 \\ Square roots of products of algebraic numbers / Peter L. Montgomery / 567 \\ A locally parameterized version of Lehmer's problem / Gary A. Ray / 573 \\ A new method for finding amicable pairs / H. J. J. te Riele / 577 \\ Generalized Fermat numbers / Hans Riesel and Anders Bj{\"o}rn / 583 \\ Evaluation of $\zeta_K(2)$ for some totally real algebraic number fields K of degree 9 / Kisao Takeuchi / 589 \\ The period of the Bell exponential integers modulo a prime / Samuel S. Wagstaff, Jr. / 595 \\ Computing invariant polynomials of $p$-adic reflection groups / Changsheng Xu / 599 \\ Author Index / 603 \\ Subject Index / 619", } @Proceedings{Mudge:1994:PTS, editor = "Trevor N. Mudge and Bruce D. Shriver", booktitle = "{Proceedings of the Twenty-Seventh Hawaii International Conference on System Sciences Vol. I: Architecture}", title = "{Proceedings of the Twenty-Seventh Hawaii International Conference on System Sciences Vol. I: Architecture}", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "various", year = "1994", ISBN = "0-8186-5050-8 (paper), 0-8186-5051-6 (microfiche)", ISBN-13 = "978-0-8186-5050-5 (paper), 978-0-8186-5051-2 (microfiche)", LCCN = "????", bibdate = "Mon Jan 13 10:02:18 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "First of five volumes. IEEE Catalog No. 94TH0607-2.", acknowledgement = ack-nhfb, } @Proceedings{Zahar:1994:ACF, editor = "R. V. M. (Ramsay Vincent Michael) Zahar", booktitle = "{Approximation and computation: a festschrift in honor of Walter Gautschi: proceedings of the Purdue conference, December 2--5, 1993}", title = "{Approximation and computation: a festschrift in honor of Walter Gautschi: proceedings of the Purdue conference, December 2--5, 1993}", volume = "119", publisher = pub-BIRKHAUSER, address = pub-BIRKHAUSER:adr, pages = "xlvi + 591", year = "1994", ISBN = "0-8176-3753-2", ISBN-13 = "978-0-8176-3753-8", LCCN = "QA221 .A634 1994", bibdate = "Wed May 9 09:01:57 MDT 2007", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/r/rice-john-r.bib; https://www.math.utah.edu/pub/bibnet/authors/s/stenger-frank.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; z3950.loc.gov:7090/Voyager", series = "International series of numerical mathematics", acknowledgement = ack-nhfb, subject = "Approximation theory; Congresses; Orthogonal polynomials; Numerical integration; Functions, Special", } @Proceedings{Knowles:1995:PSC, editor = "Simon Knowles and William H. McAllister", booktitle = "Proceedings of the 12th Symposium on Computer Arithmetic, July 19--21, 1995, Bath, England", title = "Proceedings of the 12th Symposium on Computer Arithmetic, July 19--21, 1995, Bath, England", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "xvi + 252", year = "1995", ISBN = "0-8186-7089-4 (paperback), 0-8186-7089-4 (case), 0-8186-7149-1 (microfiche), 0-8186-7089-4 (softbound), 0-7803-2949-X (casebound)", ISBN-13 = "978-0-8186-7089-3 (paperback), 978-0-8186-7089-3 (case), 978-0-8186-7149-4 (microfiche), 978-0-8186-7089-3 (softbound), 978-0-7803-2949-2 (casebound)", LCCN = "QA 76.9 C62 S95 1995", bibdate = "Sun Mar 29 08:48:20 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, } @Proceedings{Singh:1995:CRT, editor = "Avtar Singh", booktitle = "Conference record of the Twenty-Ninth Asilomar Conference on Signals, Systems \& Computers: October 30--November 1, 1995 Pacific Grove, California", title = "Conference record of the Twenty-Ninth Asilomar Conference on Signals, Systems \& Computers: October 30--November 1, 1995 Pacific Grove, California", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "various", year = "1995", ISBN = "0-8186-7370-2", ISBN-13 = "978-0-8186-7370-2", LCCN = "TK7801 .A83 1995", bibdate = "Sun Mar 29 08:51:26 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "Two volumes.", acknowledgement = ack-nhfb, } @Book{Bartschat:1996:CAP, editor = "Klaus Bartschat", booktitle = "Computational Atomic Physics: Electron and Positron Collisions with Atoms and Ions", title = "Computational Atomic Physics: Electron and Positron Collisions with Atoms and Ions", publisher = pub-SV, address = pub-SV:adr, pages = "xviii + 249 + 13", year = "1996", DOI = "https://doi.org/10.1007/978-3-642-61010-3", ISBN = "3-540-60179-1 (hardcover), 3-642-61010-2 (e-book), 3-642-64655-7 (paperback)", ISBN-13 = "978-3-540-60179-1 (hardcover), 978-3-642-61010-3 (e-book), 978-3-642-64655-3 (paperback)", LCCN = "QC794.6.C6 C655 1996", bibdate = "Fri Apr 25 14:30:14 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "https://link.springer.com/book/10.1007/978-3-642-61010-3", acknowledgement = ack-nhfb, tableofcontents = "Front Matter / i--xvii \\ Electron-Atom Scattering Theory: An Overview / Klaus Bartschat / 1--14 \\ Core Potentials for Quasi One--Electron Systems / Klaus Bartschat / 15--26 \\ Energies and Oscillator Strengths Using Configuration Interaction Wave Functions / Alan Hibbert / 27--64 \\ The Distorted-Wave Method for Elastic Scattering and Atomic Excitation / Don H. Madison, Klaus Bartschat / 65--86 \\ Distorted-Wave Methods for Ionization / Ian E. McCarthy, Xixiang Zhang / 87--100 \\ The Close--Coupling Approximation / R. P. McEachran / 101--135 \\ The R-Matrix Method / P. G. Burke, M. P. Scott / 137--159 \\ Momentum-Space Convergent-Close-Coupling Method for a Model e--H Scattering Problem / Igor Bray, Andris Stelbovics / 161--180 \\ The Calculation of Spherical Bessel and Coulomb Functions / A. R. Barnett / 181--202 \\ Scattering Amplitudes for Electron--Atom Scattering / Klaus Bartschat / 203--217 \\ Density Matrices: Connection Between Theory and Experiment / Klaus Bartschat / 219--246 \\ Back Matter / 247--249", } @Proceedings{LakshmanYN:1996:IPI, editor = "{Lakshman Y.N.}", booktitle = "{ISSAC '96: Proceedings of the 1996 International Symposium on Symbolic and Algebraic Computation, July 24--26, 1996, Zurich, Switzerland}", title = "{ISSAC '96: Proceedings of the 1996 International Symposium on Symbolic and Algebraic Computation, July 24--26, 1996, Zurich, Switzerland}", publisher = pub-ACM, address = pub-ACM:adr, pages = "xvii + 313", year = "1996", ISBN = "0-89791-796-0", ISBN-13 = "978-0-89791-796-4", LCCN = "QA 76.95 I59 1996", bibdate = "Thu Mar 12 08:00:14 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, sponsor = "ACM; Special Interest Group in Symbolic and Algebraic Manipulation (SIGSAM). ACM; Special Interest Group on Numerical Mathematics (SIGNUM).", } @Book{Berggren:1997:PSB, editor = "Lennart Berggren and Jonathan M. Borwein and Peter B. Borwein", booktitle = "Pi, a source book", title = "Pi, a source book", publisher = pub-SV, address = pub-SV:adr, pages = "xix + 716", year = "1997", DOI = "https://doi.org/10.1007/978-1-4757-2736-4", ISBN = "0-387-94924-0, 1-4757-2736-4 (e-book), 1-4757-2738-0 (print), 3-540-94924-0", ISBN-13 = "978-0-387-94924-6, 978-1-4757-2736-4 (e-book), 978-1-4757-2738-8 (print), 978-3-540-94924-4", LCCN = "QA484 .P5 1997", bibdate = "Fri Sep 2 17:41:50 MDT 2022", bibsource = "fsz3950.oclc.org:210/WorldCat; https://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib; https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/pi.bib; z3950.loc.gov:7090/Voyager", abstract = "The aim of this book is to provide a complete history of pi from the dawn of mathematical time to the present. The story of pi reflects the most seminal, the most serious and sometimes the silliest aspects of mathematics, and a surprising amount of the most important mathematics and mathematicians have contributed to its unfolding. Pi is one of the few concepts in mathematics whose mention evokes a response of recognition and interest in those not concerned professionally with the subject. Yet, despite this, no source book on pi has been published. One of the beauties of the literature on pi is that it allows for the inclusion of very modern, yet still accessible, mathematics. Mathematicians and historians of mathematics will find this book indispensable. Teachers at every level from the seventh grade onward will find here ample resources for anything from special topic courses to individual talks and special student projects. The literature on pi included in this source book falls into three classes: first a selection of the mathematical literature of four millennia, second a variety of historical studies or writings on the cultural meaning and significance of the number, and third, a number of treatments on pi that are fanciful, satirical and/or whimsical.", acknowledgement = ack-nhfb, ORCID-numbers = "Borwein, Jonathan/0000-0002-1263-0646", subject = "Pi; Pi (Le nombre); Pi.; Pi (le nombre)", tableofcontents = "Preface / v \\ \\ Acknowledgments / ix \\ \\ Introduction / xvii \\ \\ 1. The Rhind Mathematical Papyrus-Problem 50 ($\approx$ 1650 B.C.) / A problem dealing with the area of a round field of given diameter / 1 \\ \\ 2. Engels. Quadrature of the Circle in Ancient Egypt (1977) / A conjectural explanation of how the mathematicians of ancient Egypt approximated the area of a circle / 3 \\ \\ 3. Archimedes. Measurement of a Circle ($\approx$ 250 BC) / The seminal work in which Archimedes presents the first true algorithm for $\pi$ / 7 \\ \\ 4. Phillips. Archimedes the Numerical Analyst (1981) / A summary of Archimedes' work on the computation of $\pi$ using modern notation / 15 \\ \\ 5. Lam and Ang. Circle Measurements in Ancient China (1986) / This paper discusses and contains a translation of Liu Hui's (3rd century) method for evaluating $\pi$ and also examines values for $\pi$ given by Zu Chongzhi (429--500) / 20 \\ \\ 6. The Ban{\=u} M{\=u}s{\=a}: The Measurement of Plane and Solid Figures ($\approx$ 850) / This extract gives an explicit statement and proof that the ratio of the circumference to the diameter is constant / 36 \\ \\ 7. M{\=a}dhava. The Power Series for Arctan and Pi ($\approx$ 1400) / These theorems by a fifteenth century Indian mathematician give Gregory's series for arctan with remainder terms and Leibniz's series for $\pi$ / 45 \\ \\ 8. Hope-Jones. Ludolph (or Ludolff or Lucius) van Ceulen (1938) / Correspondence about van Ceulen's tombstone in reference to it containing some digits of $\pi$ / 51 \\ \\ 9. Vi{\'e}te. Variorum de Rebus Mathematicis Reponsorum Liber VII (1593) / Two excerpts. One containing the first infinite expression of $\pi$, obtained by relating the area of a regular $2n$-gon to that of a regular $n$-gon / 53 \\ \\ 10. Wallis. Computation of $\pi$ by Successive Interpolations (1655) / How Wallis derived the infinite product for $\pi$ that bears his name / 68 \\ \\ 11. Wallis. Arithmetica Infinitorum (1655) / An excerpt including Prop. 189, 191 and an alternate form of the result that gives Wm. Brounker's continued fraction expression for $4/\pi$ / 78 \\ \\ 12. Huygens. De Circuli Magnitudine Inventa (1724) / Huygens's proof of W. Snell's discovery of improvements in Archimedes' method of estimating the lengths of circular arcs / 81 \\ \\ 13. Gregory. Correspondence with John Collins (1671) / A letter to Collins in which he gives his series for arctangent, carried to the ninth power. / 87 \\ \\ 14. Roy. The Discovery of the Series Formula for $\pi$ by Leibniz, Gregory, and Nilakantha (1990) / A discussion of the discovery of the series $\pi/4 = 1 - 1/3 + 1/5, \cdots{}$ / 92 \\ \\ 15. Jones. The First Use of $\pi$ for the Circle Ratio (1706) / An excerpt from Jones' book, the Synopsis Palmariorum Matheseos: or, a New Introduction to the Mathematics, London, 1706 / 108 \\ \\ 16. Newton. Of the Method of Fluxions and Infinite Series (1737) / An excerpt giving Newton's calculation of $\pi$ to 16 decimal places / 110 \\ \\ 17. Euler. Chapter 10 of Introduction to Analysis of the Infinite (On the Use of the Discovered Fractions to Sum Infinite Series) (1748) / This includes many of Euler's infinite series for $\pi$ and powers of $\pi$ / 112 \\ \\ 18. Lambert. M{\'e}moire Sur Quelques Propri{\'e}t{\'e}s Remarquables Des Quantit{\'e}s Transcendentes Circulaires et Logarithmiques (1761) / An excerpt from Lambert's original proof of the irrationality of $\pi$ / 129 \\ \\ 19. Lambert. Irrationality of $\pi$ (1969) / A translation and Struik's discussion of Lambert's proof of the irrationality of $\pi$ / 141 \\ \\ 20. Shanks. Contributions to Mathematics Comprising Chiefly of the Rectification of the Circle to 607 Places of Decimals (1853) / Pages from Shank's report of his monumental hand calculation of $\pi$ / 147 \\ \\ 21. Hermite. Sur La Fonction Exponentielle (1873) / The first proof of the transcendence of $e$ / 162 \\ \\ 22. Lindemann. Ueber die Zahl $\pi$ (1882) / The first proof of the transcendence of $\pi$ / 194 \\ \\ 23. Weierstrass. Zu Lindemann's Abhandlung ``Uber die Ludolphsche Zahl'' (1885) / Weierstrass' proof of the transcendence of $\pi$ / 207 \\ \\ 24. Hilbert. Ueber die Trancendenz der Zahlen $e$ und $\pi$ (1893) / Hilbert's short and elegant simplification of the transcendence proofs for $e$ and $\pi$ / 226 \\ \\ 25. Goodwin. Quadrature of the Circle (1894) / The dubious origin of the attempted legislation of the value of $\pi$ in Indiana / 230 \\ \\ 26. Edington. House Bill No. 246, Indiana State Legislature, 1897 (1935) / A summary of the action taken by the Indiana State Legislature to fix the value of $\pi$ (including a copy of the actual bill that was proposed) / 231 \\ \\ 27. Singmaster. The Legal Values of Pi (1985) / A history of the attempt by Indiana to legislate the value of $\pi$ / 236 \\ \\ 28. Ramanujan. Squaring the Circle (1913) / A geometric approximation to $\pi$ / 240 \\ \\ 29. Ramanujan. Modular Equations and Approximations to $\pi$ (1914) / Ramanujan's seminal paper on $\pi$ that includes a number of striking series and algebraic approximations / 241 \\ \\ 30. Watson. The Marquis and the Land Agent: A Tale of the Eighteenth Century (1933) / A Presidential address to the Mathematical Association in which the author gives an account of ``some of the elementary work on arcs and ellipses and other curves which led up to the idea of inverting an elliptic integral, and so laying the foundations of elliptic functions and doubly periodic functions generally.'' / 258 \\ \\ 31. Ballantine. The Best (?) Formula for Computing $\pi$ to a Thousand Places (1939) / An early attempt to orchestrate the calculation of $\pi$ more cleverly / 271 \\ \\ 32. Birch. An Algorithm for Construction of Arctangent Relations (1946) / The object of this note is to express $\pi / 4 $ as a sum of arctan relations in powers of 10 / 274 \\ \\ 33. Niven. A Simple Proof that $\pi$ Is Irrational (1947) / A very concise proof of the irrationality of $\pi$ / 276 \\ \\ 34. Reitwiesner. An ENIAC Determination of $\pi$ and $e$ to 2000 Decimal Places (1950) / One of the first computer-based computations / 277 \\ \\ 35. Schepler. The Chronology of Pi (1950) / A fairly reliable outline of the history of $\pi$ from 3000 BC to 1949 / 282 \\ \\ 36. Mahler. On the Approximation of $\pi$ (1953) / ``The aim of this paper is to determine an explicit lower bound free of unknown constants for the distance of $\pi$ from a given rational or algebraic number'' / 306 \\ \\ 37. Wrench, Jr. The Evolution of Extended Decimal Approximations to $\pi$ (1960) / A history of the calculation of the digits of $\pi$ to 1960 \\ \\ 38. Shanks and Wrench, Jr. Calculation of $\pi$ to 100,000 Decimals (1962) / A landmark computation of $\pi$ to more than 100,000 places / 326 \\ \\ 39. Sweeny. On the Computation of Euler's Constant (1963) / The computation of Euler's constant to 3566 decimal places / 350 \\ \\ 40. Baker. Approximations to the Logarithms of Certain Rational Numbers (1964) / The main purpose of this deep and fundamental paper is to ``deduce results concerning the accuracy with which the natural logarithms of certain rational numbers may be approximated by rational numbers, or, more generally, by algebraic numbers of bounded degree.'' / 359 \\ \\ 41. Adams. Asymptotic Diophantine Approximations to $E$ (1966) / An asymptotic estimate for the rational approximation to $e$ which disproves the conjecture that $e$ behaves like almost all numbers in this respect / 368 \\ \\ 42. Mahler. Applications of Some Formulae by Hermite to the Approximations of Exponentials of Logarithms (1967) / An important extension of Hilbert's approach to the study of transcendence / 372 \\ \\ 43. Eves. In Mathematical Circles; A Selection of Mathematical Stories and Anecdotes (excerpt) (1969) / A collection of mathematical stories and anecdotes about $\pi$ / 400 \\ \\ 44. Eves. Mathematical Circles Revisited; A Second Collection of Mathematical Stories and Anecdotes (excerpt) (1971) / A further collection of mathematical stories and anecdotes about $\pi$ / 402 \\ \\ 45. Todd. The Lemniscate Constants (1975) / A unifying account of some of the methods used for computing the lemniscate constants / 412 \\ \\ 46. Salamin. Computation of r Using Arithmetic-Geometric Mean (1976) / The first quadratically converging algorithm for $\pi$ based on Gauss's AGM and on Legendre's relation for elliptic integrals / 418 \\ \\ 47. Brent. Fast Multiple-Precision Evaluation of Elementary Functions (1976) / ``This paper contains the `Gauss-Legendre' method and some different algorithms for log and exp (using Landen transformations).'' / 424 \\ \\ 48. Beukers. A Note on the Irrationality of $\zeta(2)$ and $\zetq(3)$ (1979) / A short and elegant recasting of Ap{\'e}ry's proof of the irrationality of $\zeta(3)$ (and $\zeta(2)$) / 434 \\ \\ 49. van der Poorten. A Proof that Euler Missed \ldots{} Ap{\'e}ry's Proof of the Irrationality of $\zeta(3)$ (1979) / An illuminating account of Ap{\'e}ry's astonishing proof of the irrationality of $\zeta(3)$ / 439 \\ \\ 50. Brent and McMillan. Some New Algorithms for High-Precision Computation of Euler's Constant (1980) / Several new algorithms for high precision calculation of Euler's constant, including one which was used to compute 30,100 decimal places / 448 \\ \\ 51. Apostol. A Proof that Euler Missed: Evaluating $\zeta(2)$ the Easy Way (1983) / This note shows that one of the double integrals considered by Beukers ([48] in the table of contents) can be used to establish directly that $\zeta(2) = \pi / 6$ / 456 \\ \\ 52. O'Shaughnessy. Putting God Back in Math (1983) / An article about the Institute of Pi Research, an organization that ``pokes fun at creationists by pointing out that even the Bible makes mistakes.'' / 458 \\ \\ 53. Stern. A Remarkable Approximation to $\pi$ (1985) / Justification of the value of $\pi$ in the Bible through numerological interpretations / 460 \\ \\ 54. Newman and Shanks. On a Sequence Arising in Series for $\pi$ (1984) / More connections between $\pi$ and modular equations / 462 \\ \\ 55. Cox. The Arithmetic-Geometric Mean of Gauss (1984) / An extensive study of the complex analytic properties of the AGM / 481 \\ \\ 56. Borwein and Borwein. The Arithmetic-Geometric Mean and Fast Computation of Elementary Functions (1984) / The relationship between the AGM iteration and fast computation of elementary functions (one of the by-products is an algorithm for $\pi$) / 537 \\ \\ 57. Newman. A Simplified Version of the Fast Algorithms of Brent and Salamin (1984) / Elementary algorithms for evaluating $e^x$ and $\pi$ using the Gauss AGM without explicit elliptic function theory / 553 \\ \\ 58. Wagon. Is Pi Normal? (1985) / A discussion of the conjecture that $\pi$ has randomly distributed digits / 557 \\ \\ 59. Keith. Circle Digits: A Self-Referential Story (1986) / A mnemonic for the first 402 decimal places of $\pi$ / 560 \\ \\ 60. Bailey. The Computation of $\pi$ to 29,360,000 Decimal Digits Using Borweins' Quartically Convergent Algorithm (1988) / The algorithms used, both for $\pi$ and for performing the required multiple-precision arithmetic / 562 \\ \\ 61. Kanada. Vectorization of Multiple-Precision Arithmetic Program and 201,326,000 Decimal Digits of 1 Calculation (1988) / Details of the computation and statistical tests of the first 200 million digits of $\pi$ / 576 \\ \\ 62. Borwein and Borwein. Ramanujan and Pi (1988) / This article documents Ramanujan's life, his ingenious approach to calculating $\pi$, and how his approach is now incorporated into modern computer algorithms / 588 \\ \\ 63. Chudnovsky and Chudnovsky. Approximations and Complex Multiplication According to Ramanujan (1988) / This excerpt describes ``Ramanujan's original quadratic period--quasiperiod relations for elliptic curves with complex multiplication and their applications to representations of fractions of $\pi$ and other logarithms in terms of rapidly convergent nearly integral (hypergeometric) series.'' / 596 \\ \\ 64. Borwein, Borwein and Bailey. Ramanujan, Modular Equations, and Approximations to Pi or How to Compute One Billion Digits of Pi (1989) / An exposition of the computation of $\pi$ using mathematics rooted in Ramanujan's work / 623 \\ \\ 65. Borwein, Borwein and Dilcher. Pi, Euler Numbers, and Asymptotic Expansions (1989) / An explanation as to why the slowly convergent Gregory series for $\pi$, truncated at 500,000 terms, gives $\pi$ to 40 places with only the 6th, 17th, 18th, and 29th places being incorrect / 642 \\ \\ 66. Beukers, B{\'e}zivin, and Robba. An Alternative Proof of the Lindemann--Weierstrass Theorem (1990) / The Lindemann--Weierstrass theorem as a by-product of a criterion for rationality of solutions of differential equations / 649 \\ \\ 67. Webster. The Tail of Pi (1991) / Various anecdotes about $\pi$ from the 14th annual IMO Lecture to the Royal Society / 654 \\ \\ 68. Eco. An excerpt from Foucault's Pendulum (1993) / ``The unnumbered perfection of the circle itself.'' / 658 \\ \\ 69. Keith. Pi Mnemonics and the Art of Constrained Writing (1996) / A mnemonic for $\pi$ based on Edgar Allen Poe's poem ``The Raven.'' / 659 \\ \\ 70. Bailey, Borwein, and Plouffe. On the Rapid Computation of Various Polylogarithmic Constants (1996) / A fast method for computing individual digits of $\pi$ in base 2 / 663 \\ Appendix I --- On the Early History of Pi / 677 \\ \\ Appendix II --- A Computational Chronology of Pi / 683 \\ \\ Appendix III --- Selected Formulae for Pi / 686 \\ \\ Bibliography / 690 \\ \\ Credits / 697 \\ \\ Index / 701", } @Proceedings{Boisvert:1997:QNS, editor = "Ronald F. Boisvert", booktitle = "Quality of Numerical Software: Assessment and Enhancement. {Proceedings of the IFIP TC2/WG2.5 Working Conference on the Quality of Numerical Software, Assessment and Enhancement, Oxford, United Kingdom, 8--12 July 1996}", title = "Quality of Numerical Software: Assessment and Enhancement. {Proceedings of the IFIP TC2/WG2.5 Working Conference on the Quality of Numerical Software, Assessment and Enhancement, Oxford, United Kingdom, 8--12 July 1996}", publisher = pub-CHAPMAN-HALL, address = pub-CHAPMAN-HALL:adr, pages = "vii + 384", year = "1997", ISBN = "0-412-80530-8", ISBN-13 = "978-0-412-80530-1", LCCN = "QA297 .I35 1996", bibdate = "Fri Jul 09 05:58:30 2004", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, } @Proceedings{Lang:1997:ISC, editor = "Tomas Lang and Jean-Michel Muller and Naofumi Takagi", booktitle = "13th {IEEE} Symposium on Computer Arithmetic: proceedings, July 6--9, 1997, Asilomar, California, {USA}", title = "13th {IEEE} Symposium on Computer Arithmetic: proceedings, July 6--9, 1997, Asilomar, California, {USA}", volume = "13", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "xiii + 291", year = "1997", ISBN = "0-8186-7846-1, 0-8186-7847-X, 0-8186-7848-8", ISBN-13 = "978-0-8186-7846-2, 978-0-8186-7847-9, 978-0-8186-7848-6", ISSN = "1063-6889", LCCN = "QA76.9.C62 S95 1997", bibdate = "Fri Mar 27 09:56:17 MST 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "IEEE Computer Society order number PR07846. IEEE Order Plan catalog number 97CB36091.", series = "Symposium on Computer Arithmetic", acknowledgement = ack-nhfb, sponsor = "IEEE.", } @Proceedings{Thiele:1997:IIC, editor = "Lothar Thiele and others", booktitle = "{IEEE International Conference on Application-Specific Systems, Architectures and Processors: proceedings, July 14--16, 1997, Z{\"u}rich, Switzerland}", title = "{IEEE International Conference on Application-Specific Systems, Architectures and Processors: proceedings, July 14--16, 1997, Z{\"u}rich, Switzerland}", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "xii + 540", year = "1997", ISBN = "0-8186-7959-X, 0-8186-7960-3, 0-8186-7958-1", ISBN-13 = "978-0-8186-7959-9, 978-0-8186-7960-5, 978-0-8186-7958-2", LCCN = "TK7874.6 .I57 1997eb; TK7874.6 .I57 1997; TK7874.6 .I58 1997", bibdate = "Sun Mar 4 21:13:29 MST 2007", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; melvyl.cdlib.org:210/CDL90", acknowledgement = ack-nhfb, meetingname = "International Conference on Application-Specific Systems, Architectures, and Processors (11th: 1997: Z{\"u}rich, Switzerland)", remark = "IEEE Computer Society Press order number PR07958. IEEE catalog number 97TB100177", subject = "Array processors; Congresses; Signal processing; Digital techniques; Application-specific integrated circuits", } @Proceedings{Anonymous:1998:TIC, editor = "Anonymous", booktitle = "{Tricomi's ideas and contemporary applied mathematics: (Roma, 28--29 novembre --- Torino, 1--2 dicembre 1997)}", title = "{Tricomi's ideas and contemporary applied mathematics: (Roma, 28--29 novembre --- Torino, 1--2 dicembre 1997)}", volume = "147", publisher = "Accademia Nazionale dei Lincei", address = "Roma, Italy", pages = "322", year = "1998", ISSN = "0391-805X", LCCN = "QA299.6", bibdate = "Fri May 31 16:34:38 MDT 2024", bibsource = "fsz3950.oclc.org:210/WorldCat; https://www.math.utah.edu/pub/bibnet/authors/g/gautschi-walter.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", series = "Atti dei convegni Lincei", acknowledgement = ack-nhfb, } @Proceedings{Rusev:1998:TMS, editor = "Petur Rusev and I. Dimovski and Virginia Kiryakova", booktitle = "{Transformation methods and special functions, Varna '96: second international workshop: proceedings}", title = "{Transformation methods and special functions, Varna '96: second international workshop: proceedings}", publisher = "Institute of Mathematics and Informatics, Bulgarian Academy of Sciences", address = "Sofia, Bulgaria", pages = "vi + 613", year = "1998", ISBN = "954-8986-05-1", ISBN-13 = "978-954-8986-05-2", LCCN = "????", bibdate = "Thu Dec 1 11:08:47 MST 2011", bibsource = "fsz3950.oclc.org:210/WorldCat https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, meetingname = "International Workshop ``Transform Methods and Special Functions'' (2nd: 1996: Varna, Bulgaria)", remark = "``This second edition of the Workshops 'TM and SF' has been devoted to the 100th anniversary of the eminent Bulgarian mathematician Nikola Obreshkoff (1896-1963)''--T.p. verso.", subject = "Transformations (Mathematics); Differential equations; Integral equations", tableofcontents = "On a generalization of the theorem on two constants / G. Adamczyk \\ On a construction of the solutions of some elliptic equations with generalized coefficients / A. Antonevich \\ Some non-metrizable spaces of harmonic functions / G. Balikov, I. Dimitrov \\ Duhamel-type representations of the solutions of non-local boundary value problems for the fractional diffusion-wave equation / E. Bazhlekova \\ Pointwise convergence for Hankel transform / J. Betancour, L. Rodr{\'i}guez-Mesa \\ On fractional order continuity, integrability and derivability of real functions / B. Bonilla, J. Trujillo, M. Rivero \\ On the root functions of a nonlocal Sturm--Liouville problem / N. Bozhinov \\ Mellin transform theory and the role of its differential and integral operators / P. Butzer, S. Jansche \\ Exact solution of some systems of non-selfadjoint partial differential equations / D. Callebaut \\ Numerical computation of Lame functions / H.-J. Dobner, S. Ritter \\ Some extremal problems for $p$-valent alpha-convex functions / J. Dziok \\ Extension of a result on the convolution product of distributions / B. Fisher, A. Kilicman \\ Simple algorithms for approximations of generalized elliptic-type integrals / M. A. El-Gabali, Shyam L. Kalla \\ On Zernicke polynomials / H.-J. Glaeske \\ Tomato salad problem in spherical stereology / R. Gorenflo \\ On existence of solutions of ordinary differential equations of fractional order / N. Hayek \ldots{} [et al.] \\ Tauberian theorem for distributions / J. Jel{\'i}nek \\ Radiation field integrals and their evaluation techniques / Shyam L. Kalla, H. G. Khajah \\ On the product of distributions / A. Kami{\'n}ski \\ On isotopies of algebras and triple systems / N. Kamiya \\ Univalence criteria connected with arithmetic and geometric means: I / S. Kanas, A. Lecko \\ Global and causal solutions of fractional linear differential equations / S. Kempfle, H. Beyer \\ On the applications of Mikusinski's operational calculus to the controllability of dynamical systems / W. Kierat, K. Sk{\'o}rnik \\ Application of fractional calculus to solve Abel--Volterra nonlinear and linear integral equations / A. Kilbas, M. Saigo \\ Note on linear operators and fractional calculus operators in the univalent function theory / Yong Chan Kim \\ Application of the generalized Mikhaylov criterion / S. Krasinska \\ On some classes of holomorphic functions in the half-plane / A. Lazi{\'n}ska \\ Expansions in series of Legendre functions / E. R. Love, M. Hunter \\ Intersections with Gronwall methods / W. Luh \\ Applications of fractional calculus in mechanics / F. Mainardi \\ Automorphisms in the commutant of the integration operator in spaces of Lebesgue integrable functions / S. Mincheva \\ Applications of fractional calculus operators to univalent functions / S. Owa \\ Application of orthogonal polynomials to solution of fractional integral equations / I. Podlubny \\ Remark on Watson transform / A. Prudnikov, U. Sk{\'o}rnik \\ Generalized operators of fractional integro-differentiation in meaning of M. Saigo and their applications / O. Repin \\ Fractional integrals and wavelet transformations / B. Rubin, D. Ryabogin, E. Shamir \\ More generalization of fractional calculus / M. Saigo, N. Maeda \\ On certain subclasses of analytic functions involving a linear operator / H. Saitoh \\ On some sequence spaces / E. Savas \\ Class of integro-differential equations via fractional calculus / N. Shawagfeh \\ On some extreme points of the unit ball / J. Sokol, W. Szumny \\ Hyper-Bessel operators, differential equations, functions, and integral transforms of 4th order / S. Spirova \\ Some operational techniques in the theory of special functions / H. M. Srivastava \\ Convolution in the theory of univalent functions / J. Stankiewicz \\ Some extensions of the Rolle and Gauss--Lucas theorems / T. Stoyanov \\ On infinitely divisible probability distributions and integral equations / K. Takano \\ Generating functions related to pairs of inverse functions / R. Tremblay, B. J. Fug{\`e}re \\ Integral transforms connected with the group representations / N. Tretyakova \\ On some integral operators in the clas of functions with negative coefficients / L. Trojnar-Spelina \\ On a new generalized Taylor's formula / J. Trujillo, M. Rivero, B. Bonilla \\ Some properties of the finite Laplace transform / M. Valbuena, L. Galue, I. Ali \\ Airy integral transform and the Paley--Wiener theorem / Vu Kim Tuan \\ On starlike functions related with hyperbolic regions / A. Wi{\'s}niowska \\ Editorial: Nikola Obreshkoff (1896--1963): biographical data and 100 selected papers of Acad. N. Obreshkoff \\ Obreshkoff's generalization of Descartes rule / P. Rusev \\ Obrechkoff's generalization of the Laplace and Meijer transformations: origins and recent developments / I. Dimovski, V. Kiryakova \\ Longstanding conjecture failed? / V. Kiryakova \\ Afterthoughts on interpretation of fractional derivatives and integrals / R. Gorenflo \\ Modelling viscous damped oscillations by fractional differential operators / S. Kempfle \\ Considerations on fractional calculus: interpretations and applications / F. Mainardi \\ Introduction to the fractional calculus and some applications / K. Oldham", } @Book{Bultheel:1999:ORF, editor = "Adhemar Bultheel and Pablo Gonzales-Vera and Erik Hendriksen and Olav Njastad", booktitle = "Orthogonal Rational Functions", title = "Orthogonal Rational Functions", volume = "5", publisher = pub-CAMBRIDGE, address = pub-CAMBRIDGE:adr, pages = "xiv + 407", year = "1999", DOI = "https://doi.org/10.1017/CBO9780511530050", ISBN = "0-521-65006-2 (hardcover)", ISBN-13 = "978-0-521-65006-9 (hardcover)", LCCN = "QA404.5 .O75 1999", bibdate = "Tue Mar 24 21:04:21 MDT 2009", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcw.bib; z3950.loc.gov:7090/Voyager", series = "Cambridge monographs on applied and computational mathematics", URL = "http://www.loc.gov/catdir/description/cam029/98011646.html; http://www.loc.gov/catdir/toc/cam024/98011646.html", acknowledgement = ack-nhfb, subject = "functions, orthogonal; functions of complex variables", tableofcontents = "1. Preliminaries / 15--41 \\ 2. The fundamental spaces / 42--63 \\ 3. The kernel functions / 64--73 \\ 4. Recurrence and second kind functions / 74--105 \\ 5. Para-orthogonality and quadrature / 106--120 \\ 6. Interpolation / 121--148 \\ 7. Density of the rational functions / 149--160 \\ 8. Favard theorems / 161--172 \\ 9. Convergence / 173--238 \\ 10. Moment problems / 239--256 \\ 11. The boundary case / 257--341 \\ 12. Some applications / 342--388 \\ Conclusion / 389--392 \\ Bibliography/ 393--404 \\ Index / 405--407", } @Proceedings{IEEE:1999:PIF, editor = "IEEE", booktitle = "Proceedings of the {IEEE} Forum on Research and Technology Advances in Digital Libraries, May 19--21, 1999, Baltimore, Maryland", title = "Proceedings of the {IEEE} Forum on Research and Technology Advances in Digital Libraries, May 19--21, 1999, Baltimore, Maryland", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "xi + 217", year = "1999", ISBN = "0-7695-0219-9", ISBN-13 = "978-0-7695-0219-9", LCCN = "ZA4080 .F67 1999", bibdate = "Fri Jul 09 06:32:32 2004", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, xxeditor = "Frances M. Titsworth", } @Proceedings{Koren:1999:ISC, editor = "Israel Koren and Peter Kornerup", booktitle = "14th {IEEE} Symposium on Computer Arithmetic: proceedings: April 14--16, 1999, Adelaide, Australia", title = "14th {IEEE} Symposium on Computer Arithmetic: proceedings: April 14--16, 1999, Adelaide, Australia", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "xi + 274", year = "1999", ISBN = "0-7803-5609-8, 0-7695-0116-8, 0-7695-0118-4", ISBN-13 = "978-0-7803-5609-2, 978-0-7695-0116-1, 978-0-7695-0118-5", ISSN = "1063-6889", LCCN = "QA76.6 .S887 1999", bibdate = "Mon Feb 7 07:28:26 MST 2000", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "IEEE Computer Society Order Number PR00116. IEEE Order Plan Catalog Number 99CB36336.", URL = "http://computer.org/conferen/home/arith/; http://www.ecs.umass.edu/ece/arith14/program.html", acknowledgement = ack-nhfb, annote = "Also known as ARITH-14.", source = "Computer arithmetic", sponsor = "IEEE.", } @Proceedings{Luk:1999:PSA, editor = "Franklin T. Luk", booktitle = "Proceedings of {SPIE: Advanced signal processing algorithms, architectures, and implementations IX: 19--21 July, 1999, Denver, Colorado}", title = "Proceedings of {SPIE: Advanced signal processing algorithms, architectures, and implementations IX: 19--21 July, 1999, Denver, Colorado}", volume = "3807", publisher = pub-SPIE, address = pub-SPIE:adr, pages = "ix + 648", year = "1999", ISBN = "0-8194-3293-8", ISBN-13 = "978-0-8194-3293-3", LCCN = "TK5102.5 .A3325 1999; TK5102.5 .A3173 1999eb; TK5102.9 .A37 1999; TK5102.5; TS510 .S63", bibdate = "Mon Mar 5 07:43:43 MST 2007", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; melvyl.cdlib.org:210/CDL90", acknowledgement = ack-nhfb, subject = "Signal processing; Digital techniques; Congresses; Algorithms; Computer architecture", } @Book{Berggren:2000:PSB, editor = "Lennart Berggren and Jonathan Borwein and Peter Borwein", booktitle = "Pi: a source book", title = "Pi: a source book", publisher = pub-SV, address = pub-SV:adr, edition = "Second", pages = "xx + 736", year = "2000", DOI = "https://doi.org/10.1007/978-1-4757-3240-5", ISBN = "0-387-98946-3 (hardcover)", ISBN-13 = "978-0-387-98946-4 (hardcover)", LCCN = "QA484 .P5 2000", MRclass = "11-00 (01A05 01A75 11-03)", MRnumber = "1746004", bibdate = "Wed Aug 10 11:09:47 2016", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib; https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/master.bib; https://www.math.utah.edu/pub/tex/bib/mathcw.bib; https://www.math.utah.edu/pub/tex/bib/pi.bib", acknowledgement = ack-nhfb, author-dates = "Jonathan Michael Borwein (20 May 1951--2 August 2016)", libnote = "Not yet in my library.", ORCID-numbers = "Borwein, Jonathan/0000-0002-1263-0646", subject = "Pi (mathematical constant)", tableofcontents = "Preface / v \\ \\ Preface to the Second Edition / viii \\ Acknowledgments / ix \\ \\ Introduction / xvii \\ \\ 1. The Rhind Mathematical Papyrus-Problem 50 ($\approx$ 1650 B.C.) / A problem dealing with the area of a round field of given diameter / 1 \\ \\ 2. Engels. Quadrature of the Circle in Ancient Egypt (1977) / A conjectural explanation of how the mathematicians of ancient Egypt approximated the area of a circle / 3 \\ \\ 3. Archimedes. Measurement of a Circle ($\approx$ 250 BC) / The seminal work in which Archimedes presents the first true algorithm for $\pi$ / 7 \\ \\ 4. Phillips. Archimedes the Numerical Analyst (1981) / A summary of Archimedes' work on the computation of $\pi$ using modern notation / 15 \\ \\ 5. Lam and Ang. Circle Measurements in Ancient China (1986) / This paper discusses and contains a translation of Liu Hui's (3rd century) method for evaluating $\pi$ and also examines values for $\pi$ given by Zu Chongzhi (429--500) / 20 \\ \\ 6. The Ban{\=u} M{\=u}s{\=a}: The Measurement of Plane and Solid Figures ($\approx$ 850) / This extract gives an explicit statement and proof that the ratio of the circumference to the diameter is constant / 36 \\ \\ 7. M{\=a}dhava. The Power Series for Arctan and Pi ($\approx$ 1400) / These theorems by a fifteenth century Indian mathematician give Gregory's series for arctan with remainder terms and Leibniz's series for $\pi$ / 45 \\ \\ 8. Hope-Jones. Ludolph (or Ludolff or Lucius) van Ceulen (1938) / Correspondence about van Ceulen's tombstone in reference to it containing some digits of $\pi$ / 51 \\ \\ 9. Vi{\'e}te. Variorum de Rebus Mathematicis Reponsorum Liber VII (1593) / Two excerpts. One containing the first infinite expression of $\pi$, obtained by relating the area of a regular $2n$-gon to that of a regular $n$-gon / 53 \\ \\ 10. Wallis. Computation of $\pi$ by Successive Interpolations (1655) / How Wallis derived the infinite product for $\pi$ that bears his name / 68 \\ \\ 11. Wallis. Arithmetica Infinitorum (1655) / An excerpt including Prop. 189, 191 and an alternate form of the result that gives Wm. Brounker's continued fraction expression for $4/\pi$ / 78 \\ \\ 12. Huygens. De Circuli Magnitudine Inventa (1724) / Huygens's proof of W. Snell's discovery of improvements in Archimedes' method of estimating the lengths of circular arcs / 81 \\ \\ 13. Gregory. Correspondence with John Collins (1671) / A letter to Collins in which he gives his series for arctangent, carried to the ninth power. / 87 \\ \\ 14. Roy. The Discovery of the Series Formula for $\pi$ by Leibniz, Gregory, and Nilakantha (1990) / A discussion of the discovery of the series $\pi/4 = 1 - 1/3 + 1/5, \cdots{}$ / 92 \\ \\ 15. Jones. The First Use of $\pi$ for the Circle Ratio (1706) / An excerpt from Jones' book, the Synopsis Palmariorum Matheseos: or, a New Introduction to the Mathematics, London, 1706 / 108 \\ \\ 16. Newton. Of the Method of Fluxions and Infinite Series (1737) / An excerpt giving Newton's calculation of $\pi$ to 16 decimal places / 110 \\ \\ 17. Euler. Chapter 10 of Introduction to Analysis of the Infinite (On the Use of the Discovered Fractions to Sum Infinite Series) (1748) / This includes many of Euler's infinite series for $\pi$ and powers of $\pi$ / 112 \\ \\ 18. Lambert. M{\'e}moire Sur Quelques Propri{\'e}t{\'e}s Remarquables Des Quantit{\'e}s Transcendentes Circulaires et Logarithmiques (1761) / An excerpt from Lambert's original proof of the irrationality of $\pi$ / 129 \\ \\ 19. Lambert. Irrationality of $\pi$ (1969) / A translation and Struik's discussion of Lambert's proof of the irrationality of $\pi$ / 141 \\ \\ 20. Shanks. Contributions to Mathematics Comprising Chiefly of the Rectification of the Circle to 607 Places of Decimals (1853) / Pages from Shank's report of his monumental hand calculation of $\pi$ / 147 \\ \\ 21. Hermite. Sur La Fonction Exponentielle (1873) / The first proof of the transcendence of $e$ / 162 \\ \\ 22. Lindemann. Ueber die Zahl $\pi$ (1882) / The first proof of the transcendence of $\pi$ / 194 \\ \\ 23. Weierstrass. Zu Lindemann's Abhandlung ``Uber die Ludolphsche Zahl'' (1885) / Weierstrass' proof of the transcendence of $\pi$ / 207 \\ \\ 24. Hilbert. Ueber die Trancendenz der Zahlen $e$ und $\pi$ (1893) / Hilbert's short and elegant simplification of the transcendence proofs for $e$ and $\pi$ / 226 \\ \\ 25. Goodwin. Quadrature of the Circle (1894) / The dubious origin of the attempted legislation of the value of $\pi$ in Indiana / 230 \\ \\ 26. Edington. House Bill No. 246, Indiana State Legislature, 1897 (1935) / A summary of the action taken by the Indiana State Legislature to fix the value of $\pi$ (including a copy of the actual bill that was proposed) / 231 \\ \\ 27. Singmaster. The Legal Values of Pi (1985) / A history of the attempt by Indiana to legislate the value of $\pi$ / 236 \\ \\ 28. Ramanujan. Squaring the Circle (1913) / A geometric approximation to $\pi$ / 240 \\ \\ 29. Ramanujan. Modular Equations and Approximations to $\pi$ (1914) / Ramanujan's seminal paper on $\pi$ that includes a number of striking series and algebraic approximations / 241 \\ \\ 30. Watson. The Marquis and the Land Agent: A Tale of the Eighteenth Century (1933) / A Presidential address to the Mathematical Association in which the author gives an account of ``some of the elementary work on arcs and ellipses and other curves which led up to the idea of inverting an elliptic integral, and so laying the foundations of elliptic functions and doubly periodic functions generally.'' / 258 \\ \\ 31. Ballantine. The Best (?) Formula for Computing $\pi$ to a Thousand Places (1939) / An early attempt to orchestrate the calculation of $\pi$ more cleverly / 271 \\ \\ 32. Birch. An Algorithm for Construction of Arctangent Relations (1946) / The object of this note is to express $\pi / 4 $ as a sum of arctan relations in powers of 10 / 274 \\ \\ 33. Niven. A Simple Proof that $\pi$ Is Irrational (1947) / A very concise proof of the irrationality of $\pi$ / 276 \\ \\ 34. Reitwiesner. An ENIAC Determination of $\pi$ and $e$ to 2000 Decimal Places (1950) / One of the first computer-based computations / 277 \\ \\ 35. Schepler. The Chronology of Pi (1950) / A fairly reliable outline of the history of $\pi$ from 3000 BC to 1949 / 282 \\ \\ 36. Mahler. On the Approximation of $\pi$ (1953) / ``The aim of this paper is to determine an explicit lower bound free of unknown constants for the distance of $\pi$ from a given rational or algebraic number'' / 306 \\ \\ 37. Wrench, Jr. The Evolution of Extended Decimal Approximations to $\pi$ (1960) / A history of the calculation of the digits of $\pi$ to 1960 \\ \\ 38. Shanks and Wrench, Jr. Calculation of $\pi$ to 100,000 Decimals (1962) / A landmark computation of $\pi$ to more than 100,000 places / 326 \\ \\ 39. Sweeny. On the Computation of Euler's Constant (1963) / The computation of Euler's constant to 3566 decimal places / 350 \\ \\ 40. Baker. Approximations to the Logarithms of Certain Rational Numbers (1964) / The main purpose of this deep and fundamental paper is to ``deduce results concerning the accuracy with which the natural logarithms of certain rational numbers may be approximated by rational numbers, or, more generally, by algebraic numbers of bounded degree.'' / 359 \\ \\ 41. Adams. Asymptotic Diophantine Approximations to $E$ (1966) / An asymptotic estimate for the rational approximation to $e$ which disproves the conjecture that $e$ behaves like almost all numbers in this respect / 368 \\ \\ 42. Mahler. Applications of Some Formulae by Hermite to the Approximations of Exponentials of Logarithms (1967) / An important extension of Hilbert's approach to the study of transcendence / 372 \\ \\ 43. Eves. In Mathematical Circles; A Selection of Mathematical Stories and Anecdotes (excerpt) (1969) / A collection of mathematical stories and anecdotes about $\pi$ / 400 \\ \\ 44. Eves. Mathematical Circles Revisited; A Second Collection of Mathematical Stories and Anecdotes (excerpt) (1971) / A further collection of mathematical stories and anecdotes about $\pi$ / 402 \\ \\ 45. Todd. The Lemniscate Constants (1975) / A unifying account of some of the methods used for computing the lemniscate constants / 412 \\ \\ 46. Salamin. Computation of r Using Arithmetic-Geometric Mean (1976) / The first quadratically converging algorithm for $\pi$ based on Gauss's AGM and on Legendre's relation for elliptic integrals / 418 \\ \\ 47. Brent. Fast Multiple-Precision Evaluation of Elementary Functions (1976) / ``This paper contains the `Gauss-Legendre' method and some different algorithms for log and exp (using Landen transformations).'' / 424 \\ \\ 48. Beukers. A Note on the Irrationality of $\zeta(2)$ and $\zetq(3)$ (1979) / A short and elegant recasting of Ap{\'e}ry's proof of the irrationality of $\zeta(3)$ (and $\zeta(2)$) / 434 \\ \\ 49. van der Poorten. A Proof that Euler Missed \ldots{} Ap{\'e}ry's Proof of the Irrationality of $\zeta(3)$ (1979) / An illuminating account of Ap{\'e}ry's astonishing proof of the irrationality of $\zeta(3)$ / 439 \\ \\ 50. Brent and McMillan. Some New Algorithms for High-Precision Computation of Euler's Constant (1980) / Several new algorithms for high precision calculation of Euler's constant, including one which was used to compute 30,100 decimal places / 448 \\ \\ 51. Apostol. A Proof that Euler Missed: Evaluating $\zeta(2)$ the Easy Way (1983) / This note shows that one of the double integrals considered by Beukers ([48] in the table of contents) can be used to establish directly that $\zeta(2) = \pi / 6$ / 456 \\ \\ 52. O'Shaughnessy. Putting God Back in Math (1983) / An article about the Institute of Pi Research, an organization that ``pokes fun at creationists by pointing out that even the Bible makes mistakes.'' / 458 \\ \\ 53. Stern. A Remarkable Approximation to $\pi$ (1985) / Justification of the value of $\pi$ in the Bible through numerological interpretations / 460 \\ \\ 54. Newman and Shanks. On a Sequence Arising in Series for $\pi$ (1984) / More connections between $\pi$ and modular equations / 462 \\ \\ 55. Cox. The Arithmetic-Geometric Mean of Gauss (1984) / An extensive study of the complex analytic properties of the AGM / 481 \\ \\ 56. Borwein and Borwein. The Arithmetic-Geometric Mean and Fast Computation of Elementary Functions (1984) / The relationship between the AGM iteration and fast computation of elementary functions (one of the by-products is an algorithm for $\pi$) / 537 \\ \\ 57. Newman. A Simplified Version of the Fast Algorithms of Brent and Salamin (1984) / Elementary algorithms for evaluating $e^x$ and $\pi$ using the Gauss AGM without explicit elliptic function theory / 553 \\ \\ 58. Wagon. Is Pi Normal? (1985) / A discussion of the conjecture that $\pi$ has randomly distributed digits / 557 \\ \\ 59. Keith. Circle Digits: A Self-Referential Story (1986) / A mnemonic for the first 402 decimal places of $\pi$ / 560 \\ \\ 60. Bailey. The Computation of $\pi$ to 29,360,000 Decimal Digits Using Borweins' Quartically Convergent Algorithm (1988) / The algorithms used, both for $\pi$ and for performing the required multiple-precision arithmetic / 562 \\ \\ 61. Kanada. Vectorization of Multiple-Precision Arithmetic Program and 201,326,000 Decimal Digits of 1 Calculation (1988) / Details of the computation and statistical tests of the first 200 million digits of $\pi$ / 576 \\ \\ 62. Borwein and Borwein. Ramanujan and Pi (1988) / This article documents Ramanujan's life, his ingenious approach to calculating $\pi$, and how his approach is now incorporated into modern computer algorithms / 588 \\ \\ 63. Chudnovsky and Chudnovsky. Approximations and Complex Multiplication According to Ramanujan (1988) / This excerpt describes ``Ramanujan's original quadratic period--quasiperiod relations for elliptic curves with complex multiplication and their applications to representations of fractions of $\pi$ and other logarithms in terms of rapidly convergent nearly integral (hypergeometric) series.'' / 596 \\ \\ 64. Borwein, Borwein and Bailey. Ramanujan, Modular Equations, and Approximations to Pi or How to Compute One Billion Digits of Pi (1989) / An exposition of the computation of $\pi$ using mathematics rooted in Ramanujan's work / 623 \\ \\ 65. Borwein, Borwein and Dilcher. Pi, Euler Numbers, and Asymptotic Expansions (1989) / An explanation as to why the slowly convergent Gregory series for $\pi$, truncated at 500,000 terms, gives $\pi$ to 40 places with only the 6th, 17th, 18th, and 29th places being incorrect / 642 \\ \\ 66. Beukers, B{\'e}zivin, and Robba. An Alternative Proof of the Lindemann--Weierstrass Theorem (1990) / The Lindemann--Weierstrass theorem as a by-product of a criterion for rationality of solutions of differential equations / 649 \\ \\ 67. Webster. The Tail of Pi (1991) / Various anecdotes about $\pi$ from the 14th annual IMO Lecture to the Royal Society / 654 \\ \\ 68. Eco. An excerpt from Foucault's Pendulum (1993) / ``The unnumbered perfection of the circle itself.'' / 658 \\ \\ 69. Keith. Pi Mnemonics and the Art of Constrained Writing (1996) / A mnemonic for $\pi$ based on Edgar Allen Poe's poem ``The Raven.'' / 659 \\ \\ 70. Bailey, Borwein, and Plouffe. On the Rapid Computation of Various Polylogarithmic Constants (1996) / A fast method for computing individual digits of $\pi$ in base 2 / 663 \\ Appendix I --- On the Early History of Pi / 677 \\ \\ Appendix II --- A Computational Chronology of Pi / 683 \\ \\ Appendix III --- Selected Formulae for Pi / 686 \\ \\ Appendix IV --- Translations of Vi{\`e}te and Huygens / 690 \\ Bibliography / 711 \\ \\ Credits / 717 \\ \\ Index / 721", } @Proceedings{Cocolicchio:2000:ASF, editor = "Decio Cocolicchio and G. Dattoli and H. M. Srivastava", booktitle = "{Advanced special functions and applications: proceedings of the workshop: Melfi (PZ), Italy, 9--12 May 1999}", title = "{Advanced special functions and applications: proceedings of the workshop: Melfi (PZ), Italy, 9--12 May 1999}", volume = "1", publisher = "Aracne", address = "Roma, Italy", edition = "1.", pages = "336", year = "2000", ISBN = "88-7999-265-X", ISBN-13 = "978-88-7999-265-7", LCCN = "QA351 .A38 2000", bibdate = "Sat Oct 30 19:16:34 MDT 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; z3950.loc.gov:7090/Voyager", series = "Proceedings of the Melfi school on advanced topics in mathematics and physics", acknowledgement = ack-nhfb, subject = "Functions, Special; Congresses", } @Proceedings{Dunkl:2000:PIW, editor = "Charles Dunkl and Mourad Ismail and Roderick Wong", booktitle = "{Proceedings of the international workshop, special functions: Hong Kong, 21--25 June 1999}", title = "{Proceedings of the international workshop, special functions: Hong Kong, 21--25 June 1999}", publisher = pub-WORLD-SCI, address = pub-WORLD-SCI:adr, pages = "xi + 438", year = "2000", ISBN = "981-02-4393-6", ISBN-13 = "978-981-02-4393-7", LCCN = "QA351 .P76 2000", bibdate = "Fri Jul 09 06:30:25 2004", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, } @Proceedings{Sprague:2000:PAH, editor = "Ralph H. Sprague", booktitle = "{Proceedings of the 33rd Annual Hawaii International Conference on System Sciences: abstracts and CD-ROM of full papers: January 4--7, 2000, Maui, Hawaii}", title = "{Proceedings of the 33rd Annual Hawaii International Conference on System Sciences: abstracts and CD-ROM of full papers: January 4--7, 2000, Maui, Hawaii}", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "liv + 259", year = "2000", ISBN = "0-7695-0493-0, 0-7695-0494-9, 0-7695-0495-7", ISBN-13 = "978-0-7695-0493-3, 978-0-7695-0494-0, 978-0-7695-0495-7", LCCN = "TA168 .H37 2000; TA168 .H37 2000xeb; TA168", bibdate = "Sun Mar 4 21:23:42 MST 2007", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; melvyl.cdlib.org:210/CDL90", acknowledgement = ack-nhfb, meetingname = "Hawaii International Conference on System Sciences (33rd: 2000: Maui, Hawaii)", remark = "IEEE Computer Society Order Number: PR00493", subject = "Systems engineering; Congresses; Information theory; Electronic data processing; System design", } @Proceedings{Banerji:2001:SFS, editor = "P. K. Banerji", booktitle = "{Special functions: selected articles: proceedings of the First National Conference of the Society for Special Functions and their Applications, March 3--4, 2000, Jodhpur, India}", title = "{Special functions: selected articles: proceedings of the First National Conference of the Society for Special Functions and their Applications, March 3--4, 2000, Jodhpur, India}", publisher = "Published by Scientific Publishers (India) for the Society for Special Functions and their Applications", address = "Jodhpur, India", pages = "258", year = "2001", ISBN = "81-7233-267-X", ISBN-13 = "978-81-7233-267-9", LCCN = "QA351 .S665 2001", bibdate = "Sat Oct 30 19:13:10 MDT 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; z3950.loc.gov:7090/Voyager", acknowledgement = ack-nhfb, subject = "Functions, Special; Congresses", } @Proceedings{Burgess:2001:ISC, editor = "N. Burgess and L. Ciminiera", booktitle = "{15th IEEE Symposium on Computer Arithmetic: ARITH-15 2001: proceedings: Vail, Colorado, 11--13 June, 2001}", title = "{15th IEEE Symposium on Computer Arithmetic: ARITH-15 2001: proceedings: Vail, Colorado, 11--13 June, 2001}", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "xii + 285", year = "2001", ISBN = "0-7695-1150-3; 0-7695-1152-X", ISBN-13 = "978-0-7695-1150-4; 978-0-7695-1152-8", ISSN = "1063-6889", LCCN = "QA76.9.C62 S95 2001", bibdate = "Fri May 03 14:20:49 2002", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", note = "IEEE order no. PR01150.", price = "US\$145", acknowledgement = ack-nhfb, keywords = "ARITH-15", xxnote = "Check dates: 11--13 or 11--17??", xxtitle = "Computer Arithmetic: Papers presented at the {15th IEEE Symposium on Computer Arithmetic (Arith-15 2001), 11--17 June, 2001, Vail, CO}", } @Book{Garvan:2001:SCN, editor = "Frank (Frank G.) Garvan and Mourad Ismail", booktitle = "Symbolic Computation, Number Theory, Special Functions, Physics, and Combinatorics", title = "Symbolic Computation, Number Theory, Special Functions, Physics, and Combinatorics", volume = "4", publisher = pub-KLUWER, address = pub-KLUWER:adr, pages = "x + 283", year = "2001", ISBN = "1-4020-0101-0", ISBN-13 = "978-1-4020-0101-7", LCCN = "QA295 .S86 2001", bibdate = "Sat Oct 30 17:31:50 MDT 2010", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; prodorbis.library.yale.edu:7090/voyager", series = "Developments in mathematics", acknowledgement = ack-nhfb, subject = "q-series; Congresses; Algebra; Data processing; Number theory; Functions, Special; Mathematical physics; Combinatorial analysis", tableofcontents = "Preface \\ Participants \\ Gaussian hypergeometric series and combinatorial congruences / Scott Ahlgren / 1--12 \\ A double bounded key identity for Gollnitz's (BIG) partition theorem / Krishnaswami Alladi and Alexander Berkovich / 13--32 \\ Engel expansions of q-series by computer algebra / George E. Andrews, Arnold Knopfmacher and Peter Paule / [and others] / 33--57 \\ Sums of squares and the preservation of modularity under congruence restrictions / Paul T. Bateman, Boris A. Datskovsky and Marvin I. Knopp / 59--71 \\ On the transformation formula for the Dedekind eta-function / Bruce C. Berndt and K. Venkatachaliengar / 73--77 \\ Experiments and discoveries in q-trigonometry / R. Wm. Gosper / 79--105 \\ Algebraic consequences of Jacobi's two- and four-square theorems / Michael D. Hirschhorn and James A. McGowan / 107--132 \\ The Borweins' Cubic Theta Functions and q-Elliptic Functions / Richard Lewis, Zhi-Guo Liu / 133--145 \\ Some Eisenstein Series Identities Associated with the Borwein Functions / Zhi-Guo Liu / 147--169 \\ Hankel Determinants of Eisenstein Series / Stephen C. Milne / 171--188 \\ Jacobi's Identity and Two K3-Surfaces / Maki Murata / 189--198 \\ $q$-Random Matrix Ensembles / K. A. Muttalib, Y. Chen, M. E. H. Ismail / 199--221 \\ Differential Endomorphisms for Modular Forms On $\Gamma_0(4)$ / Ken Ono / 223--229 \\ On the Asymptotics of Takeuchi Numbers / Thomas Prellberg / 231--242 \\ Fine-Tuning Zeilberger's Algorithm / Axel Riese / 243--254 \\ Gaussian Integrals and the Rogers--Ramanujan Identities / D. Stanton / 255--265 \\ Some Remarks on a Product Expansion / M. V. Subbarao, A. Verma / 267--283 \\ Back Matter / 285--285", } @Book{Lide:2001:CEM, editor = "David R. Lide", booktitle = "A Century of Excellence in Measurements, Standards, and Technology", title = "A Century of Excellence in Measurements, Standards, and Technology", volume = "958", publisher = pub-NIST, address = pub-NIST:adr, pages = "ix + 386", year = "2001", bibdate = "Fri Jul 09 06:29:11 2004", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", series = "NIST special publication", acknowledgement = ack-nhfb, } @Proceedings{Siafarikas:2001:PFI, editor = "Panayiotis D. Siafarikas and Theodore Seio Chihara", booktitle = "{Proceedings of the Fifth International Symposium on Orthogonal Polynomials, Special Functions and their Applications: Patras, Greece, 20--24 September 1999}", title = "{Proceedings of the Fifth International Symposium on Orthogonal Polynomials, Special Functions and their Applications: Patras, Greece, 20--24 September 1999}", volume = "133(1/2)", publisher = pub-ELSEVIER, address = pub-ELSEVIER:adr, pages = "xxvii + 705", year = "2001", ISSN = "0377-0427 (print), 1879-1778 (electronic)", LCCN = "QA76 J86 v. 133, no. 1/2", bibdate = "Sat Oct 30 19:08:06 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", series = "Journal of computational and applied mathematics", acknowledgement = ack-nhfb, } @Proceedings{Borrione:2002:TIW, editor = "Dominique Borrione", booktitle = "{Third International Workshop on the ACL2 Theorem Prover and its Applications (ACL2-2002), April 8--9, 2002, in Grenoble, France. Presentations, affiliated with ETAPS 2002}", title = "{Third International Workshop on the ACL2 Theorem Prover and its Applications (ACL2-2002), April 8--9, 2002, in Grenoble, France. Presentations, affiliated with ETAPS 2002}", publisher = "????", address = "????", pages = "????", year = "2002", ISBN = "????", ISBN-13 = "????", LCCN = "????", bibdate = "Sat Jun 25 12:28:18 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "http://www.cs.utexas.edu/users/moore/acl2/workshop-2002/", acknowledgement = ack-nhfb, } @Book{Lide:2002:CEM, editor = "David R. Lide", booktitle = "A Century of Excellence in Measurements, Standards, and Technology", title = "A Century of Excellence in Measurements, Standards, and Technology", publisher = pub-CRC, address = pub-CRC:adr, pages = "ix + 386", year = "2002", ISBN = "0-8493-1247-7", ISBN-13 = "978-0-8493-1247-2", LCCN = "QC100.U6 .C46 2002", bibdate = "Fri Jul 09 06:29:11 2004", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "Republication of \cite{Lide:2001:CEM}.", URL = "http://www.loc.gov/catdir/toc/fy031/2002283707.html", acknowledgement = ack-nhfb, tableofcontents = "Introduction / 1 \\ 1901--1930 \\ The Absolute Measurement of Inductance / 5 \\ Determination of the Constants of Total Radiation From a Black Body / 7 \\ The Testing of Thermal Insulators / 10 \\ Precipitation Hardening of Metal Alloys / 14 \\ Construction and Operation of a Simple Homemade Radio Receiving Outfit / 16 \\ Methods for Standardizing and Testing Precision Gage Blocks / 19 \\ Recommended Minimum Requirements for Small Dwelling Construction / 22 \\ Visibility of Radiant Energy / 25 \\ Test of the Severity of Building Fires / 28 \\ Calculation of Compounds in Portland Cements / 33 \\ Development of the Visual-Type Airway Radio-Beacon System / 38 \\ 1931--1950 \\ A Hydrogen Isotope of Mass 2 / 43 \\ Air Flow and Turbulence in Boundary Layers / 46 \\ Thermodynamic Properties of Water and Steam for Power Generation / 49 \\ Absolute Pressure Calibrations of Microphones / 53 \\ Absolute Determination of the Ampere / 56 \\ Radio Proximity Fuzes / 59 \\ Stability of Double-Walled Manganin Resistors / 63 \\ Manufacture of Paper for War Maps and Other Applications / 66 \\ Transmission of Sound Waves in Gases at Low Pressures / 69 \\ Atomic Energy Levels and Other Spectroscopic Data / 73 \\ Iteration Method for the Solution of the Eigenvalue Problem of Linear Differential and Integral Operators / 77 \\ 1951--1960 \\ Methods of Conjugate Gradients for Solving Linear Systems / 81 \\ Computer Development at the National Bureau of Standards / 86 \\ Thermal Converters as AC-DC Transfer Standards for Current and Voltage Measurements at Audio Frequencies / 90 \\ Selected Values of Chemical Thermodynamic Properties / 93 \\ Applied Inorganic Analysis / 97 \\ The Diamond Anvil Pressure Cell / 100 \\ Polymer Crystallization With Folded Chains / 104 \\ Cryogenic Engineering / 107 \\ Reversal of the Parity Conservation Law in Nuclear Physics / 111 \\ 1961--1970 \\ Effects of Configuration Interaction on Intensities and Phase Shifts / 116 \\ Electromagnetic Waves in Stratified Media / 120 \\ ``Second Breakdown'' in Transistors / 123 \\ Stress Relaxation With Finite Strain / 126 \\ Realistic Evaluation of the Precision and Accuracy of Instrument Calibration Systems / 129 \\ Experimental Statistics / 132 \\ Handbook of Mathematical Functions / 135 \\ Paths, Trees, and Flowers / 140 \\ Concepts, Terminology, and Notation for Optical Modulation / 145 \\ Theory of Light Scattering in Fluids / 149 \\ Scaling Analysis of Thermodynamic Properties in the Critical Region of Fluids / 152 \\ Resonance Tunneling of Field Emitted Electrons Through Adsorbates on Metal Surfaces / 155 \\ Quantitative Electron Probe Microanalysis / 160 \\ Limits for Qualitative Detection and Quantitative Determination / 164 \\ Traceability: An Evolving Concept / 167 \\ Code for Information Interchange --- ASCII / 172 \\ Consumer Information Series / 174 \\ Theory of Isoperibol Calorimetry for Laser Power and Energy Measurement / 178 \\ Influence of Water on Crack Growth in Glass / 181 \\ Phase Equilibria Diagrams / 184 \\ Determination of Reduced Cells in Crystallography / 188 \\ 1971--1980 \\ Speed of Light From Direct Frequency and Wavelength Measurements / 191 \\ Connecting Visible Wavelength Standards With X Rays and $\gamma$ Rays / 194 \\ Laser Cooling of Atoms / 200 \\ Spin-Polarized Electrons / 203 \\ Needs for Radioactivity Standards and Measurements in Different Fields / 209 \\ The Topografiner: An Instrument for Measuring Surface Microtopography / 214 \\ Electron-Stimulated Desorption / 219 \\ Photochemistry of Small Molecules / 224 \\ Role of Standard Reference Materials in Measurement Systems / 227 \\ Metrology and Standardization to Assist Industrializing Economies / 230 \\ Publications Taking Us Toward a Metric America / 234 \\ A New Approach to Manipulator Control: The Cerebellar Model Articulation Controller / 237 \\ Three Dimensional Metrology / 241 \\ Initial Graphics Exchange Specifications / 246 \\ Data Encryption Standard / 250 \\ OMNIDATA and the Computerization of Scientific Data / 254 \\ FORTRAN Test Programs / 258 \\ Design and Evaluation Criteria for Energy Conservation in New Buildings / 260 \\ Computer Program for Heating and Cooling Loads in Buildings / 266 \\ Methods for Testing and Rating the Performance of Heating and Air Conditioning Systems / 270 \\ System for Fire Safety Evaluation of Health Care Facilities / 275 \\ Estimation of Rate of Heat Release by Means of Oxygen Consumption Measurements / 280 \\ Probability-Based Load Criteria for Structural Design / 283 \\ Resistivity--Dopant Density Relationship for Phosphorus-Doped Silicon / 289 \\ 1981--1990 \\ Critical Data for Critical Needs / 291 \\ Materials at Low Temperatures / 294 \\ Optical Fiber Characterization / 297 \\ Quasicrystals / 300 \\ Protein Crystallography by Joint X-Ray and Neutron Diffraction / 303 \\ Strain Effects in Superconducting Compounds / 306 \\ Dental Research at the National Bureau of Standards / 309 \\ Handbook for Standard Reference Materials Users / 313 \\ A Practical Josephson Voltage Standard at One Volt / 315 \\ Plane-Wave Scattering-Matrix Theory of Antennas and Antenna--Antenna Interactions / 319 \\ The Automated Manufacturing Research Facility / 322 \\ Submicrometer Linewidth Metrology / 328 \\ Observation of Atoms Laser-Cooled Below the Doppler Limit / 331 \\ Laser-Excited Hot-Electron Induced Desorption / 334 \\ Measurement of the Universal Gas Constant Using an Acoustic Resonator / 339 \\ Thermal and Oxidative Degradation of Polymers / 344 \\ HAZARD I: Software for Fire Hazard Assessment / 347 \\ Analysis of the Catastrophic Rupture of a Pressure Vessel / 350 \\ Curing Those Uncontrollable Fits of Interaction / 353 \\ Baldrige Criteria for Performance Excellence / 357 \\ 1991--2000 \\ The Advanced Technology Program / 359 \\ NIST Manufacturing Extension Partnership / 363 \\ Questions and Answers on Quality / 366 \\ Uniformity in Weights and Measures Laws and Regulations / 368 \\ Certification of 10 $\mu$m Diameter Polystyrene Spheres (``Space Beads') / 371 \\ Bose--Einstein Condensation in a Dilute Atomic Vapor / 375 \\ Index / 379", } @Proceedings{Anonymous:2003:CRN, editor = "Anonymous", booktitle = "5th Conference on Real Numbers and Computers 2003 --- {RNC5}, Lyon, France, September 2003", title = "5th Conference on Real Numbers and Computers 2003 --- {RNC5}, Lyon, France, September 2003", publisher = "????", address = "????", pages = "????", year = "2003", ISBN = "????", ISBN-13 = "????", LCCN = "????", bibdate = "Sat Jun 25 14:57:33 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, } @Proceedings{Koelink:2003:OPS, editor = "Erik Koelink and Walter {Van Assche}", booktitle = "Orthogonal polynomials and special functions: {Leuven 2002}", title = "Orthogonal polynomials and special functions: {Leuven 2002}", volume = "1817", publisher = pub-SV, address = pub-SV:adr, pages = "x + 249", year = "2003", ISBN = "3-540-40375-2", ISBN-13 = "978-3-540-40375-3", ISSN = "0075-8434 (print), 1617-9692 (electronic)", LCCN = "33 33-06 33C 68W", bibdate = "Sat Oct 30 17:00:03 MDT 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; z3950.bibsys.no:2100/BIBSYS", series = "Lecture notes in mathematics", acknowledgement = ack-nhfb, subject = "functions, special; orthogonal polynomials", } @Book{Berggren:2004:PSB, editor = "Lennart Berggren and Jonathan Borwein and Peter Borwein", booktitle = "Pi: a source book", title = "Pi: a source book", publisher = pub-SV, address = pub-SV:adr, edition = "Third", pages = "xx + 797", year = "2004", DOI = "https://doi.org/10.1007/978-1-4757-4217-6", ISBN = "0-387-20571-3", ISBN-13 = "978-0-387-20571-7", MRclass = "11-00 (01A05 01A75 11-03)", MRnumber = "2065455", MRreviewer = "F. Beukers", bibdate = "Wed Aug 10 11:09:47 2016", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib; https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, author-dates = "Jonathan Michael Borwein (20 May 1951--2 August 2016)", ORCID-numbers = "Borwein, Jonathan/0000-0002-1263-0646", remark = "CECM Preprint 2003:210.", tableofcontents = "Preface to the Third Edition / v \\ Preface to the Second Edition / vi \\ Preface / vii \\ Acknowledgments / x \\ Introduction / xvii \\ 1. The Rhind Mathematical Papyrus --- Problem 50 ($\approx$1650 B.C.) / A problem dealing with the area of a round field of given diameter / 1 \\ 2. Engels / Quadrature of the Circle in Ancient Egypt (1977) / A conjectural explanation of how the mathematicians of ancient Egypt approximated the area of a circle / 3 \\ 3. Archimedes / Measurement of a Circle --- (-250 B.C.) / The seminal work in which Archimedes presents the first true algorithm for $ \pi $ / 7 \\ 4. Phillips / Archimedes the Numerical --- Analyst (1981) / A summary of Archimedes' work on the computation of $ \pi $ using modem notation / 15 \\ 5. Lam and Ang / Circle Measurements in Ancient China (1986) / This paper discusses and contains a translation of Liu Hui's (3rd century) method for evaluating $ \pi $ and also examines values for $ \pi $ given by Zu Chongzhi (429--500) / 20 \\ 6. The Ban{\=u} M{\=u}s{\=a}: The Measurement of Plane and Solid Figures (--850) / This extract gives an explicit statement and proof that the ratio of the circumference to the diameter is constant / 36 \\ 7. M{\=a}dhava / The Power Series for Arctan and Pi (-1400) / These theorems by a fifteenth century Indian mathematician give Gregory's series for arctan with remainder terms and Leibniz's series for $ \pi $ / 45 \\ 8. Hope-Jones / Ludolph (or Ludolff or Lucius) van Ceulen (1938) / Correspondence about van Ceulen's tombstone in reference to it containing some digits of $ \pi $ / 51 \\ 9. Vi{\`e}te / \booktitle{Variorum de Rebus Mathematicis Reponsorum Liber VII} (1593) / Two excerpts. One containing the first infinite expression of $ \pi $, obtained by relating the area of a regular $2n$-gon to that of a regular $n$-gon / 53 \\ 10. Wallis. Computation of $ \pi $ by Successive Interpolations (1655) / How Wallis derived the infinite product for $ \pi $ that bears his name / 68 \\ 11. Wallis / \booktitle{Arithmetica Infinitorum} (1655) / An excerpt including Prop. 189, 191 and an alternate form of the result that gives Wm. Brounker's continued fraction expression for $ 4 / \pi$ / ?? \\ 12. Huygens / \booktitle{De Circuli Magnitudine Inventa} (1654) / Huygens's demonstration of how to triple the number of correct decimals over those in Archimedes' estimate of $ \pi $ / 81 13. Gregory / Correspondence with John Collins (1671) / A letter to Collins in which he gives his series for arctangent, carried to the ninth power / 87 \\ 14. Roy / The Discovery of the Series Formula for $ \pi $ by Leibniz, Gregory, and Nilakantha (1990) / A discussion of the discovery of the series $ \pi / 4 = 1 - 1/3 + 1/5 - \cdots{} $ / 92 \\ 15. Jones / The First Use of $ \pi $ for the Circle Ratio (1706) / An excerpt from Jones' book, the \booktitle{Synopsis Palmariorum Matheseos: or, a New Introduction to the Mathematics}, London, 1706 / 108 \\ 16. Newton / Of the Method of Fluxions and Infinite Series (1737) / An excerpt giving Newton's calculation of $ \pi $ to 16 decimal places / 110 \\ 17. Euler / Chapter 10 of \booktitle{Introduction to Analysis of the Infinite (On the Use of the Discovered Fractions to Sum Infinite Series)} (1748) / This includes many of Euler's infinite series for $ \pi $ and powers of $ \pi $ / 112 \\ 18. Lambert / \booktitle{M{\'e}moire Sur Quelques Propri{\'e}t{\'e}s Remarquables Des Quantit{\'e}s Transcendentes Circulaires et Logarithmiques} (1761) / An excerpt from Lambert's original proof of the irrationality of $ \pi $ / 129 19. Lambert / Irrationality of $ \pi $ (1969) / A translation and Struik's discussion of Lambert's proof of the irrationality of $ \pi $ / 141 \\ 20. Shanks / Contributions to Mathematics Comprising Chiefly of the Rectification of the Circle to 607 Places of Decimals (1853) / Pages from Shanks's report of his monumental hand calculation of $ \pi $ / 147 \\ 21. Hermite / \booktitle{Sur La Fonction Exponentielle} (1873) / The first proof of the transcendence of $ e $ / 162 \\ 22. Lindemann / \booktitle{Ueber die Zahl $ \pi $} (1882) / The first proof of the transcendence of $ \pi $ / 194 23. Weierstrass / \booktitle{Zu Lindemann's Abhandlung ``{\"U}ber die Ludolphsche Zahl''} (1885) / Weierstrass' proof of the transcendence of $ \pi $ / 207 24. Hilbert / \booktitle{Ueber die Transzendenz der Zahlen $ e $ und $ \pi $} (1893) / Hilbert's short and elegant simplification of the transcendence proofs for $ e $ and $ \pi $ / 226 25. Goodwin / Quadrature of the Circle (1894) / The dubious origin of the attempted legislation of the value of $ \pi $ in Indiana / 230 \\ 26. Edington / House Bill No. 246, Indiana State Legislature, 1897 (1935) / A summary of the action taken by the Indiana State Legislature to fix the value of $ \pi $ (including a copy of the actual bill that was proposed) / 231 \\ 27. Singmaster / The Legal Values of Pi (1985) / A history of the attempt by Indiana to legislate the value of $ \pi $ / 236 \\ 28. Ramanujan / Squaring the Circle (1913) / A geometric approximation to $ \pi $ / 240 \\ 29. Ramanujan / Modular Equations and Approximations to $ \pi $ (1914) / Ramanujan's seminal paper on pi that includes a number of striking series and algebraic approximations / 241 \\ 30. Watson / The Marquis and the Land Agent: A Tale of the Eighteenth Century (1933) / A Presidential address to the Mathematical Association in which the author gives an account of ``some of the elementary work on arcs and ellipses and other curves which led up to the idea of inverting an elliptic integral, and so laying the foundations of elliptic functions and doubly periodic functions generally.'' / ?? \\ 31. Ballantine / The Best (?) Formula for Computing $ \pi $ to a Thousand Places (1939) / An early attempt to orchestrate the calculation of $ \pi $ more cleverly / 271 \\ 32. Birch / An Algorithm for Construction of Arctangent Relations (1946) / The object of this note is to express $ \pi / 4$ as a sum of arctan relations in powers of 10 / 274 \\ 33. Niven / A Simple Proof that $ \pi $ is Irrational (1947) / A very concise proof of the irrationality of $ \pi $ / 276 \\ 34. Reitwiesner / An ENIAC Determination of $ \pi $ and $ e $ to 2000 Decimal Places (1950) / One of the first computer-based computations / 277 \\ 35. Schepler / The Chronology of Pi (1950) / A fairly reliable outline of the history of $ \pi $ from 3000 B.C. to 1949 / 282 \\ 36. Mahler / On the Approximation of $ \pi $ (1953) / ``The aim of this paper is to determine an explicit lower bound free of unknown constants for the distance of $ \pi $ from a given rational or algebraic number.'' / 306 \\ 37. Wrench, Jr. / The Evolution of Extended Decimal Approximations to $ \pi $ (1960) / A history of the calculation of the digits of $ \pi $ to 1960 / 319 \\ 38. Shanks and Wrench, Jr. / Calculation of $ \pi $ to 100,000 Decimals (1962) / A landmark computation of $ \pi $ to more than 100,000 places / 326 39. Sweeny / On the Computation of Euler's Constant (1963) / The computation of Euler's constant to 3566 decimal places / 350 40. Baker / Approximations to the Logarithms of Certain Rational Numbers (1964) / The main purpose of this deep and fundamental paper is to ``deduce results concerning the accuracy with which the natural logarithms of certain rational numbers may be approximated by rational numbers, or, more generally, by algebraic numbers of bounded degree.'' / 359 \\ 41. Adams / Asymptotic Diophantine Approximations to e (1966) / An asymptotic estimate for the rational approximation to $ e $ which disproves the conjecture that $ e $ behaves like almost all numbers in this respect / 368 \\ 42. Mahler / Applications of Some Formulae by Hermite to the Approximations of Exponentials of Logarithms (1967) / An important extension of Hilbert's approach to the study of transcendence / 372 43. Eves / In Mathematical Circles; A Selection of Mathematical Stories and Anecdotes (excerpt) (1969) / A collection of mathematical stories and anecdotes about $ \pi $ / 456 \\ 44. Eves / Mathematical Circles Revisited; A Second Collection of Mathematical Stories and Anecdotes (excerpt) (1971) / A further collection of mathematical stories and anecdotes about $ \pi $ / 402 45. Todd / The Lemniscate Constants (1975) / A unifying account of some of the methods used for computing the lemniscate constants / 412 \\ 46. Salamin / Computation of $ \pi $ Using Arithmetic--Geometric Mean (1976) / The first quadratically converging algorithm for $ \pi $ based on Gauss's AGM and on Legendre's relation for elliptic integrals / 418 \\ 47. Brent / Fast Multiple-Precision Evaluation of Elementary Functions (1976) / ``This paper contains the `Gauss--Legendre' method and some different algorithms for $\log$ and $\exp$ (using Landen transformations).'' / 424 \\ 48. Beukers / A Note on the Irrationality of $ \zeta(2) $ and $ \zeta(3) $ (1979) / A short and elegant recasting of Apery's proof of the irrationality of $\zeta(3)$ (and $\zeta(2)$) / 434 \\ 49. van der Poorten / A Proof that Euler Missed \ldots{} Apery's Proof of the Irrationality of $\zeta (3)$ (1979) / An illuminating account of Apery's astonishing proof of the irrationality of $\zeta (3)$ / 439 \\ 50. Brent and McMillan / Some New Algorithms for High-Precision Computation of Euler's Constant (1980) / Several new algorithms for high-precision calculation of Euler's constant, including one which was used to compute 30,100 decimal places / 448 \\ 51. Apostol / A Proof that Euler Missed: Evaluating $\zeta(2)$ the Easy Way (1983) / This note shows that one of the double integrals considered by Beukers ([48] in the table of contents) can be used to establish directly that $\zeta(2) = \pi^2 / 6$ / 456 \\ 52. O'Shaughnessy / Putting God Back in Math (1983) / An article about the Institute of Pi Research, an organization that ``pokes fun at creationists by pointing out that even the Bible makes mistakes.'' / 458 \\ 53. Stern / A Remarkable Approximation to $ \pi $ (1985) / Justification of the value of $ \pi $ in the Bible through numerological interpretations / 460 \\ 54. Newman and Shanks / On a Sequence Arising in Series for $ \pi $ (1984) / More connections between $ \pi $ and modular equations / 462 \\ 55. Cox / The Arithmetic--Geometric Mean of Gauss (1984) / An extensive study of the complex analytic properties of the AGM / 481 \\ 56. Borwein and Borwein / The Arithmetic--Geometric Mean and Fast Computation of Elementary Functions (1984) / The relationship between the AGM iteration and fast computation of elementary functions (one of the by-products is an algorithm for $ \pi $) / 537 57. Newman / A Simplified Version of the Fast Algorithms of Brent and Salamin (1984) / Elementary algorithms for evaluating $ e^x $ and $ \pi $ using the Gauss AGM without explicit elliptic function theory / 553 \\ 58. Wagon / Is Pi Normal? (1985) / A discussion of the conjecture that $ \pi $ has randomly distributed digits / 557 \\ 59. Keith / Circle Digits: A Self-Referential Story (1986) / A mnemonic for the first 402 decimal places of $ \pi $ / 560 \\ 60. Bailey / The Computation of $ \pi $ to 29,360,000 Decimal Digits Using Borwein's Quartically Convergent Algorithm (1988) / The algorithms used, both for $ \pi $ and for performing the required multiple-precision arithmetic / 562 \\ 61. Kanada / Vectorization of Multiple-Precision Arithmetic Program and 201,326,000 Decimal Digits of $ \pi $ Calculation (1988) / Details of the computation and statistical tests of the first 200 million digits of $ \pi $ / 576 \\ 62. Borwein and Borwein / Ramanujan and Pi (1988) / This article documents Ramanujan's life, his ingenious approach to calculating $ \pi $, and how his approach is now incorporated into modern computer algorithms / 588 \\ 63. Chudnovsky and Chudnovsky / Approximations and Complex Multiplication According to Ramanujan (1988) / This excerpt describes ``Ramanujan's original quadratic period--quasiperiod relations for elliptic curves with complex multiplication and their applications to representations of fractions of $ \pi $ and other logarithms in terms of rapidly convergent nearly integral (hypergeometric) series.'' / 596 \\ 64. Borwein, Borwein and Bailey / Ramanujan, Modular Equations, and Approximations to Pi or How to Compute One Billion Digits of Pi (1989) / An exposition of the computation of $ \pi $ using mathematics rooted in Ramanujan's work / 623 \\ 65. Borwein, Borwein and Dilcher / Pi, Euler Numbers, and Asymptotic Expansions (1989) / An explanation as to why the slowly convergent Gregory series for $ \pi $, truncated at 500,000 terms, gives $ \pi $ to 40 places with only the 6th, 17th, 18th, and 29th places being incorrect / 642 \\ 66. Beukers, Bezivin, and Robba / An Alternative Proof of the Lindemann--Weierstrass Theorem (1990) / The Lindemann--Weierstrass theorem as a by-product of a criterion for rationality of solutions of differential equations / 649 \\ 67. Webster / The Tale of Pi (1991) / Various anecdotes about $ \pi $ from the 14th annual IMO Lecture to the Royal Society / 654 \\ 68. Eco / An excerpt from Foucault's Pendulum (1993) / ``The unnumbered perfection of the circle itself.'' / 658 \\ 69. Keith / Pi Mnemonics and the Art of Constrained Writing (1996) / A mnemonic for $ \pi $ based on Edgar Allen Poe's poem ``The Raven.'' / 659 \\ 70. Bailey, Borwein, and Plouffe / On the Rapid Computation of Various Polylogarithmic Constants (1997) / A fast method for computing individual digits of $ \pi $ in base 2 / 663 \\ Appendix I --- On the Early History of Pi / 677 \\ Appendix II --- A Computational Chronology of Pi / 683 \\ Appendix III --- Selected Formulae for Pi / 686 \\ Appendix IV --- Translations of Viele and Huygens / 690 \\ Bibliography / 710 \\ Credits / 717 \\ A Pamphlet on Pi / 721 \\ Contents / 723 \\ 1. Pi and Its Friends / 725 \\ 2. Normality of Numbers / 741 \\ 3. Historia Cyclometrica / 753 \\ 4. Demotica Cyclometrica / 771 \\ References / 779 \\ Index / 783", } @Proceedings{Frougny:2004:RCR, editor = "Christiane Frougny and Vasco Brattka and Norbert M{\"u}ller", booktitle = "{RNC'6, 6th Conference on Real Numbers and Computers: Nov 15--17, 2004, Dagstuhl, Germany}", title = "{RNC'6, 6th Conference on Real Numbers and Computers: Nov 15--17, 2004, Dagstuhl, Germany}", publisher = "Universita{\"a}t Trier, Fachbereich IV, Mathematik, Informatik", address = "Trier, Germany", bookpages = "216 + i", pages = "216 + i", year = "2004", ISSN = "0944-0488", ISSN-L = "0944-0488", bibdate = "Thu Apr 28 05:55:01 2022", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", note = "Forschungsbericht Nr. 04-8.", URL = "http://www.informatik.uni-trier.de/Reports/TR-08-2004; http://www.informatik.uni-trier.de/Reports/TR-08-2004/rnc6-complete.pdf", acknowledgement = ack-nhfb, keywords = "base conversion; decimal floating-point arithmetic", tableofcontents = "Introduction / Christiane Frougny / 1--4 \\ Invited Lecture: New ideas and results for solving Differential equations symbolically [abstract only] / Benno Fuchssteiner / 5--5 \\ Invited Lecture: A survey of Integer Relations algorithms and rational numbers [abstract only] / Simon Plouffe / 6--6 \\ Invited Lecture: Real Numbers and Robustness in Computational Geometry / Stefan Schirra / 7--21 \\ Bridging the gap between formal specification and bit-level floating-point arithmetic / Sylvie Boldo / 22--36 \\ Automata, Borel functions and real numbers in Pisot base / B. Cagnard, P. Simonnet / 37--54 \\ Generating formally certified bounds on values and round-off errors / Marc Daumas, Guillaume Melquiond / 55--70 \\ A proven correctly rounded logarithm in double-precision / Florent de Dinechin, Catherine Loirat, Jean-Michel Muller / 71--85 \\ A comparison of polynomial evaluation schemes / L. Fousse, S. Schmitt / 86--102 \\ A comparison of real and complex pseudozero sets for polynomials with real coefficients / Stef Graillat, Philippe Langlois / 103--112 \\ On Intermediate Precision Required for Correctly-Rounding Decimal-to-Binary Floating-Point Conversion / Michel Hack / 113--134 \\ The Generic Multiple-Precision Floating-Point Addition With Exact Rounding (as in the MPFR Library) / Vincent Lef{\`e}vre / 135--145 \\ Software Division and Square Root Using Goldschmidt's Algorithms / Peter Markstein / 146--157 \\ A Fast Algorithm for Julia Sets of Hyperbolic Rational Functions / R. Rettinger / 158--171 \\ An extension of Chaitin's halting probability $\Omega$ to measurement operator in infinite dimensional quantum system / Kohtaro Tadaki / 172--191 \\ On the Hierarchy of $\Delta_2^0$-Real Numbers / Xizhong Zheng / 192--215 \\ Trierer Forschungsberichte Mathematik / Informatik [one page list of reports] / 1--1 (216--216)", } @Proceedings{Wahdan:2004:IHE, editor = "Abdel-Moniem Wahdan", booktitle = "{ICEEC'04: 2004 International Conference on Electrical, Electronic and Computer Engineering: proceedings: 5--7 September, 2004, Cairo, Egypt}", title = "{ICEEC'04: 2004 International Conference on Electrical, Electronic and Computer Engineering: proceedings: 5--7 September, 2004, Cairo, Egypt}", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "xlv + 954", year = "2004", ISBN = "0-7803-8575-6", ISBN-13 = "978-0-7803-8575-7", LCCN = "TK7801 .I5125 2004", bibdate = "Tue Jul 19 08:01:02 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; melvyl.cdlib.org:210/CDL90", note = "IEEE catalog number 04EX893.", acknowledgement = ack-nhfb, subject = "Electric engineering; Congresses; Electronics; Congresses; Computer engineering; Congresses", } @Proceedings{IEEE:2005:PIS, editor = "{IEEE}", booktitle = "{Proceedings of the 17th IEEE Symposium on Computer Arithmetic, ARITH-17, June 27--29, 2005, Cape Cod, Massachusetts, USA}", title = "{Proceedings of the 17th IEEE Symposium on Computer Arithmetic, ARITH-17, June 27--29, 2005, Cape Cod, Massachusetts, USA}", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "????", year = "2005", ISBN = "????", ISBN-13 = "????", LCCN = "????", bibdate = "Tue Jun 21 19:02:16 2005", bibsource = "http://arith17.polito.it/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, xxnote = "Not yet published: check editor??", } @Book{Ismail:2005:TAS, editor = "Mourad E. H. Ismail and Erik Koelink", booktitle = "Theory and Applications of Special Functions: a Volume Dedicated to {Mizan Rahman}", title = "Theory and Applications of Special Functions: a Volume Dedicated to {Mizan Rahman}", volume = "13", publisher = pub-SV, address = pub-SV:adr, pages = "x + 491", year = "2005", ISBN = "0-387-24231-7 (hardcover), 0-387-24233-3 (e-book)", ISBN-13 = "978-0-387-24231-6 (hardcover), 978-0-387-24233-0(e-book)", LCCN = "QA351 .T44 2005", bibdate = "Sat Oct 30 07:35:31 MDT 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; z3950.loc.gov:7090/Voyager", series = "Developments in mathematics", URL = "http://www.loc.gov/catdir/enhancements/fy0663/2005275626-d.html; http://www.loc.gov/catdir/toc/fy0605/2005275626.html", abstract = "This book, dedicated to Mizan Rahman, is made up of a collection of articles on various aspects of q-series and special functions. It also includes an article by Askey, Ismail, and Koelink on Rahman's mathematical contributions and how they influenced the recent upsurge in the subject. Theory and Applications of Special Functions is intended for researchers and graduate students in special functions, algebraic combinatorics, quantum groups, and integrable systems.", acknowledgement = ack-nhfb, subject = "Mathematics; Special Functions; Functions, special; Integral Transforms; Approximations and Expansions; Integral Transforms, Operational Calculus", tableofcontents = "Mizan Rahman, his mathematics and literary writings / Richard Askey, Mourad E. H. Ismail and Erik Koelink \\ On the completeness of sets of $q$-Bessel function $J\nu^{(3)}(x; q)$ / L. D. Abreu and J. Bustoz \\ $\alpha$-Gaussian polynomials and finite Rogers-Ramanujan identities / George E. Andrews \\ On a generalized gamma convolution related to the $q$-calculus / Christian Berg \\ Ramanujan and cranks / Bruce C. Berndt, Heng Huat Chan, Song Heng Chan and Wen-Chin Liaw \\ Saalschutz chain reactions and multiple $q$-series transformations / Chu Wenchang \\ Painleve equations and associated polynomials / Peter A. Clarkson \\ Zeta functions of Heisenberg graphs over finite rings / Michelle DeDeo, Maria Martinez, Archie Medrano, Marvin Minei, Harold Stark and Audrey Terras \\ $q$-Analogues of some multivariable biorthogonal polynomials / George Gasper and Mizan Rahman \\ ^?? : Some systems of multivariable orthogonal Askey-Wilson polynomials / George Gasper and Mizan Rahman \\ Continuous Hahn functions as Clebsch--Gordan coefficients / Wolter Groenevelt, Erik Koelink and Hjalmar Rosengren \\ New proofs of some $q$-series results / Mourad E. H. Ismail and Ruiming Zhang \\ Little $q$-Jacobi functions of complex order / Kevin W. J. Kadell \\ Second addition formula for continuous $q$-ultraspherical polynomials / Tom H. Koornwinder \\ Bilateral series involving basic hypergeometric functions / Hjalmar Rosengren \\ Hilbert space asymptotics of a class of orthonormal polynomials on a bounded interval / S. N. M. Ruijsenaars \\ Abel--Rothe type generalizations of Jacobi's triple product identity / Michael Schlosser \\ Summable sums of hypergeometric series / D. Stanton \\ Askey--Wilson functions and quantum groups / Jasper V. Stokman \\ Analog of the Cauchy--Hadamard formula for expansions in $q$-polynomials /Remarks on some basic hypergeometric series / Changgui Zhang", } @Proceedings{Vassiliadis:2005:IIC, editor = "Stamatis Vassiliadis and Nikitas J. Dimopoulos and Sanjay Vishnu Rajopadhye", booktitle = "{16th IEEE International Conference on Application-Specific Systems, Architectures, and Processors: ASAP 2005: 23--25 July 2005, Samos, Greece}", title = "{16th IEEE International Conference on Application-Specific Systems, Architectures, and Processors: ASAP 2005: 23--25 July 2005, Samos, Greece}", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "xiii + 419", year = "2005", ISBN = "0-7695-2407-9", ISBN-13 = "978-0-7695-2407-8", LCCN = "TK7874.6 .I58 2005", bibdate = "Sun Mar 4 21:53:56 MST 2007", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; melvyl.cdlib.org:210/CDL90", acknowledgement = ack-nhfb, meetingname = "International Conference on Application-Specific Systems, Architectures, and Processors (16th: 2005: Samos, Greece)", subject = "Array processors; Congresses; Signal processing; Digital techniques; Application specific integrated circuits", } @Proceedings{zuCastell:2005:ILO, editor = "Wolfgang zu Castell and Frank Filbir and Brigitte Forster", booktitle = "{Inzell lectures on orthogonal polynomials}", title = "{Inzell lectures on orthogonal polynomials}", publisher = "Nova Science", address = "Hauppauge, NY, USA", pages = "x + 199", year = "2005", ISBN = "1-59454-108-6", ISBN-13 = "978-1-59454-108-7", LCCN = "QA404.5 .I595 2005", bibdate = "Sat Oct 30 17:16:08 MDT 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; prodorbis.library.yale.edu:7090/voyager", note = "Lectures from the Inzell Summer School on Orthogonal Polynomials, Harmonic Analysis, Approximation, and Applications, held in Inzell, Germany, September 17--21, 2001.", series = "Advances in the theory of special functions and orthogonal polynomials", acknowledgement = ack-nhfb, remark = "Canonical moments, orthogonal polynomials with applications to statistics / Holger Dette \\ Discrete commutative hypergroups / Rupert Lasser \\ Orthogonal polynomials and Banach algebras / Ryszard Szwarc \\ Lecture notes on orthogonal polynomials of several variables / Yuan Xu", subject = "Orthogonal polynomials", } @Proceedings{Anonymous:2006:PCR, editor = "Anonymous", booktitle = "{Proceedings of the 7th Conference on Real Numbers and Computers (RNC 7) LORIA, Nancy, France, July 10--12, 2006}", title = "{Proceedings of the 7th Conference on Real Numbers and Computers (RNC 7) LORIA, Nancy, France, July 10--12, 2006}", publisher = "????", address = "????", pages = "????", year = "2006", ISBN = "????", ISBN-13 = "????", LCCN = "????", bibdate = "Tue Jun 27 10:26:43 2006", bibsource = "http://rnc7.loria.fr/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, } @Proceedings{Menezes:2006:PAS, editor = "Ronaldo Menezes", booktitle = "{Proceedings of the 44th annual Southeast Regional Conference 2006: Melbourne, Florida, March 10--12, 2006}", title = "{Proceedings of the 44th annual Southeast Regional Conference 2006: Melbourne, Florida, March 10--12, 2006}", publisher = pub-ACM, address = pub-ACM:adr, pages = "823", year = "2006", ISBN = "1-59593-315-8 (print)", ISBN-13 = "978-1-59593-315-7 (print)", LCCN = "QA75.5 A184 2006 E", bibdate = "Sat Oct 9 15:04:24 MDT 2010", bibsource = "fsz3950.oclc.org:210/WorldCat; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, subject = "Computer-assisted instruction; Congresses; Database management; Electronic data processing", } @Proceedings{Brown:2007:PIS, editor = "C. W. Brown", booktitle = "{Proceedings of the 2007 International Symposium on Symbolic and Algebraic Computation, July 29--August 1, 2007, University of Waterloo, Waterloo, Ontario, Canada}", title = "{Proceedings of the 2007 International Symposium on Symbolic and Algebraic Computation, July 29--August 1, 2007, University of Waterloo, Waterloo, Ontario, Canada}", publisher = pub-ACM, address = pub-ACM:adr, pages = "????", year = "2007", ISBN = "1-59593-743-9 (print), 1-59593-742-0 (CD-ROM)", ISBN-13 = "978-1-59593-743-8 (print), 978-1-59593-742-1 (CD-ROM)", LCCN = "QA76.5 S98 2007", bibdate = "Fri Jun 20 08:53:37 2008", bibsource = "https://www.math.utah.edu/pub/tex/bib/axiom.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/maple-extract.bib", note = "ACM order number 505070.", acknowledgement = ack-nhfb, } @Proceedings{Holzapfel:2007:AGA, editor = "Rolf-Peter Holzapfel and A. Muhammed Uludag and Masaaki Yoshida", booktitle = "{Arithmetic and geometry around hypergeometric functions: lecture notes of a CIMPA Summer School held at Galatasaray University, Istanbul, Turkey, June 13--25, 2005}", title = "{Arithmetic and geometry around hypergeometric functions: lecture notes of a CIMPA Summer School held at Galatasaray University, Istanbul, Turkey, June 13--25, 2005}", volume = "235", publisher = pub-BIRKHAUSER, address = pub-BIRKHAUSER:adr, pages = "viii + 437", year = "2007", ISBN = "3-7643-8283-X", ISBN-13 = "978-3-7643-8283-4", LCCN = "QA245 .S86 2005", bibdate = "Sat Oct 30 21:12:24 MDT 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; z3950.loc.gov:7090/Voyager", series = "Progress in mathematics", URL = "http://www.loc.gov/catdir/enhancements/fy0913/2006939568-d.html; http://www.loc.gov/catdir/enhancements/fy0913/2006939568-t.html", acknowledgement = ack-nhfb, subject = "Algebra; Congresses.; Geometry, Algebraic; Congresses; Number theory", } @Proceedings{Iske:2007:AAP, editor = "Armin Iske and Jeremy Levesley", booktitle = "{Algorithms for Approximation: Proceedings of the 5th International Conference, Chester, July 2005}", title = "{Algorithms for Approximation: Proceedings of the 5th International Conference, Chester, July 2005}", publisher = pub-SV, address = pub-SV:adr, pages = "300", year = "2007", DOI = "https://doi.org/10.1007/978-3-540-46551-5", ISBN = "3-540-46551-0, 3-540-33283-9", ISBN-13 = "978-3-540-46551-5, 978-3-540-33283-1", LCCN = "QA221 .A44 2007", bibdate = "Thu Dec 1 09:41:19 MST 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; z3950.bibsys.no:2100/BIBSYS", acknowledgement = ack-nhfb, subject = "Mathematics; Special Functions; Functions, special; Engineering mathematics; Mathematics of Computing; Approximations and Expansions; Computer science; Computational Mathematics and Numerical Analysis; Appl. Mathematics / Computational Methods of Engineering", } @Proceedings{Kornerup:2007:PIS, editor = "Peter Kornerup and Jean-Michel Muller", booktitle = "{Proceedings of the 18th IEEE Symposium on Computer Arithmetic, June 25--27, 2007, Montpellier, France}", title = "{Proceedings of the 18th IEEE Symposium on Computer Arithmetic, June 25--27, 2007, Montpellier, France}", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "xii + 269", year = "2007", ISBN = "0-7695-2854-6", ISBN-13 = "978-0-7695-2854-0", ISSN = "1063-6889", LCCN = "QA76.9.C62", bibdate = "Tue Jun 27 10:26:43 2006", bibsource = "http://www.lirmm.fr/arith18/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; odin2.bib.sdu.dk:210/Horizon", URL = "http://www.lirmm.fr/arith18/", acknowledgement = ack-nhfb, keywords = "ARITH-18", } @Book{Cerone:2008:AIS, editor = "Pietro Cerone and Sever Silvestru Dragomir", booktitle = "Advances in Inequalities for Special Functions", title = "Advances in Inequalities for Special Functions", publisher = "Nova Science Publishers", address = "New York, NY, USA", pages = "170", year = "2008", ISBN = "1-60021-919-5 (hardcover), 1-60692-621-7 (e-book)", ISBN-13 = "978-1-60021-919-1 (hardcover), 978-1-60692-621-5 (e-book)", LCCN = "QA351 .A375 2008", bibdate = "Fri Oct 18 16:18:25 MDT 2024", bibsource = "fsz3950.oclc.org:210/WorldCat; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", series = "Advances in mathematical inequalities", URL = "http://catalog.hathitrust.org/api/volumes/oclc/159822357.html", acknowledgement = ack-nhfb, subject = "Functions, Special; Inequalities (Mathematics); Fonctions sp{\'e}ciales; In{\'e}galit{\'e}s (Math{\'e}matiques); MATHEMATICS; Calculus; Mathematical Analysis; Functions, Special; Inequalities (Mathematics)", tableofcontents = "Special functions approximations and bounds via integral representation / P. Cerone / 1 \\ Inequalities for positive Dirichlet series / P. Cerone and S. S. Dragomir / 37 \\ Monotonicity of the mean value function of normalized Bessel functions of first kind / Stamatis Koumandos / 67 \\ Sturm theory for some classes of Sturm--Liouville equations and inequalities and monotonicity properties for the zeros of Bessel functions / Andrea Laforgia and Pierpaolo Natalini / 73 \\ Inequalities for the Gamma function via convexity / Milan Merkle / 81 \\ Some inequalities for hyperharmonic series / Istv{\'a}n Mez{\H{o}} / 101 \\ The Hermite--Hadamard inequalities for double Dirichlet averages and their applications to special functions / Edward Neuman / 107 \\ On new inequalities involving convex functions / B. G. Pachpatte / 119 \\ On growth rates of Weierstrass $\wp'(z)$ and $\wp(z)$ / Tibor K. Pog{\'a}ny / 125 \\ On certain special functions of number theory and mathematical analysis / J{\'o}zsef S{\'a}ndor / 133 \\ On the operator $\oplus^k_B$ related to the Bessel-wave equation and Laplacian--Bessel / Mehmet Zeki Sarikaya and H{\"u}seyin Yildirim / 149 \\ Inequalities for Walsh polynomials with semi-monotone coefficients of higher order / {\v{Z}}ivorad Tomovski / 161 \\ Index / 169", } @Proceedings{Dominici:2008:SFO, editor = "Diego Dominici and Robert S. Maier", booktitle = "{Special functions and orthogonal polynomials: AMS Special Session on Special Functions and Orthogonal Polynomials, April 21--22, 2007, Tucson, Arizona}", title = "{Special functions and orthogonal polynomials: AMS Special Session on Special Functions and Orthogonal Polynomials, April 21--22, 2007, Tucson, Arizona}", volume = "471", publisher = pub-AMS, address = pub-AMS:adr, pages = "v + 218", year = "2008", ISBN = "0-8218-4650-7", ISBN-13 = "978-0-8218-4650-6", LCCN = "????", bibdate = "Sat Oct 30 17:30:10 MDT 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; z3950.bibsys.no:2100/BIBSYS", series = "Contemporary mathematics", acknowledgement = ack-nhfb, } @Proceedings{IEEE:2008:ICA, editor = "{IEEE}", booktitle = "{2008 International Conference on Application-Specific Systems, Architectures and Processors: Leuven, Belgium, 2--4 July 2008}", title = "{2008 International Conference on Application-Specific Systems, Architectures and Processors: Leuven, Belgium, 2--4 July 2008}", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "xiv + 309 + 12", year = "2008", ISBN = "1-4244-1897-6 (paperback), 1-4244-1898-4", ISBN-13 = "978-1-4244-1897-8 (paperback), 978-1-4244-1898-5", LCCN = "????", bibdate = "Mon Feb 10 07:31:38 MST 2020", bibsource = "fsz3950.oclc.org:210/WorldCat; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/prng.bib", note = "IEEE catalog number CFP08063-PRT.", URL = "http://www.gbv.de/dms/tib-ub-hannover/631855815.pdf; https://ieeexplore.ieee.org/servlet/opac?punumber=4569858", acknowledgement = ack-nhfb, remark = "Kongress auch zitiert als: ASAP 08. Parallel als Online-Ausg. erschienen. ASAP 08.", tableofcontents = "ASAP08 Conference proceedings / c1--c1 / doi: 10.1109/ASAP.2008.4580199 \\ ASAP08 Conference proceedings / c2--c2 / doi: 10.1109/ASAP.2008.4580200 \\ Frontmatter and table of contents / c1--xiii / doi: 10.1109/ASAP.2008.4580202 \\ ASAP Organizing and Steering Committees / ix \\ ASAP Technical Program Committee / x \\ Keynote 1: Security and Opportunities for Application-Specific Processors / Ruby B. Lee / xii \\ Keynote 2: Art of Application-Specific Processor Design: Great Artists use Good Tools / Gert Goossens / xiv \\ Session 1: Application-Specific Processor Instruction Sets / 1 \\ Copyright notice / i--i / doi: 10.1109/ASAP.2008.4580197 \\ Copyright notice / ii--ii / doi: 10.1109/ASAP.2008.4580198 \\ K. Atasu, O. Mencer, W. Luk, C. Ozturan and G. Dundar / Fast custom instruction identification by convex subgraph enumeration / 1--6 / doi: 10.1109/ASAP.2008.4580145 \\ Y. Hilewitz, C. Lauradoux and R. B. Lee / Bit matrix multiplication in commodity processors / 7--12 / doi: 10.1109/ASAP.2008.4580146 \\ M. Alle et al. / Synthesis of application accelerators on Runtime Reconfigurable Hardware / 13--18 / doi: 10.1109/ASAP.2008.4580147 \\ A. Amaricai, M. Vladutiu, M. Udrescu, L. Prodan and O. Boncalo / Floating point multiplication rounding schemes for interval arithmetic / 19--24 / doi: 10.1109/ASAP.2008.4580148 \\ S. Balasubramanian, H. W. Carter, A. Bogdanov, A. Rupp and Jintai Ding / Fast multivariate signature generation in hardware: The case of rainbow / 25--30 / doi: 10.1109/ASAP.2008.4580149 \\ M. Hosseinabady and J. Nunez-Yanez / Fault-tolerant dynamically reconfigurable NoC-based SoC / 31--36 / doi: 10.1109/ASAP.2008.4580150 \\ T. Lorunser et al. / Security Processor with Quantum Key Distribution / 37--42 / doi: 10.1109/ASAP.2008.4580151 \\ P. K. Meher and J. C. Patra / Fully-pipelined efficient architectures for FPGA realization of discrete Hadamard transform / 43--48 / doi: 10.1109/ASAP.2008.4580152 \\ R. Rajore, G. Garga, H. S. Jamadagni and S. K. Nandy / Reconfigurable Viterbi decoder on mesh connected multiprocessor architecture / 49--54 / doi: 10.1109/ASAP.2008.4580153 \\ T. Ramdas, G. K. Egan, D. Abramson and K. K. Baldridge / Run-time thread sorting to expose data-level parallelism / 55--60 / doi: 10.1109/ASAP.2008.4580154 \\ S. Jovanovic, C. Tanougast and S. Weber / A New High-Performance Scalable Dynamic Interconnection for FPGA-based Reconfigurable Systems / 61--66 / doi: 10.1109/ASAP.2008.4580155 \\ D. Dickin and L. Shannon / Extending the SIMPPL SoC architectural framework to support application-specific architectures on multi-FPGA platforms / 67--72 / doi: 10.1109/ASAP.2008.4580156 \\ A. E. Kiasari, S. Hessabi and H. Sarbazi-Azad / PERMAP: A performance-aware mapping for application-specific SoCs / 73--78 / doi: 10.1109/ASAP.2008.4580157 \\ A. C. Atici, L. Batina, Junfeng Fan, I. Verbauwhede and S. Berna Ors Yalcin / Low-cost implementations of NTRU for pervasive security / 79--84 / doi: 10.1109/ASAP.2008.4580158 \\ M. Knezzevic, K. Sakiyama, Y. K. Lee and I. Verbauwhede / On the high-throughput implementation of RIPEMD-160 hash algorithm / 85--90 / doi: 10.1109/ASAP.2008.4580159 \\ Wang Haixin, Bai Guoqiang and Chen Hongyi / Zodiac: System architecture implementation for a high-performance Network Security Processor / 91--96 / doi: 10.1109/ASAP.2008.4580160 \\ P. K. Meher / Efficient systolization of cyclic convolution for systolic implementation of sinusoidal transforms / 97--101 / doi: 10.1109/ASAP.2008.4580161 \\ D. B. Thomas and W. Luk / Resource efficient generators for the floating-point uniform and exponential distributions / 102--107 / doi: 10.1109/ASAP.2008.4580162 \\ I. L. Dalal, D. Stefan and J. Harwayne-Gidansky / Low discrepancy sequences for Monte Carlo simulations on reconfigurable platforms / 108--113 / doi: 10.1109/ASAP.2008.4580163 \\ Y. Vanderperren and W. Dehaene / A subsampling pulsed UWB demodulator based on a flexible complex SVD / 114--119 / doi: 10.1109/ASAP.2008.4580164 \\ J. Divyasree, H. Rajashekar and K. Varghese / Dynamically reconfigurable regular expression matching architecture / 120--125 / doi: 10.1109/ASAP.2008.4580165 \\ J. Khan, S. Niar, A. Menhaj, Y. Elhillali and J. L. Dekeyser / An MPSoC architecture for the Multiple Target Tracking application in driver assistant system / 126--131 / doi: 10.1109/ASAP.2008.4580166 \\ Wangyuan Zhang and Tao Li / Managing multi-core soft-error reliability through utility-driven cross domain optimization / 132--137 / doi: 10.1109/ASAP.2008.4580167 \\ S. Braganza and M. Leeser / An efficient implementation of a phase unwrapping kernel on reconfigurable hardware / 138--143 / doi: 10.1109/ASAP.2008.4580168 \\ H. Flatt, S. Blume, S. Hesselbarth, T. Schunemann and P. Pirsch / A parallel hardware architecture for connected component labeling based on fast label merging / 144--149 / doi: 10.1109/ASAP.2008.4580169 \\ Yuki Kobayashi, M. Jayapala, P. Raghavan, F. Catthoor and Masaharu Imai / Operation shuffling over cycle boundaries for low energy L0 clustering / 150--155 / doi: 10.1109/ASAP.2008.4580170 \\ V. Kundeti, Yunsi Fei and S. Rajasekaran / An efficient digital circuit for implementing Sequence Alignment algorithm in an extended processor / 156--161 / doi: 10.1109/ASAP.2008.4580171 \\ B. K. Mohanty and P. K. Meher / Concurrent systolic architecture for high-throughput implementation of 3-dimensional discrete wavelet transform / 162--166 / doi: 10.1109/ASAP.2008.4580172 \\ S. Mirzaei, A. Irturk, R. Kastner, B. T. Weals and R. E. Cagley / Design space exploration of a cooperative MIMO receiver for reconfigurable architectures / 167--172 / doi: 10.1109/ASAP.2008.4580173 \\ Mao Nakajima and Minoru Watanabe / Dynamic holographic reconfiguration on a four-context ODRGA / 173--178 / doi: 10.1109/ASAP.2008.4580174 \\ F. Pardo, P. Lopez and D. Cabello / FPGA-based hardware accelerator of the heat equation with applications on infrared thermography / 179--184 / doi: 10.1109/ASAP.2008.4580175 \\ M. Rahmati, M. S. Sadri and M. A. Naeini / FPGA based singular value decomposition for image processing applications / 185--190 / doi: 10.1109/ASAP.2008.4580176 \\ A. Jacob, J. Buhler and R. D. Chamberlain / Accelerating Nussinov RNA secondary structure prediction with systolic arrays on FPGAs / 191--196 / doi: 10.1109/ASAP.2008.4580177 \\ J. Lee, L. Shannon, M. J. Yedlin and G. F. Margrave / A multi-FPGA application-specific architecture for accelerating a floating point Fourier Integral Operator / 197--202 / doi: 10.1109/ASAP.2008.4580178 \\ K. F. C. Yiu, Chun Hok Ho, N. Grbric, Yao Lu, Xiaoxiang Shi and W. Luk / Reconfigurable acceleration of microphone array algorithms for speech enhancement / 203--208 / doi: 10.1109/ASAP.2008.4580179 \\ Yang Sun, Yuming Zhu, M. Goel and J. R. Cavallaro / Configurable and scalable high throughput turbo decoder architecture for multiple 4G wireless standards / 209--214 / doi: 10.1109/ASAP.2008.4580180 \\ M. B. S. Tavares, S. Kunze, E. Matus and G. P. Fettweis / Architecture and VLSI realization of a high-speed programmable decoder for LDPC convolutional codes / 215--220 / doi: 10.1109/ASAP.2008.4580181 \\ D. Llorente, K. Karras, T. Wild and A. Herkersdorf / Buffer allocation for advanced packet segmentation in Network Processors / 221--226 / doi: 10.1109/ASAP.2008.4580182 \\ A. Vazquez and E. Antelo / New insights on Ling adders / 227--232 / doi: 10.1109/ASAP.2008.4580183 \\ N. Brisebarre, F. de Dinechin and J. Muller / Integer and floating-point constant multipliers for FPGAs / 239--244 / doi: 10.1109/ASAP.2008.4580184 \\ N. Brisebarre, S. Chevillard, M. D. Ercegovac, J. Muller and S. Torres / An efficient method for evaluating polynomial and rational function approximations / 233--238 / doi: 10.1109/ASAP.2008.4580185 \\ A. Garcia, M. Berekovic and T. Vander Aa / Mapping of the AES cryptographic algorithm on a Coarse-Grain reconfigurable array processor / 245--250 / doi: 10.1109/ASAP.2008.4580186 \\ J. Nimmy et al. / RECONNECT: A NoC for polymorphic ASICs using a low overhead single cycle router / 251--256 / doi: 10.1109/ASAP.2008.4580187 \\ M. Mbaye, N. Belanger, Y. Savaria and S. Pierre / Loop-oriented metrics for exploring an application-specific architecture design-space / 257--262 / doi: 10.1109/ASAP.2008.4580188 \\ S. K. Dash and T. Srikanthan / Rapid estimation of instruction cache hit rates using loop profiling / 263--268 / doi: 10.1109/ASAP.2008.4580189 \\ Xuan Guan and Yunsi Fei / Reducing power consumption of embedded processors through register file partitioning and compiler support / 269--274 / doi: 10.1109/ASAP.2008.4580190 \\ A. Tumeo, M. Monchiero, G. Palermo, F. Ferrandi and D. Sciuto / Lightweight DMA management mechanisms for multiprocessors on FPGA / 275--280 / doi: 10.1109/ASAP.2008.4580191 \\ P. de Langen and B. Juurlink / Memory copies in multi-level memory systems / 281--286 / doi: 10.1109/ASAP.2008.4580192 \\ R. Adrsha, Mythri, S. K. Nandy and R. Narayan / Architecture of a polymorphic ASIC for interoperability across multi-mode H.264 decoders / 287--292 / doi: 10.1109/ASAP.2008.4580193 \\ R. R. Osorio and J. D. Bruguera / An FPGA architecture for CABAC decoding in manycore systems / 293--298 / doi: 10.1109/ASAP.2008.4580194 \\ A. Guntoro and M. Glesner / Novel approach on lifting-based DWT and IDWT processor with multi-context configuration to support different wavelet filters / 299--304 / doi: 10.1109/ASAP.2008.4580195 \\ B. K. Mohanty and P. K. Meher / Throughput-scalable hybrid-pipeline architecture for multilevel lifting 2-D DWT of JPEG 2000 coder / 305--309 / doi: 10.1109/ASAP.2008.4580196 \\ Author index / 310--321 / doi: 10.1109/ASAP.2008.4580201", } @Proceedings{Bruguera:2009:PIS, editor = "Javier D. Bruguera and Marius Cornea and Debjit DasSarma and John Harrison", booktitle = "{Proceedings of the 19th IEEE Symposium on Computer Arithmetic, June 8--10, 2009, Portland, Oregon, USA}", title = "{Proceedings of the 19th IEEE Symposium on Computer Arithmetic, June 8--10, 2009, Portland, Oregon, USA}", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "xi + 235", year = "2009", ISBN = "0-7695-3670-0, 1-4244-4329-6", ISBN-13 = "978-0-7695-3670-5, 978-1-4244-4329-1", ISSN = "1063-6889", LCCN = "QA76.6 .S887 2009", bibdate = "Fri Jun 12 12:24:37 2009", bibsource = "http://www.ac.usc.es/arith19/; https://www.math.utah.edu/pub/tex/bib/cryptography2000.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/master.bib; https://www.math.utah.edu/pub/tex/bib/mathcw.bib", URL = "http://www.ac.usc.es/arith19/", acknowledgement = ack-nhfb, keywords = "ARITH-19", tableofcontents = "Keynote Talk \\ Anton: A Specialized Machine for Millisecond-Scale Molecular Dynamics Simulations of Proteins / David E. Shaw / 3 \\ Session 1: Algorithms and Number Systems \\ Efficient Data Structure and Algorithms for Sparse Integers, Sets and Predicates / Jean E. Vuillemin / 7 \\ A Dual-Purpose Real/Complex Logarithmic Number System ALU / Mark G. Arnold and Sylvain Collange / 15 \\ Selected RNS Bases for Modular Multiplication / J. C. Bajard, M. Kaihara, and T. Plantard / 25 \\ Invited Talk \\ A Historical Perspective on Computer Arithmetic / Stanley Mazor / 35 \\ Session 2: Arithmetic Hardware \\ Higher Radix Squaring Operations Employing Left-to-Right Dual Recoding / David W. Matula / 39 \\ Advanced Clockgating Schemes for Fused-Multiply-Add-Type Floating-Point Units / Jochen Preiss, Maarten Boersma, and Silvia Melitta Mueller / 48 \\ Unified Approach to the Design of Modulo-$(2^n \pm 1)$ Adders Based on Signed-LSB Representation of Residues / Ghassem Jaberipur and Behrooz Parhami / 57 \\ Session 3: Finite Fields and Cryptography \\ Subquadratic Space Complexity Multiplier for a Class of Binary Fields Using Toeplitz Matrix Approach / M. A. Hasan and C. Negre / 67 \\ Hybrid Binary-Ternary Joint Form and Its Application in Elliptic Curve / Cryptography / Jithra Adikari, Vassil Dimitrov, and Laurent Imbert / 76 \\ Polynomial Multiplication over Finite Fields Using Field Extensions and Interpolation / Murat Cenk, Cetin Kaya Koc, and Ferruh Ozbudak / 84 \\ Session 4: Mathematical Software \\ A New Binary Floating-Point Division Algorithm and Its Software Implementation on the ST231 Processor / Claude-Pierre Jeannerod, Herve Knochel, Christophe Monat, Guillaume Revy, and Gilles Villard / 95 \\ Fast and Accurate Bessel Function Computation / John Harrison / 104 \\ Implementation Specific Verification of Divide and Square Root Instructions / Elena Guralnik, Ariel J. Birnbaum, Anatoly Koyfinan, and Avi Kaplan / 114 \\ Session 5: Decimal Hardware \\ A Decimal Floating-Point Adder with Decoded Operands and a Decimal Leading-Zero Anticipator / Liang-Kai Wang and Michael J. Schulte / 125 \\ A High-Performance Significand BCD Adder with IEEE 754-2008 Decimal Rounding / Alvaro Vazquez and Elisardo Antelo / 135 \\ Fully Redundant Decimal Arithmetic / Saeid Gorgin and Ghassem Jaberipur / 145 \\ Session 6: Floating-Point Techniques \\ On the Computation of Correctly-Rounded Sums / P. Kornerup, V. Lefevre, N. Louvet, and J. M. Muller / 155 \\ Multi-operand Floating-Point Addition / Alexandre F. Tenca / 161 \\ Certified and Fast Computation of Supremum Norms of Approximation Errors / Sylvain Chevillard, Mioara Jolde{\c{s}}, and Christoph Lauter / 169 \\ Session 7: Decimal Transcendentals \\ Computation of Decimal Transcendental Functions Using the CORDIC Algorithm / {\'A}lvaro V{\'a}zquez, Julio Villalba, and Elisardo Antelo / 179 \\ Decimal Transcendentals via Binary / John Harrison / 187 \\ A 32-bit Decimal Floating-Point Logarithmic Converter / Dongdong Chen, Yu Zhang, Younhee Choi, Moon Ho Lee, and Seok-Bum Ko / 195 \\ Special Session on Automated Synthesis of Arithmetic Operations \\ Datapath Synthesis for Standard-Cell Design / Reto Zimmermann / 207 \\ Design Space Exploration for Power-Efficient Mixed-Radix Ling Adders / Chung-Kuan Cheng / 212 \\ Challenges in Automatic Optimization of Arithmetic Circuits / Ajay K. Verma, Philip Brisk, and Paolo Ienne / 213 \\ Panel on Decimal Arithmetic in Industry \\ Energy and Delay Improvement via Decimal Floating Point Units / Hossam A. H. Fahmy, Ramy Raafat, Amira M. Abdel-Majeed, Rodina Samy, Torek ElDeeb, and Yasmin Farouk / 221 \\ IEEE 754-2008 Decimal Floating-Point for Intel Architecture Processors / Marius Cornea / 225 \\ Special Session on Interval Arithmetic \\ IEEE Interval Standard Working Group --- P1788: Current Status / William Edmonson and Guillaume Melquiond / 231 \\ Author Index", } @Proceedings{Fukuda:2010:MSI, editor = "Komei Fukuda and Joris van der Hoeven and Michael Joswig and Nobuki Takayama", booktitle = "{Mathematical software --- ICMS 2010: third International Congress on Mathematical Software, K{\=o}be, Japan, September 13--17, 2010: proceedings}", title = "{Mathematical software --- ICMS 2010: third International Congress on Mathematical Software, K{\=o}be, Japan, September 13--17, 2010: proceedings}", volume = "6327", publisher = pub-SV, address = pub-SV:adr, pages = "xvi + 368", year = "2010", DOI = "https://doi.org/10.1007/978-3-642-15582-6", ISBN = "3-642-15581-2 (paperback), 3-642-15582-0 (e-book)", ISBN-13 = "978-3-642-15581-9 (paperback), 978-3-642-15582-6 (e-book)", LCCN = "QA76.95 .I5654 2010", bibdate = "Sat Aug 9 14:06:27 MDT 2014", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/lncs.bib; https://www.math.utah.edu/pub/tex/bib/lncs2010a.bib; https://www.math.utah.edu/pub/tex/bib/magma.bib; https://www.math.utah.edu/pub/tex/bib/maple-extract.bib; https://www.math.utah.edu/pub/tex/bib/mathematica.bib; https://www.math.utah.edu/pub/tex/bib/matlab.bib; z3950.loc.gov:7090/Voyager", series = ser-LNCS, URL = "http://link.springer.com/book/10.1007/978-3-642-15582-6", acknowledgement = ack-nhfb, subject = "Mathematics; Data processing; Congresses; Computer software", tableofcontents = "Plenary \\ Computational Discrete Geometry / Thomas C. Hales / 1--3 \\ Exploiting Structured Sparsity in Large Scale Semidefinite Programming Problems / Masakazu Kojima / 4--9 \\ Reliable and Efficient Geometric Computing / Kurt Mehlhorn / 10--11 \\ The Sage Project: Unifying Free Mathematical Software to Create a Viable Alternative to Magma, Maple, Mathematica and MATLAB / Bur{\c{c}}in Er{\"o}cal, William Stein / 12--27 \\ Computation of Special Functions (Invited) \\ Sollya: An Environment for the Development of Numerical Codes / Sylvain Chevillard, Mioara Jolde , Christoph Lauter / 28--31 \\ Validated Special Functions Software / Annie Cuyt, Franky Backeljauw, Stefan Becuwe, Joris Van Deun / 32--34 \\ The Dynamic Dictionary of Mathematical Functions (DDMF) / Alexandre Benoit, Fr{\'e}d{\'e}ric Chyzak, Alexis Darrasse, Stefan Gerhold, Marc Mezzarobba, Bruno Salvy / 35--41 \\ Reliable Computing with GNU MPFR / Paul Zimmermann / 42--45 \\ Computational Group Theory (Invited) \\ Simplicial Cohomology of Smooth Orbifolds in GAP / Mohamed Barakat, Simon G{\"o}rtzen / 46--49 \\ Computing Polycyclic Quotients of Finitely (L-)Presented Groups via Groebner Bases / Bettina Eick, Max Horn / 50--53 \\ Constructive Membership Testing in Black-Box Classical Groups / Sophie Ambrose, Scott H. Murray, Cheryl E. Praeger, Csaba Schneider / 54--57 \\ Computational Group Theory (Contributed) \\ Towards High-Performance Computational Algebra with GAP / Reimer Behrends, Alexander Konovalov, Steve Linton, Frank L{\"u}beck, Max Neunh{\"o}effer / 58--61 \\ An Improvement of a Function Computing Normalizers for Permutation Groups / Izumi Miyamoto / 62--68 \\ A GAP Package for Computation with Coherent Configurations / Dmitrii V. Pasechnik, Keshav Kini / 69--72 \\ Computer Algebra (Invited) \\ CoCoALib: A C++ Library for Computations in Commutative Algebra \ldots{} and Beyond / John Abbott, Anna M. Bigatti / 73--76 \\ LinBox Founding Scope Allocation, Parallel Building Blocks, and Separate Compilation / Jean-Guillaume Dumas, Thierry Gautier, Cl{\'e}ment Pernet, B. David Saunders / 77--83 \\ FGb: A Library for Computing Gr{\"o}bner Bases / Jean-Charles Faug{\`e}re / 84--87 \\ Fast Library for Number Theory: An Introduction / William B. Hart / 88--91 \\ Exact Numeric Computation for Algebraic and Geometric Computation (Invited) \\ Controlled Perturbation for Certified Geometric Computing with Fixed-Precision Arithmetic / Dan Halperin / 92--95 \\ Exact Numeric Computation for Algebraic and Geometric Computation (Invited) \\ Exact Geometric and Algebraic Computations in CGAL / Menelaos I. Karavelas / 96--99 \\ On Solving Systems of Bivariate Polynomials / Fabrice Rouillier / 100--104 \\ Accurate and Reliable Computing in Floating-Point Arithmetic / Siegfried M. Rump / 105--108 \\ Exact Numeric Computation for Algebraic and Geometric Computation (Contributed) \\ Deferring Dag Construction by Storing Sums of Floats Speeds-Up Exact Decision Computations Based on Expression Dags / Marc M{\"o}rig / 109--120 \\ The Design of Core 2: A Library for Exact Numeric Computation in Geometry and Algebra / Jihun Yu, Chee Yap, Zilin Du, Sylvain Pion, Herv{\'e} Br{\"o}nnimann / 121--141 \\ Formal Proof (Invited) \\ Introducing HOL Zero / Mark Adams / 142--143 \\ Euler s Polyhedron Formula in mizar / Jesse Alama / 144--147 \\ Building a Library of Mechanized Mathematical Proofs: Why Do It? and What Is It Like to Do? / R. D. Arthan / 148--148 \\ Linear Programs for the Kepler Conjecture / Thomas C. Hales / 149--151 \\ A Formal Proof of Pick s Theorem / John Harrison / 152--154 \\ Formal Proof (Contributed) \\ Evaluation of Automated Theorem Proving on the Mizar Mathematical Library / Josef Urban, Krystof Hoder, Andrei Voronkov / 155--166 \\ Geometry and Visualization (Invited) \\ On Local Deformations of Planar Quad-Meshes / Tim Hoffmann / 167--169 \\ Construction of Harmonic Surfaces with Prescribed Geometry / Matthias Weber / 170--173 \\ Geometry and Visualization (Contributed) \\ A Library of OpenGL-Based Mathematical Image Filters / Martin von Gagern, Christian Mercat / 174--185 \\ MD-jeep: An Implementation of a Branch and Prune Algorithm for Distance Geometry Problems / Antonio Mucherino, Leo Liberti, Carlile Lavor / 186--197 \\ TADD: A Computational Framework for Data Analysis Using Discrete Morse Theory / Jan Reininghaus, David G{\"u}nther, Ingrid Hotz, Steffen Prohaska, Hans-Christian Hege / 198--208 \\ Groebner Bases and Applications (Invited) \\ Introduction to Normaliz 2.5 / Winfried Bruns, Bogdan Ichim, Christof S{\"o}ger / 209--212 \\ Computer Algebra Methods in Tropical Geometry / Thomas Markwig / 213--216 \\ Groebner Bases and Applications (Contributed) \\ A New Desingularization Algorithm for Binomial Varieties in Arbitrary Characteristic / Roc{\'\i}o Blanco / 217--220 \\ An Algorithm of Computing Inhomogeneous Differential Equations for Definite Integrals / Hiromasa Nakayama, Kenta Nishiyama / 221--232 \\ Groebner Bases and Applications (Contributed) \\ New Algorithms for Computing Primary Decomposition of Polynomial Ideals / Masayuki Noro / 233--244 \\ An Automated Confluence Proof for an Infinite Rewrite System Parametrized over an Integro-Differential Algebra / Loredana Tec, Georg Regensburger, Markus Rosenkranz, Bruno Buchberger / 245--248 \\ Operadic Gr{\"o}bner Bases: An Implementation / Vladimir Dotsenko, Mikael Vejdemo-Johansson / 249--252 \\ Number Theoretical Software (Invited) \\ Magma - A Tool for Number Theory / John Cannon, Steve Donnelly, Claus Fieker, Mark Watkins / 253--255 \\ Number Theoretical Software (Contributed) \\ Enumerating Galois Representations in Sage / Craig Citro, Alexandru Ghitza / 256--259 \\ NZMATH 1.0 / Satoru Tanaka, Naoki Ogura, Ken Nakamula, Tetsushi Matsui, Shigenori Uchiyama / 260--269 \\ Software for Optimization and Polyhedral Computation (Invited) \\ Removing Redundant Quadratic Constraints / David Adjiashvili, Michel Baes, Philipp Rostalski / 270--281 \\ Traversing Symmetric Polyhedral Fans / Anders Nedergaard Jensen / 282--294 \\ C++ Tools for Exploiting Polyhedral Symmetries / Thomas Rehn, Achill Sch{\"u}rmann / 295--298 \\ isl: An Integer Set Library for the Polyhedral Model / Sven Verdoolaege / 299--302 \\ Software for Optimization and Polyhedral Computation (Contributed) \\ The Reformulation-Optimization Software Engine / Leo Liberti, Sonia Cafieri, David Savourey / 303--314 \\ Generating Smooth Lattice Polytopes / Christian Haase, Benjamin Lorenz, Andreas Paffenholz / 315--328 \\ Reliable Computation (Invited) \\ Mathemagix: Towards Large Scale Programming for Symbolic and Certified Numeric Computations / Gr{\'e}goire Lecerf / 329--332 \\ Complex Inclusion Functions in the CoStLy C++ Class Library / Markus Neher / 333--336 \\ Standardized Interval Arithmetic and Interval Arithmetic Used in Libraries / Nathalie Revol / 337--341 \\ Reliable Computation (Contributed) \\ Efficient Evaluation of Large Polynomials / Charles E. Leiserson, Liyun Li, Marc Moreno Maza, Yuzhen Xie / 342--353 \\ Communicating Functional Expressions from Mathematica to C-XSC / Evgenija D. Popova, Walter Kr{\"a}mer / 354--365 \\ Author Index / 367--368", } @Book{Olver:2010:NHM, editor = "Frank W. J. Olver and Daniel W. Lozier and Ronald F. Boisvert and Charles W. Clark", key = "NIST", booktitle = "{NIST} Handbook of Mathematical Functions", title = "{NIST} Handbook of Mathematical Functions", publisher = pub-CAMBRIDGE, address = pub-CAMBRIDGE:adr, pages = "xv + 951", year = "2010", ISBN = "0-521-19225-0", ISBN-13 = "978-0-521-19225-5", LCCN = "QA331 .N57 2010", bibdate = "Sat May 15 09:08:09 2010", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/l/lanczos-cornelius.bib; https://www.math.utah.edu/pub/bibnet/authors/w/wigner-eugene.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/master.bib; https://www.math.utah.edu/pub/tex/bib/mathcw.bib", price = "US\$99.00", URL = "http://dlmf.nist.gov/; http://www.cambridge.org/9780521140638", acknowledgement = ack-nhfb, remark = "Includes a DVD with a searchable PDF of each chapter.", tableofcontents = "1. Algebraic and analytic methods [Ranjan Roy, Frank W. J. Olver, Richard A. Askey and Roderick S. C. Wong] \\ 2. Asymptotic approximations [Frank W. J. Olver and Roderick S. C. Wong] \\ 3. Numerical methods [Nico M. Temme] \\ 4. Elementary functions [Ranjan Roy and Frank W. J. Olver] \\ 5. Gamma function [Richard A. Askey and Ranjan Roy] \\ 6. Exponential, logarithmic, sine and cosine integrals [Nico M. Temme] \\ 7. Error functions, Dawson's and Fresnel integrals [Nico M. Temme] \\ 8. Incomplete gamma and related functions [Richard B. Paris] \\ 9. Airy and related functions [Frank W. J. Olver] \\ 10. Bessel functions [Frank W. J. Olver and Leonard C. Maximon] \\ 11. Struve and related functions [Richard B. Paris] \\ 12. Parabolic cylinder functions [Nico M. Temme] \\ 13. Confluent hypergeometric functions [Adri B. Olde Daalhuis] \\ 14. Legendre and related functions [T. Mark Dunster] \\ 15. Hypergeometric function [Adri B. Olde Daalhuis] \\ 16. Generalized hypergeometric functions and Meijer G-function [Richard A. Askey and Adri B. Olde Daalhuis] \\ 17. q-Hypergeometric and related functions [George E. Andrews] \\ 18. Orthogonal polynomials [Tom H. Koornwinder, Roderick S. C. Wong, Roelof Koekoek and Rene F. Swarttouw] \\ 19. Elliptic integrals [Bille C. Carlson] \\ 20. Theta functions [William P. Reinhardt and Peter L. Walker] \\ 21. Multidimensional theta functions [Bernard Deconinck] \\ 22. Jacobian elliptic functions [William P. Reinhardt and Peter L. Walker] \\ 23. Weierstrass elliptic and modular functions [William P. Reinhardt and Peter L. Walker] \\ 24. Bernoulli and Euler polynomials [Karl Dilcher] \\ 25. Zeta and related functions [Tom M. Apostol] \\ 26. Combinatorial analysis [David M. Bressoud] \\ 27. Functions of number theory [Tom M. Apostol] \\ 28. Mathieu functions and Hill's equation [Gerhard Wolf] \\ 29. Lam{\'e} functions [Hans Volkmer] \\ 30. Spheroidal wave functions [Hans Volkmer] \\ 31. Heun functions [Brian D. Sleeman and Vadim Kuznetsov] \\ 32. Painlev{\'e} transcendents [Peter A. Clarkson] \\ 33. Coulomb functions [Ian J. Thompson] \\ 34. 3j, 6j, 9j symbols [Leonard C. Maximon] \\ 35. Functions of matrix argument [Donald St. P. Richards] \\ 36. Integrals with coalescing saddles [Michael V. Berry and Chris Howls]", } @Book{Polyanin:2011:CHM, editor = "A. D. (Andrei Dmitrievich) Polyanin and A. I. Chernoutisan", booktitle = "A Concise Handbook of Mathematics, Physics, and Engineering Sciences", title = "A Concise Handbook of Mathematics, Physics, and Engineering Sciences", publisher = pub-CHAPMAN-HALL-CRC, address = pub-CHAPMAN-HALL-CRC:adr, pages = "xxviii + 1097", year = "2011", DOI = "https://doi.org/10.1201/b10276", ISBN = "0-429-13137-2, 1-282-90264-4, 1-4398-0639-X (hardcover), 1-4398-0640-3 (PDF)", ISBN-13 = "978-0-429-13137-0, 978-1-282-90264-0, 978-1-4398-0639-5 (hardcover), 978-1-4398-0640-1 (PDF)", LCCN = "QA40 .C65 2010", bibdate = "Wed Jun 12 15:24:00 MDT 2024", bibsource = "fsz3950.oclc.org:210/WorldCat; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", abstract = "\booktitle{A Concise Handbook of Mathematics, Physics, and Engineering Sciences} takes a practical approach to the basic notions, formulas, equations, problems, theorems, methods, and laws that most frequently occur in scientific and engineering applications and university education. The authors pay special attention to issues that many engineers and students find difficult to understand. The first part of the book contains chapters on arithmetic, elementary and analytic geometry, algebra, differential and integral calculus, functions of complex variables, integral transforms, ordinary and partial differential equations. \ldots{} .", acknowledgement = ack-nhfb, remark = "Chapter authors: A. I. Chernoutisan, A. V. Egorov, A. V. Manzhirov, A. D. Polyanin, V. D. Polyanin, V. A. Popov, B. V. Putyatin, Yu. V. Repina, V. M. Safrai, A. I. Zhurov.", subject = "Mathematics; Handbooks, manuals, etc; Physics; Engineering; Math{\'e}matiques; Guides, manuels, etc; Physique; Ing{\'e}nierie; Engineering; Mathematics; Physics", tableofcontents = "Preface \\ Editors \\ \\ Part I: Mathematics \\ M1: Arithmetic and Elementary Algebra \\ M2: Elementary Functions \\ M3: Elementary Geometry \\ M4: Analytic Geometry \\ M5: Algebra \\ M6: Limits and Derivatives \\ M7: Integrals \\ M8: Series \\ M9: Functions of Complex Variable \\ M10: Integral Transforms \\ M11: Ordinary Differential Equations \\ M12: Partial Differential Equations \\ M13: Special Functions and Their Properties \\ M14: Probability Theory \\ \\ Part II: Physics \\ P1: Physical Foundations of Mechanics \\ P2: Molecular Physics and Thermodynamics \\ P3: Electrodynamics \\ P4: Oscillations and Waves \\ P5: Optics \\ P6: Quantum Mechanics. Atomic Physics \\ P7: Quantum Theory of Crystals \\ P8: Elements of Nuclear Physics \\ \\ Part III: Elements of Applied and Engineering Sciences \\ E1: Dimensions and Similarity \\ E2: Mechanics of Point Particles and Rigid Bodies \\ E3: Elements of Strength of Materials \\ E4: Hydrodynamics \\ E5: Mass and Heat Transfer \\ E6: Electrical Engineering \\ E7: Empirical and Engineering Formulas and Criteria for Their Applicability \\ \\ Part IV: Supplements \\ S1: Integrals \\ S2: Integral Transforms \\ S3: Orthogonal Curvilinear Systems of Coordinates \\ S4: Ordinary Differential Equations \\ S5: Some Useful Electronic Mathematical Resources \\ S6: Physical Tables \\ S7: Periodic Table \\ Index", } @Proceedings{Schost:2011:IPI, editor = "{\'E}ric Schost and Ioannis Z. Emiris", booktitle = "{ISSAC 2011: Proceedings of the 2011 International Symposium on Symbolic and Algebraic Computation, June 7--11, 2011, San Jose, CA, USA}", title = "{ISSAC 2011: Proceedings of the 2011 International Symposium on Symbolic and Algebraic Computation, June 7--11, 2011, San Jose, CA, USA}", publisher = pub-ACM, address = pub-ACM:adr, pages = "362 (est.)", year = "2011", ISBN = "1-4503-0675-6", ISBN-13 = "978-1-4503-0675-1", LCCN = "QA76.95 .I59 2011", bibdate = "Fri Mar 14 12:24:11 2014", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/issac.bib; https://www.math.utah.edu/pub/tex/bib/maple-extract.bib; https://www.math.utah.edu/pub/tex/bib/mathematica.bib", acknowledgement = ack-nhfb, } @Proceedings{Schwarz:2011:PIS, editor = "Eric Schwarz and Vojin G. Oklobdzija", booktitle = "{Proceedings of the 20th IEEE Symposium on Computer Arithmetic, July 25--27, 2011, T{\"u}bingen, Germany}", title = "{Proceedings of the 20th IEEE Symposium on Computer Arithmetic, July 25--27, 2011, T{\"u}bingen, Germany}", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "xix + 253", year = "2011", ISBN = "0-7695-4318-9, 1-4244-9457-5", ISBN-13 = "978-0-7695-4318-5, 978-1-4244-9457-6", LCCN = "QA76.6", bibdate = "Sat Aug 20 09:19:17 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/master.bib; https://www.math.utah.edu/pub/tex/bib/mathcw.bib", acknowledgement = ack-nhfb, keywords = "ARITH-20", tableofcontents = "Foreword / ix \\ Dedication / x \\ Steering Committee / xv \\ Symposium Committee / xvi \\ Program Committee / xvii \\ Additional Reviewers / xviii \\ Corporate Sponsors / xix \\ Session 1: Keynote Talk: Chair: Eric Schwarz and Vojin G. Oklobdzija \\ High Intelligence Computing: The New Era of High Performance Computing / Ralf Fischer / 3 \\ Session 2: Multiple-Precision Algorithms: Chair: Marius Cornea \\ Short Division of Long Integers / David Harvey and Paul Zimmermann / 7 \\ High Degree Toom'n'Half for Balanced and Unbalanced Multiplication / Marco Bodrato / 15 \\ Augmented Precision Square Roots and 2-D Norms, and Discussion on Correctly Rounding sqrt($x^2 + y^2$) / Nicolas Brisebarre, Mioara Jolde{\c{s}}, Peter Kornerup, Erik Martin-Dorel, and Jean-Michel Muller / 23 \\ Session 3: Transcendental Methods: Chair: Naofumi Takagi \\ Towards a Quaternion Complex Logarithmic Number System / Mark G. Arnold, John Cowles, Vassilis Paliouras, and Ioannis Kouretas / 33 \\ ROM-less LNS / R. Che Ismail and J. N. Coleman / 43 \\ Composite Iterative Algorithm and Architecture for q-th Root Calculation / Alvaro Vazquez and Javier D. Bruguera / 52 \\ On the Fixed-Point Accuracy Analysis and Optimization of FFT Units with CORDIC Multipliers / Omid Sarbishei and Katarzyna Radecka / 62 \\ Session 4: Special Session on Industrial Practices: Chair: Mike Schulte \\ Self Checking in Current Floating-Point Units / Daniel Lipetz and Eric Schwarz / 73 \\ How to Square Floats Accurately and Efficiently on the ST231 Integer Processor/ Claude-Pierre Jeannerod, Jingyan Jourdan-Lu, Christophe Monat, and Guillaume Revy / 77 \\ A 1.5 Ghz VLIW DSP CPU with Integrated Floating Point and Fixed Point Instructions in 40 nm CMOS / Timothy Anderson, Due Bui, Shriram Moharil, Soujanya Narnur, Mujibur Rahman, Anthony Lell, Eric Biscondi, Ashish Shrivastava, Peter Dent, Mingjian Yan, and Hasan Mahmood / 82 \\ The POWER7 Binary Floating-Point Unit / Maarten Boersma, Michael Kroner, Christophe Layer, Petra Leber, Silvia M. Muller, and Kerstin Schelm / 87 \\ Session 5: Addition: Chair: Alberto Nannarelli \\ Accelerating Computations on FPGA Carry Chains by Operand Compaction / Thomas B. Preus{\ss}er, Martin Zabel, and Rainer G. Spallek / 95 \\ Fast Ripple-Carry Adders in Standard-Cell CMOS VLSI / Neil Burgess / 103 \\ A Family of High Radix Signed Digit Adders / Saeid Gorgin and Ghassem Jaberipur / 112 \\ Session 6: Floating-Point Units: Chair: Javier Bruguera \\ Fused Multiply-Add Microarchitecture Comprising Separate Early-Normalizing Multiply and Add Pipelines / David R. Lutz / 123 \\ Latency Sensitive FMA Design / Sameh Galal and Mark Horowitz / 129 \\ The IBM zEnterprise-196 Decimal Floating-Point Accelerator / Steven Carlough, Adam Collura, Silvia Mueller, and Michael Kroener / 139 \\ Session 7: Division, Square-Root and Reciprocal Square-Root: Chair: Peter Kornerup \\ Radix-8 Digit-by-Rounding: Achieving High-Performance Reciprocals, Square Roots, and Reciprocal Square Roots / J. Adam Butts, Ping Tak Peter Tang, Ron O. Dror, and David E. Shaw / 149 \\ Tight Certification Techniques for Digit-by-Rounding Algorithms with Application to a New 1/sqrt(x) Design / Ping Tak Peter Tang, J. Adam Butts, Ron O. Dror, and David E. Shaw / 159 \\ Radix-16 Combined Division and Square Root Unit / Alberto Nannarelli / 169 \\ A Prescale-Lookup-Postscale Additive Procedure for Obtaining a Single Precision Ulp Accurate Reciprocal / David W. Matula and Mihai T. Panu / 177 \\ Session 8: Special Session on High Performance Arithmetic for FPGA's: Chair: Martin Langhammer \\ Teraflop FPGA Design / Martin Langhammer / 187 \\ The Arithmetic Operators You Will Never See in a Microprocessor / Florent de Dinechin / 189 \\ Accelerating Large-Scale HPC Applications Using FPGAs / Rob Dimond, Sebastien Racaniere, and Oliver Pell / 191 \\ Session 9: Arithmetic Algorithms for Cryptography: Chair: David Matula \\ A General Approach for Improving RNS Montgomery Exponentiation Using Pre-processing / Filippo Gandino, Fabrizio Lamberti, Paolo Montuschi, and Jean-Claude Bajard / 195 \\ Bit-Sliced Binary Normal Basis Multiplication / Billy Bob Brumley and Dan Page / 205 \\ Efficient SIMD Arithmetic Modulo a Mersenne Number / Joppe W. Bos, Thorsten Kleinjung, Arjen K. Lenstra, and Peter L. Montgomery / 213 \\ Session 10: Tools for Formal Certified Code: Chair: Martin Schmookler \\ Automatic Generation of Code for the Evaluation of Constant Expressions at Any Precision with a Guaranteed Error Bound / Sylvain Chevillard / 225 \\ Automatic Generation of Fast and Certified Code for Polynomial Evaluation / Christophe Mouilleron and Guillaume Revy / 233 \\ Flocq: A Unified Library for Proving Floating-Point Algorithms in Coq / Sylvie Boldo and Guillaume Melquiond / 243 \\ Author Index / 253", } @Book{Hwu:2012:GCG, editor = "Wen-mei Hwu", booktitle = "{GPU} computing gems", title = "{GPU} computing gems", publisher = "Morgan Kaufmann", address = "Boston, MA", edition = "Jade", pages = "xvi + 541 + 16", year = "2012", ISBN = "0-12-385963-8 (hardback)", ISBN-13 = "978-0-12-385963-1 (hardback)", LCCN = "T385 .G6875 2012", bibdate = "Sat Feb 8 18:16:05 MST 2014", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/matlab.bib; https://www.math.utah.edu/pub/tex/bib/prng.bib; z3950.loc.gov:7090/Voyager", series = "Applications of GPU computing series", abstract = "Since the introduction of CUDA in 2007, more than 100 million computers with CUDA capable GPUs have been shipped to end users. GPU computing application developers can now expect their application to have a mass market. With the introduction of OpenCL in 2010, researchers can now expect to develop GPU applications that can run on hardware from multiple vendors.", acknowledgement = ack-nhfb, subject = "Graphics processing units; Programming; Imaging systems; Computer graphics; Image processing; Digital techniques", tableofcontents = "Part 1: Parallel Algorithms and Data Structures --- Paulius Micikevicius, NVIDIA \\ 1 Large-Scale GPU Search \\ 2 Edge v. Node Parallelism for Graph Centrality Metrics \\ 3 Optimizing parallel prefix operations for the Fermi architecture \\ 4 Building an Efficient Hash Table on the GPU \\ 5 An Efficient CUDA Algorithm for the Maximum Network Flow Problem \\ 6 On Improved Memory Access Patterns for Cellular Automata Using CUDA \\ 7 Fast Minimum Spanning Tree Computation on Large Graphs \\ 8 Fast in-place sorting with CUDA based on bitonic sort \\ Part 2: Numerical Algorithms --- Frank Jargstorff, NVIDIA \\ 9 Interval Arithmetic in CUDA \\ 10 Approximating the erfinv Function \\ 11 A Hybrid Method for Solving Tridiagonal Systems on the GPU \\ 12 LU Decomposition in CULA \\ 13 GPU Accelerated Derivative-free Optimization \\ Part 3: Engineering Simulation --- Peng Wang, NVIDIA \\ 14 Large-scale gas turbine simulations on GPU clusters \\ 15 GPU acceleration of rarefied gas dynamic simulations \\ 16 Assembly of Finite Element Methods on Graphics Processors \\ 17 CUDA implementation of Vertex-Centered, Finite Volume CFD methods on Unstructured Grids with Flow Control Applications \\ 18 Solving Wave Equations on Unstructured Geometries \\ 19 Fast electromagnetic integral equation solvers on graphics processing units (GPUs) \\ Part 4: Interactive Physics for Games and Engineering Simulation --- Richard Tonge, NVIDIA \\ 20 Solving Large Multi-Body Dynamics Problems on the GPU \\ 21 Implicit FEM Solver in CUDA \\ 22 Real-time Adaptive GPU multi-agent path planning \\ Part 5: Computational Finance --- Thomas Bradley, NVIDIA \\ 23 High performance finite difference PDE solvers on GPUs for financial option pricing \\ 24 Identifying and Mitigating Credit Risk using Large-scale Economic Capital Simulations \\ 25 Financial Market Value-at-Risk Estimation using the Monte Carlo Method \\ Part 6: Programming Tools and Techniques --- Cliff Wooley, NVIDIA \\ 26 Thrust: A Productivity-Oriented Library for CUDA \\ 27 GPU Scripting and Code Generation with PyCUDA \\ 28 Jacket: GPU Powered MATLAB Acceleration \\ 29 Accelerating Development and Execution Speed with Just In Time GPU Code Generation \\ 30 GPU Application Development, Debugging, and Performance Tuning with GPU Ocelot \\ 31 Abstraction for AoS and SoA Layout in C++ \\ 32 Processing Device Arrays with C++ Metaprogramming \\ 33 GPU Metaprogramming: A Case Study in Biologically-Inspired Machine Vision \\ 34 A Hybridization Methodology for High-Performance Linear Algebra Software for GPUs \\ 35 Dynamic Load Balancing using Work-Stealing \\ 36 Applying software-managed caching and CPU/GPU task scheduling for accelerating dynamic workloads", } @Book{Jonasson:2012:APS, author = "Kristj{\'a}n J{\'o}nasson", booktitle = "Applied Parallel and Scientific Computing: {10th international conference, PARA 2010, Reykjav{\'\i}k, Iceland, June 6--9, 2010: revised selected papers, Part 1}", title = "Applied Parallel and Scientific Computing: {10th international conference, PARA 2010, Reykjav{\'\i}k, Iceland, June 6--9, 2010: revised selected papers, Part 1}", volume = "7133", publisher = pub-SV, address = pub-SV:adr, pages = "xxvii + 339 + 155", year = "2012", DOI = "https://doi.org/10.1007/978-3-642-28151-8", ISBN = "3-642-28150-8 (print), 3-642-28145-1 (e-book), 3-642-28151-6 (e-book)", ISBN-13 = "978-3-642-28145-7, 978-3-642-28151-8", LCCN = "QA76.642 .P37 2010", bibdate = "Fri Apr 25 14:47:05 MDT 2025", bibsource = "fsz3950.oclc.org:210/WorldCat; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", series = ser-LNCS, acknowledgement = ack-nhfb, tableofcontents = "Front Matter \\ Keynote Papers and General Topics \\ Keynote Papers \\ On Aggressive Early Deflation in Parallel Variants of the QR Algorithm / Bo K{\aa}gstr{\"o}m, Daniel Kressner, Meiyue Shao / 1--10 \\ Limits to Nonlinear Inversion / Klaus Mosegaard / 11--21 \\ Cache Blocking / Fred G. Gustavson / 22--32 \\ General Topics \\ Cloud Computing \\ A Model for Efficient Onboard Actualization of an Instrumental Cyclogram for the Mars MetNet Mission on a Public Cloud Infrastructure / Jose Luis V{\'a}zquez-Poletti, Gonzalo Barderas, Ignacio M. Llorente, Pilar Romero / 33--42 \\ HPC Algorithms \\ Impact of Asynchronism on GPU Accelerated Parallel Iterative Computations / Sylvain Contassot-Vivier, Thomas Jost, St{\'e}phane Vialle / 43--53 \\ Simulation of Seismic Waves Propagation in Multiscale Media: Impact of Cavernous\slash Fractured Reservoirs / Victor Kostin, Vadim Lisitsa, Galina Reshetova, Vladimir Tcheverda / 54--64 \\ Improvements of a Fast Parallel Poisson Solver on Irregular Domains / Andreas Adelmann, Peter Arbenz, Yves Ineichen / 65--74 \\ Distributed Java Programs Initial Mapping Based on Extremal Optimization / Eryk Laskowski, Marek Tudruj, Ivanoe De Falco, Umberto Scafuri, Ernesto Tarantino, Richard Olejnik / 75--85 \\ HPC Programming Tools \\ Software Environment for Parallel Optimization of Complex Systems / Ewa Niewiadomska-Szynkiewicz, Michal Marks / 86--96 \\ Parallel Programming in Morpho / Snorri Agnarsson / 97--107 \\ Extending Distributed Shared Memory for the Cell Broadband Engine to a Channel Model / Kenneth Skovhede, Morten N. Larsen, Brian Vinter / 108--118 \\ Global Asynchronous Parallel Program Control for Multicore Processors / Janusz Borkowski, Marek Tudruj, Adam Smyk, Damian Kopanski / 119--130 \\ HPC in Meteorology \\ Highly Scalable Dynamic Load Balancing in the Atmospheric Modeling System COSMO-SPECS+FD4 / Matthias Lieber, Verena Gr{\"u}tzun, Ralf Wolke, Matthias S. M{\"u}ller, Wolfgang E. Nagel / 131--141 \\ Interactive Weather Simulation and Visualization on a Display Wall with Many-Core Compute Nodes / B{\aa}rd Fjukstad, Tor-Magne Stien Hagen, Daniel St{\o}dle, Phuong Hoai Ha, John Markus Bj{\o}rndalen, Otto Anshus / 142--151 \\ Parallel Numerical Algorithms \\ The Algorithm of Multiple Relatively Robust Representations for Multi-core Processors / Matthias Petschow, Paolo Bientinesi / 152--161 \\ Parallelization of Multilevel ILU Preconditioners on Distributed-Memory Multiprocessors / Jos{\'e} I. Aliaga, Matthias Bollh{\"o}fer, Alberto F. Mart{\'\i}n, Enrique S. Quintana-Ort{\'\i} / 162--172 \\ CUDA 2D Stencil Computations for the Jacobi Method / Jos{\'e} Mar{\'\i}a Cecilia, Jos{\'e} Manuel Garc{\'\i}a, Manuel Ujald{\'o}n / 173--183 \\ Streaming Model Computation of the FDTD Problem / Adam Smyk, Marek Tudruj / 184--192 \\ Numerical Aspects of Spectral Segmentation on Polygonal Grids / Anna Matsekh, Alexei Skurikhin, Lakshman Prasad, Edward Rosten / 193--203 \\ General Topics \\ Parallel Numerical Algorithms \\ Parallel Kriging Algorithm for Unevenly Spaced Data / Jacek Strzelczyk, Stanislawa Porzycka / 204--212 \\ Parallel Computing in Physics \\ Parallel Particle-in-Cell Monte-Carlo Algorithm for Simulation of Gas Discharges under PVM and MPI / Christoph Schwanke, Andreas Pflug, Michael Siemers, Bernd Szyszka / 213--219 \\ Monte Carlo Simulations of Spin Systems on Multi-core Processors / Marco Guidetti, Andrea Maiorano, Filippo Mantovani, Marcello Pivanti, Sebastiano F. Schifano, Raffaele Tripiccione / 220--230 \\ Software Environment for Market Balancing Mechanisms Development, and Its Application to Solving More General Problems in Parallel Way / Mariusz Kamola / 231--241 \\ The Development of Fully Coupled Simulation Software by Reusing Segregated Solvers / Mika Malinen / 242--248 \\ Implementation and Evaluation of Quadruple Precision BLAS Functions on GPUs / Daichi Mukunoki, Daisuke Takahashi / 249--259 \\ Aggregated Pumping Station Operation Planning Problem (APSOP) for Large Scale Water Transmission System / Jacek B aszczyk, Krzysztof Malinowski, Alnoor Allidina / 260--269 \\ PerPI: A Tool to Measure Instruction Level Parallelism / Bernard Goossens, Philippe Langlois, David Parello, Eric Petit / 270--281 \\ InterCell: A Software Suite for Rapid Prototyping and Parallel Execution of Fine Grained Applications / Jens Gustedt, St{\'e}phane Vialle, Herv{\'e} Frezza-Buet, D havh Boumba Sitou, Nicolas Fressengeas, Jeremy Fix / 282--292 \\ PAS2P Tool, Parallel Application Signature for Performance Prediction / Alvaro Wong, Dolores Rexachs, Emilio Luque / 293--302 \\ A Software Tool for Federated Simulation of Wireless Sensor Networks and Mobile Ad Hoc Networks / Ewa Niewiadomska-Szynkiewicz, Andrzej Sikora / 303--313 \\ HPC Software Engineering / Performance Engineering of GemsFDTD Computational Electromagnetics Solver / Ulf Andersson, Brian J. N. Wylie / 314--324 \\ Scheduling Architecture Supported Regions in Parallel Programs / Marek Tudruj, {\L}ukasz Ma{\'s}ko / 325--336 \\ Back Matter", } @Book{Arfken:2013:MMP, author = "George B. (George Brown) Arfken and Hans-J{\"u}rgen Weber and Frank E. Harris", booktitle = "Mathematical Methods for Physicists: a Comprehensive Guide", title = "Mathematical Methods for Physicists: a Comprehensive Guide", publisher = pub-ELSEVIER-ACADEMIC, address = pub-ELSEVIER-ACADEMIC:adr, edition = "Seventh", pages = "xiii + 1205", year = "2013", ISBN = "0-12-384654-4 (hardcover)", ISBN-13 = "978-0-12-384654-9 (hardcover)", LCCN = "QA37.3 .A74 2013", bibdate = "Thu May 3 08:02:53 MDT 2012", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/h/harris-frank-e.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/master.bib; https://www.math.utah.edu/pub/tex/bib/numana2010.bib; jenson.stanford.edu:2210/unicorn", acknowledgement = ack-nhfb, subject = "Mathematical analysis; Mathematical physics", tableofcontents = "Preface / xi--xiii \\ 1: Mathematical Preliminaries / 1--82 \\ 2: Determinants and Matrices / 83--121 \\ 3: Vector Analysis / 123--203 \\ 4: Tensors and Differential Forms / 205--249 \\ 5: Vector Spaces / 251--297 \\ 6: Eigenvalue Problems / 299--328 \\ 7: Ordinary Differential Equations / 329--380 \\ 8: Sturm--Liouville Theory / 381--399 \\ 9: Partial Differential Equations / 401--445 \\ 10: Green's Functions / 447--467 \\ 11: Complex Variable Theory / 469--550 \\ 12: Further Topics in Analysis / 551--598 \\ 13: Gamma Function / 599--641 \\ 14: Bessel Functions / 643--713 \\ 15: Legendre Functions / 715--772 \\ 16: Angular Momentum / 773--814 \\ 17: Group Theory / 815--870 \\ 18: More Special Functions / 871--933 \\ 19: Fourier Series / 935--962 \\ 20: Integral Transforms / 963--1046 \\ 21: Integral Equations / 1047--1079 \\ 22: Calculus of Variations / 1081--1124 \\ 23: Probability and Statistics / 1125--1179 \\ Index / 1181--1205", } @Proceedings{IEEE:2013:PIS, editor = "{IEEE}", booktitle = "{Proceedings of the 21st IEEE Symposium on Computer Arithmetic, Austin, Texas, USA, 8--10 April 2013}", title = "{Proceedings of the 21st IEEE Symposium on Computer Arithmetic, Austin, Texas, USA, 8--10 April 2013}", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "xvi + 229", year = "2013", ISBN = "0-7695-4957-8", ISBN-13 = "978-0-7695-4957-6", ISSN = "1063-6889", LCCN = "QA76.9.C62 S95 2013", bibdate = "Sat Aug 01 08:03:11 2013", bibsource = "https://www.math.utah.edu/pub/tex/bib/cryptography2010.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, keywords = "computer arithmetic units; correctness proofs; cryptography; domain specific designs; error analysis; exascale computing; floating point arithmetic; floating-point error analysis; formal verification; function approximation; modular arithmetic; theorem proving; verification", } @Proceedings{Hong:2014:MSI, editor = "Hoon Hong and Chee Yap", booktitle = "Mathematical Software --- {ICMS 2014: 4th International Conference, Seoul, South Korea, August 5--9, 2014, Proceedings}", title = "Mathematical Software --- {ICMS 2014: 4th International Conference, Seoul, South Korea, August 5--9, 2014, Proceedings}", volume = "8592", publisher = pub-SV, address = pub-SV:adr, pages = "xxxii + 735", year = "2014", DOI = "https://doi.org/10.1007/978-3-662-44199-2", ISBN = "3-662-44198-5 (paperback), 3-662-44199-3 (e-book)", ISBN-13 = "978-3-662-44198-5 (paperback), 978-3-662-44199-2 (e-book)", LCCN = "QA76.9.M35", bibdate = "Sat Sep 23 09:59:48 2023", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/gnu.bib; https://www.math.utah.edu/pub/tex/bib/magma.bib; https://www.math.utah.edu/pub/tex/bib/maple-extract.bib; https://www.math.utah.edu/pub/tex/bib/mathematica.bib; https://www.math.utah.edu/pub/tex/bib/texbook3.bib", acknowledgement = ack-nhfb, tableofcontents = "Front Matter \\ Invited Talks \\ Experimental Computation and Visual Theorems / Jonathan M. Borwein / 1--8 \\ Soft Math --- Math Soft / Bruno Buchberger / 9--15 \\ Mathematical Theory Exploration \\ Flyspecking Flyspeck / Mark Adams / 16--20 \\ Symbolic Computing Package for Mathematica for Versatile Manipulation of Mathematical Expressions / Youngjoo Chung / 21--25 \\ Representing, Archiving, and Searching the Space of Mathematical Knowledge / Mihnea Iancu, Michael Kohlhase, Corneliu Prodescu / 26--30 \\ Early Examples of Software in Mathematical Knowledge Management / Patrick Ion / 31--35 \\ Discourse-Level Parallel Markup and Meaning Adoption in Flexiformal Theory Graphs / Michael Kohlhase, Mihnea Iancu / 36--40 \\ Complexity Analysis of the Bivariate Buchberger Algorithm in Theorema / Alexander Maletzky, Bruno Buchberger / 41--48 \\ Theorema 2.0: A System for Mathematical Theory Exploration / Wolfgang Windsteiger / 49--52 \\ Computational Group Theory \\ New Approaches in Black Box Group Theory / Alexandre Borovik, {\c{S}}{\"u}kr{\"u} Yal{\c{c}}{\i}nkaya / 53--58 \\ A GAP Package for Computing with Real Semisimple Lie Algebras / Heiko Dietrich, Paolo Faccin, Willem A. de Graaf / 59--66 \\ Bacterial Genomics and Computational Group Theory: The BioGAP Package for GAP / Attila Egri-Nagy, Andrew R. Francis, Volker Gebhardt / 67--74 \\ SgpDec: Cascade (De)Compositions of Finite Transformation Semigroups and Permutation Groups / Attila Egri-Nagy, James D. Mitchell, Chrystopher L. Nehaniv / 75--82 \\ Approximating Generators for Integral Arithmetic Groups / Bettina Eick / 83--86 \\ Software for Groups: Theory and Practice / Alexander Hulpke / 87--91 \\ Computation of Genus 0 Belyi Functions / Mark van Hoeij, Raimundas Vidunas / 92--98 \\ On Computation of the First Baues--Wirsching Cohomology of a Freely-Generated Small Category / Yasuhiro Momose, Yasuhide Numata / 99--105 \\ Coding Theory \\ Codes over a Non Chain Ring with Some Applications / Aysegul Bayram, Elif Segah Oztas, Irfan Siap / 106--110 \\ On the Weight Enumerators of the Projections of the 2-adic Golay Code of Length 24 to $\mathbb{Z}_{2^e}$ / Sunghyu Han / 111--114 \\ Coding Theory \\ Computer Based Reconstruction of Binary Extremal Self-dual Codes of Length 32 / Jon-Lark Kim / 115--118 \\ Magma Implementation of Decoding Algorithms for General Algebraic Geometry Codes / Kwankyu Lee / 119--123 \\ Reversible Codes and Applications to DNA / Elif Segah Oztas, Irfan Siap, Bahattin Yildiz / 124--128 \\ Computational Topology \\ javaPlex: A Research Software Package for Persistent (Co)Homology / Henry Adams, Andrew Tausz, Mikael Vejdemo-Johansson / 129--136 \\ PHAT --- Persistent Homology Algorithms Toolbox / Ulrich Bauer, Michael Kerber, Jan Reininghaus, Hubert Wagner / 137--143 \\ Computing Persistence Modules on Commutative Ladders of Finite Type / Emerson G. Escolar, Yasuaki Hiraoka / 144--151 \\ Heuristics for Sphere Recognition / Michael Joswig, Frank H. Lutz, Mimi Tsuruga / 152--159 \\ CAPD::RedHom v2 --- Homology Software Based on Reduction Algorithms / Mateusz Juda, Marian Mrozek / 160--166 \\ The Gudhi Library: Simplicial Complexes and Persistent Homology / Cl{\'e}ment Maria, Jean-Daniel Boissonnat, Marc Glisse, Mariette Yvinec / 167--174 \\ Numerical Algebraic Geometry \\ Bertini_real: Software for One- and Two-Dimensional Real Algebraic Sets / Daniel A. Brake, Daniel J. Bates, Wenrui Hao, Jonathan D. Hauenstein, Andrew J. Sommese, CharlesW. Wampler / 175--182 \\ Hom4PS-3: A Parallel Numerical Solver for Systems of Polynomial Equations Based on Polyhedral Homotopy Continuation Methods / Tianran Chen, Tsung-Lin Lee, Tien-Yien Li / 183--190 \\ Geometry \\ CGAL --- Reliable Geometric Computing for Academia and Industry / Eric Berberich / 191--197 \\ Implementing the $L_\infty$ Segment Voronoi Diagram in CGAL and Applying in VLSI Pattern Analysis / Panagiotis Cheilaris, Sandeep Kumar Dey, Maria Gabrani, Evanthia Papadopoulou / 198--205 \\ BULL! --- The Molecular Geometry Engine Based on Voronoi Diagram, Quasi-Triangulation, and Beta-Complex / Deok-Soo Kim, Youngsong Cho, Jae-Kwan Kim, Joonghyun Ryu, Mokwon Lee, Jehyun Cha et al. / 206--213 \\ Integrating Circumradius and Area Formulae for Cyclic Pentagons / Shuichi Moritsugu / 214--221 \\ Computer Aided Geometry / Douglas Navarro Guevara, Adrian Navarro Alvarez / 222--229 \\ The Sustainability of Digital Educational Resources / Yongsheng Rao, Ying Wang, Yu Zou, Jingzhong Zhang / 230--234 \\ A Touch-Operation-Based Dynamic Geometry System: Design and Implementation / Wei Su, Paul S. Wang, Chuan Cai, Lian Li / 235--239 \\ OpenGeo: An Open Geometric Knowledge Base / Dongming Wang, Xiaoyu Chen, Wenya An, Lei Jiang, Dan Song / 240--245 \\ Curves and Surfaces \\ On Computing a Cell Decomposition of a Real Surface Containing Infinitely Many Singularities / Daniel J. Bates, Daniel A. Brake, Jonathan D. Hauenstein, Andrew J. Sommese, Charles W. Wampler / 246--252 \\ Robustly and Efficiently Computing Algebraic Curves and Surfaces / Eric Berberich / 253--260 \\ Computing the Orthogonal Projection of Rational Curves onto Rational Parameterized Surface by Symbolic Methods / Zhiwang Gan, Meng Zhou / 261--268 \\ Isotopic $\epsilon$-Approximation of Algebraic Curves / Kai Jin / 269--276 \\ Isotopic Arrangement of Simple Curves: An Exact Numerical Approach Based on Subdivision / Jyh-Ming Lien, Vikram Sharma, Gert Vegter, Chee Yap / 277--282 \\ Quantified Reasoning \\ Real Quantifier Elimination in the RegularChains Library / Changbo Chen, Marc Moreno Maza / 283--290 \\ Software for Quantifier Elimination in Propositional Logic / Eugene Goldberg, Panagiotis Manolios / 291--294 \\ Quantifier Elimination for Linear Modular Constraints / Ajith K. John, Supratik Chakraborty / 295--302 \\ Skolemization Modulo Theories / Konstantin Korovin, Margus Veanes / 303--306 \\ Incremental QBF Solving by DepQBF / Florian Lonsing, Uwe Egly / 307--314 \\ NLCertify: A Tool for Formal Nonlinear Optimization / Victor Magron / 315--320 \\ Special Functions and Concrete Mathematics \\ Developing Linear Algebra Packages on Risa/Asir for Eigenproblems / Katsuyoshi Ohara, Shinichi Tajima, Akira Terui / 321--324 \\ Mathematical Software for Modified Bessel Functions / Juri Rappoport / 325--332 \\ BetaSCP2: A Program for the Optimal Prediction of Side-Chains in Proteins / Joonghyun Ryu, Mokwon Lee, Jehyun Cha, Chanyoung Song, Deok-Soo Kim / 333--340 \\ Computation of an Improved Lower Bound to Giuga's Primality Conjecture / Matthew Skerritt / 341--345 \\ An Extension and Efficient Calculation of the Horner's Rule for Matrices / Shinichi Tajima, Katsuyoshi Ohara, Akira Terui / 346--351 \\ Groebner Bases \\ What Is New in CoCoA? / John Abbott, Anna Maria Bigatti / 352--358 \\ Maximizing Likelihood Function for Parameter Estimation in Point Clouds via Groebner Basis / Joseph Awange, B{\'e}la Pal{\'a}ncz, Robert Lewis / 359--366 \\ Groebner Basis in Geodesy and Geoinformatics / Joseph Awange, B{\'e}la Pal{\'a}ncz, Robert Lewis / 367--373 \\ Groebner Bases in Theorema / Bruno Buchberger, Alexander Maletzky / 374--381 \\ Effective Computation of Radical of Ideals and Its Application to Invariant Theory / Amir Hashemi / 382--389 \\ Generic and Parallel Groebner Bases in JAS / Heinz Kredel / 390--397 \\ Application of Groebner Basis Methodology to Nonlinear Mechanics Problems / Y. Jane Liu, John Peddieson / 398--405 \\ Software for Discussing Parametric Polynomial Systems: The Gr{\"o}bner Cover / Antonio Montes, Michael Wibmer / 406--413 \\ An Algorithm for Computing Standard Bases by Change of Ordering via Algebraic Local Cohomology / Katsusuke Nabeshima, Shinichi Tajima / 414--418 \\ Verification of Gr{\"o}bner Basis Candidates / Masayuki Noro, Kazuhiro Yokoyama / 419--424 \\ Triangular Decompositions of Polynomial Systems \\ Cylindrical Algebraic Decomposition in the RegularChains Library / Changbo Chen, Marc Moreno Maza / 425--433 \\ Hierarchical Comprehensive Triangular Decomposition / Zhenghong Chen, Xiaoxian Tang, Bican Xia / 434--441 \\ A Package for Parametric Matrix Computations / Robert M. Corless, Steven E. Thornton / 442--449 \\ Choosing a Variable Ordering for Truth-Table Invariant Cylindrical Algebraic Decomposition by Incremental Triangular Decomposition / Matthew England, Russell Bradford, James H. Davenport, David Wilson / 450--457 \\ Using the Regular Chains Library to Build Cylindrical Algebraic Decompositions by Projecting and Lifting / Matthew England, David Wilson, Russell Bradford, James H. Davenport / 458--465 \\ An Improvement of Rosenfeld--Gr{\"o}bner Algorithm / Amir Hashemi, Zahra Touraji / 466--471 \\ Doing Algebraic Geometry with the RegularChains Library / Parisa Alvandi, Changbo Chen, Steffen Marcus, Marc Moreno Maza, {\'E}ric Schost, Paul Vrbik / 472--479 \\ On Multivariate Birkhoff Rational Interpolation / Peng Xia, Bao-Xin Shang, Na Lei / 480--483 \\ Computing Moore--Penrose Inverses of Ore Polynomial Matrices / Yang Zhang / 484--491 \\ Parametric Polynomial Systems \\ Software Using the Gr{\"o}bner Cover for Geometrical Loci Computation and Classification / Miguel A. Ab{\'a}nades, Francisco Botana, Antonio Montes, Tom{\'a}s Recio / 492--499 \\ Using Maple's RegularChains Library to Automatically Classify Plane Geometric Loci / Francisco Botana, Tom{\'a}s Recio / 500--503 \\ Solving Parametric Polynomial Systems by RealComprehensiveTriangularize / Changbo Chen, Marc Moreno Maza / 504--511 \\ QE Software Based on Comprehensive Gr{\"o}bner Systems / Ryoya Fukasaku / 512--517 \\ SyNRAC: A Toolbox for Solving Real Algebraic Constraints / Hidenao Iwane, Hitoshi Yanami, Hirokazu Anai / 518--522 \\ An Algorithm for Computing Tjurina Stratifications of $\mu$-Constant Deformations by Using Local Cohomology Classes with Parameters / Katsusuke Nabeshima, Shinichi Tajima / 523--530 \\ An Implementation Method of Boolean Gr{\"o}bner Bases and Comprehensive Boolean Gr{\"o}bner Bases on General Computer Algebra Systems / Akira Nagai, Shutaro Inoue / 531--536 \\ A Method to Determine if Two Parametric Polynomial Systems Are Equal / Jie Zhou, Dingkang Wang / 537--544 \\ Mathematical Web/Mobile Interfaces and Visualization \\ An Implementation Method of a CAS with a Handwriting Interface on Tablet Devices / Mitsushi Fujimoto / 545--548 \\ New Way of Explanation of the Stochastic Interpretation of Wave Functions and Its Teaching Materials Using KETpic / Kenji Fukazawa / 549--553 \\ {IFSGen4\LaTeX}: Interactive Graphical User Interface for Generation and Visualization of Iterated Function Systems in {\LaTeX} / Akemi G{\'a}lvez, Kiyoshi Kitahara, Masataka Kaneko / 554--561 \\ GNU {\TeX}MACS: towards a Scientific Office Suite / Massimiliano Gubinelli, Joris van der Hoeven, Fran{\c{c}}ois Poulain, Denis Raux / 562--569 \\ Computer Software Program for Representation and Visualization of Free-Form Curves through Bio-inspired Optimization Techniques / Andr{\'e}s Iglesias, Akemi G{\'a}lvez / 570--577 \\ On Some Attempts to Verify the Effect of Using High-Quality Graphics in Mathematics Education / Kiyoshi Kitahara, Tadashi Takahashi, Masataka Kaneko / 578--585 \\ Math Web Search Interfaces and the Generation Gap of Mathematicians / Andrea Kohlhase / 586--593 \\ Practice with Computer Algebra Systems in Mathematics Education and Teacher Training Courses / Hideyo Makishita / 594--600 \\ Development of Visual Aid Materials in Teaching the Bivariate Normal Distributions / Toshifumi Nomachi, Toshihiko Koshiba, Shunji Ouchi / 601--606 \\ Creating Interactive Graphics for Mathematics Education Utilizing KETpic / Shunji Ouchi, Yoshifumi Maeda, Kiyoshi Kitahara, Naoki Hamaguchi / 607--613 \\ A Tablet-Compatible Web-Interface for Mathematical Collaboration / Marco Pollanen, Jeff Hooper, Bruce Cater, Sohee Kang / 614--620 \\ Development and Evaluation of a Web-Based Drill System to Master Basic Math Formulae Using a New Interactive Math Input Method / Shizuka Shirai, Tetsuo Fukui / 621--628 \\ Generating Data of Mathematical Figures for 3D Printers with KETpic and Educational Impact of the Printed Models / Setsuo Takato, Naoki Hamaguchi, Haiduke Sarafian / 629--634 \\ A Touch-Based Mathematical Expression Editor / Wei Su, Paul S. Wang, Lian Li / 635--640 \\ Establishment of KETpic Programming Styles for Drawing / Satoshi Yamashita, Yoshifumi Maeda, Hisashi Usui, Kiyoshi Kitahara, Hideyo Makishita, Kazushi Ahara / 641--646 \\ General Session \\ Integration of Libnormaliz in CoCoALib and CoCoA 5 / John Abbott, Anna Maria Bigatti, Christof S{\"o}ger / 647--653 \\ Elements of Design for Containers and Solutions in the LinBox Library / Brice Boyer, Jean-Guillaume Dumas, Pascal Giorgi, Cl{\'e}ment Pernet, B. David Saunders / 654--662 \\ Recent Developments in Normaliz / Winfried Bruns, Christof S{\"o}ger / 663--668 \\ The Basic Polynomial Algebra Subprograms / Changbo Chen, Svyatoslav Covanov, Farnam Mansouri, Marc Moreno Maza, Ning Xie, Yuzhen Xie / 669--676 \\ Function Interval Arithmetic / Jan Duracz, Amin Farjudian, Michal Kone{\v{c}}n{\'y}, Walid Taha / 677--684 \\ Generating Optimized Sparse Matrix Vector Product over Finite Fields / Pascal Giorgi, Bastien Vialla / 685--690 \\ swMATH --- An Information Service for Mathematical Software / Gert-Martin Greuel, Wolfram Sperber / 691--701 \\ MathLibre: Modifiable Desktop Environment for Mathematics / Tatsuyoshi Hamada / 702--705 \\ Software Packages for Holonomic Gradient Method / Tamio Koyama, Hiromasa Nakayama, Katsuyoshi Ohara, Tomonari Sei, Nobuki Takayama / 706--712 \\ Metalibm: A Mathematical Functions Code Generator / Olga Kupriianova, Christoph Lauter / 713--717 \\ From Calculus to Algorithms without Errors / Norbert M{\"u}ller, Martin Ziegler / 718--724 \\ Dense Arithmetic over Finite Fields with the CUMODP Library / Sardar Anisul Haque, Xin Li, Farnam Mansouri, Marc Moreno Maza, Wei Pan, Ning Xie / 725--732 \\ Back Matter / / 733--735", } @Book{Higham:2015:PCA, editor = "Nicholas J. Higham and Mark R. Dennis and Paul Glendinning and Paul A. Martin and Fadil Santosa and Jared Tanner", booktitle = "The {Princeton} Companion to Applied Mathematics", title = "The {Princeton} Companion to Applied Mathematics", publisher = pub-PRINCETON, address = pub-PRINCETON:adr, pages = "994 (est.)", year = "2015", ISBN = "0-691-15039-7 (hardcover)", ISBN-13 = "978-0-691-15039-0 (hardcover)", LCCN = "QA155 .P75 2015", bibdate = "Wed Sep 9 05:32:49 MDT 2015", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/h/higham-nicholas-john.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/prng.bib; z3950.loc.gov:7090/Voyager", acknowledgement = ack-nhfb, subject = "Algebra; Mathematics; Mathematical models", tableofcontents = "Preface / ix \\ Contributors / xiii \\ Part I: Introduction to Applied Mathematics \\ I.1 What Is Applied Mathematics? / 1 \\ I.2 The Language of Applied Mathematics / 8 \\ I.3 Methods of Solution / 27 \\ I.4 Algorithms / 40 \\ I.5 Goals of Applied Mathematical Research / 48 \\ I.6 The History of Applied Mathematics / 55 \\ Part II: Concepts \\ II.1 Asymptotics / 81 \\ II.2 Boundary Layer / 82 \\ II.3 Chaos and Ergodicity / 82 \\ II.4 Complex Systems / 83 \\ II.5 Conformal Mapping / 84 \\ II.6 Conservation Laws / 86 \\ II.7 Control / 88 \\ II.8 Convexity / 89 \\ II.9 Dimensional Analysis and Scaling / 90 \\ II.10 The Fast Fourier Transform / 94 \\ II.11 Finite Differences / 95 \\ II.12 The Finite-Element Method / 96 \\ II.13 Floating-Point Arithmetic / 96 \\ II.14 Functions of Matrices / 97 \\ II.15 Function Spaces / 99 \\ II.16 Graph Theory / 101 \\ II.17 Homogenization / 103 \\ II.18 Hybrid Systems / 103 \\ II.19 Integral Transforms and Convolution / 104 \\ II.20 Interval Analysis / 105 \\ II.21 Invariants and Conservation Laws / 106 \\ II.22 The Jordan Canonical Form / 112 \\ II.23 Krylov Subspaces / 113 \\ II.24 The Level Set Method / 114 \\ II.25 Markov Chains / 116 \\ II.26 Model Reduction / 117 \\ II.27 Multiscale Modeling / 119 \\ II.28 Nonlinear Equations and Newton's Method / 120 \\ II.29 Orthogonal Polynomials / 122 \\ II.30 Shocks / 122 \\ II.31 Singularities / 124 \\ II.32 The Singular Value Decomposition / 126 \\ II.33 Tensors and Manifolds / 127 \\ II.34 Uncertainty Quantification / 131 \\ II.35 Variational Principle / 134 \\ II.36 Wave Phenomena / 134 \\ Part III: Equations, Laws, and Functions of Applied Mathematics \\ III.1 Benford's Law / 135 \\ III.2 Bessel Functions / 137 \\ III.3 The Black--Scholes Equation / 137 \\ III.4 The Burgers Equation / 138 \\ III.5 The Cahn--Hilliard Equation / 138 \\ III.6 The Cauchy--Riemann Equations / 139 \\ III.7 The Delta Function and Generalized Functions / 139 \\ III.8 The Diffusion Equation / 142 \\ III.9 The Dirac Equation / 142 \\ III.10 Einstein's Field Equations / 144 \\ III.11 The Euler Equations / 146 \\ III.12 The Euler--Lagrange Equations / 147 \\ III.13 The Gamma Function / 148 \\ III.14 The Ginzburg--Landau Equation / 148 \\ III.15 Hooke's Law / 149 \\ III.16 The Korteweg--de Vries Equation / 150 \\ III.17 The Lambert $W$ Function / 151 \\ III.18 Laplace's Equation / 155 \\ III.19 The Logistic Equation / 156 \\ III.20 The Lorenz Equations / 158 \\ III.21 Mathieu Functions / 159 \\ III.22 Maxwell's Equations / 160 \\ III.23 The Navier--Stokes Equations / 162 \\ III.24 The Painlev{\'e} Equations / 163 \\ III.25 The Riccati Equation / 165 \\ III.26 Schr{\"o}dinger's Equation / 167 \\ III.27 The Shallow-Water Equations / 167 \\ III.28 The Sylvester and Lyapunov Equations / 168 \\ III.29 The Thin-Film Equation / 169 \\ III.30 The Tricomi Equation / 170 \\ III.31 The Wave Equation / 171 \\ Part IV: Areas of Applied Mathematics \\ IV.1 Complex Analysis / 173 \\ IV.2 Ordinary Differential Equations / 181 \\ IV.3 Partial Differential Equations / 190 \\ IV.4 Integral Equations / 200 \\ IV.5 Perturbation Theory and Asymptotics / 208 \\ IV.6 Calculus of Variations / 218 \\ IV.7 Special Functions / 227 \\ IV.8 Spectral Theory / 236 \\ IV.9 Approximation Theory / 248 \\ IV.10 Numerical Linear Algebra and Matrix Analysis / 263 \\ IV.11 Continuous Optimization (Nonlinear and Linear Programming) / 281 \\ IV.12 Numerical Solution of Ordinary Differential Equations / 293 \\ IV.13 Numerical Solution of Partial Differential Equations / 306 \\ IV.14 Applications of Stochastic Analysis / 319 \\ IV.15 Inverse Problems / 327 \\ IV.16 Computational Science / 335 \\ IV.17 Data Mining and Analysis / 350 \\ IV.18 Network Analysis / 360 \\ IV.19 Classical Mechanics / 374 \\ IV.20 Dynamical Systems / 383 \\ IV.21 Bifurcation Theory / 393 \\ IV.22 Symmetry in Applied Mathematics / 402 \\ IV.23 Quantum Mechanics / 411 \\ IV.24 Random-Matrix Theory / 419 \\ IV.25 Kinetic Theory / 428 \\ IV.26 Continuum Mechanics / 446 \\ IV.27 Pattern Formation / 458 \\ IV.28 Fluid Dynamics / 467 \\ IV.29 Magnetohydrodynamics / 476 \\ IV.30 Earth System Dynamics / 485 \\ IV.31 Effective Medium Theories / 500 \\ IV.32 Mechanics of Solids / 505 \\ IV.33 Soft Matter / 516 \\ IV.34 Control Theory / 523 \\ IV.35 Signal Processing / 533 \\ IV.36 Information Theory / 545 \\ IV.37 Applied Combinatorics and Graph Theory / 552 \\ IV.38 Combinatorial Optimization / 564 \\ IV.39 Algebraic Geometry / 570 \\ IV.40 General Relativity and Cosmology / 579 \\ Part V: Modeling \\ V.1 The Mathematics of Adaptation (Or the Ten Avatars of Vishnu) / 591 \\ V.2 Sport / 598 \\ V.3 Inerters / 604 \\ V.4 Mathematical Biomechanics / 609 \\ V.5 Mathematical Physiology / 616 \\ V.6 Cardiac Modeling / 623 \\ V.7 Chemical Reactions / 627 \\ V.8 Divergent Series: Taming the Tails / 634 \\ V.9 Financial Mathematics / 640 \\ V.10 Portfolio Theory / 648 \\ V.11 Bayesian Inference in Applied Mathematics / 658 \\ V.12 A Symmetric Framework with Many Applications / 661 \\ V.13 Granular Flows / 665 \\ V.14 Modern Optics / 673 \\ V.15 Numerical Relativity / 680 \\ V.16 The Spread of Infectious Diseases / 687 \\ V.17 The Mathematics of Sea Ice / 694 \\ V.18 Numerical Weather Prediction / 705 \\ V.19 Tsunami Modeling / 712 \\ V.20 Shock Waves / 720 \\ V.21 Turbulence / 724 \\ Part VI: Example Problems \\ VI.1 Cloaking / 733 \\ VI.2 Bubbles / 735 \\ VI.3 Foams / 737 \\ VI.4 Inverted Pendulums / 741 \\ VI.5 Insect Flight / 743 \\ VI.6 The Flight of a Golf Ball / 746 \\ VI.7 Automatic Differentiation / 749 \\ VI.8 Knotting and Linking of Macromolecules / 752 \\ VI.9 Ranking Web Pages / 755 \\ VI.10 Searching a Graph / 757 \\ VI.11 Evaluating Elementary Functions / 759 \\ VI.12 Random Number Generation / 761 \\ VI.13 Optimal Sensor Location in the Control of Energy-Efficient Buildings / 763 \\ VI.14 Robotics / 767 \\ VI.15 Slipping, Sliding, Rattling, and Impact: Nonsmooth Dynamics and Its Applications / 769 \\ VI.16 From the $N$-Body Problem to Astronomy and Dark Matter / 771 \\ VI.17 The $N$-Body Problem and the Fast Multipole Method / 775 \\ VI.18 The Traveling Salesman Problem / 778 \\ Part VII: Application Areas \\ VII.1 Aircraft Noise / 783 \\ VII.2 A Hybrid Symbolic--Numeric Approach to Geometry Processing and Modeling / 787 \\ VII.3 Computer-Aided Proofs via Interval Analysis / 790 \\ VII.4 Applications of Max-Plus Algebra / 795 \\ VII.5 Evolving Social Networks, Attitudes, and Beliefs --- and Counterterrorism / 800 \\ VII.6 Chip Design / 804 \\ VII.7 Color Spaces and Digital Imaging / 808 \\ VII.8 Mathematical Image Processing / 813 \\ VII.9 Medical Imaging / 816 \\ VII.10 Compressed Sensing / 823 \\ VII.11 Programming Languages: An Applied Mathematics View / 828 \\ VII.12 High-Performance Computing / 839 \\ VII.13 Visualization / 843 \\ VII.14 Electronic Structure Calculations (Solid State Physics) / 847 \\ VII.15 Flame Propagation / 852 \\ VII.16 Imaging the Earth Using Green's Theorem / 857 \\ VII.17 Radar Imaging / 860 \\ VII.18 Modeling a Pregnancy Testing Kit / 864 \\ VII.19 Airport Baggage Screening with X-Ray Tomography / 866 \\ VII.20 Mathematical Economics / 868 \\ VII.21 Mathematical Neuroscience / 873 \\ VII.22 Systems Biology / 879 \\ VII.23 Communication Networks / 883 \\ VII.24 Text Mining / 887 \\ VII.25 Voting Systems / 891 \\ Part VIII: Final Perspectives \\ VIII.1 Mathematical Writing / 897 \\ VIII.2 How to Read and Understand a Paper / 903 \\ VIII.3 How to Write a General Interest Mathematics Book / 906 \\ VIII.4 Workflow / 912 \\ VIII.5 Reproducible Research in the Mathematical Sciences / 916 \\ VIII.6 Experimental Applied Mathematics / 925 \\ VIII.7 Teaching Applied Mathematics / 933 \\ VIII.8 Mediated Mathematics: Representations of Mathematics in Popular Culture and Why These Matter / 943 \\ VIII.9 Mathematics and Policy / 953 \\ Index / 963", } @Proceedings{Muller:2015:ISC, editor = "Jean-Michel Muller and Arnaud Tisserand and Julio Villalba", booktitle = "{2015 IEEE 22nd Symposium on Computer Arithmetic (ARITH 2015) Lyon, France, 22--24 June 2015}", title = "{2015 IEEE 22nd Symposium on Computer Arithmetic (ARITH 2015) Lyon, France, 22--24 June 2015}", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "xii + 176", year = "2015", ISBN = "1-4799-8665-8, 1-4799-8663-1", ISBN-13 = "978-1-4799-8665-1, 978-1-4799-8663-7", ISSN = "1063-6889", LCCN = "QA76.9.C62 S95 2015", bibdate = "Sat Aug 01 08:03:11 2015", bibsource = "https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/mathcw.bib", URL = "https://ieeexplore.ieee.org/servlet/opac?punumber=7193754", acknowledgement = ack-nhfb, keywords = "computer arithmetic units; correctness proofs; cryptography; domain specific designs; error analysis; exascale computing; floating point arithmetic; floating-point error analysis; formal verification; function approximation; modular arithmetic; theorem proving; verification", } @Proceedings{Greuel:2016:MSI, editor = "Gert-Martin Greuel", booktitle = "{Mathematical Software --- ICMS 2016: 5th International Conference, Berlin, Germany, July 11--14, 2016: proceedings}", title = "{Mathematical Software --- ICMS 2016: 5th International Conference, Berlin, Germany, July 11--14, 2016: proceedings}", volume = "9725", publisher = pub-SV, address = pub-SV:adr, pages = "xxiv + 532", year = "2016", DOI = "https://doi.org/10.1007/978-3-319-42432-3", ISBN = "3-319-42431-9 (print), 3-319-42432-7 (electronic)", ISBN-13 = "978-3-319-42431-6 (print), 978-3-319-42432-3 (electronic)", ISSN = "0302-9743 (print), 1611-3349 (electronic)", ISSN-L = "0302-9743", LCCN = "QA76.9.M35", bibdate = "Mon Feb 5 08:28:37 MST 2018", bibsource = "fsz3950.oclc.org:210/WorldCat; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/numana2010.bib", series = ser-LNCS # "\slash " # ser-LNAI, URL = "http://zbmath.org/?q=an:1342.68017", abstract = "This book constitutes the proceedings of the 5th International Conference on Mathematical Software, ICMS 2015, held in Berlin, Germany, in July 2016. The 68 papers included in this volume were carefully reviewed and selected from numerous submissions. The papers are organized in topical sections named: univalent foundations and proof assistants; software for mathematical reasoning and applications; algebraic and toric geometry; algebraic geometry in applications; software of polynomial systems; software for numerically solving polynomial systems; high-precision arithmetic, effective analysis, and special functions; mathematical optimization; interactive operation to scientific artwork and mathematical reasoning; information services for mathematics: software, services, models, and data; semDML: towards a semantic layer of a world digital mathematical library; miscellanea.", acknowledgement = ack-nhfb, } @Proceedings{Montuschi:2016:ISC, editor = "Paolo Montuschi and Michael Schulte and Javier Hormigo and Stuart Oberman and Nathalie Revol", booktitle = "{2016 IEEE 23nd Symposium on Computer Arithmetic (ARITH 2016), Santa Clara, California, USA, 10--13 July 2016}", title = "{2016 IEEE 23nd Symposium on Computer Arithmetic (ARITH 2016), Santa Clara, California, USA, 10--13 July 2016}", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "xxi + 182", year = "2016", ISBN = "1-5090-1615-5", ISBN-13 = "978-1-5090-1615-0", ISSN = "1063-6889", LCCN = "QA76.9.C62 S95 2016", bibdate = "Fri Dec 16 15:16:45 2016", bibsource = "https://www.math.utah.edu/pub/tex/bib/cryptography2000.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/mathcw.bib", URL = "https://ieeexplore.ieee.org/servlet/opac?punumber=7562813", acknowledgement = ack-nhfb, keywords = "computer arithmetic units; correctness proofs; cryptography; domain specific designs; error analysis; exascale computing; floating point arithmetic; floating-point error analysis; formal verification; function approximation; modular arithmetic; theorem proving; verification", } @Proceedings{Burgess:2017:ISC, editor = "Neil Burgess and Javier Bruguera and Florent de Dinechin", booktitle = "{24th IEEE Symposium on Computer Arithmetic (ARITH 24), London, UK, 24--26 July 2017}", title = "{2017 IEEE 24th Symposium on Computer Arithmetic (ARITH 24), London, UK, 24--26 July 2017}", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "xii + 198", year = "2017", ISBN = "1-5386-1966-0 (print), 1-5386-1965-2, 1-5386-1964-4", ISBN-13 = "978-1-5386-1966-7 (print), 978-1-5386-1965-0, 978-1-5386-1964-3", ISSN = "1063-6889", LCCN = "QA76.9.C62 S95 2017", bibdate = "Fri Nov 17 10:14:11 2017", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/h/higham-nicholas-john.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/gnu.bib", URL = "https://ieeexplore.ieee.org/servlet/opac?punumber=8019911", acknowledgement = ack-nhfb, keywords = "computer arithmetic units; correctness proofs; cryptography; domain specific designs; error analysis; exascale computing; floating point arithmetic; floating-point error analysis; formal verification; function approximation; modular arithmetic; theorem proving; verification", } @Proceedings{Tenca:2018:PIS, editor = "Alexandre Tenca and Naofumi Takagi", booktitle = "Proceedings of the {25th International Symposium on Computer Arithmetic, 25--27 June 2018 Amherst, MA, USA}", title = "Proceedings of the {25th International Symposium on Computer Arithmetic, 25--27 June 2018 Amherst, MA, USA}", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "17 + 152", month = jun, year = "2018", DOI = "https://doi.org/10.1109/ARITH.2018.8464697", ISBN = "1-5386-2612-8 (USB), 1-5386-2665-9", ISBN-13 = "978-1-5386-2612-2 (USB), 978-1-5386-2613-9, 978-1-5386-2665-8", ISSN = "2576-2265", LCCN = "QA76.9.C62", bibdate = "Fri Jan 31 08:05:31 2020", bibsource = "https://www.math.utah.edu/pub/tex/bib/cryptography2000.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", note = "IEEE catalog number CFP18121-USB.", abstract = "Presents the title page of the proceedings record.", acknowledgement = ack-nhfb, subject = "ARITH-25; Computer arithmetic; Congresses; Computer programming; Floating-point arithmetic; Computer arithmetic and logic units", } @Proceedings{Takagi:2019:ISC, editor = "Naofumi Takagi and Sylvie Boldo and Martin Langhammer", booktitle = "{2019 IEEE 26th Symposium on Computer Arithmetic ARITH-26 (2019), Kyoto, Japan, 10--12 June 2019}", title = "{2019 IEEE 26th Symposium on Computer Arithmetic ARITH-26 (2019), Kyoto, Japan, 10--12 June 2019}", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "15 + 220", month = jun, year = "2019", DOI = "https://doi.org/10.1109/ARITH.2019.00001", ISBN = "1-72813-366-1", ISBN-13 = "978-1-72813-366-9", ISSN = "1063-6889", ISSN-L = "1063-6889", bibdate = "Fri Jan 31 08:18:07 2020", bibsource = "https://www.math.utah.edu/pub/tex/bib/cryptography2000.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", abstract = "Presents the title page of the proceedings record.", acknowledgement = ack-nhfb, keywords = "ARITH-26", } @Proceedings{Bigatti:2020:MSI, editor = "Anna Maria Bigatti and Jacques Carette and James H. Davenport and Michael Joswig and Timo de Wolff", booktitle = "Mathematical Software --- {ICMS 2020: 7th International Conference, Braunschweig, Germany, July 13--16, 2020, Proceedings}", title = "Mathematical Software --- {ICMS 2020: 7th International Conference, Braunschweig, Germany, July 13--16, 2020, Proceedings}", publisher = pub-SV, address = pub-SV:adr, pages = "xxiii + 494", year = "2020", DOI = "https://doi.org/10.1007/978-3-030-52200-1", bibdate = "Sat Sep 23 06:50:01 2023", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/julia.bib; https://www.math.utah.edu/pub/tex/bib/macaulay2.bib; https://www.math.utah.edu/pub/tex/bib/matlab.bib; https://www.math.utah.edu/pub/tex/bib/python.bib; https://www.math.utah.edu/pub/tex/bib/texbook3.bib", acknowledgement = ack-nhfb, tableofcontents = "Front Matter / / i--xxiii \\ Gr{\"o}bner Bases in Theory and Practice \\ Front Matter / / 1--1 A Design and an Implementation of an Inverse Kinematics Computation in Robotics Using Gr{\"o}bner Bases / Noriyuki Horigome, Akira Terui, Masahiko Mikawa / 3--13 \\ Real Algebraic Geometry \\ Front Matter / / 15--15 \\ Curtains in CAD: Why Are They a Problem and How Do We Fix Them? / Akshar Nair, James Davenport, Gregory Sankaran / 17--26 \\ Chordality Preserving Incremental Triangular Decomposition and Its Implementation / Changbo Chen / 27--36 \\ Algebraic Geometry via Numerical Computation \\ Front Matter / / 37--37 \\ $\mathbb{Q}(\sqrt{-3})$-Integral Points on a Mordell Curve / Francesca Bianchi / 39--50 \\ A Numerical Approach for Computing Euler Characteristics of Affine Varieties / Xiaxin Li, Jose Israel Rodriguez, Botong Wang / 51--60 \\ Evaluating and Differentiating a Polynomial Using a Pseudo-witness Set / Jonathan D. Hauenstein, Margaret H. Regan / 61--69 \\ Computational Algebraic Analysis \\ Front Matter / / 71--71 \\ Algorithms for Pfaffian Systems and Cohomology Intersection Numbers of Hypergeometric Integrals / Saiei-Jaeyeong Matsubara-Heo, Nobuki Takayama / 73--84 \\ Software for Number Theory and Arithmetic Geometry \\ Front Matter / / 85--85 \\ Computations with Algebraic Surfaces / Andreas-Stephan Elsenhans, J{\"o}rg Jahnel / 87--93 \\ Evaluating Fractional Derivatives of the Riemann Zeta Function / Ricky E. Farr, Sebastian Pauli, Filip Saidak / 94--101 \\ Groups and Group Actions \\ Front Matter / / 103--103 \\ Towards Efficient Normalizers of Primitive Groups / Sergio Siccha / 105--114 \\ Homomorphic Encryption and Some Black Box Attacks / Alexandre Borovik, {\c{S}}{\"u}kr{\"u} Yal{\c{c}}{\i}nkaya / 115--124 \\ Nilpotent Quotients of Associative $\mathbb{Z}$-Algebras and Augmentation Quotients of Baumslag--Solitar Groups / Tobias Moede / 125--130 \\ The GAP Package LiePRing / Bettina Eick, Michael Vaughan-Lee / 131--140 \\ The Classification Problem in Geometry \\ Front Matter / / 141--141 \\ Classifying Simplicial Dissections of Convex Polyhedra with Symmetry / Anton Betten, Tarun Mukthineni / 143--152 \\ Classification Results for Hyperovals of Generalized Quadrangles / Bart De Bruyn / 153--161 \\ Isomorphism and Invariants of Parallelisms of Projective Spaces / Svetlana Topalova, Stela Zhelezova / 162--172 \\ Classification of Linear Codes by Extending Their Residuals / Stefka Bouyuklieva, Iliya Bouyukliev / 173--180 \\ The Program Generation in the Software Package QextNewEdition / Iliya Bouyukliev / 181--189 \\ Polyhedral Methods in Geometry and Optimization \\ Front Matter / / 191--191 \\ Algebraic Polytopes in Normaliz / Winfried Bruns / 193--201 \\ Real Tropical Hyperfaces by Patchworking in polymake / Michael Joswig, Paul Vater / 202--211 \\ Practical Volume Estimation of Zonotopes by a New Annealing Schedule for Cooling Convex Bodies / Apostolos Chalkis, Ioannis Z. Emiris, Vissarion Fisikopoulos / 212--221 \\ Slack Ideals in Macaulay2 / Antonio Macchia, Amy Wiebe / 222--231 \\ Hyperplane Arrangements in polymake / Lars Kastner, Marta Panizzut / 232--240 \\ A Convex Programming Approach to Solve Posynomial Systems / Marianne Akian, Xavier Allamigeon, Marin Boyet, St{\'e}phane Gaubert / 241--250 \\ Univalent Mathematics: Theory and Implementation \\ Front Matter / / 251--251 \\ Equality Checking for General Type Theories in Andromeda 2 / Andrej Bauer, Philipp G. Haselwarter, Anja Petkovi / 253--259 \\ Artificial Intelligence and Mathematical Software \\ Front Matter / / 261--261 \\ GeoLogic --- Graphical Interactive Theorem Prover for Euclidean Geometry / Miroslav Ol{\v{s}}{\'a}k / 263--271 \\ A Formalization of Properties of Continuous Functions on Closed Intervals / Yaoshun Fu, Wensheng Yu / 272--280 \\ Variable Ordering Selection for Cylindrical Algebraic Decomposition with Artificial Neural Networks / Changbo Chen, Zhangpeng Zhu, Haoyu Chi / 281--291 \\ Applying Machine Learning to Heuristics for Real Polynomial Constraint Solving / Christopher W. Brown, Glenn Christopher Daves / 292--301 \\ A Machine Learning Based Software Pipeline to Pick the Variable Ordering for Algorithms with Polynomial Inputs / Dorian Florescu, Matthew England / 302--311 \\ Databases in Mathematics \\ Front Matter / / 313--313 \\ FunGrim: A Symbolic Library for Special Functions / Fredrik Johansson / 315--323 \\ Accelerating Innovation Speed in Mathematics by Trading Mathematical Research Data \\ Front Matter / / 325--325 \\ Operational Research Literature as a Use Case for the Open Research Knowledge Graph / Mila Runnwerth, Markus Stocker, S{\"o}ren Auer / 327--334 \\ Making Presentation Math Computable: Proposing a Context Sensitive Approach for Translating {\LaTeX} to Computer Algebra Systems / Andr{\'e} Greiner-Petter, Moritz Schubotz, Akiko Aizawa, Bela Gipp / 335--341 \\ Employing C++ Templates in the Design of a Computer Algebra Library / Alexander Brandt, Robert H. C. Moir, Marc Moreno Maza / 342--352 \\ Mathematical World Knowledge Contained in the Multilingual Wikipedia Project / Dennis Tobias Halbach / 353--361 \\ Archiving and Referencing Source Code with Software Heritage / Roberto Di Cosmo / 362--373 \\ The Jupyter Environment for Computational Mathematics \\ Front Matter / / 375--375 \\ Polymake.jl: A New Interface to polymake / Marek Kaluba, Benjamin Lorenz, Sascha Timme / 377--385 \\ Web Based Notebooks for Teaching, an Experience at Universidad de Zaragoza / Miguel Angel Marco Buzunariz / 386--392 \\ Phase Portraits of Bi-dimensional Zeta Values / Olivier Bouillot / 393--405 \\ Prototyping Controlled Mathematical Languages in Jupyter Notebooks / Jan Frederik Schaefer, Kai Amann, Michael Kohlhase / 406--415 \\ General Session \\ Front Matter / / 417--417 \\ Method to Create Multiple Choice Exercises for Computer Algebra System / Tatsuyoshi Hamada, Yoshiyuki Nakagawa, Makoto Tamura / 419--425 \\ A Flow-Based Programming Environment for Geometrical Construction / Kento Nakamura, Kazushi Ahara / 426--431 \\ MORLAB --- A Model Order Reduction Framework in MATLAB and Octave / Peter Benner, Steffen W. R. Werner / 432--441 \\ FlexRiLoG --- A SageMath Package for Motions of Graphs / Georg Grasegger, Jan Legersk{\'y} / 442--450 \\ Markov Transition Matrix Analysis of Mathematical Expression Input Models / Francis Quinby, Seyeon Kim, Sohee Kang, Marco Pollanen, Michael G. Reynolds, Wesley S. Burr / 451--461 \\ Certifying Irreducibility in $\mathbb{Z}[ ]$ / John Abbott / 462--472 \\ A Content Dictionary for In-Object Comments / Lars Hellstr{\"o}m / 473--481 \\ Implementing the Tangent Graeffe Root Finding Method / Joris van der Hoeven, Michael Monagan / 482--492 \\ Back Matter / / 493--494", } @Proceedings{Cornea:2020:ISC, editor = "Marius Cornea and Weiqiang Liu and Arnaud Tisserand", booktitle = "{2020 27th IEEE Symposium on Computer Arithmetic: ARITH 2020: proceedings: Portland, Oregon, USA, 7--10 June 2020}", title = "{2020 27th IEEE Symposium on Computer Arithmetic: ARITH 2020: proceedings: Portland, Oregon, USA, 7--10 June 2020}", publisher = pub-IEEE, address = pub-IEEE:adr, year = "2020", DOI = "https://doi.org/10.1109/ARITH48897.2020", ISBN = "1-72817-120-2, 1-72817-121-0", ISBN-13 = "978-1-72817-120-3, 978-1-72817-121-0", LCCN = "????", bibdate = "Wed Jul 7 06:23:45 MDT 2021", bibsource = "fsz3950.oclc.org:210/WorldCat; https://www.math.utah.edu/pub/tex/bib/benfords-law.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "https://ieeexplore.ieee.org/servlet/opac?punumber=9146973", acknowledgement = ack-nhfb, } @Book{Ismail:2020:ESF, editor = "Mourad H. Ismail and Walter Van Assche", booktitle = "Encyclopedia of Special Functions: the {Askey--Bateman Project}. Volume 1, Univariate Orthogonal Polynomials", title = "Encyclopedia of Special Functions: the {Askey--Bateman Project}. Volume 1, Univariate Orthogonal Polynomials", publisher = pub-CAMBRIDGE, address = pub-CAMBRIDGE:adr, pages = "xiv + 388", year = "2020", DOI = "https://doi.org/10.1017/9780511979156", ISBN = "0-511-97915-0 (e-book), 0-521-19742-2 (hardcover), 1-108-76433-9 (e-book)", ISBN-13 = "978-0-511-97915-6 (e-book), 978-0-521-19742-7 (hardcover), 978-1-108-76433-9 (e-book)", LCCN = "QA351 .E63 2020", bibdate = "Fri Nov 10 17:02:40 MST 2023", bibsource = "fsz3950.oclc.org:210/WorldCat; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", abstract = "This is the first of three volumes that form the Encyclopedia of Special Functions, an extensive update of the Bateman Manuscript Project. Volume 1 contains most of the material on orthogonal polynomials, from the classical orthogonal polynomials of Hermite, Laguerre and Jacobi to the Askey--Wilson polynomials, which are the most general basic hypergeometric orthogonal polynomials. Separate chapters cover orthogonal polynomials on the unit circle, zeros of orthogonal polynomials and matrix orthogonal polynomials, with detailed results about matrix-valued Jacobi polynomials. A chapter on moment problems provides many examples of indeterminate moment problems. A thorough bibliography rounds off what will be an essential reference.", acknowledgement = ack-nhfb, subject = "Functions, Special; Encyclopedias; Fonctions sp{\'e}ciales; Encyclop{\'e}dies; Functions, Special", tableofcontents = "Frontmatter / i--iv \\ Contents / v--viii \\ Contributors / ix--x \\ Preface / xi--xiv \\ 1: Preliminaries / Mourad E. H. Ismail / 1--15 \\ 2: General Orthogonal Polynomials / Mourad E. H. Ismail / 16--50 \\ 3: Jacobi and Related Polynomials / Mourad E. H. Ismail / 51--99 \\ 4: Recursively Defined Polynomials / Mourad E. H. Ismail / 100--118 \\ 5: Wilson and Related Polynomials / Mourad E. H. Ismail / 119--128 \\ 6: Discrete Orthogonal Polynomials / Mourad E. H. Ismail / 129--156 \\ 7: Some $q$-Orthogonal Polynomials / Mourad E. H. Ismail / 157--177 \\ 8: The Askey--Wilson Family of Polynomials / Mourad E. H. Ismail / 178--198 \\ 9: Orthogonal Polynomials on the Unit Circle / Leonid Golinskii / 199--241 \\ 10: Zeros of Orthogonal Polynomials / Andrea Laforgia and Martin E. Muldoon / 242--268 \\ 11: The Moment Problem / Christian Berg and Jacob S. Christiansen / 269--306 \\ 12: Matrix--Valued Orthogonal Polynomials and Differential Equations / Antonio J. Dur{\'a}n and F. Alberto Gr{\"u}nbaum / 307--333 \\ 13: Some Families of Matrix--Valued Jacobi Orthogonal Polynomials / F. Alberto Gr{\"u}nbaum, I. Pacharoni and J. A. Tirao / 334--356 \\ References / 357--384 \\ Index / 385--388", } @Book{Koornwinder:2020:ESF, editor = "T. H. Koornwinder and Jasper V. Stokman", booktitle = "Encyclopedia of Special Functions: the {Askey--Bateman Project}. Volume 2. Multivariable Special Functions", title = "Encyclopedia of Special Functions: the {Askey--Bateman Project}. Volume 2. Multivariable Special Functions", publisher = pub-CAMBRIDGE, address = pub-CAMBRIDGE:adr, pages = "xii + 427", year = "2020", DOI = "https://doi.org/10.1017/9780511777165", ISBN = "0-511-77716-7 (e-book), 1-107-00373-3 (hardcover)", ISBN-13 = "978-0-511-77716-5 (e-book), 978-1-107-00373-6 (hardcover)", LCCN = "QA351 .E63 2021", bibdate = "Fri Nov 10 17:39:45 MST 2023", bibsource = "fsz3950.oclc.org:210/WorldCat; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, tableofcontents = "Frontmatter / i--iv \\ Contents / v--viii \\ List of Contributors / ix--x \\ Preface / xi--xii \\ 1: General Overview of Multivariable Special Functions / T. H. Koornwinder, J. V. Stokman / 1--18 \\ 2: Orthogonal Polynomials of Several Variables / Yuan Xu / 19--78 \\ 3: Appell and Lauricella Hypergeometric Functions / K. Matsumoto / 79--100 \\ 4: A-Hypergeometric Functions / N. Takayama / 101--121 \\ 5: Hypergeometric and Basic Hypergeometric Series and Integrals Associated with Root Systems / M. J. Schlosser / 122--158 \\ 6: Elliptic Hypergeometric Functions Associated with Root Systems / H. Rosengren, S. O. Warnaar / 159--186 \\ 7: Dunkl Operators and Related Special Functions / C. F. Dunkl / 187--216 \\ 8: Jacobi Polynomials and Hypergeometric Functions Associated with Root Systems / G. J. Heckman, E. M. Opdam / 217--257 \\ 9: Macdonald--Koornwinder Polynomials / J. V. Stokman / 258--313 \\ 10: Combinatorial Aspects of Macdonald and Related Polynomials / J. Haglund / 314--367 \\ 11: Knizhnik--Zamolodchikov-Type Equations, Selberg Integrals and Related Special Functions / V. Tarasov, A. Varchenko / 368--401 \\ 12: $9 j$--Coefficients and Higher / J. Van der Jeugt / 402--419 \\ Index / 420--428", } @Proceedings{IEEE:2021:ISC, editor = "{IEEE}", booktitle = "{2021 IEEE 28th Symposium on Computer Arithmetic: ARITH 2021: virtual conference, 14--16 June 2021: proceedings}", title = "{2021 IEEE 28th Symposium on Computer Arithmetic: ARITH 2021: virtual conference, 14--16 June 2021: proceedings}", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "????", year = "2021", DOI = "https://doi.org/10.1109/ARITH51176.2021", ISBN = "1-66542-293-9 (print), 1-66544-648-X (e-book)", ISBN-13 = "978-1-66542-293-2 (print), 978-1-66544-648-8 (e-book)", LCCN = "????", bibdate = "Thu Sep 21 10:36:52 MDT 2023", bibsource = "fsz3950.oclc.org:210/WorldCat; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetransemergtopcomput.bib; https://www.math.utah.edu/pub/tex/bib/risc-v.bib", acknowledgement = ack-nhfb, keywords = "ARITH-28", meetingname = "IEEE International Symposium on Computer Arithmetic 28. 2021", remark = "The 28th IEEE Symposium on Computer Arithmetic --- ARITH 2021 --- originally scheduled in Turin, Italy, is held in June 2021 as a virtual conference due to the uncertainty of the world health and travel situation.", } @Proceedings{IEEE:2022:ISC, editor = "{IEEE}", booktitle = "{2022 IEEE 29th Symposium on Computer Arithmetic: ARITH 2022: virtual conference, 12--14 September 2022: proceedings}", title = "{2022 IEEE 29th Symposium on Computer Arithmetic: ARITH 2022: virtual conference, 12--14 September 2022: proceedings}", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "????", year = "2022", DOI = "https://doi.org/10.1109/ARITH54963.2022", ISBN = "1-66547-827-6, 1-66547-828-4", ISBN-13 = "978-1-66547-827-4, 978-1-66547-828-1", LCCN = "????", bibdate = "Thu Sep 21 10:14:25 2023", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/risc-v.bib", acknowledgement = ack-nhfb, keywords = "ARITH-29", meetingname = "IEEE Symposium on Computer Arithmetic 29. 2022", } @Proceedings{IEEE:2023:PIS, editor = "{IEEE}", booktitle = "Proceedings: {2023 IEEE 30th Symposium on Computer Arithmetic: ARITH 2023, 4--6 September 2023 Portland, United States}", title = "Proceedings: {2023 IEEE 30th Symposium on Computer Arithmetic: ARITH 2023, 4--6 September 2023 Portland, United States}", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "167", year = "2023", ISBN-13 = "979-83-503-1923-1 (print), 979-83-503-1922-4 (electronic)", LCCN = "????", bibdate = "Wed May 08 09:18:10 2024", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/risc-v.bib", acknowledgement = ack-nhfb, keywords = "ARITH-30", } @Proceedings{IEEE:2024:PIS, editor = "{IEEE}", booktitle = "Proceedings: {2024 IEEE 31st Symposium on Computer Arithmetic: ARITH 2024, 10--12 June 2024, M{\'a}laga, Spain}", title = "Proceedings: {2024 IEEE 31st Symposium on Computer Arithmetic: ARITH 2024, 10--12 June 2024, M{\'a}laga, Spain}", publisher = pub-IEEE, address = pub-IEEE:adr, bookpages = "147", pages = "147", year = "2024", DOI = "https://doi.org/10.1109/ARITH61463.2024", ISBN-13 = "979-83-503-8432-1, 979-83-503-8433-8", ISSN = "2576-2265 (electronic), 1063-6889 (print-on-demand)", LCCN = "QA76.9.C62 .I578 2024", bibdate = "Thu Nov 13 11:32:36 2025", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/c/clenshaw-charles-w.bib; https://www.math.utah.edu/pub/bibnet/authors/h/higham-nicholas-john.bib; https://www.math.utah.edu/pub/bibnet/authors/o/olver-frank-w-j.bib; https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "https://ieeexplore.ieee.org/xpl/conhome/10579097/proceeding", acknowledgement = ack-nhfb, keywords = "ARITH 2024; ARITH-31", } @Proceedings{IEEE:2025:PIS, editor = "{IEEE}", booktitle = "Proceedings: {2025 IEEE 32nd Symposium on Computer Arithmetic: ARITH 2025, 4--7 May 2025 El Paso, [TX,] USA}", title = "Proceedings: {2025 IEEE 32nd Symposium on Computer Arithmetic: ARITH 2025, 4--7 May 2025 El Paso, [TX,] USA}", publisher = pub-IEEE, address = pub-IEEE:adr, bookpages = "xiv + 161", pages = "xiv + 161", year = "2025", DOI = "https://doi.org/10.1109/ARITH64983.2025", ISBN-13 = "979-83-315-2159-2, 979-83-315-2160-8", LCCN = "QA76.9.C62 .I578 2025", bibdate = "Thu Nov 13 12:47:07 2025", bibsource = "https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/risc-v.bib", URL = "https://ieeexplore.ieee.org/xpl/conhome/11037935/proceeding", acknowledgement = ack-nhfb, keywords = "ARITH 2025; ARITH-32", }