%%% -*-BibTeX-*- %%% ==================================================================== %%% BibTeX-file{ %%% author-1 = "Norbert Juffa", %%% author-2 = "Nelson H. F. Beebe", %%% version = "3.730", %%% date = "01 October 2024", %%% time = "07:14:16 MDT", %%% filename = "fparith.bib", %%% address-1 = "2445 Mission College Blvd., %%% Santa Clara, CA 95054, %%% USA", %%% address-2 = "University of Utah %%% Department of Mathematics, 110 LCB %%% 155 S 1400 E RM 233 %%% Salt Lake City, UT 84112-0090 %%% USA", %%% telephone-1 = "+1 408 727 1885", %%% telephone-2 = "+1 801 581 5254", %%% FAX-1 = "+1 408 727 1265", %%% FAX-2 = "+1 801 581 4148", %%% URL-2 = "https://www.math.utah.edu/~beebe", %%% checksum = "64245 211519 971884 9807031", %%% email-1 = "norbert at iit.com (Internet)", %%% email-2 = "beebe at math.utah.edu, beebe at acm.org, %%% beebe at computer.org (Internet)", %%% codetable = "ISO/ASCII", %%% keywords = "bibliography; BibTeX; constant-time %%% arithmetic; fixed-point arithmetic; %%% floating-point arithmetic; floating-point %%% exceptions; floating-point summation; %%% floating-point verification; IEEE 754 %%% arithmetic; inexact; Infinity; integer %%% arithmetic; logarithmic number system (LNS) %%% arithmetic; NaN (Not a Number); overflow; %%% posit arithmetic; residue number system; %%% rounding algorithms; Sets-Of-Real-Numbers %%% (SORN) arithmetic; subnormal; underflow; unum %%% arithmetic; zero divide", %%% license = "public domain", %%% supported = "yes", %%% docstring = "[From Norbert Juffa:] This is a bibliography %%% of material on floating-point arithmetic %%% that I came up with while doing research on %%% a floating-point package of his own. I %%% don't claim it to be anywhere near %%% complete. The material listed is only what %%% I myself possess. %%% %%% My main interest was in software based, %%% binary floating-point arithmetic on a %%% microprocessor, so you won't find much %%% material about the hardware used in %%% floating-point arithmetic (e.g., adders, %%% carry propagation schemes, higher radix %%% representation for multiplication and %%% division, etc.) in this list. There is also %%% not too much on non-binary floating-point %%% arithmetic. %%% %%% For most fields covered in this %%% bibliography, the important or historically %%% relevant articles should be included. There %%% is also some material on integer arithmetic %%% in this list as some of the methods used %%% with integer arithmetic contain interesting %%% ideas that may be useful in the realization %%% of a floating-point arithmetic package. %%% %%% Also, depending on the type of %%% microprocessor used, one may need to %%% implement integer multiplication and %%% division for use in the floating-point %%% package, so articles about this topic are %%% included as well. %%% %%% As I am German, there is a bit of material %%% in German in this bibliography. However, %%% English translations are provided for all %%% non-English titles. %%% %%% Thanks to the people who have helped me with %%% previous versions of this document by sending %%% me papers or additional references: Steven %%% Sommars (sesv at research.bell-labs.com), Jim %%% Kiernan (jmk at teak.cray.com), Warren %%% Ferguson (ferguson at seas.smu.edu), Nhuan %%% Doduc (ndoduc at framentec.fr), K. C. Ng %%% (kwok.ng at eng.sun.com). %%% %%% [From Nelson H. F. Beebe:] I took Norbert %%% Juffa's unmarked-up bibliography data, %%% converted it to BibTeX form, and then more %%% than doubled the size of the bibliography %%% by supplying additional entries on %%% floating-point arithmetic extracted from %%% my own extensive bibliography collections, %%% and from several online journal databases, %%% including the Compendex, OCLC, IEEE INSPEC, %%% MathSciNet, and UnCover databases. In August %%% 2010, access to an electronic copy of the %%% bibliography of Sterbenz's 1974 book allowed %%% further checks to make sure that all of his %%% references to early work are included here. %%% %%% The computation of sums of floating-point %%% numbers is a fundamental step in most %%% numerical computations. The special phrase %%% ``accurate floating-point summation'' appears %%% in the keywords values of publications that %%% address this important problem. %%% %%% The Web site %%% %%% http://www.acsel-lab.com/arithmetic/ %%% %%% archives papers from the IEEE conferences %%% on computer arithmetic (ARITH-1, ARITH-2, %%% ...). Keyword values for those papers %%% contain ARITH-n to more easily identify them. %%% %%% In \cite[p. 94]{Buchholz:1962:PCS}, S. G. %%% Campbell remarks: %%% %%% ``To avoid difficulties of limited range %%% and scaling in fixed-point arithmetic, %%% G. R. Stibitz in the early 1940's %%% proposed an automatic scaling procedure, %%% called {\emph floating-point arithmetic}, %%% which was incorporated in the Bell %%% Telephone Laboratories' Model V Relay %%% Computer. A similar procedure was %%% developed, apparently independently, for %%% the Harvard Mark II computer. Automatic %%% scaling was a controversial subject for %%% quite a few years. Many opposed it on the %%% basis that the programmer could not be %%% relieved of the responsibility of knowing %%% the size of his numbers and that %%% programmed scaling would give him better %%% control over significance. Nevertheless, %%% as early as 1950, users began to %%% incorporate automatic scaling on %%% fixed-point computers by means of %%% subroutines, first on the %%% plugboard-controlled CPC (Card Programmed %%% Calculator) and later on stored-program %%% machines. Then, after it had thus proved %%% its usefulness, floating-point arithmetic %%% was provided as a built-in feature, %%% starting with the IBM 704 and NORC %%% computers, and this gave an enormous %%% increase in speed over the %%% subroutines. Today floating-point %%% operation is available, at least as an %%% option, on all computers intended to be %%% used full- or part-time on substantial %%% scientific computing applications. In %%% view of the almost universal use of %%% floating-point arithmetic, it is %%% remarkable that there is very little %%% literature on the subject.'' %%% %%% We now know that Leonardo Torres y Quevedo %%% first suggested floating-point arithmetic on %%% paper in 1914 in Spain %%% \cite{TorresyQuevedo:1982:EAD}. Konrad Zuse %%% reinvented it independently in Germany in the %%% early 1936, and had it operational in the Z3, %%% an electromechanical computer, in 1941 %%% \cite{Zuse:1970:CML,Ceruzzi:1981:ECK, %%% Randell:1982:ODC,Schwarz:1981:EYC, %%% Zuse:1982:MAE,Zuse:1982:OCD,Zuse:1984:CML, %%% Zuse:1993:CML,Smiley:2010:MWI}. Zuse's work %%% remained unknown in the USA until about 1948. %%% F. L. Bauer's foreword to Zuse's %%% autobiography \cite{Zuse:1993:CML} suggests a %%% bust of Zuse engraved %%% %%% ``Creator of the first fully automated, %%% program-controlled and freely %%% programmable computer using binary %%% floating-point calculation. It was %%% operational in 1941.'' %%% %%% The experience with floating-point arithmetic %%% by George Stibitz on the Bell Labs Model V %%% General Purpose Relay Calculator (1945, but %%% floating-point arithmetic proposed in 1940 %%% Stibitz memo) \cite{Alt:1948:BTLa}, by %%% Howard Aiken on the Harvard Mark II (1947), %%% and by Gerald Alway and Jim Wilkinson on the %%% Pilot ACE in Britain (1946--1948) %%% \cite{Wilkinson:1975:PAA}, encouraged the %%% provision of floating-point arithmetic in the %%% American and British commercialization of %%% scientific computers in the 1950s. However, %%% Zuse's use of Infinity and Indefinite %%% predated by more than 15 years the Infinity %%% of the IBM 7030 Stretch (1956--1958, with %%% first customer delivery, to Los Alamos %%% Laboratory, in April 1961), and the Infinity %%% and Indefinite of the CDC 6400/6600/7600 %%% (1960--1963, with first customer delivery, to %%% Lawrence Livermore Laboratory, in September %%% 1964) %%% \cite{Thornton:1970:DCC,Thornton:1980:CP}. I %%% [NHFB] have so far been unable to find any %%% definitive published statement about whether %%% Zuse's ideas were known to, and inspired, the %%% CDC and IBM architects, or whether they %%% rediscovered them independently. %%% %%% It is our hope that this bibliography will %%% help to remedy the deficiency noted in the %%% last sentence of Campbell's remark, even %%% though the first edition of this bibliography %%% in 1994 appeared 32 years after the wonderful %%% book about IBM's Project Stretch, and it took %%% 48 years for that book to be included here. %%% %%% At version 3.730, the year coverage looked %%% like this: %%% %%% 1703 ( 1) 1811 ( 0) 1919 ( 0) %%% 1708 ( 0) 1816 ( 0) 1924 ( 2) %%% 1709 ( 0) 1817 ( 0) 1925 ( 1) %%% 1710 ( 0) 1818 ( 0) 1926 ( 2) %%% 1711 ( 0) 1819 ( 0) 1927 ( 1) %%% 1712 ( 1) 1820 ( 0) 1928 ( 0) %%% 1714 ( 0) 1822 ( 0) 1930 ( 2) %%% 1716 ( 0) 1824 ( 0) 1932 ( 3) %%% 1717 ( 0) 1825 ( 0) 1933 ( 1) %%% 1718 ( 0) 1826 ( 0) 1934 ( 1) %%% 1719 ( 0) 1827 ( 0) 1935 ( 1) %%% 1720 ( 0) 1828 ( 0) 1936 ( 4) %%% 1721 ( 0) 1829 ( 0) 1937 ( 3) %%% 1722 ( 0) 1830 ( 0) 1938 ( 4) %%% 1723 ( 0) 1831 ( 0) 1939 ( 2) %%% 1724 ( 0) 1832 ( 0) 1940 ( 3) %%% 1725 ( 0) 1833 ( 0) 1941 ( 1) %%% 1726 ( 1) 1834 ( 0) 1942 ( 3) %%% 1727 ( 0) 1835 ( 0) 1943 ( 2) %%% 1728 ( 0) 1836 ( 0) 1944 ( 1) %%% 1729 ( 0) 1837 ( 1) 1945 ( 2) %%% 1730 ( 0) 1838 ( 0) 1946 ( 8) %%% 1731 ( 0) 1839 ( 0) 1947 ( 10) %%% 1732 ( 0) 1840 ( 0) 1948 ( 7) %%% 1733 ( 0) 1841 ( 0) 1949 ( 4) %%% 1734 ( 0) 1842 ( 0) 1950 ( 4) %%% 1735 ( 0) 1843 ( 0) 1951 ( 10) %%% 1736 ( 0) 1844 ( 0) 1952 ( 7) %%% 1737 ( 0) 1845 ( 0) 1953 ( 4) %%% 1738 ( 0) 1846 ( 0) 1954 ( 7) %%% 1739 ( 0) 1847 ( 0) 1955 ( 9) %%% 1740 ( 0) 1848 ( 0) 1956 ( 10) %%% 1741 ( 0) 1849 ( 0) 1957 ( 17) %%% 1742 ( 0) 1850 ( 0) 1958 ( 17) %%% 1743 ( 0) 1851 ( 0) 1959 ( 26) %%% 1744 ( 0) 1852 ( 0) 1960 ( 23) %%% 1745 ( 0) 1853 ( 0) 1961 ( 31) %%% 1746 ( 0) 1854 ( 0) 1962 ( 32) %%% 1747 ( 0) 1855 ( 0) 1963 ( 25) %%% 1748 ( 0) 1856 ( 0) 1964 ( 31) %%% 1749 ( 0) 1857 ( 0) 1965 ( 57) %%% 1750 ( 0) 1858 ( 0) 1966 ( 39) %%% 1751 ( 0) 1859 ( 0) 1967 ( 46) %%% 1752 ( 0) 1860 ( 0) 1968 ( 35) %%% 1753 ( 0) 1861 ( 0) 1969 ( 55) %%% 1754 ( 0) 1862 ( 1) 1970 ( 58) %%% 1755 ( 0) 1863 ( 0) 1971 ( 64) %%% 1756 ( 0) 1864 ( 0) 1972 ( 62) %%% 1757 ( 0) 1865 ( 0) 1973 ( 83) %%% 1758 ( 0) 1866 ( 0) 1974 ( 50) %%% 1759 ( 0) 1867 ( 0) 1975 ( 103) %%% 1760 ( 0) 1868 ( 0) 1976 ( 57) %%% 1761 ( 0) 1869 ( 0) 1977 ( 78) %%% 1762 ( 0) 1870 ( 0) 1978 ( 107) %%% 1763 ( 0) 1871 ( 0) 1979 ( 80) %%% 1764 ( 0) 1872 ( 0) 1980 ( 111) %%% 1765 ( 0) 1873 ( 0) 1981 ( 123) %%% 1766 ( 0) 1874 ( 0) 1982 ( 89) %%% 1767 ( 0) 1875 ( 1) 1983 ( 155) %%% 1768 ( 0) 1876 ( 0) 1984 ( 104) %%% 1769 ( 0) 1877 ( 0) 1985 ( 150) %%% 1770 ( 0) 1878 ( 0) 1986 ( 98) %%% 1771 ( 0) 1879 ( 1) 1987 ( 146) %%% 1772 ( 0) 1880 ( 0) 1988 ( 160) %%% 1773 ( 0) 1881 ( 1) 1989 ( 192) %%% 1774 ( 0) 1882 ( 0) 1990 ( 156) %%% 1775 ( 0) 1883 ( 0) 1991 ( 234) %%% 1776 ( 0) 1884 ( 0) 1992 ( 187) %%% 1777 ( 0) 1885 ( 0) 1993 ( 201) %%% 1778 ( 0) 1886 ( 0) 1994 ( 179) %%% 1779 ( 0) 1887 ( 0) 1995 ( 169) %%% 1780 ( 0) 1888 ( 1) 1996 ( 186) %%% 1781 ( 0) 1889 ( 0) 1997 ( 140) %%% 1782 ( 0) 1890 ( 0) 1998 ( 133) %%% 1783 ( 0) 1891 ( 0) 1999 ( 199) %%% 1784 ( 0) 1892 ( 1) 2000 ( 155) %%% 1785 ( 0) 1893 ( 2) 2001 ( 211) %%% 1786 ( 0) 1894 ( 0) 2002 ( 168) %%% 1787 ( 0) 1895 ( 0) 2003 ( 153) %%% 1788 ( 0) 1896 ( 0) 2004 ( 187) %%% 1789 ( 0) 1897 ( 1) 2005 ( 183) %%% 1790 ( 0) 1898 ( 1) 2006 ( 134) %%% 1791 ( 0) 1899 ( 0) 2007 ( 157) %%% 1792 ( 0) 1900 ( 0) 2008 ( 85) %%% 1793 ( 0) 1901 ( 0) 2009 ( 125) %%% 1794 ( 0) 1902 ( 0) 2010 ( 96) %%% 1795 ( 0) 1903 ( 0) 2011 ( 144) %%% 1796 ( 0) 1904 ( 0) 2012 ( 84) %%% 1797 ( 0) 1905 ( 0) 2013 ( 92) %%% 1798 ( 0) 1906 ( 0) 2014 ( 59) %%% 1799 ( 0) 1907 ( 0) 2015 ( 112) %%% 1800 ( 0) 1908 ( 0) 2016 ( 102) %%% 1801 ( 0) 1909 ( 1) 2017 ( 137) %%% 1802 ( 0) 1910 ( 1) 2018 ( 106) %%% 1803 ( 0) 1911 ( 0) 2019 ( 164) %%% 1804 ( 0) 1912 ( 0) 2020 ( 117) %%% 1805 ( 0) 1913 ( 0) 2021 ( 115) %%% 1806 ( 0) 1914 ( 0) 2022 ( 123) %%% 1807 ( 0) 1915 ( 1) 2023 ( 135) %%% 1808 ( 0) 1916 ( 0) 2024 ( 38) %%% 1809 ( 0) 1917 ( 0) 2025 ( 1) %%% 19xx ( 2) %%% 20xx ( 2) %%% %%% Article: 3707 %%% Book: 375 %%% InBook: 2 %%% InCollection: 73 %%% InProceedings: 2070 %%% Manual: 69 %%% MastersThesis: 148 %%% Misc: 268 %%% Periodical: 3 %%% PhdThesis: 87 %%% Proceedings: 397 %%% TechReport: 403 %%% Unpublished: 9 %%% %%% Total entries: 7611 %%% %%% This bibliography is sorted by year, and %%% within each year, by author and title key, %%% with ``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.

@Preamble{ "\def \toenglish #1\endtoenglish{[{\em English:} #1\unskip]} " # "\ifx \undefined \booktitle \def \booktitle #1{{{\em #1}}} \fi" # "\ifx \undefined \circled \def \circled #1{(#1)} \fi" # "\ifx \undefined \mathbb \def \mathbb #1{{\bf #1}} \fi" # "\ifx \undefined \mathcal \def \mathcal #1{{\cal #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 \r \let \r = \ocirc \fi" # "\ifx \undefined \reg \def \reg {\circled{R}} \fi" # "\ifx \undefined \texttt \def \texttt #1{{\tt #1}} \fi" # "\ifx \undefined \TM \def \TM {${}^{\sc TM}$} \fi" # "\hyphenation{ Cor-dic COR-DIC Gleit-komma Gleit-komma-zahl-en Gleit-komma-pro-zess-or Gleit-komma-for-mat Mikro-rech-en-tech-nik Mikro-rech-ner-sys-tem-en Mikro-rech-ner Mi-yo-shi Quad-rat-wurz-eln Semi-nu-mer-i-cal Swartz-lan-der }" }

%%% ==================================================================== %%% Acknowledgement abbreviations:

@String{ack-jg= "Joe Grcar, email: \path=jfgrcar@lbl.gov="} @String{ack-jh= "John Harrison, email: \path=johnh@ichips.intel.com="} @String{ack-jr= "Jon Rokne, Department of Computer Science, The University of Calgary, 2500 University Drive N.W., Calgary, Alberta T2N 1N4, Canada"} @String{ack-kr= "Karin Remington, National Institute of Standards and Technology, Building 820, Room 365, Gaithersburg, MD 20899, USA, Tel: +1 301 975-5119, FAX: +1 301 990-4127, e-mail: \path|karin@cam.nist.gov|"} @String{ack-mfc= "Michael F. Cowlishaw, IBM UK (MP5), PO Box 31, Birmingham Road, Warwick, CV34 5JL, UK, e-mail: \path|mfc@uk.ibm.com|, URL: \path|http://www2.hursley.ibm.com/decimal|"} @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, FAX: +1 801 581 4148, 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-nj= "Norbert Juffa, 2445 Mission College Blvd. Santa Clara, CA 95054 USA email: \path=norbert@iit.com="} @String{ack-pb= "Preston Briggs, Tera Computer Company, 2815 Eastlake East, Seattle, WA 98102, USA, Tel: +1 206 325-0800, e-mail: \path|preston@tera.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="} @String{ack-sfo= "Stuart F. Oberman, e-mail: \path=stuart.oberman@amd.com="}

%%% ==================================================================== %%% Institute 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= "University of California"} @String{inst-BERKELEY:adr= "Berkeley, CA, USA"} @String{inst-BERKELEY-CPAM= "Center for Pure and Applied Mathematics, University of California"} @String{inst-BERKELEY-CPAM:adr= "Berkeley, CA, USA"} @String{inst-BERKELEY-CS= "Department of Computer Science, University of California"} @String{inst-BERKELEY-CS:adr= "Berkeley, CA, USA"} @String{inst-BERKELEY-EECS= "Department of Electrical Engineering and Computer Science, University of California, Berkeley"} @String{inst-BERKELEY-EECS:adr= "Berkeley, CA, USA"} @String{inst-BERKELEY-MATH-EECS= "Department of Mathematics and Department of Electrical Engineering and Computer Science, University of California, Berkeley"} @String{inst-BERKELEY-MATH-EECS:adr= "Berkeley, CA, USA"} @String{inst-CS-PURDUE= "Department of Computer Science, Purdue University"} @String{inst-CS-PURDUE:adr= "West Lafayette, IN 47907-2107, 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-ETH= "ETH Z{\"u}rich"} @String{inst-ETH:adr= "Z{\"u}rich, Switzerland"} @String{inst-HP= "Hewlett--Packard Corporation"} @String{inst-HP:adr= "Palo Alto, CA, USA"} @String{inst-INST-ADV-STUDY= "Institute for Advanced Study"} @String{inst-INST-ADV-STUDY:adr= "Princeton, NJ, USA"} @String{inst-INTEL= "Intel Corporation"} @String{inst-INTEL:adr= "Santa Clara, CA, USA"} @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-NLRC= "NASA Langley Research Center"} @String{inst-NLRC:adr= "Hampton, VA, USA"} @String{inst-NPL= "National Physical Laboratory"} @String{inst-NPL:adr= "Teddington, Middlesex TW11 0LW, UK"} @String{inst-MATHWORKS= "The MathWorks, Inc."} @String{inst-MATHWORKS:adr= "3 Apple Hill Drive, Natick, MA 01760-2098, USA"} @String{inst-MRC-WISCONSIN= "Mathematics Research Center, University of Wisconsin, Madison"} @String{inst-MRC-WISCONSIN:adr= "Madison, WI, 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"} @String{inst-UT-CS= "Department of Computer Science, University of Tennessee, Knoxville"} @String{inst-UT-CS:adr= "Knoxville, TN 37996, USA"}

%%% ==================================================================== %%% Journal abbreviations:

@String{j-ACTA-INFO= "Acta Informatica"} @String{j-ACTA-NUMERICA= "Acta Numerica"} @String{j-ACM-COMM-COMP-ALGEBRA= "ACM Communications in Computer Algebra"} @String{j-ACM-J-EXP-ALGORITHMICS= "ACM Journal of Experimental Algorithmics"} @String{j-ADA-LETT= "Ada Letters"} @String{j-ADV-COMPUT-MATH= "Advances in Computational Mathematics"} @String{j-ADV-SOFT-SCI-TECH= "Advances in software science and technology"} @String{j-ADV-THEORY-SIMUL= "Advanced Theory and Simulations"} @String{j-ALGORITHMICA= "Algorithmica"} @String{j-ALGORITHMS-BASEL= "Algorithms ({Basel})"} @String{j-ALTA-FREQ= "Alta frequenza"} @String{j-AM-J-MATH= "American Journal of Mathematics"} @String{j-AM-SCI= "American Scientist"} @String{j-AMER-J-PHYSICS= "American Journal of Physics"} @String{j-AMER-MATH-MONTHLY= "American Mathematical Monthly"} @String{j-AMER-STAT= "The American Statistician"} @String{j-ANN-HIST-COMPUT= "Annals of the History of Computing"} @String{j-ANN-MATH-ARTIF-INTELL= "Annals of Mathematics and Artificial Intelligence"} @String{j-ANN-MATH-STAT= "Annals of Mathematical Statistics"} @String{j-ANN-NUM-MATH= "Annals of Numerical Mathematics"} @String{j-APL-QUOTE-QUAD= "ACM SIGAPL APL Quote Quad"} @String{j-APPL-ALGEBRA-ENG-COMMUN-COMPUT= "Applicable algebra in engineering, communication and computing"} @String{j-APPL-MATH-COMP= "Applied Mathematics and Computation"} @String{j-APPL-MATH-LETT= "Applied Mathematics Letters"} @String{j-APPL-NUM-MATH= "Applied Numerical Mathematics"} @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-ASTA-ADV-STAT-ANAL= "AStA. Advances in Statistical Analysis"} @String{j-ASTRON-J= "Astronomical Journal"} @String{j-AUSTRALIAN-COMP-J= "Australian Computer Journal"} @String{j-AUTO-CTL-COMP-SCI= "Automatic Control and Computer Sciences"} @String{j-AUTOMATION-REMOTE-CTL= "Automation and Remote Control"} @String{j-BELL-LABS-RECORD= "Bell Laboratories Record"} @String{j-BELL-SYST-TECH-J= "The Bell System Technical Journal"} @String{j-BER-WISSENSCHAFTGESCH= "{Berichte zur Wissenschaftsgeschichte}"} @String{j-BIOL-CYBERN= "Biological Cybernetics"} @String{j-BIOMETRICS= "Biometrics"} @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-HIST-SCI= "British Journal for the History of Science"} @String{j-BULL-AMS= "Bulletin of the American Mathematical Society"} @String{j-BULL-CALCUTTA-MATH-SOC= "Bulletin of the Calcutta Mathematical Society"} @String{j-BYTE= "Byte Magazine"} @String{j-C-R-ACAD-BULGARE-SCI= "Comptes rendus de l'Acad{\'e}mie bulgare des sciences"} @String{j-C-R-ACAD-SCI-PARIS-SER-I-MATH= "Comptes Rendus des S{\'e}ances de l'Acad{\'e}mie des Sciences. S{\'e}rie I. Math{\'e}matique"} @String{j-CACM= "Communications of the Association for Computing Machinery"} @String{j-CAD= "Computer Aided Design"} @String{j-CCCUJ= "C/C++ Users Journal"} @String{j-CCPE= "Concurrency and Computation: Prac\-tice and Experience"} @String{j-CGF= "Com{\-}pu{\-}ter Graphics Forum"} @String{j-COED= "CoED"} @String{j-COLLEGE-MATH-J= "College Mathematics Journal"} @String{j-COMBIN-PROBAB-COMPUT= "Combinatorics, Probability and Computing"} @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-DESIGN= "Computer Design"} @String{j-COMP-EDU-J= "Computers in education journal"} @String{j-COMP-GEOM= "Computational Geometry. Theory and Applications"} @String{j-COMP-GEOSCI= "Computers and Geosciences"} @String{j-COMP-J= "The Computer Journal"} @String{j-COMP-LANG-MAG= "Computer Language Magazine"} @String{j-COMP-LANGS= "Computer Languages"} @String{j-COMP-PHYS-COMM= "Computer Physics Communications"} @String{j-COMP-SURV= "ACM Computing Surveys"} @String{j-COMPUT-AIDED-ENG= "Computer-aided engineering: CAE"} @String{j-COMPUT-EDGE= "ComputingEdge"} @String{j-COMPUT-ELECTR-ENG= "Computers and Electrical Engineering"} @String{j-COMPUT-MATH-APPL= "Computers and Mathematics with Applications"} @String{j-COMPUT-PHYS= "Computers in Physics"} @String{j-COMPUT-SCI-ENG= "Computing in Science and Engineering"} @String{j-COMPUT-SECUR= "Computers \& Security"} @String{j-COMPUTE= "Compute"} @String{j-COMPUTER= "Computer"} @String{j-COMPUTING= "Computing: Archiv fur informatik und numerik"} @String{j-COMPUTING-SUPPLEMENTUM= "Computing. 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Professional Group on Electronic Computers"} @String{j-TRANS-SOC-COMP-SIM= "Transactions of the Society for Computer Simulation"} @String{j-TRETS= "ACM Transactions on Reconfigurable Technology and Systems"} @String{j-TUGboat= "TUGboat (Journal of the {\TeX} Users Group)"} @String{j-WALL-ST-J= "Wall Street Journal"} @String{j-ZASTOS-MAT= "Zastosowania Matematyki"} @String{j-Z-ANGE-MATH-MECH= "Zeitschrift f{\"u}r Angewandte Mathematik und Mechanik"} @String{j-Z-ANGE-MATH-PHYS= "{Zeitschrift f{\"u}r Angewandte Mathematik und Physik = Journal of Applied Mathematics and Physics}"}

%%% ==================================================================== %%% Publishers and their addresses:

@String{pub-A-K-PETERS= "A. K. Peters, Ltd."} @String{pub-A-K-PETERS:adr= "Natick, MA, USA"} @String{pub-ACADEMIC= "Academic Press"} @String{pub-ACADEMIC:adr= "New York, NY, USA"} @String{pub-ACM= "ACM Press"} @String{pub-ACM:adr= "New York, NY 10036, USA"} @String{pub-AFIPS= "AFIPS Press"} @String{pub-AFIPS:adr= "Montvale, NJ, USA"} @String{pub-AIP= "American Institute of Physics"} @String{pub-AIP:adr= "Woodbury, NY, USA"} @String{pub-AKADEMIE-VERLAG= "Akademie-Verlag"} @String{pub-AKADEMIE-VERLAG:adr= "Berlin, Germany"} @String{pub-AMS= "American Mathematical Society"} @String{pub-AMS:adr= "Providence, RI, USA"} @String{pub-ANSI= "American National Standards Institute"} @String{pub-ANSI:adr= "1430 Broadway, New York, NY 10018, USA"} @String{pub-AW= "Ad{\-d}i{\-s}on-Wes{\-l}ey"} @String{pub-AW:adr= "Reading, MA, USA"} @String{pub-BALTZER= "J. C. Baltzer AG, Scientific Publishing Company"} @String{pub-BALTZER:adr= "Basel, Switzerland"} @String{pub-BENCUM= "Benjamin/Cummings Pub. Co."} @String{pub-BENCUM:adr= "Redwood City, CA, USA"} @String{pub-BIB-INST= "Bibliographisches Institut"} @String{pub-BIB-INST:adr= "Mannheim, Germany"} @String{pub-BIRKHAUSER-BOSTON= "Birkh{\"a}user Boston Inc."} @String{pub-BIRKHAUSER-BOSTON:adr= "Cambridge, MA, USA"} @String{pub-BIRKHAUSER= "Birkh{\"a}user"} @String{pub-BIRKHAUSER:adr= "Cambridge, MA, USA; Berlin, Germany; Basel, Switzerland"} @String{pub-BRADY= "Robert J. Brady Co."} @String{pub-BRADY:adr= "Bowie, MD 20715, USA"} @String{pub-CAMBRIDGE= "Cambridge University Press"} @String{pub-CAMBRIDGE:adr= "Cambridge, UK"} @String{pub-CHAPMAN-HALL= "Chapman and Hall, Ltd."} @String{pub-CHAPMAN-HALL:adr= "London, UK"} @String{pub-CLARENDON= "Clarendon Press"} @String{pub-CLARENDON:adr= "Oxford, UK"} @String{pub-CRC= "CRC Press"} @String{pub-CRC:adr= "2000 N.W. Corporate Blvd., Boca Raton, FL 33431-9868, USA"} @String{pub-CSLI= "CSLI Publications"} @String{pub-CSLI:adr= "Stanford, CA, USA"} @String{pub-DOUBLEDAY= "Doubleday"} @String{pub-DOUBLEDAY:adr= "New York, NY, USA"} @String{pub-DOVER= "Dover"} @String{pub-DOVER:adr= "New York, NY, USA"} @String{pub-DOWDEN= "Dowden, Hutchinson and Ross"} @String{pub-DOWDEN:adr= "Stroudsburg, PA, USA"} @String{pub-DP= "Digital Press"} @String{pub-DP:adr= "12 Crosby Drive, Bedford, MA 01730, USA"} @String{pub-ELLIS-HORWOOD= "Ellis Horwood"} @String{pub-ELLIS-HORWOOD:adr= "New York, NY, USA"} @String{pub-ELSEVIER= "Elsevier"} @String{pub-ELSEVIER:adr= "Amsterdam, The Netherlands"} @String{pub-ELSEVIER-MORGAN-KAUFMANN= "Elsevier\slash Morgan Kaufmann"} @String{pub-ELSEVIER-MORGAN-KAUFMANN:adr= "San Francisco, CA, USA"} @String{pub-ENH= "Elsevier North-Holland, Inc."} @String{pub-ENH:adr= "New York, NY, USA"} @String{pub-FSF= "{Free Software Foundation, Inc.}"} @String{pub-FSF:adr= "51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA, Tel: (617) 876-3296"} @String{pub-GRUYTER= "Walter de Gruyter"} @String{pub-GRUYTER:adr= "Berlin, Germany"} @String{pub-HANSER= "Carl Hanser"} @String{pub-HANSER:adr= "M{\"u}nchen, Germany"} @String{pub-HARVARD= "Harvard University Press"} @String{pub-HARVARD:adr= "Cambridge, MA, USA"} @String{pub-HAYDEN-BOOK= "Hayden Book Co."} @String{pub-HAYDEN-BOOK:adr= "Rochelle Park, NJ, USA"} @String{pub-HMSO= "HMSO"} @String{pub-HMSO:adr= "London, UK"} @String{pub-HRW= "Holt, Reinhart, and Winston"} @String{pub-HRW:adr= "New York, NY, USA"} @String{pub-IBM= "IBM Corporation"} @String{pub-IBM:adr= "San Jose, CA, USA"} @String{pub-IBM-WATSON= "IBM Thomas J. Watson Research Center"} @String{pub-IBM-WATSON:adr= "Yorktown Heights, NY, USA"} @String{pub-IEC= "International Electrotechnical Commission"} @String{pub-IEC:adr= "3, rue de Varemb{\'e}, PO Box 131, CH-1211 Geneva 20, Switzerland. Telephone: +41 22 919 02 11. Telefax: +41 22 919 03 00. E-mail: {\tt info@\penalty0iec.\penalty0ch}. URL: {\tt http:\penalty0//\penalty0www.\penalty0iec.\penalty0ch}"} @String{pub-IEE= "IEE (Institution of Electrical Engineers)"} @String{pub-IEE:adr= "London, UK"} @String{pub-IEEE= "IEEE Computer Society Press"} @String{pub-IEEE:adr= "1109 Spring Street, Suite 300, Silver Spring, MD 20910, USA"} @String{pub-IEEE-STD= "IEEE"} @String{pub-IEEE-STD:adr= "New York, NY, USA"} @String{pub-IMACS= "IMACS"} @String{pub-IMACS:adr= "Department of Computer Science, Rutgers University, New Brunswick, NJ"} @String{pub-INTEL= "Intel Corporation"} @String{pub-INTEL:adr= "Santa Clara, CA, USA"} @String{pub-ISO= "International Organization for Standardization"} @String{pub-ISO:adr= "Geneva, Switzerland"} @String{pub-IWT= "IWT"} @String{pub-IWT:adr= "Vaterstetten, Germany"} @String{pub-JOSEPH-HENRY= "Joseph Henry Press"} @String{pub-JOSEPH-HENRY:adr= "Washington, DC, USA"} @String{pub-JW= "John Wiley"} @String{pub-JW:adr= "New York, NY, USA"} @String{pub-KLUWER= "Kluwer Academic Publishers Group"} @String{pub-KLUWER:adr= "Norwell, MA, USA, and Dordrecht, The Netherlands"} @String{pub-MAC= "Macmillan Publishing Company"} @String{pub-MAC:adr= "New York, NY, USA"} @String{pub-MATH-ASSOC-AMER= "Mathematical Association of America"} @String{pub-MATH-ASSOC-AMER:adr= "Washington, DC, USA"} @String{pub-MCGRAW-HILL= "Mc{\-}Graw-Hill"} @String{pub-MCGRAW-HILL:adr= "New York, NY, USA"} @String{pub-MILLER-FREEMAN= "Miller Freeman Publications"} @String{pub-MILLER-FREEMAN:adr= "San Francisco, CA, USA"} @String{pub-MIT= "The MIT Press"} @String{pub-MIT:adr= "Cambridge, MA"} @String{pub-MORGAN-KAUFMANN= "Morgan Kaufmann Publishers"} @String{pub-MORGAN-KAUFMANN:adr= "Los Altos, CA 94022, USA"} @String{pub-MORGAN-KAUFMANN-ELSEVIER= "Morgan Kaufmann\slash Elsevier"} @String{pub-MORGAN-KAUFMANN-ELSEVIER:adr= "Waltham, MA, USA"} @String{pub-MOTOROLA= "Motorola Corporation"} @String{pub-MOTOROLA:adr= "Phoenix, AZ, USA"} @String{pub-MT= "M\&T Books"} @String{pub-MT:adr= "M\&T Publishing, Inc., 501 Galveston Drive, Redwood City, CA 94063, USA"} @String{pub-NASA= "National Aeronautics and Space Administration"} @String{pub-NASA:adr= "Washington, DC, USA"} @String{pub-NORTH-HOLLAND= "North-Hol{\-}land"} @String{pub-NORTH-HOLLAND:adr= "Amsterdam, The Netherlands"} @String{pub-OLDENBOURG= "R. Oldenbourg Verlag"} @String{pub-OLDENBOURG:adr= "Munich, Germany and Vienna, Austria"} @String{pub-ORA= "O'Reilly \& {Associates, Inc.}"} @String{pub-ORA:adr= "981 Chestnut Street, Newton, MA 02164, USA"} @String{pub-ORACLE= "Oracle Press (division of Osborne McGraw-Hill)"} @String{pub-ORACLE:adr= "Berkeley, CA, USA"} @String{pub-OXFORD= "Oxford University Press"} @String{pub-OXFORD:adr= "Walton Street, Oxford OX2 6DP, UK"} @String{pub-PENGUIN= "Penguin"} @String{pub-PENGUIN:adr= "London, UK"} @String{pub-PERGAMON= "Pergamon Press"} @String{pub-PERGAMON:adr= "New York, NY, USA"} @String{pub-PH= "Pren{\-}tice-Hall"} @String{pub-PH:adr= "Upper Saddle River, NJ 07458, USA"} @String{pub-PHPTR= "Pren{\-}tice-Hall PTR"} @String{pub-PHPTR:adr= "Upper Saddle River, NJ 07458, USA"} @String{pub-PITMAN= "Pitman Publishing Ltd."} @String{pub-PITMAN:adr= "London, UK"} @String{pub-PLENUM= "Plenum Press"} @String{pub-PLENUM:adr= "New York, NY, USA"} @String{pub-PROJECT-GUTENBERG= "Project Gutenberg"} @String{pub-PROJECT-GUTENBERG:adr= "P.O. Box 2782, Champaign, IL 61825-2782, USA"} @String{pub-PRINCETON= "Princeton University Press"} @String{pub-PRINCETON:adr= "Princeton, NJ, USA"} @String{pub-R-E-KRIEGER= "Robert E. Krieger Publishing Company"} @String{pub-R-E-KRIEGER:adr= "Huntington, NY, USA"} @String{pub-RESTON= "Reston Publishing Co. Inc."} @String{pub-RESTON:adr= "Reston, VA, USA"} @String{pub-SF= "Scott, Foresman and Company"} @String{pub-SF:adr= "Glenview, IL, USA"} @String{pub-SIAM= "Society for Industrial and Applied Mathematics"} @String{pub-SIAM:adr= "Philadelphia, PA, USA"} @String{pub-SIEMENS= "Siemens Aktiengesellschaft"} @String{pub-SIEMENS:adr= "Berlin and Munich, Germany"} @String{pub-SPIE= "Society of Photo-optical Instrumentation Engineers (SPIE)"} @String{pub-SPIE:adr= "Bellingham, WA, USA"} @String{pub-SPRINGER-WIEN= "Spring{\-}er"} @String{pub-SPRINGER-WIEN:adr= "Wien / New York"} @String{pub-SRNI= "Software Research Northwest, Inc."} @String{pub-SRNI:adr= "17710 100th Avenue SW, Vashon Island, WA 98070, USA"} @String{pub-SUN= "Sun Microsystems"} @String{pub-SUN:adr= "2550 Garcia Avenue, Mountain View, CA 94043, USA"} @String{pub-SV= "Spring{\-}er-Ver{\-}lag"} @String{pub-SV:adr= "Berlin, Germany~/ Heidelberg, Germany~/ London, UK~/ etc."} @String{pub-SYNGRESS= "Syngress Publishing, Inc."} @String{pub-SYNGRESS:adr= "Rockland, MA, USA"} @String{pub-TEUBNER= "B. G. Teubner"} @String{pub-TEUBNER:adr= "Stuttgart, Germany"} @String{pub-TEWI= "te-wi"} @String{pub-TEWI:adr= "M{\"u}nchen, Germany"} @String{pub-TOMASH= "Tomash Publishers"} @String{pub-TOMASH:adr= "Los Angeles, CA, USA"} @String{pub-USENIX= "USENIX"} @String{pub-USENIX-EL-CERRITO:adr= "P.O. Box 7, El Cerrito 94530, CA, USA"} @String{pub-USENIX:adr= "San Francisco, CA, USA"} @String{pub-USGPO= "United States Government Printing Office"} @String{pub-USGPO:adr= "Washington, DC, USA"} @String{pub-VEB-VERLAG-TECHNIK= "VEB Verlag Technik"} @String{pub-VEB-VERLAG-TECHNIK:adr= "Berlin, Germany"} @String{pub-VIKING= "Viking"} @String{pub-VIKING:adr= "New York, NY, USA"} @String{pub-WCB= "William C. Brown Company Publishers"} @String{pub-WCB:adr= "Dubuque, IA, USA"} @String{pub-WEITEK= "Weitek Corporation"} @String{pub-WEITEK:adr= "1060 E. Arques Ave., Sunnyvale, CA 94086-BRM-9759, USA"} @String{pub-WESTERN-PERIODICALS= "Western Periodicals Co.,"} @String{pub-WESTERN-PERIODICALS:adr= "North Hollywood, CA"} @String{pub-WILEY= "Wiley"} @String{pub-WILEY:adr= "New York, NY, USA"} @String{pub-WILEY-INTERSCIENCE= "Wiley-In{\-}ter{\-}sci{\-}ence"} @String{pub-WILEY-INTERSCIENCE: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-LNCS= "Lecture Notes in Computer Science"} @String{ser-LNCSE= "Lecture Notes in Computational Science and Engineering"} @String{ser-PROJECT-GUTENBERG= "Project Gutenberg"} @String{ser-SIGPLAN= "ACM SIGPLAN Notices"}

%%% ==================================================================== %%% Bibliography entries, sorted by year, and within each year, by %%% citation label, with ``bibsort -byyear''. %%% %%% NB: Some journals do not indicate a month for each issue, and a few %%% have volumes, but no issue numbers, or issues without volumes. Such %%% cases are identified by xxmonth, xxnumber, and xxvolume entries with a %%% value of "(none)", so that searches can still be used to identify %%% other entries where such information is still missing. %%% %%% ISBNs were introduced in 1972, so books published before that do not %%% have ISBN entries, unless the publisher supplied one in a post-1971 %%% printing. A few books, and many conference proceedings, published %%% since 1972, lack ISBNs; these are identified with xxISBN values of %%% (none), again to facilitate searching for missing values. %%% %%% TO DO: Add entries for more of these US patents on floating-point %%% arithmetic: %%% %%% http://patft.uspto.gov/netacgi/nph-Parser?Sect1=PTO2&Sect2=HITOFF&u=%2Fnetahtml%2FPTO%2Fsearch-adv.htm&r=0&p=1&f=S&l=50&Query=ttl%2F%28floating+and+point+and+arithmetic%29%0D%0A%0D%0A&d=PALL %%% Searched on 17 October 2019 for "ttl/(floating and point and arithmetic)" %%% %%% 1 10,416,962 Decimal and binary floating point arithmetic calculations %%% 2 10,318,241 Fixed-point and floating-point arithmetic operator circuits in specialized processing blocks %%% 3 10,289,413 Hybrid analog-digital floating point number representation and arithmetic %%% 4 10,216,480 Shift and divide operations using floating-point arithmetic %%% 5 10,216,479 Apparatus and method for performing arithmetic operations to accumulate floating-point numbers %%% 6 10,175,944 Mixed-precision floating-point arithmetic circuitry in specialized processing blocks %%% 7 10,073,676 Reduced floating-point precision arithmetic circuitry %%% 8 10,042,606 Fixed-point and floating-point arithmetic operator circuits in specialized processing blocks %%% 9 9,904,514 Fused floating-point arithmetic circuitry %%% 10 9,665,346 Performing arithmetic operations using both large and small floating point values %%% %%% 11 9,600,234 Floating-point arithmetic device, semiconductor device and information processing system %%% 12 8,984,041 Performing arithmetic operations using both large and small floating point values %%% 13 8,909,690 Performing arithmetic operations using both large and small floating point values %%% 14 8,793,533 Method and device for performing failsafe hardware-independent floating-point arithmetic %%% 15 8,694,567 Method and apparatus for arithmetic operation on a value represented in a floating-point format %%% 16 8,615,542 Multi-function floating point arithmetic pipeline %%% 17 8,166,092 Arithmetic device for performing division or square root operation of floating point number and arithmetic method therefor %%% 18 8,117,426 System and apparatus for group floating-point arithmetic operations %%% 19 7,949,696 Floating-point number arithmetic circuit for handling immediate values %%% 20 7,730,287 Method and software for group floating-point arithmetic operations %%% %%% 21 7,530,061 Programmatic access to the widest mode floating-point arithmetic supported by a processor %%% 22 7,392,274 Multi-function floating point arithmetic pipeline %%% 23 7,188,133 Floating point number storage method and floating point arithmetic device %%% 24 7,069,288 Floating point system with improved support of interval arithmetic %%% 25 6,965,908 Multi-function floating point arithmetic pipeline %%% 26 6,785,701 Apparatus and method of performing addition and rounding operation in parallel for floating-point arithmetic logical unit %%% 27 6,748,587 Programmatic access to the widest mode floating-point arithmetic supported by a processor %%% 28 6,571,264 Floating-point arithmetic device %%% 29 6,388,672 Graphic translate engine, floating point arithmetic unit and floating point multiply-add calculation unit %%% 30 6,314,442 Floating-point arithmetic unit which specifies a least significant bit to be incremented %%% %%% 31 6,205,461 Floating point arithmetic logic unit leading zero count using fast approximate rounding %%% 32 6,199,089 Floating point arithmetic logic unit rounding using at least one least significant bit %%% 33 6,185,593 Method and apparatus for parallel normalization and rounding technique for floating point arithmetic operations %%% 34 6,175,847 Shifting for parallel normalization and rounding technique for floating point arithmetic operations %%% 35 6,173,299 Method and apparatus for selecting an intermediate result for parallel normalization and rounding technique for floating point arithmetic operations %%% 36 6,151,615 Method and apparatus for formatting an intermediate result for parallel normalization and rounding technique for floating point arithmetic operations %%% 37 6,094,668 Floating point arithmetic unit including an efficient close data path %%% 38 6,088,715 Close path selection unit for performing effective subtraction within a floating point arithmetic unit %%% 39 6,085,212 Efficient method for performing close path subtraction in a floating point arithmetic unit %%% 40 6,085,211 Logic circuit and floating-point arithmetic unit %%% %%% 41 6,085,208 Leading one prediction unit for normalizing close path subtraction results within a floating point arithmetic unit %%% 42 6,078,939 Apparatus useful in floating point arithmetic %%% 43 5,931,896 Floating point addition and subtraction arithmetic circuit performing preprocessing of addition or subtraction operation rapidly %%% 44 5,931,895 Floating-point arithmetic processing apparatus %%% 45 5,901,076 Ripple carry shifter in a floating point arithmetic unit of a microprocessor %%% 46 5,831,884 Apparatus for performing arithmetic operation of floating point numbers capable of improving speed of operation by performing canceling prediction operation in parallel %%% 47 5,771,183 Apparatus and method for computation of sticky bit in a multi-stage shifter used for floating point arithmetic %%% 48 5,748,516 Floating point processing unit with forced arithmetic results %%% 49 5,726,926 Shifter for shifting floating point number utilizing arithmetic operation of redundant binary number, and adder containing the same %%% 50 5,677,861 Arithmetic apparatus for floating-point numbers %%% %%% 51 5,633,818 Method and apparatus for performing floating point arithmetic operation and rounding the result thereof %%% 52 5,631,859 Floating point arithmetic unit having logic for quad precision arithmetic %%% 53 5,568,412 Rounding-off method and apparatus of floating point arithmetic apparatus for addition/subtraction %%% 54 5,550,768 Rounding normalizer for floating point arithmetic operations %%% 55 5,515,308 Floating point arithmetic unit using modified Newton-Raphson technique for division and square root %%% 56 5,434,809 Method and apparatus for performing floating point arithmetic operation and rounding the result thereof %%% 57 5,432,727 Apparatus for computing a sticky bit for a floating point arithmetic unit %%% 58 5,375,196 Rapid line drawing in computer graphics employing floating-point arithmetic %%% 59 5,359,548 Floating-point arithmetic system having multiple-input addition and subtraction means %%% 60 5,341,321 Floating point arithmetic unit using modified Newton-Raphson technique for division and square root %%% %%% 61 5,313,415 Method and apparatus for performing floatinga point arithmetic operation and rounding the result thereof %%% 62 5,303,175 Floating point arithmetic unit %%% 63 5,301,137 Circuit for fixed point or floating point arithmetic operations %%% 64 5,241,493 Floating point arithmetic unit with size efficient pipelined multiply-add architecture %%% 65 5,222,037 Floating-point processor for performing an arithmetic operation on fixed-point part data with high speed rounding of a result %%% 66 5,212,662 Floating point arithmetic two cycle data flow %%% 67 5,212,661 Apparatus for performing floating point arithmetic operation and rounding the result thereof %%% 68 5,200,916 Mantissa processing circuit of floating point arithmetic apparatus for addition and subtraction %%% 69 5,128,889 Floating-point arithmetic apparatus with compensation for mantissa truncation %%% 70 5,121,351 Floating point arithmetic system %%% %%% 71 5,075,882 Normalizing circuit of floating-point arithmetic circuit for two input data %%% 72 4,999,803 Floating point arithmetic system and method %%% 73 4,999,802 Floating point arithmetic two cycle data flow %%% 74 4,991,130 Normalization control system for floating point arithmetic operation %%% 75 4,961,162 Multiprocessing system for performing floating point arithmetic operations %%% 76 4,908,788 Shift control signal generation circuit for floating-point arithmetic operation %%% 77 4,896,286 Floating-point arithmetic apparatus %%% 78 4,872,132 Method and means for block floating point arithmetic %%% 79 4,839,846 Apparatus for performing floating point arithmetic operations and rounding the result thereof %%% 80 4,811,272 Apparatus and method for an extended arithmetic logic unit for expediting selected floating point operations %%% %%% 81 4,796,218 Arithmetic circuit capable of executing floating point operation and fixed point operation %%% 82 4,796,217 Rounding unit for use in arithmetic processing of floating point data %%% 83 4,780,842 Cellular processor apparatus capable of performing floating point arithmetic operations %%% 84 4,758,972 Precision rounding in a floating point arithmetic unit %%% 85 4,683,547 Special accumulate instruction for multiple floating point arithmetic units which use a putaway bus to enhance performance %%% 86 4,649,508 Floating-point arithmetic operation system %%% 87 4,620,292 Arithmetic logic unit for floating point data and/or fixed point data %%% 88 4,562,553 Floating point arithmetic system and method with rounding anticipation %%% 89 4,217,657 Floating point arithmetic control %%% 90 4,161,784 Microprogrammable floating point arithmetic unit capable of performing arithmetic operations on long and short operands %%% %%% 91 4,130,879 Apparatus for performing floating point arithmetic operations using submultiple storage %%% 92 3,829,673 FLOATING POINT ARITHMETIC UNIT ADAPTED FOR CONVERTING A COMPUTER TO FLOATING POINT ARITHMETIC %%% 93 3,701,976 FLOATING POINT ARITHMETIC UNIT FOR A PARALLEL PROCESSING COMPUTER %%% 94 3,193,669 708/505 %%% 95 3,037,701 708/495 %%% 96 2,965,297 708/503 708/504 708/671

@Article{Leibniz:1703:EAB, author = "G.-W. Leibniz", title = "Explication de {l'Arithm{\'e}tique} Binaire. ({French}) [{Explanation} of binary arithmetic]", journal = "M{\'e}moires de math{\'e}matique et de physique de l Acad{\'e}mie royale des sciences", volume = "??", number = "??", pages = "85--89", month = "????", year = "1703", bibdate = "Fri Mar 17 08:48:17 2023", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", note = "Leibniz is often credited with the invention of the binary number system, but there is other work from his era, and detailed analysis of Leibniz's use of binary numbers. See \cite{Greve:1966:HLR,Glaser:1969:HMN,Glaser:1971:HBO,Lautz:1979:JLD,Glaser:1981:HBO,Esmay:2017:HNS,Strickland:2022:LBI,Zlatopolski:2023:PAV}", URL = "https://hal.archives-ouvertes.fr/ads-00104781/document", acknowledgement = ack-nhfb, language = "French", } @Book{Pelicano:1712:APQ, author = "Wenceslao Josepho Pelicano", title = "Arithmeticus Perfectus Qui tria numerare nescit, \ldots{}. ({Latin}) [{A} Perfect Arithmetic for who does not know how to count to three]", publisher = "????", address = "Prague, Czechoslovakia", pages = "????", year = "1712", bibdate = "Fri Mar 17 08:56:14 2023", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "https://play.google.com/books/reader?id=cNxdAAAAcAAJ", acknowledgement = ack-nhfb, longtitle = "Arithmeticus Perfectus Qui tria numerare nescit, sen Arithmetica dualis in qua Numerando non proceditur, nisi ad duo, \& tamen omnes questiones Arithmetic{\ae} negoti{\^o} facile enodar possunt ad perilsturem equite Mathematic{\ae}. ({Latin}) [{A} Perfect Arithmetic for who does not know how to count to three, is a dual arithmetic in which he does not proceed to numbering except to two, and yet all the questions of arithmetic business can easily be entangled to the peril of the mathematician]", } @Article{Colson:1726:SAN, author = "John {Colson, F.R.S.}", title = "A Short Account of Negativo-Affirmative Arithmetick", journal = j-PHILOS-TRANS-R-SOC-LOND, volume = "34", number = "392--398", pages = "161--173", month = "????", year = "1726", CODEN = "PTRSAV", DOI = "https://doi.org/10.1098/rstl.1726.0032", ISSN = "0370-2316 (print), 2053-9207 (electronic)", bibdate = "Tue Jul 28 14:21:05 2015", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "http://arith22.gforge.inria.fr/slides/s2-ercegovac.pdf", acknowledgement = ack-nhfb, fjournal = "Philosophical transactions of the Royal Society of London", journal-URL = "http://rsta.royalsocietypublishing.org/", remark = "Early work on signed-digit arithmetic, reported by Milo{\v{s}} Ercegovac in a talk at the ARITH'22 conference, Lyon, France, June 2015.", } @Unpublished{Babbage:1837:MPC, author = "Charles Babbage", title = "On the mathematical powers of the calculating engine", month = dec, year = "1837", bibdate = "Wed Oct 13 11:09:59 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", note = "Manuscript held by Museum of the History of Science, Oxford, UK. Reprinted in \cite[\S 2.1]{Randell:1982:ODC}.", acknowledgement = ack-nhfb, } @Book{Nystrom:1862:PNS, author = "John W. (John William) Nystrom", title = "Project of a new system of arithmetic, weight, measure and coins, proposed to be called the tonal system, with sixteen to the base", publisher = "J. B. Lippincott and Co.", address = "Philadelphia, PA, USA", pages = "106", year = "1862", LCCN = "QC96 .N95", bibdate = "Sat Oct 29 10:28:27 MDT 2016", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; z3950.loc.gov:7090/Voyager", URL = "http://unifoundry.com/tonal/; https://catalog.hathitrust.org/Record/011602816; https://lccn.loc.gov/04025433", acknowledgement = ack-nhfb, author-dates = "Johan Vilhelm Nystr{\"o}m (1824--1885)", remark = "Thanks to Paul Hardy of Unifoundry in San Diego, CA, USA, for information about Nystrom's promotion of hexadecimal arithmetic, and for creating a Web site about Nystrom's tonal system, as well as developing Unicode-compatible fonts to represent the additional digits needed by the system.", subject = "Weights and measures; Numeration; base-16 arithmetic; hexadecimal arithmetic", } @Article{Anonymous:1875:AOM, author = "Anonymous", title = "The Arithmetical Operations of Multiplication and Division", journal = j-SCI-AMER, volume = "32", number = "3", pages = "41--42", day = "16", month = jan, year = "1875", CODEN = "SCAMAC", DOI = "https://doi.org/10.1038/scientificamerican01161875-41", ISSN = "0036-8733 (print), 1946-7087 (electronic)", ISSN-L = "0036-8733", bibdate = "Fri May 17 10:29:24 MDT 2013", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/sciam1870.bib", URL = "http://www.nature.com/scientificamerican/journal/v32/n3/pdf/scientificamerican01161875-41.pdf", acknowledgement = ack-nhfb, fjournal = "Scientific American", journal-URL = "http://www.nature.com/scientificamerican", } @Article{Anonymous:1879:TRA, author = "Anonymous", title = "Three Rules for Abbreviating Multiplication", journal = j-SCI-AMER, volume = "41", number = "12", pages = "184--184", day = "20", month = sep, year = "1879", CODEN = "SCAMAC", DOI = "https://doi.org/10.1038/scientificamerican09201879-184", ISSN = "0036-8733 (print), 1946-7087 (electronic)", ISSN-L = "0036-8733", bibdate = "Fri May 17 10:36:54 MDT 2013", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/sciam1870.bib", URL = "http://www.nature.com/scientificamerican/journal/v41/n12/pdf/scientificamerican09201879-184.pdf", acknowledgement = ack-nhfb, fjournal = "Scientific American", journal-URL = "http://www.nature.com/scientificamerican", } @Article{Newcomb:1881:NFU, author = "Simon Newcomb", title = "Note on the frequency of use of the different digits in natural numbers", journal = j-AM-J-MATH, volume = "4", number = "1/4", pages = "39--40", year = "1881", CODEN = "AJMAAN", ISSN = "0002-9327 (print), 1080-6377 (electronic)", ISSN-L = "0002-9327", bibdate = "Thu Feb 15 16:35:24 2007", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "http://links.jstor.org/sici?sici=0002-9327%281881%294%3A1%2F4%3C39%3ANOTFOU%3E2.0.CO%3B2-K", abstract = "That the ten digits do not occur with equal frequency must be evident to any one making much use of logarithmic tables, and noticing how much faster the first pages wear out than the last ones. The first significant figure is oftener 1 than any other digit, and the frequency diminishes up to 9.", acknowledgement = ack-nhfb, fjournal = "American Journal of Mathematics", journal-URL = "https://www.jstor.org/journal/amerjmath", keywords = "Benford's Law; Law of Anomalous Numbers; Zipf's Law", remark = "From p. 40: ``The law of probability of the occurrence of numbers is such that all mantissas of their logarithms are equally probable.''", } @Article{Holman:1888:DPM, author = "Silas W. Holman", title = "Discussion of the precision of measurement", journal = "Technol. Q.", volume = "1", number = "??", pages = "194--207", month = "????", year = "1888", bibdate = "Mon May 21 17:12:03 2018", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, remark = "Cited in \cite[ref. 14]{Carter:2013:ESF}.", } @Book{Holman:1892:DPM, author = "Silas W. Holman", title = "Discussion of the Precision of Measurements: With Examples Taken Mainly From Physics And Electrical Engineering", publisher = "Ferris Brothers Printers", address = "New York, NY, USA", pages = "????", year = "1892", bibdate = "Mon May 21 17:14:39 2018", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, remark = "Cited in \cite[ref. 15]{Carter:2013:ESF}.", } @Article{Anonymous:1893:IDb, author = "Anonymous", title = "The Instantaneous Divider", journal = j-SCI-AMER, volume = "68", number = "21", pages = "325--325", day = "27", month = may, year = "1893", CODEN = "SCAMAC", DOI = "https://doi.org/10.1038/scientificamerican05271893-325a", ISSN = "0036-8733 (print), 1946-7087 (electronic)", ISSN-L = "0036-8733", bibdate = "Wed May 22 19:04:00 MDT 2013", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/sciam1890.bib", URL = "http://www.nature.com/scientificamerican/journal/v68/n21/pdf/scientificamerican05271893-325a.pdf", acknowledgement = ack-nhfb, fjournal = "Scientific American", journal-URL = "http://www.nature.com/scientificamerican", } @Article{Felt:1893:MA, author = "Dorr E. Felt", title = "Mechanical Arithmetic", journal = j-SCI-AMER, volume = "69", number = "20", pages = "309--310", day = "11", month = nov, year = "1893", CODEN = "SCAMAC", DOI = "https://doi.org/10.1038/scientificamerican11111893-309b", ISSN = "0036-8733 (print), 1946-7087 (electronic)", ISSN-L = "0036-8733", bibdate = "Wed May 22 19:04:25 MDT 2013", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/sciam1890.bib", URL = "http://www.nature.com/scientificamerican/journal/v69/n20/pdf/scientificamerican11111893-309b.pdf", acknowledgement = ack-nhfb, fjournal = "Scientific American", journal-URL = "http://www.nature.com/scientificamerican", } @Article{Aley:1897:DES, author = "Robert J. Aley", title = "A Device for Extracting the Square Root of Certain Surd Quantities", journal = j-AMER-MATH-MONTHLY, volume = "4", number = "8/9", pages = "204--208", month = aug # "\slash " # sep, year = "1897", CODEN = "AMMYAE", ISSN = "0002-9890 (print), 1930-0972 (electronic)", ISSN-L = "0002-9890", bibdate = "Mon Jun 28 12:37:08 MDT 1999", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "American Mathematical Monthly", journal-URL = "https://www.jstor.org/journals/00029890.htm", } @Article{Anonymous:1898:OFA, author = "Anonymous", title = "Our Fingers as an Aid in Multiplication", journal = j-SCI-AMER, volume = "79", number = "17", pages = "265--266", day = "22", month = oct, year = "1898", CODEN = "SCAMAC", DOI = "https://doi.org/10.1038/scientificamerican10221898-265", ISSN = "0036-8733 (print), 1946-7087 (electronic)", ISSN-L = "0036-8733", bibdate = "Wed May 22 19:08:57 MDT 2013", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/sciam1890.bib", URL = "http://www.nature.com/scientificamerican/journal/v79/n17/pdf/scientificamerican10221898-265.pdf", acknowledgement = ack-nhfb, fjournal = "Scientific American", journal-URL = "http://www.nature.com/scientificamerican", } @Article{Ludgate:1909:PAM, author = "P. E. Ludgate", title = "On a proposed analytical machine", journal = j-SCI-PROC-ROY-DUBLIN-SOC, volume = "12", number = "9", pages = "77--91", year = "1909", CODEN = "SPRDAP", ISSN = "0371-2303", bibdate = "Wed Oct 13 11:14:08 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", note = "Reprinted in \cite[\S 2.4]{Randell:1982:ODC}.", acknowledgement = ack-nhfb, fjournal = "Scientific proceedings of the Royal Dublin Society", } @Article{Babbage:1910:BBA, author = "H. P. Babbage", title = "{Babbage}: {Babbage}'s analytical engine", journal = j-MONTHLY-NOT-ROY-ASTRON-SOC, volume = "70", number = "??", pages = "517--526, 645", year = "1910", CODEN = "MNRAA4", ISSN = "0035-8711 (print), 1365-2966 (electronic)", ISSN-L = "0035-8711", bibdate = "Wed Oct 13 11:12:08 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", note = "Reprinted in \cite[\S 2.3]{Randell:1982:ODC}.", acknowledgement = ack-nhfb, fjournal = "Monthly Notices of the Royal Astronomical Society", journal-URL = "https://academic.oup.com/mnras", } @Article{TorresyQuevedo:1915:EAS, author = "L. {Torres y Quevedo}", title = "Essais sur l'automatique. Sa definition. {{\'E}}tendue th{\'e}orique de ses applications ({French}) [{Essays} on automation. {Its} definition. {Theoretical} extent of its applications]", journal = j-REV-GEN-SCI-PURES-APPL, volume = "??", number = "??", pages = "601--611", day = "15", month = nov, year = "1915", ISSN = "0370-7431", bibdate = "Wed Oct 13 11:15:08 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", note = "Reprinted in \cite[\S 2.5]{Randell:1982:ODC}. Translated by Mr. R. Basu.", acknowledgement = ack-nhfb, fjournal = "Revue G{\'e}n{\'e}rale des Sciences Pures et Appliqu{\'e}es", language = "French", } @Article{Barrow:1924:QDD, author = "D. F. Barrow", title = "Questions and Discussions: Discussions: On Taking Square Roots of Integers", journal = j-AMER-MATH-MONTHLY, volume = "31", number = "10", pages = "482--484", month = dec, year = "1924", CODEN = "AMMYAE", ISSN = "0002-9890 (print), 1930-0972 (electronic)", ISSN-L = "0002-9890", bibdate = "Mon Jun 28 12:37:29 MDT 1999", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "American Mathematical Monthly", journal-URL = "https://www.jstor.org/journals/00029890.htm", } @Article{Smith:1924:FPA, author = "David Eugene Smith", title = "The First Printed Arithmetic ({Treviso}, 1478)", journal = j-ISIS, volume = "6", number = "3", pages = "311--331", month = "????", year = "1924", CODEN = "ISISA4", ISSN = "0021-1753 (print), 1545-6994 (electronic)", ISSN-L = "0021-1753", bibdate = "Tue Jul 30 21:57:26 MDT 2013", bibsource = "http://www.jstor.org/action/showPublication?journalCode=isis; http://www.jstor.org/stable/i211087; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/isis1920.bib", URL = "http://www.jstor.org/stable/224315", acknowledgement = ack-nhfb, fjournal = "Isis", journal-URL = "http://www.jstor.org/journal/isis", } @Book{Karpinski:1925:HA, author = "Louis Charles Karpinski", title = "The History of Arithmetic", publisher = "Rand McNally \& Company", address = "New York, NY, USA", pages = "xi + 200", year = "1925", bibdate = "Fri Nov 28 18:09:05 2003", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-mfc # " and " # ack-nhfb, } @Article{Cajori:1926:BRB, author = "Florian Cajori", title = "Book Review: {{\booktitle{The History of Arithmetic}} by Louis Charles Karpinski}", journal = j-ISIS, volume = "8", number = "1", pages = "231--232", month = feb, year = "1926", CODEN = "ISISA4", ISSN = "0021-1753 (print), 1545-6994 (electronic)", ISSN-L = "0021-1753", bibdate = "Tue Jul 30 21:57:36 MDT 2013", bibsource = "http://www.jstor.org/action/showPublication?journalCode=isis; http://www.jstor.org/stable/i211094; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/isis1920.bib", URL = "http://www.jstor.org/stable/223706", acknowledgement = ack-nhfb, fjournal = "Isis", journal-URL = "http://www.jstor.org/journal/isis", } @Article{Smith:1926:FGC, author = "David Eugene Smith", title = "The First Great Commercial Arithmetic", journal = j-ISIS, volume = "8", number = "1", pages = "41--49", month = feb, year = "1926", CODEN = "ISISA4", ISSN = "0021-1753 (print), 1545-6994 (electronic)", ISSN-L = "0021-1753", bibdate = "Tue Jul 30 21:57:36 MDT 2013", bibsource = "http://www.jstor.org/action/showPublication?journalCode=isis; http://www.jstor.org/stable/i211094; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/isis1920.bib", URL = "http://www.jstor.org/stable/223673", acknowledgement = ack-nhfb, fjournal = "Isis", journal-URL = "http://www.jstor.org/journal/isis", } @Article{Cajori:1927:EAP, author = "Florian Cajori", title = "The Earliest Arithmetic Published in {America}", journal = j-ISIS, volume = "9", number = "3", pages = "391--401", month = dec, year = "1927", CODEN = "ISISA4", ISSN = "0021-1753 (print), 1545-6994 (electronic)", ISSN-L = "0021-1753", bibdate = "Tue Jul 30 21:57:45 MDT 2013", bibsource = "http://www.jstor.org/action/showPublication?journalCode=isis; http://www.jstor.org/stable/i214411; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/isis1920.bib", URL = "http://www.jstor.org/stable/330806", acknowledgement = ack-nhfb, fjournal = "Isis", journal-URL = "http://www.jstor.org/journal/isis", } @Article{J:1930:RPRb, author = "R. A. J.", title = "Recent Publications: Reviews: {{\em Standard Table of Square Roots}}, by {L. M. Milne-Thomson}", journal = j-AMER-MATH-MONTHLY, volume = "37", number = "6", pages = "314--314", month = jun # "\slash " # jul, year = "1930", CODEN = "AMMYAE", ISSN = "0002-9890 (print), 1930-0972 (electronic)", ISSN-L = "0002-9890", bibdate = "Mon Jun 28 12:35:55 MDT 1999", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "American Mathematical Monthly", journal-URL = "https://www.jstor.org/journals/00029890.htm", } @Article{Laughlin:1930:LND, author = "Harry H. Laughlin", title = "Large-Number Division by Calculating Machine", journal = j-AMER-MATH-MONTHLY, volume = "37", number = "6", pages = "287--293", month = jun # "\slash " # jul, year = "1930", CODEN = "AMMYAE", ISSN = "0002-9890 (print), 1930-0972 (electronic)", ISSN-L = "0002-9890", bibdate = "Mon Jun 28 12:35:55 MDT 1999", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "American Mathematical Monthly", journal-URL = "https://www.jstor.org/journals/00029890.htm", } @Article{Garver:1932:QDNb, author = "Raymond Garver", title = "Questions, Discussions, and Notes: a Square Root Method and Continued Fractions", journal = j-AMER-MATH-MONTHLY, volume = "39", number = "9", pages = "533--535", month = nov, year = "1932", CODEN = "AMMYAE", ISSN = "0002-9890 (print), 1930-0972 (electronic)", ISSN-L = "0002-9890", bibdate = "Mon Jun 28 12:36:47 MDT 1999", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "American Mathematical Monthly", journal-URL = "https://www.jstor.org/journals/00029890.htm", } @Article{Polachek:1932:MMU, author = "H. Polachek", title = "A method of multiplication used by {Saadia Gaon} in the 10th century", journal = j-SCRIPTA-MATH, volume = "1", number = "??", pages = "245--246", month = "????", year = "1932", ISSN = "0036-9713", bibdate = "Thu Oct 26 11:15:25 2023", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/scripta-math.bib", acknowledgement = ack-nhfb, ajournal = "Scripta Math.", fjournal = "Scripta Mathematica: A Quarterly Journal Devoted to the Philosophy, History, and Expository Treatment of Mathematics", jfm = "59.0019.03", ZBmath = "2542288", } @Article{Simons:1932:IGA, author = "T. E. Simons", title = "{Isaac Greenwood}'s arithmetic", journal = j-SCRIPTA-MATH, volume = "1", number = "??", pages = "262--264", month = "????", year = "1932", ISSN = "0036-9713", bibdate = "Thu Oct 26 11:15:25 2023", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/scripta-math.bib", acknowledgement = ack-nhfb, ajournal = "Scripta Math.", fjournal = "Scripta Mathematica: A Quarterly Journal Devoted to the Philosophy, History, and Expository Treatment of Mathematics", jfm = "59.0026.02", ZBmath = "2542316", } @Book{Couffignal:1933:MCL, author = "L. Couffignal", title = "Les machines {\`a} calculer, leurs principes, leur {\'e}volution. ({French}) [{Calculating} machines, their principles, their evolution]", publisher = "Gauthier-Villars", address = "Paris, France", year = "1933", bibdate = "Wed Oct 13 11:19:11 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", note = "Extracts reprinted in \cite[\S 3.2]{Randell:1982:ODC}. Translated by Mr. R. Basu.", acknowledgement = ack-nhfb, language = "French", } @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/fparith.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "American Mathematical Monthly", journal-URL = "https://www.jstor.org/journals/00029890.htm", } @Article{Simons:1935:EEC, author = "L. G. Simons", title = "An early eighteenth century {American} readyreckoner", journal = j-SCRIPTA-MATH, volume = "3", number = "??", pages = "94--96", month = "????", year = "1935", ISSN = "0036-9713", bibdate = "Thu Oct 26 11:15:25 2023", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/scripta-math.bib", acknowledgement = ack-nhfb, ajournal = "Scripta Math.", fjournal = "Scripta Mathematica: A Quarterly Journal Devoted to the Philosophy, History, and Expository Treatment of Mathematics", jfm = "61.0020.03", ZBmath = "2531128", } @Article{Calvert:1936:DDS, author = "H. R. Calvert", title = "Decimal Division of Scales before the Metric System", journal = j-ISIS, volume = "25", number = "2", pages = "433--436", month = sep, year = "1936", CODEN = "ISISA4", ISSN = "0021-1753 (print), 1545-6994 (electronic)", ISSN-L = "0021-1753", bibdate = "Tue Jul 30 21:21:25 MDT 2013", bibsource = "http://www.jstor.org/action/showPublication?journalCode=isis; http://www.jstor.org/stable/i211115; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/isis1930.bib", URL = "http://www.jstor.org/stable/225380", acknowledgement = ack-nhfb, fjournal = "Isis", journal-URL = "http://www.jstor.org/journal/isis", } @Book{LeVita:1936:ALI, author = "Maurice H. {Le Vita}", title = "An Arithmetic of Life Insurance", publisher = "Life Office Management Association", address = "New York", pages = "xii + 132", year = "1936", LCCN = "HG8781 .L43", bibdate = "Sat Aug 12 14:42:20 MDT 2023", bibsource = "fsz3950.oclc.org:210/WorldCat; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "http://hdl.handle.net/2027/mdp.39015017290688; https://babel.hathitrust.org/cgi/pt?id=mdp.39015017290688", abstract = "We have determined this item to be in the public domain according to US copyright law through information in the bibliographic record and/or US copyright renewal records. The digital version is available for all educational uses worldwide. Please contact HathiTrust staff at hathitrust-help <at> umich.edu with any questions about this item. Life insurance--Mathematics. (LCCN)36034103. (OCoLC)ocm03161649. HG8781 .L43. HG 8781 .L66. Http://hdl.handle.net/2027/mdp.39015017290688.", acknowledgement = ack-nhfb, remark = "While not about computer arithmetic, this book is about practical applications of arithmetic in the life insurance industry. It considers compound interest calculations, present values of financial instruments, mortality tables, insurance premiums, bond returns, and so on. In Table 1, which shows the growth of an investment at 3 percent over 10 years, the author [amusingly] comments ``Mathematicians have already worked out the multiplications and have incorporated them into a table. With the use of this table, the result can be obtained by a single multiplication.''. Although the MARC record indicates the book is in the public domain, the URL only permits download of a single page at a time, although it is possible to page through the book.", tableofcontents = "Preface / vii \\ Index to Tables / xi \\ Notes on the Solution of Problems / xii \\ \\ I. Interest and Discount / 1 \\ Interest \\ Discount \\ Examples \\ Observations \\ Problems \\ \\ II. The Mortality Table and Easy Problems in Rate Calculation at Age 90 / 6 \\ Natural or Net One-Year Term Premium \\ Net One-Year Pure Endowment \\ Net Single Premium Whole Life \\ Fundamental Principle \\ Observations \\ Problems \\ \\ III. Single Premium Contracts --- Age 35 / 14 \\ Mortality Table \\ (Net) Single Premium 10-Year Term \\ (Net) Single Premium 10-Year Pure Endowment \\ (Net) Single Premium 10-Year Endowment \\ (Net) Single Premium Whole Life \\ (Net) Single Premium 10-Year Temporary Annuity \\ (Net) Single Premium Life Annuity \\ \\ IV. Development of Net Annual Premiums / 22 \\ Extension of Use of Discount \\ (Net) Annual Premium, Whole Life Policy \\ (Net) Annual Premium, 10-Payment Life \\ (Net) Annual Premium, 10-Year Term \\ (Net) Annual Premium, 10-Year Endowment \\ The ``Practical' and `More Exact' Premium \\ Alternative Method of Obtaining Net Annual Premiums \\ Problems \\ \\ V. More About Premiums / 30 \\ The Yearly Renewable Term Policy \\ Loading the Premium \\ Partial or Instalment Premiums \\ Problems \\ \\ VI. Net Level Premium Reserves / 34 \\ 10-Year Term Policy \\ 10-Year Endowment Policy \\ Whole Life Policy \\ 10-Payment Life Policy \\ Yearly Renewable Term \\ Problems \\ \\ VII. Reserve Fund, Mortality Fund and Cost of Insurance / 44 \\ Reserve Credit \\ Payment of Death Claims \\ The Reserve Fund and Mortality Fund \\ Cost of Insurance \\ Problems \\ \\ VIII. Full Preliminary Term Valuation \\ Gross Level Annual Premiums \\ Preliminary Term \\ Terminal Reserves Policy P \\ Full Preliminary Term Valuation \\ Modified Preliminary Term \\ Problems \\ \\ IX. Initial, Mean, and Calendar Year Reserves \\ \\ Initial Reserve \\ Mean Reserve \\ Calendar Year Reserve \\ Initial and Mean Reserves, F.P.T. Basis \\ Problems \\ \\ X. Non-Foreiture Options; Cash VALUE; Paid-Up and Extended Insurance; Loan Value / 61 \\ The Cash Value \\ What Happens When Policy is Surrendered \\ Non-Forfeiture Options (Option 1, Option 2, and Option 3) \\ Cash Values under F.P.T. Basis of Valuation \\ Loan Values \\ Problems \\ \\ XI. Profits to a Life Insurance Company / 67 \\ Actual and Expected Mortality \\ Actual and Expected Expenses \\ Actual and Expected Interest \\ Gain and Loss for the Year \\ Arithmetical Statement of Financial Status \\ Problems \\ \\ XII. Distribution of Surplus --- Dividend Options / 73 \\ Allocating Surplus to Policyholders \\ Share of Gain from Mortality, Loading and Interest \\ Dividend Formula \\ Variations in Dividend Formula \\ Dividend Options (Options 1 to 6) \\ Withdrawing the Dividend or Surrendering the Dividend Credit \\ Problems \\ \\ XIII. Further Problems in Interest and Discount / 79 \\ Fundamental Principle \\ Annuity Certain (Present Value of) \\ Annuity Certain (Amount of) \\ Interest Compounded Oftener than Once a Year \\ Value of a Bond \\ Problems \\ \\ XIV. Deferred Annuity --- Certain and Continuous Annuity --- Optional Modes of Settlement / 91 \\ Deferred Life Annuity \\ ``Combination'' Annuity Contract \\ Optional Modes of Settlement \\ Problems \\ \\ XV. Mortality Rates and Mortality Tables / 94 \\ Rate of Mortality \\ Building the Table \\ Changing Number at Initial Age \\ Selection and Select Tables \\ Indicating the Age (for Mortality Analysis) \\ Select Table \\ Ultimate Tables and Select and Ultimate Tables \\ Problems \\ \\ XVI. More About Reserves / 103 \\ Fundamental Principle Amended \\ Corollary to Amended Principle \\ Prospective and Retrospective Method of Obtaining the Reserve \\ Reserves on Single Premium Policies \\ Reserves on Substandard Policies \\ Percentage Mortality Rating Table \\ Problems. \\ \\ XVII. Modified Preliminary Term Valuation / 109 \\ M.P.T. Basis of Valuation \\ Method A: M.P.T. Whole Life Basis \\ Method B: M.P.T. 20-Payment Life Basis \\ Method C: M.P.T. Original Plan Basis \\ Illinois Standard System of Valuation \\ Ohio Standard System of Valuation \\ New Jersey Standard System of Valuation \\ Problems \\ \\ Appendices: \\ I. Miscellaneous Examples / 118 \\ II. Chart 3B --- Single Premium Life Insurance (Age 35) / 121 \\ III. Chart 5B -- Single Premium Life Annuity (Age 35) / 122 \\ IV. Answers to Problems and Miscellaneous Examples 123 \\ Index / 127", usmarc540 = "[US Library of Congress record]: We have determined this item to be in the public domain according to US copyright law through information in the bibliographic record and/or US copyright renewal records. The digital version is available for all educational uses worldwide.", } @Article{Phillips:1936:BC, author = "E. W. Phillips", title = "Binary calculation", journal = j-J-INST-ACTUARIES, volume = "67", number = "??", pages = "187--221", year = "1936", ISSN = "0020-2681", bibdate = "Wed Oct 13 11:33:16 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", note = "Reprinted in \cite[\S 7.1]{Randell:1982:ODC}.", acknowledgement = ack-nhfb, fjournal = "Journal of the Institute of Actuaries", } @Misc{Zuse:1936:VSD, author = "K. Zuse", title = "{Verfahren zur selbstt{\"a}tigen Durchfuhrung von Rechnungen mit Hilfe von Rechenmaschinen}. ({German}) [{Procedure} for automatic execution of calculations by calculating machines]", howpublished = "German patent application Z23624.", day = "11", month = apr, year = "1936", bibdate = "Wed Oct 13 11:22:03 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", note = "Reprinted in \cite[\S 4.1]{Randell:1982:ODC}.", acknowledgement = ack-nhfb, language = "German", } @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/fparith.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "American Mathematical Monthly", journal-URL = "https://www.jstor.org/journals/00029890.htm", } @Book{Millikan:1937:MMP, author = "Robert A. Millikan and Duane E. Roller and Earnest C. Watson", title = "Mechanics, Molecular Physics, Heat, and Sound", publisher = pub-MIT, address = pub-MIT:adr, pages = "????", year = "1937", bibdate = "Mon May 21 17:16:04 2018", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", note = "See Appendix: Significant Figures and Notations by Powers of Ten.", acknowledgement = ack-nhfb, remark = "Cited in \cite[ref. 22]{Carter:2013:ESF}.", } @MastersThesis{Shannon:1937:SAR, author = "Claude Elwood Shannon", title = "A Symbolic Analysis of Relay and Switching Circuits", type = "{Master of Science}", school = "Department of Electrical Engineering, MIT", address = "Cambridge, MA, USA", pages = "72", day = "10", month = aug, year = "1937", bibdate = "Thu Nov 18 10:35:20 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", note = "Not submitted until 1940", URL = "http://dspace.mit.edu/bitstream/handle/1721.1/11173/34541425.pdf", acknowledgement = ack-nhfb, author-dates = "April 30, 1916--February 24, 2001", remark = "Smiley \cite[p. 91]{Smiley:2010:MWI} claims: ``He [Shannon] also had the insight, like Atanasoff, that the binary arithmetic that relay switches represented would simplify information systems. His master's thesis, written when he as twenty-one and published when he was twenty-two, is considered to be one of the most important, if not the most important, master's thesis of the twentieth century.''\par Pages 59--61 of the thesis are a section called ``Electric Adder to the Base Two'', and pages 62--68, ``A Factor Table Machine'', describe a machine that will print a table of factors and primes of all the integers from 1 to 100,000,000. Shannon notes on page 68: ``As to the practicality of such a device, it might be said that J. P. Kulik spent 20 years in constructing a table of primes up to 100,000,000 and when finished it was found to contain so many errors that it was not worth publishing. The machine described here could probably be made to handle 5 numbers per second so that the table would require only about 2 months to construct.''", } @Article{Benford:1938:LAN, author = "Frank Benford", title = "The Law of Anomalous Numbers", journal = j-PROC-AMER-PHIL-SOC, volume = "78", number = "4", pages = "551--572", month = mar, year = "1938", CODEN = "PAPCAA", ISSN = "0003-049X (print), 2326-9243 (electronic)", ISSN-L = "0003-049X", bibdate = "Thu Feb 15 16:28:28 2007", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "http://links.jstor.org/sici?sici=0003-049X%2819380331%2978%3A4%3C551%3ATLOAN%3E2.0.CO%3B2-G", abstract = "It has been observed that the first pages of a table of common logarithms show more wear than do the last pages, indicating that more used numbers begin with the digit 1 than with the digit 9. A compilation of some 20,000 first digits taken from widely divergent sources shows that there is a logarithmic distribution of first digits when the numbers are composed of four or more digits. An analysis of the numbers from different sources shows that the numbers taken from unrelated subjects, such as a group of newspaper items, show a much better agreement with a logarithmic distribution than do numbers from mathematical tabulations or other formal data. There is here the peculiar fact that numbers that individually are without relationship are, when considered in large groups, in good agreement with a distribution law---hence the name ``Anomalous Numbers.''\par A further analysis of the data shows a strong tendency for bodies of numerical data to fall into geometric series. If the series is made up of numbers containing three or more digits the first digits form a logarithmic series. If the numbers contain only single digits the geometric relation still holds but the simple logarithmic relation no longer applies.\par An equation is given showing the frequencies of first digits in the different orders of numbers 1 to 10, 10 to 100, etc.\par The equation also gives the frequency of digits in the second, third + place of a multi-digit number, and it is shown that the same law applies to reciprocals.\par There are many instances showing that the geometric series, or the logarithmic law, has long been recognized as a common phenomenon in factual literature and in the ordinary affairs of life. The wire gauge and drill gauge of the mechanic, the magnitude scale of the astronomer and the sensory response curves of the psychologist are all particular examples of a relationship that seems to extend to all human affairs. The Law of Anomalous Numbers is thus a general probability law of widespread application.", acknowledgement = ack-nhfb, fjournal = "Proceedings of the {American Philosophical Society} held at {Philadelphia} for promoting useful knowledge", journal-URL = "http://www.jstor.org/journal/procamerphilsoci", keywords = "Benford's Law; Law of Anomalous Numbers; Zipf's Law", remark = "From \cite{Logan:1978:FDP}: ``Benford's paper was published in 1938 in a journal of rather limited circulation and not usually read by mathematicians. It so happened that it was immediately followed in the same issue by a physics paper which became of some importance for secret nuclear work during World War II [H. A. Bethe, M. E. Rose, and L. P. Smith, `The Multiple Scattering of Electrons', Proc. Amer. Phil. Soc. 78(4), 573--585 (1938)]. That is why Benford's paper caught the attention of physicists in the early 1940's and was much discussed. This led to the notes in Nature by Goudsmit and Furry [3] and Furry and Hurwitz [4] containing an effort to explain Benford's law. We considered it at that time merely a welcome diversion and did not expect that over thirty papers would be devoted to this subject in subsequent years.'' The 2006 bibliography \cite{Hurlimann:2006:BLB} cites 325 publications about Benford's Law.", } @PhdThesis{Couffignal:1938:AMA, author = "L. Couffignal", title = "Sur l'analyse m{\'e}canique. Application aux machines {\`a} calculer et aux calculs de la m{\'e}canique c{\'e}leste. ({French}) [{On} mechanical analysis. Application to calculating machines and to calculation in celestial mechanics]", publisher = "Gauthier-Villars", school = "Facult{\'e} des Sciences de Paris", address = "Paris, France", year = "1938", bibdate = "Wed Oct 13 11:17:22 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", note = "Extracts reprinted in \cite[\S 2.7]{Randell:1982:ODC}. Translated by Mr. R. Basu.", acknowledgement = ack-nhfb, language = "French", } @Book{Hardy:1938:ITN, author = "G. H. (Godfrey Harold) Hardy and E. M. (Edward Maitland) Wright", title = "An Introduction to the Theory of Numbers", publisher = pub-OXFORD, address = pub-OXFORD:adr, pages = "xvi + 403", year = "1938", LCCN = "QA241 .H28", bibdate = "Fri Nov 30 06:49:15 2007", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, } @Article{Shannon:1938:SAR, author = "Claude E. Shannon", title = "A Symbolic Analysis of Relay and Switching Circuits", journal = j-TRANS-AMER-INST-ELEC-ENG, volume = "57", number = "??", pages = "713--723", month = dec, year = "1938", CODEN = "TAEEA5", ISSN = "0096-3860", bibdate = "Sat Nov 20 08:54:12 2010", bibsource = "http://www2.research.att.com/~njas/doc/shannonbio.html; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, fjournal = "Transactions of the American Institute of Electrical Engineers", journal-URL = "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=6413714", remark = "This is Shannon's first published paper (his Master's thesis). In 1940, it received the Alfred Noble Prize of the combined engineering societies of the United States, an award given each year to a person not over thirty for a paper published in one of the journals of the participating societies. H. H. Goldstine \cite{Goldstine:1972:CPN} called this work ``one of the most important master's theses ever written \ldots{} a landmark in that it helped to change digital circuit design from an art to a science.''", } @Article{Jager:1939:AAD, author = "Robert Jager and Boyd C. Patterson", title = "The Artificial Arithmetick in Decimals of {Robert Jager} ({London}, 1651)", journal = j-ISIS, volume = "31", number = "1", pages = "25--31", month = nov, year = "1939", CODEN = "ISISA4", ISSN = "0021-1753 (print), 1545-6994 (electronic)", ISSN-L = "0021-1753", bibdate = "Tue Jul 30 21:26:49 MDT 2013", bibsource = "http://www.jstor.org/action/showPublication?journalCode=isis; http://www.jstor.org/stable/i302217; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/isis1930.bib", URL = "http://www.jstor.org/stable/226014", acknowledgement = ack-nhfb, fjournal = "Isis", journal-URL = "http://www.jstor.org/journal/isis", } @Article{K:1939:BRBd, author = "M. G. K.", title = "Book Review: {{\booktitle{Duodecimal Arithmetic}}, by George S. Terry}", journal = j-J-R-STAT-SOC, volume = "102", number = "2", pages = "299--300", month = "????", year = "1939", DOI = "https://doi.org/10.2307/2980013", ISSN = "0952-8385", bibdate = "Sat Jan 24 11:18:08 MST 2015", bibsource = "http://www.jstor.org/stable/i349540; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/jrss-a-1930.bib", URL = "http://www.jstor.org/stable/2980013", acknowledgement = ack-nhfb, fjournal = "Journal of the Royal Statistical Society", journal-URL = "http://www.jstor.org/journals/09528385.html", } @TechReport{Atanasoff:1940:CMS, author = "J. V. Atanasoff", title = "Computing machine for the solution of large systems of linear algebraic equations", type = "Unpublished memorandum", institution = "Iowa State College", address = "Ames, IA, USA", month = aug, year = "1940", bibdate = "Wed Oct 13 11:36:11 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", note = "Reprinted in \cite[\S 7.2]{Randell:1982:ODC}.", acknowledgement = ack-nhfb, } @Misc{Bush:1940:AM, author = "V. Bush", title = "Arithmetical Machine", howpublished = "Vannevar Bush Papers, Container 18, Folder: Caldwell, Samuel, 1939--1940", year = "1940", bibdate = "Wed Oct 13 11:37:32 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", note = "Reprinted in \cite[\S 7.3]{Randell:1982:ODC}. Copyright interest in the unpublished writings of Vannevar Bush has been dedicated to the public.", acknowledgement = ack-nhfb, } @TechReport{Stibitz:1940:C, author = "G. R. Stibitz", title = "Computer", type = "Unpublished memorandum", institution = "Bell Telephone Laboratories", address = "New York, NY, USA", year = "1940", bibdate = "Wed Oct 13 11:30:38 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", note = "Reprinted in \cite[\S 6.1]{Randell:1982:ODC}.", acknowledgement = ack-nhfb, } @MastersThesis{Berry:1941:DED, author = "Clifford Edward Berry", title = "Design of electrical data recording and reading mechanism", type = "{M.S.} thesis", school = "Iowa State College", address = "Ames, IA, USA", pages = "32", year = "1941", bibdate = "Thu Nov 18 11:18:18 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, author-dates = "1918--1963", historical-note = "From http://www.lib.iastate.edu/arch/rgrp/5-2-1-1.html: ``John Vincent Atanasoff received his M.S. (1926) in Mathematics from Iowa State College (University) and received his Ph.D. (1930) in Theoretical Physics from the University of Wisconsin-Madison. He returned to Iowa State in 1930 as Assistant Professor in mathematics and physics and was promoted to Associate Professor (1936). Atanasoff began developing concepts for an electronic computing machine in 1937. It was shortly thereafter that he, along with graduate student Clifford Berry, started work on the world's first electronic digital computer. The computer would later be named the Atanasoff-Berry Computer (ABC). Work on the machine stopped at the start of World War II in 1941 and the ABC was never patented. Atanasoff and Berry were both called to support the war effort and left Iowa State. By the end of the decade, the ABC was removed from the basement of Physics Hall and dismantled. Atanasoff and the ABC were part of a major court case between Honeywell, Inc. and Sperry Rand Corporation which occurred 1967-1973. The case involved the ENIAC patent which covered basic ownership rights to the design of electronic digital computers. During the trial, the judge concluded that the invention of the ENIAC was derived from the work of John Vincent Atanasoff at Iowa State University. A replica of the Atanasoff-Berry Computer was completed and unveiled to the public in 1996. The ABC replica was constructed by a team of Iowa State scientists at the Ames Laboratory and was exhibited at museums throughout the country over the next several years.''", remark = "This thesis may be one of the earliest surviving records of the ABC computer built by John Atanasoff and Clifford Berry in the late 1930s. The one-of-a-kind ABC was destroyed in 1948, and most of its parts were lost. Also cited in ``Charles W. Bradley Collection on the ENIAC Trial, 1930--1966'', (found in http://discover.lib.umn.edu/): CWB as an attorney for the group retained by Honeywell in the Honeywell v. Sperry Rand ENIAC trial. The ABC had a 50-bit word.", } @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/fparith.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", } @TechReport{Mauchly:1942:UHS, author = "J. W. Mauchly", title = "The use of high speed vacuum tube devices for calculating", type = "Privately circulated memorandum", institution = "Moore School of Electrical Engineering, University of Pennsylvania", address = "Philadelphia, PA, USA", month = aug, year = "1942", bibdate = "Wed Oct 13 11:52:54 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", note = "Reprinted in \cite[\S 7.6]{Randell:1982:ODC}.", acknowledgement = ack-nhfb, } @TechReport{Rajchman:1942:REP, author = "J. A. Rajchman and G. A. Morton and A. W. Vance", title = "Report on Electronic Predictors for Anti-Aircraft Fire Control", institution = "Research Laboratories, R.C.A. Manufacturing Company, Inc.", address = "Camden, NJ, USA", month = apr, year = "1942", bibdate = "Sat Feb 8 10:28:55 2020", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", note = "Reprinted in \cite[\S 7.4]{Randell:1982:ODC}.", acknowledgement = ack-nhfb, } @Article{Crawford:1943:DNS, author = "W. S. H. Crawford", title = "Discussions and Notes: Square Roots from a Table of Cosines", journal = j-AMER-MATH-MONTHLY, volume = "50", number = "3", pages = "190--191", month = mar, year = "1943", CODEN = "AMMYAE", ISSN = "0002-9890 (print), 1930-0972 (electronic)", ISSN-L = "0002-9890", bibdate = "Mon Jun 28 12:36:57 MDT 1999", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "American Mathematical Monthly", journal-URL = "https://www.jstor.org/journals/00029890.htm", } @TechReport{Rademacher:1943:MTI, author = "Hans Rademacher", title = "Mathematical Topics of Interest in {PX}, Part Two: Summary of Articles Dealing with Rounding off Errors", type = "{PX} Report", number = "14", institution = "Moore School of Electrical Engineering, Office of the Director Records, University of Pennsylvania", address = "Philadelphia, PA, USA", day = "30", month = nov, year = "1943", bibdate = "Fri Jun 15 17:39:28 2018", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, remark = "Cited on page 34 of Haigh, Priestley, and Rope, \booktitle{ENIAC in Action} (2016) (ISBN 0-262-03398-4) as an annotated bibliography of rounding errors.", } @Article{Goudsmit:1944:SFN, author = "S. A. Goudsmit and W. H. Furry", title = "Significant Figures of Numbers in Statistical Tables", journal = j-NATURE, volume = "154", number = "3921", pages = "800--801", day = "23", month = dec, year = "1944", CODEN = "NATUAS", DOI = "https://doi.org/10.1038/154800a0", ISSN = "0028-0836 (print), 1476-4687 (electronic)", ISSN-L = "0028-0836", bibdate = "Sun Sep 18 11:57:19 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "http://www.nature.com/nature/journal/v154/n3921/pdf/154800a0.pdf", abstract = "A rough qualitative explanation of this fact can easily be given. If we consider tables in which the entries become rarer the larger they are, we can draw the obvious conclusion that in any interval, say, between 10 and 99, or 10,000 and 99,999, there are more entries on the small side than on the large side.", acknowledgement = ack-nhfb, fjournal = "Nature", journal-URL = "http://www.nature.com/nature/archive/", keywords = "Benford's Law; Law of Anomalous Numbers; Zipf's Law", } @Article{Furry:1945:DND, author = "W. H. Furry and Henry Hurwitz", title = "Distribution of Numbers and Distribution of Significant Figures", journal = j-NATURE, volume = "155", number = "3924", pages = "52--53", day = "13", month = jan, year = "1945", CODEN = "NATUAS", DOI = "https://doi.org/10.1038/155052a0", ISSN = "0028-0836 (print), 1476-4687 (electronic)", ISSN-L = "0028-0836", bibdate = "Sun Sep 18 11:51:37 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/benfords-law.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "http://www.nature.com/nature/journal/v155/n3924/pdf/155052a0.pdf", acknowledgement = ack-nhfb, fjournal = "Nature", journal-URL = "http://www.nature.com/nature/archive/", keywords = "Benford's Law; Law of Anomalous Numbers; Zipf's Law", } @TechReport{vonNeumann:1945:FDR, author = "John von Neumann", title = "First Draft of a Report on the {EDVAC}", institution = "University of Pennsylvania", day = "30", month = jun, year = "1945", bibdate = "Mon Jun 06 19:17:03 2005", bibsource = "ftp://ftp.math.utah.edu/pub/mirrors/ftp.ira.uka.de/bibliography/Ai/alife.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", note = "Reprinted in \cite[\S 8.2]{Randell:1982:ODC}.", acknowledgement = ack-nhfb, remark = "This is the report that got von Neumann's name associated with the serial, stored-program, general purpose, digital architecture upon which 99.99\% of all computers today are based.", xxnote = "Report prepared for U.S. Army Ordinance Department under Contract W-670-ORD-4926. Reprinted in \cite[pp.~177--246]{Stern:1981:EUA}, \cite[pp.~399--413]{Randell:1982:ODC}, \cite{vonNeumann:1993:FDR}, and \cite{Laplante:1996:GPC}.", } @Article{Aiken:1946:ASC, author = "H. H. Aiken and G. M. Hopper", title = "The {Automatic Sequence Controlled Calculator}", journal = j-ELECTR-ENG, volume = "65", number = "??", pages = "384--391, 449--454, 522--528", year = "1946", CODEN = "ELENAC", ISSN = "0095-9197", bibdate = "Wed Oct 13 11:26:29 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", note = "Reprinted in \cite[\S 5.2]{Randell:1982:ODC}.", acknowledgement = ack-nhfb, fjournal = "Electrical Engineering (American Institute of Electrical Engineers)", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=6413714", } @TechReport{Burks:1946:PDL, author = "Arthur W. Burks and Herman H. Goldstine and John von Neumann", title = "Preliminary discussion of the logical design of an electronic computing instrument", institution = inst-INST-ADV-STUDY, address = inst-INST-ADV-STUDY:adr, pages = "42", day = "28", month = jun, year = "1946", MRnumber = "MR22442", bibdate = "Wed Oct 13 08:17:48 2004", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/v/von-neumann-john.bib; https://www.math.utah.edu/pub/mirrors/ftp.ira.uka.de/bibliography/Math/computer.arithmetic.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/Theory/arith.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", note = "Report to the U.S. Army Ordnance Department under contract W-36-034-OKD-7481. Reprinted in \cite[Paper 2]{Taub:1963:JNCa}, \cite{Bell:1971:CSR}, \cite[pp. 221--259]{Swartzlander:1976:CDD} and \cite[pp. 97--146]{Aspray:1987:PJN}", URL = "https://deepblue.lib.umich.edu/handle/2027.42/3972; https://grch.com.ar/docs/p1/Apuntes/eng/Logical%20Design%20of%20an%20Electronic%20Computing%20Instrument.pdf", acknowledgement = ack-nhfb, author-dates = "John von Neumann (28 December 1903--8 February 1957)", remark = "Discusses floating-point versus fixed-point computation, and concludes that floating-point is probably not justifiable. They wrote:\par ``There appear to be two major purposes in a `floating' decimal point system both of which arise from the fact that the number of digits in a word is a constant fixed by design considerations for each particular machine. The first of these purposes is to retain in a sum or product as many significant digits as possible and the second of these is to free the human operator from the burden of estimating and inserting into a problem `scale factors' --- multiplicative constants which serve to keep numbers within the limits of the machine.\par There is, of course, no denying the fact that human time is consumed in arranging for the introduction of suitable scale factors. We only argue that the time so consumed is a very small percentage of the total time we will spend in preparing an interesting problem for our machine. The first advantage of the floating point is, we feel, somewhat illusory. In order to have such a floating point, one must waste memory capacity which could otherwise be used for carrying more digits per word. It would therefore seem to us not at all clear whether the modest advantages of a floating binary point offset the loss of memory capacity and the increased complexity of the arithmetic and control circuits.''", } @Article{Cesareo:1946:RI, author = "O. Cesareo", title = "The Relay Interpolator", journal = j-BELL-LABS-RECORD, volume = "23", number = "??", pages = "457--460", year = "1946", CODEN = "BLRCAB", ISSN = "0005-8564", bibdate = "Wed Oct 13 11:31:47 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", note = "Reprinted in \cite[\S 6.2]{Randell:1982:ODC}.", acknowledgement = ack-nhfb, fjournal = "Bell Laboratories Record", } @Article{Comrie:1946:BDC, author = "L. J. Comrie", title = "{Babbage}'s Dream Come True", journal = j-NATURE, volume = "158", number = "4017", pages = "567--568", day = "26", month = oct, year = "1946", CODEN = "NATUAS", DOI = "https://doi.org/10.1038/158567a0", ISSN = "0028-0836 (print), 1476-4687 (electronic)", ISSN-L = "0028-0836", bibdate = "Wed Aug 9 07:30:09 2023", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/b/babbage-charles.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", abstract = "The black mark earned by the government of the day more than a hundred years ago for its failure to see Charles Babbage's difference engine brought to a successful conclusion has still to be wiped out. It is not too much to say that it cost Britain the leading place in the art of mechanical computing. Babbage then conceived and worked on his `analytical engine', designed to store numbers and operate on them according to a sequence of processes conveyed to the machine by cards similar to those used in the Jacquard loom. This, however, was never completed.", acknowledgement = ack-nhfb, fjournal = "Nature", journal-URL = "http://www.nature.com/nature/archive/", remark = "The title in the Nature archive index is completely wrong: it says ``A Manual of Operation for the Automatic Sequence Controlled Calculator''. After the critical introduction [see the abstract], the article goes on to describe IBM's gift to Harvard of the Automatic Sequence Controlled Calculator. It reports about the ASCC: ``The machine contains seventy-two storage counters, capable of holding twenty-three digits and a sign. \ldots{} When working to the full 23-figure capacity of the machine, multiplication takes about six seconds, and division twice as long; additions and subtractions are done at the rate of three a second, whatever their length.'' It then describes the computation of logarithms via the Taylor series expansion of $ \log (1 + x) $, and reports that it can do exponential, trigonometric, and hyperbolic functions as well.", } @TechReport{Dreyer:1946:REM, author = "H.-J. Dreyer and A. Walther", title = "{Der Rechenautomat Ipm. Entwicklung Mathematischer Instrumente in Deutschland 1939 bis 1945}. ({German}) [{The} {Ipm} Calculator. {The} development of mathematical instruments in {Germany} 1939--1945]", type = "{Bericht}", number = "A3", institution = "Institut f{\"u}r Praktische Mathematik, Technische Hochschule", address = "Darmstadt, West Germany", day = "19", month = aug, year = "1946", bibdate = "Wed Oct 13 11:20:15 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", note = "Reprinted in \cite[\S 3.3]{Randell:1982:ODC}. Translated by Mr. and Mrs. P. Jones", acknowledgement = ack-nhfb, language = "German", } @Article{Goldstine:1946:ENI, author = "H. H. Goldstine and Adele Goldstine", title = "The {Electronic Numerical Integrator and Computer (ENIAC)}", journal = j-MATH-TABLES-OTHER-AIDS-COMPUT, volume = "2", number = "15", pages = "97--110", month = jul, year = "1946", CODEN = "MTTCAS", ISSN = "0891-6837", bibdate = "Tue Oct 13 08:44:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; JSTOR database", note = "Reprinted in \cite[\S 7.7]{Randell:1982:ODC}.", acknowledgement = ack-nhfb, ajournal = "Math. Tables Other Aids Comput.", fjournal = "Mathematical Tables and Other Aids to Computation", journal-URL = "http://www.ams.org/mcom/", } @TechReport{Goldstine:1947:PCPa, author = "Herman H. Goldstine and John von Neumann", title = "Planning and coding of problems for an electronic computing instrument. {Part 1}, Vol. 1", type = "Technical report", institution = inst-INST-ADV-STUDY, address = inst-INST-ADV-STUDY:adr, pages = "??", month = "????", year = "1947", bibdate = "Fri Nov 16 16:35:14 2012", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/v/von-neumann-john.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, } @TechReport{Goldstine:1947:PCPb, author = "Herman H. Goldstine and John von Neumann", title = "Planning and coding of problems for an electronic computing instrument. {Part 2}, Vol. 1", type = "Technical report", number = "1", institution = inst-INST-ADV-STUDY, address = inst-INST-ADV-STUDY:adr, pages = "69", day = "1", month = apr, year = "1947", MRnumber = "MR22443", bibdate = "Wed Jun 01 17:41:33 2005", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/v/von-neumann-john.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", note = "Report prepared for U.S. Army Ordnance Department under contract W-36-034-OKD-7481. Reprinted in \cite[80--151]{Taub:1963:JNCa}. Knuth \cite[p. 278]{Knuth:1998:SA} cites pp. 142--151 of this report as the first published treatment of double-precision arithmetic on digital computers.", acknowledgement = ack-nhfb, } @TechReport{Goldstine:1947:PCPc, author = "Herman H. Goldstine and John von Neumann", title = "Planning and coding of problems for an electronic computing instrument. {Part 1}, Vol. 2", type = "Technical report", institution = inst-INST-ADV-STUDY, address = inst-INST-ADV-STUDY:adr, pages = "??", month = "????", year = "1947", bibdate = "Fri Nov 16 16:35:14 2012", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/v/von-neumann-john.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, } @TechReport{Goldstine:1947:PCPd, author = "Herman H. Goldstine and John von Neumann", title = "Planning and coding of problems for an electronic computing instrument. {Part 2}, Vol. 2", type = "Technical report", institution = inst-INST-ADV-STUDY, address = inst-INST-ADV-STUDY:adr, pages = "??", month = "????", year = "1947", bibdate = "Fri Nov 16 16:35:14 2012", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/v/von-neumann-john.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, } @TechReport{Goldstine:1947:PCPe, author = "Herman H. Goldstine and John von Neumann", title = "Planning and coding of problems for an electronic computing instrument. {Part 1}, Vol. 3", type = "Technical report", institution = inst-INST-ADV-STUDY, address = inst-INST-ADV-STUDY:adr, pages = "??", month = "????", year = "1947", bibdate = "Fri Nov 16 16:35:14 2012", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/v/von-neumann-john.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, } @TechReport{Goldstine:1947:PCPf, author = "Herman H. Goldstine and John von Neumann", title = "Planning and coding of problems for an electronic computing instrument. {Part 2}, Vol. 3", type = "Technical report", institution = inst-INST-ADV-STUDY, address = inst-INST-ADV-STUDY:adr, pages = "??", month = "????", year = "1947", bibdate = "Fri Nov 16 16:35:14 2012", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/v/von-neumann-john.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, } @Book{HUCL:1946:MOA, author = "{Harvard University. Computation Laboratory}", title = "A Manual of Operation for the {Automatic Sequence Controlled Calculator}", publisher = pub-HARVARD, address = pub-HARVARD:adr, pages = "561", year = "1946", LCCN = "QA3 .H3 v.1", bibdate = "Mon Nov 4 07:46:57 MST 2002", bibsource = "https://www.math.utah.edu/pub/tex/bib/annhistcomput.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", series = "Its Annals v. 1", acknowledgement = ack-nhfb, keywords = "electronic digital computers; mathematics --- bibliography", } @TechReport{Turing:1946:PEC, author = "A. M. Turing", title = "Proposed Electronic Calculator", type = "Report", number = "E882", institution = inst-NPL, address = inst-NPL:adr, pages = "48", year = "1946", bibdate = "Fri Nov 25 16:22:27 2005", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/t/turing-alan-mathison.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "http://www.emula3.com/docs/Turing_Report_on_ACE.pdf", acknowledgement = ack-nhfb, author-dates = "Alan Mathison Turing (23 June 1912--7 June 1954)", xxnote = "Reprinted in \cite{Turing:1972:MTO}.", } @Article{Juley:1947:BC, author = "J. Juley", title = "The Ballistic Computer", journal = j-BELL-LABS-RECORD, volume = "24", number = "??", pages = "5--9", year = "1947", CODEN = "BLRCAB", ISSN = "0005-8564", bibdate = "Wed Oct 13 11:32:38 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", note = "Reprinted in \cite[\S 6.3]{Randell:1982:ODC}.", acknowledgement = ack-nhfb, fjournal = "Bell Laboratories Record", } @InProceedings{Mauchly:1947:PPE, author = "J. W. Mauchly", booktitle = "{Proceedings of a Symposium on Large Scale Digital Calculating Machinery, 7--10 January 1947}", title = "Preparation of problems for {EDVAC}-type machines", publisher = "Harvard University Press", address = "Cambridge, MA, USA", year = "1947", bibdate = "Wed Oct 13 11:55:58 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", note = "Reprinted in {\em Annals of the Computation Laboratory of Harvard University}, {\bf 16}, 203--207 (1948). Reprinted in \cite[\S 8.2]{Randell:1982:ODC}.", acknowledgement = ack-nhfb, } @Article{Richeson:1947:FAP, author = "A. W. Richeson", title = "The First Arithmetic Printed in {English}", journal = j-ISIS, volume = "37", number = "1--2", pages = "47--56", month = may, year = "1947", CODEN = "ISISA4", ISSN = "0021-1753 (print), 1545-6994 (electronic)", ISSN-L = "0021-1753", bibdate = "Tue Jul 30 21:27:09 MDT 2013", bibsource = "http://www.jstor.org/action/showPublication?journalCode=isis; http://www.jstor.org/stable/i302230; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/isis1940.bib", URL = "http://www.jstor.org/stable/226161", acknowledgement = ack-nhfb, fjournal = "Isis", journal-URL = "http://www.jstor.org/journal/isis", remark = "From the second paragraph: ``The earliest printed work in English to discuss arithmetic, \booktitle{The Myrrour of the World}, came from the press of William Caxton in 1481. This work, however, contained only one page devoted entirely to arithmetic. It is not until 1537 that we find another book printed in English devoted entirely to arithmetic and dealing with the Hindu--Arabic system of numbers. This is an anonymous publication, from a press located at St. Albans, with the title \booktitle{An Introduction for to Lerne to Recken with the Pen, or with the Counters},'' This book went through eight editions between 1537 and 1629. The article quotes book sections on integer division.", } @Article{vonNeumann:1947:NIM, author = "John von Neumann and Herman H. Goldstine", title = "Numerical Inverting of Matrices of High Order", journal = j-BULL-AMS, volume = "53", number = "11", pages = "1021--1099", month = nov, year = "1947", CODEN = "BAMOAD", ISSN = "0002-9904 (print), 1936-881X (electronic)", ISSN-L = "0002-9904", MRclass = "65.0X", MRnumber = "MR0024235 (9,471b)", MRreviewer = "E. Bodewig", bibdate = "Thu Nov 8 14:49:36 2007", 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/bibnet/subjects/acc-stab-num-alg.bib; https://www.math.utah.edu/pub/mirrors/ftp.ira.uka.de/bibliography/Math/acc-stab-num-alg.bib; https://www.math.utah.edu/pub/mirrors/ftp.ira.uka.de/bibliography/Math/auto.diff.bib; https://www.math.utah.edu/pub/mirrors/ftp.ira.uka.de/bibliography/Math/computer.arithmetic.1.bib; https://www.math.utah.edu/pub/mirrors/ftp.ira.uka.de/bibliography/Math/Matrix.bib; https://www.math.utah.edu/pub/mirrors/ftp.ira.uka.de/bibliography/Math/sparse.linear.systems.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", note = "See \cite{Goldstine:1951:NIM} for Part II. Reprinted in \cite[vol. 5, pp. 479--557]{Taub:1961:JNCa}.", URL = "http://projecteuclid.org/euclid.bams/1183511222", ZMnumber = "0031.31402", acknowledgement = ack-nhfb # " and " # ack-jg, author-dates = "John von Neumann (28 December 1903--8 February 1957)", fjournal = "Bulletin of the American Mathematical Society", journal-URL = "http://www.ams.org/journals/bull/all_issues.html", kwds = "nla, linear system, rounding error", referred = "[Markov1999a].", remark = "This may be the first journal publication about the use of double-precision arithmetic on computers, used for the accumulation of inner products and matrix products. On pp. 1035--1038, the authors show that when the sum of $m$ products of two $s$-digit numbers is accumulated in precision $ 2 s $, with rounding to $s$ digits only after the final sum is complete, then the rounding error is essentially one unit in the last place, instead of $m$ times that value. Cited in \cite{Sterbenz:1974:FPC}.", } @Article{Alt:1948:BTLa, author = "Franz L. Alt", title = "A {Bell Telephone Laboratories}' Computing Machine---{I}", journal = j-MATH-TABLES-OTHER-AIDS-COMPUT, volume = "3", number = "21", pages = "1--13", month = jan, year = "1948", CODEN = "MTTCAS", ISSN = "0891-6837", bibdate = "Tue Oct 13 08:44:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; JSTOR database", note = "Reprinted in \cite[\S 6.4]{Randell:1982:ODC}.", acknowledgement = ack-nhfb, ajournal = "Math. Tables Other Aids Comput.", fjournal = "Mathematical Tables and Other Aids to Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Eckert:1948:EC, author = "W. J. Eckert", title = "Electrons and computation", journal = j-SCI-MONTHLY, volume = "67", number = "5", pages = "315--323", month = nov, year = "1948", CODEN = "SCMOAA", ISSN = "0096-3771 (print), 2327-7513 (electronic)", bibdate = "Wed Oct 13 11:27:40 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", note = "Reprinted in \cite[\S 5.3]{Randell:1982:ODC}.", acknowledgement = ack-nhfb, fjournal = "The Scientific Monthly", journal-URL = "http://www.jstor.org/journal/sciemont", } @InProceedings{Rademacher:1948:AEP, author = "Hans A. Rademacher", title = "On the Accumulation of Errors in Processes of Integration on High-Speed Calculating Machines", crossref = "Anonymous:1948:PSL", pages = "176--187", year = "1948", bibdate = "Mon Jun 18 11:03:56 2018", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, remark = "From page 186, remark by John W. Mauchly: ``I just wanted to remark that this paper is probably one of the first to appear as a result of the interaction between computing machines and mathematics. I think many of us have been looking forward with certainty to the day when the advent of computing machines would influence the course of mathematics, and there are still many directions in which we can look forward to further such impacts. This reaction is not, of course, one way. Mathematics should also influence the computing machine.''", } @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; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, } @Article{Turing:1948:REM, author = "A. M. Turing", title = "Rounding-Off Errors in Matrix Processes", journal = j-QUART-J-MECH-APPLIED-MATH, volume = "1", pages = "287--308", month = sep, year = "1948", CODEN = "QJMMAV", ISSN = "0033-5614 (print), 1464-3855 (electronic)", ISSN-L = "0033-5614", MRclass = "65.0X", MRnumber = "MR0028100 (10,405c)", MRreviewer = "E. Bodewig", bibdate = "Sat Nov 19 12:09:58 2005", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/a/turing-alan-mathison.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/fparith.bib", note = "Reprinted in \cite{Turing:1992:PM} with summary and notes (including corrections)", URL = "http://turing.ecs.soton.ac.uk/browse.php/B/18", ZMnumber = "0033.28501", acknowledgement = ack-nhfb, fjournal = "Quarterly Journal of Mechanics and Applied Mathematics", journal-URL = "http://qjmam.oxfordjournals.org/content/by/year", mynote = "The notes are not very good. They mainly correct errors and fill in the gaps of derivations. Much better would have been higher level comments from a numerical analyst, e.g. on where Turing's ideas/predictions were wrong.", } @Article{Williams:1948:EDC, author = "F. C. Williams and T. Kilburn", title = "Electronic digital computers", journal = j-NATURE, volume = "162", number = "4117", pages = "487--487", day = "25", month = sep, year = "1948", CODEN = "NATUAS", DOI = "https://doi.org/10.1038/162487a0", ISSN = "0028-0836 (print), 1476-4687 (electronic)", ISSN-L = "0028-0836", bibdate = "Wed Oct 13 12:00:02 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", note = "Reprinted in \cite[\S 8.4]{Randell:1982:ODC}.", acknowledgement = ack-nhfb, fjournal = "Nature", journal-URL = "http://www.nature.com/nature/archive/", remark = "Letter submitted 3 August 1948. Describes a small prototype digital computer with 32 31-bit words for instructions, data, and working storage.", } @Article{Davis:1949:MM, author = "Harry M. Davis", title = "Mathematical Machines", journal = j-SCI-AMER, volume = "180", number = "4", pages = "28--39", month = apr, year = "1949", CODEN = "SCAMAC", DOI = "https://doi.org/10.1038/scientificamerican0449-28", ISSN = "0036-8733 (print), 1946-7087 (electronic)", ISSN-L = "0036-8733", bibdate = "Sat May 18 16:12:10 MDT 2013", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/sciam1940.bib", URL = "http://www.jstor.org/stable/24967160; http://www.nature.com/scientificamerican/journal/v180/n4/pdf/scientificamerican0449-28.pdf", acknowledgement = ack-nhfb, fjournal = "Scientific American", journal-URL = "http://www.nature.com/scientificamerican", remark = "Description of counting devices before 1950, including an introduction to the binary number system.", } @Article{Hartree:1949:NSR, author = "Douglas R. (Douglas Rayner) Hartree", title = "Note On Systematic Roundoff Errors in Numerical Integration", journal = "Journal of Research of the National Bureau of Standards", volume = "42", number = "??", pages = "62--??", month = "????", year = "1949", bibdate = "Wed Feb 14 19:07:37 2007", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, } @Article{Huskey:1949:PCP, author = "H. D. Huskey", title = "On the Precision of a Certain Procedure of Numerical Integration", journal = j-J-RES-NATL-BUR-STAND, volume = "42", number = "1", pages = "57--62", month = jan, year = "1949", CODEN = "JRNBAG", DOI = "https://doi.org/10.6028/jres.042.005", ISSN = "0091-0635 (print), 2376-5305 (electronic)", ISSN-L = "0091-0635", bibdate = "Fri Aug 20 09:32:34 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", note = "With an appendix by Douglas R. Hartree.", abstract = "An example of numerical integration is given that shows very systematic effects in the less significant digits. This lack of randomness gives rounding-off errors that exceed the predicted standard deviation by a factor of three. The example considered in this paper shows that systematic rounding-off errors can occur in numerical in integration, irrespective of the number of digits kept in the contributions to the integral. In the appendix this phenomenon is examined, and criteria are set up to detect the cases in which it may arise to a serious extent.", acknowledgement = ack-nhfb, journal-URL = "https://www.nist.gov/nist-research-library/journal-research-nist/past-papers", remark = "Cited in \cite{Sterbenz:1974:FPC}.", } @TechReport{Tukey:1949:TRA, author = "John W. Tukey and M. F. Freeman", title = "Transformation related to the angular and the square-root", type = "SRG Memorandum report", number = "24", institution = inst-PRINCETON, address = inst-PRINCETON:adr, pages = "??", year = "1949", 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/fparith.bib", acknowledgement = ack-nhfb, } @Book{Stifler:1950:HSC, editor = "W. W. {Stifler, Jr.}", title = "High-speed computing devices", publisher = pub-MCGRAW-HILL, address = pub-MCGRAW-HILL:adr, pages = "xiii + 451", year = "1950", LCCN = "QA75 .E5", bibdate = "Fri Aug 20 10:18:44 MDT 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; z3950.loc.gov:7090/Voyager", acknowledgement = ack-nhfb, remark = "Supervised by Charles Brown Tompkins and C. H. Wakelin, Engineering Research Associates. The contents of this volume were first assembled in the form of a report to the Office of Naval Research, prepared under a provision of contract N6-ONR-240, Task 1. Cited in \cite{Sterbenz:1974:FPC}.", subject = "computers", } @Article{Harrison:1950:BDC, author = "J. O. {Harrison, Jr.}", title = "Binary--Decimal Conversion on a Desk Calculator (in Automatic Computing Machinery; Discussions)", journal = j-MATH-TABLES-OTHER-AIDS-COMPUT, volume = "4", number = "32", pages = "231--234", month = oct, year = "1950", 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/fparith.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp1950.bib; JSTOR database", URL = "http://www.jstor.org/stable/2002500", acknowledgement = ack-nhfb, ajournal = "Math. Tables Other Aids Comput.", fjournal = "Mathematical Tables and Other Aids to Computation", journal-URL = "http://www.ams.org/mcom/", } @InProceedings{Wilkes:1950:E, author = "M. V. Wilkes and W. Renwick", booktitle = "{Report of a Conference on High Speed Automatic Calculating Machines, 22--25 June 1949}", title = "The {EDSAC}", publisher = "University Mathematical Laboratory", address = "Cambridge, UK", pages = "9--11", month = jan, year = "1950", bibdate = "Wed Oct 13 12:00:41 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", note = "Reprinted in \cite[\S 8.5]{Randell:1982:ODC}.", acknowledgement = ack-nhfb, } @InProceedings{Worsley:1950:ED, author = "B. H. Worsley", booktitle = "Report of a Conference on High Speed Automatic Calculating Machines, {22--25 June 1949}", title = "The {EDSAC} demonstration", publisher = "University Mathematical Laboratory", address = "Cambridge, UK", pages = "12--16", month = jan, year = "1950", bibdate = "Wed Oct 13 11:39:09 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", note = "Reprinted in \cite[\S 8.6]{Randell:1982:ODC}.", acknowledgement = ack-nhfb, } @Misc{Anonymous:1951:R, author = "Anonymous", title = "{RECIPROOT}", howpublished = "Manchester University Electronic Computer Web document.", day = "9", month = jul, year = "1951", bibdate = "Sat Aug 19 09:57:33 2023", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", note = "To calculate square roots and reciprocal square roots.", URL = "https://0x5f37642f.com/documents/ManchesterRecipRoot.pdf", acknowledgement = ack-nhfb, remark = "Code used on Ferranti Mark I.", } @Article{Booth:1951:SBM, author = "A. D. Booth", title = "A signed binary multiplication technique", journal = j-QUART-J-MECH-APPLIED-MATH, volume = "4", number = "2", pages = "236--240", month = "????", year = "1951", CODEN = "QJMMAV", ISSN = "0033-5614 (print), 1464-3855 (electronic)", ISSN-L = "0033-5614", bibdate = "Thu Apr 2 08:38:35 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-sfo # " and " # ack-nhfb, fjournal = "Quarterly Journal of Mechanics and Applied Mathematics", journal-URL = "http://qjmam.oxfordjournals.org/content/by/year", } @InCollection{Brown:1951:HRR, author = "G. W. Brown", title = "History of {RAND}'s random digits", crossref = "Householder:1951:MCM", pages = "31--32", year = "1951", bibdate = "Sun Sep 18 12:02:43 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "http://www.rand.org/pubs/papers/2008/P113.pdf", acknowledgement = ack-nhfb, } @Article{Gill:1951:PSS, author = "S. Gill", title = "A process for the step-by-step integration of differential equations in an automatic digital computing machine", journal = j-PROC-CAMBRIDGE-PHIL-SOC, volume = "47", pages = "96--108", year = "1951", CODEN = "PCPSA4", ISSN = "0008-1981", bibdate = "Tue Aug 28 05:54:12 2001", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, fjournal = "Proceedings of the Cambridge Philosophical Society. Mathematical and physical sciences", journal-URL = "http://journals.cambridge.org/action/displayJournal?jid=PSP", keywords = "accurate floating-point summation; floating-point arithmetic; rounding errors", remark = "Higham \cite{Higham:1993:AFP} comments ``In 1951 Gill [8] noticed that the rounding error in the sum of two numbers could be estimated by subtracting one of the numbers from the sum, and he made use of this estimate in a Runge--Kutta code in a program library for the EDSAC computer.''", } @Article{Goldstine:1951:NIM, author = "Herman H. Goldstine and John von Neumann", title = "Numerical Inverting of Matrices of High Order. {II}", journal = j-PROC-AM-MATH-SOC, volume = "2", pages = "188--202", year = "1951", CODEN = "PAMYAR", ISSN = "0002-9939 (print), 1088-6826 (electronic)", ISSN-L = "0002-9939", MRclass = "65.0X", MRnumber = "MR0041539 (12,861b)", MRreviewer = "F. J. Murray", bibdate = "Thu Nov 8 14:49:46 2007", 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/bibnet/subjects/acc-stab-num-alg.bib; https://www.math.utah.edu/pub/mirrors/ftp.ira.uka.de/bibliography/Math/acc-stab-num-alg.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", note = "See \cite{vonNeumann:1947:NIM} for Part I. Reprinted in \cite[Paper 15, pp. 558--572]{Taub:1963:JNCa}.", URL = "http://www.jstor.org/view/00029939/di970628/97p0185x/0", ZMnumber = "043.12301", acknowledgement = ack-nhfb, author-dates = "John von Neumann (28 December 1903--8 February 1957)", fjournal = "Proceedings of the American Mathematical Society", journal-URL = "http://www.ams.org/journals/proc", } @Article{MacMillan:1951:FDC, author = "Donald B. MacMillan and Richard H. Stark", title = "``Floating Decimal'' Calculation on the {IBM} Card Programmed Electronic Calculator (in Automatic Computing Machinery; Discussions)", journal = j-MATH-TABLES-OTHER-AIDS-COMPUT, volume = "5", number = "34", pages = "86--92", month = apr, year = "1951", 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/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/", } @InProceedings{Rademacher:1951:AEP, author = "Hans Rademacher", booktitle = "Proceedings of a Symposium on Large-scale Digital Calculating Machinery", title = "On the accumulation of errors in processes of integration on high-speed calculating machines", publisher = pub-HARVARD, address = pub-HARVARD:adr, pages = "176--185", year = "1951", bibdate = "Mon Jun 18 06:50:41 2018", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, } @Article{Shirley:1951:BNB, author = "John W. Shirley", title = "Binary Numeration before {Leibniz}", journal = j-AMER-J-PHYSICS, volume = "19", number = "8", pages = "452--454", month = nov, year = "1951", CODEN = "AJPIAS", DOI = "https://doi.org/10.1119/1.1933042", ISSN = "0002-9505 (print), 1943-2909 (electronic)", ISSN-L = "0002-9505", bibdate = "Sat Feb 8 10:29:20 2020", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "http://scitation.aip.org/content/aapt/journal/ajp/19/8/10.1119/1.1933042", abstract = "Though it is frequently stated that binary numeration was first formally proposed by Leibniz as an illustration of his dualistic philosophy, the mathematical papers of Thomas Hariot (1560--1621) show clearly that Hariot not only experimented with number systems, but also understood clearly the theory and practice of binary numeration nearly a century before Leibniz's time.", acknowledgement = ack-nhfb, fjournal = "American Journal of Physics", journal-URL = "http://scitation.aip.org/content/aapt/journal/ajp", remark = "Submitted 14 November 1950.", } @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/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", } @InProceedings{Andrews:1952:RBL, author = "E. G. Andrews", editor = "J. C. McPherson", booktitle = "Proceedings of the {AIEE-IRE '51}: Papers and discussions presented at the December 10--12, 1951, joint {AIEE-IRE} computer conference, Philadelphia, {PA}: Review of electronic digital computers", title = "A review of the {Bell Laboratories}' digital computer developments", publisher = pub-ACM, address = pub-ACM:adr, bookpages = "vi + 114", pages = "101--105", year = "1952", DOI = "https://doi.org/10.1145/1434770.1434787", LCCN = "????", bibdate = "Fri Dec 03 09:17:58 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, keywords = "Bell Laboratories Model 1--5 relay computers; complex fixed-point decimal arithmetic; complex floating-point decimal arithmetic", remark-1 = "From page 101: ``The Bell Telephone Laboratories have designed and built seven digital computers.''", remark-2 = "From page 101: ``George R. Stibitz \ldots{} designed and \ldots{} called it a ``complex number computer.''", remark-3 = "From page 101: ``This computer operated with binary coded decimal notation, with the decimal digits 0 to 9 being represented by the binary numbers 0011 to 1100. The input and output information consisted of eight place numbers, but the calculator carried operations out to ten places, the two extra places being used to improve accuracy when accumulating the results of several problems.''", remark-4 = "From page 104: ``Besides these advantages in operation ease, another has been developed but it has not yet been incorporated into a computer as far as is known. This is the self-correcting code for numerical data. Dr. R. W. Hamming devised this principle of coding. A computer with this feature would show an unattended operation performance superior to anything we know today.'' [Refers to ``Error Detecting and Error Correcting Codes'', The Bell System Technical Journal {\bf 29}(2) 147--160, April 1960, doi:10.1002/j.1538-7305.1950.tb00463.x].", } @Article{Davis:1952:ARS, author = "K. Davis and R. Biddulph and S. Balashek", title = "Automatic Recognition of Spoken Digits", journal = j-J-ACOUST-SOC-AM, volume = "24", number = "6", pages = "637--642", month = nov, year = "1952", CODEN = "JASMAN", ISSN = "0001-4966", ISSN-L = "0001-4966", bibdate = "Fri Nov 28 16:38:51 2003", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-mfc # " and " # ack-nhfb, fjournal = "Journal of the Acoustical Society of America", journal-URL = "http://scitation.aip.org/content/asa/journal/jasa", } @Article{Hammersley:1952:CSS, author = "J. M. Hammersley", title = "The computation of sums of squares and products on a desk calculator", journal = j-BIOMETRICS, volume = "8", number = "??", pages = "156--168", year = "1952", CODEN = "BIOMB6", DOI = "https://doi.org/10.2307/3001930", ISSN = "0006-341X (print), 1541-0420 (electronic)", ISSN-L = "0006-341X", MRclass = "65.0X", MRnumber = "48908", MRreviewer = "E. Lukacs", bibdate = "Thu Jun 25 17:01:03 2020", bibsource = "https://www.math.utah.edu/pub/tex/bib/biometrics1950.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, ajournal = "Biometrics", fjournal = "Biometrics", journal-URL = "http://www.jstor.org/journal/biometrics", } @Article{Irani:1952:SMT, author = "Rida A. K. Irani", title = "A sexagesimal multiplication table in the {Arabic} alphabetical system", journal = j-SCRIPTA-MATH, volume = "18", number = "??", pages = "92--93", month = "????", year = "1952", ISSN = "0036-9713", ISSN-L = "0036-9713", MRclass = "01.0X", MRnumber = "49109", bibdate = "Thu Oct 26 11:15:25 2023", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/scripta-math.bib", ZMnumber = "0046.00102", acknowledgement = ack-nhfb, ajournal = "Scripta Math.", fjournal = "Scripta Mathematica: A Quarterly Journal Devoted to the Philosophy, History, and Expository Treatment of Mathematics", ZBmath = "3070983", } @Article{Michaelson:1952:BA, author = "R. L. Michaelson", title = "Binary Arithmetic", journal = j-INC-STAT, volume = "3", number = "1", pages = "35--40", month = feb, year = "1952", CODEN = "????", DOI = "https://doi.org/10.2307/2986591", ISSN = "1466-9404", ISSN-L = "1466-9404", bibdate = "Thu Jan 22 18:10:18 MST 2015", bibsource = "http://www.jstor.org/stable/i349863; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/jrss-d-1950.bib", URL = "http://www.jstor.org/stable/2986591", acknowledgement = ack-nhfb, fjournal = "The Incorporated Statistician", journal-URL = "http://www.jstor.org/journals/14669404.html", } @Article{Morrill:1952:SEM, author = "C. D. Morrill and R. V. Baum", title = "A Stabilized Electronic Multiplier", journal = j-TRANS-IRE-PROF-GROUP-ELEC-COMPUT, volume = "EC-1", number = "??", pages = "52--59", month = dec, year = "1952", CODEN = "????", ISSN = "????", bibdate = "Fri Jul 15 15:20:53 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", acknowledgement = ack-nhfb, fjournal = "Transactions of the I.R.E. Professional Group on Electronic Computers", } @InProceedings{Sheldon:1952:ICP, author = "J. W. Sheldon and L. Tatum", booktitle = "Review of Electronic Digital Computers. {Joint AIEE--IRE Computer Conference. 10--12 December 1951}", title = "The {IBM} card-programmed electronic calculator", publisher = "American Institute of Electrical Engineers", address = "New York, NY, USA", pages = "30--36", year = "1952", bibdate = "Wed Oct 13 11:28:45 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", note = "Reprinted in \cite[\S 5.4]{Randell:1982:ODC}.", acknowledgement = ack-nhfb, } @Article{Brooker:1953:FOE, author = "R. A. Brooker and D. J. Wheeler", title = "Floating Operations on the {EDSAC} (in Automatic Computing Machinery; Discussions)", journal = j-MATH-TABLES-OTHER-AIDS-COMPUT, volume = "7", number = "41", pages = "37--47", month = jan, year = "1953", CODEN = "MTTCAS", DOI = "https://doi.org/10.1090/S0025-5718-1953-0052901-3", 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/fparith.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp1950.bib; JSTOR database", abstract = "The difficulties which arise when programming calculations for large automatic calculating machines which have a fixed decimal point are discussed. This leads to a consideration of the possibility of using floating decimal arithmetic for certain kinds of calculations. A method by which floating decimal arithmetic can be carried out with any fixed decimal-point machine is outlined and the scheme adopted for use with the EDSAC is described in detail.", acknowledgement = ack-nhfb, ajournal = "Math. Tables Other Aids Comput.", fjournal = "Mathematical Tables and Other Aids to Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "floating-point decimal arithmetic", remark-01 = "From page 38: ``The problem [of programmer-controlled numeric scaling] does not arise with machines designed to operate directly with numbers expressed in the floating radix form. Numbers in this form are represented by $a \cdot r^p$. The first machine of this kind was the Bell Telephone Laboratories Relay Computer Model V(1). This is a decimal machine (that is, $r = 10$) in which $1 > |a| > 0.1$, $19 > p > -19$ and $a$ is expressed to an accuracy of seven significant figures. Since this was completed all important relay machines have been equipped with similar facilities. No electronic machine of this kind has yet been built but we would remark that in our opinion an electronic machine provided with a floating point arithmetical unit would be a powerful computing instrument even if it had a relatively slow store, a magnetic drum, for example.''", remark-02 = "From page 39: ``Two long and two short storage locations are set aside to form a kind of `arithmetical unit.' One long location holds the numerical part of a number and one short location holds the exponent. Together they form the {\em floating decimal accumulator}. In a similar fashion the other long location and the other short location form the {\em floating decimal register}.''", remark-03 = "From page 40: ``the floating decimal accumulator is then `cleared' by replacing the number held in it by zero, that is, by the special number $010^{-63}$.''", remark-04 = "From page 40: `The use of two separate storage locations for the floating decimal accumulator allows the range and accuracy of numbers held therein to be greater than those held in a single storage location elsewhere. This enables products to be accumulated without loss of accuracy due to intermediate rounding-off errors.''", remark-05 = "From the conclusion on page 46: ``From a direct comparison it would seem that the floating 'orders,' other than those used for reading and writing, are about 60 times as slow as the machine orders and hence that a programme using the interpretive subroutine would be slower by the same factor. This is not altogether true because in such a programme fewer orders' are needed than would otherwise be necessary as there are no scale factors to deal with and the techniques for counting and for the modification of orders' have been streamlined. Moreover, the time taken by the C auxiliaries is about the same as that taken by the corresponding subroutine in fixed decimal-point working. These factors vary from problem to problem but our experience has shown that the reduction in speed varies from about 20 to 1 to about 4 to 1. The reduction of the time taken to code a problem has to be experienced to be believed!''", } @Book{IBM:1953:POT, author = "{IBM Corporation}", title = "Principles of Operation: Type 701 and Associated Equipment", publisher = pub-IBM, address = pub-IBM:adr, pages = "103", year = "1953", bibdate = "Wed Sep 14 23:17:49 1994", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, } @Article{Samelson:1953:ORR, author = "Klaus Samelson and Friedrich L. Bauer", title = "{Optimale Rechengenauigkeit bei Rechenanlagen mit gleitendem Komma}. ({German}) [{Optimal} calculation accuracy for calculators with a floating point]", journal = j-Z-ANGE-MATH-PHYS, volume = "4", number = "4", pages = "312--316", month = jul, year = "1953", CODEN = "ZAMPDB", DOI = "https://doi.org/10.1007/bf02074638", ISSN = "0044-2275 (print), 1420-9039 (electronic)", ISSN-L = "0044-2275", bibdate = "Sun Aug 16 18:04:35 2020", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", abstract = "Digital automatic computing machines with floating binary point are investigated with respect to their accuracy. It is concluded that special devices can be designed without difficulty, which guarantee the highest accuracy obtainable with a fixed number of digits. For binary computers, a modification of the usual conversion routine is necessary, which is discussed in detail.", acknowledgement = ack-nhfb, fjournal = "{Zeitschrift f{\"u}r Angewandte Mathematik und Physik = Journal of Applied Mathematics and Physics}", journal-URL = "http://link.springer.com/journal/33", keywords = "number base conversion", language = "German", } @InCollection{Stiefel:1953:MCA, author = "E. Stiefel", booktitle = "Les machines {\`a} calculer et la pens{\'e}e humaine", title = "La machine {\`a} calculer arithm{\'e}tique ``{Z4}'' de l'{Ecole} {Polytechnique} {F}{\'e}d{\'e}rale {\`a} {Zurich} ({Suisse}) et son application {\`a} la r{\'e}solution d'une {\'e}quation aux d{\'e}riv{\'e}es partielles de type elliptique. ({French}) [{The} arithmetic calculator ``{Z4}'' of the {Swiss Federal Polytechnical University in Zurich (Switzerland)} and its application to the solution of an elliptical type partial derivative equation]", publisher = "Centre National de la Recherche Scientifique", address = "Paris, France", pages = "33--40", year = "1953", MRclass = "65.0X", MRnumber = "0066093", 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/fparith.bib", series = "Colloques internationaux du Centre National de la Recherche Scientifique, no 37", acknowledgement = ack-nhfb, author-dates = "Eduard Stiefel (21 April 1909--25 November 1978)", language = "French", } @Article{Backus:1954:ISS, author = "J. W. Backus", title = "The {IBM 701 Speedcoding} System", journal = j-J-ACM, volume = "1", number = "1", pages = "4--6", month = jan, year = "1954", CODEN = "JACOAH", ISSN = "0004-5411 (print), 1557-735X (electronic)", ISSN-L = "0004-5411", bibdate = "Fri Nov 04 00:18:27 1994", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "http://community.computerhistory.org/scc/projects/FORTRAN/paper/p4-backus.pdf", abstract = "The IBM 701 Speedcoding System is a set of instructions which causes the 701 to behave like a three-address floating point calculator. Let us call this the Speedcoding calculator. In addition to operating in floating point, this Speedcoding calculator has extremely convenient means for getting information into the machine and for printing results; it has an extensive set of operations to make the job of programming as easy as possible. Speedcoding also provides automatic address modification, flexible tracing, convenient use of auxiliary storage, and built-in checking.", acknowledgement = ack-nhfb, fjournal = "Journal of the ACM", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J401", } @Manual{Burroughs:1954:DH, author = "{Burroughs Corporation.ElectroData}", title = "Datatron handbooks", organization = "The Division", address = "Pasadena, CA, USA", year = "1954", bibdate = "Sat Feb 24 15:01:45 MST 1996", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", note = "12 volumes in 1.", acknowledgement = ack-nhfb, keywords = "Datatron (Computer)", remark = "Datatron programming and coding manual --- Datatron operation manual --- Central computer handbook --- Paper tape system handbook --- Card converter Model 500 handbook --- Magnetic tape system handbook --- Cardatron system handbook --- Model 560 DATAFILE handbook --- Control console and consolette handbook --- Floating point control unit handbook --- External switching unit handbook --- Tape preparation unit handbook.", } @Article{Freeman:1954:TSA, author = "H. Freeman and E. Parsons", title = "Time-Sharing Analog Multiplier ({TSAM})", journal = j-TRANS-IRE-PROF-GROUP-ELEC-COMPUT, volume = "EC-3", number = "1", pages = "11--17", month = mar, year = "1954", CODEN = "????", ISSN = "????", bibdate = "Fri Jul 15 15:20:53 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", acknowledgement = ack-nhfb, fjournal = "Transactions of the I.R.E. Professional Group on Electronic Computers", remark = "Cited in US Patent 3,043,516 (10 July 1962): Time Summing Device for Division, Multiplication, Root Taking and Interpolation, by Harold W. Abbott and Vernon P. Mathis.", } @Article{Gorn:1954:AAC, author = "Saul Gorn", title = "The Automatic Analysis and Control of Computing Errors", journal = j-J-SIAM, volume = "2", number = "2", pages = "69--81", month = jun, year = "1954", CODEN = "JSIMAV", ISSN = "0368-4245 (print), 1095-712X (electronic)", bibdate = "Thu Oct 15 18:16:06 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "Journal of the Society for Industrial and Applied Mathematics", journal-URL = "http://epubs.siam.org/loi/smjmap.1", } @Article{Kovach:1954:AMU, author = "L. D. Kovach and W. Comley", title = "An Analog Multiplier Using Thyrite", journal = j-TRANS-IRE-PROF-GROUP-ELEC-COMPUT, volume = "EC-3", number = "2", pages = "42--45", month = jun, year = "1954", CODEN = "????", ISSN = "????", bibdate = "Sun Jul 17 09:57:13 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", acknowledgement = ack-nhfb, fjournal = "Transactions of the I.R.E. Professional Group on Electronic Computers", } @Article{Mayer:1954:ODF, author = "M. A. Mayer and B. M. Gordon and R. N. Nicola", title = "An operational digital feedback divider", journal = j-IRE-TRANS-ELEC-COMPUT, volume = "EC-3", number = "1", pages = "17--20", month = mar, year = "1954", CODEN = "IRELAO", ISSN = "0367-9950", bibdate = "Sun Jul 17 09:52:58 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", acknowledgement = ack-nhfb, fjournal = "IRE Transactions on Electronic Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=5407885", } @Article{Moshman:1954:GPR, author = "Jack Moshman", title = "The Generation of Pseudo-Random Numbers on a Decimal Calculator", journal = j-J-ACM, volume = "1", number = "2", pages = "88--91", month = apr, year = "1954", CODEN = "JACOAH", ISSN = "0004-5411 (print), 1557-735X (electronic)", ISSN-L = "0004-5411", bibdate = "Fri Nov 04 00:18:27 1994", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, fjournal = "Journal of the ACM", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J401", keywords = "decimal floating-point arithmetic", } @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", 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 = "Thu Feb 08 16:17:25 2018", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp1950.bib", acknowledgement = ack-nhfb, ajournal = "Math. Tables Other Aids Comput.", fjournal = "Mathematical Tables and Other Aids to Computation", journal-URL = "http://www.ams.org/mcom/", 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{Crockett:1955:GMM, author = "J. B. Crockett and H. Chernoff", title = "Gradient Methods of Maximization", journal = j-PAC-J-MATH, volume = "5", number = "??", pages = "33--50", year = "1955", CODEN = "PJMAAI", ISSN = "0030-8730 (print), 1945-5844 (electronic)", ISSN-L = "0030-8730", bibdate = "Fri Aug 20 08:54:24 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, fjournal = "Pacific Journal of Mathematics", journal-URL = "http://msp.org/pjm", remark = "Cited in \cite{Sterbenz:1974:FPC}.", } @Book{Eckert:1955:FFS, author = "W. J. (Wallace John) Eckert and Rebecca Bradley Jones", title = "Faster, faster; a simple description of a giant electronic calculator and the problems it solves", publisher = pub-IBM, address = pub-IBM:adr, pages = "160", year = "1955", LCCN = "QA76 .E25", bibdate = "Fri Aug 20 08:57:56 MDT 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; z3950.loc.gov:7090/Voyager", acknowledgement = ack-nhfb, author-dates = "1902--1971", remark = "Cited in \cite{Sterbenz:1974:FPC}.", subject = "Computers", xxaddress = pub-MCGRAW-HILL:adr, xxpublisher = pub-MCGRAW-HILL, } @Book{Hastings:1955:ADC, author = "Cecil B. {Hastings, Jr.} and Jeanne T. Hayward and James P. {Wong, Jr.}", title = "Approximations for Digital Computers", publisher = pub-PRINCETON, address = pub-PRINCETON:adr, pages = "viii + 201", year = "1955", LCCN = "QA76 .H33", bibdate = "Mon Sep 30 14:51:50 1996", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nj, } @Article{Lenaerts:1955:ASR, author = "E. H. Lenaerts", title = "Automatic Square Rooting", journal = j-ELECTRON-ENG, volume = "27", number = "??", pages = "287--289", month = jul, year = "1955", CODEN = "ELEGAP", ISSN = "0013-4902", 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/fparith.bib", acknowledgement = ack-nj, fjournal = "Electronic Engineering", } @Article{Parsons:1955:SDC, author = "Frances L. Parsons", title = "A Simple Desk-Calculator Method for Checking Binary Results of Digital Computer Arithmetic Operations", journal = j-J-ACM, volume = "2", number = "3", pages = "205--207", month = jul, year = "1955", CODEN = "JACOAH", ISSN = "0004-5411 (print), 1557-735X (electronic)", ISSN-L = "0004-5411", bibdate = "Wed Nov 09 01:00:34 1994", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, fjournal = "Journal of the ACM", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J401", } @Book{Richards:1955:AOD, author = "Richard Kohler Richards", title = "Arithmetic Operations in Digital Computers", publisher = "D. Van Nostrand", address = "New York, NY, USA", pages = "iv + 397", year = "1955", LCCN = "QA75 .R5 1955", bibdate = "Fri Nov 28 18:46:37 2003", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, } @Article{Robertson:1955:TCM, author = "J. E. Robertson", title = "Two's Complement Multiplication in Binary Parallel Digital Computers", journal = j-IRE-TRANS-ELEC-COMPUT, volume = "EC-4", number = "3", pages = "118--119", month = sep, year = "1955", CODEN = "IRELAO", DOI = "https://doi.org/10.1109/IRETELC.1955.5407911", ISSN = "0367-9950", bibdate = "Thu Jun 30 15:10:39 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", URL = "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5407911", acknowledgement = ack-nhfb, fjournal = "IRE Transactions on Electronic Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=5407885", } @Book{Stiefel:1955:RID, author = "Eduard Stiefel", title = "{Rechenautomaten im Dienste der Technik. Erfahrungen mit dem Zuse-Rechenautomaten Z4}. ({German}) [{Calculating} machines in the service of technology. {Experience} with the {Zuse}-calculator {Z4}]", volume = "45", publisher = "Westdeutscher Verlag", address = "Cologne and Opladen, West Germany", pages = "29--45; Diskussion 47--65", year = "1955", MRclass = "68.0X", MRnumber = "0073311", MRreviewer = "H. H. Goldstine", 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/fparith.bib", series = "Arbeitsgemeinschaft f{\"{u}}r Forschung des Landes Nordrhein-Westfalen. Heft", acknowledgement = ack-nhfb, author-dates = "Eduard Stiefel (21 April 1909--25 November 1978)", language = "German", } @Article{Estrin:1956:NHS, author = "G. Estrin and B. Gilchrist and J. H. Pomerene", title = "A Note on High-Speed Digital Multiplication", journal = j-IRE-TRANS-ELEC-COMPUT, volume = "EC-5", number = "3", pages = "140--140", month = sep, year = "1956", CODEN = "IRELAO", DOI = "https://doi.org/10.1109/TEC.1956.5219936", ISSN = "0367-9950", bibdate = "Thu Jun 30 15:46:27 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", URL = "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5219936", acknowledgement = ack-nhfb, fjournal = "IRE Transactions on Electronic Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=5407885", } @Book{Hildebrand:1956:INA, author = "Francis Begnaud Hildebrand", title = "Introduction to Numerical Analysis", publisher = pub-MCGRAW-HILL, address = pub-MCGRAW-HILL:adr, pages = "511", year = "1956", LCCN = "QA300 .H5", bibdate = "Fri Aug 20 09:19:58 MDT 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; z3950.loc.gov:7090/Voyager", series = "International series in pure and applied mathematics", acknowledgement = ack-nhfb, remark = "See also second edition \cite{Hildebrand:1974:INA,Hildebrand:1987:INA}.", subject = "Numerical analysis", tableofcontents = "1. Introduction (significant figures, random errors, error bounds, etc.) \\ 2. Interpolation with divided differences \\ 3. Lagrangian methods \\ 4. Finite-difference interpolation \\ 5. Operations with finite differences \\ 6. Numerical solution of ordinary differential equations \\ 7. Least-squares polynomial approximation \\ 8. Gaussian quadrature and related topics \\ 9. Approximations of various types \\ 10. Numerical solution of equations \\ Appendix: Justification of the Crout reduction", } @TechReport{Lazarus:1956:MI, author = "R. B. Lazarus and N. Metropolis and W. Orvedahl and J. H. Richardson and W. {Spack, Jr.} and R. L. Bivins and J. V. Caulfield and I. Kral and A. F. Malmberg and G. T. McKinley and R. E. Williamson", title = "{MANIAC II}", type = "Report", number = "LA-2083", institution = inst-LASL, address = inst-LASL:adr, pages = "54", day = "1", month = oct, year = "1956", bibdate = "Tue Mar 20 11:14:15 2007", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/m/metropolis-nicholas.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "https://sgp.fas.org/othergov/doe/lanl/lib-www/la-pubs/00320765.pdf", acknowledgement = ack-nhfb, remark-1 = "OSTI ID: 4382471", remark-2 = "From page 9: ``The Maniac II \ldots{} has 12,288 48-bit words. It is asynchronous and has no clock. Its speed is that appropriate to (1) a memory cycle of 8 microseconds, (2) a basic add time of 6 microseconds, and (3) a shift time of about 1.3 microseconds per stage. The average multiply time is about 160 microseconds.''", remark-3 = "From page 14, the 48-bit floating-point word consists of a 1-bit sign, a 3-bit exponent-of-65536, a 1-bit sign of the exponent, and a 43-bit fraction in [0,1). The range of nonzero numbers is (0x1p-155, 0x1p112), or roughly (2e-47, 5e33). The report notes: ``Although such a large base implies of possibility of as many as 15 lead[ing] zeros, the large word size of 48 bits guarantees adequate significance.''", } @Article{Lilamand:1956:TDM, author = "M. Lejet Lilamand", title = "A Time-Division Multiplier", journal = j-IRE-TRANS-ELEC-COMPUT, volume = "EC-5", number = "1", pages = "26--34", month = mar, year = "1956", CODEN = "IRELAO", DOI = "https://doi.org/10.1109/TEC.1956.5219789", ISSN = "0367-9950", bibdate = "Thu Jun 30 15:46:18 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", URL = "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5219789", acknowledgement = ack-nhfb, fjournal = "IRE Transactions on Electronic Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=5407885", } @Article{Perkins:1956:EPC, author = "Robert Perkins", title = "{EASIAC}, {A} Pseudo-Computer", journal = j-J-ACM, volume = "3", number = "2", pages = "65--72", month = apr, year = "1956", CODEN = "JACOAH", ISSN = "0004-5411 (print), 1557-735X (electronic)", ISSN-L = "0004-5411", bibdate = "Fri Nov 04 23:46:26 1994", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", abstract = "One of the primary functions of the MIDAC installation at the University of Michigan is the instruction of beginners in the various aspects of digital machine use including programming and coding. \ldots{} In conducting these courses it was soon found to be extremely difficult, in five or six instruction periods, to bring a complete newcomer up to the point where he can code and check out on MIDAC anything more than a rather trivial routine. As might be expected the difficulty centers around problems of scaling, instruction modification and binary representation. \ldots{} To alleviate these problems it was decided that a new computer was needed: one designed to make programming easier. At the cost of some of MIDAC's speed and capacity plus two or three man-months of programming time EASIAC, the EASy Instruction Automatic Computer, was realized as a translation-interpretation program in MIDAC.", acknowledgement = ack-nhfb, fjournal = "Journal of the ACM", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J401", keywords = "decimal floating-point arithmetic", remark = "Early example of a decimal floating-point machine.", } @Article{Robertson:1956:NCD, author = "J. E. Robertson", title = "A new class of digital division methods", journal = j-IRE-TRANS-ELEC-COMPUT, volume = "EC-5", number = "??", pages = "65--73", month = jun, year = "1956", CODEN = "IRELAO", ISSN = "0367-9950", bibdate = "Fri Nov 09 19:14:45 2007", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, fjournal = "IRE Transactions on Electronic Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=5407885", } @Article{Stegun:1956:PC, author = "Irene A. Stegun and Milton Abramowitz", title = "Pitfalls in computation", journal = "Journal of the Society for Industrial and Applied Mathematics", volume = "4", pages = "207--219", year = "1956", bibdate = "Tue Aug 28 06:16:28 2001", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, remark = "Cited in \cite{Sterbenz:1974:FPC}.", } @Article{Sydnor:1956:AMS, author = "R. L. Sydnor and T. R. O'Meara and J. Strathman", title = "Analog Multipliers and Squarers Using a Multigrid Modulator", journal = j-IRE-TRANS-ELEC-COMPUT, volume = "EC-5", number = "2", pages = "82--85", month = jun, year = "1956", CODEN = "IRELAO", DOI = "https://doi.org/10.1109/TEC.1956.5219804", ISSN = "0367-9950", bibdate = "Thu Jun 30 15:46:21 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", URL = "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5219804", acknowledgement = ack-nhfb, fjournal = "IRE Transactions on Electronic Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=5407885", } @Article{Weinberger:1956:OMA, author = "A. Weinberger and J. L. Smith", title = "A One-Microsecond Adder Using One-Megacycle Circuitry", journal = j-IRE-TRANS-ELEC-COMPUT, volume = "EC-5", number = "??", pages = "65--73", month = jun, year = "1956", CODEN = "IRELAO", ISSN = "0367-9950", bibdate = "Fri Nov 09 19:38:57 2007", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, fjournal = "IRE Transactions on Electronic Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=5407885", } @Article{Wolontis:1956:CFD, author = "V. M. Wolontis", title = "A Complete Floating-Decimal Interpretive System for the {IBM 650 Magnetic Drum Calculator}", journal = "IBM Technical Newsletter", volume = "??", number = "11", pages = "xxi + 63", month = mar, year = "1956", bibdate = "Mon Sep 04 13:55:28 2023", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "https://babel.hathitrust.org/cgi/pt?id=coo.31924003945759&seq=7", acknowledgement = ack-nhfb, } @Book{Alt:1957:EDC, editor = "Franz L. Alt", title = "Electronic Digital Computers: Their Use in Science and Engineering", publisher = pub-ACADEMIC, address = pub-ACADEMIC:adr, pages = "x + 335", year = "1957", bibdate = "Sun Jun 17 18:53:23 2018", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, remark = "Section 2.3 discusses number representation and decimal versus binary arithmetic.", xxnote = "Check page count: incomplete view in Google books. Price bibliography says 1958.", } @Article{Ercoli:1957:EDO, author = "Paolo Ercoli and Roberto Vacca", title = "Errors Due to Overflow in Arithmetic Operations Particularly as Regards {FINAC} Electronic Computer", journal = j-J-ACM, volume = "4", number = "4", pages = "450--455", month = oct, year = "1957", CODEN = "JACOAH", ISSN = "0004-5411 (print), 1557-735X (electronic)", ISSN-L = "0004-5411", bibdate = "Thu Dec 08 09:28:48 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", note = "See letter \cite{Ercoli:1960:LEE}.", acknowledgement = ack-nhfb, fjournal = "Journal of the ACM", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J401", keywords = "floating-point arithmetic; overflow", } @Article{Gini:1957:SFD, author = "Corrado Gini", title = "Sulla frequenza delle cifre iniziali dei numeri osservati. ({Italian}) [{On} the frequency of initial digits of observed numbers]", journal = "Bull. Inst. Internat. Stat.", volume = "35", number = "??", pages = "57--76", year = "1957", bibdate = "Sun Sep 18 11:54:25 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", note = "29th session, 2nd delivery, Rio de Janerio.", URL = "", acknowledgement = ack-nhfb, keywords = "Benford's Law; Law of Anomalous Numbers; Zipf's Law", language = "Italian", } @Article{Herzel:1957:SDD, author = "Amato Herzel", title = "Sulla distribuzione della cifre iniziali dei numeri statistici. ({Italian}) [{On} the frequency of initial digits of statistical numbers]", journal = "Atti dell XV e XVII Riunione, Societa Italiana di Statistica", volume = "??", number = "??", pages = "??--??", year = "1957", bibdate = "Sun Sep 18 12:00:07 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, keywords = "Benford's Law; Law of Anomalous Numbers; Zipf's Law", language = "Italian", } @Article{Howe:1957:TRA, author = "R. M. Howe and E. G. Gilbert", title = "Trigonometric Resolution in Analog Computers by Means of Multiplier Elements", journal = j-IRE-TRANS-ELEC-COMPUT, volume = "EC-6", number = "2", pages = "86--92", month = jun, year = "1957", CODEN = "IRELAO", DOI = "https://doi.org/10.1109/TEC.1957.5221576", ISSN = "0367-9950", bibdate = "Thu Jun 30 16:07:27 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", URL = "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5221576", acknowledgement = ack-nhfb, fjournal = "IRE Transactions on Electronic Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=5407885", } @Article{Kalbfell:1957:EAM, author = "David C. Kalbfell", title = "An Electronic Analog Multiplier", journal = j-IRE-TRANS-ELEC-COMPUT, volume = "EC-6", number = "2", pages = "100--103", month = jun, year = "1957", CODEN = "IRELAO", DOI = "https://doi.org/10.1109/TEC.1957.5221579", ISSN = "0367-9950", bibdate = "Thu Jun 30 16:07:27 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", URL = "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5221579", acknowledgement = ack-nhfb, fjournal = "IRE Transactions on Electronic Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=5407885", } @Article{Kogbetliantz:1957:CEN, author = "E. G. 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", ISSN = "0018-8646 (print), 2151-8556 (electronic)", ISSN-L = "0018-8646", bibdate = "Tue Sep 06 18:04:49 1994", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nj, fjournal = "IBM Journal of Research and Development", journal-URL = "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=5288520", } @Article{Lehman:1957:HSD, author = "M. Lehman", title = "High-Speed Digital Multiplication", journal = j-IRE-TRANS-ELEC-COMPUT, volume = "EC-6", number = "3", pages = "204--205", month = sep, year = "1957", CODEN = "IRELAO", DOI = "https://doi.org/10.1109/TEC.1957.5222025", ISSN = "0367-9950", bibdate = "Thu Jun 30 16:07:34 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", URL = "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5222025", acknowledgement = ack-nhfb, fjournal = "IRE Transactions on Electronic Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=5407885", } @Article{Luke:1957:CLZ, author = "Y. L. Luke", title = "On the Computation of $ \log {Z} $ and $ \arctan {Z} $", journal = j-MATH-TABLES-OTHER-AIDS-COMPUT, volume = "11", number = "??", pages = "16--18", month = "????", year = "1957", CODEN = "MTTCAS", ISSN = "0891-6837", bibdate = "Thu Sep 1 10:16:09 1994", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nj, ajournal = "Math. Tables Other Aids Comput.", fjournal = "Mathematical Tables and Other Aids to Computation", journal-URL = "http://www.ams.org/mcom/", } @TechReport{Metze:1957:SPO, author = "G. Metze", title = "A Study of Parallel One's Complement Arithmetic Units with Separate Carry or Borrow Storage", type = "Report", number = "81", institution = "Digital Computer Laboratory, University of Illinois", address = "Urbana, IL, USA", pages = "77", year = "1957", bibdate = "Thu Nov 18 09:07:07 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, remark = "Flores' book \cite[p. 485]{Flores:1963:LCA} cites this report, and says that its conclusion is that the disadvantages of one's complement arithmetic outweigh the advantages.", } @Article{Murphy:1957:PIA, author = "R. W. Murphy", title = "A Positive-Integer Arithmetic for Data Processing", journal = j-IBM-JRD, volume = "1", number = "2", pages = "158--170", month = apr, year = "1957", CODEN = "IBMJAE", DOI = "https://doi.org/10.1147/rd.12.0158", ISSN = "0018-8646 (print), 2151-8556 (electronic)", ISSN-L = "0018-8646", MRclass = "68.0X", MRnumber = "18,939d", bibdate = "Tue Sep 11 15:35:33 MDT 2012", bibsource = "http://www.research.ibm.com/journal/; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ibmjrd.bib", URL = "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5392735; http://www.research.ibm.com/journal/rd/012/ibmrd0102G.pdf", abstract = "It is hypothesized that positive numbers suffice for the expression of quantities in accounting. New arithmetic operations are devised that yield non-negative results in computation, and the applicability of these operations to data processing is studied. These operations permit a wide variety of functions to be computed with fewer and less complex steps and imply the feasibility of constructing less complex data-processing machines.", acknowledgement = ack-nhfb, book-URL = "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=5288520", fjournal = "IBM Journal of Research and Development", journal-URL = "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=5288520", reviewer = "C. C. Gotlieb", } @Article{Pawlak:1957:UEN, author = "Z. Pawlak", title = "Use of expansions with a negative basis in the arithmometer of a digital computer", journal = "Bull. Acad. Pol. Sci., Ser. Sci. Tech.", volume = "5", number = "??", pages = "232--236", month = "????", year = "1957", bibdate = "Thu Nov 06 05:56:32 2014", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, keywords = "negative base", } @TechReport{Taylor:1957:CCA, author = "W. Bruce Taylor", title = "{COIN} (Compile-Interpreter): an automatic programming, fixed and floating-point library of subroutines for the {ERA} 1103 computer", institution = "Operations Research Office, Johns Hopkins University", address = "Bethesda, MD, USA", pages = "43", year = "1957", bibdate = "Sat Feb 24 15:01:45 MST 1996", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", series = "Staff paper / Johns Hopkins University, Operations Research Office; ORO-SP-32 Staff paper (Johns Hopkins University. Operations Research Office); ORO-SP-32.", acknowledgement = ack-nhfb, keywords = "Military art and science --- Data processing.; Military research --- Data processing.", remark = "Cover title. ``Operating under contract with the Department of the Army.'' ``Project COMPLAB.'' ``October 1957.'' ORO-SP-32", } @Article{Wadel:1957:NBN, author = "L. B. Wadel", title = "Negative base number systems", journal = j-IRE-TRANS-ELEC-COMPUT, volume = "EC-6", number = "??", pages = "123--??", month = jun, year = "1957", CODEN = "IRELAO", ISSN = "0367-9950", bibdate = "Thu Nov 06 05:53:55 2014", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, fjournal = "IRE Transactions on Electronic Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=5407885", keywords = "negative base", } @Article{Walker:1957:EMA, author = "R. M. Walker and D. E. Rosenheim and P. A. Lewis and A. G. Anderson", title = "An Experimental 50-Megacycle Arithmetic Unit", journal = j-IBM-JRD, volume = "1", number = "3", pages = "257--278", month = jul, year = "1957", CODEN = "IBMJAE", DOI = "https://doi.org/10.1147/rd.13.0257", ISSN = "0018-8646 (print), 2151-8556 (electronic)", ISSN-L = "0018-8646", bibdate = "Tue Sep 11 15:35:35 MDT 2012", bibsource = "http://www.research.ibm.com/journal/; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ibmjrd.bib", URL = "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5392687; http://www.research.ibm.com/journal/rd/013/ibmrd0103H.pdf", abstract = "An experimental 50-megacycle arithmetic unit has been built which performs a repetitive multiplication program and checks the results for errors. The unit uses pulse circuitry which has been developed to perform digital operations at a 50-megacycle pulse-repetition rate. This paper describes the arithmetic system and the circuits which perform the required functions. These circuits include a full binary adder, a phase-locked frequency divider which provides a 3.125-megacycle secondary timing source, a reshaping and retiming circuit using germanium diodes and capacitive storage, a high-speed shift register, a high-speed indicator register, and a binary word generator.\par Various novel features of a digital system operating at these high speeds are described. These include the use of coaxial delay lines for the distribution of signals and as storage elements, and the use of secondary emission tubes in amplifier and multivibrator circuits.\par In a 50-megacycle system the interdependence of the space and time dimensions is marked, and although this introduces problems which are not ordinarily encountered in computing systems, it may be used advantageously to provide features such as the variable-phase clock system used in the arithmetic unit.\par The performance and reliability of the arithmetic unit are discussed as well as the reliability of the components and circuits which make up the system. Although the techniques and circuitry discussed here have been applied only to a relatively simple arithmetic unit, it is felt that they could be useful in a variety of high-speed computing and measurements applications.", acknowledgement = ack-nhfb, book-URL = "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=5288520", fjournal = "IBM Journal of Research and Development", journal-URL = "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=5288520", } @Article{Weibel:1957:EAM, author = "Erich S. Weibel", title = "An Electronic Analog Multiplier Using Carriers", journal = j-IRE-TRANS-ELEC-COMPUT, volume = "EC-6", number = "1", pages = "30--34", month = mar, year = "1957", CODEN = "IRELAO", DOI = "https://doi.org/10.1109/TEC.1957.5221558", ISSN = "0367-9950", bibdate = "Thu Jun 30 16:07:23 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", URL = "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5221558", acknowledgement = ack-nhfb, fjournal = "IRE Transactions on Electronic Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=5407885", } @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} \url{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", } @TechReport{Anonymous:1958:ARM, author = "Anonymous", title = "Analysis and research memorandum 294", type = "{AR} Memo", number = "294 (AD 207929)", institution = "Massachusetts Institute of Technology, Naval Supersonic Laboratory", address = "Cambridge, MA, USA", pages = "7", day = "23", month = oct, year = "1958", bibdate = "Sat Feb 24 15:01:45 MST 1996", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, keywords = "Error functions.; Floating-point arithmetic.", } @Article{Bemer:1958:MMS, author = "Robert W. Bemer", title = "A Machine Method for Square-Root Computation", journal = j-CACM, volume = "1", number = "1", pages = "6--7", month = jan, year = "1958", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Thu Sep 15 18:35:46 1994", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nj, fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", } @Article{Bemer:1958:SMC, author = "R. W. Bemer", title = "A Subroutine Method for Calculating Logarithms", journal = j-CACM, volume = "1", number = "5", pages = "5--7", month = may, year = "1958", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Thu Sep 1 10:16:09 1994", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nj, fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", } @Article{Couleur:1958:BBD, author = "J. F. Couleur", title = "{BIDEC} --- a Binary-to-Decimal or Decimal-to-Binary Converter", journal = j-IRE-TRANS-ELEC-COMPUT, volume = "EC-7", pages = "313--316", year = "1958", CODEN = "IRELAO", ISSN = "0367-9950", bibdate = "Fri Nov 28 16:28:12 2003", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-mfc # " and " # ack-nhfb, fjournal = "IRE Transactions on Electronic Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=5407885", keywords = "decimal floating-point arithmetic", } @Article{Delury:1958:CAN, author = "Daniel B. Delury", title = "Computation with Approximate Numbers", journal = j-MATH-TEACH, volume = "51", pages = "521--530", month = nov, year = "1958", ISSN = "0025-5769 (print), 2330-0582 (electronic)", bibdate = "Fri Nov 28 11:47:47 2003", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, fjournal = "The Mathematics Teacher", } @Article{Ershov:1958:PAO, author = "Andrei P. Ershov", title = "On Programming of Arithmetic Operations", journal = j-CACM, volume = "1", number = "8", pages = "3--6", month = aug, year = "1958", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Wed Jul 14 15:48:21 MDT 2004", bibsource = "http://dblp.uni-trier.de/db/journals/cacm/cacm1.html#Ershov58; http://portal.acm.org/; 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", oldlabel = "Ershov58", XMLdata = "ftp://ftp.informatik.uni-trier.de/pub/users/Ley/bib/records.tar.gz#journals/cacm/Ershov58", } @Article{Gower:1958:NIM, author = "J. C. Gower", title = "A Note on an Iterative Method for Root Extraction", journal = j-COMP-J, volume = "1", pages = "142--143", year = "1958", CODEN = "CMPJA6", ISSN = "0010-4620 (print), 1460-2067 (electronic)", ISSN-L = "0010-4620", bibdate = "Thu Sep 15 18:36:55 1994", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nj, fjournal = "The Computer Journal", journal-URL = "http://comjnl.oxfordjournals.org/", xxmonth = "(none)", xxnumber = "(none)", } @Article{Hammer:1958:PST, author = "Franz Hammer", title = "{Nicht Pascal sondern der T{\"u}binger Professor Wilhelm Schickard erfand die Rechenmaschine!}. ({German}) [{Not Pascal}, but the {T{\"u}bingen} professor {William Schickard}, invented the calculator!]", journal = "{B{\"u}romarkt}", volume = "20", number = "??", pages = "1023--1025", month = "????", year = "1958", ISSN = "0007-3148", ISSN-L = "0007-3148", bibdate = "Sat Jul 27 11:05:26 2013", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/scicontext.bib", acknowledgement = ack-nhfb, language = "German", } @Article{Kogbetliantz:1958:CANa, author = "E. G. 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", ISSN = "0018-8646 (print), 2151-8556 (electronic)", ISSN-L = "0018-8646", bibdate = "Thu Sep 1 10:16:10 1994", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nj, fjournal = "IBM Journal of Research and Development", journal-URL = "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=5288520", } @Article{Kogbetliantz:1958:CANb, author = "E. G. 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", ISSN = "0018-8646 (print), 2151-8556 (electronic)", ISSN-L = "0018-8646", bibdate = "Thu Sep 1 10:16:10 1994", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nj, fjournal = "IBM Journal of Research and Development", journal-URL = "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=5288520", } @Article{Metropolis:1958:SDC, author = "N. Metropolis and R. L. Ashenhurst", title = "Significant Digit Computer Arithmetic", journal = j-IRE-TRANS-ELEC-COMPUT, volume = "EC-7", number = "4", pages = "265--267", month = dec, year = "1958", CODEN = "IRELAO", DOI = "https://doi.org/10.1109/TEC.1958.5222657", ISSN = "0367-9950", bibdate = "Wed Feb 14 19:23:38 2007", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/m/metropolis-nicholas.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", URL = "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5222657", abstract = "The usual floating point arithmetic makes error analysis difficult. This paper describes an alternative system which offers a means of analyzing floating point calculations more effectively and which also possesses certain advantages from an equipment standpoint.", acknowledgement = ack-nhfb, fjournal = "IRE Transactions on Electronic Computers", journal-URL = "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=5407885", keywords = "MANIAC III", remark = "Cited in \cite{Sterbenz:1974:FPC}.", } @Article{Robertson:1958:NCDa, author = "J. E. Robertson", title = "A new class of digital division methods", journal = j-IRE-TRANS-ELEC-COMPUT, volume = "EC-7", number = "3", pages = "88--92", month = sep, year = "1958", CODEN = "IRELAO", ISSN = "0367-9950", bibdate = "Thu Apr 2 08:38:35 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-sfo # " and " # ack-nhfb, fjournal = "IRE Transactions on Electronic Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=5407885", } @Article{Robertson:1958:NCDb, author = "James E. Robertson", title = "A New Class of Digital Division Methods", journal = j-IRE-TRANS-ELEC-COMPUT, volume = "EC-7", number = "3", pages = "218--222", month = sep, year = "1958", CODEN = "IRELAO", DOI = "https://doi.org/10.1109/TEC.1958.5222579", ISSN = "0367-9950", bibdate = "Thu Jul 14 15:56:43 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", URL = "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5222579", acknowledgement = ack-nhfb, fjournal = "IRE Transactions on Electronic Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=5407885", } @Article{Schmid:1958:TFQ, author = "Hermann Schmid", title = "A Transistorized Four-Quadrant Time-Division Multiplier with an Accuracy of 0.1 Per Cent", journal = j-IRE-TRANS-ELEC-COMPUT, volume = "EC-7", number = "1", pages = "41--47", month = mar, year = "1958", CODEN = "IRELAO", DOI = "https://doi.org/10.1109/TEC.1958.5222094", ISSN = "0367-9950", bibdate = "Thu Jul 14 15:56:43 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", URL = "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5222094", acknowledgement = ack-nhfb, fjournal = "IRE Transactions on Electronic Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=5407885", } @Article{Sisson:1958:IDR, author = "Roger L. Sisson", title = "An Improved Decimal Redundancy Check", journal = j-CACM, volume = "1", number = "5", pages = "10--12", month = may, year = "1958", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Mon Jan 22 07:22:12 MST 2001", bibsource = "http://dblp.uni-trier.de/db/journals/cacm/cacm1.html#Sisson58; https://www.math.utah.edu/pub/tex/bib/fparith.bib", abstract = "As more emphasis is placed on improving the accuracy of data fed into automatic computing systems, more emphasis will be placed on redundancy checking of predictable fields within the input. Two systems (at least) of checking a field of decimal digits have been proposed. In both of these it is assumed that the field to be checked is all numeric and that the redundancy must be of only one digit.", acknowledgement = ack-nhfb, fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "decimal floating-point arithmetic", oldlabel = "Sisson58", XMLdata = "ftp://ftp.informatik.uni-trier.de/pub/users/Ley/bib/records.tar.gz#journals/cacm/Sisson58", } @Article{Tocher:1958:TMD, author = "K. D. Tocher", title = "Techniques of Multiplication and Division for Automatic Binary Computers", journal = j-QUART-J-MECH-APPLIED-MATH, volume = "11", number = "3", pages = "364--384", month = "????", year = "1958", CODEN = "QJMMAV", ISSN = "0033-5614 (print), 1464-3855 (electronic)", ISSN-L = "0033-5614", bibdate = "Thu Apr 2 08:38:35 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-sfo # " and " # ack-nhfb, fjournal = "Quarterly Journal of Mechanics and Applied Mathematics", journal-URL = "http://qjmam.oxfordjournals.org/content/by/year", } @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", 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; http://portal.acm.org/; 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", oldlabel = "Wadey58", XMLdata = "ftp://ftp.informatik.uni-trier.de/pub/users/Ley/bib/records.tar.gz#journals/cacm/Wadey58", } @Article{Ashenhurst:1959:UFP, author = "Robert L. Ashenhurst and Nicholas Metropolis", title = "Unnormalized Floating Point Arithmetic", journal = j-J-ACM, volume = "6", number = "3", pages = "415--428", month = jul, year = "1959", CODEN = "JACOAH", DOI = "https://doi.org/10.1145/320986.320996", ISSN = "0004-5411 (print), 1557-735X (electronic)", ISSN-L = "0004-5411", MRclass = "68.00", MRnumber = "MR0105833 (21 \#4568)", MRreviewer = "H. H. Goldstine", bibdate = "Fri Dec 08 13:06:24 1995", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/m/metropolis-nicholas.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/jacm.bib", ZMnumber = "0121.12102", abstract = "Algorithms for floating point computer arithmetic are described, in which fractional parts are not subject to the usual normalization convention. These algorithms give results in a form which furnishes some indication of their degree of precision. An analysis of one-stage error propagation is developed for each operation; a suggested statistical model for long-run error propagation is also set forth.", acknowledgement = ack-nhfb, fjournal = "Journal of the Association for Computing Machinery", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J401", } @Article{Buchholz:1959:FFC, author = "Wilfried Buchholz", title = "Fingers or Fists? (The Choice of Decimal or Binary Representation)", journal = j-CACM, volume = "2", number = "12", pages = "3--11", month = "????", year = "1959", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Mon Jan 22 06:28:45 MST 2001", bibsource = "http://dblp.uni-trier.de/db/journals/cacm/cacm2.html#Buchholz59; 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 = "decimal floating-point arithmetic", oldlabel = "Buchholz59", XMLdata = "ftp://ftp.informatik.uni-trier.de/pub/users/Ley/bib/records.tar.gz#journals/cacm/Buchholz59", xxnote = "Check author first name: article has W. Buchholz; I think it should be Werner, not Wilfried.", } @Article{Carr:1959:EAF, author = "John W. {Carr III}", title = "Error Analysis in Floating Point Arithmetic", journal = j-CACM, volume = "2", number = "5", pages = "10--15", month = may, year = "1959", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Wed Jul 14 15:48:23 MDT 2004", bibsource = "http://dblp.uni-trier.de/db/journals/cacm/cacm2.html#Carr59a; http://portal.acm.org/; 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", oldlabel = "Carr59a", XMLdata = "ftp://ftp.informatik.uni-trier.de/pub/users/Ley/bib/records.tar.gz#journals/cacm/Carr59a", } @InCollection{Carr:1959:PC, author = "John W. {Carr III}", editor = "Eugene M. Grabbe and Simon Ramo and Dean E. Wooldridge", booktitle = "Handbook of Automation, Computation, and Control", title = "Programming and Coding", publisher = pub-WILEY, address = pub-WILEY:adr, bookpages = "????", year = "1959", LCCN = "TJ213 .G72", bibdate = "Fri Aug 20 08:41:16 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; z3950.loc.gov:7090/Voyager", note = "Chapter 2.", acknowledgement = ack-nhfb, remark = "1958--1961: Volume 1. Control fundamentals. Volume 2. Computers and data processing. Volume 3. Systems and components. Cited in \cite{Sterbenz:1974:FPC}.", subject = "automation; mathematics; computers; operations research", } @Article{Daggett:1959:DBC, author = "D. H. Daggett", title = "Decimal-Binary Conversions in {CORDIC}", journal = j-IRE-TRANS-ELEC-COMPUT, volume = "EC-8", number = "5", pages = "335--339", month = sep, year = "1959", CODEN = "IRELAO", ISSN = "0367-9950", bibdate = "Thu Sep 08 08:14:00 1994", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nj, fjournal = "IRE Transactions on Electronic Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=5407885", keywords = "decimal floating-point arithmetic", } @Article{Ercoli:1959:BAD, author = "Paolo Ercoli and Roberto Vacca", title = "Binary Arithmetic for Discretely Variable Word Length in a Serial Computer", journal = j-CACM, volume = "2", number = "4", pages = "13--15", month = apr, year = "1959", CODEN = "CACMA2", 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/cacm2.html#ErcoliV59; http://portal.acm.org/; 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", oldlabel = "ErcoliV59", XMLdata = "ftp://ftp.informatik.uni-trier.de/pub/users/Ley/bib/records.tar.gz#journals/cacm/ErcoliV59", } @Article{Forsythe:1959:RNR, author = "George E. Forsythe", title = "Reprint of a Note on Rounding-Off Errors", journal = j-SIAM-REVIEW, volume = "1", number = "1", pages = "66--67", month = jan, year = "1959", CODEN = "SIREAD", DOI = "https://doi.org/10.1137/1001011", ISSN = "0036-1445 (print), 1095-7200 (electronic)", ISSN-L = "0036-1445", MRclass = "65.00", MRnumber = "MR0099119 (20 \#5563)", bibdate = "Thu Mar 27 09:04:24 MDT 2014", bibsource = "http://epubs.siam.org/toc/siread/1/1; https://www.math.utah.edu/pub/bibnet/authors/f/forsythe-george-elmer.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/fparith.bib; https://www.math.utah.edu/pub/tex/bib/siamreview.bib", ZMnumber = "0168.14002", acknowledgement = ack-nhfb, fjournal = "SIAM Review", journal-URL = "http://epubs.siam.org/sirev", keywords = "numerical analysis; random rounding-off procedure", onlinedate = "January 1959", received = "7 February 1958", remark = "From a footnote on the cover page: ``This note was written in June 1950 at the National Bureau of Standards, Los Angeles, and a few copies were distributed in multilithed form by that organization as an appendix to a reproduction of a well-known article by Rademacher [1]. The note was abstracted in Math. Reviews, Vol. 12 (1951), p. 208. Because he receives frequent requests for copies of the note, which are unavailable, the author has asked the editors to reprint it in essentially its original form.''", } @Article{Garner:1959:RMS, author = "Harvey L. Garner", title = "A Ring Model for the Study of Multiplication for Complement Codes", journal = j-IRE-TRANS-ELEC-COMPUT, volume = "EC-8", number = "1", pages = "25--30", month = mar, year = "1959", CODEN = "IRELAO", DOI = "https://doi.org/10.1109/TEC.1959.5222757", ISSN = "0367-9950", bibdate = "Thu Jul 14 15:56:44 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", URL = "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5222757", acknowledgement = ack-nhfb, fjournal = "IRE Transactions on Electronic Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=5407885", } @Article{Garner:1959:RNS, author = "Harvey L. Garner", title = "The Residue Number System", journal = j-IRE-TRANS-ELEC-COMPUT, volume = "EC-8", number = "2", pages = "140--147", month = jun, year = "1959", CODEN = "IRELAO", DOI = "https://doi.org/10.1109/TEC.1959.5219515", ISSN = "0367-9950", bibdate = "Thu Jul 14 15:56:44 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", URL = "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5219515", acknowledgement = ack-nhfb, fjournal = "IRE Transactions on Electronic Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=5407885", } @InProceedings{Gray:1959:NFP, author = "H. L. Gray and C. Harrison", booktitle = "Proceedings of the Eastern Joint Computer Conference, Boston, {MA}, December 1--3, 1959", title = "Normalized floating-point arithmetic with an index of significance", volume = "16", publisher = pub-AFIPS, address = pub-AFIPS:adr, pages = "244--248", year = "1959", LCCN = "TK7885.A1 J6", bibdate = "Tue Oct 09 09:43:31 2007", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, } @InProceedings{Henrici:1959:TES, author = "Peter Henrici", editor = "????", booktitle = "Proceedings of the International Conference on Information Processing, {UNESCO}", title = "Theoretical and Experimental Studies on the Accumulation of Error in the Numerical Solution of Initial Value Problems for Systems of Ordinary Differential Equations", publisher = "????", address = "????", pages = "36--43", year = "1959", LCCN = "????", bibdate = "Fri Aug 20 09:11:32 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, remark = "Cited in \cite{Sterbenz:1974:FPC}.", } @Article{Kogbetliantz:1959:CSC, author = "E. G. 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", ISSN = "0018-8646 (print), 2151-8556 (electronic)", ISSN-L = "0018-8646", 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/fparith.bib", acknowledgement = ack-nj, fjournal = "IBM Journal of Research and Development", journal-URL = "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=5288520", } @Article{Lucal:1959:AOD, author = "Harold M. Lucal", title = "Arithmetic Operations for Digital Computers Using a Modified Reflected Binary Code", journal = j-IRE-TRANS-ELEC-COMPUT, volume = "EC-8", number = "4", pages = "449--458", month = dec, year = "1959", CODEN = "IRELAO", DOI = "https://doi.org/10.1109/TEC.1959.5222057", ISSN = "0367-9950", bibdate = "Thu Jul 14 15:56:45 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", URL = "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5222057", acknowledgement = ack-nhfb, fjournal = "IRE Transactions on Electronic Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=5407885", } @Article{Muller:1959:CMG, author = "Mervin E. Muller", title = "A Comparison of Methods for Generating Normal Deviates on Digital Computers", journal = j-J-ACM, volume = "6", number = "3", pages = "376--383", month = jul, year = "1959", CODEN = "JACOAH", ISSN = "0004-5411 (print), 1557-735X (electronic)", ISSN-L = "0004-5411", bibdate = "Tue Nov 1 09:33:26 1994", bibsource = "ftp://ftp.ira.uka.de/pub/bibliography/Distributed/QLD/Pre.1970.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, descriptors = "RVG", fjournal = "Journal of the ACM", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J401", remark = "Cited in \cite{Sterbenz:1974:FPC}.", } @Article{Pawlak:1959:EDC, author = "Z. Pawlak", title = "An electronic digital computer based on the `$ - 2 $' system", journal = "Bull. Acad. Pol. Sci., Ser. Sci. Tech.", volume = "7", number = "??", pages = "713--722", month = "????", year = "1959", bibdate = "Thu Nov 06 05:58:31 2014", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, keywords = "negative base", } @Article{Pfeiffer:1959:FQM, author = "Paul E. Pfeiffer", title = "A Four-Quadrant Multiplier Using Triangular Waves, Diodes, Resistors, and Operational Amplifiers", journal = j-IRE-TRANS-ELEC-COMPUT, volume = "EC-8", number = "2", pages = "222--227", month = jun, year = "1959", CODEN = "IRELAO", DOI = "https://doi.org/10.1109/TEC.1959.5219526", ISSN = "0367-9950", bibdate = "Thu Jul 14 15:56:45 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", URL = "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5219526", acknowledgement = ack-nhfb, fjournal = "IRE Transactions on Electronic Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=5407885", } @Article{Rothstein:1959:RBN, author = "Jerome Rothstein", title = "Residues of Binary Numbers Modulo Three", journal = j-IRE-TRANS-ELEC-COMPUT, volume = "EC-8", number = "2", pages = "229--229", month = jun, year = "1959", CODEN = "IRELAO", DOI = "https://doi.org/10.1109/TEC.1959.5219529", ISSN = "0367-9950", bibdate = "Thu Jul 14 15:56:45 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", URL = "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5219529", acknowledgement = ack-nhfb, fjournal = "IRE Transactions on Electronic Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=5407885", } @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", 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; http://portal.acm.org/; https://www.math.utah.edu/pub/tex/bib/fparith.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{Sheridan:1959:ATC, author = "Peter B. Sheridan", title = "The Arithmetic Translator Compiler of the {IBM FORTRAN Automatic Coding System}", journal = j-CACM, volume = "2", number = "2", pages = "9--21", month = feb, year = "1959", CODEN = "CACMA2", 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/cacm2.html#Sheridan59; http://ei.cs.vt.edu/~cs4304/bibliography.html; http://portal.acm.org/; 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", oldlabel = "Sheridan59", XMLdata = "ftp://ftp.informatik.uni-trier.de/pub/users/Ley/bib/records.tar.gz#journals/cacm/Sheridan59", } @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/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/", remark = "This 1/4-page note comments that if $ \sqrt {x + i y} = u + i v $, then $ u = \sqrt {[\sqrt {x^2 + y^2} + x] / 2} $ and $ v = \sqrt {[\sqrt {x^2 + y^2} - x] / 2} $, where both inner and outer square roots should be evaluated to twice working precision.", } @Article{Taranto:1959:BCF, author = "Donald Taranto", title = "Binary Conversion, with Fixed Decimal Precision, of a Decimal Fraction", journal = j-CACM, volume = "2", number = "7", pages = "27--27", month = jul, year = "1959", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Mon Jan 22 07:25:37 MST 2001", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; http://dblp.uni-trier.de/db/journals/cacm/cacm2.html#Taranto59; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nj, fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "decimal floating-point arithmetic", oldlabel = "Taranto59", XMLdata = "ftp://ftp.informatik.uni-trier.de/pub/users/Ley/bib/records.tar.gz#journals/cacm/Taranto59", } @Article{Volder:1959:CTC, author = "J. E. Volder", title = "The {CORDIC} Trigonometric Computing Technique", journal = j-IRE-TRANS-ELEC-COMPUT, volume = "EC-8", number = "5", pages = "330--334", month = sep, year = "1959", CODEN = "IRELAO", ISSN = "0367-9950", 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/fparith.bib", acknowledgement = ack-nj, 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", ISSN = "0010-4620 (print), 1460-2067 (electronic)", ISSN-L = "0010-4620", bibdate = "Fri Sep 29 08:55:07 MDT 2000", bibsource = "Compiler/semantics.bib; http://www3.oup.co.uk/computer_journal/hdb/Volume_01/Issue_04/; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "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/", } @Article{Wilkinson:1959:EZIa, author = "J. H. Wilkinson", title = "The evaluation of the zeros of ill-conditioned polynomials. {I}", journal = j-NUM-MATH, volume = "1", pages = "150--166", month = dec, year = "1959", CODEN = "NUMMA7", DOI = "https://doi.org/10.1007/BF01386381", ISSN = "0029-599X (print), 0945-3245 (electronic)", ISSN-L = "0029-599X", MRclass = "65.00", MRnumber = "MR0109435 (22 \#321)", MRreviewer = "H. Rutishauser", bibdate = "Mon Oct 18 01:28:20 MDT 1999", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/w/wilkinson-james-hardy.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, fjournal = "Numerische Mathematik", journal-URL = "http://link.springer.com/journal/211", } @Article{Wilkinson:1959:EZIb, author = "J. H. Wilkinson", title = "The evaluation of the zeros of ill-conditioned polynomials. {II}", journal = j-NUM-MATH, volume = "1", pages = "167--180", month = dec, year = "1959", CODEN = "NUMMA7", DOI = "https://doi.org/10.1007/BF01386382", ISSN = "0029-599X (print), 0945-3245 (electronic)", ISSN-L = "0029-599X", MRclass = "65.00", MRnumber = "MR0109435 (22 \#321)", MRreviewer = "H. Rutishauser", bibdate = "Mon Oct 18 01:28:20 MDT 1999", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/w/wilkinson-james-hardy.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, fjournal = "Numerische Mathematik", journal-URL = "http://link.springer.com/journal/211", } @MastersThesis{Young:1959:SSM, author = "John Mackay Young", title = "A study of some methods of numerical integration using floating point arithmetic", type = "Thesis ({M.S. in Mathematics})", school = "Texas A.\&M. College", address = "College Station, TX, USA", pages = "50", year = "1959", bibdate = "Sat Feb 24 15:01:45 MST 1996", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, } @Article{Bockstaele:1960:NFA, author = "P. Bockstaele", title = "Notes on the First Arithmetics Printed in {Dutch} and {English}", journal = j-ISIS, volume = "51", number = "3", pages = "315--321", month = sep, year = "1960", CODEN = "ISISA4", ISSN = "0021-1753 (print), 1545-6994 (electronic)", ISSN-L = "0021-1753", bibdate = "Tue Jul 30 21:28:03 MDT 2013", bibsource = "http://www.jstor.org/action/showPublication?journalCode=isis; http://www.jstor.org/stable/i302265; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/isis1960.bib", URL = "http://www.jstor.org/stable/226510", acknowledgement = ack-nhfb, fjournal = "Isis", journal-URL = "http://www.jstor.org/journal/isis", } @Article{Brown:1960:EDC, author = "David T. Brown", title = "Error Detecting and Correcting Binary Codes for Arithmetic Operations", journal = j-IRE-TRANS-ELEC-COMPUT, volume = "EC-9", number = "3", pages = "333--337", month = sep, year = "1960", CODEN = "IRELAO", DOI = "https://doi.org/10.1109/TEC.1960.5219855", ISSN = "0367-9950", bibdate = "Thu Jul 14 15:34:12 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", URL = "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5219855", acknowledgement = ack-nhfb, fjournal = "IRE Transactions on Electronic Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=5407885", } @Article{Cohen:1960:EAF, author = "Eckford Cohen", title = "The elementary arithmetical functions", journal = j-SCRIPTA-MATH, volume = "25", number = "??", pages = "221--228", month = "????", year = "1960", ISSN = "0036-9713", ISSN-L = "0036-9713", MRclass = "10.00", MRnumber = "118694", MRreviewer = "H. Bergstr{\"{o}}m", bibdate = "Thu Oct 26 11:15:25 2023", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/scripta-math.bib", ZMnumber = "0116.27101", acknowledgement = ack-nhfb, ajournal = "Scripta Math.", fjournal = "Scripta Mathematica: A Quarterly Journal Devoted to the Philosophy, History, and Expository Treatment of Mathematics", xxpages = "221--227", ZBmath = "3189455", } @Article{Croy:1960:IAD, author = "John E. Croy", title = "Improved Arrangement of a Decimal Multiplier", journal = j-IRE-TRANS-ELEC-COMPUT, volume = "EC-9", number = "2", pages = "263--263", month = jun, year = "1960", CODEN = "IRELAO", DOI = "https://doi.org/10.1109/TEC.1960.5219830", ISSN = "0367-9950", bibdate = "Thu Jul 14 15:34:11 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", URL = "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5219830", acknowledgement = ack-nhfb, fjournal = "IRE Transactions on Electronic Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=5407885", } @Article{Ercoli:1960:LEE, author = "Paolo Ercoli", title = "Letter to the {Editor}: {Errors} Due to Overflow in Arithmetic Operations", journal = j-CACM, volume = "3", number = "12", pages = "A9--A9", month = dec, year = "1960", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Fri Nov 25 18:19:29 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/fparith.bib", note = "See \cite{Ercoli:1957:EDO}.", acknowledgement = ack-nhfb, fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", } @Article{Gurzi:1960:HSM, author = "Fred Gurzi", title = "A High-Speed Multiplication Process for Digital Computers", journal = j-CACM, volume = "3", number = "4", pages = "241--245", month = apr, year = "1960", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Fri Nov 25 18:19:26 MST 2005", bibsource = "http://dblp.uni-trier.de/db/journals/cacm/cacm3.html#Gurzi60; 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/fparith.bib", acknowledgement = ack-nhfb, fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", oldlabel = "Gurzi60", XMLdata = "ftp://ftp.informatik.uni-trier.de/pub/users/Ley/bib/records.tar.gz#journals/cacm/Gurzi60", } @TechReport{Jensen:1960:CIF, author = "B. A. Jensen", title = "Coding instructions for floating point trigonometric, inverse trigonometric hyperbolic and exponential functions", type = "Group report", number = "30G-0009", institution = "Massachusetts Institute of Technology, Lincoln Laboratory", address = "Lexington, MA, USA", pages = "7", year = "1960", bibdate = "Sat Feb 24 15:01:45 MST 1996", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, keywords = "Trigonometrical functions --- Computer programs.", remark = "Cover title. ``Millstone Hill Radar Station.'' ``1 November 1960.'' Air Force Contract", } @Article{Kanner:1960:NUA, author = "Herbert Kanner", title = "A Note on the Use of the Abacus in Number Conversion", journal = j-CACM, volume = "3", number = "3", pages = "167--167", month = mar, year = "1960", CODEN = "CACMA2", DOI = "https://doi.org/10.1145/367149.367166", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Fri Nov 25 18:19:26 MST 2005", bibsource = "http://dblp.uni-trier.de/db/journals/cacm/cacm3.html#Kanner60; 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/fparith.bib", abstract = "Conversion of integers in both directions between the base ten and any desired base can be accomplished with remarkable simplicity by use of the abacus. In fact, the conversion procedures are far more cumbersome on the electric desk calculator, for reasons which will be apparent.", acknowledgement = ack-nhfb, fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "number base conversion", oldlabel = "Kanner60", XMLdata = "ftp://ftp.informatik.uni-trier.de/pub/users/Ley/bib/records.tar.gz#journals/cacm/Kanner60", } @Article{Knuth:1960:INS, author = "Donald E. Knuth", title = "An Imaginary Number System", journal = j-CACM, volume = "3", number = "4", pages = "245--247", month = apr, year = "1960", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", MRclass = "65.99", MRnumber = "23\#B554", MRreviewer = "C. B. Haselgrove", bibdate = "Fri Nov 25 18:19:26 MST 2005", bibsource = "http://dblp.uni-trier.de/db/journals/cacm/cacm3.html#Knuth60; 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/fparith.bib; MathSciNet database", note = "Errata, {\sl Commun\-i\-ca\-tions of the ACM\/ \bf 4} (August 1961), 355 \cite{Knuth:1961:EIN}. See \cite{Nadler:1961:DSR} for extension to division and square root.", abstract = "For centuries the decimal number system reigned supreme, except, perhaps, among the Mayan Indians, until the advent of digital computers brought the binary and octal systems into the limelight. This paper introduces another number system which may prove useful for manipulating complex numbers on machines.", acknowledgement = ack-nhfb, fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", oldlabel = "Knuth60", XMLdata = "ftp://ftp.informatik.uni-trier.de/pub/users/Ley/bib/records.tar.gz#journals/cacm/Knuth60", ZMreviewer = "C. B. Haselgrove", } @Article{Lowan:1960:PREa, author = "Arnold N. Lowan", title = "On the Propagation of Round-Off Errors in the Numerical Integration of the Heat Equation", journal = j-MATH-COMPUT, volume = "14", number = "70", pages = "139--146", 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/fparith.bib; JSTOR database", acknowledgement = ack-nhfb, ajournal = "Math. Comput.", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Lowan:1960:PREb, author = "Arnold N. Lowan", title = "On the Propagation of Round-Off Errors in the Numerical Treatment of the Wave Equation", journal = j-MATH-COMPUT, volume = "14", number = "71", pages = "223--228", month = jul, 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/fparith.bib; JSTOR database", acknowledgement = ack-nhfb, ajournal = "Math. Comput.", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Pawlak:1960:ODC, author = "Z. Pawlak", title = "The organization of a digital computer based on the `$ - 2 $' system", journal = "Bull. Acad. Pol. Sci., Ser. Tech. Sci.", volume = "8", number = "??", pages = "253--258", month = "????", year = "1960", bibdate = "Thu Nov 06 06:00:08 2014", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, keywords = "negative base", } @Article{Perlin:1960:HPC, author = "I. E. Perlin and J. R. Garrett", title = "High Precision Calculation of $ \operatorname {Arcsin} x $, $ \operatorname {Arccos} x $, and $ \operatorname {Arctan} x $ (in {Technical Notes and Short Papers})", journal = j-MATH-COMPUT, volume = "14", number = "71", pages = "270--274", month = jul, 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 = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; https://www.math.utah.edu/pub/tex/bib/fparith.bib; JSTOR database", acknowledgement = ack-nj # " and " # ack-nhfb, ajournal = "Math. Comput.", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Perry:1960:CBF, author = "C. Perry", title = "Conversion between floating point representations", journal = j-CACM, volume = "3", year = "1960", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", MRclass = "352.68.00", MRnumber = "22\#8714", bibdate = "Fri Dec 8 08:24:14 1995", bibsource = "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", } @Article{Pope:1960:MPA, author = "D. A. Pope and M. L. Stein", title = "Multiple Precision Arithmetic", journal = j-CACM, volume = "3", number = "12", pages = "652--654", month = dec, year = "1960", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Thu Sep 1 10:15:08 1994", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nj, fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", } @InCollection{Reitwiesner:1960:BA, author = "G. W. Reitwiesner", title = "Binary arithmetic", crossref = "Alt:1960:AC", pages = "231--308", year = "1960", bibdate = "Sat May 18 14:36:56 2002", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, } @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", 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/fparith.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{Silver:1960:LER, author = "Roland Silver", title = "Letter to the {Editor}: Rounding in Floating-Point Arithmetic", journal = j-CACM, volume = "3", number = "12", pages = "A9--A9", month = dec, year = "1960", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Fri Nov 25 18:19:29 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/fparith.bib", acknowledgement = ack-nhfb, fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "floating-point arithmetic; rounding errors", } @Article{Sklansky:1960:CSA, author = "J. Sklansky", title = "Conditional Sum Addition Logic", journal = j-TRANS-IRE, volume = "EC-9", number = "2", pages = "226--230", month = jun, year = "1960", bibdate = "Fri Nov 09 19:17:08 2007", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, fjournal = "Transactions of the IRE", } @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", 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/fparith.bib", note = "See \cite{Sarafyan:1959:NMC}.", 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{Wadey:1960:FA, author = "W. G. Wadey", title = "Floating-Point Arithmetics", journal = j-J-ACM, volume = "7", number = "2", pages = "129--139", month = apr, year = "1960", CODEN = "JACOAH", ISSN = "0004-5411 (print), 1557-735X (electronic)", ISSN-L = "0004-5411", MRclass = "65.00 (68.00)", MRnumber = "22\#6090", bibdate = "Fri Dec 08 14:44:46 1995", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", abstract = "Three types of floating-point arithmetics with error control are discussed and compared with conventional floating-point arithmetic. General multiplication and division shift criteria are derived (for any base) for Metropolis-style arithmetics. The limitations and most suitable range of application for each arithmetic are discussed.", acknowledgement = ack-nhfb, fjournal = "Journal of the ACM", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J401", remark = "Cited in \cite{Sterbenz:1974:FPC}.", reviewer = "C. B. Haselgrove", xxmonth = "none", xxnumber = "none", } @Article{Wilkinson:1960:EAF, author = "J. H. Wilkinson", title = "Error Analysis of Floating-Point Computation", journal = j-NUM-MATH, volume = "2", pages = "319--340", month = dec, year = "1960", CODEN = "NUMMA7", ISSN = "0029-599X (print), 0945-3245 (electronic)", ISSN-L = "0029-599X", MRclass = "65.00", MRnumber = "MR0116477 (22 \#7264)", MRreviewer = "C. B. Haselgrove", bibdate = "Fri May 14 17:18:08 2010", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/w/wilkinson-james-hardy.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, fjournal = "Numerische Mathematik", journal-URL = "http://link.springer.com/journal/211", } @Article{Amble:1961:AFP, author = "O. Amble and Jan V. Garwick", title = "On the Accuracy of Floating Point Computers {[BIT {\bf 1}(2), 1961, pp. 87--88]}", journal = j-NORDISK-TIDSKR-INFORM-BEHAND, volume = "1", number = "3", pages = "220--222", month = sep, year = "1961", CODEN = "BITTEL, NBITAB", DOI = "https://doi.org/10.1007/BF01933434", 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=3; https://www.math.utah.edu/pub/tex/bib/bit.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", note = "See \cite{Garwick:1961:AFP,Garwick:1961:RAF}.", URL = "http://www.springerlink.com/openurl.asp?genre=article&issn=0006-3835&volume=1&issue=3&spage=220", acknowledgement = ack-nhfb, journal-URL = "http://link.springer.com/journal/10543", keywords = "floating-point accuracy; floating-point arithmetic", xxpages = "220--221??", } @TechReport{Anonymous:1961:MCM, author = "Anonymous", title = "Modern Computing Methods", type = "Notes on Applied Science", number = "16", institution = "National Physical Laboratory", address = "Her Majesty's Stationery Office, London", year = "1961", bibdate = "Fri Aug 20 10:05:25 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, remark = "Cited in \cite{Sterbenz:1974:FPC}.", } @Article{Avizienis:1961:SDN, author = "Algirdas Avi{\v{z}}ienis", title = "Signed-Digit Number Representations for Fast Parallel Arithmetic", journal = j-IRE-TRANS-ELEC-COMPUT, volume = "EC-10", number = "3", pages = "389--400", month = sep, year = "1961", CODEN = "IRELAO", DOI = "https://doi.org/10.1109/TEC.1961.5219227", ISSN = "0367-9950", bibdate = "Thu Jul 14 14:45:04 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", URL = "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5219227", acknowledgement = ack-nhfb, fjournal = "IRE Transactions on Electronic Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=5407885", } @Article{Cheney:1961:DCB, author = "Philip Warren Cheney", title = "A Digital Correlator Based on the Residue Number System", journal = j-IRE-TRANS-ELEC-COMPUT, volume = "EC-10", number = "1", pages = "63--70", month = mar, year = "1961", CODEN = "IRELAO", DOI = "https://doi.org/10.1109/TEC.1961.5219154", ISSN = "0367-9950", bibdate = "Thu Jul 14 14:45:02 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", URL = "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5219154", abstract = "A system design for a digital correlator based on the application of the residue number system for computation is presented. Areas of investigation include sampling, analog-to-residue conversion, logical design of the arithmetic units, residue-to-analog conversion, and modes of operation of the proposed digital correlator. The advantages of speed of computation and simplicity of logic due to the use of a residue number system are shown to result in a significantly faster and simpler system than if a conventional number system were used. The resulting digital correlator is designed for megacycle sampling and computation with a 0.1 per cent system precision.", acknowledgement = ack-nhfb, fjournal = "IRE Transactions on Electronic Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=5407885", } @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{Clarkson:1961:DMI, author = "W. K. Clarkson", title = "A Divisionless Method of Integer Conversion", journal = j-CACM, volume = "4", number = "7", pages = "315--316", month = jul, year = "1961", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Thu Sep 08 08:14:59 1994", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nj, fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", } @Article{Cox:1961:NMP, author = "Albert G. Cox and H. A. Luther", title = "A Note on Multiple Precision Arithmetic", journal = j-CACM, volume = "4", number = "8", pages = "353--353", month = aug, year = "1961", CODEN = "CACMA2", DOI = "https://doi.org/10.1145/366678.366693", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Sat Dec 03 14:07:44 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/fparith.bib", abstract = "Since computers have means to detect overflow on addition or subtraction, this can be used in a multiple precision addition-subtraction subroutine to obviate a sign analysis. Consider all integers in the computer to be expressed ``radix t'', that is, in the form $ a = a_0 + a_1 t + \cdots + a_n t^n $ where $ a_i $ are of like sign and have magnitudes less than $t$. $t$ is a positive integer determined by the word length of the computer. Consider the expression $ c = a \pm b = (a_0 \pm b_0) + (a_1 \pm b_1)t + \cdots + (a_n \pm b_n)t^n$. Here $ a_i \pm b_i$ may exceed the storage capacity of one word or may have a sign different from $c$.", acknowledgement = ack-nhfb, fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", } @Article{Croy:1961:RTM, author = "John E. Croy", title = "Rapid Technique of Manual or Machine Binary-to-Decimal Integer Conversion Using Decimal Radix Arithmetic", journal = j-IRE-TRANS-ELEC-COMPUT, volume = "EC-10", number = "4", pages = "777--777", month = dec, year = "1961", CODEN = "IRELAO", DOI = "https://doi.org/10.1109/TEC.1961.5219286", ISSN = "0367-9950", bibdate = "Thu Jul 14 14:45:05 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", URL = "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5219286", acknowledgement = ack-nhfb, fjournal = "IRE Transactions on Electronic Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=5407885", } @Article{Freiman:1961:SAC, author = "C. V. Freiman", title = "Statistical analysis of certain binary division algorithms", journal = j-IRE-PROC, volume = "49", number = "1", pages = "91--103", month = jan, year = "1961", bibdate = "Mon Sep 16 16:12:21 2002", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-sfo # " and " # ack-nhfb, fjournal = "IRE Proceedings", xxnote = "Check author name: I have found this paper cited both as Freiman and as Frieman?? Hennessy and Patterson \cite{Hennessy:2002:CAQ} have Freiman.", } @TechReport{Garner:1961:RNS, author = "H. L. Garner and R. F. Arnold and B. C. Benson and C. G. Brockus and R. J. Gonzalez and D. P. Rozenberg", title = "Residue number systems for computers", type = "{ASD} Technical Report", number = "61-483", institution = "Electronic Technology Laboratory, The University of Michigan", address = "Ann Arbor, MI, USA", month = oct, year = "1961", bibdate = "Thu Nov 18 09:27:06 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "http://deepblue.lib.umich.edu/bitstream/2027.42/5023/4/bac2784.0001.001.txt", abstract = "The purpose of the research performed under this contract was to investigate the feasibility of residue number systems in their applications to digital computers. The problems of such an application are the ones of magnitude determination, sign determination, overflow, scaling, and division. These problems are not independent, but are found to be quite interrelated. A theoretical treatment of residue number systems is given which lays the foundation for a unified study of the complete problem. Treatments of an organizational nature are given which deal with multiplication, division, and scaling. The matter of correlating the theoretical and organizational studies to physical realizations involving networks is treated also. The question of whether the residue number system can be successfully applied to general purpose computers is still an open one. Their application to special purpose machines is considered both feasible and practical.", acknowledgement = ack-nhfb, remark = "This report represents the results of research performed by the group at The University of Michigan under the direction of Professor H. L. Garner. Concurrently, research on the same subject was being conducted at Harvard University under the direction of Professor Howard Aiken, and at the Lockheed Missile System Division under the direction of Dr. Richard Tanaka. There was a considerable exchange of information among the above groups during the course of the research effort. The efforts attained exhibit little overlap, rather they are complementary. A portion of this report was extracted from the doctoral dissertation of D. P. Rozenberg. His work was supported by this contract, and led to the Ph.D..", } @Article{Garwick:1961:AFP, author = "Jan V. Garwick", title = "The Accuracy of Floating Point Computers", journal = j-NORDISK-TIDSKR-INFORM-BEHAND, volume = "1", number = "2", pages = "87--88", month = jun, year = "1961", CODEN = "BITTEL, NBITAB", DOI = "https://doi.org/10.1007/BF01939221", 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=2; https://www.math.utah.edu/pub/tex/bib/bit.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", note = "See also comment and reply \cite{Amble:1961:AFP,Garwick:1961:RAF}.", URL = "http://www.springerlink.com/openurl.asp?genre=article&issn=0006-3835&volume=1&issue=2&spage=87", acknowledgement = ack-nhfb, journal-URL = "http://link.springer.com/journal/10543", keywords = "floating-point accuracy; floating-point arithmetic", } @Article{Garwick:1961:RAF, author = "J. V. Garwick", title = "Reply to {``On the Accuracy of Floating Point Computers'' [BIT {\bf 1}(3), 1961, pp. 220--221]}", journal = j-NORDISK-TIDSKR-INFORM-BEHAND, volume = "1", number = "3", pages = "222--222", year = "1961", 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/fparith.bib", note = "See \cite{Garwick:1961:AFP,Amble:1961:AFP}.", acknowledgement = ack-nhfb, journal-URL = "http://link.springer.com/journal/10543", keywords = "floating-point accuracy; floating-point arithmetic", } @InProceedings{Henderson:1961:RCE, author = "D. S. Henderson", booktitle = "{Proceedings of the 1961 16th ACM national meeting}", title = "Residue class error checking codes", publisher = pub-ACM, address = pub-ACM:adr, pages = "132.101--132.104", year = "1961", DOI = "https://doi.org/10.1145/800029.808563", bibdate = "Thu Aug 07 19:04:58 2008", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", abstract = "Error checking codes based on residues have the advantage over parity-based codes of monitoring arithmetic as well as transmission operations. Codes may be devised to detect burst-type errors, to correct single errors in a binary arithmetic operation, to correct burst errors and finally to correct any single-digit error in a decimal operation.", acknowledgement = ack-nhfb, keywords = "decimal arithmetic", } @Article{Kettel:1961:AAM, author = "E. Kettel and W. Schneider", title = "An Accurate Analog Multiplier and Divider", journal = j-IRE-TRANS-ELEC-COMPUT, volume = "EC-10", number = "2", pages = "269--272", month = jun, year = "1961", CODEN = "IRELAO", DOI = "https://doi.org/10.1109/TEC.1961.5219198", ISSN = "0367-9950", bibdate = "Thu Jul 14 14:45:03 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", URL = "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5219198", acknowledgement = ack-nhfb, fjournal = "IRE Transactions on Electronic Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=5407885", } @Article{Knuth:1961:EIN, author = "Donald E. Knuth", title = "Errata: {``An imaginary number system''}", journal = j-CACM, volume = "4", number = "8", pages = "355--355", month = aug, year = "1961", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Tue Jan 09 07:10:22 1996", bibsource = "https://www.math.utah.edu/pub/tex/bib/cacm1960.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", note = "See \cite{Knuth:1960:INS}.", acknowledgement = ack-nhfb, fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", } @Article{Kovatch:1961:HEA, author = "G. Kovatch and W. E. Meserve", title = "The {Hall-Effect} Analog Multiplier", journal = j-IRE-TRANS-ELEC-COMPUT, volume = "EC-10", number = "3", pages = "512--515", month = sep, year = "1961", CODEN = "IRELAO", DOI = "https://doi.org/10.1109/TEC.1961.5219240", ISSN = "0367-9950", bibdate = "Thu Jul 14 14:45:04 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", URL = "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5219240", acknowledgement = ack-nhfb, fjournal = "IRE Transactions on Electronic Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=5407885", } @Article{Lehman:1961:STH, author = "M. Lehman and N. Burla", title = "Skip Techniques for High-Speed Carry-Propagation in Binary Arithmetic Units", journal = j-IRE-TRANS-ELEC-COMPUT, volume = "EC-10", number = "4", pages = "691--698", month = dec, year = "1961", CODEN = "IRELAO", DOI = "https://doi.org/10.1109/TEC.1961.5219274", ISSN = "0367-9950", bibdate = "Thu Jul 14 14:45:04 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", URL = "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5219274", acknowledgement = ack-nhfb, fjournal = "IRE Transactions on Electronic Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=5407885", } @Article{MacSorley:1961:HSA, author = "O. L. MacSorley", title = "High-speed arithmetic in binary computers", journal = j-IRE-PROC, volume = "49", number = "??", pages = "67--91", month = jan, year = "1961", bibdate = "Sat May 18 14:34:55 2002", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", note = "Reprinted in \cite{Swartzlander:1976:CDD}.", acknowledgement = ack-nhfb, fjournal = "IRE Proceedings", } @Book{Morrison:1961:CBP, editor = "Philip Morrison and Emily Morrison", title = "{Charles Babbage} on the principles and development of the calculator: and other seminal writings", publisher = pub-DOVER, address = pub-DOVER:adr, pages = "xxxviii + 400", year = "1961", ISBN = "0-486-24691-4 (paperback)", ISBN-13 = "978-0-486-24691-8 (paperback)", LCCN = "QA75 .C52 1961", bibdate = "Sat Jan 12 22:30:15 MST 2013", bibsource = "clio-db.cc.columbia.edu:7090/Voyager; https://www.math.utah.edu/pub/bibnet/authors/b/babbage-charles.bib; https://www.math.utah.edu/pub/tex/bib/adabooks.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", price = "US\$7.95", acknowledgement = ack-nhfb, subject = "Calculators; Babbage, Charles", subject-dates = "Charles Babbage (26 December 1791--18 October 1871)", tableofcontents = "Chapters from Passages from the life of a philosopher \\ Selections from Babbage's calculating engines \\ Appendix of miscellaneous papers", } @Article{Nadler:1961:DSR, author = "Morton Nadler", title = "Division and square root in the quater-imaginary number system", journal = j-CACM, volume = "4", number = "4", pages = "192--193", month = apr, year = "1961", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", MRclass = "65.99", MRnumber = "23\#B555", 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/fparith.bib", note = "See \cite{Knuth:1960:INS}.", acknowledgement = ack-nhfb, fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "$\sqrt(x)$; elementary functions", ZMreviewer = "C. B. Haselgrove", } @Article{Pinkham:1961:DFS, author = "Roger S. Pinkham", title = "On the Distribution of First Significant Digits", journal = j-ANN-MATH-STAT, volume = "32", number = "4", pages = "1223--1230", month = dec, year = "1961", CODEN = "AASTAD", ISSN = "0003-4851 (print), 2168-8990 (electronic)", ISSN-L = "0003-4851", bibdate = "Thu Feb 15 15:07:57 2007", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "http://links.jstor.org/sici?sici=0003-4851%28196112%2932%3A4%3C1223%3AOTDOFS%3E2.0.CO%3B2-T", abstract = "It has been noticed by astute observers that well used tables of logarithms are invariably dirtier at the front than at the back. Upon reflection one is led to inquire whether there are more physical constants with low order first significant digits than high. Actual counts by Benford [2] show that not only is this the case but that it seems to be an empirical truth that whenever one has a large body of physical data, Farmer's Almanac, Census Reports, Chemical Rubber Handbook, etc., the proportion of these data with first significant digit $n$ or less is approximately $ \log_{10}(n + 1) $. Any reader formerly unaware of this ``peculiarity'' will find an actual sampling experiment wondrously tantalizing. Thus, for example, approximately $ 0.7 $ of the physical constants in the Chemical Rubber Handbook begin with $4$ or less ($ \log_{10}(4 + 1) = 0.699 $). This is to be contrasted with the widespread intuitive evaluation $ 4 / 9 $ ths.", acknowledgement = ack-nhfb, fjournal = "Annals of Mathematical Statistics", journal-URL = "http://projecteuclid.org/all/euclid.aoms/", keywords = "Benford's Law; Law of Anomalous Numbers; Zipf's Law", remark = "From p. 1223: ``\ldots{} the proportion of these data with first significant digit $n$ or less is approximately $ \log_{10}(n + 1) $.''\par \ldots{} ``The only distribution for first significant digits which is invariant under scale change of the underlying distribution is $ \log_{10}(n + 1) $. Contrary to suspicion this is a non-trivial mathematical result, for the variable $n$ is discrete.''.\par From p. 1230: ``The foregoing results bear on questions of round-off in computing machines. Since $ d(u v) = u d v + v d u $, the error resulting from multiplying two rounded numbers will be governed primarily by the first significant digits of the two numbers being multiplied. Now the distribution of first significant digits, favoring as it does low order digits, tends to produce less error than would be the case if first significant digits were uniform as has sometimes been assumed.''\par Cited in \cite{Sterbenz:1974:FPC}.", } @Article{Rabinowitz:1961:MPD, author = "Philip Rabinowitz", title = "Multiple-Precision Division", journal = j-CACM, volume = "4", number = "2", pages = "98--98", month = feb, year = "1961", CODEN = "CACMA2", DOI = "https://doi.org/10.1145/366105.366171", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Thu Sep 08 00:29:25 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/fparith.bib", acknowledgement = ack-nj # " and " # ack-nhfb, fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", } @Article{Saltman:1961:RCT, author = "Roy G. Saltman", title = "Reducing Computing Time for Synchronous Binary Division", journal = j-IRE-TRANS-ELEC-COMPUT, volume = "EC-10", number = "2", pages = "169--174", month = jun, year = "1961", CODEN = "IRELAO", DOI = "https://doi.org/10.1109/TEC.1961.5219186", ISSN = "0367-9950", bibdate = "Thu Jul 14 14:45:03 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", URL = "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5219186", acknowledgement = ack-nhfb, fjournal = "IRE Transactions on Electronic Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=5407885", } @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", 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/fparith.bib; JSTOR database", acknowledgement = ack-nj # " and " # ack-nhfb, ajournal = "Math. Comput.", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Spielberg:1961:RPS, author = "K. Spielberg", title = "Representation of Power Series in Terms of Polynomials, Rational Approximations and Continued Fractions", journal = j-J-ACM, volume = "8", number = "4", pages = "613--627", month = oct, year = "1961", CODEN = "JACOAH", ISSN = "0004-5411 (print), 1557-735X (electronic)", ISSN-L = "0004-5411", bibdate = "Sat Dec 09 14:01:44 1995", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nj, fjournal = "Journal of the ACM", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J401", } @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/fparith.bib; JSTOR database", acknowledgement = ack-nhfb, ajournal = "Math. Comput.", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @TechReport{Weik:1961:TSD, author = "Martin H. Weik", title = "A Third Survey of Domestic Electronic Digital Computing Systems", type = "Report", number = "1115", institution = "Ballistic Research Laboratories", address = "Aberdeen Proving Ground, MD, USA", pages = "1131", month = mar, year = "1961", bibdate = "Fri Nov 28 19:37:51 2003", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", abstract = "Based on the results of a third survey, the engineering and programming characteristics of two hundred twenty-two different electronic digital computing systems are given. The data are presented from the point of view of application, numerical and arithmetic characteristics, input, output and storage systems, construction and checking features, power, space, weight, and site preparation and personnel requirements, production records, cost and rental rates, sale and lease policy, reliability, operating experience, and time availability, engineering modifications and improvements and other related topics. An analysis of the survey data, fifteen comparative tables, a discussion of trends, a revised bibliography, and a complete glossary of computer engineering and programming terminology are included.", acknowledgement = ack-mfc # " and " # ack-nhfb, } @Article{Wilson:1961:ARB, author = "J. B. Wilson and R. S. Ledley", title = "An Algorithm for Rapid Binary Division", journal = j-IRE-TRANS-ELEC-COMPUT, volume = "EC-10", number = "4", pages = "662--670", month = dec, year = "1961", CODEN = "IRELAO", DOI = "https://doi.org/10.1109/TEC.1961.5219271", ISSN = "0367-9950", bibdate = "Thu Jul 14 14:45:04 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", URL = "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5219271", acknowledgement = ack-nhfb, fjournal = "IRE Transactions on Electronic Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=5407885", } @InProceedings{Ashenhurst:1962:MIA, author = "R. L. Ashenhurst", title = "The {Maniac III} Arithmetic System", crossref = "AFIPS:1962:APS", pages = "192--202", year = "1962", bibdate = "Wed Feb 14 17:14:21 2007", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, } @Book{Buchholz:1962:PCS, editor = "Werner Buchholz", title = "Planning a Computer System: {Project Stretch}", publisher = pub-MCGRAW-HILL, address = pub-MCGRAW-HILL:adr, pages = "xvii + 322", year = "1962", LCCN = "1876", bibdate = "Fri Nov 19 10:02:31 MST 2010", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/t/tukey-john-w.bib; https://www.math.utah.edu/pub/tex/bib/annhistcomput.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ibmjrd.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib; library.ox.ac.uk:210/ADVANCE", note = "This important book is the primary description of the influential IBM 7030 Stretch computer, written by its architects. See also a detailed critical review \cite{Strachey:1962:BRP}.", URL = "http://ed-thelen.org/comp-hist/IBM-7030-Planning-McJones.pdf", acknowledgement = ack-nhfb, remark = "The text of the book is in the public domain, with the permission of the author in 2003. See \cite{MacKenzie:1991:IAL} for a remark about the noisy mode for floating-point arithmetic in the IBM 7030 Stretch. That mode is first mentioned on page 25 of this book, and described in detail on page 102, which states: ``By definition of ordinary normalized FLP operations, numbers are frequently extended on the right by attaching zeros. During addition the n-digit operand that is not preshifted is extended with n zeros, so as to provide the extra positions to which the preshifted operand can be added. Any operand or result that is shifted left to be normalized requires a corresponding number of zeros to be shifted in at the right. Both sets of zeros tend to produce numbers smaller in absolute value than they would have been if more digits had been carried. In the noisy mode these numbers are simply extended with 1's instead of zeros (1's in a binary machine, 9's in a decimal machine). So all numbers tend to be too large in absolute value. The true value, if there had been no significance loss, should lie between these two extremes. Hence, two runs, one made without and one made with the noisy mode, should show differences in result that indicate which digits may have been affected by significance loss. The principal weakness of the noisy-mode procedure is that it requires two runs for the same problem. A much less important weakness is that the loss of significance cannot be guaranteed to show up --- it merely has a very high probability of showing up --- whereas built-in significance checks can be made slightly pessimistic, so that actual significance loss will not be greater than indicated. On the other hand, little extra hardware and no extra storage are required for the noisy-mode approach. Furthermore, significance loss is relatively rare, so that running a problem twice when Significance loss is suspected does not pose a serious problem. What is serious is the possibility of unsuspected significance loss. In discussions of significance two points are often overlooked. The first of these is trivial: the best way of ensuring significant results is to use an adequate number of fraction digits. The second is almost equally mundane: for a given procedure, normalized FLP arithmetic will ordinarily produce the greatest precision possible for the number of fraction digits used. Normalized FLP arithmetic has been criticized with respect to significance loss, because such loss is not indicated by the creation of leading zeros, as it is with fixed-point arithmetic. In other words, the contention is not that normalized FLP arithmetic is more prone to significance loss than equivalent fixed-point arithmetic, which would be untrue, but that an equivalent indication of such loss is not provided. Loss of significance, however, is also a serious problem in fixed-point arithmetic; multiplication and division do not handle it at all correctly by means of leading zeros. (In particular, fixed-point multiplication may lead to serious or even total significance loss, which would not have occurred with normalized FLP arithmetic: and although leading zeros in addition and subtraction of fixed-point operands do give correct significance indications, the use of other operations and of built-in scaling loops frequently destroys entirely the leading-zeros method of counting significance.)''", subject = "Computer architecture", tableofcontents = "Foreword v \\ Preface vii \\ 1. Project Stretch 1 \\ [by W. Buchholz] \\ 2. Architectural Philosophy 5 \\ [by F. P. Brooks, Jr.] \\ 2.1. The Two Objectives of Project Stretch 5 \\ 2.2. Resources 6 \\ 2.3. Guiding Principles 7 \\ 2.4. Contemporary Trends in Computer Architecture 10 \\ 2.5. Hindsight 15 \\ 3. System Summary of IBM 7030 17 \\ [by W. Buchholz] \\ 3.1. System Organization 17 \\ 3.2. Memory Units 17 \\ 3.3. Index Memory 19 \\ 3.4. Special Registers 19 \\ 3.5. Input and Output Facilities 19 \\ 3.6. High-speed Disk Units 20 \\ 3.7. Central Processing Unit 20 \\ 3.8. Instruction Controls 21 \\ 3.9. Index-arithmetic Unit 21 \\ 3.10. Instruction Look-ahead 21 \\ 3.11. Arithmetic Unit 22 \\ 3.12. Instruction Set 24 \\ 3.13. Data Arithmetic 24 \\ 3.14. Radix-conversion Operations 27 \\ 3.15. Connective Operations 27 \\ 3.16. Index-arithmetic Operations 27 \\ 3.17. Branching Operations 28 \\ 3.18. Transmission Operations 28 \\ 3.19. Input-Output Operations 29 \\ 3.20. New Features 29 \\ 3.21. Performance 32 \\ 4. Natural Data Units 33 \\ [by G. P. Blaauw, F. P. Brooks, Jr., and W. Buchholz] \\ 4.1. Lengths and Structures of Natural Data Units 33 \\ 4.2. Procedures for Specifying Natural Data Units 36 \\ 4.3. Data Hierarchies 39 \\ 4.4. Classes of Operations 40 \\ 5. Choosing a Number Base 42 \\ [by W. Buchholz] \\ 5.1. Introduction 42 \\ 5.2. Information Content 45 \\ 5.3. Arithmetic Speed 49 \\ 5.4. Numerical Data 50 \\ 5.5. Nonnumerical Data 51 \\ 5.6. Addresses 52 \\ 5.7. Transformation 53 \\ 5.8. Partitioning of Memory 54 \\ 5.9. Program Interpretation 56 \\ 5.10. Other Number Bases 58 \\ 5.11. Conclusion 58 \\ 6. Character Set 60 \\ [by R. W. Bemer and W. Buchholz] \\ 6.1. Introduction 60 \\ 6.2. Size of Set 62 \\ 6.3. Subsets 62 \\ 6.4. Expansion of Set 63 \\ 6.5. Code 63 \\ 6.6. Parity Bit 66 \\ 6.7. Sequence 66 \\ 6.8. Blank 67 \\ 6.9. Decimal Digits 68 \\ 6.10. Typewriter Keyboard 68 \\ 6.11. Adjacency 69 \\ 6.12. Uniqueness 69 \\ 6.13. Signs 70 \\ 6.14. Tape-recording Convention 71 \\ 6.15. Card-punching Convention 71 \\ 6.16. List of 7030 Character Set 72 \\ 7. Variable-field-length Operation 75 \\ [by G. P. Blaauw, F. P. Brooks, Jr., and W. Buchholz] \\ 7.1. Introduction 75 \\ 7.2. Addressing of Variable-field-length Data 76 \\ 7.3. Field Length 77 \\ 7.4. Byte Size 78 \\ 7.5. Universal Accumulator 79 \\ 7.6. Accumulator Operand 79 \\ 7.7. Binary and Decimal Arithmetic 80 \\ 7.8. Integer Arithmetic 81 \\ 7.9. Numerical Signs 82 \\ 7.10. Indicators 84 \\ 7.11. Arithmetical Operations 85 \\ 7.12. Radix-conversion Operation 87 \\ 7.13. Logical Connectives of Two Variables 87 \\ 7.14. Connective Operations 89 \\ 8. Floating-point Operation 92 \\ [by S. G. Campbell] \\ General Discussion \\ 8.1. Problems of Fixed-point Arithmetic 92 \\ 8.2. Floating-point Arithmetic 94 \\ 8.3. Normalization 97 \\ 8.4. Floating-point Singularities 98 \\ 8.5. Range and Precision 99 \\ 8.6. Round-off Error 100 \\ 8.7. Significance Checks 101 \\ 8.8. Forms of Floating-point Arithmetic 103 \\ 8.9. Structure of Floating-point Data 104 \\ Floating-point Features of the 7030 \\ 8.10. Floating-point Instruction Format 106 \\ 8.11. Floating-point Data Formats 106 \\ 8.12. Singular Floating-point Numbers 108 \\ 8.13. Indicators 112 \\ 8.14. Universal Accumulator 113 \\ 8.15. Fraction Arithmetic 114 \\ 8.16. Floating-point-arithmetic Operations 114 \\ 8.17. Fixed-point Arithmetic Using Unnormalized \\ Floating-point Operations 118 \\ 8.18. Special Functions and Forms of Arithmetic 119 \\ 8.19. Multiple-precision Arithmetic 119 \\ 8.20. General Remarks 121 \\ 9. Instruction Formats 122 \\ [by W. Buchholz] \\ 9.1. Introduction 122 \\ 9.2. Earlier Instruction Languages 122 \\ 9.3. Evolution of the Single-address Instruction 124 \\ 9.4. Implied Addresses 125 \\ 9.5. Basic 7030 Instruction Formats 126 \\ 9.6. Instruction Efficiency 127 \\ 9.7. The Simplicity of Complexity 131 \\ 9.8. Relationship to Automatic Programming Languages 132 \\ 10. Instruction Sequencing 133 \\ [by F. P. Brooks, Jr.] \\ 10.1. Modes of Instruction Sequencing 133 \\ 10.2. Instruction Counter 134 \\ 10.3. Unconditional Branching 135 \\ 10.4. Conditional Branching 136 \\ 10.5. Program-interrupt System 136 \\ 10.6. Components of the Program-interrupt System 137 \\ 10.7. Examples of Program-interrupt Techniques 140 \\ 10.8. Execute Instructions 146 \\ 10.9. Execute Operations in the 7030 148 \\ 11. Indexing 150 \\ [by G. P. Blaauw] \\ 11.1. Introduction 150 \\ 11.2. Indexing Functions 151 \\ 11.3. Instruction Format for Indexing 155 \\ 11.4. Incrementing 157 \\ 11.5. Counting 159 \\ 11.6. Advancing by One 161 \\ 11.7. Progressive Indexing 161 \\ 11.8. Data Transmission 162 \\ 11.9. Data Ordering 163 \\ 11.10. Refilling 165 \\ 11.11. Indirect Addressing and Indirect Indexing 167 \\ 11.12. Indexing Applications 169 \\ 11.13. Record-handling Applications 172 \\ 11.14. File Maintenance 175 \\ 11.15. Subroutine Control 177 \\ 11.16. Conclusion 178 \\ 12. Input-Output Control 179 \\ [by W. Buchholz] \\ 12.1. A Generalized Approach to Connecting \\ Input-Output and External Storage 179 \\ 12.2. Input-Output Instructions 180 \\ 12.3. Defining the Memory Area 181 \\ 12.4. Writing and Reading 182 \\ 12.5. Controlling and Locating 183 \\ 12.6. An Alternative Approach 184 \\ 12.7. Program Interruptions 184 \\ 12.8. Buffering 180 \\ 12.9. Interface 188 \\ 12.10. Operator Control of Input-Output Units 190 \\ 13. Multiprogramming 192 \\ [by E. F. Codd, E. S. Lowry, E. McDonough, and C. A. Scalzi] \\ 13.1. Introduction 192 \\ 13.2. Multiprogramming Requirements 193 \\ 13.3. 7030 Features that Assist Multiprogramming 195 \\ 13.4. Programmed Logic 197 \\ 13.5. Concluding Remarks 200 \\ 13.6. References 201 \\ 14. The Central Processing Unit 202 \\ [by E. Bloch] \\ 14.1. Concurrent System Operation 202 \\ 14.2. Concurrency within the Central Processing Unit 204 \\ 14.3. Data Flow 204 \\ 14.4. Arithmetic Unit 208 \\ 14.5. Checking 216 \\ 14.6. Component Count 216 \\ 14.7. Performance 217 \\ 14.8. Circuits 218 \\ 14.9. Packaging 223 \\ 15. The Look-ahead Unit 228 \\ [by R. S. Balance, J. Cocke, and H. G. Kolsky] \\ 15.1. General Description 228 \\ 15.2. Timing-simulation Program 230 \\ 15.3. Description of the Look-ahead Unit 238 \\ 15.4. Forwarding 240 \\ 15.5. Counter Sequences 241 \\ 15.6. Recovery after Interrupt 246 \\ 15.7. A Look-back at the Look-ahead 247 \\ 16. The Exchange 248 \\ [by W. Buchholz] \\ 16.1. General Description 248 \\ 16.2. Starting a WRITE or READ Operation 250 \\ 16.3. Data Transfer during Writing 250 \\ 16.4. Data Transfer during Reading 251 \\ 16.5. Terminating a WRITE or READ Operation 252 \\ 16.6. Multiple Operations 252 \\ 16.7. CONTROL and LOCATE Operations 252 \\ 16.8. Interrogating the Control Word 253 \\ 16.9. Forced Termination 253 \\ 17. A Nonarithmetical System Extension 254 \\ [by S. G. Campbell, P. S. Herwitz, and J. H. Pomerene] \\ 17.1. Nonarithmetical Processing 254 \\ 17.2. The Set-up Mode 258 \\ 17.3. Byte-sequence Formation 259 \\ 17.4. Pattern Selection 260 \\ 17.5. Transformation Facilities 261 \\ 17.6. Statistical Aids 263 \\ 17.7. The BYTE-BY-BYTE Instruction 263 \\ 17.8. Monitoring for Special Conditions 264 \\ 17.9. Instruction Set 265 \\ 17.10. Collating Operations 266 \\ 17.11. Table Look-up Operations 267 \\ 17.12. Example 267 \\ Appendix A. Summary Data 273 \\ A.1. List of the Larger IBM Stored-program Computers 273 \\ A.2. Instruction Formats 275 \\ A.3. List of Registers and Special Addresses 276 \\ A.4. Summary of Operations and Modifiers 277 \\ A.5. Summary of Indicators 287 \\ Appendix B. Programming Examples 292 \\ Notation 292 \\ B.1. Polynomial Evaluation 295 \\ B.2. Cube-root Extraction 296 \\ B.3. Matrix Multiplication 298 \\ B.4. Conversion of Decimal Numbers to a \\ Floating-point Normalized Vector 299 \\ B.5. Editing a Typed Message 301 \\ B.6. Transposition of a Large Bit Matrix 303 \\ Index 305", } @InCollection{Campbell:1962:FPO, author = "S. G. Campbell", editor = "Werner Buchholz", booktitle = "Planning a Computer System: {Project Stretch}", title = "Floating-Point Operation", publisher = pub-MCGRAW-HILL, address = pub-MCGRAW-HILL:adr, bookpages = "322", pages = "92--121", year = "1962", LCCN = "QA76.8.I2 I5", bibdate = "Wed Feb 14 17:24:27 2007", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; z3950.loc.gov:7090/Voyager", acknowledgement = ack-nhfb, subject = "IBM 7030 (Computer); IBM Stretch", } @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", 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/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", URL = "http://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", } @TechReport{Descloux:1962:REF, author = "J. Descloux", title = "Remarks on errors in first order iterative processes with floating-point computers", institution = "University of Illinois Graduate College, Digital Computer Laboratory", address = "Urbana, IL, USA", pages = "7", year = "1962", bibdate = "Sat Feb 24 15:01:45 MST 1996", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", series = "Report / University of Illinois, Digital Computer Laboratory; no. 113 Report (University of Illinois (Urbana-Champaign campus). Digital Computer Laboratory); no. 113.", acknowledgement = ack-nhfb, keywords = "Electronic digital computers.; Numerical calculations.", remark = "Supported by: National Science Foundation under grant G16489. ``March 22, 1962.''", } @Article{Fischler:1962:TRA, author = "M. A. Fischler and E. A. Poe", title = "Threshold Realization of Arithmetic Circuits", journal = j-IRE-TRANS-ELEC-COMPUT, volume = "EC-11", number = "2", pages = "287--288", month = apr, year = "1962", CODEN = "IRELAO", DOI = "https://doi.org/10.1109/TEC.1962.5219366", ISSN = "0367-9950", bibdate = "Thu Jul 14 09:11:50 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", URL = "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5219366", acknowledgement = ack-nhfb, fjournal = "IRE Transactions on Electronic Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=5407885", } @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", } @Article{Grau:1962:FNR, author = "A. A. Grau", title = "On a floating-point number representation for use with algorithmic languages", journal = j-CACM, volume = "5", number = "3", pages = "160--161", month = mar, year = "1962", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", MRclass = "68.00", MRnumber = "26\#4506", bibdate = "Fri Dec 8 08:24:14 1995", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", abstract = "Algorithmic languages, such as ALGOL, make provision for two types of numbers, real and integer, which are usually implemented on the computer by means of floating-point and fixed-point numbers respectively. The concepts real and integer, however, are taken from mathematics, where the set of integers forms a proper subset of the set of real numbers. In implementation a real problem is posed by the fact that the set of fixed-point numbers is not a proper subset of the set of floating-point numbers; this problem becomes very apparent in attempts to implement ALGOL 60. Furthermore, the {\em one\/} mathematical operation of addition is implemented in the machine by one of {\em two\/} machine operations, fixed-point addition or floating-point addition.", acknowledgement = ack-nhfb, fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", reviewer = "R. L. Ashenhurst", } @Article{Guffin:1962:CSL, author = "Ronald M. Guffin", title = "A Computer for Solving Linear Simultaneous Equations Using the Residue Number System", journal = j-IRE-TRANS-ELEC-COMPUT, volume = "EC-11", number = "2", pages = "164--173", month = apr, year = "1962", CODEN = "IRELAO", DOI = "https://doi.org/10.1109/TEC.1962.5219349", ISSN = "0367-9950", bibdate = "Thu Jul 14 09:11:49 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", URL = "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5219349", acknowledgement = ack-nhfb, fjournal = "IRE Transactions on Electronic Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=5407885", } @Book{Hamming:1962:NMS, author = "R. W. (Richard Wesley) Hamming", title = "Numerical methods for scientists and engineers", publisher = pub-MCGRAW-HILL, address = pub-MCGRAW-HILL:adr, pages = "411", year = "1962", LCCN = "QA297 .H28", bibdate = "Fri Aug 20 09:12:08 MDT 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; z3950.loc.gov:7090/Voyager", series = "International series in pure and applied mathematics", acknowledgement = ack-nhfb, remark = "Cited in \cite{Sterbenz:1974:FPC}.", subject = "Numerical analysis; Electronic digital computers", } @Article{Jones:1962:FPF, author = "F. B. Jones and A. W. Wymore", title = "Floating Point Feature on the {IBM Type 1620}", journal = j-IBM-TDB, volume = "05-62", pages = "43--46", month = may, year = "1962", CODEN = "IBMTAA", ISSN = "0018-8689", bibdate = "Fri Nov 28 11:26:02 2003", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, fjournal = "IBM Technical Disclosure Bulletin", keywords = "decimal floating-point arithmetic", remark = "The IBM 1620 normalized variable-precision arithmetic provides up to 100 decimal digits with an exponent range of $ \pm 99 $.", } @Article{Karatsuba:1962:MMN, author = "A. Karatsuba and Y. Ofman", title = "Multiplication of multidigit numbers on automata", journal = j-DOKL-AKAD-NAUK, volume = "145", number = "??", pages = "293--294", month = "????", year = "1962", CODEN = "DANKAS", ISSN = "0002-3264", bibdate = "Tue Jul 05 10:35:11 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, fjournal = "Doklady Akademii nauk SSSR", journal-URL = "http://istina.msu.ru/journals/366838/", } @Article{Keir:1962:DOD, author = "Y. A. Keir and P. W. Cheney and M. Tannenbaum", title = "Division and overflow detection in residue number systems", journal = j-IRE-TRANS-ELEC-COMPUT, volume = "EC-11", number = "4", pages = "501--507", month = aug, year = "1962", CODEN = "IRELAO", DOI = "https://doi.org/10.1109/TEC.1962.5219389", ISSN = "0367-9950", MRclass = "68.00", MRnumber = "MR0150990 (27 \#976)", MRreviewer = "N. H. Choksy", bibdate = "Thu Jul 14 09:11:51 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", URL = "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5219389", acknowledgement = ack-nhfb, fjournal = "IRE Transactions on Electronic Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=5407885", } @Article{Kesner:1962:FPA, author = "O. Kesner", title = "Floating-point arithmetic in {COBOL}", journal = j-CACM, volume = "5", number = "5", pages = "269--271", month = may, year = "1962", CODEN = "CACMA2", DOI = "https://doi.org/10.1145/367710.367739", 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/fparith.bib", acknowledgement = ack-nhfb, fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", } @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/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 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{Lake:1962:LEH, author = "G. T. Lake", title = "Letter to the Editor: Hardware Conversion of Decimal and Binary Numbers", journal = j-CACM, volume = "5", number = "9", pages = "468--469", month = sep, year = "1962", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Fri Nov 28 18:16:30 2003", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-mfc # " and " # ack-nhfb, fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "decimal floating-point arithmetic", } @Article{Lynch:1962:WBD, author = "W. C. Lynch", title = "On a Wired-In Binary-to-Decimal Conversion Scheme", journal = j-CACM, volume = "5", number = "3", pages = "159--159", month = mar, year = "1962", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Fri Nov 28 18:21:41 2003", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-mfc # " and " # ack-nhfb, fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "decimal floating-point arithmetic", } @Article{MacSorley:1962:RBA, author = "O. L. MacSorley", title = "Review: \booktitle{An Algorithm for Rapid Binary Division}, by {J. B. Wilson and R. S. Ledley}", journal = j-IRE-TRANS-ELEC-COMPUT, volume = "EC-11", number = "3", pages = "420--420", month = jun, year = "1962", CODEN = "IRELAO", ISSN = "0367-9950", bibdate = "Sat Jul 16 20:52:08 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", acknowledgement = ack-nhfb, fjournal = "IRE Transactions on Electronic Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=5407885", } @Article{McGee:1962:BM, author = "W. C. McGee", title = "A Binary Multiplication", journal = j-SIAM-REVIEW, volume = "4", number = "3", pages = "256--256", month = "????", year = "1962", CODEN = "SIREAD", DOI = "https://doi.org/10.1137/1004066", ISSN = "0036-1445 (print), 1095-7200 (electronic)", ISSN-L = "0036-1445", bibdate = "Thu Mar 27 09:04:54 MDT 2014", bibsource = "http://epubs.siam.org/toc/siread/4/3; https://www.math.utah.edu/pub/tex/bib/fparith.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 = "July 1962", } @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", ISSN = "0018-8646 (print), 2151-8556 (electronic)", ISSN-L = "0018-8646", 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/fparith.bib", acknowledgement = ack-nj, fjournal = "IBM Journal of Research and Development", journal-URL = "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=5288520", } @Article{Metze:1962:CBD, author = "Gernot Metze", title = "A Class of Binary Divisions Yielding Minimally Represented Quotients", journal = j-IRE-TRANS-ELEC-COMPUT, volume = "EC-11", number = "6", pages = "761--764", month = dec, year = "1962", CODEN = "IRELAO", DOI = "https://doi.org/10.1109/TEC.1962.5219460", ISSN = "0367-9950", bibdate = "Thu Jul 14 09:11:53 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", URL = "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5219460", acknowledgement = ack-nhfb, fjournal = "IRE Transactions on Electronic Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=5407885", } @Article{Mitchell:1962:CMD, author = "John N. Mitchell", title = "Computer Multiplication and Division Using Binary Logarithms", journal = j-IRE-TRANS-ELEC-COMPUT, volume = "EC-11", number = "4", pages = "512--517", month = aug, year = "1962", CODEN = "IRELAO", DOI = "https://doi.org/10.1109/TEC.1962.5219391", ISSN = "0367-9950", bibdate = "Thu Jul 14 09:11:51 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", URL = "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5219391", abstract = "A method of computer multiplication and division is proposed which uses binary logarithms. The logarithm of a binary number may be determined approximately from the number itself by simple shifting and counting. A simple add or subtract and shift operation is all that is required to multiply or divide. Since the logarithms used are approximate there can be errors in the result. An error analysis is given and a means of reducing the error for the multiply operation is show", acknowledgement = ack-nhfb, fjournal = "IRE Transactions on Electronic Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=5407885", } @Article{Rozier:1962:DBC, author = "Charles P. Rozier", title = "Decimal-to-Binary Conversion Using Octal Radix Arithmetic", journal = j-IRE-TRANS-ELEC-COMPUT, volume = "EC-11", number = "5", pages = "708--709", month = oct, year = "1962", CODEN = "IRELAO", DOI = "https://doi.org/10.1109/TEC.1962.5219436", ISSN = "0367-9950", bibdate = "Thu Jul 14 09:11:52 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", URL = "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5219436", acknowledgement = ack-nhfb, fjournal = "IRE Transactions on Electronic Computers", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=5407885", } @Article{Shanks:1962:CD, author = "Daniel Shanks and John W. {Wrench, Jr.}", title = "Calculation of $ \pi $ to 100,000 Decimals", journal = j-MATH-COMPUT, volume = "16", number = "77", pages = "76--99", month = jan, 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/fparith.bib; JSTOR database", acknowledgement = ack-nhfb, ajournal = "Math. Comput.", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Misc{Sierra:1962:FDP, author = "Huberto M. Sierra", title = "Floating Decimal Point Arithmetic Control Means for Calculator", howpublished = "US Patent 3,037,701", day = "5", month = jun, year = "1962", bibdate = "Sat Mar 24 05:59:45 2018", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "https://patents.google.com/patent/US3037701A", abstract = "The invention relates to digital computers and more particularly to digital computers for making calculations with numbers having a wide range of values with a limited number of digit storing positions.", acknowledgement = ack-nhfb, remark = "US Patent filed 21 November 1956, granted to IBM on 5 June 1962, expired 5 June 1979 (17 years after grant). This may be the earliest US patent on floating-point arithmetic.", } @Article{Smith:1962:ACDa, author = "Robert L. Smith", title = "{Algorithm 116}: {Complex} Division", journal = j-CACM, volume = "5", number = "8", pages = "435--435", month = aug, year = "1962", CODEN = "CACMA2", DOI = "https://doi.org/10.1145/368637.368661", 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/fparith.bib", acknowledgement = ack-nhfb, fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", } @Article{Spielberg:1962:PCF, author = "Kurt Spielberg", title = "Polynomial and Continued-Fraction Approximations for Logarithmic Functions", journal = j-MATH-COMPUT, volume = "16", number = "78", pages = "205--217", 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 = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; https://www.math.utah.edu/pub/tex/bib/fparith.bib; JSTOR database", acknowledgement = ack-nj # " and " # ack-nhfb, ajournal = "Math. Comput.", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Strachey:1962:BRP, author = "Christopher Strachey", title = "Book Reviews: {{\booktitle{Planning a Computer System: Project Stretch}}. Edited by Werner Buchholz. 322 pp.(London: McGraw-Hill)}", journal = j-COMP-J, volume = "5", number = "2", pages = "152--153", month = aug, year = "1962", CODEN = "CMPJA6", DOI = "https://doi.org/10.1093/comjnl/5.2.152", ISSN = "0010-4620 (print), 1460-2067 (electronic)", ISSN-L = "0010-4620", bibdate = "Wed Sep 25 09:59:25 2024", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/t/tukey-john-w.bib; https://www.math.utah.edu/pub/tex/bib/annhistcomput.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ibmjrd.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", note = "See \cite{Buchholz:1962:PCS}.", acknowledgement = ack-nhfb, fjournal = "The Computer Journal", journal-URL = "http://comjnl.oxfordjournals.org/", keywords = "IBM 7030 (Stretch); IBM 7950 (Harvest)", } @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}.", } @InCollection{Zuse:1962:ERE, author = "K. Zuse", editor = "W. Hoffman", booktitle = "{Digitale Informationswandler}", title = "{Entwicklungslinien einer Rechenger{\"a}te-Entwicklung von der Mechanik zur Elektronik}. ({German}) [Lines of development of computing equipment development from mechanics to electronics]", publisher = "Vieweg \& Sohn, GmbH", address = "Braunschweig, West Germany", pages = "508--532", year = "1962", bibdate = "Wed Oct 13 11:23:04 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", note = "Reprinted in \cite[\S 4.3]{Randell:1982:ODC}. Translated by Mr. and Mrs. P. Jones.", acknowledgement = ack-nhfb, language = "German", } @Article{Bemer:1963:NRT, author = "R. W. Bemer", title = "A Note on Range Transformations for Square Root and Logarithm", journal = j-CACM, volume = "6", number = "6", pages = "306--307", 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/fparith.bib", acknowledgement = ack-nhfb, fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "$\log(x)$; $\sqrt(x)$; elementary functions", } @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", 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/fparith.bib", acknowledgement = ack-nj, fjournal = "Numerische Mathematik", journal-URL = "http://link.springer.com/journal/211", xxmonth = "(none)", xxnumber = "(none)", } @Article{Daly:1963:HSA, author = "W. G. Daly and J. F. Kruy", title = "A High-Speed Arithmetic Unit Using Tunnel Diodes", journal = j-IEEE-TRANS-ELEC-COMPUT, volume = "EC-12", number = "5", pages = "503--511", month = oct, year = "1963", CODEN = "IEECA8", DOI = "https://doi.org/10.1109/PGEC.1963.263644", ISSN = "0367-7508", bibdate = "Thu Jul 14 08:10:42 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", URL = "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4037964", acknowledgement = ack-nhfb, fjournal = "IEEE Transactions on Electronic Computers", } @Article{Descloux:1963:NRE, author = "J. Descloux", title = "Note on the Round-Off Errors in Iterative Processes", journal = j-MATH-COMPUT, volume = "17", number = "81", pages = "18--27", 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/fparith.bib; JSTOR database", acknowledgement = ack-nhfb, ajournal = "Math. Comput.", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Dietmeyer:1963:CPN, author = "Donald L. Dietmeyer", title = "Conversion from Positive to Negative and Imaginary Radix", journal = j-IEEE-TRANS-ELEC-COMPUT, volume = "EC-12", number = "1", pages = "20--22", month = feb, year = "1963", CODEN = "IEECA8", DOI = "https://doi.org/10.1109/PGEC.1963.263391", ISSN = "0367-7508", bibdate = "Thu Jul 14 08:10:36 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", URL = "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4037765", acknowledgement = ack-nhfb, fjournal = "IEEE Transactions on Electronic Computers", keywords = "imaginary base; negative base", } @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", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", 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/fparith.bib", acknowledgement = ack-nj, fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", } @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}) $, at the expense of three divisions. More accurate approximations are given which require in addition a single multiplication.", acknowledgement = ack-nhfb # " and " # ack-nj, fjournal = "The Computer Journal", journal-URL = "http://comjnl.oxfordjournals.org/", } @Book{Flores:1963:LCA, author = "I. Flores", title = "The Logic of Computer Arithmetic", publisher = pub-PH, address = pub-PH:adr, pages = "xii + 493", year = "1963", LCCN = "QA76.5 .F46", bibdate = "Thu Sep 1 10:12:51 1994", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nj, } @Article{Geier:1963:ACD, author = "A. Geier and I. Sturz", title = "The approximate calculation of a definite integral with automatic choice of integration steps (intervals). The program for the computer {$ {\rm MECIPTI} $} in a floating point regime. ({Romanian})", journal = "An. Univ. Timi{\c{s}}oara Ser. {\c{S}}ti. Mat.-Fiz.", volume = "1", pages = "133--139", year = "1963", MRclass = "68.00", MRnumber = "30\#3588", bibdate = "Fri Dec 08 13:52:47 1995", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, reviewer = "R. N. Goss", } @Article{Goldstein:1963:SAD, author = "Max Goldstein", title = "Significance arithmetic on a digital computer", journal = j-CACM, volume = "6", number = "3", pages = "111--117", month = mar, year = "1963", CODEN = "CACMA2", DOI = "https://doi.org/10.1145/366274.366339", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Fri Nov 25 18:19:45 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/fparith.bib", abstract = "The 7090 at NYU has been modified to include a ``Significance Mode'' of operation which is intended to facilitate the identification of significant bits in the results of floating-point arithmetic operations. The manner in which floating-point arithmetic is handled in this mode is discussed. Several numerical experiments using this mode are described and comparisons are made with the ordinary ``normalized mode.'' Examples include power series evaluation, linear equations solution, determinant evaluation and matrix inversion.", 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/fparith.bib", note = "See \cite{Knuth:1962:EPC}.", acknowledgement = ack-nhfb, fjournal = "Communications of the ACM", journal-URL = "http://dl.acm.org/pub.cfm?id=J782", } @Article{Krishnamurthy:1963:CMD, author = "E. V. Krishnamurthy", title = "On Computer Multiplication and Division Using Binary Logarithms", journal = j-IEEE-TRANS-ELEC-COMPUT, volume = "EC-12", number = "3", pages = "319--320", month = jun, year = "1963", CODEN = "IEECA8", DOI = "https://doi.org/10.1109/PGEC.1963.263572", ISSN = "0367-7508", bibdate = "Thu Jul 14 08:10:39 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", URL = "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4037874", acknowledgement = ack-nhfb, fjournal = "IEEE Transactions on Electronic Computers", } @Article{Lehman:1963:MAB, author = "M. Lehman", title = "The Minimization of Assimilations in Binary Carry-Storage Arithmetic Units", journal = j-IEEE-TRANS-ELEC-COMPUT, volume = "EC-12", number = "4", pages = "409--410", month = aug, year = "1963", CODEN = "IEECA8", DOI = "https://doi.org/10.1109/PGEC.1963.263479", ISSN = "0367-7508", bibdate = "Thu Jul 14 08:10:40 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", URL = "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4037921", acknowledgement = ack-nhfb, fjournal = "IEEE Transactions on Electronic Computers", } @Article{Lehmer:1963:RDO, author = "D. H. Lehmer", title = "{R63--17} Division and Overflow Detection in Residue Number Systems", journal = j-IEEE-TRANS-ELEC-COMPUT, volume = "EC-12", number = "1", pages = "36--37", month = feb, year = "1963", CODEN = "IEECA8", DOI = "https://doi.org/10.1109/PGEC.1963.263493", ISSN = "0367-7508", bibdate = "Thu Jul 14 08:10:36 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", URL = "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4037786; http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=4037753; http://ieeexplore.ieee.org/xpl/tocresult.jsp?isnumber=4037754", acknowledgement = ack-nhfb, fjournal = "IEEE Transactions on Electronic Computers", keywords = "residue arithmetic; residue number system", } @Article{Lindamood:1963:MCO, author = "George E. Lindamood and George Shapiro", title = "Magnitude Comparison and Overflow Detection in Modular Arithmetic Computers", journal = j-SIAM-REVIEW, volume = "5", number = "4", pages = "342--350", month = "????", year = "1963", CODEN = "SIREAD", DOI = "https://doi.org/10.1137/1005095", ISSN = "0036-1445 (print), 1095-7200 (electronic)", ISSN-L = "0036-1445", bibdate = "Thu Mar 27 09:05:07 MDT 2014", bibsource = "http://epubs.siam.org/toc/siread/5/4; https://www.math.utah.edu/pub/tex/bib/fparith.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 1963", } @Article{Metropolis:1963:BOU, author = "N. Metropolis and R. L. Ashenhurst", title = "Basic Operations in an Unnormalized Arithmetic System", journal = j-IEEE-TRANS-ELEC-COMPUT, volume = "EC-12", number = "6", pages = "896--904", month = dec, year = "1963", CODEN = "IEECA8", DOI = "https://doi.org/10.1109/PGEC.1963.263593", ISSN = "0367-7508", bibdate = "Thu Jul 14 08:10:44 MDT 2011", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/m/metropolis-nicholas.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", URL = "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4038037", ZMnumber = "0124.07910", abstract = "A particular set of unnormalized arithmetic operations termed ``basic'' are described, in the context of the University of Chicago Maniac III Computer. Each basic operation involves three operand words and generates two result words, all in unnormalized floating point format. The use of these operations in the implementation of multi-precision arithmetic is explained; in particular, it is demonstrated that multiprecision division can be effected in a nontentative manner with their aid.", acknowledgement = ack-nhfb, fjournal = "IEEE Transactions on Electronic Computers", journal-URL = "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=4037753", keywords = "MANIAC III", remark = "Cited in \cite{Sterbenz:1974:FPC}.", } @Book{Mood:1963:ITS, author = "Alexander McFarlane Mood and Franklin A. Graybill", title = "Introduction to the theory of statistics", publisher = pub-MCGRAW-HILL, address = pub-MCGRAW-HILL:adr, edition = "Second", pages = "443", year = "1963", LCCN = "HA29 .M75 1963", bibdate = "Fri Aug 20 10:02:03 MDT 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; z3950.loc.gov:7090/Voyager", series = "McGraw-Hill series in probability and statistics", acknowledgement = ack-nhfb, remark = "Cited in \cite{Sterbenz:1974:FPC}.", subject = "Mathematical statistics", } @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/fparith.bib", acknowledgement = ack-nhfb, fjournal = "Proceedings of the IEEE", journal-URL = "http://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{Stroud:1963:MPF, author = "A. H. Stroud and D. Secrest", title = "A multiple-precision floating-point interpretive program for the {Control Data} 1604", journal = j-COMP-J, volume = "6", number = "1", pages = "62--66", month = apr, year = "1963", CODEN = "CMPJA6", ISSN = "0010-4620 (print), 1460-2067 (electronic)", ISSN-L = "0010-4620", bibdate = "Fri Sep 29 08:55:45 MDT 2000", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; http://www3.oup.co.uk/computer_journal/hdb/Volume_06/Issue_01/; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "http://www3.oup.co.uk/computer_journal/hdb/Volume_06/Issue_01/060062.sgm.abs.html; http://www3.oup.co.uk/computer_journal/hdb/Volume_06/Issue_01/tiff/62.tif; http://www3.oup.co.uk/computer_journal/hdb/Volume_06/Issue_01/tiff/63.tif; http://www3.oup.co.uk/computer_journal/hdb/Volume_06/Issue_01/tiff/64.tif; http://www3.oup.co.uk/computer_journal/hdb/Volume_06/Issue_01/tiff/65.tif; http://www3.oup.co.uk/computer_journal/hdb/Volume_06/Issue_01/tiff/66.tif", acknowledgement = ack-nhfb # " and " # ack-nj, fjournal = "The Computer Journal", journal-URL = "http://comjnl.oxfordjournals.org/", } @Article{Svoboda:1963:AD, author = "A. Svoboda", title = "An algorithm for division", journal = j-INF-PROCESS-MACH, volume = "9", number = "??", pages = "25--33", month = "????", year = "1963", CODEN = "IPRMDD", ISSN = "0373-885X", bibdate = "Thu Apr 2 08:38:35 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-sfo # " and " # ack-nhfb, fjournal = "Information processing machines", } @InProceedings{Wilkinson:1963:PSA, author = "J. H. Wilkinson", title = "Plane rotations in floating-point arithmetic", crossref = "Metropolis:1963:PFS", pages = "185--198", year = "1963", MRclass = "65.35", MRnumber = "28\#1744", bibdate = "Fri Dec 8 08:24:14 1995", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, reviewer = "A. S. Householder", } @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", note = "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}.", 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 = "http://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.", } @Manual{Anonymous:1964:PPF, author = "Anonymous", title = "{PINT}: {Purdue} floating point interpretive system: for the {RPC 4000 General Precision} electronic computer", organization = "General Precision, Inc.", address = "West Lafayette, IN, USA", pages = "45", year = "1964", bibdate = "Mon May 06 10:24:31 2002", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, keywords = "Electronic digital computers --- Programming.; Purdue University School of Electrical Engineering; RPC 4000 (Computer)", remark = "Program no. H1-02.0", } @Article{Ashenhurst:1964:FEU, author = "R. L. Ashenhurst", title = "Function Evaluation in Unnormalized Arithmetic", journal = j-J-ACM, volume = "11", number = "2", pages = "168--187", month = apr, year = "1964", CODEN = "JACOAH", ISSN = "0004-5411 (print), 1557-735X (electronic)", ISSN-L = "0004-5411", bibdate = "Thu Nov 03 22:33:52 1994", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, fjournal = "Journal of the ACM", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J401", } @Article{Baer:1964:MPA, author = "Robert M. Baer and Martin G. Redlich", title = "Multiple-precision arithmetic and the exact calculation of the $ 3 - j $, $ 6 - j $ and $ 9 - j $ symbols", journal = j-CACM, volume = "7", number = "11", pages = "657--659", month = nov, year = "1964", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", MRclass = "65.25", MRnumber = "31\#865", bibdate = "Mon Oct 24 09:33:02 MDT 2005", bibsource = "http://portal.acm.org/; https://www.math.utah.edu/pub/tex/bib/fparith.bib", abstract = "Described in this paper is a system of general-purpose multiple-precision fixed-point routines and their use in subroutines which calculate exactly the quantum-mechanical 3-j, 6-j and 9-j symbols of large arguments.", acknowledgement = ack-nhfb, fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", } @MastersThesis{Bookhart:1964:SFP, author = "Thomas Woodward Bookhart", title = "A study of floating point arithmetic", type = "Thesis ({M.S. in Math.})", school = "Georgia Institute of Technology", address = "Atlanta, GA, USA", pages = "51", year = "1964", bibdate = "Sat Feb 24 15:01:45 MST 1996", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, keywords = "Electronic digital computers.; Number theory.; Numerical calculations --- Computer programs.", } @Article{Brooker:1964:PPS, author = "R. A. Brooker", title = "A programming package for some general modes of arithmetic", journal = j-CACM, volume = "7", number = "2", pages = "119--127", month = feb, year = "1964", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Mon Oct 24 09:33:00 MDT 2005", bibsource = "http://portal.acm.org/; https://www.math.utah.edu/pub/tex/bib/fparith.bib", abstract = "An interpretive programming package is described for computation with operands which may be real, complex, single or double precision, or real multiple precision. It also performs operations on matrices formed from these elements. A simple language structure is used to describe the computation.", acknowledgement = ack-nhfb, fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", } @Manual{Burroughs:1964:BBI, author = "{Burroughs Corporation}", title = "{Burroughs B5500} Information Processing Systems Reference Manual", organization = "Burroughs Corporation", address = "Detroit, MI, USA", year = "1964", bibdate = "Wed Nov 22 21:15:11 2006", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, keywords = "decimal floating-point arithmetic", remark = "The B5500 uses a decimal integer or fixed-point coefficient of 21 or 22 digits, with an exponent range of $ \pm 63 $.", } @Article{Cody:1964:DPS, author = "W. J. {Cody, Jr.}", title = "Double-Precision Square Root for the {CDC-3600}", journal = j-CACM, volume = "7", number = "12", pages = "715--718", month = dec, year = "1964", CODEN = "CACMA2", DOI = "https://doi.org/10.1145/355588.365122", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Thu Sep 1 10:15:43 1994", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; https://www.math.utah.edu/pub/bibnet/authors/c/cody-william-j.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", abstract = "In January of 1960, the late Hans J. Maehly completed a summary of approximations to the elementary functions for the CDC-1604 computer. The approximations and techniques suggested by Maehly are equally applicable to the second large computer in the CDC line, the 3600. Unlike the 1604, however, the 3600 has built-in double-precision floating-point arithmetic. The present work, largely inspired by the successes of Maehly and his associates, concerns the extension of one of Maehly's ideas to a double-precision subroutine for the 3600.", acknowledgement = ack-nj, fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "$sqrt(x)$; CDC 3600; elementary functions", } @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/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", URL = "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4038119", acknowledgement = ack-nhfb, fjournal = "IEEE Transactions on Electronic Computers", } @Article{Eve:1964:EP, author = "J. Eve", title = "The Evaluation of Polynomials", journal = j-NUM-MATH, volume = "6", pages = "17--21", year = "1964", CODEN = "NUMMA7", ISSN = "0029-599X (print), 0945-3245 (electronic)", ISSN-L = "0029-599X", bibdate = "Fri Sep 16 10:21:47 1994", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nj, fjournal = "Numerische Mathematik", journal-URL = "http://link.springer.com/journal/211", xxmonth = "(none)", xxnumber = "(none)", } @MastersThesis{Goldschmidt:1964:ADC, author = "Robert E. Goldschmidt", title = "Applications of Division by Convergence", type = "Thesis ({M.S.})", school = "Department of Electrical Engineering, Massachusetts Institute of Technology", address = "Cambridge, MA, USA", pages = "44", month = jun, year = "1964", bibdate = "Thu Apr 2 08:38:35 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "https://dspace.mit.edu/handle/1721.1/11113", acknowledgement = ack-sfo # " and " # ack-nhfb, advisor = "Alfred K. Susskind", remark = "This thesis introduced the quadratically convergent Goldschmidt algorithm for iterative division, as an alternative to Newton--Raphson iteration: see \cite{Flynn:1966:VHS, Anderson:1967:ISMb, Flynn:1970:DFI, Ito:1995:EIAb, Oberman:1996:FIR, Ercegovac:1999:IGD, Ercegovac:2000:IGD, Gallagher:2000:FTN, Even:2003:PEA, Markstein:2004:SDS, Even:2005:PEA, Piso:2006:OMD, Schulte:2007:FPD, Kong:2008:RMI, Piso:2008:FOS, Piso:2008:NRA, Ruiz:2008:EIR, Kong:2010:RMR, Kong:2011:GDM, Piso:2011:VLG, Panhaleux:2012:CFP} for improvements and error analysis. For computing $ Q = N / D $, Newton--Raphson starts with $ Y \approx 1 / D $, and iterates $ Y \leftarrow (2 - Y D) Y $, or equivalently, $ Y \leftarrow Y - (Y (Y D - 1)) $. With the same starting point, Goldschmidt instead iterates $ N \leftarrow N Y $, $ D \leftarrow D Y $, $ Y = 2 - D $, which converges to $ Y = 1 $. When parallel multipliers are available, the two multiplies can be done at the same time, whereas Newton--Raphson requires two sequential multiplies. In addition, the update $ 2 D $ can be computed in hardware by a fast bitwise two's complement operation, instead of a slower floating-point subtraction. The Goldschmidt algorithm has been used in CPUs of the IBM S/360 model 91 and the AMD K7. Unfortunately, unlike Newton--Raphson, each iteration of the Goldschmidt algorithm increases rounding error. Numerical tests of the Goldschmidt algorithm show worst-case errors in ulps roughly equal to the number of iterations. Without an FMA, the Newton--Raphson worst case errors are about 1.7 ulps, and with an FMA, about 1.0 ulps. In hardware in both cases, a few bits of extra intermediate precision can lead to correctly rounded results. Without extra precision, additional fixups are needed for correct rounding.", } @Article{Gram:1964:RZF, author = "Christian Gram", title = "On the Representation of Zero in Floating-Point Arithmetic", journal = j-NORDISK-TIDSKR-INFORM-BEHAND, volume = "4", number = "3", pages = "156--161", month = sep, year = "1964", CODEN = "BITTEL, NBITAB", DOI = "https://doi.org/10.1007/BF01956026", ISSN = "0006-3835 (print), 1572-9125 (electronic)", ISSN-L = "0006-3835", MRclass = "68.00", MRnumber = "29\#5412", bibdate = "Fri Dec 8 08:24:14 1995", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "http://www.springerlink.com/openurl.asp?genre=article&issn=0006-3835&volume=4&issue=3&spage=156", acknowledgement = ack-nhfb, journal-URL = "http://link.springer.com/journal/10543", keywords = "floating-point arithmetic; rounding errors", } @Article{Gregory:1964:FAN, author = "Robert T. Gregory and James L. Raney", title = "Floating-point arithmetic with $ 84 $-bit numbers", journal = j-CACM, volume = "7", pages = "10--13", year = "1964", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", MRclass = "68.00", MRnumber = "30\#2706", bibdate = "Fri Dec 8 08:24:14 1995", bibsource = "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", } @Book{Henrici:1964:ENA, author = "Peter Henrici", title = "Elements of numerical analysis", publisher = pub-WILEY, address = pub-WILEY:adr, pages = "xv + 328", year = "1964", LCCN = "QA297 .H4", bibdate = "Fri Aug 20 09:17:19 MDT 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; z3950.loc.gov:7090/Voyager", acknowledgement = ack-nhfb, remark = "Based on the author's lecture notes for a course entitled {\em Numerical Mathematical Analysis\/} at the University of California, Los Angeles. Cited in \cite{Sterbenz:1974:FPC}.", subject = "Numerical analysis", } @Article{Jacobsohn:1964:SFM, author = "D. Jacobsohn", title = "A Suggestion for a Fast Multiplier", journal = j-IEEE-TRANS-ELEC-COMPUT, volume = "EC-13", number = "6", pages = "754--754", month = dec, year = "1964", CODEN = "IEECA8", DOI = "https://doi.org/10.1109/PGEC.1964.263942", ISSN = "0367-7508", bibdate = "Thu Jul 14 06:57:06 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", URL = "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4038318", acknowledgement = ack-nhfb, fjournal = "IEEE Transactions on Electronic Computers", } @Book{Khinchin:1964:CF, author = "Aleksandr Iakovlevich Khinchin", title = "Continued fractions", publisher = "P. Noordhoff", address = "Groningen, The Netherlands", pages = "101", year = "1964", LCCN = "QA295 .K513 1964a", bibdate = "Fri Nov 30 06:55:53 MST 2007", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; z3950.loc.gov:7090/Voyager", note = "Translated by Peter Wynn", 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", } @Article{Klokacev:1964:RNF, author = "I. V. Kloka{\v{c}}ev", title = "A refinement of the normalized floating point number notation on digital computers. ({Russian})", journal = "{\v{Z}}. Vy{\v{c}}isl. Mat. i Mat. Fiz.", volume = "4", pages = "192--194", year = "1964", MRclass = "68.00", MRnumber = "28\#4694", bibdate = "Fri Dec 8 08:24:14 1995", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, language = "Russian", } @Article{Kundu:1964:TMD, author = "P. Kundu and S. Banerji", title = "Transistorized Multiplier and Divider and Its Applications", journal = j-IEEE-TRANS-ELEC-COMPUT, volume = "EC-13", number = "3", pages = "288--295", month = jun, year = "1964", CODEN = "IEECA8", DOI = "https://doi.org/10.1109/PGEC.1964.263919", ISSN = "0367-7508", bibdate = "Thu Jul 14 06:57:00 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", URL = "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4038157", acknowledgement = ack-nhfb, fjournal = "IEEE Transactions on Electronic Computers", } @Article{Lamson:1964:DAD, author = "Roger C. Lamson", title = "A Division Algorithm for a Digital Differential Analyzer", journal = j-IEEE-TRANS-ELEC-COMPUT, volume = "EC-13", number = "1", pages = "54--55", month = feb, year = "1964", CODEN = "IEECA8", DOI = "https://doi.org/10.1109/PGEC.1964.263838", ISSN = "0367-7508", bibdate = "Thu Jul 14 06:56:57 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", URL = "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4038079", acknowledgement = ack-nhfb, fjournal = "IEEE Transactions on Electronic Computers", } @Article{Lynn:1964:REM, author = "M. Stuart Lynn", title = "On the Round-Off Error in the Method of Successive Over-Relaxation", journal = j-MATH-COMPUT, volume = "18", number = "85", pages = "36--49", month = jan, 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/fparith.bib; JSTOR database", acknowledgement = ack-nhfb, ajournal = "Math. Comput.", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Book{Maley:1964:MDC, author = "Gerald A. Maley and Edward J. Skiko", title = "Modern digital computers", publisher = pub-PH, address = pub-PH:adr, pages = "xiv + 216", year = "1964", LCCN = "QA76.5 .M192", bibdate = "Thu Nov 18 10:04:00 MST 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; z3950.loc.gov:7090/Voyager", series = "Prentice-Hall electrical engineering series", acknowledgement = ack-nhfb, keywords = "binary arithmetic", subject = "Electronic digital computers", } @Book{McCracken:1964:NMF, author = "Daniel D. McCracken and William S. Dorn", title = "Numerical methods and {FORTRAN} programming, with applications in engineering and science", publisher = pub-WILEY, address = pub-WILEY:adr, pages = "xii + 457", year = "1964", bibdate = "Sat Jan 27 13:40:57 MST 1996", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, keywords = "Numerical analysis --- Data processing.; {Fortran} (Computer program language)", remark = "Cited in \cite{Sterbenz:1974:FPC}.", } @Article{Merrill:1964:IDC, author = "Roy D. Merrill", title = "Improving Digital Computer Performance Using Residue Number Theory", journal = j-IEEE-TRANS-ELEC-COMPUT, volume = "EC-13", number = "2", pages = "93--101", month = apr, year = "1964", CODEN = "IEECA8", DOI = "https://doi.org/10.1109/PGEC.1964.263776; https://doi.org/10.1109/PGEC.1964.263777", ISSN = "0367-7508", bibdate = "Thu Jul 14 06:56:58 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", URL = "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4038105; http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=4037753; http://ieeexplore.ieee.org/xpl/tocresult.jsp?isnumber=4038099", acknowledgement = ack-nhfb, fjournal = "IEEE Transactions on Electronic Computers", keywords = "residue arithmetic; residue number system", summary = "Residue arithmetic has the interesting characteristic that in multiplication, addition and subtraction any digit in the result is dependent only on its two corresponding operand digits. Consequently, for these operations, residue arithmetic is \ldots{}", } @Article{Miller:1964:ESD, author = "R. H. Miller", title = "An example in ``significant-digit'' arithmetic", journal = j-CACM, volume = "7", number = "1", pages = "21--21", month = jan, year = "1964", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Mon Oct 24 09:33:00 MDT 2005", bibsource = "http://portal.acm.org/; https://www.math.utah.edu/pub/tex/bib/fparith.bib", abstract = "Different methods of handling the summing process for the geometric series are shown to give results indicating widely differing significances when carried out in a machine incorporating ``significant-digit'' arithmetic.", acknowledgement = ack-nhfb, fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", remark = "Cited in \cite{Sterbenz:1974:FPC}.", } @MastersThesis{Moss:1964:RDC, author = "George Joseph {Moss, Jr.}", title = "Recording digital counter with floating point output format", type = "Thesis ({M.S.})", school = "University of Maryland", address = "College Park, MD, USA", pages = "155", year = "1964", bibdate = "Sat Feb 24 15:01:45 MST 1996", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, } @Book{Rice:1964:AFV, author = "John R. Rice", title = "The Approximation of Functions", volume = "1", publisher = pub-AW, address = pub-AW:adr, pages = "various", year = "1964", LCCN = "QA221 .R5 V.1-2", bibdate = "Fri Dec 08 13:02:52 1995", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nj, } @Article{Santos:1964:BVB, author = "J. Santos and H. Arango", title = "Base $3$ vs Base $2$ Synchronous Arithmetic Units", journal = j-IEEE-TRANS-ELEC-COMPUT, volume = "EC-13", number = "5", pages = "608--609", month = oct, year = "1964", CODEN = "IEECA8", DOI = "https://doi.org/10.1109/PGEC.1964.263734", ISSN = "0367-7508", bibdate = "Thu Jul 14 06:57:04 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", URL = "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4038256", acknowledgement = ack-nhfb, fjournal = "IEEE Transactions on Electronic Computers", } @Article{Stein:1964:DCM, author = "M. L. Stein", title = "Divide-and-Correct Methods for Multiple Precision Division", journal = j-CACM, volume = "7", number = "8", pages = "472--474", month = aug, year = "1964", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Thu Sep 1 10:15:08 1994", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nj, fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", } @MastersThesis{Sweo:1964:SFP, author = "David Ernest Sweo", title = "A study of floating point arithmetic with the residue number system", type = "Thesis ({M.S.})", school = "UCLA - Engineering", address = "Los Angeles, CA, USA", pages = "89", year = "1964", bibdate = "Sat Feb 24 15:01:45 MST 1996", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, keywords = "Congruences and residues.; Electronic digital computers --- Programming.", } @Article{Wallace:1964:SFM, author = "C. S. Wallace", title = "A Suggestion for a Fast Multiplier", journal = j-IEEE-TRANS-ELEC-COMPUT, volume = "EC-13", number = "1", pages = "14--17", month = feb, year = "1964", CODEN = "IEECA8", DOI = "https://doi.org/10.1109/PGEC.1964.263830", ISSN = "0367-7508", bibdate = "Thu Jul 14 06:56:56 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", URL = "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4038071", acknowledgement = ack-nhfb, fjournal = "IEEE Transactions on Electronic Computers", } @Article{Wolfe:1964:RTE, author = "Jack M. Wolfe", title = "Reducing Truncation Errors by Programming", journal = j-CACM, volume = "7", number = "6", pages = "355--356", month = jun, year = "1964", CODEN = "CACMA2", DOI = "https://doi.org/10.1145/512274.512287", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Fri Nov 25 18:19:54 MST 2005", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; 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/fparith.bib", abstract = "In accumulating a sum such as in a numerical integration with a large number of intervals, the sum itself becomes much larger than the individual addends. This may produce a less accurate sum as the number of intervals is increased. Separate variables can be established as accumulators to hold partial sums within various distinct intervals. Thus, the extensive successive truncations are eliminated.", acknowledgement = ack-nhfb # " and " # ack-nj, fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "floating-point arithmetic; truncation errors", remark = "This algorithm is a precursor of Kahan's compensated summation algorithm \cite{Kahan:1965:PFR}; see \cite[p. 384]{Lange:2020:NDF} for a discussion.", } @Article{Ashenhurst:1965:EEC, author = "R. L. Ashenhurst and N. Metropolis", title = "Error Estimation in Computer Calculation", journal = j-AMER-MATH-MONTHLY, volume = "72", number = "2 (Part 2)", pages = "47--58", year = "1965", CODEN = "AMMYAE", ISSN = "0002-9890 (print), 1930-0972 (electronic)", ISSN-L = "0002-9890", MRclass = "65.80", MRnumber = "MR0192671 (33 \#896)", MRreviewer = "James H. Wilkinson", bibdate = "Fri Jan 12 11:37:56 1996", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/m/metropolis-nicholas.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", note = "Reprinted in Department of Economics, Graduate School of Business, University of Chicago, Center for Mathematical Studies in Business and Economics, number 45.", ZMnumber = "0216.49602", acknowledgement = ack-jr, fjournal = "The American Mathematical Monthly", journal-URL = "https://www.jstor.org/journals/00029890.htm", keywords = "error estimation; floating-point arithmetic", } @InProceedings{Ashenhurst:1965:EIU, author = "R. L. Ashenhurst", editor = "L. B. Rall", booktitle = "Error in digital computation", title = "Experimental investigation of unnormalized arithmetic", publisher = pub-WILEY, address = pub-WILEY:adr, pages = "3--37", year = "1965", bibdate = "Thu Nov 8 14:49:59 2007", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, keywords = "ARI FLP", } @InProceedings{Ashenhurst:1965:TAE, author = "R. L. Ashenhurst", editor = "L. B. Rall", booktitle = "Error in digital computation", title = "Techniques for automatic error monitoring and control", publisher = pub-WILEY, address = pub-WILEY:adr, pages = "43--59", year = "1965", bibdate = "Tue Oct 9 09:42:56 2007", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, keywords = "ARI FLP", } @Article{Atrubin:1965:ODR, author = "A. J. Atrubin", title = "A One-Dimensional Real-Time Iterative Multiplier", journal = j-IEEE-TRANS-ELEC-COMPUT, volume = "EC-14", number = "3", pages = "394--399", month = jun, year = "1965", CODEN = "IEECA8", DOI = "https://doi.org/10.1109/PGEC.1965.264145", ISSN = "0367-7508", bibdate = "Thu Jul 14 06:26:33 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", URL = "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4038457", acknowledgement = ack-nhfb, fjournal = "IEEE Transactions on Electronic Computers", } @Article{Blum:1965:EAP, author = "B. I. Blum", title = "An Extended Arithmetic Package", journal = j-CACM, volume = "8", number = "5", pages = "318--320", month = may, year = "1965", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Thu Sep 8 08:20:53 1994", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nj, fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", } @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/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", URL = "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4038605", acknowledgement = ack-nhfb, fjournal = "IEEE Transactions on Electronic Computers", } @Article{Dadda:1965:SSP, author = "Luigi Dadda", title = "Some Schemes for Parallel Multipliers", journal = j-ALTA-FREQ, volume = "34", number = "??", pages = "349--356", month = mar, year = "1965", CODEN = "ALFRAJ", ISSN = "0002-6557", ISSN-L = "0002-6557", bibdate = "Fri Nov 09 10:47:42 2007", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", note = "Reprinted in \cite[pages 118--225]{Swartzlander:1990:CAa}.", URL = "http://en.wikipedia.org/wiki/Dadda_multiplier; http://en.wikipedia.org/wiki/luigi_dadda", acknowledgement = ack-nhfb, author-dates = "29 April 1923--26 October 2012", fjournal = "Alta frequenza", remark = "According to an ARITH'21 conference attendee, this is a recommended article for understanding the Wallace and Dadda multipliers.", } @Article{Deiters:1965:ODD, author = "Robert M. Deiters", title = "Optimum Design of a Diode Squarer by Applying the Criterion of Square Root of the Integral of Per Cent Error Squared", journal = j-IEEE-TRANS-ELEC-COMPUT, volume = "EC-14", number = "3", pages = "456--463", month = jun, year = "1965", CODEN = "IEECA8", DOI = "https://doi.org/10.1109/PGEC.1965.264153", ISSN = "0367-7508", bibdate = "Thu Jul 14 06:26:34 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", URL = "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4038465", acknowledgement = ack-nhfb, fjournal = "IEEE Transactions on Electronic Computers", } @Article{Dodd:1965:RSB, author = "George G. Dodd", title = "Remarks on Simulation of {Boolean} Functions", journal = j-CACM, volume = "8", number = "8", pages = "517--517", 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 = "ftp://ftp.ira.uka.de/pub/bibliography/Compiler/bevan.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/fparith.bib", abstract = "Recently M. Morris Mano presented a method for performing Boolean OR, AND and NOT operations by means of arithmetic and conditional transfer operations in a decimal computer lacking builtin logical instructions.", acknowledgement = ack-nhfb, checked = "19940407", fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "Boolean operations; decimal arithmetic", sjb = "Commenting on \cite{Mano:1965:PSB}.", } @Article{Earle:1965:LCS, author = "J. G. Earle", title = "Latched Carry-Save Adder", journal = j-IBM-TDB, volume = "7", number = "??", pages = "909--910", month = mar, year = "1965", CODEN = "IBMTAA", ISSN = "0018-8689", bibdate = "Fri Nov 09 10:49:36 2007", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, fjournal = "IBM Technical Disclosure Bulletin", } @InCollection{Garner:1965:NSA, author = "H. L. Garner", title = "Number systems and arithmetic", crossref = "Alt:1965:AC", volume = "6", pages = "131--194", year = "1965", bibdate = "Sat May 18 14:18:19 2002", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, } @Article{Garner:1965:RID, author = "H. L. Garner", title = "{R65-22} Improving Digital Computer Performance Using Residue Number Theory", journal = j-IEEE-TRANS-ELEC-COMPUT, volume = "EC-14", number = "2", pages = "277--277", month = apr, year = "1965", CODEN = "IEECA8", DOI = "https://doi.org/10.1109/PGEC.1965.263975", ISSN = "0367-7508", bibdate = "Thu Jul 14 06:26:23 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", URL = "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4038430; http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=4037753; http://ieeexplore.ieee.org/xpl/tocresult.jsp?isnumber=4038385", acknowledgement = ack-nhfb, fjournal = "IEEE Transactions on Electronic Computers", keywords = "residue arithmetic; residue number system", } @Article{Hammel:1965:RLC, author = "D. Hammel", title = "{R65-54} The Logic of Computer Arithmetic", journal = j-IEEE-TRANS-ELEC-COMPUT, volume = "EC-14", number = "4", pages = "670--670", month = aug, year = "1965", CODEN = "IEECA8", DOI = "https://doi.org/10.1109/PGEC.1965.264031", ISSN = "0367-7508", bibdate = "Thu Jul 14 06:26:40 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", URL = "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4038535", acknowledgement = ack-nhfb, fjournal = "IEEE Transactions on Electronic Computers", } @Article{Hammer:1965:BRBa, author = "Preston C. Hammer", title = "Book Review: {{\booktitle{Experimental Arithmetic, High Speed Computing and Mathematics}} by N. C. Metropolis; A. H. Taub; John Todd; C. B. Tompkins}", journal = j-TECHNOMETRICS, volume = "7", number = "1", pages = "82--82", month = feb, year = "1965", CODEN = "TCMTA2", DOI = "https://doi.org/10.2307/1266139", ISSN = "0040-1706 (print), 1537-2723 (electronic)", ISSN-L = "0040-1706", bibdate = "Sat Jun 21 13:17:45 MDT 2014", bibsource = "http://www.jstor.org/journals/00401706.html; http://www.jstor.org/stable/i254241; https://www.math.utah.edu/pub/bibnet/authors/m/metropolis-nicholas.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/technometrics1960.bib", URL = "http://www.jstor.org/stable/1266139", acknowledgement = ack-nhfb, fjournal = "Technometrics", journal-URL = "http://www.jstor.org/journals/00401706.html", } @Article{Hammer:1965:BRBb, author = "Preston C. Hammer", title = "Book Review: {{\booktitle{Experimental Arithmetic, High Speed Computing and Mathematics}} by N. C. Metropolis, A. H. Taub, John Todd, and C. B. Tompkins}", journal = j-TECHNOMETRICS, volume = "7", number = "1", pages = "82--82", month = feb, year = "1965", CODEN = "TCMTA2", ISSN = "0040-1706 (print), 1537-2723 (electronic)", ISSN-L = "0040-1706", bibdate = "Tue Mar 20 13:44:52 2007", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/m/metropolis-nicholas.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/technometrics1960.bib", URL = "http://links.jstor.org/sici?sici=0040-1706%28196502%297%3A1%3C82%3AEAHSCA%3E2.0.CO%3B2-9", acknowledgement = ack-nhfb, fjournal = "Technometrics", journal-URL = "http://www.jstor.org/journals/00401706.html", } @Article{Hamming:1965:NLB, author = "R. W. Hamming and W. L. Mammel", title = "A Note on the Location of the Binary Point in a Computing Machine", journal = j-IEEE-TRANS-ELEC-COMPUT, volume = "EC-14", number = "2", pages = "260--261", month = apr, year = "1965", CODEN = "IEECA8", DOI = "https://doi.org/10.1109/PGEC.1965.264258", ISSN = "0367-7508", bibdate = "Thu Jul 14 06:26:22 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/benfords-law.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", URL = "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4038414", acknowledgement = ack-nhfb, fjournal = "IEEE Transactions on Electronic Computers", remark = "The authors consider the low-level multiplication circuit efficiency of placing the binary point before or after the first bit. If the leading bit is equally likely to be a 0 or a 1, then their analysis shows that it is better to place the point before the leading bit. However, they report that a more likely distribution is logarithmic (as predicted by Benford's Law, although the Newcomb / Benford work is neither mentioned nor cited), in which case there is no advantage for either choice of placement of the binary point. The conclude that it would be humane to place it after the leading digit, by analogy with how people learn decimal arithmetic.", } @Article{Ikebe:1965:NTP, author = "Yasuhiko Ikebe", title = "Note on Triple-Precision Floating-Point Arithmetic with 132-Bit Numbers", journal = j-CACM, volume = "8", number = "3", pages = "175--177", month = mar, year = "1965", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", MRclass = "68.00", MRnumber = "30\#2707", bibdate = "Fri Dec 08 14:32:51 1995", bibsource = "ftp://ftp.ira.uka.de/pub/bibliography/Compiler/bevan.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", abstract = "In a recent paper, Gregory and Raney described a technique for double-precision floating-point arithmetic. A similar technique can be developed for triple-precision floating-point arithmetic and its is the purpose of this note to describe this technique. Only the multiplication and the division algorithms are described, since the addition-subtraction algorithm can be obtained by a trivial modification of the algorithm in Gregory's and Raney's paper.", acknowledgement = ack-nhfb, checked = "19940404", fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", } @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/fparith.bib; JSTOR database", acknowledgement = ack-nj # " and " # ack-nhfb, ajournal = "Math. Comput.", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @InCollection{Kahan:1965:FPO, author = "W. Kahan", booktitle = "Programmer's Reference Manual", title = "The Floating-Point Over\slash Underflow Trap Routine {{\tt FPTRP}}", publisher = "Institute of Computer Science, University of Toronto", address = "Toronto, Ontario, Canada", year = "1965", LCCN = "????", bibdate = "Wed Feb 14 19:11:11 2007", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/k/kahan-william-m.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", note = "Section 4.1.", acknowledgement = ack-nhfb, remark = "Cited in \cite{Sterbenz:1974:FPC}.", } @Article{Kahan:1965:PFR, author = "W. Kahan", title = "Pracniques: {Further} Remarks on Reducing Truncation Errors", journal = j-CACM, volume = "8", number = "1", pages = "40--40", month = jan, year = "1965", CODEN = "CACMA2", DOI = "https://doi.org/10.1145/363707.363723", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Thu Sep 08 00:18:17 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/bibnet/authors/k/kahan-william-m.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/bit.bib; https://www.math.utah.edu/pub/tex/bib/cacm1960.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nj # " and " # ack-nhfb, fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "floating-point arithmetic; truncation errors", remark = "Gives summation error estimate formula `s = fl(a + b); error = (a - s) + b', also discovered independently by \cite{Moller:1965:QDP}. Extends Gill's earlier fixed-point result \cite{Gill:1951:PSS}. See improvement \cite{Ozawa:1993:SAE}.", } @Article{Kanner:1965:NBC, author = "Herbert Kanner", title = "Number Base Conversion in Significant Digit Arithmetic", journal = j-J-ACM, volume = "12", number = "2", pages = "242--246", month = apr, year = "1965", CODEN = "JACOAH", ISSN = "0004-5411 (print), 1557-735X (electronic)", ISSN-L = "0004-5411", bibdate = "Thu Nov 03 08:47:50 1994", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, fjournal = "Journal of the ACM", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J401", remark = "Cited in \cite{Sterbenz:1974:FPC}.", } @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/fparith.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", } @Article{Krishnamurthy:1965:DCM, author = "E. V. Krishnamurthy", title = "On a Divide-and-Correct Method for Variable Precision Division", journal = j-CACM, volume = "8", number = "3", pages = "179--181", month = mar, year = "1965", CODEN = "CACMA2", DOI = "https://doi.org/10.1145/363791.363829", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", MRclass = "68.00", MRnumber = "30\#2709", bibdate = "Thu Sep 1 10:15:08 1994", bibsource = "ftp://ftp.ira.uka.de/pub/bibliography/Compiler/bevan.bib; 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/fparith.bib", abstract = "Described in this paper is a divide-and-correct method for variable precision division in digital computers. Unlike the earlier methods of Stein and Pope, the present method uses a suitably rounded form of the normalized divisor for getting an estimate of the quotient characters. This result is a correction of at most plus or minus one to the estimate, to obtain the exact quotient character. It is believed that this method will be widely applicable for division operations in variable word-length character-oriented machines.", acknowledgement = ack-nj, fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "correct rounding; floating-point arithmetic", } @TechReport{Lederer:1965:FPP, author = "E. Lederer", title = "Floating-point-Pegasus, {FLPPEG}", type = "{ISD-Bericht}", number = "12", institution = "ISD, Inst. f{\"u}r Statik u. Dynamik d. Luft- u. Raumfahrtkonstruktionen, Universit{\"a}t Stuttgart", address = "Stuttgart, Germany", pages = "5", year = "1965", bibdate = "Thu May 09 09:05:51 2002", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, } @InProceedings{Lehman:1965:SAT, author = "M. Lehman", title = "Serial Arithmetic Techniques", crossref = "AFIPS:1965:FJC", pages = "715--725", year = "1965", bibdate = "Tue Jan 08 23:08:35 2008", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, } @MastersThesis{Liddiard:1965:DPF, author = "Lawrence Anthony Liddiard", title = "Double precision floating point arithmetic", type = "Thesis ({M.S.})", school = "University of Minnesota", address = "????", pages = "various", year = "1965", bibdate = "Sat Feb 24 15:01:45 MST 1996", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, keywords = "Calculators.; Numerical calculations.", } @Book{Lyusternik:1965:HCE, author = "L. A. Lyusternik and O. A. Chervonenkis and A. R. Yanpolski", title = "Handbook for Computing Elementary Functions", publisher = pub-PERGAMON, address = pub-PERGAMON:adr, pages = "xiii + 251", year = "1965", LCCN = "QA221.L513", bibdate = "Thu Sep 1 10:13:16 1994", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; https://www.math.utah.edu/pub/tex/bib/fparith.bib", note = "Translated from the Russian by G. J. Tee. Translation edited by K. L. Stewart.", acknowledgement = ack-nj, } @Article{Mano:1965:PSB, author = "M. Morris Mano", title = "Pracniques: {Simulation} of {Boolean} functions in a decimal computer", journal = j-CACM, volume = "8", number = "1", pages = "39--40", month = jan, 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 = "ftp://ftp.ira.uka.de/pub/bibliography/Compiler/bevan.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/fparith.bib", note = "See remarks \cite{Dodd:1965:RSB}.", acknowledgement = ack-nhfb, checked = "19940304", fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", sjb = "Describes how to perform Boolean operations by means of arithmetic and conditional transfer operations on a decimal computer lacking any built in logical instructions. See also \cite{Dodd:cacm:65}.", } @Article{Martin:1965:SHS, author = "A. R. Martin and A. B. Rosenstein", title = "A Shiftrix for High-Speed Multiplication", journal = j-IEEE-TRANS-ELEC-COMPUT, volume = "EC-14", number = "4", pages = "639--643", month = aug, year = "1965", CODEN = "IEECA8", DOI = "https://doi.org/10.1109/PGEC.1965.264010", ISSN = "0367-7508", bibdate = "Thu Jul 14 06:26:39 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", URL = "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4038521", acknowledgement = ack-nhfb, fjournal = "IEEE Transactions on Electronic Computers", } @InProceedings{Metropolis:1965:AIE, author = "N. Metropolis", title = "Analysis of Inherent Errors in Matrix Decomposition Using Unnormalized Arithmetic", crossref = "Kalenich:1965:IPP", volume = "2", pages = "441--442", year = "1965", bibdate = "Wed Feb 14 19:21:38 2007", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/m/metropolis-nicholas.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", ZMnumber = "0161.35502", acknowledgement = ack-nhfb, remark = "Cited in \cite{Sterbenz:1974:FPC}.", } @Article{Metropolis:1965:AUA, author = "N. Metropolis", title = "Algorithms in unnormalized arithmetic. {I}. {Recurrence} relations", journal = j-NUM-MATH, volume = "7", number = "2", pages = "104--112", month = apr, year = "1965", CODEN = "NUMMA7", ISSN = "0029-599X (print), 0945-3245 (electronic)", ISSN-L = "0029-599X", MRclass = "68.00", MRnumber = "MR0178589 (31 \#2846)", MRreviewer = "G. Dahlquist", bibdate = "Wed Nov 14 19:04:40 2007", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/m/metropolis-nicholas.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", note = "See erratum \cite{Metropolis:1965:BAU}.", ZMnumber = "0128.36301", acknowledgement = ack-nhfb, fjournal = "Numerische Mathematik", journal-URL = "http://link.springer.com/journal/211", } @Article{Metropolis:1965:BAU, author = "N. Metropolis", title = "Berichtigung: {Algorithms} in unnormalized arithmetic. {I}. {Recurrence} relations", journal = j-NUM-MATH, volume = "7", number = "4", pages = "354", month = aug, year = "1965", CODEN = "NUMMA7", ISSN = "0029-599X (print), 0945-3245 (electronic)", ISSN-L = "0029-599X", MRclass = "Contributed Item", MRnumber = "MR1553943", bibdate = "Wed Nov 14 19:04:40 2007", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/m/metropolis-nicholas.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, fjournal = "Numerische Mathematik", journal-URL = "http://link.springer.com/journal/211", } @Article{Metropolis:1965:RCU, author = "N. Metropolis and R. L. Ashenhurst", title = "Radix Conversion in an Unnormalized Arithmetic System", journal = j-MATH-COMPUT, volume = "19", number = "91", pages = "435--441", 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/bibnet/authors/m/metropolis-nicholas.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; JSTOR database", URL = "http://links.jstor.org/sici?sici=0025-5718%28196507%2919%3A91%3C435%3ARCIAUA%3E2.0.CO%3B2-D", ZMnumber = "0146.14607", acknowledgement = ack-nhfb, ajournal = "Math. Comput.", 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/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", URL = "http://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/fparith.bib", acknowledgement = ack-nj, fjournal = "Numerische Mathematik", journal-URL = "http://link.springer.com/journal/211", xxmonth = "(none)", xxnumber = "(none)", } @Article{Moller:1965:NQD, author = "Ole M{\o}ller", title = "Note on Quasi Double-Precision", journal = j-NORDISK-TIDSKR-INFORM-BEHAND, volume = "5", number = "4", pages = "251--255", year = "1965", CODEN = "BITTEL, NBITAB", ISSN = "0006-3835 (print), 1572-9125 (electronic)", ISSN-L = "0006-3835", bibdate = "Sat Nov 14 09:14:57 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", note = "See \cite{Moller:1965:QDP}.", acknowledgement = ack-nj # " and " # ack-nhfb, journal-URL = "http://link.springer.com/journal/10543", } @Article{Moller:1965:QDP, author = "Ole M{\o}ller", title = "Quasi Double-Precision in Floating Point Addition", journal = j-NORDISK-TIDSKR-INFORM-BEHAND, volume = "5", number = "1", pages = "37--50", month = mar, year = "1965", CODEN = "BITTEL, NBITAB", DOI = "https://doi.org/10.1007/BF01975722", ISSN = "0006-3835 (print), 1572-9125 (electronic)", ISSN-L = "0006-3835", MRclass = "65.80", MRnumber = "MR0181130 (31 \#5359)", 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/fparith.bib", note = "See also \cite{Moller:1965:NQD}.", URL = "http://www.springerlink.com/openurl.asp?genre=article&issn=0006-3835&volume=5&issue=1&spage=37", acknowledgement = ack-nj # " and " # ack-nhfb, journal-URL = "http://link.springer.com/journal/10543", keywords = "accurate floating-point summation; floating-point arithmetic; rounding errors", remark = "Gives summation error estimate formula `s = fl(a + b); error = (a - s) + b', also discovered independently by \cite{Kahan:1965:PFR}. Extends Gill's earlier fixed-point result \cite{Gill:1951:PSS}.", } @InCollection{Moore:1965:AACa, author = "Ramon E. Moore", title = "The automatic analysis and control of error in digital computing based on the use of interval numbers", crossref = "Rall:1965:EDCa", chapter = "2", bookpages = "????", pages = "61--130", year = "1965", MRclass = "65.61 (65.80)", MRnumber = "MR0176614 (31 \#886)", MRreviewer = "T. E. Hull", bibdate = "Thu Jun 20 10:51:40 2002", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "http://interval.louisiana.edu/Moores_early_papers/Moore_in_Rall_V1.pdf", acknowledgement = ack-nhfb, remark = "Cited in \cite{Sterbenz:1974:FPC}.", } @InCollection{Moore:1965:AACb, author = "Ramon E. Moore", title = "Automatic local coordinate transformations to reduce the growth of error bounds in interval computation of solutions of ordinary differential equations", crossref = "Rall:1965:EDCb", chapter = "2", pages = "103--140", year = "1965", MRclass = "65.80 (65.60)", MRnumber = "MR0185839 (32 \#3299)", MRreviewer = "T. E. Hull", bibdate = "Thu Jun 20 10:51:40 2002", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "http://interval.louisiana.edu/Moores_early_papers/Moore_in_Rall_V2.pdf", acknowledgement = ack-nhfb, remark = "Cited in \cite{Sterbenz:1974:FPC}.", } @Article{Morrison:1965:MCC, author = "D. R. Morrison", title = "A Method for Computing Certain Inverse Functions", journal = j-MATH-TABLES-OTHER-AIDS-COMPUT, volume = "10", number = "??", pages = "202--208", month = "????", year = "1965", CODEN = "MTTCAS", ISSN = "0891-6837", bibdate = "Thu Sep 1 10:16:10 1994", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nj, ajournal = "Math. Tables Other Aids Comput.", fjournal = "Mathematical Tables and Other Aids to Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Nathan:1965:CM, author = "Amos Nathan", title = "The Cascade Multiplier", journal = j-IEEE-TRANS-ELEC-COMPUT, volume = "EC-14", number = "2", pages = "243--247", month = apr, year = "1965", CODEN = "IEECA8", DOI = "https://doi.org/10.1109/PGEC.1965.264250", ISSN = "0367-7508", bibdate = "Thu Jul 14 06:26:22 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", URL = "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4038406", acknowledgement = ack-nhfb, fjournal = "IEEE Transactions on Electronic Computers", } @Article{Penney:1965:BSC, author = "Walter Penney", title = "A {``Binary''} System for Complex Numbers", journal = j-J-ACM, volume = "12", number = "2", pages = "247--248", month = apr, year = "1965", CODEN = "JACOAH", ISSN = "0004-5411 (print), 1557-735X (electronic)", ISSN-L = "0004-5411", bibdate = "Thu Nov 03 08:47:50 1994", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, fjournal = "Journal of the ACM", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J401", } @Book{Ralston:1965:FCN, author = "Anthony Ralston", title = "A first course in numerical analysis", publisher = pub-MCGRAW-HILL, address = pub-MCGRAW-HILL:adr, pages = "xix + 578", year = "1965", LCCN = "QA297 .R3", bibdate = "Fri Aug 20 10:13:32 MDT 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; z3950.loc.gov:7090/Voyager", series = "International series in pure and applied mathematics", acknowledgement = ack-nhfb, remark = "See also second edition \cite{Ralston:1978:FCN,Ralston:2001:FCN}. Cited in \cite{Sterbenz:1974:FPC}.", subject = "Numerical analysis", } @Article{Riordan:1965:UAT, author = "R. H. S. Riordan and R. R. A. Morton", title = "The Use of Analog Techniques in Binary Arithmetic Units", journal = j-IEEE-TRANS-ELEC-COMPUT, volume = "EC-14", number = "1", pages = "29--35", month = feb, year = "1965", CODEN = "IEECA8", DOI = "https://doi.org/10.1109/PGEC.1965.264051", ISSN = "0367-7508", bibdate = "Thu Jul 14 06:26:20 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", URL = "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4038346", acknowledgement = ack-nhfb, fjournal = "IEEE Transactions on Electronic Computers", } @Article{Ross:1965:RTE, author = "D. R. Ross", title = "Reducing Truncation Errors Using Cascading Accumulators", journal = j-CACM, volume = "8", number = "1", pages = "32--33", month = jan, year = "1965", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Sun Sep 18 22:55:00 1994", bibsource = "ftp://ftp.ira.uka.de/pub/bibliography/Compiler/bevan.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", abstract = "When accumulating a large number of quantities as in numerical integration, the sum itself may become much larger than the individual addends. This results in truncation error. Much of this error can be eliminated using cascading accumulators as noted in a recent article by Wolfe. A simpler and slightly more flexible algorithm is presented which deals also with the case of negative addends.", acknowledgement = ack-nhfb, checked = "19940304", fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "accurate floating-point summation", sjb = "Refers to \cite{Wolfe:cacm:196,Wilhelm:1977:GMPb}.", } @Article{Schreiber:1965:BRB, author = "Alvin L. Schreiber", title = "Book Review: {{\booktitle{A Binary Multiplication}} (W. C. McGee)}", journal = j-SIAM-REVIEW, volume = "7", number = "1", pages = "134--136", month = "????", year = "1965", CODEN = "SIREAD", DOI = "https://doi.org/10.1137/1007016", ISSN = "0036-1445 (print), 1095-7200 (electronic)", ISSN-L = "0036-1445", bibdate = "Thu Mar 27 09:05:20 MDT 2014", bibsource = "http://epubs.siam.org/toc/siread/7/1; https://www.math.utah.edu/pub/tex/bib/fparith.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 1965", } @Article{Smith:1965:ASO, author = "Francis J. Smith", title = "An Algorithm for Summing Orthogonal Polynomial Series and their Derivatives with Applications to Curve-Fitting and Interpolation", journal = j-MATH-COMPUT, volume = "19", number = "89", pages = "33--36", 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/fparith.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp1960.bib; JSTOR database", acknowledgement = ack-nhfb, ajournal = "Math. Comput.", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Specker:1965:CAL, author = "W. H. Specker", title = "A Class of Algorithms for $ \ln x $, $ \exp x $, $ \sin x $, $ \cos x $, $ \tan^{-1} x $, and $ \cot^{-1} x $", journal = j-IEEE-TRANS-ELEC-COMPUT, volume = "EC-14", number = "1", pages = "85--86", month = feb, year = "1965", CODEN = "IEECA8", DOI = "https://doi.org/10.1109/PGEC.1965.264066", ISSN = "0367-7508", bibdate = "Thu Jul 14 06:26:21 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", URL = "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4038361", acknowledgement = ack-nhfb, fjournal = "IEEE Transactions on Electronic Computers", } @Article{Swarztrauber:1965:LED, author = "P. N. Swarztrauber", title = "Letter to the {Editor}: On the Double-Precision Square Root Routine", journal = j-CACM, volume = "8", number = "4", pages = "202", month = apr, year = "1965", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Wed Aug 31 14:02:19 1994", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; https://www.math.utah.edu/pub/tex/bib/cacm1960.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "$\sqrt(x)$; elementary functions; floating-point arithmetic", } @Article{Sweeney:1965:AFP, author = "D. W. Sweeney", title = "An analysis of floating-point addition", journal = j-IBM-SYS-J, volume = "4", number = "1", pages = "31--42", year = "1965", CODEN = "IBMSA7", DOI = "https://doi.org/10.1147/sj.41.0031", ISSN = "0018-8670", bibdate = "Thu Sep 15 18:42:42 1994", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ibmsysj.bib", note = "Reprinted in \cite[pp.~317--328]{Swartzlander:1990:CAa}.", URL = "http://www.research.ibm.com/journal/sj/041/ibmsjIVRID.pdf", acknowledgement = ack-nj # "\slash " # ack-nhfb, fjournal = "IBM Systems Journal", remark = "Page 34, containing a critical table, was originally missing from the PDF file, and this was finally corrected by IBM in November 2008.", xxmonth = "(none)", } @Article{Winograd:1965:TRP, author = "Shmuel Winograd", title = "On the Time Required to Perform Addition", journal = j-J-ACM, volume = "12", number = "2", pages = "277--285", month = apr, year = "1965", CODEN = "JACOAH", ISSN = "0004-5411 (print), 1557-735X (electronic)", ISSN-L = "0004-5411", bibdate = "Wed Nov 14 19:04:40 2007", bibsource = "ftp://ftp.ira.uka.de/pub/bibliography/Parallel/Multi.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, fjournal = "Journal of the ACM", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J401", } @Article{Arango:1966:FCP, author = "H. Arango and J. Santos", title = "A Fast Carry-Propagation Circuit for Base $3$ Signed Non redundant Arithmetic", journal = j-IEEE-TRANS-ELEC-COMPUT, volume = "EC-15", number = "2", pages = "254--255", month = apr, year = "1966", CODEN = "IEECA8", DOI = "https://doi.org/10.1109/PGEC.1966.264311", ISSN = "0367-7508", bibdate = "Thu Jul 14 05:46:31 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", URL = "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4038725", acknowledgement = ack-nhfb, fjournal = "IEEE Transactions on Electronic Computers", } @Article{Brooker:1966:MFA, author = "R. A. Brooker and J. S. Rohl and S. R. Clark", title = "The main features of {Atlas Autocode}", journal = j-COMP-J, volume = "8", number = "4", pages = "303--310", month = jan, year = "1966", CODEN = "CMPJA6", ISSN = "0010-4620 (print), 1460-2067 (electronic)", ISSN-L = "0010-4620", bibdate = "Fri Sep 29 08:56:10 MDT 2000", bibsource = "http://www3.oup.co.uk/computer_journal/hdb/Volume_08/Issue_04/; https://www.math.utah.edu/pub/tex/bib/fparith.bib", note = "See comment \cite{Clark:1966:CMP}.", URL = "http://www3.oup.co.uk/computer_journal/hdb/Volume_08/Issue_04/080303.sgm.abs.html; http://www3.oup.co.uk/computer_journal/hdb/Volume_08/Issue_04/tiff/303.tif; http://www3.oup.co.uk/computer_journal/hdb/Volume_08/Issue_04/tiff/304.tif; http://www3.oup.co.uk/computer_journal/hdb/Volume_08/Issue_04/tiff/305.tif; http://www3.oup.co.uk/computer_journal/hdb/Volume_08/Issue_04/tiff/306.tif; http://www3.oup.co.uk/computer_journal/hdb/Volume_08/Issue_04/tiff/307.tif; http://www3.oup.co.uk/computer_journal/hdb/Volume_08/Issue_04/tiff/308.tif; http://www3.oup.co.uk/computer_journal/hdb/Volume_08/Issue_04/tiff/309.tif; http://www3.oup.co.uk/computer_journal/hdb/Volume_08/Issue_04/tiff/310.tif", acknowledgement = ack-nhfb, fjournal = "The Computer Journal", journal-URL = "http://comjnl.oxfordjournals.org/", } @MastersThesis{Chang:1966:DHR, author = "Henry Kwan-cheung Chang", title = "Design of a high rate floating point digital accumulator", type = "Thesis ({M.S. in Engineering})", school = "University of California, Davis", address = "Davis, CA, USA", pages = "60", year = "1966", bibdate = "Sat Feb 24 15:01:45 MST 1996", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, keywords = "Dissertations, Academic --- California --- Engineering.", } @Article{Chartres:1966:ACP, author = "Bruce A. Chartres", title = "Automatic Controlled Precision Calculations", journal = j-J-ACM, volume = "13", number = "3", pages = "386--403", month = jul, year = "1966", CODEN = "JACOAH", ISSN = "0004-5411 (print), 1557-735X (electronic)", ISSN-L = "0004-5411", bibdate = "Fri Nov 28 16:23:06 2003", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", abstract = "Recent developments in computer design and error analysis have made feasible the use of variable precision arithmetic and the preparation of programs that automatically determine their own precision requirements. Such programs enable the user to specify the accuracy he wants, and yield answers guaranteed to lie within the bounds prescribed. A class of such programs, called ``contracting error programs'', is defined in which the precision is determined by prescribing error bounds on the data. A variant of interval arithmetic is defined which enables a limited class of algorithms to be programmed as contracting error programs. A contracting error program for the solution of simultaneous linear equations is described, demonstrating the application of the idea to a wider class of problems.", acknowledgement = ack-mfc # " and " # ack-nhfb, fjournal = "Journal of the ACM", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J401", } @Article{Clark:1966:CMP, author = "S. R. Clark and W. F. Lunnon", title = "Correspondence: Multiple precision arithmetic (real and complex)", journal = j-COMP-J, volume = "9", number = "2", pages = "174--174", month = aug, year = "1966", CODEN = "CMPJA6", ISSN = "0010-4620 (print), 1460-2067 (electronic)", ISSN-L = "0010-4620", bibdate = "Fri Sep 29 08:56:14 MDT 2000", bibsource = "http://www3.oup.co.uk/computer_journal/hdb/Volume_09/Issue_02/; https://www.math.utah.edu/pub/tex/bib/fparith.bib", note = "See \cite{Brooker:1966:MFA}.", URL = "http://www3.oup.co.uk/computer_journal/hdb/Volume_09/Issue_02/tiff/174.tif", acknowledgement = ack-nhfb, fjournal = "The Computer Journal", journal-URL = "http://comjnl.oxfordjournals.org/", } @Article{Clark:1966:MPA, author = "S. R. Clark and W. F. Lunnon", title = "Multiple precision arithmetic in {Atlas Autocode}", journal = j-COMP-J, volume = "9", number = "2", pages = "174--174", month = aug, year = "1966", CODEN = "CMPJA6", DOI = "https://doi.org/10.1093/comjnl/9.2.174", ISSN = "0010-4620 (print), 1460-2067 (electronic)", ISSN-L = "0010-4620", bibdate = "Tue Dec 4 14:47:36 MST 2012", bibsource = "http://comjnl.oxfordjournals.org/content/9/2.toc; https://www.math.utah.edu/pub/tex/bib/compj1960.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "http://comjnl.oxfordjournals.org/content/9/2/174.full.pdf+html", acknowledgement = ack-nhfb, fjournal = "The Computer Journal", journal-URL = "http://comjnl.oxfordjournals.org/", } @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 = "??", pages = "127--132", month = "????", year = "1966", CODEN = "CMPTA2", ISSN = "0010-485X (print), 1436-5057 (electronic)", ISSN-L = "0010-485X", 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/fparith.bib", acknowledgement = ack-nj, fjournal = "Computing: Archiv f{\"u}r informatik und numerik", journal-URL = "http://link.springer.com/journal/607", } @Article{Flehinger:1966:PRI, author = "B. J. Flehinger", title = "On the Probability that a Random Integer Has Initial Digit `{A}'", journal = j-AMER-MATH-MONTHLY, volume = "73", number = "??", pages = "1056--1061", year = "1966", CODEN = "AMMYAE", ISSN = "0002-9890 (print), 1930-0972 (electronic)", ISSN-L = "0002-9890", bibdate = "Wed Feb 14 18:53:58 2007", bibsource = "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", remark = "Probably November or December issue; missing from amermathmonthly1960.bib.", } @Article{Flynn:1966:VHS, author = "M. J. Flynn", title = "Very high-speed computing systems", journal = j-PROC-IEEE, volume = "54", number = "12", pages = "1901--1909", year = "1966", CODEN = "IEEPAD", DOI = "https://doi.org/10.1109/proc.1966.5273", ISSN = "0018-9219 (print), 1558-2256 (electronic)", ISSN-L = "0018-9219", bibdate = "Wed Dec 13 08:39:04 2023", bibsource = "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", keywords = "floating-point arithmetic; Goldschmidt division", } @Article{Garner:1966:ECA, author = "Harvey L. Garner", title = "Error Codes for Arithmetic Operations", journal = j-IEEE-TRANS-ELEC-COMPUT, volume = "EC-15", number = "5", pages = "763--770", month = oct, year = "1966", CODEN = "IEECA8", DOI = "https://doi.org/10.1109/PGEC.1966.264566", ISSN = "0367-7508", bibdate = "Thu Jul 14 05:46:38 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", URL = "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4038884", acknowledgement = ack-nhfb, fjournal = "IEEE Transactions on Electronic Computers", } @Article{Grau:1966:BRB, author = "A. A. Grau", title = "Book Review: {{\booktitle{Rounding Errors in Algebraic Processes}} (J. H. Wilkinson)}", journal = j-SIAM-REVIEW, volume = "8", number = "3", pages = "397--398", month = "????", year = "1966", CODEN = "SIREAD", DOI = "https://doi.org/10.1137/1008087", ISSN = "0036-1445 (print), 1095-7200 (electronic)", ISSN-L = "0036-1445", bibdate = "Thu Mar 27 09:05:37 MDT 2014", bibsource = "http://epubs.siam.org/toc/siread/8/3; https://www.math.utah.edu/pub/bibnet/authors/w/wilkinson-james-hardy.bib; https://www.math.utah.edu/pub/tex/bib/fparith.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 = "July 1966", } @Article{Gregory:1966:DAU, author = "Robert T. Gregory", title = "On the Design of the Arithmetic Unit of a Fixed-Word-Length Computer from the Standpoint of Computational Accuracy", journal = j-IEEE-TRANS-ELEC-COMPUT, volume = "EC-15", number = "2", pages = "255--257", month = apr, year = "1966", CODEN = "IEECA8", DOI = "https://doi.org/10.1109/PGEC.1966.264312", ISSN = "0367-7508", bibdate = "Thu Jul 14 05:46:31 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", URL = "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4038726", acknowledgement = ack-nhfb, fjournal = "IEEE Transactions on Electronic Computers", } @Book{Greve:1966:HLR, author = "J. Greve and H. Gumin and E. Hochsteller", title = "{Herrn von Leibniz' Rechnung mit Null und Ein}. ({German}) [{Mister von Leibniz'} calculation with zero and one]", publisher = "{Siemens AG}", address = "Berlin and M{\"u}nchen, West Germany", pages = "59", year = "1966", bibdate = "Fri Mar 17 08:31:51 2023", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, keywords = "Gottfried Wilhelm Freiherr von Leibnitz", language = "German", } @InProceedings{Harding:1966:ISF, author = "L. J. {Harding, Jr.}", editor = "????", booktitle = "{SHARE} {XXVII}, Toronto, Canada, August 1966", title = "Idiosyncracies of {System\slash 360} Floating-Point", publisher = "????", address = "????", pages = "??--??", year = "1966", LCCN = "????", bibdate = "Wed Feb 14 19:04:14 2007", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, } @Misc{Harding:1966:MSF, author = "L. J. {Harding, Jr.}", title = "Modifications of {System\slash 360} Floating-Point", number = "SSD 157, C4470", howpublished = "SHARE Secretarial Distribution", pages = "11--27", month = "????", year = "1966", bibdate = "Wed Feb 14 19:05:45 2007", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, } @Article{Henrici:1966:TPM, author = "Peter Henrici", title = "Test of probabilistic models for the propagation of roundoff errors", journal = j-CACM, volume = "9", number = "6", pages = "409--410", month = jun, year = "1966", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Fri Nov 25 18:20:07 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/fparith.bib", acknowledgement = ack-nhfb, fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "floating-point arithmetic; rounding errors", } @Article{Hull:1966:TPM, author = "T. E. Hull and J. R. Swenson", title = "Tests of Probabilistic Models for Propagation of Roundoff Errors", journal = j-CACM, volume = "9", number = "2", pages = "108--113", month = feb, year = "1966", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", MRclass = "65.80", MRnumber = "32\#3297", bibdate = "Fri Nov 25 18:20:04 MST 2005", bibsource = "ftp://ftp.ira.uka.de/pub/bibliography/Theory/gvl.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/fparith.bib", abstract = "In any prolonged computation it is generally assumed that the accumulated effect of roundoff errors is in some sense statistical. The purpose of this paper is to give precise descriptions of certain probabilistic models for roundoff error, and then to describe a series of experiments for testing the validity of these models. It is concluded that the models are in general very good. Discrepancies are both rare and mild. The test techniques can also be used to experiment with various types of special arithmetic.", acknowledgement = ack-nhfb, fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "decimal floating-point arithmetic; rounding errors", } @Manual{IBM:1966:ISM, key = "IBM", title = "{IBM System\slash 360 Model 91}: Functional Characteristics", publisher = pub-IBM, address = pub-IBM:adr, year = "1966", bibdate = "Fri Aug 20 09:34:43 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", note = "File No. S360-01, Form A22-6907-2.", acknowledgement = ack-nhfb, remark = "Cited in \cite{Sterbenz:1974:FPC}.", } @Book{Isaacson:1966:ANM, author = "Eugene Isaacson and Herbert Bishop Keller", title = "Analysis of numerical methods", publisher = pub-WILEY, address = pub-WILEY:adr, pages = "xv + 541", year = "1966", LCCN = "QA297 .I8", bibdate = "Fri Aug 20 09:37:14 MDT 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; z3950.loc.gov:7090/Voyager", acknowledgement = ack-nhfb, remark = "Cited in \cite{Sterbenz:1974:FPC}. Reprinted in \cite{Isaacson:1994:ANM}.", subject = "Numerical analysis", } @Misc{Kahan:1966:ISS, author = "W. Kahan", title = "{7094 II} System Support for Numerical Analysis", howpublished = "SHARE Secretary Distribution 159, C4537", pages = "1--54", year = "1966", bibdate = "Wed Nov 07 08:39:33 2007", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, remark = "Cited in \cite{Sterbenz:1974:FPC}.", } @InCollection{Kogbetliantz:1966:GEF, author = "E. G. Kogbetliantz", title = "Generation of Elementary Functions", crossref = "Ralston:1966:MMD", pages = "7--35", year = "1966", bibdate = "Sat Dec 09 14:09:27 1995", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nj, } @Article{Kuki:1966:EGS, author = "H. Kuki and E. Hanson and J. J. Ortega and J. C. Butcher and P. G. Anderson", title = "Evaluation Guidelines {SHARE Numerical Analysis Project (N.A.P.)}", journal = "{SHARE Secretary Distribution}", volume = "SSD 150, part II", number = "C4304", pages = "1--42", year = "1966", bibdate = "Fri Aug 20 09:45:16 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, remark = "Cited in \cite{Sterbenz:1974:FPC}.", } @Article{Lam:1966:COG, author = "Lay-yong Lam", title = "On the {Chinese} origin of the {Galley} method of arithmetical division", journal = j-BRITISH-J-HIST-SCI, volume = "3", number = "part 1, 9", pages = "66--69", year = "1966", CODEN = "BJHSAT", DOI = "https://doi.org/10.1017/S0007087400000200", ISSN = "0007-0874 (print), 1474-001X (electronic)", ISSN-L = "0007-0874", MRclass = "01.10", MRnumber = "0204243 (34 \#4087)", MRreviewer = "I. N. Veselovski{\u\i}", bibdate = "Wed Sep 22 18:01:23 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; MathSciNet database", ZMnumber = "0143.24202", acknowledgement = ack-nhfb, fjournal = "British Journal for the History of Science", journal-URL = "http://journals.cambridge.org/action/displayJournal?jid=BJH", keywords = "history", } @Article{Mancino:1966:MPF, author = "O. G. Mancino", title = "Multiple precision floating-point conversion from decimal-to-binary and vice versa", journal = j-CACM, volume = "9", number = "5", pages = "347--348", month = may, year = "1966", CODEN = "CACMA2", DOI = "https://doi.org/10.1145/355592.365635", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Fri Nov 25 18:20:06 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/fparith.bib", abstract = "Decimal-to-binary and binary-to-decimal floating-point conversion is often performed by using a table of the powers $ 10^i $ ($i$ a positive integer) for converting from base $ 10$ to base $2$, and by using a table of the coefficients of a polynomial approximation of $ 10^x (0 \leq x < 1)$ for converting from base $2$ to base $ 10$. These tables occupy a large storage region in the case of a nonsingle precision conversion. This paper shows that a single small table suffices for a floating-point conversion from decimal to binary, and vice versa, in any useful precision.", acknowledgement = ack-nhfb, fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "floating-point arithmetic; Multiple precision arithmetic; number base conversion", } @Unpublished{Mazor:1966:FSI, author = "Stan Mazor", title = "{Fairchild Symbol II} Decimal Floating Point Unit", year = "1966", bibdate = "Wed Nov 22 21:18:53 2006", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", note = "Cited in \cite[p. 106]{Cowlishaw:2003:DFP}.", acknowledgement = ack-nhfb, keywords = "decimal floating-point arithmetic", } @Book{Moore:1966:IA, author = "Ramon E. Moore", title = "Interval analysis", publisher = pub-PH, address = pub-PH:adr, pages = "xi + 145", year = "1966", LCCN = "QA297 .M63", bibdate = "Sat Feb 14 08:15:54 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, remark = "Cited in \cite{Sterbenz:1974:FPC}.", } @Article{Nickel:1966:NFA, author = "K. Nickel", title = "{{\"U}ber die Notwendigkeit einer Fehlerschranken- Arithmetik f{\"u}r Rechenautomaten}", journal = j-NUM-MATH, volume = "9", pages = "69--79", year = "1966", CODEN = "NUMMA7", ISSN = "0029-599X (print), 0945-3245 (electronic)", ISSN-L = "0029-599X", bibdate = "Fri Jan 12 11:37:56 1996", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-jr, fjournal = "Numerische Mathematik", journal-URL = "http://link.springer.com/journal/211", } @Book{Parker:1966:SNS, author = "Francis D. Parker", title = "The structure of number systems", publisher = pub-PH, address = pub-PH:adr, pages = "xi + 137", year = "1966", LCCN = "QA241 .P3; QA241 .P239", bibdate = "Fri Nov 9 19:10:47 MST 2007", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; melvyl.cdlib.org:210/CDL90", series = "Teachers' mathematics reference series", acknowledgement = ack-nhfb, subject = "Number theory", } @Book{Richards:1966:EDS, author = "R. K. (Richard Kohler) Richards", title = "Electronic Digital Systems", publisher = pub-WILEY, address = pub-WILEY:adr, pages = "ix + 637", year = "1966", LCCN = "TK7888.3 .R52", bibdate = "Mon Oct 24 05:33:53 MDT 2016", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; z3950.loc.gov:7090/Voyager", acknowledgement = ack-nhfb, author-dates = "1921--", remark = "According to \cite[pp. 168--170]{Smiley:2010:MWI}, this was the first book to draw attention to the pre-World War II work by John Atanasoff (1903--1995) and Clifford Berry (1918--1963) at Iowa State College of Agricultural and Mechanic Arts (renamed on 4 July 1959 to Iowa State University of Science and Technology), and to name their computer the ABC. Richards is quoted with ``The ancestry of all electronic digital systems appears to be traceable to a computer which will be called the Atanasoff--Berry Computer.'' The book provided important evidence in the priority dispute and courtroom battles over who invented the computer (Atanasoff and Berry versus Mauchly and Eckert). Berry died in unresolved circumstances: suicide or murder?", subject = "Electronic digital computers", tableofcontents = "1: History and Introduction / 1 \\ 2: Theory of Digital Systems / 50 \\ 3: The Stored-Program Concept / 109 \\ 4: Automatic Programming / 293 \\ 5: Digital Data Transmission / 362 \\ 6: Combined Analog and Digital Techniques / 451 \\ 7: Telephone and Message Switching Systems / 481 \\ 8: The Relationship between Thinking and Digital Systems / 519 \\ 9: Miscellaneous Digital Systems / 541 \\ 10: Digital System Reliability / 571 \\ 11: The Automatic Design of Digital Systems / 610 \\ Index / 631", } @Article{Saidan:1966:EEA, author = "A. S. Saidan", title = "The Earliest Extant {Arabic} Arithmetic: {{\booktitle{Kitab al-Fusul fi al Hisab al-Hindi}}} of {Abu al-Hasan, Ahmad ibn Ibrahim al-Uqlidisi}", journal = j-ISIS, volume = "57", number = "4", pages = "475--490", month = "Winter", year = "1966", CODEN = "ISISA4", ISSN = "0021-1753 (print), 1545-6994 (electronic)", ISSN-L = "0021-1753", bibdate = "Tue Jul 30 21:22:16 MDT 2013", bibsource = "http://www.jstor.org/action/showPublication?journalCode=isis; http://www.jstor.org/stable/i211145; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/isis1960.bib", URL = "http://www.jstor.org/stable/228518", acknowledgement = ack-nhfb, fjournal = "Isis", journal-URL = "http://www.jstor.org/journal/isis", } @Article{Smith:1966:CP, author = "John Smith", title = "The Challenge of {Pi}", journal = j-IEEE-SPECTRUM, volume = "3", number = "10", pages = "5--5", month = oct, year = "1966", CODEN = "IEESAM", DOI = "https://doi.org/10.1109/MSPEC.1966.5217340", ISSN = "0018-9235 (print), 1939-9340 (electronic)", ISSN-L = "0018-9235", bibdate = "Wed Jan 15 08:45:04 2020", bibsource = "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/pi.bib", acknowledgement = ack-nhfb, fjournal = "IEEE Spectrum", journal-URL = "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=6", keywords = "Computer errors; Digital arithmetic; Educational institutions; Hardware; Physics computing; Power engineering computing; Programming profession; Registers; Testing; Upper bound", remark = "Report of a computation of $ \pi $ to 17,935 places using base-1,000,000 arithmetic. A footnote reports ``The AIL result to 17,940 places was identical with the reference to 17,935 places. Reference: D. Shanks and J. W. Wrench, Jr., `Calculation of Pi to 100,000 Decimals', Mathematics of Computation, January 1962, Vol. 16, No. 77, pp. 67--99.''", } @Article{Spielberg:1966:CEU, author = "K. Spielberg", title = "Computation of $ e^x $ with the use of large tables", journal = j-IBM-SYS-J, volume = "5", number = "2", pages = "102--114", year = "1966", CODEN = "IBMSA7", ISSN = "0018-8670", bibdate = "Thu Sep 15 18:43:03 1994", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nj, fjournal = "IBM Systems Journal", xxmonth = "(none)", } @Article{Tienari:1966:SPM, author = "M. Tienari and V. Suokonautio", title = "A Set of Procedures Making Real Arithmetic of Unlimited Accuracy Possible Within {Algol 60}", journal = j-NORDISK-TIDSKR-INFORM-BEHAND, volume = "6", number = "4", pages = "332--338", month = jul, year = "1966", CODEN = "BITTEL, NBITAB", DOI = "https://doi.org/10.1007/BF01966093", ISSN = "0006-3835 (print), 1572-9125 (electronic)", ISSN-L = "0006-3835", bibdate = "Mon Sep 12 08:07:38 1994", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "http://www.springerlink.com/openurl.asp?genre=article&issn=0006-3835&volume=6&issue=4&spage=332", acknowledgement = ack-nj, journal-URL = "http://link.springer.com/journal/10543", } @Article{vanWijngaarden:1966:NAI, author = "A. {van Wijngaarden}", title = "Numerical Analysis as an Independent Science", journal = j-NORDISK-TIDSKR-INFORM-BEHAND, volume = "6", number = "1", pages = "66--81", month = mar, year = "1966", CODEN = "BITTEL, NBITAB", DOI = "https://doi.org/10.1007/BF01939551", 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=6&issue=1; https://www.math.utah.edu/pub/tex/bib/bit.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "http://www.springerlink.com/openurl.asp?genre=article&issn=0006-3835&volume=6&issue=1&spage=66", abstract = "The paper describes how a number of well-known mathematical concepts ought to be modified in order to make sense within the scope of numerical analysis. It is also shown how obvious difficulties can be overcome in a logical way. All algorithms suggested are given as ALGOL procedures.", acknowledgement = ack-nhfb, journal-URL = "http://link.springer.com/journal/10543", remark = "Cited in \cite{Sterbenz:1974:FPC}.", } @Article{Watson:1966:SCC, author = "R. W. Watson and C. W. Hastings", title = "Self-Checked Computation using Residue Arithmetic", journal = j-PROC-IEEE, volume = "54", number = "12", pages = "1920--1931", month = dec, year = "1966", CODEN = "IEEPAD", ISSN = "0018-9219 (print), 1558-2256 (electronic)", ISSN-L = "0018-9219", bibdate = "Fri Nov 09 19:37:52 2007", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, fjournal = "Proceedings of the IEEE", journal-URL = "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=5", } @Article{Whitney:1966:PCR, author = "D. E. Whitney", title = "Propagation and control of roundoff error in the matrix exponential method", journal = j-PROC-IEEE, volume = "54", number = "10", pages = "1483--1484", month = oct, year = "1966", CODEN = "IEEPAD", ISSN = "0018-9219 (print), 1558-2256 (electronic)", ISSN-L = "0018-9219", bibdate = "Sat Jul 16 11:25:06 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, fjournal = "Proceedings of the IEEE", journal-URL = "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=5", } @TechReport{Wiegel:1966:MBA, author = "R. E. Wiegel", title = "Methods of Binary Addition", type = "Technical report", number = "195", institution = "Department of Computer Science, University of Illinois", address = "Urbana, IL, USA", month = feb, year = "1966", bibdate = "Fri Nov 09 19:40:18 2007", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, } @Article{Adams:1967:SCP, author = "Duane A. Adams", title = "A stopping criterion for polynomial root finding", journal = j-CACM, volume = "10", number = "10", pages = "655--658", month = oct, year = "1967", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", MRclass = "65.50", MRnumber = "39\#2314", 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/fparith.bib", abstract = "When searching for the root of a polynomial, it is generally difficult to know just when to accept a number as an adequate approximation to the root. In this paper an algorithm is presented which allows one to terminate the iteration process on the basis of calculated bounds for the roundoff error which occurs in evaluating the polynomial. This stopping criterion has been tested on numerous examples and has been found to serve as a satisfactory means for accepting a complex number as a zero of a real polynomial.", acknowledgement = ack-nhfb, fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "floating-point arithmetic; rounding errors", } @Article{Anderson:1967:ISMb, author = "S. F. Anderson and J. G. Earle and R. E. Goldschmidt and D. M. Powers", title = "The {IBM System\slash 360 Model 91}: Floating-point execution unit", journal = j-IBM-JRD, volume = "11", number = "1", pages = "34--53", month = jan, year = "1967", CODEN = "IBMJAE", DOI = "https://doi.org/10.1147/rd.111.0034", ISSN = "0018-8646 (print), 2151-8556 (electronic)", ISSN-L = "0018-8646", bibdate = "Tue Sep 11 15:36:09 MDT 2012", bibsource = "http://www.research.ibm.com/journal/; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ibmjrd.bib", URL = "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5392016", abstract = "The principal requirement for the Model 91 floating-point execution unit was that it be designed to support the instruction-issuing rate of the processor. The chosen solution was to develop separate, instruction-oriented algorithms for the add, multiply, and divide functions. Linked together by the floating-point instruction unit, the multiple execution units provide concurrent instruction execution at the burst rate of one instruction per cycle.", acknowledgement = ack-nhfb # "\slash " # ack-nj, book-URL = "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=5288520", fjournal = "IBM Journal of Research and Development", journal-URL = "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=5288520", } @TechReport{Clark:1967:PSF, author = "N. A. Clark and W. J. Cody and K. E. Hillstrom and E. A. Thieleker", title = "Performance Statistics of the {FORTRAN IV (H)} Library for the {IBM System\slash 360}", type = "Technical Report", number = "ANL-7231", institution = inst-ANL, address = inst-ANL:adr, year = "1967", bibdate = "Fri Aug 20 08:47:24 2010", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/c/cody-william-j.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", note = "Reprinted in SHARE Secretary Distribution, SDD 169, C4473, pp. 12 46.", acknowledgement = ack-nhfb, remark = "Cited in \cite{Sterbenz:1974:FPC}.", } @Article{Cody:1967:CFI, author = "W. J. Cody", title = "Critique of the {FORTRAN IV (H)} Library for the {IBM System\slash 360}", journal = "{SHARE Secretary Distribution}", volume = "SSD 169", number = "C4473", pages = "4--11", year = "1967", bibdate = "Fri Aug 20 08:52:04 2010", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/c/cody-william-j.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, remark = "Cited in \cite{Sterbenz:1974:FPC}.", } @InProceedings{Cody:1967:IMD, author = "W. J. Cody", title = "The influence of machine design on numerical algorithms", crossref = "AFIPS:1967:ACP", pages = "305--309", year = "1967", bibdate = "Sat Sep 24 00:18:32 1994", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/c/cody-william-j.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, } @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{Crisansan-Zverca:1967:PED, author = "Mariana Cri{\c{s}}an-Zverca", title = "Pseudo-operations of an electronic digital machine with floating point. ({Romanian})", journal = j-STUD-CERCET-MAT, volume = "19", pages = "1413--1424", year = "1967", ISSN = "0039-4068, 0567-6401", MRclass = "68.00", MRnumber = "40\#3753", bibdate = "Fri Dec 8 08:24:14 1995", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, fjournal = "Studii {\c{s}}i cercet{\u{a}}ri Matematice", } @Article{Curry:1967:ART, author = "E. Curry", title = "The analysis of round-off and truncation errors in a hybrid control system", journal = j-IEEE-TRANS-AUTOMAT-CONTR, volume = "12", number = "5", pages = "601--604", month = oct, year = "1967", CODEN = "IETAA9", ISSN = "0018-9286 (print), 1558-2523 (electronic)", ISSN-L = "0018-9286", bibdate = "Sat Jul 16 11:25:03 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, fjournal = "IEEE Transactions on Automatic Control", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=9", summary = "The round-off and truncation errors of a digital-analog (hybrid) servo system are studied in terms of the least upper bound and second moment, or variance. The study was applied to the azimuth function of a guided missile launcher system and a \ldots{}", } @Article{DeRegt:1967:NRA, author = "M. P. DeRegt", title = "Negative radix arithmetic", journal = j-COMP-DESIGN, volume = "6", number = "??", pages = "52--63", month = may, year = "1967", CODEN = "CMPDAM", ISSN = "0010-4566", bibdate = "Thu Nov 06 06:07:24 2014", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, fjournal = "Computer Design", keywords = "negative base", } @Article{Ferrari:1967:DMU, author = "Domenico Ferrari", title = "A Division Method Using a Parallel Multiplier", journal = j-IEEE-TRANS-ELEC-COMPUT, volume = "EC-16", number = "2", pages = "224--226", month = apr, year = "1967", CODEN = "IEECA8", DOI = "https://doi.org/10.1109/PGEC.1967.264580", ISSN = "0367-7508", bibdate = "Wed Jul 13 21:15:02 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", note = "See correction \cite{Ferrari:1969:CDM}.", URL = "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4039036", abstract = "The use of a parallel multiplier for performing high-speed binary division requires that an algorithm be devised that obtains the quotient by means of multiplications and additions. Furthermore, its hardware implementation must be as simple and as fast as possible. A suitable algorithm, which applies to a first approximation to the reciprocal of the divisor, has already been proposed [1]. A similar algorithm is presented in this paper. The comparison between the two methods for equal numbers of multiplications shows that the latter is more accurate. Conversely, a given accuracy can often be obtained with a higher speed. The generation of a piecewise-linear initial approximation is also discussed.", acknowledgement = ack-nhfb, fjournal = "IEEE Transactions on Electronic Computers", } @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", } @Article{Fike:1967:RAO, author = "C. T. Fike", title = "A Rational Approximation Optimal by {Moursund}'s Criterion (letter to the editor)", journal = j-CACM, volume = "10", number = "11", pages = "683--684", month = nov, year = "1967", 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/fparith.bib", acknowledgement = ack-nj, fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", } @Article{Filho:1967:AGF, author = "A. M. S. Filho and G. Schwachheim", title = "Algorithm 309: Gamma Function with Arbitrary Precision", journal = j-CACM, volume = "10", number = "8", pages = "511--512", month = aug, year = "1967", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Thu Sep 1 10:16:10 1994", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nj, fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", } @Book{Forsythe:1967:CSL, author = "George E. Forsythe and Cleve B. Moler", title = "Computer Solution of Linear Algebraic Systems", publisher = pub-PH, address = pub-PH:adr, pages = "xi + 148", year = "1967", LCCN = "QA297 .F57 1967", MRclass = "65.35", MRnumber = "MR0219223 (36 \#2306)", MRreviewer = "N. Gastinel", bibdate = "Tue May 25 08:52:32 2010", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/f/forsythe-george-elmer.bib; https://www.math.utah.edu/pub/bibnet/authors/m/moler-cleve-b.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/fparith.bib; https://www.math.utah.edu/pub/tex/bib/gvl.bib", ZMnumber = "0154.40401", acknowledgement = ack-nhfb, keywords = "numerical analysis; software", remark = "Translated to Russian (1969), Japanese (1969), and German (1971). Cited in \cite{Sterbenz:1974:FPC}.", subject = "numerical analysis; data processing; matrices", tableofcontents = "Reader's background and purpose of book \\ Vector and matrix norms \\ Diagonal form of a matrix under orthogonal equivalence \\ Proof of diagonal-form theorem \\ Types of computational problems in linear algebra \\ Types of matrices encountered impractical problems \\ Sources of computational problems of linear algebra \\ Condition of a linear system \\ Gaussian elimination and LU decomposition \\ Need for interchanging rows \\ Scaling equations and unknowns \\ The Crout and Doolittle variants \\ Iterative improvement \\ Computing the determinant \\ Nearly singular matrices \\ Algol 60 program \\ Fortran, extended Algol, and PL/1 programs \\ Matrix inversion \\ An example: Hilbert matrices \\ Floating-point round-off analysis \\ Rounding error in Gaussian elimination \\ Convergence of iterative improvement \\ Positive definite matrices; band matrices \\ Iterative methods for solving linear systems \\ Nonlinear systems of equations", } @Article{Freeman:1967:CMS, author = "Herbert Freeman", title = "Calculation of Mean Shift for a Binary Multiplier Using $2$, $3$, or $4$ Bits at a Time", journal = j-IEEE-TRANS-ELEC-COMPUT, volume = "EC-16", number = "6", pages = "864--866", month = dec, year = "1967", CODEN = "IEECA8", DOI = "https://doi.org/10.1109/PGEC.1967.264752", ISSN = "0367-7508", bibdate = "Wed Jul 13 21:15:06 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", URL = "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4039205", acknowledgement = ack-nhfb, fjournal = "IEEE Transactions on Electronic Computers", } @Article{Freiman:1967:CDU, author = "C. V. Freiman and others", title = "Composite Division Unit", journal = j-IBM-TDB, volume = "9", number = "8", pages = "994--995", month = jan, year = "1967", CODEN = "IBMTAA", ISSN = "0018-8689", bibdate = "Tue Jan 08 22:40:53 2008", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, fjournal = "IBM Technical Disclosure Bulletin", } @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/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{Goldberg:1967:BED, author = "I. Bennett Goldberg", title = "$ 27 $ Bits Are Not Enough For $8$-Digit Accuracy", journal = j-CACM, volume = "10", number = "2", pages = "105--106", month = feb, year = "1967", CODEN = "CACMA2", DOI = "https://doi.org/10.1145/363067.363112", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Fri Nov 25 18:20:11 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/fparith.bib", abstract = "From the inequality $ 10^8 < 2^{27} $, we are likely to conclude that we can represent 8-digit decimal floating-point numbers accurately by 27-bit [binary] floating-point numbers. However, we need 28 significant bits to represent some 8-digit numbers accurately. In general, we can show that if $ 10^p < 2^{q - 1} $, then $q$ significant bits are always enough for $p$-digit decimal accuracy. Finally, we can define a compact 27-bit floating-point representation that will give 28 significant bits, for numbers of practical importance.", acknowledgement = ack-nj # " and " # ack-mfc # " and " # ack-nhfb, fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "decimal floating-point arithmetic", received = "August 1966 (revised October 1966)", } @Book{Gschwind:1967:DDC, author = "H. W. Gschwind", title = "Design of Digital Computers: An Introduction", publisher = pub-SV, address = pub-SV:adr, pages = "viii + 530", year = "1967", LCCN = "TK7888.3 .G72", bibdate = "Fri Nov 09 18:56:39 2007", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, } @Misc{Hertz:1967:CHF, author = "Theodore M. Hertz", title = "Computer Having Floating Point Multiplication", howpublished = "US Patent 3,304,417.", day = "14", month = feb, year = "1967", bibdate = "Sat Mar 24 06:05:12 2018", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", note = "Filed 1 October 1962.", URL = "https://patentimages.storage.googleapis.com/47/86/c8/897ec6540ff584/US3304417.pdf; https://patents.google.com/patent/US3304417", abstract = "Prior art digital computers utilized fixed point arithmetic processes in executing computer operations. In performing fixed point arithmetic processes a computer assumes the binary point (analogous to a decimal point) to be between the sign of a number and the most significant digit of the number. Thus the number is considered to have an absolute value of less than one. As explained in patent application filed September 24, 1962, Serial No. 225,676, now abandoned, for a ``Computer Having Floating Point Addition and Floating Point Subtraction'' invented by me, in order to execute fixed point addition and subtraction operations, numbers to be operated upon by a computer must first be appropriately scaled prior to computation in order to obtain the scaling of the numbers is accomplished prior to entering them into the computer or by shifting them through programming afterwards. Fixed point multiplication needs to have numbers scale prior to performing computations with them. In multiplication, shifting of the result through programming is desirable in order to prevent loss of significant portions which should be retained for use in later computations. Particularly is it true where after a multiplication operation has been performed, the product must be added or subtracted from or with other numbers, in which case scaling would first have to be done before the additional computation could be made.", acknowledgement = ack-nhfb, } @Article{Howell:1967:MPA, author = "K. M. Howell", title = "Multiple precision arithmetic techniques", journal = j-COMP-J, volume = "9", number = "4", pages = "383--387", month = feb, year = "1967", CODEN = "CMPJA6", ISSN = "0010-4620 (print), 1460-2067 (electronic)", ISSN-L = "0010-4620", bibdate = "Fri Sep 29 08:56:19 MDT 2000", bibsource = "http://www3.oup.co.uk/computer_journal/hdb/Volume_09/Issue_04/; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "http://www3.oup.co.uk/computer_journal/hdb/Volume_09/Issue_04/090383.sgm.abs.html; http://www3.oup.co.uk/computer_journal/hdb/Volume_09/Issue_04/tiff/383.tif; http://www3.oup.co.uk/computer_journal/hdb/Volume_09/Issue_04/tiff/384.tif; http://www3.oup.co.uk/computer_journal/hdb/Volume_09/Issue_04/tiff/385.tif; http://www3.oup.co.uk/computer_journal/hdb/Volume_09/Issue_04/tiff/386.tif; http://www3.oup.co.uk/computer_journal/hdb/Volume_09/Issue_04/tiff/387.tif", acknowledgement = ack-nhfb, fjournal = "The Computer Journal", journal-URL = "http://comjnl.oxfordjournals.org/", } @Article{Jarden:1967:EAL, author = "Dov Jarden", title = "Existence of Arbitrarily Long Sequences of Consecutive Members in Arithmetic Progressions Divisible by Arbitrarily Many Different Primes", journal = j-FIB-QUART, volume = "5", number = "3", pages = "280--??", month = oct, year = "1967", CODEN = "FIBQAU", ISSN = "0015-0517", ISSN-L = "0015-0517", bibdate = "Thu Oct 20 18:04:54 MDT 2011", bibsource = "http://www.fq.math.ca/5-3.html; https://www.math.utah.edu/pub/tex/bib/fibquart.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "http://www.fq.math.ca/Scanned/5-3/jarden1.pdf", acknowledgement = ack-nhfb, ajournal = "Fib. Quart", fjournal = "The Fibonacci Quarterly. Official Organ of the Fibonacci Association", journal-URL = "http://www.fq.math.ca/", } @Article{Krishnamurthy:1967:NRD, author = "E. V. Krishnamurthy and S. K. Nandi", title = "On the Normalization Requirement of Divisor in Divide-and-Correct Methods", journal = j-CACM, volume = "10", number = "12", pages = "809--813", month = dec, year = "1967", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Thu Sep 1 10:15:08 1994", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nj, fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", } @Article{Kuki:1967:CAE, author = "H. Kuki", title = "Comments on the {ANL} Evaluation of the {OS\slash 360 FORTRAN} Math Function Library", journal = "{SHARE Secretary Distribution}", volume = "SSD 169", number = "C4773", pages = "47--53", year = "1967", bibdate = "Fri Aug 20 09:42:04 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, remark = "Cited in \cite{Sterbenz:1974:FPC}.", } @Article{Lewis:1967:CFP, author = "H. R. {Lewis, Jr.} and E. J. {Stovall, Jr.}", title = "Comments on a Floating-Point Version of {Nordsieck}'s Scheme for the Numerical Integration of Differential Equations", journal = j-MATH-COMPUT, volume = "21", number = "98", pages = "157--161", 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/fparith.bib; JSTOR database", acknowledgement = ack-nhfb, ajournal = "Math. Comput.", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Long:1967:LAS, author = "C. T. Long and J. H. Jordan", title = "A Limited Arithmetic on Simple Continued Fractions", journal = j-FIB-QUART, volume = "5", number = "2", pages = "113--128", month = apr, year = "1967", CODEN = "FIBQAU", ISSN = "0015-0517", ISSN-L = "0015-0517", bibdate = "Thu Oct 20 18:04:52 MDT 2011", bibsource = "http://www.fq.math.ca/5-2.html; https://www.math.utah.edu/pub/tex/bib/fibquart.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "http://www.fq.math.ca/Scanned/5-2/long.pdf", acknowledgement = ack-nhfb, ajournal = "Fib. Quart", fjournal = "The Fibonacci Quarterly. Official Organ of the Fibonacci Association", journal-URL = "http://www.fq.math.ca/", } @Article{Mandelbaum:1967:CLS, author = "David Mandelbaum", title = "A Comparison of Linear Sequential Circuits and Arithmetic Sequences", journal = j-IEEE-TRANS-ELEC-COMPUT, volume = "EC-16", number = "2", pages = "151--157", month = apr, year = "1967", CODEN = "IEECA8", DOI = "https://doi.org/10.1109/PGEC.1967.264810", ISSN = "0367-7508", bibdate = "Wed Jul 13 21:15:01 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", URL = "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4039022", acknowledgement = ack-nhfb, fjournal = "IEEE Transactions on Electronic Computers", } @Article{Massell:1967:RAP, author = "E. Massell", title = "{R67-41} An Analog Photoresistive Multiplier", journal = j-IEEE-TRANS-ELEC-COMPUT, volume = "EC-16", number = "3", pages = "380--380", month = jun, year = "1967", CODEN = "IEECA8", DOI = "https://doi.org/10.1109/PGEC.1967.264730", ISSN = "0367-7508", bibdate = "Wed Jul 13 21:15:03 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", note = "See comment \cite{Azgapetian:1968:CAP}.", URL = "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4039094", acknowledgement = ack-nhfb, fjournal = "IEEE Transactions on Electronic Computers", } @InProceedings{Matula:1967:BCM, author = "David W. Matula", booktitle = "{Proceedings of the AFIPS 1967 Spring Joint Computer Conference}", title = "Base Conversion Mappings", volume = "30", publisher = pub-AFIPS, address = pub-AFIPS:adr, pages = "311--318", year = "1967", bibdate = "Wed Feb 14 19:18:24 2007", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, remark = "Cited in \cite{Sterbenz:1974:FPC}.", } @Book{McCalla:1967:INM, author = "Thomas Richard McCalla", title = "Introduction to Numerical Methods and {Fortran} Programming", publisher = pub-WILEY, address = pub-WILEY:adr, pages = "xiii + 359", month = jan, year = "1967", ISBN = "0-471-58125-9", ISBN-13 = "978-0-471-58125-3", LCCN = "QA297 .M25", bibdate = "Sat Sep 27 17:58:26 MDT 1997", bibsource = "http://www.amazon.com/exec/obidos/ISBN=0471581259/wholesaleproductA/; http://www.cbooks.com/; https://www.math.utah.edu/pub/tex/bib/fparith.bib; z3950.loc.gov:7090/Voyager", price = "US\$27.95", URL = "http://www.cbooks.com/sqlnut/SP/search/gtsumt?source=&isbn=0471581259", acknowledgement = ack-nhfb, keywords = "Electronic digital computers --- Programming.; FORTRAN (Computer program language); Numerical analysis.", remark = "Cited in \cite{Sterbenz:1974:FPC}.", } @Article{McKeeman:1967:RER, author = "W. M. McKeeman", title = "Representation Error for Real Numbers in Binary Computer Arithmetic", journal = j-IEEE-TRANS-ELEC-COMPUT, volume = "EC-16", number = "5", pages = "682--683", month = oct, year = "1967", CODEN = "IEECA8", DOI = "https://doi.org/10.1109/PGEC.1967.264781", ISSN = "0367-7508", bibdate = "Wed Jul 13 21:15:05 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", URL = "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4039164", abstract = "Real numbers can be represented in a binary computer by the form $ i \cdot B^e $ where $i$ is the integer part, $B$ the base, and $e$ the exponent. The accuracy of the representation will depend upon the number of bits allocated to the integer part and exponent part as well as what base is chosen. If $ L(i) $ and $ L(e) $ are the number of bits allocated to the magnitudes of the integer and exponent parts and we define $ I = 2^{L(i)} $ and $ E = 2^{L(e)} $, the exponent range is given by $ B^{\pm E} $, the maximum relative representation error is given by $ B / 2 I $, and the average relative representation error is given by $ (B - 1) / (4 I \ln B) $. The formulas provide quantitative comparison for the effectiveness of alternative formats for real number representations.", acknowledgement = ack-nhfb, fjournal = "IEEE Transactions on Electronic Computers", remark = "McKeeman discusses the maximum and average relative errors in binary (with and without a hidden bit), octal, and hexadecimal bases for floating-point arithmetic with a 48-bit computer word. Errors are smallest with binary and a hidden bit.", } @Article{Menzel:1967:AUA, author = "M. Menzel and N. Metropolis", title = "Algorithms in unnormalized arithmetic. {II}. {Unrestricted} polynomial evaluation", journal = j-NUM-MATH, volume = "10", number = "5", pages = "451--462", month = nov, year = "1967", CODEN = "NUMMA7", DOI = "https://doi.org/10.1007/BF02162878", ISSN = "0029-599X (print), 0945-3245 (electronic)", ISSN-L = "0029-599X", MRclass = "65.80", MRnumber = "MR0255093 (40 \#8300)", bibdate = "Wed Nov 14 19:04:40 2007", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/m/metropolis-nicholas.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", ZMnumber = "0225.65059", acknowledgement = ack-nhfb, fjournal = "Numerische Mathematik", journal-URL = "http://link.springer.com/journal/211", } @Book{Minsky:1967:CFI, author = "Marvin Lee Minsky", title = "Computation: Finite and Infinite Machines", publisher = pub-PH, address = pub-PH:adr, pages = "xvii + 317", year = "1967", LCCN = "QA267 .M55", bibdate = "Thu Oct 17 06:05:56 2019", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/t/turing-alan-mathison.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, author-dates = "1927--2016", tableofcontents = "Physical machines and their abstract counterparts \\ Part 1: Finite-state machines \\ Neural networks. Automata made up of parts \\ The memories of events in finite-state machines \\ Part 2: Infinite machines \\ Computability, effective procedures, and algorithms. Infinite machines \\ Turing machines \\ Universal Turing machines \\ Limitations of effective computability: some problems not solvable by instruction-obeying machines \\ The computable real numbers \\ The relations between Turing machines and recursive functions \\ Models similar to digital computers \\ Part 3: Symbol-manipulation systems and computability \\ The symbol-manipulation systems of post \\ Post's normal-form theorem \\ Very simple bases for computability \\ Solutions to selected problems", } @Article{Moler:1967:IRF, author = "C. B. Moler", title = "Iterative Refinement in Floating Point", journal = j-J-ACM, volume = "14", number = "2", pages = "316--321", month = apr, year = "1967", CODEN = "JACOAH", ISSN = "0004-5411 (print), 1557-735X (electronic)", ISSN-L = "0004-5411", bibdate = "Tue Nov 1 09:43:17 1994", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nj, fjournal = "Journal of the ACM", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J401", keywords = "ir, nla, lud, iterative refinement", } @Article{Mosteller:1967:DSR, author = "Frederick Mosteller and Cleo Youtz and Douglas Zahn", title = "The Distribution of Sums of Rounded Percentages", journal = j-DEMOGRAPHY, volume = "4", number = "2", pages = "850--858", month = jun, year = "1967", CODEN = "????", ISSN = "0070-3370 (print), 1533-7790 (electronic)", ISSN-L = "0070-3370", bibdate = "Thu Nov 24 08:03:16 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/benfords-law.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", note = "See \cite{Diaconis:1979:RP} for further work.", URL = "http://muse.jhu.edu/journals/dem/; http://www.biomedsearch.com/nih/Distribution-Sums-Rounded-Percentages/21318695.html; http://www.jstor.org/stable/2060324", abstract = "When percentages are computed for counts in several categories or for several positive measurements0 each taken as a fraction of their sum, the rounded percentages often fail to add to 100 percent. We investigate how frequently this failure occurs and what the distributions of sums of rounded percentages are for (1) an empirical set of data, (2) the multinomial distribution in small samples, (3) spacings between points dropped on an interval --- the broken-stick model; and (4) for simulation for several categories. The several methods produce similar distributions.We find that the probability that the sum of rounded percentages adds to exactly 100 percent is certain for two categories, about three-fourths for three categories, about two-thirds for four categories, and about [Formula: see text] for larger numbers of categories, c, on the average when categories are not improbable.", acknowledgement = ack-nhfb, fjournal = "Demography", pubmedid = "21318695", } @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{Nandi:1967:STD, author = "Salil K. Nandi and E. V. Krishnamurthy", title = "A simple technique for digital division", journal = j-CACM, volume = "10", number = "5", pages = "299--301", 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/fparith.bib", acknowledgement = ack-nhfb, fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", } @Article{Sandberg:1967:FPR, author = "I. W. Sandberg", title = "Floating-Point-Roundoff Accumulation in Digital-Filter Realizations", journal = j-BELL-SYST-TECH-J, volume = "46", number = "8", pages = "1775--1791", month = oct, year = "1967", CODEN = "BSTJAN", ISSN = "0005-8580", bibdate = "Tue Nov 9 11:15:55 MST 2010", bibsource = "http://bstj.bell-labs.com/oldfiles/year.1967/BSTJ.1967.4608.html; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "http://bstj.bell-labs.com/BSTJ/images/Vol46/bstj46-8-1775.pdf", abstract = "In this paper, several results are presented concerning the effects of roundoff in the floating-point realization of a general discrete filter governed ideally by a stable difference equation of the form\par $$ w_n = \sum_{k = 0}^M b_k x_{n - k} - \sum {k = 1}^N a_k w_{n - k}, \qquad n \geq N \qquad (1) $$ \par in which $ \{ w_n \} $ and $ \{ x_n \} $ are output and input sequences, respectively.\par In particular, for a large class of filters it is proved that there is a function $ f(K) $ with $ f(K) \to 0 $ as $ K \to \infty $ and a constant $c$, both dependent on the $ b_k $, the $ a_k $, the order in which the products on the right side of (1) are summed in the machine, and $t$, the number of bits allotted to the mantissa, such that\par $$ \langle e \rangle_K \leq c \langle y \rangle_K + f(K) $$ \par for all $ K \geq N $, in which, with the computed output sequence of the realized filter,\par $$ \langle y \rangle_K = \left (\frac {1}{K + 1} \sum_{n = 0}^K |y_n|^2 \right)^{1 / 2} $$ \par and\par $$ \langle e \rangle_K = \left (\frac {1}{K + 1} \sum_{n = 0}^K |w_n - y_n|^2 \right)^{1 / 2} $$ \par Bounds on $ f(K) $ and $c$ are given that are not difficult to evaluate, and which, in many realistic cases, are informative. For example, for the second-order bandpass filter:\par $$ w_n = x_n - a_1 w_{n - 1} - a_2 w_{n - 2}, \qquad n \geq 2 \qquad (2) $$ \par with $ a_1 $ and $ a_2 $ chosen so that its poles arc at approximately $ \pm 45^\circ $ and at distance approximately (but not less than) $ 0.001 $ from the unit circle, we find that $c$, an upper bound on the ``asymptotic output error-to-signal ratio'', is not greater than $ 0.58 \times 10^{-4} $, assuming that $ t = 27 $, that the terms on the right side of (2) are summed in the machine in the order indicated (from right to left), and that the $ x_n $ in (2) are machine numbers. If the $ x_n $ are not machine numbers, and hence must be quantized before processing, then $ c \leq 0.76 \times 10^{-4} $.\par In addition to error bounds, an inequality is derived which, if satisfied, rules out certain types of generally undesirable behavior such as self-sustained output limit cycles due to roundoff effects. This inequality is satisfied for the example described above.", acknowledgement = ack-nhfb, fjournal = "The Bell System Technical Journal", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)1538-7305/issues/", } @Article{Sasaki:1967:ASR, author = "Akio Sasaki", title = "Addition and Subtraction in the Residue Number System", journal = j-IEEE-TRANS-ELEC-COMPUT, volume = "EC-16", number = "2", pages = "157--164", month = apr, year = "1967", CODEN = "IEECA8", DOI = "https://doi.org/10.1109/PGEC.1967.264811", ISSN = "0367-7508", bibdate = "Wed Jul 13 21:15:01 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", URL = "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4039023; http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=4037753; http://ieeexplore.ieee.org/xpl/tocresult.jsp?isnumber=4039015", acknowledgement = ack-nhfb, fjournal = "IEEE Transactions on Electronic Computers", keywords = "residue arithmetic; residue number system", summary = "Improved residue expression and new arithmetic algorithms for addition and subtraction are proposed. In the proposed system positive and negative integers of any magnitude can be handled regardless of the particular choice of the set of relatively \ldots{}", } @Book{Szabo:1967:RAA, author = "Nicholas S. Szab{\'o} and Richard I. Tanaka", title = "Residue arithmetic and its applications to computer technology", publisher = pub-MCGRAW-HILL, address = pub-MCGRAW-HILL:adr, pages = "xvi + 236", year = "1967", LCCN = "QA247.35 .S95", bibdate = "Fri Jun 24 16:02:00 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; z3950.loc.gov:7090/Voyager", series = "McGraw-Hill series in information processing and computers", acknowledgement = ack-nhfb, remark = "Based on the authors' Report on residue (modular) arithmetic survey.", subject = "Modular arithmetic; Algorithms; Computer programming", } @Article{Tomasulo:1967:EAE, author = "R. M. Tomasulo", title = "An Efficient Algorithm for Exploiting Multiple Arithmetic Units", journal = j-IBM-JRD, volume = "11", number = "1", pages = "25--33", month = jan, year = "1967", CODEN = "IBMJAE", ISSN = "0018-8646 (print), 2151-8556 (electronic)", ISSN-L = "0018-8646", bibdate = "Fri Aug 26 10:27:10 1994", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, fjournal = "IBM Journal of Research and Development", journal-URL = "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=5288520", } @Book{Wilkinson:1967:BZW, author = "J. H. Wilkinson", title = "Bledy Zaokragle{\'n} w Procesach Algebraicznych. ({Polish}) [{Rounding} errors in algebraic Processes]", publisher = "PWW", address = "Warszawa, Poland", pages = "????", year = "1967", bibdate = "Thu Aug 25 09:28:02 2011", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/w/wilkinson-james-hardy.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", note = "Polish translation of \cite{Wilkinson:1963:REA}", acknowledgement = ack-nhfb, language = "Polish", } @Article{Winograd:1967:TRP, author = "Shmuel Winograd", title = "On the Time Required to Perform Multiplication", journal = j-J-ACM, volume = "14", number = "4", pages = "793--802", month = oct, year = "1967", CODEN = "JACOAH", DOI = "https://doi.org/10.1145/321420.321438", ISSN = "0004-5411 (print), 1557-735X (electronic)", ISSN-L = "0004-5411", bibdate = "Sat Feb 8 10:29:30 2020", bibsource = "ftp://ftp.ira.uka.de/pub/bibliography/Parallel/Multi.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, fjournal = "Journal of the ACM", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J401", } @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{Atkins:1968:HRD, author = "D. E. Atkins", title = "Higher-radix division using estimates of the divisor and partial remainders", journal = j-IEEE-TRANS-COMPUT, volume = "C-17", number = "10", pages = "925--934", month = oct, year = "1968", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/TC.1968.226439", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Thu Apr 2 08:38:35 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-sfo # " and " # ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", } @TechReport{Azen:1968:DMS, author = "S. Azen and S. Derr", title = "On the Distribution of the Most Significant Hexadecimal Digit", type = "Technical Report", number = "RM 5496 PR", institution = "Rand Report", address = "Santa Monica, CA, USA", year = "1968", bibdate = "Wed Feb 14 17:22:53 2007", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, } @Article{Azgapetian:1968:CAP, author = "V. Azgapetian", title = "Comment on {``An Analog Photoresistive Multiplier''}", journal = j-IEEE-TRANS-COMPUT, volume = "C-17", number = "2", pages = "188--188", month = feb, year = "1968", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/TC.1968.229083", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Wed Jul 13 17:40:47 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", note = "See \cite{Massell:1967:RAP}.", URL = "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=1687312", acknowledgement = ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", } @Article{Brennan:1968:FTA, author = "J. F. Brennan", title = "The Fastest Time of Addition and Multiplication", journal = "IBM Research Reports", volume = "4", number = "1", pages = "??--??", month = "????", year = "1968", bibdate = "Fri Nov 09 10:43:37 2007", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, } @Article{Dietmeyer:1968:GPI, author = "D. L. Dietmeyer and J. R. Duley", title = "Generating prime implicants via ternary encoding and decimal arithmetic", journal = j-CACM, volume = "11", number = "7", pages = "520--523", month = jul, year = "1968", CODEN = "CACMA2", DOI = "https://doi.org/10.1145/363397.363565", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Thu Aug 07 17:25:16 2008", 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/fparith.bib", abstract = "Decimal arithmetic, ternary encoding of cubes, and topological considerations are used in an algorithm to obtain the extremals and prime implicants of Boolean functions. The algorithm, which has been programmed in the FORTRAN language, generally requires less memory than other minimization procedures, and treats DON'T CARE terms in an efficient manner.", acknowledgement = ack-nhfb, fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "cubical complexes; extremal; minimization; prime implicants; switching function; ternary encoding", } @Article{Elliott:1968:EAA, author = "David Elliott", title = "Error analysis of an algorithm for summing certain finite series", journal = j-J-AUSTRALIAN-MATH-SOC, volume = "8", number = "2", pages = "213--221", month = may, year = "1968", CODEN = "JAUMAX", DOI = "https://doi.org/10.1017/s1446788700005267", ISSN = "0004-9735 (print), 2059-9234 (electronic)", ISSN-L = "0004-9735", MRclass = "65.25", MRnumber = "0230452", MRreviewer = "U. C. Guha", bibdate = "Fri Feb 9 15:20:35 2018", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/c/clenshaw-charles-w.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, fjournal = "Journal of the Australian Mathematical Society", remark = "Analysis of Clenshaw algorithm for Chebyshev summation \cite{Clenshaw:1955:NSC}, and report of region of instability of that procedure", } @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/fparith.bib", acknowledgement = ack-nj, } @Article{Fraser:1968:AUA, author = "M. Fraser and N. Metropolis", title = "Algorithms in unnormalized arithmetic. {III}. {Matrix} inversion", journal = j-NUM-MATH, volume = "12", number = "5", pages = "416--428", month = dec, year = "1968", CODEN = "NUMMA7", ISSN = "0029-599X (print), 0945-3245 (electronic)", ISSN-L = "0029-599X", MRclass = "65.80", MRnumber = "MR0255094 (40 \#8301)", bibdate = "Wed Nov 14 19:04:40 2007", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/m/metropolis-nicholas.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", ZMnumber = "0184.37503", acknowledgement = ack-nhfb, fjournal = "Numerische Mathematik", journal-URL = "http://link.springer.com/journal/211", } @Article{Harman:1968:ADI, author = "M. G. Harman", title = "An Attempt to Design an Improved Multiplication System", journal = j-IEEE-TRANS-COMPUT, volume = "C-17", number = "11", pages = "1090--1090", month = nov, year = "1968", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/TC.1968.226864", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Wed Jul 13 17:40:51 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", URL = "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=1687268", acknowledgement = ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", } @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, } @Book{IBM:1968:ISP, author = "{IBM Corporation}", title = "{IBM System}\slash 360 Principles of Operation", publisher = pub-IBM, address = pub-IBM:adr, edition = "Eighth", pages = "175", year = "1968", LCCN = "QA76.8.I12 I59 1968", bibdate = "Wed Sep 14 23:12:39 1994", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, } @Misc{Kahan:1968:ISS, author = "W. Kahan", title = "{7094-II} system support for numerical analysis", howpublished = "SHARE Secretarial Distribution SSD-159.", year = "1968", bibdate = "Mon Sep 16 16:14:28 2002", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, } @InProceedings{Kaneko:1968:PSA, author = "T. Kaneko and B. Liu", title = "Round-off error of floating-point digital filters", crossref = "Anonymous:1968:PSA", pages = "219--227", year = "1968", MRclass = "65.80", MRnumber = "41\#7878", bibdate = "Fri Dec 8 08:24:14 1995", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, } @Article{Matula:1968:BCT, author = "David W. Matula", title = "The Base Conversion Theorem", journal = j-PROC-AM-MATH-SOC, volume = "19", number = "3", pages = "716--723", month = jun, year = "1968", CODEN = "PAMYAR", DOI = "https://doi.org/10.1090/s0002-9939-1968-0234908-9", ISSN = "0002-9939 (print), 1088-6826 (electronic)", ISSN-L = "0002-9939", bibdate = "Fri Apr 21 07:26:39 2000", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", abstract = "A demonstration that, in converting numbers from base $ \beta $ to base $ \nu $, where $ \beta^i \neq \nu^j $ for any positive integers $ i, j $, a one-to-one mapping exists only if $ \nu^{m - 1} \geq \beta^n - 1 $. Here $m$ is the number of digits in base $ \nu $ and $n$ that in base $ \beta $.\par The implications of this result are discussed in the author's earlier paper [Proc. Amer. Fed. Information Processing Soc. 30 (1967), 311--318].", acknowledgement = ack-nhfb, fjournal = "Proceedings of the American Mathematical Society", journal-URL = "http://www.ams.org/journals/proc", received = "9 February 1967", remark = "Cited in \cite{Sterbenz:1974:FPC}.", } @Article{Matula:1968:C, author = "David W. Matula", title = "In-and-out conversions", journal = j-CACM, volume = "11", number = "1", pages = "47--50", month = jan, year = "1968", CODEN = "CACMA2", DOI = "https://doi.org/10.1145/362851.362887", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", MRclass = "65.80", MRnumber = "39\#2360", bibdate = "Fri Nov 25 18:20:17 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/fparith.bib", note = "See also \cite{Goldberg:1967:BED,Matula:1968:BCT}", abstract = "By an in-and-out conversion we mean that a floating-point number in one base is converted into a floating-point number in another base and then converted back to a floating-point number in the original base. For all combinations of rounding and truncation conversions the question is considered of how many significant digits are needed in the intermediate base to allow such in-and-out conversions to return the original number (when possible), or at least significant digit.", acknowledgement = ack-nhfb # "\slash " # ack-nj, fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "floating-point numbers; number base conversion; rounding error; significance; truncation error", received = "April 1967 (revised August 1967)", remark = "Cited in \cite{Sterbenz:1974:FPC}.", } @Article{Metropolis:1968:ANA, author = "N. Metropolis", title = "Algorithms in un-normalized arithmetic: Polynomial evaluation and matrix decomposition", journal = "Colloques internationaux, Centre National de la Recherche Scientifique, Paris", volume = "165", pages = "293--303", year = "1968", bibdate = "Wed Nov 14 19:04:40 2007", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/m/metropolis-nicholas.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", ZMnumber = "0207.15804", acknowledgement = ack-nhfb, classmath = "*65H05 (Single nonlinear equations (numerical methods)) 65F05 (Direct methods for linear systems)", } @Article{Nathan:1968:IVS, author = "A. Nathan and J. Molcho", title = "Improved Voltage Selector and Cascade Multiplier Circuits", journal = j-IEEE-TRANS-COMPUT, volume = "C-17", number = "4", pages = "380--382", month = apr, year = "1968", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/TC.1968.229386", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Wed Jul 13 17:40:48 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", URL = "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=1687350", acknowledgement = ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", } @InProceedings{Nickel:1968:EBC, author = "K. Nickel", booktitle = "Proceedings of the {IFIP} Congress", title = "Error Bounds and Computer Arithmetic", publisher = pub-NORTH-HOLLAND, address = pub-NORTH-HOLLAND:adr, pages = "54--60", year = "1968", LCCN = "????", bibdate = "Fri Aug 20 10:09:47 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, remark = "Cited in \cite{Sterbenz:1974:FPC}.", } @Article{Nievergelt:1968:CCP, author = "J. Nievergelt", title = "Computers and computing --- Past present Future", journal = j-IEEE-SPECTRUM, volume = "5", number = "1", pages = "57--61", month = jan, year = "1968", CODEN = "IEESAM", DOI = "https://doi.org/10.1109/MSPEC.1968.5215633", ISSN = "0018-9235 (print), 1939-9340 (electronic)", ISSN-L = "0018-9235", bibdate = "Wed Jan 15 09:30:58 2020", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeespectrum1960.bib", acknowledgement = ack-nhfb, fjournal = "IEEE Spectrum", journal-URL = "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=6", keywords = "Computer science; Concurrent computing; Digital arithmetic; Hardware; History; Instruments; Maintenance; Manufacturing; Programming; Relays", } @Article{Padegs:1968:SAS, author = "A. Padegs", title = "Structural aspects of the {SYSTEM\slash 360 Model 85}, Part {III}: Extensions to floating-point architecture", journal = j-IBM-SYS-J, volume = "7", number = "1", pages = "22--29", year = "1968", CODEN = "IBMSA7", ISSN = "0018-8670", bibdate = "Tue Mar 19 17:38:46 1996", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, fjournal = "IBM Systems Journal", remark = "Cited in \cite{Sterbenz:1974:FPC}.", } @Article{Phillips:1968:EME, author = "G. M. Phillips", title = "Estimate of the maximum error in best polynomial approximations", journal = j-COMP-J, volume = "11", pages = "110--111", year = "1968", CODEN = "CMPJA6", ISSN = "0010-4620 (print), 1460-2067 (electronic)", ISSN-L = "0010-4620", bibdate = "Thu Sep 15 18:39:47 1994", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nj, fjournal = "The Computer Journal", journal-URL = "http://comjnl.oxfordjournals.org/", xxmonth = "(none)", xxnumber = "(none)", } @Article{Rao:1968:ECL, author = "T. R. N. Rao", title = "Error-Checking Logic for Arithmetic-Type Operations of a Processor", journal = j-IEEE-TRANS-COMPUT, volume = "C-17", number = "9", pages = "845--849", month = sep, year = "1968", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/TC.1968.229144", 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/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", URL = "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=1687471", acknowledgement = ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", } @TechReport{Ross:1968:UMF, author = "Richard D. Ross", title = "{University of Mississippi} floating point subroutines ({UMFS})", institution = "Computer Center, University of Mississippi", address = "University", pages = "31", year = "1968", bibdate = "Sat Feb 24 15:01:45 MST 1996", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, keywords = "Electronic data processing.", remark = "Cover title.", } @Article{Sasaki:1968:BIA, author = "A. Sasaki", title = "The Basis for Implementation of Additive Operations in the Residue Number System", journal = j-IEEE-TRANS-COMPUT, volume = "C-17", number = "11", pages = "1066--1073", month = nov, year = "1968", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/PGEC.1967.264810; https://doi.org/10.1109/TC.1968.226466", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Wed Jul 13 17:40:51 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", URL = "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=1687264; http://ieeexplore.ieee.org/xpl/tocresult.jsp?isnumber=35582", acknowledgement = ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", keywords = "residue arithmetic; residue number system", summary = "A new residue number system algebra has been previously proposed by the author. The algebra has solved an essential theoretical barrier in the residue number system and has enabled one to pursue additive operations in the residue \ldots{}", } @Article{Schmookler:1968:HSB, author = "M. S. Schmookler", title = "High Speed Binary to Decimal Conversion", journal = j-IEEE-TRANS-COMPUT, volume = "C-17", pages = "506--508", year = "1968", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/TC.1968.226913", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Fri Nov 28 19:09:15 2003", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", abstract = "This note describes several methods of performing fast, efficient, binary-to-decimal conversion. With a modest amount of circuitry, an order of magnitude speed improvement can is obtained. This achievement offers a unique advantage to general-purpose computers requiring special hardware to translate between binary and decimal numbering systems.", acknowledgement = ack-mfc # " and " # ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", keywords = "decimal floating-point arithmetic", } @Book{Scott:1968:OET, author = "Th{\'e}odore G. Scott", title = "Ordinateurs {\'e}lectroniques, techniques de programmation: Computer programming techniques", publisher = "Tournai, Ed. Gamma", address = "Paris, France", pages = "662", year = "1968", bibdate = "Thu May 09 09:52:34 2002", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", note = "Cinq volumes. Version fran{\c{c}}aise de Juliette Charbonneau-Kohiyama.", acknowledgement = ack-nhfb, contents = "1. Pr{\'e}sentation du Tutac. Etude du registre - base et de ses fonctions. Le registre base. - 2. Les sous programmes. L'entr{\'e}e. La sorite. - 3. La bande magn{\'e}tique. La d{\'e}tection des erreurs. - 4. Les calculs en virgule flottante. L'{\'e}dition des programmes. - 5. La programmation symbolique. Le language Cobol. Fortran et Algol", } @Manual{Smith:1968:CC, author = "Smith, Robert E. (Robert Elijah)", title = "Competence course", organization = "Control Data Corp.", address = "Minneapolis, MN, USA", pages = "vi + 296", year = "1968", bibdate = "Sat Feb 24 15:01:45 MST 1996", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, keywords = "Electronic digital computers --- Programming.; Floating-point arithmetic.; FORTRAN (Computer program language)", remark = "``Computer programming using FORTRAN and card inputs without key punching!'' Cover title: Competence course, FORTRAN computer programming. At head of title: School computer-use plan.", } @Book{Stuart:1968:FP, author = "Fredric Stuart", title = "{Fortran} programming", publisher = pub-WILEY, address = pub-WILEY:adr, pages = "xix + 353", year = "1968", ISBN = "0-471-83477-7", ISBN-13 = "978-0-471-83477-9", LCCN = "QA76.5 .S8", bibdate = "Sat Jan 27 13:40:57 MST 1996", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, keywords = "Electronic digital computers --- Programming.; {Fortran} (Computer program language)", remark = "Cited in \cite{Sterbenz:1974:FPC}, but given year 1969.", } @Article{Tung:1968:DAS, author = "Chin Tung", title = "A Division Algorithm for Signed-Digit Arithmetic", journal = j-IEEE-TRANS-COMPUT, volume = "C-17", number = "9", pages = "887--889", month = sep, year = "1968", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/TC.1968.229150", 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/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", URL = "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=1687477", acknowledgement = ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", } @Article{Urabe:1968:RED, author = "Minoru Urabe", title = "Roundoff Error Distribution in Fixed-Point Multiplication and a Remark about the Rounding Rule", journal = j-SIAM-J-NUMER-ANAL, volume = "5", number = "2", pages = "202--210", month = jun, year = "1968", 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/fparith.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "SIAM Journal on Numerical Analysis", journal-URL = "http://epubs.siam.org/sinum", remark = "Cited in \cite{Sterbenz:1974:FPC}.", } @TechReport{Veltkamp:1968:APV, author = "G. W. Veltkamp", title = "{ALGOL} procedures voor het berekenen van een inwendig product in dubbele precisie. ({Dutch}) [{ALGOL} procedures for calculating an inner product in double precision]", type = "Technical report", number = "22", institution = "RC-Informatie, Technische Hogeschool Eindhoven", address = "Eindhoven, The Netherlands", year = "1968", bibdate = "Tue Dec 26 14:53:03 2023", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, language = "Dutch", } @Article{Vitenko:1968:OAA, author = "{\=I}. V. V{\=\i}ten'{k}o", title = "Optimal algorithms for addition and multiplication on machines with floating point. ({Russian})", journal = "{\v{Z}}. Vy{\v{c}}isl. Mat. i Mat. Fiz", volume = "8", pages = "1076--1084", year = "1968", MRnumber = "40\#2250", bibdate = "Sat Dec 16 16:45:14 1995", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, language = "Russian", reviewer = "J. Kuntzmann", } @TechReport{Yohe:1968:CPA, author = "J. M. Yohe", title = "Computer Programming for Accuracy", type = "{MRC} Technical Summary Report", number = "866", institution = inst-MRC-WISCONSIN, address = inst-MRC-WISCONSIN:adr, month = apr, year = "1968", bibdate = "Sun Dec 30 21:42:55 2007", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", abstract = "In most computations, it is tacitly assumed that the results produced by the program are 'accurate enough'. This assumption is not always valid. In this report, we discuss several possible sources of error in digital computation, and we list several steps which can be taken to guard against some types of error and determine an upper bound for the effects of other types. We illustrate these techniques with our experience in writing a program to locate zeros of the Riemann zeta function.", acknowledgement = ack-nhfb, } @Article{Anonymous:1969:VPD, author = "Anonymous", title = "Video Probability Distributions", journal = j-IEEE-SPECTRUM, volume = "6", number = "4", pages = "5--5", month = apr, year = "1969", CODEN = "IEESAM", DOI = "https://doi.org/10.1109/MSPEC.1969.5214042", ISSN = "1939-9340", ISSN-L = "0018-9235", bibdate = "Wed Jan 15 11:58:06 2020", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeespectrum1960.bib", acknowledgement = ack-nhfb, fjournal = "IEEE Spectrum", journal-URL = "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=6", keywords = "Arithmetic; Circuit noise; Detectors; Distribution functions; Gaussian noise; Probability distribution; Signal to noise ratio; Tail; Testing; Voltage", } @InProceedings{Babuska:1969:NSM, author = "Ivo Babu{\v{s}}ka", title = "Numerical stability in mathematical analysis", crossref = "Morrell:1969:IPP", pages = "11--23", year = "1969", bibdate = "Tue Nov 22 05:36:22 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, keywords = "accurate floating-point summation", } @Article{Banerji:1969:SDR, author = "D. K. Banerji and J. A. Brzozowski", title = "Sign Detection in Residue Number Systems", journal = j-IEEE-TRANS-COMPUT, volume = "C-18", number = "4", pages = "313--320", month = apr, year = "1969", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/PGEC.1967.264810; https://doi.org/10.1109/T-C.1969.222658", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Wed Jul 13 16:55:18 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", URL = "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=1671251; http://ieeexplore.ieee.org/xpl/tocresult.jsp?isnumber=35034", abstract = "This paper is concerned with the sign detection problem in residue number systems. The proposed solution is applicable only to nonredundant systems. It is shown that under rather general conditions an explicit, closed formula for the sign function can be obtained. In a special case, when one of the moduli is $2$, the sign function becomes an EXCLUSIVE-OR function. A sign detection algorithm is proposed and methods of implementing the algorithm are presented.", acknowledgement = ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", keywords = "residue arithmetic; residue number system", } @Manual{BrinchHansen:1969:RCR, author = "Per {Brinch Hansen}", title = "{RC-4000 Computer} Reference Manual", number = "RCSL No: 55-D1", organization = "A/S Regnecentralen", address = "Copenhagen, Denmark", pages = "85", month = jun, year = "1969", bibdate = "Sat Mar 09 14:14:25 2019", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "http://bitsavers.org/pdf/regnecentralen/RC_4000_Reference_Manual_Jun69.pdf; https://www.math.utah.edu/~beebe/RC-4000", acknowledgement = ack-nhfb, remark = "The RC-4000 has a 24-bit word, and supplies 12-bit and 24-bit integer arithmetic, and 48-bit floating-point arithmetic. The latter has a signed 36-bit normalized two's-complement binary fraction stored at the left of a double word, followed by a 12-bit two's-complement signed exponent. The representable range is therefore [-0x1p-2049, +0x1p2047), or about [-1.547e-617, 1.615e+161]. The sign, and zero value, can be determined by examining only the first word. Its floating-point arithmetic can exactly represent 35-bit integer arithmetic.", tableofcontents = "1. RC 4000 Specifications / 9 \\ 2. Design Considerations / 11 \\ 2.1. Word Length / 11 \\ 2.2. Register Structure / 11 \\ 2.3. Address Modification / 11 \\ 2.4. Monitor Control / 12 \\ 2.5. Input/Output Control / 13 \\ 3. Data and Instruction Formats / 15 \\ 3.1. Data Formats / 15 \\ 3.2. Storage Addressing / 15 \\ 3.3. Working Registers / 16 \\ 3.4. Instruction Format / 17 \\ 3.5. Address Modify Instruction / 18 \\ 4. Integer Arithmetic / 19 \\ 4.1. Number Representation / 19 \\ 4.2. Byte Arithmetic / 19 \\ 4.3. Multiplication and Division / 20 \\ 4.4. Overflow and Carry Indication / 21 \\ 5. Floating-Point Arithmetic / 22 \\ 5.1. Number Representation / 22 \\ 5.2. Arithmetic Operations / 23 \\ 5.3. Normalization and Rounding / 23 \\ 5.4. Precision Modes / 24 \\ 5.5. Underflow, Overflow, and Non-Normalized Operands / 25 \\ 5.6. Number Conversion / 26 \\ 5.7. Exact Arithmetic with Floating-Point Instructions / 26 \\ 6. Protection System / 28 \\ 6.1. Storage Protection / 28 \\ 6.2. Privileged Instructions / 29 \\ 6.3. Summary of Protection System / 29 \\ 6.4. An Example of Protected Areas / 30 \\ 7. Interruption System / 32 \\ 7.1. Interruption Logic / 32 \\ 7.2. Interruption Conditions / 33 \\ 8. Low-Speed Data Channel / 35 \\ 8.1. Main Characteristics / 35 \\ 8.2. Input/Output Instruction / 35 \\ 8.3. Channel Operation / 36 \\ 8.4. Disconnected and Busy Indication / 37 \\ 8.5. Device Commands / 37 \\ 8.6. Read / 37 \\ 8.7. Sense / 37 \\ 8.8. Write / 38 \\ 8.9. Control / 39 \\ 8.10. Summary of Low-Speed Channel / 39 \\ 9. High-Speed Data Channel / 41 \\ 10. Standard Peripheral Devices / 43 \\ 10.1. Console Devices / 43 \\ 10.2. Interval Timer / 43 \\ 11. Operator Control Panel / 44 \\ 11.1. Indicators and Control Keys / 44 \\ 11.2. Machine Errors / 44 \\ 11.3. Reset Control / 44 \\ 11.4. Start Control / 45 \\ 11.5. Autoload Control / 45 \\ 11.6. Local/Remote Indication / 46 \\ 12. Technical Control Panel / 47 \\ 12.1. Operating Modes / 47 \\ 12.2. Instruction Step Keys / 47 \\ 12.3. Register Setting and Display / 47 \\ 12.4. Microinstruction Selection and Display / 48 \\ 12.5. Parity Control / 48 \\ 13. Instruction Set / 49 \\ 14. Definition of Instructions / 51 \\ 14.1. Algol Notation / 51 \\ 14.2. Register Structure / 51 \\ 14.3. Elementary Operations / 54 \\ 14.4. Control Panel Functions / 55 \\ 14.5. Instruction Fetch Cycle / 55 \\ 14.6. Protection Procedures / 56 \\ 14.7. Arithmetic Procedures / 57 \\ 14.8. Instruction Execution / 58 \\ Appendix / 79 \\ A.1. Reserved Storage Locations / 79 \\ A.2. Numeric Instruction Codes / 80 \\ A.3. Instruction Execution Times / 81 \\ Index / 83", } @Article{Brown:1969:CB, author = "W. S. Brown and P. L. Richman", title = "The Choice of Base", journal = j-CACM, volume = "12", number = "10", pages = "560--561", month = oct, year = "1969", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", MRclass = "68.00", MRnumber = "43\#5755", bibdate = "Mon Mar 24 21:38:51 MST 1997", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; https://www.math.utah.edu/pub/tex/bib/fparith.bib", abstract = "A digital computer is considered, whose memory words are composed of $N$ $r$-state devices plus two sign bits (two state devices). The choice of base $ \beta $ for the internal representation floating-point numbers on such a computer is discussed. It is shown that in a certain sense $ \beta = r $ is best.", acknowledgement = ack-nj # " and " # ack-nhfb, classcodes = "C5230 (Digital arithmetic methods)", corpsource = "Bell Telephone Labs. Inc., Murray Hill, NJ, USA", fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "accuracy; base choice; digital arithmetic; floating-point; number representations", } @InProceedings{Clark:1969:SCE, author = "N. W. Clark and W. J. Cody", title = "Self-contained exponentiation", crossref = "AFIPS:1969:ACPb", volume = "35", pages = "701--706", year = "1969", bibdate = "Thu Sep 1 10:16:10 1994", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nj, } @InProceedings{Cody:1969:PTF, author = "W. J. Cody", title = "Performance testing of function subroutines", crossref = "AFIPS:1969:ACPa", volume = "34", pages = "759--763", year = "1969", bibdate = "Thu Sep 1 10:15:07 1994", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nj, } @Article{Duke:1969:DFP, author = "K. A. Duke", title = "Decimal Floating Point Processor", journal = j-IBM-TDB, volume = "11-69", pages = "862--862", month = nov, year = "1969", CODEN = "IBMTAA", ISSN = "0018-8689", bibdate = "Fri Nov 28 11:29:48 2003", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, fjournal = "IBM Technical Disclosure Bulletin", keywords = "decimal floating-point arithmetic", remark = "Decimal arithmetic with unnormalized binary integers of unspecified length for coefficient and exponent.", } @MastersThesis{Duncan:1969:FFA, author = "Daniel D. A. Duncan", title = "{FLOP}: a floating-point arithmetic package", type = "Thesis ({M.S.})", school = "University of Southwestern Louisiana", address = "Lafayette, LA, USA", pages = "108", year = "1969", bibdate = "Sat Feb 24 15:01:45 MST 1996", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, keywords = "Algorithms.; Arithmetic --- Foundations.; FLOP (Computer program language).; Transformations (Mathematics).", } @Article{Dunworth:1969:ECB, author = "A. Dunworth and J. I. Roche", title = "The Error Characteristics of the Binary Rate Multiplier", journal = j-IEEE-TRANS-COMPUT, volume = "C-18", number = "8", pages = "741--745", month = aug, year = "1969", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/T-C.1969.222757", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Wed Jul 13 16:55:20 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", URL = "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=1671350", acknowledgement = ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", } @TechReport{Ehrman:1969:SFP, author = "J. R. Ehrman", title = "A Study of Floating-Point Conversions in Some {OS\slash 360} Components", number = "SDD 196, C5207", institution = "SHARE Secretary Distribution", address = "????", pages = "1--6", year = "1969", bibdate = "Wed Feb 14 18:50:19 2007", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, } @Article{Fenstad:1969:NSM, author = "Jens Erik Fenstad", title = "Non-standard models for arithmetic and analysis", journal = j-LECT-NOTES-MATH, volume = "118", pages = "30--47", year = "1969", CODEN = "LNMAA2", DOI = "https://doi.org/10.1007/BFb0060250", ISBN = "3-540-04907-X (print), 3-540-36246-0 (e-book)", ISBN-13 = "978-3-540-04907-4 (print), 978-3-540-36246-3 (e-book)", ISSN = "0075-8434 (print), 1617-9692 (electronic)", ISSN-L = "0075-8434", bibdate = "Thu May 8 17:39:14 MDT 2014", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/lnm1960.bib", URL = "http://link.springer.com/chapter/10.1007/BFb0060250/", acknowledgement = ack-nhfb, book-DOI = "https://doi.org/10.1007/BFb0060247", book-URL = "http://www.springerlink.com/content/978-3-540-36246-3", fjournal = "Lecture Notes in Mathematics", journal-URL = "http://link.springer.com/bookseries/304", } @Article{Fenwick:1969:BMO, author = "P. M. Fenwick", title = "Binary Multiplication with Overlapped Addition Cycles", journal = j-IEEE-TRANS-COMPUT, volume = "C-18", number = "1", pages = "71--74", month = jan, year = "1969", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/T-C.1969.222527", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Wed Jul 13 16:55:17 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", URL = "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=1671120", acknowledgement = ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", } @Article{Ferrari:1969:CDM, author = "D. Ferrari", title = "Correction to {``A Division Method Using a Parallel Multiplier''}", journal = j-IEEE-TRANS-COMPUT, volume = "C-18", number = "10", pages = "960--960", month = oct, year = "1969", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/T-C.1969.222555", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Wed Jul 13 16:55:21 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", note = "See \cite{Ferrari:1967:DMU}.", URL = "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=1671148", acknowledgement = ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", } @InProceedings{Field:1969:OFP, author = "J. A. Field", title = "Optimizing floating-point arithmetic via post addition shift probabilities", crossref = "AFIPS:1969:ACPa", pages = "597--??", year = "1969", bibdate = "Tue Oct 09 07:54:02 2007", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, } @Article{Flores:1969:BRB, author = "Ivan Flores", title = "Book Review: {{\booktitle{Residue Arithmetic and Its Application to Computer Technology}} (Nicholas S. Szabo and Richard I. Tanaka)}", journal = j-SIAM-REVIEW, volume = "11", number = "1", pages = "103--104", month = "????", year = "1969", CODEN = "SIREAD", DOI = "https://doi.org/10.1137/1011027", 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/fparith.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", } @Book{Froberg:1969:INA, author = "Carl Erik Fr{\"o}berg", title = "Introduction to numerical analysis", publisher = pub-AW, address = pub-AW:adr, edition = "Second", pages = "xii + 433", year = "1969", LCCN = "QA297 .F6813 1969", bibdate = "Fri Aug 20 09:03:07 MDT 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; z3950.loc.gov:7090/Voyager", acknowledgement = ack-nhfb, remark = "Translation of Swedish original {\em L{\"a}robok i numerisk analys}. Cited in \cite{Sterbenz:1974:FPC}.", subject = "Numerical analysis", } @PhdThesis{Glaser:1969:HMN, author = "Anton Glaser", title = "History of Modern Numeration Systems", type = "{Educat.D.} thesis", school = "Temple University", address = "Philadelphia, PA, USA", pages = "261", year = "1969", bibdate = "Fri Mar 17 08:18:07 2023", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", note = "See subsequent books \cite{Glaser:1971:HBO,Glaser:1981:HBO}.", URL = "https://www.proquest.com/pqdtglobal/docview/302503306/A48758ED17C34A1APQ/1", acknowledgement = ack-nhfb, remark = "ProQuest dissertation number 7016666.", } @Article{Hammersley:1969:NAP, author = "P. Hammersley", title = "Note on {Algorithm 34}: Procedures for the Basic Arithmetical Operations in Multi-Length Working", journal = j-COMP-J, volume = "12", number = "1", pages = "102--103", month = feb, year = "1969", CODEN = "CMPJA6", ISSN = "0010-4620 (print), 1460-2067 (electronic)", ISSN-L = "0010-4620", bibdate = "Fri Sep 29 08:51:33 MDT 2000", bibsource = "http://www3.oup.co.uk/computer_journal/hdb/Volume_12/Issue_01/; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "http://www3.oup.co.uk/computer_journal/hdb/Volume_12/Issue_01/tiff/102.tif; http://www3.oup.co.uk/computer_journal/hdb/Volume_12/Issue_01/tiff/103.tif", acknowledgement = ack-nhfb, fjournal = "The Computer Journal", journal-URL = "http://comjnl.oxfordjournals.org/", } @Article{Holzwarth:1969:VBB, author = "A. Holzwarth", title = "{Ein Verfahren zur Bestimmung bester Tscheb\-y\-scheff- Ap\-prox\-i\-ma\-tion\-en der Quadratwurzelfunktion} \toenglish {A Method for Determination of Best Chebyshev Approximations to the Square Root Function} \endtoenglish", journal = j-COMPUTING, volume = "4", number = "2", pages = "168--177", year = "1969", CODEN = "CMPTA2", ISSN = "0010-485X (print), 1436-5057 (electronic)", ISSN-L = "0010-485X", 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/fparith.bib", acknowledgement = ack-nj, affiliation = "T{\"u}bingen, West Germany", classification = "C4130", description = "Chebyshev approximation; numerical analysis", fjournal = "Computing: Archiv f{\"u}r informatik und numerik", journal-URL = "http://link.springer.com/journal/607", language = "German", } @Article{Howell:1969:ASLa, author = "Jo Ann Howell and Robert T. Gregory", title = "An algorithm for solving linear algebraic equations using residue arithmetic. {I}", journal = j-BIT, volume = "9", pages = "200--224", year = "1969", CODEN = "BITTEL, NBITAB", ISSN = "0006-3835 (print), 1572-9125 (electronic)", ISSN-L = "0006-3835", MRclass = "65.35", MRnumber = "MR0261775 (41 \#6388a)", bibdate = "Thu Nov 8 14:50:15 2007", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, fjournal = "BIT (Nordisk tidskrift for informationsbehandling)", journal-URL = "http://link.springer.com/journal/10543", } @Article{Howell:1969:ASLb, author = "Jo Ann Howell and Robert T. Gregory", title = "An algorithm for solving linear algebraic equations using residue arithmetic. {II}", journal = j-BIT, volume = "9", pages = "324--337", year = "1969", CODEN = "BITTEL, NBITAB", ISSN = "0006-3835 (print), 1572-9125 (electronic)", ISSN-L = "0006-3835", MRclass = "65.35", MRnumber = "MR0261776 (41 \#6388b)", MRreviewer = "J. Legras", bibdate = "Thu Nov 8 14:50:18 2007", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, fjournal = "BIT (Nordisk tidskrift for informationsbehandling)", journal-URL = "http://link.springer.com/journal/10543", } @MastersThesis{Huey:1969:DFP, author = "Ben Milton Huey", title = "Design of a floating point processor for the {PDP-9} computer", type = "Thesis ({M.S. - Electrical Engineering})", school = "University of Arizona", address = "Tucson, AZ, USA", pages = "175", year = "1969", bibdate = "Sat Feb 24 15:01:45 MST 1996", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, keywords = "Computer storage devices.; Hybrid computers.", } @Article{Huttenhoff:1969:AUC, author = "J. H. Huttenhoff and R. R. Shively", title = "Arithmetic Unit of a Computing Element in a Global, Highly Parallel Computer", journal = j-IEEE-TRANS-COMPUT, volume = "C-18", number = "8", pages = "695--698", month = aug, year = "1969", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/T-C.1969.222751", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Wed Jul 13 16:55:20 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", URL = "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=1671344", acknowledgement = ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", } @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{Kirsch:1969:ACA, author = "Arnold Kirsch", title = "An analysis of commercial arithmetic", journal = j-EDUC-STUD-MATH, volume = "1", number = "3", pages = "300--311", month = jan, year = "1969", CODEN = "EDSMAN", DOI = "https://doi.org/10.1007/BF00558315", ISSN = "0013-1954 (print), 1573-0816 (electronic)", ISSN-L = "0013-1954", bibdate = "Sat Jan 14 17:12:11 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/educstudmath.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "http://link.springer.com/accesspage/article/10.1007/BF00558315", acknowledgement = ack-nhfb, fjournal = "Educational Studies in Mathematics", journal-URL = "http://link.springer.com/journal/10649", } @MastersThesis{Knight:1969:FPS, author = "Douglas Wayne Knight", title = "A floating point software package in {BCD} format for small, fixed word length, digital computers", type = "Thesis ({M.S.})", school = "Arizona State University, Electrical Engineering", address = "Tempe, AZ, USA", pages = "151", year = "1969", bibdate = "Sat Feb 24 15:01:45 MST 1996", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, } @Book{Knuth:1969:SA, author = "Donald E. Knuth", title = "Seminumerical Algorithms", volume = "2", publisher = pub-AW, address = pub-AW:adr, pages = "xi + 624", year = "1969", ISBN = "0-201-03802-1", ISBN-13 = "978-0-201-03802-6", LCCN = "QA76.5 .K57", MRclass = "68.00 (65.00)", MRnumber = "44 \#3531", MRreviewer = "M. Muller", bibdate = "Wed Dec 15 15:47:38 1993", bibsource = "https://www.math.utah.edu/pub/tex/bib/benfords-law.bib; https://www.math.utah.edu/pub/tex/bib/cryptography.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/master.bib", price = "US\$19.75", series = "The Art of Computer Programming", acknowledgement = ack-nhfb, remark = "Benford's Law is discussed on pp. 219--229. Cited in \cite{Sterbenz:1974:FPC}.", tableofcontents = "3: Random Numbers \\ 3.1. Introduction / 1 \\ 3.2. Generating Uniform Random Numbers / 9 \\ 3.2.1. The Linear Congruential Method / 9 \\ 3.2.1.1. Choice of modulus / 11 \\ 3.2.1.2. Choice of multiplier / 15 \\ 3.2.1.3. Potency / 21 \\ 3.2.2. Other Methods / 25 \\ 3.3. Statistical Tests / 34 \\ 3.3.1. General Test Procedures for Studying Random Data / 35 \\ 3.3.2. Empirical Tests / 54 \\ *3.3.3. Theoretical Tests / 69 \\ 3.3.4. The Spectral Test / 82 \\ 3.4. Other Types of Random Quantities / 100 \\ 3.4.1. Numerical Distributions / 101 \\ 3.4.2. Random Sampling and Shuffling / 121 \\ *3.5. What is a Random Sequence? / 127 \\ 3.6. Summary / 155 \\ 4: Arithmetic \\ 4.1. Positional Number Systems / 162 \\ 4.2. Floating-Point Arithmetic / 180 \\ 4.2.1. Single-Precision Calculations / 180 \\ 4.2.2. Accuracy of Floating-Point Arithmetic / 195 \\ *4.2.3. Double-Precision Calculations / 210 \\ 4.2.4. Statistical Distribution / 218 \\ 4.3. Multiple-Precision Arithmetic / 229 \\ 4.3.1. The Classical Algorithms / 229 \\ *4.3.2. Modular Arithmetic / 248 \\ *4.3.3. How Fast Can We Multiply? / 258 \\ 4.4. Radix Conversion / 280 \\ 4.5. Rational Arithmetic / 290 \\ 4.5.1. Fractions / 290 \\ 4.5.2. The Greatest Common Divisor / 293 \\ *4.5.3. Analysis of Euclid's Algorithm / 316 \\ 4.5.4. Factoring into Primes / 339 \\ 4.6. Polynomial Arithmetic / 360 \\ 4.6.1. Division of Polynomials / 363 \\ *4.6.2. Factorization of Polynomials / 381 \\ 4.6.3. Evaluation of Powers / 398 \\ 4.6.4. Evaluation of Polynomials / 422 \\ *4.7. Manipulation of Power Series / 444 \\ Answers to Exercises / 452 \\ Appendix A: MIX / 565 \\ 1. Description of MIX / 565 \\ 2. The MIX Assembly Language / 584 \\ Appendix B: Tables of Numerical Quantities / 596 \\ 1. Fundamental Constants (decimal) / 596 \\ 2. Fundamental Constants (octal) / 597 \\ 3. Harmonic Numbers, Bernoulli Numbers, Fibonacci Numbers / 598 \\ Appendix C: Index to Notations / 600 \\ Index and Glossary / 605", xxyear = "{\noopsort{1968c}}1969", } @Book{Knuth:ACP69-2, author = "Donald E. Knuth", title = "Seminumerical Algorithms", volume = "2", publisher = pub-AW, address = pub-AW:adr, pages = "xi + 624", year = "1969", ISBN = "0-201-03802-1", ISBN-13 = "978-0-201-03802-6", LCCN = "QA76.5 .K57", MRclass = "68.00 (65.00)", MRnumber = "44 \#3531", MRreviewer = "M. Muller", bibdate = "Wed Dec 15 15:47:38 1993", bibsource = "https://www.math.utah.edu/pub/tex/bib/benfords-law.bib; https://www.math.utah.edu/pub/tex/bib/cryptography.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/master.bib", price = "US\$19.75", series = "The Art of Computer Programming", acknowledgement = ack-nhfb, remark = "Benford's Law is discussed on pp. 219--229.", tableofcontents = "3: Random Numbers \\ 3.1. Introduction / 1 \\ 3.2. Generating Uniform Random Numbers / 9 \\ 3.2.1. The Linear Congruential Method / 9 \\ 3.2.1.1. Choice of modulus / 11 \\ 3.2.1.2. Choice of multiplier / 15 \\ 3.2.1.3. Potency / 21 \\ 3.2.2. Other Methods / 25 \\ 3.3. Statistical Tests / 34 \\ 3.3.1. General Test Procedures for Studying Random Data / 35 \\ 3.3.2. Empirical Tests / 54 \\ *3.3.3. Theoretical Tests / 69 \\ 3.3.4. The Spectral Test / 82 \\ 3.4. Other Types of Random Quantities / 100 \\ 3.4.1. Numerical Distributions / 101 \\ 3.4.2. Random Sampling and Shuffling / 121 \\ *3.5. What is a Random Sequence? / 127 \\ 3.6. Summary / 155 \\ 4: Arithmetic \\ 4.1. Positional Number Systems / 162 \\ 4.2. Floating-Point Arithmetic / 180 \\ 4.2.1. Single-Precision Calculations / 180 \\ 4.2.2. Accuracy of Floating-Point Arithmetic / 195 \\ *4.2.3. Double-Precision Calculations / 210 \\ 4.2.4. Statistical Distribution / 218 \\ 4.3. Multiple-Precision Arithmetic / 229 \\ 4.3.1. The Classical Algorithms / 229 \\ *4.3.2. Modular Arithmetic / 248 \\ *4.3.3. How Fast Can We Multiply? / 258 \\ 4.4. Radix Conversion / 280 \\ 4.5. Rational Arithmetic / 290 \\ 4.5.1. Fractions / 290 \\ 4.5.2. The Greatest Common Divisor / 293 \\ *4.5.3. Analysis of Euclid's Algorithm / 316 \\ 4.5.4. Factoring into Primes / 339 \\ 4.6. Polynomial Arithmetic / 360 \\ 4.6.1. Division of Polynomials / 363 \\ *4.6.2. Factorization of Polynomials / 381 \\ 4.6.3. Evaluation of Powers / 398 \\ 4.6.4. Evaluation of Polynomials / 422 \\ *4.7. Manipulation of Power Series / 444 \\ Answers to Exercises / 452 \\ Appendix A: MIX / 565 \\ 1. Description of MIX / 565 \\ 2. The MIX Assembly Language / 584 \\ Appendix B: Tables of Numerical Quantities / 596 \\ 1. Fundamental Constants (decimal) / 596 \\ 2. Fundamental Constants (octal) / 597 \\ 3. Harmonic Numbers, Bernoulli Numbers, Fibonacci Numbers / 598 \\ Appendix C: Index to Notations / 600 \\ Index and Glossary / 605", xxyear = "{\noopsort{1968c}}1969", } @Article{Linhardt:1969:DDT, author = "R. J. Linhardt and H. S. Miller", title = "Digit-by-Digit Transcendental-Function Computation", journal = j-RCA-REV, volume = "30", number = "2", pages = "209--247", month = jun, year = "1969", CODEN = "RCARCI", ISSN = "0033-6831", bibdate = "Fri Sep 16 10:31:06 1994", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nj, fjournal = "RCA review", } @Article{Liu:1969:EAD, author = "B. Liu and T. Kaneko", title = "Error analysis of digital filters realized with floating-point arithmetic", journal = j-PROC-IEEE, volume = "57", number = "10", pages = "1735--1747", month = oct, year = "1969", CODEN = "IEEPAD", ISSN = "0018-9219 (print), 1558-2256 (electronic)", ISSN-L = "0018-9219", bibdate = "Sat Jul 16 08:40:52 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", note = "See correction \cite{Liu:1970:CEA}.", acknowledgement = ack-nhfb, fjournal = "Proceedings of the IEEE", journal-URL = "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=5", summary = "This paper calculates the error at the output of a digital filter using floating-point arithmetic operations due to roundoff accumulation and input quantization. Expressions are derived for the mean square error for each of the three canonical forms \ldots{}", } @InProceedings{Matula:1969:TAM, author = "David W. Matula", title = "Towards an Abstract Mathematical Theory of Floating-Point Arithmetic", crossref = "AFIPS:1969:ACPa", pages = "765--772", year = "1969", bibdate = "Fri Oct 19 23:01:32 2007", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, } @Article{Posnov:1969:FPR, author = "N. N. Posnov and M. K. Buza and V. K. Kravcov", title = "The floating point in a residue class number system. ({Russian})", journal = "Vestnik Beloruss. Gos. Univ. Ser. I", volume = "3", pages = "21--27", year = "1969", MRclass = "10A10", MRnumber = "45 3303e", bibdate = "Fri Dec 8 08:24:14 1995", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, language = "Russian", reviewer = "J. B. Roberts", } @Book{Rice:1969:AFV, author = "John R. Rice", title = "The Approximation of Functions", volume = "2", publisher = pub-AW, address = pub-AW:adr, pages = "various", year = "1969", LCCN = "QA221 .R5 V.1-2", bibdate = "Fri Dec 08 13:02:54 1995", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nj, } @TechReport{Rigby:1969:DFP, author = "G. W. Rigby", title = "To draw a floating point number on the graph plotter({PRP 2})", type = "Group Research report", institution = "British Steel Corporation", address = "London, UK", year = "1969", bibdate = "Thu May 09 08:32:48 2002", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, } @Article{Rosen:1969:ECH, author = "Saul Rosen", title = "Electronic Computers: a Historical Survey", journal = j-COMP-SURV, volume = "1", number = "1", pages = "7--36", month = mar, year = "1969", CODEN = "CMSVAN", DOI = "https://doi.org/10.1145/356540.356543", ISSN = "0010-4892", ISSN-L = "0360-0300", bibdate = "Mon Sep 19 18:18:44 1994", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", abstract = "The first large scale electronic computers were built in connection with university projects sponsored by government military and research organizations. Many established companies, as well as new companies, entered the computer field during the first generation, 1947-1959, in which the vacuum tube was almost universally used as the active component in the implementation of computer logic. The second generation was characterized by the transistorized computers that began to appear in 1959. Some of the computers built then and since are considered super computers; they attempt to go to the limit of current technology in terms of size, speed, and logical complexity. From 1965 onward, most new computers belong to a third generation, which features integrated circuit technology and multiprocessor multiprogramming systems.", acknowledgement = ack-nhfb, fjournal = "ACM Computing Surveys", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J204", keywords = "computer history; electronic computers; magnetic drum computers; super computers; time-sharing; transistorized computers; university computer projects; vacuum tube computers", } @Article{S:1969:BRQ, author = "D. S.", title = "Book Review: {{\em Square Roots of Integers $2$ to $ 15 $ in Various Bases $2$ to $ 10 $: $ 88062 $ Binary Digits or Equivalent} by W. A. Beyer, N. Metropolis, and J. R. Neergaard}", journal = j-MATH-COMPUT, volume = "23", number = "107", pages = "679--679", month = jul, year = "1969", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Mar 20 13:34:21 2007", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/m/metropolis-nicholas.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "http://links.jstor.org/sici?sici=0025-5718%28196907%2923%3A107%3C679%3ASROI2T%3E2.0.CO%3B2-C", acknowledgement = ack-nhfb, ajournal = "Math. Comput.", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Shea:1969:NDN, author = "Dale D. Shea", title = "On the Number of Divisions Needed in Finding the Greatest Common Divisor", journal = j-FIB-QUART, volume = "7", number = "4", pages = "337--340", month = nov, year = "1969", CODEN = "FIBQAU", ISSN = "0015-0517", ISSN-L = "0015-0517", bibdate = "Thu Oct 20 18:05:19 MDT 2011", bibsource = "http://www.fq.math.ca/7-4.html; https://www.math.utah.edu/pub/tex/bib/fibquart.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "http://www.fq.math.ca/Scanned/7-4/shea.pdf", acknowledgement = ack-nhfb, ajournal = "Fib. Quart", fjournal = "The Fibonacci Quarterly. Official Organ of the Fibonacci Association", journal-URL = "http://www.fq.math.ca/", } @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", 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/fparith.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp1960.bib; JSTOR database", acknowledgement = ack-nj # " and " # ack-nhfb, ajournal = "Math. Comput.", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Svoboda:1969:DAS, author = "A. Svoboda", title = "Decimal Adder with Signed Digit Arithmetic", journal = j-IEEE-TRANS-COMPUT, volume = "C-18", number = "3", pages = "212--215", month = mar, year = "1969", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/T-C.1969.222633", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Fri Jun 24 20:44:42 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", keywords = "decimal floating-point arithmetic", } @Article{Troelstra:1969:EA, author = "A. S. Troelstra", title = "Elementary arithmetic", journal = j-LECT-NOTES-MATH, volume = "95", pages = "12--13", year = "1969", CODEN = "LNMAA2", DOI = "https://doi.org/10.1007/BFb0080646", ISBN = "3-540-04614-3 (print), 3-540-36130-8 (e-book)", ISBN-13 = "978-3-540-04614-1 (print), 978-3-540-36130-5 (e-book)", ISSN = "0075-8434 (print), 1617-9692 (electronic)", ISSN-L = "0075-8434", bibdate = "Thu May 8 17:39:14 MDT 2014", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/lnm1960.bib", URL = "http://link.springer.com/chapter/10.1007/BFb0080646", acknowledgement = ack-nhfb, book-DOI = "https://doi.org/10.1007/BFb0080643", book-URL = "http://www.springerlink.com/content/978-3-540-36130-5", fjournal = "Lecture Notes in Mathematics", journal-URL = "http://link.springer.com/bookseries/304", } @Article{Turner:1969:CSI, author = "L. Richard Turner", title = "Comment on Some {IBM} Software", journal = "{SHARE Secretary Distribution}", volume = "SSD 199", number = "C5279", pages = "40--43", year = "1969", bibdate = "Fri Aug 20 10:23:55 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "", acknowledgement = ack-nhfb, remark = "Cited in \cite{Sterbenz:1974:FPC}.", } @Article{Turner:1969:DSC, author = "L. Richard Turner", title = "Difficulty in {\tt sin\slash cos} Routine", journal = j-SIGNUM, volume = "4", number = "3", pages = "13--13", month = oct, 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/fparith.bib", acknowledgement = ack-nhfb, fjournal = "ACM SIGNUM Newsletter", journal-URL = "https://dl.acm.org/loi/signum", remark = "Cited in \cite{Sterbenz:1974:FPC}.", } @TechReport{Turner:1969:IOC, author = "L. R. Turner", title = "Input-Output Conversion in {System\slash 360}", number = "SSD 194, C5173", institution = "SHARE Secretary Distribution", address = "????", pages = "1--8", year = "1969", bibdate = "Thu Feb 15 15:21:42 2007", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, } @Article{Usow:1969:SSC, author = "Karl H. Usow", title = "{SIGNUM} subroutine certification committee", journal = j-SIGNUM, volume = "4", number = "3", pages = "15--18", month = oct, year = "1969", CODEN = "SNEWD6", ISSN = "0163-5778 (print), 1558-0237 (electronic)", ISSN-L = "0163-5778", bibdate = "Mon Mar 5 17:26:29 MST 2007", bibsource = "http://portal.acm.org/; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, fjournal = "ACM SIGNUM Newsletter", journal-URL = "https://dl.acm.org/loi/signum", remark = "Cited in \cite{Sterbenz:1974:FPC}.", } @TechReport{Veltkamp:1969:APV, author = "G. W. Veltkamp", title = "{ALGOL} procedures voor het rekenen in dubbele lengte. ({Dutch}) [{ALGOL} procedures for double-length arithmetic]", type = "Technical report", number = "21", institution = "RC-Informatie, Technische Hogeschool Eindhoven", address = "Eindhoven, The Netherlands", year = "1969", bibdate = "Tue Dec 26 14:53:03 2023", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, language = "Dutch", } @Article{Weinstein:1969:CCR, author = "C. Weinstein and A. V. Oppenheim", title = "Correction to {``A comparison of roundoff noise in floating point and fixed point digital filter realizations''}", journal = j-PROC-IEEE, volume = "57", number = "8", pages = "1466--1466", month = aug, year = "1969", CODEN = "IEEPAD", ISSN = "0018-9219 (print), 1558-2256 (electronic)", ISSN-L = "0018-9219", bibdate = "Sat Jul 16 08:40:52 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", note = "See \cite{Weinstein:1969:CRN}.", acknowledgement = ack-nhfb, fjournal = "Proceedings of the IEEE", journal-URL = "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=5", } @Article{Weinstein:1969:CRN, author = "C. Weinstein and A. V. Oppenheim", title = "A comparison of roundoff noise in floating point and fixed point digital filter realizations", journal = j-PROC-IEEE, volume = "57", number = "6", pages = "1181--1183", month = jun, year = "1969", CODEN = "IEEPAD", ISSN = "0018-9219 (print), 1558-2256 (electronic)", ISSN-L = "0018-9219", bibdate = "Sat Jul 16 08:40:52 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", note = "See correction \cite{Weinstein:1969:CCR}.", acknowledgement = ack-nhfb, fjournal = "Proceedings of the IEEE", journal-URL = "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=5", summary = "A statistical model for roundoff noise in floating point digital filters, proposed by Kaneko and Liu, is tested experimentally for first- and second-order digital filters. Good agreement between theory and experiment is obtained. The model is used to \ldots{}", } @Article{Weinstein:1969:RNF, author = "C. Weinstein", title = "Roundoff noise in floating point fast {Fourier} transform computation", journal = j-IEEE-TRANS-AUDIO-ELECTROACOUST, volume = "17", number = "3", pages = "209--215", month = sep, year = "1969", CODEN = "ITADAS", ISSN = "0018-9278 (print), 1558-2582 (electronic)", ISSN-L = "0018-9278", bibdate = "Sat Jul 16 08:40:53 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, fjournal = "IEEE Transactions on Audio and Electroacoustics", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=8337", summary = "A statistical model for roundoff errors is used to predict output noise-to-signal ratio when a fast Fourier transform is computed using floating point arithmetic. The result, derived for the case of white input signal, is that the ratio of mean-square \ldots{}", } @Article{Whipple:1969:CHR, author = "W. L. Whipple", title = "Comments on {``Higher-Radix Division Using Estimates of the Divisor and Partial Remainders''}", journal = j-IEEE-TRANS-COMPUT, volume = "C-18", number = "2", pages = "183--183", month = feb, year = "1969", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/T-C.1969.222623", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Wed Jul 13 16:55:18 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", note = "See \cite{Atkins:1968:HRD}.", URL = "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=1671216", acknowledgement = ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", } @InProceedings{Young:1969:SCN, author = "David M. Young and Alvis E. McDonald", title = "On the surveillance and control of number range and accuracy in numerical computation (with discussion)", crossref = "Morrell:1970:IPP", pages = "145--152", year = "1969", MRclass = "65.80", MRnumber = "40\#8303", bibdate = "Thu Jan 18 13:48:21 1996", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/y/young-david-m.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, remark = "Cited in \cite{Sterbenz:1974:FPC}, but given year 1968.", } @Article{Atkins:1970:DAU, author = "D. E. Atkins", title = "Design of the Arithmetic Units of {ILLIAC III}: Use of Redundancy and Higher Radix Methods", journal = j-IEEE-TRANS-COMPUT, volume = "C-19", number = "8", pages = "720--723", month = aug, year = "1970", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/T-C.1970.223022", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Tue Oct 09 19:03:30 2007", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "http://www.acsel-lab.com/arithmetic/arith1/papers/ARITH1_Atkins.pdf", abstract = "In keeping with the experimental nature of the Illinois Pattern Recognition Computer (ILLIAC III), the arithmetic units are intended to be a practical testing ground for recent theoretical work in computer arithmetic. This paper describes the use of redundant number systems and the design of a structure with which multiplication and division are executed radix 256. The heart of the unit is the stored-sign subtracter, a recently discovered member of the family of borrow-save subtracters and carry-save adders. A cascade of these subtracters, controlled by a multiplier recoder, provides multiplication. The same structure, controlled by a ``model division'' (a quotient recoder), performs division.", acknowledgement = ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", keywords = "arithmetic unit; computer arithmetic; division; higher radix arithmetic; ILLIAC III; multiplication; redundant number systems; signed-digit subtracter; stored-sign subtracter; stored-sign subtracter. Arithmetic unit", remark = "Despite statements elsewhere that the ILLIAC III used base 256, that is incorrect. The original manuals for the machine clearly show an IBM System/360-style base-16 encoding. This paper reveals the source of the confusion: the multiplier uses a Wallace recoding scheme, first defined for radix 4, but extended for the ILLIAC III to radix 256. Thus, multiplication is done 8 bits at a time. Division is done 4 bits at a time.", } @Article{Avizienis:1970:UAB, author = "Algirdas Avi{\v{z}}ienis and Chin Tung", title = "A Universal Arithmetic Building Element ({ABE}) and Design Methods for Arithmetic Processors", journal = j-IEEE-TRANS-COMPUT, volume = "C-19", number = "8", pages = "733--745", month = aug, year = "1970", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/T-C.1970.223023", 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/fparith.bib", URL = "http://www.acsel-lab.com/arithmetic/arith1/papers/ARITH1_Avizienis.pdf", abstract = "The advent of large-scale integration of logic circuits requires the definition of digital computer structure in terms of large functional arrays of logic of very few types. This paper describes a single-package arithmetic processor called the arithmetic building element (ABE). The ABE accepts operands in either conventional or signed-digit radix-r representation and produces signed-digit results, which the ABE can reconvert to conventional form. Radix 16 is chosen for illustrations. Arrays of ABE's may be arranged to implement unit-time parallel addition, all-combinational multiplication, and more complex functions which are presently computed by subroutines. To facilitate such arithmetic design, a graph model is developed which permits a translation of the given arithmetical algorithm into an interconnection diagram of ABE's. The design procedure is illustrated by an array for polynomial evaluation. Speed, cost, and roundoff error of the array are considered. A computer program has been written for the automatic translation of the algorithm graph to an interconnection graph, and for the evaluation of the cost and speed for a given polynomial degree and a given precision requirement.", acknowledgement = ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", keywords = "ARITH-1", } @Article{Behringer:1970:BFI, author = "F. A. Behringer", title = "{Eine bedingte Form der Integral-Ungleichun von Gronwal, leichte Verallgemeinerung eins Stabilit{\"a}tsatzes von Perron und {\"u}berlauffreie Skalierung von Analogrechenschaltungen}. ({German}) [{A} conditional version of the integral inequality of {Gronwall}, a slight generalization of a stability theorem of {Perron}, and overflow-free scaling of analogue computer set-ups]", journal = j-COMPUTING, volume = "5", number = "4", pages = "333--348", year = "1970", 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/fparith.bib; INSPEC Axiom database (1968--date)", acknowledgement = ack-ec # " and " # ack-nhfb, affiliation = "Tech. Hochschule, M{\"u}nchen, West Germany", classification = "C5460", description = "analogue computer methods; differential equations", fjournal = "Computing: Archiv f{\"u}r informatik und numerik", journal-URL = "http://link.springer.com/journal/607", language = "German", } @Article{Beyer:1970:GST, author = "W. A. Beyer and N. Metropolis and J. R. Neergaard", title = "The Generalized Serial Test Applied to Expansions of Some Irrational Square Roots in Various Bases", journal = j-MATH-COMPUT, volume = "24", number = "111", pages = "745--747", month = jul, year = "1970", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "65.15", MRnumber = "MR0273773 (42 \#8649)", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; JSTOR database", URL = "http://links.jstor.org/sici?sici=0025-5718%28197007%2924%3A111%3C745%3ATGSTAT%3E2.0.CO%3B2-2", ZMnumber = "0227.10044", acknowledgement = ack-nhfb, ajournal = "Math. Comput.", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Beyer:1970:SSD, author = "W. A. Beyer and N. Metropolis and J. R. Neergaard", title = "Statistical Study of Digits of Some Square Roots of Integers in Various Bases", journal = j-MATH-COMPUT, volume = "24", number = "110", pages = "455--473", month = apr, year = "1970", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "62.70", MRnumber = "MR0272129 (42 \#7010)", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; JSTOR database", URL = "http://links.jstor.org/sici?sici=0025-5718%28197004%2924%3A110%3C455%3ASSODOS%3E2.0.CO%3B2-I", ZMnumber = "0208.19505", acknowledgement = ack-nhfb, ajournal = "Math. Comput.", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Brent:1970:ABN, author = "R. Brent", title = "On the Addition of Binary Numbers", journal = j-IEEE-TRANS-COMPUT, volume = "C-19", number = "8", pages = "758--759", month = aug, year = "1970", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/T-C.1970.223027", 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/fparith.bib", URL = "http://www.acsel-lab.com/arithmetic/arith1/papers/ARITH1_Brent.pdf", abstract = "An upper bound is derived for the time required to add numbers modulo $ 2^n $, using circuit elements with a limited fan-in and unit delay, and assuming that all numbers have the usual binary encoding. The upper bound is within a factor $ (1 + \epsilon) $ of Winograd's lower bound (which holds for all encodings), where $ \epsilon \rightarrow 0 $ as $ n \rightarrow \infty $, and only $ O(n \log n) $ circuit elements are required.", acknowledgement = ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", keywords = "addition; ARITH-1; binary numbers; computational complexity; group multiplication; logic circuits; logical design", } @TechReport{deLugish: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 = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nj, } @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/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 = "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", keywords = "ARITH-1", } @Article{Forsythe:1970:PCW, author = "George E. Forsythe", title = "Pitfalls in Computation, or Why a Math Book Isn't Enough", journal = j-AMER-MATH-MONTHLY, volume = "77", number = "9", pages = "931--956", month = nov, year = "1970", CODEN = "AMMYAE", ISSN = "0002-9890 (print), 1930-0972 (electronic)", ISSN-L = "0002-9890", MRclass = "Contributed Item", MRnumber = "MR1536090", bibdate = "Tue Nov 22 09:08:17 MST 2005", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/f/forsythe-george-elmer.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/fparith.bib; JSTOR database", ZMnumber = "0207.15401", acknowledgement = ack-nhfb, classmath = "*65G50 Roundoff error 65Y20 Complexity and performance of numerical algorithms", fjournal = "The American Mathematical Monthly", journal-URL = "https://www.jstor.org/journals/00029890.htm", } @TechReport{Gardiner:1970:SDA, author = "V. Gardiner and N. Metropolis", title = "Significant Digit Arithmetic on a {CDC 6600}", type = "Technical Report", number = "LA--4470", institution = inst-LASL, address = inst-LASL:adr, pages = "4", day = "1", month = jan, year = "1970", bibdate = "Tue Mar 20 11:46:17 2007", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/m/metropolis-nicholas.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "http://www.osti.gov/energycitations/product.biblio.jsp?osti_id=4093265&query_id=0", acknowledgement = ack-nhfb, } @Article{Good:1970:CIA, author = "Donald I. Good and Ralph L. London", title = "Computer Interval Arithmetic: Definition and Proof of Correct Implementation", journal = j-J-ACM, volume = "17", number = "4", pages = "603--612", month = oct, year = "1970", CODEN = "JACOAH", ISSN = "0004-5411 (print), 1557-735X (electronic)", ISSN-L = "0004-5411", bibdate = "Sun Nov 6 11:55:54 1994", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, fjournal = "Journal of the ACM", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J401", } @Article{Habibi:1970:FM, author = "A. Habibi and P. A. Wintz", title = "Fast Multipliers", journal = j-IEEE-TRANS-COMPUT, volume = "C-19", number = "2", pages = "153--157", month = feb, year = "1970", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/T-C.1970.222881", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Wed Jul 13 08:01:30 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput1970.bib", URL = "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=1671474", acknowledgement = ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", } @Article{Hall:1970:GPQ, author = "E. L. Hall and D. D. Lynch and S. J. Dwyer", title = "Generation of Products and Quotients Using Approximate Binary Logarithms for Digital Filtering Applications", journal = j-IEEE-TRANS-COMPUT, volume = "C-19", number = "2", pages = "97--105", month = feb, year = "1970", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/T-C.1970.222874", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Wed Jul 13 08:01:30 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput1970.bib", URL = "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=1671467", acknowledgement = ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", } @Article{Hamming:1970:DN, author = "R. W. Hamming", title = "On the Distribution of Numbers", journal = j-BELL-SYST-TECH-J, volume = "49", number = "8", pages = "1609--1625", month = oct, year = "1970", CODEN = "BSTJAN", DOI = "https://doi.org/10.1002/j.1538-7305.1970.tb04281.x", ISSN = "0005-8580 (print), 2376-7154 (electronic)", ISSN-L = "0005-8580", bibdate = "Tue Nov 9 11:15:55 MST 2010", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; http://bstj.bell-labs.com/oldfiles/year.1970/BSTJ.1970.4908.html; http://www.alcatel-lucent.com/bstj/vol49-1970/bstj-vol49-issue08.html; https://www.math.utah.edu/pub/bibnet/authors/h/hamming-richard-w.bib; https://www.math.utah.edu/pub/tex/bib/bstj1970.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "http://bstj.bell-labs.com/BSTJ/images/Vol49/bstj49-8-1609.pdf; http://www.alcatel-lucent.com/bstj/vol49-1970/articles/bstj49-8-1609.pdf", abstract = "This paper examines the distribution of the mantissas of floating-point numbers and shows how the arithmetic operations of a computer transform various distributions toward the limiting distribution $$ r(x) = 1 / (x \ln b) \qquad (1 / b \leq x \leq 1) $$ (where $b$ is the base of the number system). The paper also gives a number of applications to hardware, software, and general computing which show that this distribution is not merely an amusing curiosity. A brief examination of the distribution of exponents is included.", acknowledgement = ack-nj # " and " # ack-nhfb, fjournal = "The Bell System Technical Journal", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)1538-7305/issues/", } @Article{Hansen:1970:APN, author = "Rodney T. Hansen", title = "Arithmetic of Pentagonal Numbers", journal = j-FIB-QUART, volume = "8", number = "1", pages = "83--86", month = feb, year = "1970", CODEN = "FIBQAU", ISSN = "0015-0517", ISSN-L = "0015-0517", bibdate = "Thu Oct 20 18:05:23 MDT 2011", bibsource = "http://www.fq.math.ca/8-1.html; https://www.math.utah.edu/pub/tex/bib/fibquart.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "http://www.fq.math.ca/Scanned/8-1/hansen.pdf", acknowledgement = ack-nhfb, ajournal = "Fib. Quart", fjournal = "The Fibonacci Quarterly. Official Organ of the Fibonacci Association", journal-URL = "http://www.fq.math.ca/", } @Article{Harris:1970:NND, author = "V. C. Harris", title = "Note on the Number of Divisions Required in Finding the Greatest Common Divisor", journal = j-FIB-QUART, volume = "8", number = "1", pages = "104--??", month = feb, year = "1970", CODEN = "FIBQAU", ISSN = "0015-0517", ISSN-L = "0015-0517", bibdate = "Thu Oct 20 18:05:23 MDT 2011", bibsource = "http://www.fq.math.ca/8-1.html; https://www.math.utah.edu/pub/tex/bib/fibquart.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "http://www.fq.math.ca/Scanned/8-1/harris2.pdf", acknowledgement = ack-nhfb, ajournal = "Fib. Quart", fjournal = "The Fibonacci Quarterly. Official Organ of the Fibonacci Association", journal-URL = "http://www.fq.math.ca/", } @MastersThesis{Harvey:1970:SSP, author = "Michael P. Harvey", title = "A system study and problem solution in polynomial evaluation", type = "Thesis ({M.S.})", school = "California State College", address = "Long Beach, CA, USA", pages = "vi + 119", year = "1970", bibdate = "Sat Feb 24 15:01:45 MST 1996", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, keywords = "Electronic digital computers.; Numerical calculations.; Polynomials.", remark = "Title on spine: Automated floating point polynomial evaluation.", } @TechReport{Hillstrom:1970:PSF, author = "K. E. Hillstrom", title = "Performance Statistics for the {FORTRAN IV (H)} and {PL/I} (Version 5) Libraries in the {IBM OS\slash 360 Release 18}", type = "Report", number = "ANL-7666", institution = inst-ANL, address = inst-ANL:adr, year = "1970", bibdate = "Fri Aug 20 09:29:37 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, remark = "Cited in \cite{Sterbenz:1974:FPC}.", } @Article{Hornbuckle:1970:LMA, author = "G. D. Hornbuckle and E. I. Ancona", title = "The {LX-1} Microprocessor and Its Application to Real-Time Signal Processing", journal = j-IEEE-TRANS-COMPUT, volume = "C-19", number = "8", pages = "710--720", month = aug, year = "1970", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/T-C.1970.223021", 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/fparith.bib", URL = "http://www.acsel-lab.com/arithmetic/arith1/papers/ARITH1_Hornbuckle.pdf", acknowledgement = ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", keywords = "ARITH-1", } @Article{Howell:1970:SLE, author = "Jo Ann Howell and Robert T. Gregory", title = "Solving linear equations using residue arithmetic --- {Algorithm II}", journal = j-BIT, volume = "10", pages = "23--37", year = "1970", CODEN = "BITTEL, NBITAB", ISSN = "0006-3835 (print), 1572-9125 (electronic)", ISSN-L = "0006-3835", MRclass = "65.35", MRnumber = "MR0261777 (41 \#6389)", MRreviewer = "J. Legras", bibdate = "Thu Nov 8 14:50:19 2007", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, fjournal = "BIT (Nordisk tidskrift for informationsbehandling)", journal-URL = "http://link.springer.com/journal/10543", } @Article{Kailas:1970:AMC, author = "M. V. Kailas", title = "Another method of converting from hexadecimal to decimal", journal = j-CACM, volume = "13", number = "3", pages = "193--193", month = mar, 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/fparith.bib", abstract = "There is a simple paper-and-pencil method of converting a hexadecimal number N to decimal.", acknowledgement = ack-nhfb, classcodes = "C5230 (Digital arithmetic methods)", fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "digital arithmetic", } @Article{Knuth:1970:VNF, author = "Donald E. Knuth", title = "{Von Neumann}'s First Computer Program", journal = j-COMP-SURV, volume = "2", number = "4", pages = "247--260", month = dec, year = "1970", CODEN = "CMSVAN", DOI = "https://doi.org/10.1145/356580.356581", ISSN = "0010-4892", ISSN-L = "0360-0300", bibdate = "Sat Mar 23 15:03:00 2002", bibsource = "ftp://ftp.math.utah.edu/pub/mirrors/ftp.ira.uka.de/bibliography/Misc/compsurv.bib; ftp://ftp.math.utah.edu/pub/mirrors/ftp.ira.uka.de/bibliography/Misc/HBP/ACMCS.bib; https://www.math.utah.edu/pub/tex/bib/compsurv.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", note = "Reprinted in \cite{Aspray:1987:PJN}.", acknowledgement = ack-nhfb, fjournal = "ACM Computing Surveys", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J204", } @Article{Krishnamurthy:1970:OIS, author = "E. V. Krishnamurthy", title = "On Optimal Iterative Schemes For High-Speed Division", journal = j-IEEE-TRANS-COMPUT, volume = "C-19", number = "3", pages = "227--231", month = mar, year = "1970", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/T-C.1970.222901", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Tue Jan 08 23:09:36 2008", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", } @Article{Krishnamurthy:1970:RTT, author = "E. V. Krishnamurthy", title = "On Range-Transformation Techniques for Division", journal = j-IEEE-TRANS-COMPUT, volume = "C-19", number = "2", pages = "157--160", month = feb, year = "1970", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/T-C.1970.222882", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Wed Jul 13 08:01:30 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput1970.bib", URL = "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=1671475", acknowledgement = ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", } @Article{Ling:1970:HSC, author = "H. Ling", title = "High-Speed Computer Multiplication Using a Multiple-Bit Decoding Algorithm", journal = j-IEEE-TRANS-COMPUT, volume = "C-19", number = "8", pages = "706--709", month = aug, year = "1970", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/T-C.1970.223020", 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/fparith.bib", URL = "http://www.acsel-lab.com/arithmetic/arith1/papers/ARITH1_Ling.pdf", abstract = "This paper presents a method of performing the binary multiplication beyond the scheme of multiple ADD and SHIFT. The binary multiplication algorithm will be discussed first, followed by block decoding method, logic implementation, hardware consideration, and two examples which are at the end of the discussion.", acknowledgement = ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", keywords = "ARITH-1", } @Article{Linz:1970:AFP, author = "Peter Linz", title = "Accurate Floating-Point Summation", journal = j-CACM, volume = "13", number = "6", pages = "361--362", month = jun, year = "1970", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", MRclass = "65.25", MRnumber = "44\#3468", MRreviewer = "R. P. Brent", bibdate = "Tue Mar 25 13:26:09 MST 1997", bibsource = "Compendex database; ftp://ftp.ira.uka.de/pub/bibliography/Distributed/QLD/1970.bib; ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; https://www.math.utah.edu/pub/tex/bib/fparith.bib", abstract = "This paper describes an alternate method for summing a set of floating-point numbers. Comparison of the error bound for this method with that of the standard summation method shows that it is considerably less sensitive to propagation of round-off error.", acknowledgement = ack-nhfb # "\slash " # ack-nj, classcodes = "C5230 (Digital arithmetic methods)", corpsource = "New York University, NY, USA", descriptors = "Simulation; statistics; numeric calculation", fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", journalabr = "Commun ACM", keywords = "accurate floating-point summation; computers, data processing; digital arithmetic; floating point arithmetic; floating-point addition; mathematics; numerical methods; round-off error; round-off errors; round-off propagation; summation", remark = "Cited in \cite{Sterbenz:1974:FPC}.", } @Article{Liu:1970:CEA, author = "B. Liu and T. Kaneko", title = "Correction to {``Error analysis of digital filters realized with floating-point arithmetic''}", journal = j-PROC-IEEE, volume = "58", number = "3", pages = "376--376", month = mar, year = "1970", CODEN = "IEEPAD", ISSN = "0018-9219 (print), 1558-2256 (electronic)", ISSN-L = "0018-9219", bibdate = "Sat Jul 16 08:40:52 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", note = "See \cite{Liu:1969:EAD}.", acknowledgement = ack-nhfb, fjournal = "Proceedings of the IEEE", journal-URL = "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=5", } @Article{Long:1970:LAS, author = "C. T. Long and J. H. Jordan", title = "A Limited Arithmetic on Simple Continued Fractions --- {II}", journal = j-FIB-QUART, volume = "8", number = "2", pages = "135--157", month = mar, year = "1970", CODEN = "FIBQAU", ISSN = "0015-0517", ISSN-L = "0015-0517", bibdate = "Thu Oct 20 18:05:25 MDT 2011", bibsource = "http://www.fq.math.ca/8-2.html; https://www.math.utah.edu/pub/tex/bib/fibquart.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "http://www.fq.math.ca/Scanned/8-2/long.pdf", acknowledgement = ack-nhfb, ajournal = "Fib. Quart", fjournal = "The Fibonacci Quarterly. Official Organ of the Fibonacci Association", journal-URL = "http://www.fq.math.ca/", } @TechReport{Malcolm:1970:AFA, author = "Michael Malcolm", title = "An algorithm for floating-point accumulation of sums with small relative error", number = "STAN-CS-70-163", institution = "Computer Science Department, Stanford University", address = "Stanford, CA, USA", pages = "21", year = "1970", bibdate = "Sat Feb 24 15:01:45 MST 1996", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, keywords = "accurate floating-point summation; Algorithms.; FORTRAN (Computer program language); Numerical calculations --- Computer programs.", remark = "Reproduced by National Technical Information Service. AD 708 691.", } @MastersThesis{Marasa:1970:AAE, author = "John D. Marasa", title = "Accumulated arithmetic error in floating-point and alternative logarithmic number systems", type = "Thesis ({M.S.})", school = "Washington University, Sever Institute of Technology, Department of Applied Mathematics and Computer Science", address = "St. Louis, MO, USA", pages = "vii + 88", year = "1970", bibdate = "Sat Feb 24 15:01:45 MST 1996", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, keywords = "Programming (Mathematics)", } @Article{Matula:1970:ECA, author = "David W. Matula", title = "The emergence of computational arithmetic as a component of the computer science curriculum", journal = j-SIGCSE, volume = "2", number = "3", pages = "41--44", month = nov, year = "1970", CODEN = "SIGSD3", DOI = "https://doi.org/10.1145/873641.873650", ISSN = "0097-8418 (print), 2331-3927 (electronic)", ISSN-L = "0097-8418", bibdate = "Sun Nov 18 08:53:42 MST 2012", bibsource = "http://portal.acm.org/; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/sigcse1970.bib", 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{Matula:1970:FFP, author = "D. W. Matula", title = "A Formalization of Floating-Point Numeric Base Conversion", journal = j-IEEE-TRANS-COMPUT, volume = "C-19", number = "8", pages = "681--692", month = aug, year = "1970", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/T-C.1970.223017", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Wed Jul 13 08:01:35 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput1970.bib", URL = "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=1671610; http://www.acsel-lab.com/arithmetic/arith1/papers/ARITH1_Matula.pdf", abstract = "The process of converting arbitrary real numbers into a floating-point format is formalized as a mapping of the reals into a specified subset of real numbers. The structure of this subset, the set of $n$ significant digit base $b$ floating-point numbers, is analyzed and properties of conversion mappings are determined. For a restricted conversion mapping of the $n$ significant digit base $b$ numbers to the $m$ significant-digit base $d$ numbers, the one-to-one, onto, and order-preserving properties of the mapping are summarized. Multiple conversions consisting of a composition of individual conversion mappings are investigated and some results of the invariant points of such compound conversions are presented. The hardware and software implications of these results with regard to establishing goals and standards for floating-point formats and conversion procedures are considered.", acknowledgement = ack-nj # " and " # ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", keywords = "ARITH-1", } @Article{Mifsud:1970:MDA, author = "C. J. Mifsud", title = "A multiple-precision division algorithm", journal = j-CACM, volume = "13", number = "11", pages = "666--668", month = nov, 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/cacm1970.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", note = "See addendum \cite{Mifsud:1973:AMP}.", acknowledgement = ack-nhfb, classcodes = "C5230 (Digital arithmetic methods)", corpsource = "Mitre Corp., McLean, VA, USA", fjournal = "Communications of the ACM", journal-URL = "https://dl.acm.org/loi/cacm", keywords = "digital arithmetic", } @Article{Nickel:1970:KBS, author = "K. Nickel", title = "{Das Kahan--Babu{\v{s}}kasche Summierungsverfahren in Triplex-ALGOL 60}. ({German}) [{The Kahan--Babu{\v{s}}ka} summation method in {Triplex-ALGOL 60}]", journal = j-Z-ANGE-MATH-MECH, volume = "50", pages = "369--373", year = "1970", CODEN = "ZAMMAX", ISSN = "0044-2267 (print), 1521-4001 (electronic)", ISSN-L = "0044-2267", bibdate = "Tue Aug 28 06:12:04 2001", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, fjournal = "{Zeitschrift f{\"u}r Angewandte Mathematik und Mechanik}", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)1521-4001", keywords = "accurate floating-point summation", language = "German", } @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 = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; https://www.math.utah.edu/pub/tex/bib/fparith.bib; JSTOR database", acknowledgement = ack-nhfb # " and " # ack-nj, ajournal = "Math. Comput.", classcodes = "C4130 (Interpolation and function approximation)", corpsource = "Nagoya University, 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{Oppenheim:1970:RDF, author = "A. Oppenheim", title = "Realization of digital filters using block-floating-point arithmetic", journal = j-IEEE-TRANS-AUDIO-ELECTROACOUST, volume = "18", number = "2", pages = "130--136", month = jun, year = "1970", CODEN = "ITADAS", ISSN = "0018-9278 (print), 1558-2582 (electronic)", ISSN-L = "0018-9278", bibdate = "Sat Jul 16 08:40:52 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, fjournal = "IEEE Transactions on Audio and Electroacoustics", journal-URL = "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=8337", summary = "Recently, statistical models for the effects of roundoff noise in fixed-point and floating-point realizations of digital filters have been proposed and verified, and a comparison between these realizations presented. In this paper a structure for \ldots{}", } @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 = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; https://www.math.utah.edu/pub/tex/bib/fparith.bib; JSTOR database", acknowledgement = ack-nj # " and " # ack-nhfb, ajournal = "Math. Comput.", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Rao:1970:BEC, author = "T. R. N. Rao", title = "Biresidue Error-Correcting Codes for Computer Arithmetic", journal = j-IEEE-TRANS-COMPUT, volume = "C-19", number = "5", pages = "398--402", month = may, year = "1970", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/T-C.1970.222937", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Wed Jul 13 08:01:32 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput1970.bib", URL = "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=1671530", acknowledgement = ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", } @Article{Rao:1970:BLR, author = "T. R. N. Rao and A. K. Trehan", title = "Binary Logic for Residue Arithmetic Using Magnitude Index", journal = j-IEEE-TRANS-COMPUT, volume = "C-19", number = "8", pages = "752--757", month = aug, year = "1970", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/T-C.1970.223026", 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/fparith.bib", URL = "http://www.acsel-lab.com/arithmetic/arith1/papers/ARITH1_Rao.pdf", abstract = "We consider a residue number system using n pairwise relatively prime moduli $ m_1, \ldots {}, m_n $ to represent any integer $X$ in the range $ M / 2 \leq X > M / 2 $, when $ M = \prod m_i $. The moduli $ m_i $ are chosen to be of the $ 2 - 1 $ type, in order that the residue arithmetic can be implemented by means of binary registers and binary logic. Further, for each residue number $X$, a magnitude index $ P_x $ is maintained for all arithmetic operations. We investigate the properties of such a system and derive the addition, subtraction, multiplication, sign determination, and overflow detection algorithms. The proposed organization is found to improve the operation times for sign detection and overflow detection operations, while rendering multiplication to be a difficult operation.", acknowledgement = ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", keywords = "ARITH-1; base extension; index generation logic; magnitude index; modular adders; naturalized form; overflow detection; residue multiplication; residue number system; scale by 2; sign determination", } @Article{Robertson:1970:CBM, author = "J. E. Robertson", title = "The Correspondence Between Methods of Digital Division and Multiplier Recoding Procedures", journal = j-IEEE-TRANS-COMPUT, volume = "C-19", number = "8", pages = "692--701", month = aug, year = "1970", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/T-C.1970.223018", 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/fparith.bib", URL = "http://www.acsel-lab.com/arithmetic/arith1/papers/ARITH1_Robertson.pdf", abstract = "This paper relates previous analyses of the binary SRT division to the theory of multiplier recoding. Since each binary quotient digit has three possible values, the quotient resulting from the SRT division is in recoded form; in this paper it is shown that the recoding is a function of the divisor, and the method for determining the characteristic Boolean function of the recoding is presented. The relationship between the division and the recoding is established by scaling the division in such a way that the scaled ``visor'' becomes a constant. Higher radix results are also discussed.", acknowledgement = ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", keywords = "ARITH-1", } @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/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", pages = "??", year = "1970", 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/fparith.bib", acknowledgement = ack-nj, } @Article{Shively:1970:PFI, author = "Richard R. Shively", title = "Preface: {First IEEE Conference on Computer Arithmetic, Minneapolis, Minnesota, June 16, 1969}", journal = j-IEEE-TRANS-COMPUT, volume = "C-19", number = "8", pages = "679--680", month = aug, year = "1970", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/T-C.1970.223016", 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/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput1970.bib", URL = "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=1671609; http://www.acsel-lab.com/arithmetic/arith1/papers/ARITH1_contents.pdf; http://www.acsel-lab.com/arithmetic/arith1/papers/ARITH1_Shively.pdf", acknowledgement = ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", keywords = "ARITH-1", } @Article{Sikdar:1970:DMM, author = "K. Sikdar", title = "Determination of Multipliers Mapping an Arbitrary Integer into a Range of Certain Type", journal = j-IEEE-TRANS-COMPUT, volume = "C-19", number = "12", pages = "1221--1222", month = dec, year = "1970", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/T-C.1970.222863", ISSN = "0018-9340 (print), 1557-9956 (electronic)", ISSN-L = "0018-9340", bibdate = "Wed Jul 13 08:01:39 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput1970.bib", URL = "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=1671456", acknowledgement = ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", } @Article{Svoboda:1970:ADC, author = "A. Svoboda", title = "Adder With Distributed Control", journal = j-IEEE-TRANS-COMPUT, volume = "C-19", number = "8", pages = "749--751", month = aug, year = "1970", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/T-C.1970.223025", 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/fparith.bib", URL = "http://www.acsel-lab.com/arithmetic/arith1/papers/ARITH1_Svoboda.pdf", abstract = "An adder is described for addition of a large number of binary numbers $ x_j, j = 1, 2, \ldots {}, m $, where $ x_j = \sum_i x_{j, i} 2^i, x_{j, i} = 0, 1, i = 0, 1, \ldots {}, n - 1 $. The adder's algorithm has two parts: (1) the bits $ x_{j, i} $ are added independently for each binary order $ i : s_i = \sum_j x_{j, i} \leq m $ and the result expressed in the binary form $ s_i = \sum_k a_{i, k} 2^k, a_{i, k} = 0, 1, k = 0, 1, \ldots {}, p - 1 $ (where $ 2^{p - 1} \leq m < 2^p $) ; (2) the sum $ y = \sum_j x_j $ is formed by adding terms $ s_{i, k} {2^{i + k}} $ as contributions of the bit $ s_{i, k} $ to the total $y$. A hardware implementation of this algorithm is suggested where the sum $ s_i $ is obtained by a sequential circuit which reorders the values $ x_{j, i}' $ so that their sum $ s_i $, remains unchanged and so that after the reordering the new values $ x_{j, i}' $ obey the conditions $ x_{j + 1, i} \leq x_{j, i} $ for every $ j = 1, 2, \ldots {}, m - 1 $. The implementation with integrated circuits should be quite rewarding because the control of the circuit is done with independent control elements distributed all over the chip.", acknowledgement = ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers", journal-URL = "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12", keywords = "adder; adder for large number of numbers; ARITH-1; distributed control; reordering", } @Article{Taub:1970:ECS, author = "D. M. Taub and C. E. Owen and B. P. Day", title = "Experimental Computer for Schools", journal = "Proceedings of the IEE", volume = "117", number = "2", pages = "303--312", month = feb, year = "1970", bibdate = "Fri Nov 28 19:30:38 2003", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", abstract = "The computer is a small desk-top machine designed for teaching schoolchildren how computers work. It works in decimal notation and has a powerful instruction set which includes 3-address floating-point instructions implemented as `extracode' subroutines. Addressing can be absolute, relative or indirect. For input it uses a capacitive touch keyboard, and for output and display a perfectly normal TV receiver is used. Another input/output device is an ordinary domestic tape recorder, used mainly for long term storage of programs. To make the operation of the machine easy to follow, it can be made to stop at certain stages in the processing of an instruction and automaticaly display the contents of all registers and storage locations relevant at that time. The paper gives a description of the machine and a discussion of the factors that hav influenced its design.", acknowledgement = ack-mfc # " and " # ack-nhfb, keywords = "decimal floating-point arithmetic", remark = "IBM Schools computer with scaled integer format, six decimal digits and exponent range of $ [ - 6, 0] $.", } @Book{Thornton:1970:DCC, author = "James E. Thornton", title = "Design of a Computer: the {Control Data 6600}", publisher = pub-SF, address = pub-SF:adr, pages = "v + 181", year = "1970", ISBN = "0-07-057302-6, 0-07-057303-4", ISBN-13 = "978-0-07-057302-4, 978-0-07-057303-1", LCCN = "TK7889.C6 T5 1970", bibdate = "Wed Sep 14 22:45:51 1994", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "https://archive.computerhistory.org/resources/text/CDC/cdc.6600.thornton.design_of_a_computer_the_control_data_6600.1970.102630394.pdf", acknowledgement = ack-nhfb, remark = "See Table V: Nonstandard Floating Point Arithmetic on page 79 for the rules governing Indefinite, Infinity, and Negative Zero operands.", tableofcontents = "I: Introduction / 1 \\ A. Justification for Large Computers / 1 \\ B. Building Blocks / 4 \\ C. The Approach / 5 \\ II: Organization of the 6600 / 9 \\ A. General / 9 \\ B. Peripheral Subsystem / 10 \\ C. Central Processor-CPU / 12 \\ D. Central Storage / 15 \\ E. Extended Core Storage / 17 \\ III: Basic Circuit Properties / 19 \\ A. The Silicon Transistor / 19 \\ B. DCTL Logic Circuits / 21 \\ C. Logic Symbols / 24 \\ D. Transmission Lines / 28 \\ E. Packaging / 32 \\ IV: Central Storage System / 37 \\ A. Storage Module / 37 \\ B. Theory of Interleaved Storage / 44 \\ C. Stunt Box / 47 \\ D. Storage Bus System / 51 \\ E. Extended Core Storage / 53 \\ F. ECS Coupler and Controller / 55 \\ V: Central Processor Functional Units / 57 \\ A. Boolean Unit / 59 \\ B. Fixed Add Unit / 63 \\ C. Data Trunks / 69 \\ D. Shift Unit / 71 \\ E. Add Unit / 77 \\ F. Multiply Unit / 88 \\ G. Divide Unit / 101 \\ H. Increment Unit / 105 \\ I. Branch Unit / 111 \\ J. ECS Coupler-Controller / 114 \\ VI: Central Processor Control / 117 \\ A. Exchange Jump / 117 \\ B. Instruction Fetch / 120 \\ C. Instruction Issue / 123 \\ D. Scoreboard / 125 \\ Register Entry/Exit Control / 134 \\ F. Summary / 137 \\ VII: Peripheral Subsystem / 141 A. Peripheral Processors / 141 \\ B. Dead Start / 154 \\ C. Disk Storage / 157 \\ VIII: Systems Operation / 163 \\ A. Files / 163 \\ B. Tables / 165 \\ C. Circular Buffer for 1/0 / 166 \\ D. Job Processing / 167 \\ E. System Monitor MTR / 168 \\ F. Control Points / 169 \\ G. Summary / 171 \\ Appendix / 173 \\ Index / 177", } @Article{Tienari:1970:SMR, author = "Martti Tienari", title = "A Statistical Model of Roundoff Error for Varying Length Floating-Point Arithmetic", journal = j-BIT, volume = "10", number = "3", pages = "355--365", month = sep, year = "1970", CODEN = "BITTEL, NBITAB", DOI = "https://doi.org/10.1007/BF01934204", ISSN = "0006-3835 (print), 1572-9125 (electronic)", ISSN-L = "0006-3835", MRclass = "68.00 (65.00)", MRnumber = "42\#7108", bibdate = "Fri Dec 8 08:24:14 1995", bibsource = "https://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "http://www.springerlink.com/openurl.asp?genre=article&issn=0006-3835&volume=10&issue=3&spage=355", acknowledgement = ack-nhfb, fjournal = "BIT (Nordisk tidskrift for informationsbehandling)", journal-URL = "http://link.springer.com/journal/10543", keywords = "floating-point arithmetic; rounding errors", reviewer = "R. W. Hamming", } @Article{Tung:1970:SDD, author = "Chin Tung", title = "Signed-Digit Division Using Combinational Arithmetic Nets", journal = j-IEEE-TRANS-COMPUT, volume = "C-19", number = "8", pages = "746--748", month = aug, year = "1970", CODEN = "ITCOB4", DOI = "https://doi.org/10.1109/T-C.1970.223024", 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/fparith.bib", URL = "http://www.acsel-lab.com/arithmetic/arith1/papers/ARITH1_Tung.pdf", abstract = "To meet the challenge created by the advent of large- scale integration, a unique microelectronic arithmetic building element and combinational arithmetic nets, composed of the building elements, have been studied and proposed for arithmetic processor design. A fast division algorithm, particularly suitable for floating- point arithmetic, has also been developed for signed-digit arithmetic. This algorithm is characterized by the need of preprocessing the divisor and then exact generation of quotient digits. This paper describes the implementation of this division algorithm with the arithmetic building element and combinational arithmetic nets. The intention here is to explore the feasibility of applying large-scale integration technology to arithmetic processors.", acknowledgement = ack-nhfb, ajournal = "IEEE Trans. Comput.", fjournal = "IEEE Transactions on Computers&qu