Documentation : Scilab is Not Naive

Documentation : Scilab is Not Naive Commit Details

Date:2010-12-19 22:04:29 (7 years 6 months ago)
Author:Michael Baudin
Commit:32
Parents: 31
Message:Added variants.
Changes:
A/en_US/scripts/cdiv_polar.sci
A/en_US/scripts/cdiv_smithStewart.sci
M/en_US/scripts/cdiv.tst
M/en_US/scilabisnotnaive.bib

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en_US/scripts/cdiv.tst
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// cdiv_smith.sci : original Smith's method
exec("cdiv_smith.sci");
// cdiv_smithStewart.sci : original Smith's method, improved by Steward
exec("cdiv_smithStewart.sci");
// A Scilab port of the ANSI/ISO C algorithm
exec("cdiv_ansiisoC.sci");
// A Scilab port of the Li et al. improved Smith
exec("cdiv_smithLi.sci");
// A division based on polar form
exec("cdiv_polar.sci");
// An improved Smith method, with ideas from Stewart (prodminmax)
exec("cdiv_smith2.sci");
exec("cdiv_smith6.sci");
end
// A Scilab port of the ANSI/ISO C algorithm
exec("cdiv_ansiisoC.sci");
// A Scilab port of the Li et al. improved Smith
exec("cdiv_smithLi.sci");
exec("assert_computedigits.sci");
exec("assert_datasetread.sci");
ieee(2);
function msg = fmtdig ( x , y , expected , divfunc , headermsg , footermsg )
function [msg,digits] = fmtdig ( x , y , expected , divfunc , headermsg , footermsg )
// Performs the complex division x/y, compare to expected, and produve a message
// containing the number of digits in the real and imaginary parts.
r = divfunc ( a + %i * b , c + %i * d );
format("e",10);
path=pwd();
basis = 2;
msg=[];
dg=[];
dataset = fullfile(path,"cdiv.dataset.csv");
table = assert_datasetread ( dataset , "#" , "," , %t );
ntests = size(table,"r");
expectedR = e+%i*f;
x = a + %i * b;
y = c + %i * d;
msg = msprintf("Test #%3d/%d, %70s, ", k,ntests,sci2exp([a b c d e f]) );
msg = msg + fmtdig ( x , y , expectedR , cdiv_scilab , "Sc" , ", " );
msg = msg + fmtdig ( x , y , expectedR , cdiv_naive , "Na" , ", " );
msg = msg + fmtdig ( x , y , expectedR , cdiv_smith , "Sm" , ", " );
msg = msg + fmtdig ( x , y , expectedR , cdiv_smith2 , "S2" , ", " );
//msg = msg + fmtdig ( x , y , expectedR , cdiv_smith3 , "S3" , ", " );
msg(1) = msprintf("Test #%3d/%d, %70s, ", k,ntests,sci2exp([a b c d e f]) );
dg(k,1)=0;
j=2;
[msg(j),dg(k,j)] = fmtdig ( x , y , expectedR , cdiv_scilab , "Sc" , ", " );
j=j+1;
[msg(j),dg(k,j)] = fmtdig ( x , y , expectedR , cdiv_naive , "Na" , ", " );
j=j+1;
[msg(j),dg(k,j)] = fmtdig ( x , y , expectedR , cdiv_smith , "Sm" , ", " );
j=j+1;
// [msg(j),dg(k,j)] = fmtdig ( x , y , expectedR , cdiv_smithStewart , "St" , ", " );
// j=j+1;
//[msg(j),dg(k,j)] = fmtdig ( x , y , expectedR , cdiv_smith2 , "S2" , ", " );
//j=j+1;
//[msg(j),dg(k,j)] = fmtdig ( x , y , expectedR , cdiv_smith3 , "S3" , ", " );
//j=j+1;
if ( haveimproved ) then
msg = msg + fmtdig ( x , y , expectedR , cdiv_smith4 , "S4" , ", " );
[msg(j),dg(k,j)] = fmtdig ( x , y , expectedR , cdiv_smith4 , "S4" , ", " );
j=j+1;
end
msg = msg + fmtdig ( x , y , expectedR , cdiv_ansiisoC , "AC", ", " );
msg = msg + fmtdig ( x , y , expectedR , cdiv_smithLi , "Li" , ", " );
msg = msg + fmtdig ( x , y , expectedR , cdiv_smith5 , "S5" , ", " );
msg = msg + fmtdig ( x , y , expectedR , cdiv_smith6 , "S6" , "" );
mprintf("%s\n",msg);
[msg(j),dg(k,j)] = fmtdig ( x , y , expectedR , cdiv_ansiisoC , "AC", ", " );
j=j+1;
[msg(j),dg(k,j)] = fmtdig ( x , y , expectedR , cdiv_smithLi , "Li" , ", " );
j=j+1;
[msg(j),dg(k,j)] = fmtdig ( x , y , expectedR , cdiv_smith5 , "S5" , "" );
j=j+1;
//[msg(j),dg(k,j)] = fmtdig ( x , y , expectedR , cdiv_smith6 , "S6" , ", " );
//j=j+1;
//[msg(j),dg(k,j)] = fmtdig ( x , y , expectedR , cdiv_polar , "PO" , "" );
//j=j+1;
mprintf("%s\n",strcat(msg,""));
end
mprintf("Average:\n");
j=2;
mprintf("Sc=%d\n",mean(dg(:,j)));
j=j+1;
mprintf("Na=%d\n",mean(dg(:,j)));
j=j+1;
mprintf("Sm=%d\n",mean(dg(:,j)));
j=j+1;
//mprintf("ST=%d\n",mean(dg(:,j)));
//j=j+1;
//mprintf("S2=%d\n",mean(dg(:,j)));
//j=j+1;
//mprintf("S3=%d\n",mean(dg(:,j)));
//j=j+1;
mprintf("S4=%d\n",mean(dg(:,j)));
j=j+1;
mprintf("AC=%d\n",mean(dg(:,j)));
j=j+1;
mprintf("Li=%d\n",mean(dg(:,j)));
j=j+1;
mprintf("S5=%d\n",mean(dg(:,j)));
j=j+1;
//mprintf("S6=%d\n",mean(dg(:,j)));
//j=j+1;
//mprintf("PO=%d\n",mean(dg(:,j)));
//j=j+1;
// Discovered Issues:
// (0+%i*1 )/(0+%i*0) Scilab = Nan + Nan i Octave = NaN + Inf i Matlab = NaN + Inf i
// (1+%i*-1)/(0+%i*0) Scilab = Nan + Nan i Octave = Inf - Inf i Matlab = Inf - Inf i
en_US/scripts/cdiv_polar.sci
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// Copyright (C) 2010 - Michael Baudin
// A complex division based on polar form.
// Can be very inaccurate:
// x=-1-%i
// y=-1+%i
// cdiv_polar(x,y) : - 1.837D-16 + i (Exact = %i)
//
//
function r = cdiv_polar ( x , y )
// Put x in to polar form x=xr*exp(xa*%i)
xr = abs(x)
xa = atan(imag(x),real(x))
// Put y in to polar form y=yr*exp(ya*%i)
yr = abs(y)
ya = atan(imag(y),real(y))
// Divide : (x/y)
r = (xr/yr)*exp((xa-ya)*%i)
endfunction
en_US/scripts/cdiv_smithStewart.sci
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// Copyright (C) 2010 - Michael Baudin
// A Scilab port of Stewart's algorithm.
// References
//
// "A Note on Complex Division"
// G. W. Stewart, University of Maryland
// ACM Transactions on Mathematical Software, Vol. 11
// No 3, September 1985, Pages 238-241
//
// TODO : fix this using :
// G. W. Stewart
// Corrigendum: ``{A} Note on Complex Division''",
// TOMS,
// volume 12
// number 3
// pages = 285--285
// month = September, 1986
function r = cdiv_smithStewart ( x,y )
error("Does not work")
c = real(x)
d = imag(x)
e = real(y)
f = imag(y)
flip = %f
if ( abs(f) >= abs(e) ) then
[f,e]=switch(e,f)
[c,d]=switch(d,c)
flip=%t
end
s=1/e
t=1/(e+f*(f*s))
if ( abs(f) >= abs(s) ) then
[f,s]=switch(s,f)
end
if ( abs(d) >= abs(s) ) then
a=t*(c+s*(d*f))
elseif ( abs(d) >= abs(f) ) then
a=t*(c+d*(s*f))
else
a=t*(c+f*(s*d))
end
if ( abs(c) >= abs(s) ) then
b=t*(d+s*(c*f))
elseif ( abs(c) >= abs(f) ) then
b=t*(d+c*(s*f))
else
b=t*(d+f*(s*c))
end
if ( flip ) then
b=-b
end
// Avoid using x + %i * y, which may create
// %inf+%i*%inf, which actually multiplies 0 and %inf,
// generating an unwanted %nan.
r = complex(a,b)
endfunction
function [c,d]=switch(a,b)
c=a
d=b
endfunction
function r = cdiv_smithStewart_Other ( x,y )
c = real(x)
d = imag(x)
e = real(y)
f = imag(y)
flip = %f
if ( abs(f) >= abs(e) ) then
// Switch e and f
tmp=e
e=f
f=tmp
// Switch c and d
tmp=c
c=d
d=tmp
flip=%t
end
s=1/e
t=1/(e+f*(f*s))
if ( abs(f) >= abs(s) ) then
// Switch f and s
tmp=f
f=s
s=tmp
end
if ( abs(d) >= abs(s) ) then
a=t*(c+s*(d*f))
elseif ( abs(d) >= abs(f) ) then
a=t*(c+d*(s*f))
else
a=t*(c+f*(s*d))
end
if ( abs(c) >= abs(s) ) then
b=t*(d+s*(c*f))
elseif ( abs(c) >= abs(f) ) then
b=t*(d+c*(s*f))
else
b=t*(d+f*(s*c))
end
if ( flip ) then
b=-b
end
// Avoid using x + %i * y, which may create
// %inf+%i*%inf, which actually multiplies 0 and %inf,
// generating an unwanted %nan.
r = complex(a,b)
endfunction
en_US/scilabisnotnaive.bib
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% Copyright (C) 2008-2010 - Consortium Scilab - Digiteo - Michael Baudin
%
% This file must be used under the terms of the
% Creative Commons Attribution-ShareAlike 3.0 Unported License :
% http://creativecommons.org/licenses/by-sa/3.0/
@book{AbramowitzStegun1972,
author = {Abramowitz, M. and Stegun, I. A.},
title = {Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables},
year = {1972},
publisher= {}}
@UNPUBLISHED{wikipediaquadratic,
title = {Quadratic Equation},
note = {\url{http://en.wikipedia.org/wiki/Quadratic_equation}}
}
@UNPUBLISHED{wikipedialossofsign,
title = {Loss of significance},
note = {\url{http://en.wikipedia.org/wiki/Loss_of_significance}}
}
@UNPUBLISHED{mathworldquadratic,
title = {Quadratic Equation},
note = {\url{http://mathworld.wolfram.com/QuadraticEquation.html}}
}
@book{NumericalRecipes,
author = {W. H. Press and Saul A. Teukolsky and William T. Vetterling and Brian P. Flannery},
title = {Numerical Recipes in C, Second Edition},
year = {1992},
publisher= {}}
@book{WhatEveryComputerScientist,
author = {David Goldberg},
title = {What Every Computer Scientist Should Know About Floating-Point Arithmetic},
month={March},
year = {1991},
publisher= {Association for Computing Machinery, Inc.},
note = {\url{http://www.physics.ohio-state.edu/~dws/grouplinks/floating_point_math.pdf}},
}
@article{Nievergelt2003,
jstor_articletype = {primary_article},
title = {How (Not) to Solve Quadratic Equations},
author = {Nievergelt, Yves},
journal = {The College Mathematics Journal},
jstor_issuetitle = {},
volume = {34},
number = {2},
jstor_formatteddate = {Mar., 2003},
pages = {90--104},
url = {http://www.jstor.org/stable/3595780},
ISSN = {07468342},
abstract = {},
publisher = {Mathematical Association of America},
language = {},
year = {2003},
}
@article{Kahan2004,
title = {On the Cost of Floating-Point Computation Without Extra-Precise Arithmetic},
author = {W. Kahan},
note = {\url{http://www.cs.berkeley.edu/~wkahan/Qdrtcs.pdf}},
year = {2004},
}
@article{Smith1962,
author = {Smith,, Robert L.},
title = {Algorithm 116: Complex division},
journal = {Commun. ACM},
volume = {5},
number = {8},
year = {1962},
issn = {0001-0782},
pages = {435},
doi = {http://doi.acm.org/10.1145/368637.368661},
publisher = {ACM},
address = {New York, NY, USA},
}
@article{Jenkins1975,
author = {Jenkins,, M. A.},
title = {Algorithm 493: Zeros of a Real Polynomial [C2]},
journal = {ACM Trans. Math. Softw.},
volume = {1},
number = {2},
year = {1975},
issn = {0098-3500},
pages = {178--189},
doi = {http://doi.acm.org/10.1145/355637.355643},
publisher = {ACM},
address = {New York, NY, USA},
}
@article{214414,
author = {Stewart,, G. W.},
title = {A note on complex division},
journal = {ACM Trans. Math. Softw.},
volume = {11},
number = {3},
year = {1985},
issn = {0098-3500},
pages = {238--241},
doi = {http://doi.acm.org/10.1145/214408.214414},
publisher = {ACM},
address = {New York, NY, USA},
}
@article{567808,
author = {Li,, Xiaoye S. and Demmel,, James W. and Bailey,, David H. and Henry,, Greg and Hida,, Yozo and Iskandar,, Jimmy and Kahan,, William and Kang,, Suh Y. and Kapur,, Anil and Martin,, Michael C. and Thompson,, Brandon J. and Tung,, Teresa and Yoo,, Daniel J.},
title = {Design, implementation and testing of extended and mixed precision BLAS},
journal = {ACM Trans. Math. Softw.},
volume = {28},
number = {2},
year = {2002},
issn = {0098-3500},
pages = {152--205},
doi = {http://doi.acm.org/10.1145/567806.567808},
publisher = {ACM},
address = {New York, NY, USA},
}
@article{KAHAN1987,
author = {KAHAN, W.},
title = {Branch cuts for complex elementary functions, or much ado about nothing's sign bit.},
year = {1987},
pages = {165--211},
publisher = {In The State of the Art m Numerzcal Analysts. Proceedings of the Joint IMA/SIAM Conference, A. Iserles and M J. D Powell, Eds. Clarendon Press, Oxford, England},
}
@article{1039814,
author = {Priest,, Douglas M.},
title = {Efficient scaling for complex division},
journal = {ACM Trans. Math. Softw.},
volume = {30},
number = {4},
year = {2004},
issn = {0098-3500},
pages = {389--401},
doi = {http://doi.acm.org/10.1145/1039813.1039814},
publisher = {ACM},
address = {New York, NY, USA},
}
@UNPUBLISHED{schimdtnd,
title = {Numerical Derivatives},
note = {\url{http://fermi.la.asu.edu/PHY531/intro/node1.html}},
author = {K.E. Schmidt}
}
@book{dixmier,
author = {J. Dixmier, P. Dugac},
year = {1969},
publisher= {Gauthier-Villars}}
@article{Forsythe1966,
author = {George E. Forsythe},
title = {How Do You Solve A Quadratic Equation ?},
year = {1966},
publisher = {Technical Report No. CS40},
note = {\url{ftp://reports.stanford.edu/pub/cstr/reports/cs/tr/66/40/CS-TR-66-40.pdf}},
}
@ARTICLE{1667289,
title={Underflow and the Denormalized Numbers},
author={Coonen, J.T.},
journal={Computer},
year={1981},
month={March },
volume={14},
number={3},
pages={75-87},
doi={10.1109/C-M.1981.220382},
ISSN={0018-9162}, }
@book{artcomputerKnuthVol2,
author = {D. E. Knuth},
title = {The Art of Computer Programming, Volume 2, Seminumerical Algorithms},
year = {1998},
publisher= {Third Edition, Addison Wesley, Reading, MA}}
@Article{Wynn:1962:AAP,
author = "P. Wynn",
title = "An Arsenal of {ALGOL} Procedures for Complex Arithmetic",
journal = "BIT Numerical Mathematics",
volume = "2",
number = "4",
pages = "232--255",
month = dec,
year = "1962",
CODEN = "BITTEL, NBITAB",
DOI = "http://www.springerlink.com/openurl.asp?genre=article&id=doi:10.1007/BF01940171",
ISSN = "0006-3835 (print), 1572-9125 (electronic)",
bibdate = "Wed Jan 4 18:52:07 MST 2006",
bibsource = "ftp://ftp.math.utah.edu/pub/tex/bib/bit.bib;
http://springerlink.metapress.com/openurl.asp?genre=issue&issn=0006-3835&volume=2&issue=4",
URL = "http://www.springerlink.com/openurl.asp?genre=article&issn=0006-3835&volume=2&issue=4&spage=232",
acknowledgement = ack-nhfb,
}
@article{DBLP:journals/cacm/Friedland67,
author = {Paul Friedland},
title = {Algorithm 312: Absolute value and square root of a complex
number},
journal = {Commun. ACM},
volume = {10},
number = {10},
year = {1967},
pages = {665},
ee = {http://doi.acm.org/10.1145/363717.363780},
bibsource = {DBLP, http://dblp.uni-trier.de}
}
@article{journals/ibmrd/MolerM83,
title = {Replacing Square Roots by Pythagorean Sums.},
author = {Cleve B. Moler and Donald Morrison},
journal = {IBM Journal of Research and Development},
number = {6},
pages = {577-581},
url = {http://dblp.uni-trier.de/db/journals/ibmrd/ibmrd27.html#MolerM83},
volume = {27},
year = {1983},
description = {dblp},
date = {2002-01-03},
keywords = {dblp }
}
@techreport{900236,
author = {Lawson,, C. L. and Hanson,, R. J. and Kincaid,, D. R. and Krogh,, F. T.},
title = {Basic Linear Algebra Subprograms for FORTRAN Usage},
year = {1977},
source = {http://www.ncstrl.org:8900/ncstrl/servlet/search?formname=detail\&id=oai%3Ancstrlh%3Autexas_cs%3AUTEXAS_CS%2F%2FCS-TR-77-72},
publisher = {University of Texas at Austin},
address = {Austin, TX, USA},
}
@article{355771,
author = {Blue,, James L.},
title = {A Portable Fortran Program to Find the Euclidean Norm of a Vector},
journal = {ACM Trans. Math. Softw.},
volume = {4},
number = {1},
year = {1978},
issn = {0098-3500},
pages = {15--23},
doi = {http://doi.acm.org/10.1145/355769.355771},
publisher = {ACM},
address = {New York, NY, USA},
}
@InCollection{Cody:1971:SEF,
author = "W. J. Cody",
title = "Software for the Elementary Functions",
crossref = "Rice:1971:MS",
pages = "171--186",
year = "1971",
bibdate = "Thu Sep 15 18:56:47 1994",
bibsource = "garbo.uwasa.fi:/pc/doc-soft/fpbiblio.txt",
acknowledgement = ack-nj,
}
@techreport{ieee754-1985,
author = {Stevenson, David },
booktitle = {ANSI/IEEE Std 754-1985},
citeulike-article-id = {1677576},
journal = {ANSI/IEEE Std 754-1985},
keywords = {floating-point, para08},
month = {August},
posted-at = {2008-04-21 12:47:40},
priority = {1},
title = {IEEE standard for binary floating-point arithmetic},
url = {http://ieeexplore.ieee.org/xpls/abs\_all.jsp?arnumber=30711},
year = {1985}
}
@Book{P754:2008:ISF,
author = "{IEEE Task P754}",
title = "{IEEE 754-2008, Standard for Floating-Point
Arithmetic}",
publisher = pub-IEEE-STD,
address = pub-IEEE-STD:adr,
pages = "58",
day = "29",
month = aug,
year = "2008",
DOI = "http://doi.ieeecomputersociety.org/10.1109/IEEESTD.2008.4610935",
ISBN = "0-7381-5753-8 (paper), 0-7381-5752-X (electronic)",
ISBN-13 = "978-0-7381-5753-5 (paper), 978-0-7381-5752-8
(electronic)",
bibdate = "Thu Sep 25 09:50:30 2008",
URL = "http://ieeexplore.ieee.org/servlet/opac?punumber=4610933;
http://en.wikipedia.org/wiki/IEEE_754-2008",
abstract = "This standard specifies interchange and arithmetic
formats and methods for binary and decimal
floating-point arithmetic in computer programming
environments. This standard specifies exception
conditions and their default handling. An
implementation of a floating-point system conforming to
this standard may be realized entirely in software,
entirely in hardware, or in any combination of software
and hardware. For operations specified in the normative
part of this standard, numerical results and exceptions
are uniquely determined by the values of the input
data, sequence of operations, and destination formats,
all under user control.",
acknowledgement = ack-nhfb,
}
@article{matlab-hypot,
note={\url{http://www.mathworks.com/access/helpdesk/help/techdoc/index.html?/access/helpdesk/help/techdoc/ref/hypot.html&http://www.google.fr/search?q=hypot&ie=utf-8&oe=utf-8&aq=t&rls=org.mozilla:fr:official&client=firefox-a}},
title={Matlab - hypot : square root of sum of squares},
}
@article{fdlibm,
title = {A Freely Distributable C Math Library},
author = {Sun Microsystems, Inc.},
year = {1993},
note = {\url{http://www.netlib.org/fdlibm}},
}
@book{261217,
author = {Muller,, Jean-Michel},
title = {Elementary functions: algorithms and implementation},
year = {1997},
isbn = {0-8176-3990-X},
publisher = {Birkhauser Boston, Inc.},
address = {Secaucus, NJ, USA},
}
@UNPUBLISHED{WWWMapleSoft,
title = {MapleSoft},
author = {Maple},
note = {\url{http://www.maplesoft.com}}
}
@UNPUBLISHED{WWWMathematica,
title = {Mathematica},
author = {Wolfram Research},
note = {\url{http://www.wolfram.com/products/mathematica}}
}
@UNPUBLISHED{WWWWolframAlpha,
title = {Wolfram Alpha},
author = {Wolfram Research},
note = {\url{http://www.wolframalpha.com}}
}
@UNPUBLISHED{WWWMaxima,
title = {Maxima},
note = {\url{http://maxima.sourceforge.net/}}
}
@UNPUBLISHED{WWWScilab,
title = {Scilab},
author = {The Scilab Consortium},
note = {\url{http://www.scilab.org}}
}
@UNPUBLISHED{WWWMatlab,
title = {Matlab},
author = {The MathWorks},
note = {\url{http://www.mathworks.fr}}
}
@UNPUBLISHED{WWWOctave,
title = {Octave},
note = {\url{http://www.gnu.org/software/octave}}
}
@Book{Higham:2002:ASN,
author = "Nicholas J. Higham",
title = "Accuracy and Stability of Numerical Algorithms",
publisher = "Society for Industrial and Applied Mathematics",
address = "Philadelphia, PA, USA",
edition = "Second",
year = "2002",
pages = "xxx+680",
ISBN = "0-89871-521-0"
}
@article{JenkinsTraub1970,
jstor_articletype = {primary_article},
title = {A Three-Stage Algorithm for Real Polynomials Using Quadratic Iteration},
author = {Jenkins, M. A. and Traub, J. F.},
journal = {SIAM Journal on Numerical Analysis},
jstor_issuetitle = {},
volume = {7},
number = {4},
jstor_formatteddate = {Dec., 1970},
pages = {545--566},
url = {http://www.jstor.org/stable/2949376},
ISSN = {00361429},
abstract = {},
language = {},
year = {1970},
publisher = {Society for Industrial and Applied Mathematics},
copyright = {Copyright © 1970 Society for Industrial and Applied Mathematics},
}
@book{KelleyNewtonMethod,
author = {Kelley, C. T.},
title = {Solving nonlinear equations with Newton's method},
year = {2003},
publisher= {SIAM}}
@article{DumontetVignes1977,
title = {D\'etermination du pas optimal dans le calcul des d\'eriv\'ees sur ordinateur},
author = {Dumontet, J. and Vignes, J.},
journal = {R.A.I.R.O Analyse num\'erique},
volume = {11},
number = {1},
pages = {13--25},
url = {http://www.numdam.org/item?id=M2AN_1977__11_1_13_0},
year = {1977},
}
@article{1979SteplemanWinarsky,
jstor_articletype = {primary_article},
title = {Adaptive Numerical Differentiation},
author = {Stepleman, R. S. and Winarsky, N. D.},
journal = {Mathematics of Computation},
jstor_issuetitle = {},
volume = {33},
number = {148},
jstor_formatteddate = {Oct., 1979},
pages = {1257--1264},
url = {http://www.jstor.org/stable/2006459},
ISSN = {00255718},
abstract = {It is well known that the calculation of an accurate approximate derivative $f'(x)$ of a nontabular function $f(x)$ on a finite-precision computer by the formula $d(h) = (f(x + h) - f(x - h))/2h$ is a delicate task. If $h$ is too large, truncation errors cause poor answers, while if $h$ is too small, cancellation and other "rounding" errors cause poor answers. We will show that by using simple results on the nature of the asymptotic convergence of $d(h)$ to $f'$, a reliable numerical method can be obtained which can yield efficiently the theoretical maximum number of accurate digits for the given machine precision.},
language = {},
year = {1979},
publisher = {American Mathematical Society},
copyright = {Copyright © 1979 American Mathematical Society},
}
@book{Gill81MurrayWright,
address = {London},
author = {Gill, P. E. and Murray, W. and Wright, M. H. },
citeulike-article-id = {1055426},
keywords = {bibtex-import},
posted-at = {2007-01-20 09:02:15},
priority = {2},
publisher = {Academic Press},
title = {Practical optimization},
year = {1981}
}
@Book{MullerEtAl2010,
title = {Handbook of Floating-Point Arithmetic},
author = {Muller, Jean-Michel and Brisebarre, Nicolas and de
Dinechin, Florent and Jeannerod, Claude-Pierre and
Lef{\`e}vre, Vincent and Melquiond, Guillaume and Revol,
Nathalie and Stehl{\'e}, Damien and Torres, Serge},
publisher = {{B}irkh\"auser {B}oston },
pages = {572 },
note = {{ACM} {G}.1.0; {G}.1.2; {G}.4; {B}.2.0; {B}.2.4; {F}.2.1.,
ISBN 978-0-8176-4704-9},
year = {2010},
}
@article{Monniaux2008,
author = {Monniaux, David},
title = {The pitfalls of verifying floating-point computations},
journal = {ACM Trans. Program. Lang. Syst.},
volume = {30},
number = {3},
year = {2008},
issn = {0164-0925},
pages = {1--41},
doi = {http://doi.acm.org/10.1145/1353445.1353446},
publisher = {ACM},
address = {New York, NY, USA},
}
@techreport{CordenKreitzerIntel2009,
title = {Consistency of Floating-Point Results using the Intel Compiler or Why doesn't my application always give the same answer?},
author = {Corden, Martyn J. and Kreitzer, David},
institution = {Intel Corporation, Software Solutions Group},
year = {2009}
}
@techreport{GCCManual2008,
title = {The GNU Compiler Collection},
author = {Free Software Foundation},
year = {2008}
}
@UNPUBLISHED{Intel64IA32Architecture,
title = {Intel 64 and IA-32 Architectures Software Developer's Manual. Volume 1: Basic Architecture},
author = {Intel Corporation},
year = {2009},
note = {\url{http://www.intel.com/products/processor/manuals}}
}
@UNPUBLISHED{MPFRWeb,
author = {MPFR},
title = {The MPFR Library},
note = {\url{http://www.mpfr.org}}
}
@Article{Fousse:2007:MMP,
author = "Laurent Fousse and Guillaume Hanrot and Vincent Lef\`evre and Patrick
P\'elissier and Paul Zimmermann",
title = "{MPFR}: A Multiple-Precision Binary Floating-Point Library with Correct Rounding",
journal = "{ACM} Transactions on Mathematical Software",
volume = "33",
number = "2",
month = jun,
year = "2007",
pages = "13:1--13:15",
URL = "http://doi.acm.org/10.1145/1236463.1236468",
abstract = "This paper presents a multiple-precision binary floating-point library,
written in the ISO C language, and based on the GNU MP library. Its
particularity is to extend to arbitrary-precision ideas from the IEEE 754
standard, by providing \emph{correct rounding} and \emph{exceptions}.
We demonstrate how these strong semantics are achieved --- with no
significant slowdown with respect to other arbitrary-precision tools
--- and discuss a few applications where such a library can be useful.",
}
@UNPUBLISHED{GMPWeb,
author = {GMP},
title = {GNU Multiple Precision Arithmetic Library},
note = {\url{http://gmplib.org}}
}
@book{GolubVanLoad1996,
address = {Baltimore, MD, USA},
author = {Golub, Gene H. and Van Loan, Charles F. },
isbn = {0801854148},
publisher = {Johns Hopkins University Press},
title = {Matrix computations (3rd ed.)},
url = {http://portal.acm.org/citation.cfm?id=248979},
year = {1996}
}
@techreport{ProgrammingC,
title = {Programming languages ? C, ISO/IEC 9899:TC3},
author = {ISO/IEC},
year = {2007},
note = {\url{http://www.open-std.org/jtc1/sc22/wg14/www/docs/n1256.pdf}}
}
@UNPUBLISHED{KahanMarketing2000,
author = {William Kahan},
title = {Marketing versus Mathematics},
year = {2000},
note = {\url{www.cs.berkeley.edu/~wkahan/MktgMath.ps}}
}
@UNPUBLISHED{Li2000,
author = {X. S. Li and J. W. Demmel and D. H. Bailey and G. Henry and Y. Hida and J. Iskandar and
W. Kahan and A. Kapur and M. C. Martin and T. Tung and D. J. Yoo},
title = {Design, Implementation and Testing of Extended and Mixed Precision BLAS},
year = {2000},
note = {\url{www.netlib.org/lapack/lawnspdf/lawn149.pdf}}
}
% Copyright (C) 2008-2010 - Consortium Scilab - Digiteo - Michael Baudin
%
% This file must be used under the terms of the
% Creative Commons Attribution-ShareAlike 3.0 Unported License :
% http://creativecommons.org/licenses/by-sa/3.0/
@book{AbramowitzStegun1972,
author = {Abramowitz, M. and Stegun, I. A.},
title = {Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables},
year = {1972},
publisher= {}}
@UNPUBLISHED{wikipediaquadratic,
title = {Quadratic Equation},
note = {\url{http://en.wikipedia.org/wiki/Quadratic_equation}}
}
@UNPUBLISHED{wikipedialossofsign,
title = {Loss of significance},
note = {\url{http://en.wikipedia.org/wiki/Loss_of_significance}}
}
@UNPUBLISHED{mathworldquadratic,
title = {Quadratic Equation},
note = {\url{http://mathworld.wolfram.com/QuadraticEquation.html}}
}
@book{NumericalRecipes,
author = {W. H. Press and Saul A. Teukolsky and William T. Vetterling and Brian P. Flannery},
title = {Numerical Recipes in C, Second Edition},
year = {1992},
publisher= {}}
@book{WhatEveryComputerScientist,
author = {David Goldberg},
title = {What Every Computer Scientist Should Know About Floating-Point Arithmetic},
month={March},
year = {1991},
publisher= {Association for Computing Machinery, Inc.},
note = {\url{http://www.physics.ohio-state.edu/~dws/grouplinks/floating_point_math.pdf}},
}
@article{Nievergelt2003,
jstor_articletype = {primary_article},
title = {How (Not) to Solve Quadratic Equations},
author = {Nievergelt, Yves},
journal = {The College Mathematics Journal},
jstor_issuetitle = {},
volume = {34},
number = {2},
jstor_formatteddate = {Mar., 2003},
pages = {90--104},
url = {http://www.jstor.org/stable/3595780},
ISSN = {07468342},
abstract = {},
publisher = {Mathematical Association of America},
language = {},
year = {2003},
}
@article{Kahan2004,
title = {On the Cost of Floating-Point Computation Without Extra-Precise Arithmetic},
author = {W. Kahan},
note = {\url{http://www.cs.berkeley.edu/~wkahan/Qdrtcs.pdf}},
year = {2004},
}
@article{Smith1962,
author = {Smith,, Robert L.},
title = {Algorithm 116: Complex division},
journal = {Commun. ACM},
volume = {5},
number = {8},
year = {1962},
issn = {0001-0782},
pages = {435},
doi = {http://doi.acm.org/10.1145/368637.368661},
publisher = {ACM},
address = {New York, NY, USA},
}
@article{Jenkins1975,
author = {Jenkins,, M. A.},
title = {Algorithm 493: Zeros of a Real Polynomial [C2]},
journal = {ACM Trans. Math. Softw.},
volume = {1},
number = {2},
year = {1975},
issn = {0098-3500},
pages = {178--189},
doi = {http://doi.acm.org/10.1145/355637.355643},
publisher = {ACM},
address = {New York, NY, USA},
}
@article{214414,
author = {Stewart,, G. W.},
title = {A note on complex division},
journal = {ACM Trans. Math. Softw.},
volume = {11},
number = {3},
year = {1985},
issn = {0098-3500},
pages = {238--241},
doi = {http://doi.acm.org/10.1145/214408.214414},
publisher = {ACM},
address = {New York, NY, USA},
}
@article{567808,
author = {Li,, Xiaoye S. and Demmel,, James W. and Bailey,, David H. and Henry,, Greg and Hida,, Yozo and Iskandar,, Jimmy and Kahan,, William and Kang,, Suh Y. and Kapur,, Anil and Martin,, Michael C. and Thompson,, Brandon J. and Tung,, Teresa and Yoo,, Daniel J.},
title = {Design, implementation and testing of extended and mixed precision BLAS},
journal = {ACM Trans. Math. Softw.},
volume = {28},
number = {2},
year = {2002},
issn = {0098-3500},
pages = {152--205},
doi = {http://doi.acm.org/10.1145/567806.567808},
publisher = {ACM},
address = {New York, NY, USA},
}
@article{KAHAN1987,
author = {KAHAN, W.},
title = {Branch cuts for complex elementary functions, or much ado about nothing's sign bit.},
year = {1987},
pages = {165--211},
publisher = {In The State of the Art m Numerzcal Analysts. Proceedings of the Joint IMA/SIAM Conference, A. Iserles and M J. D Powell, Eds. Clarendon Press, Oxford, England},
}
@article{1039814,
author = {Priest,, Douglas M.},
title = {Efficient scaling for complex division},
journal = {ACM Trans. Math. Softw.},
volume = {30},
number = {4},
year = {2004},
issn = {0098-3500},
pages = {389--401},
doi = {http://doi.acm.org/10.1145/1039813.1039814},
publisher = {ACM},
address = {New York, NY, USA},
}
@UNPUBLISHED{schimdtnd,
title = {Numerical Derivatives},
note = {\url{http://fermi.la.asu.edu/PHY531/intro/node1.html}},
author = {K.E. Schmidt}
}
@book{dixmier,
author = {J. Dixmier, P. Dugac},
year = {1969},
publisher= {Gauthier-Villars}}
@article{Forsythe1966,
author = {George E. Forsythe},
title = {How Do You Solve A Quadratic Equation ?},
year = {1966},
publisher = {Technical Report No. CS40},
note = {\url{ftp://reports.stanford.edu/pub/cstr/reports/cs/tr/66/40/CS-TR-66-40.pdf}},
}
@ARTICLE{1667289,
title={Underflow and the Denormalized Numbers},
author={Coonen, J.T.},
journal={Computer},
year={1981},
month={March },
volume={14},
number={3},
pages={75-87},
doi={10.1109/C-M.1981.220382},
ISSN={0018-9162}, }
@book{artcomputerKnuthVol2,
author = {D. E. Knuth},
title = {The Art of Computer Programming, Volume 2, Seminumerical Algorithms},
year = {1998},
publisher= {Third Edition, Addison Wesley, Reading, MA}}
@Article{Wynn:1962:AAP,
author = "P. Wynn",
title = "An Arsenal of {ALGOL} Procedures for Complex Arithmetic",
journal = "BIT Numerical Mathematics",
volume = "2",
number = "4",
pages = "232--255",
month = dec,
year = "1962",
CODEN = "BITTEL, NBITAB",
DOI = "http://www.springerlink.com/openurl.asp?genre=article&id=doi:10.1007/BF01940171",
ISSN = "0006-3835 (print), 1572-9125 (electronic)",
bibdate = "Wed Jan 4 18:52:07 MST 2006",
bibsource = "ftp://ftp.math.utah.edu/pub/tex/bib/bit.bib;
http://springerlink.metapress.com/openurl.asp?genre=issue&issn=0006-3835&volume=2&issue=4",
URL = "http://www.springerlink.com/openurl.asp?genre=article&issn=0006-3835&volume=2&issue=4&spage=232",
acknowledgement = ack-nhfb,
}
@article{DBLP:journals/cacm/Friedland67,
author = {Paul Friedland},
title = {Algorithm 312: Absolute value and square root of a complex
number},
journal = {Commun. ACM},
volume = {10},
number = {10},
year = {1967},
pages = {665},
ee = {http://doi.acm.org/10.1145/363717.363780},
bibsource = {DBLP, http://dblp.uni-trier.de}
}
@article{journals/ibmrd/MolerM83,
title = {Replacing Square Roots by Pythagorean Sums.},
author = {Cleve B. Moler and Donald Morrison},
journal = {IBM Journal of Research and Development},
number = {6},
pages = {577-581},
url = {http://dblp.uni-trier.de/db/journals/ibmrd/ibmrd27.html#MolerM83},
volume = {27},
year = {1983},
description = {dblp},
date = {2002-01-03},
keywords = {dblp }
}
@techreport{900236,
author = {Lawson,, C. L. and Hanson,, R. J. and Kincaid,, D. R. and Krogh,, F. T.},
title = {Basic Linear Algebra Subprograms for FORTRAN Usage},
year = {1977},
source = {http://www.ncstrl.org:8900/ncstrl/servlet/search?formname=detail\&id=oai%3Ancstrlh%3Autexas_cs%3AUTEXAS_CS%2F%2FCS-TR-77-72},
publisher = {University of Texas at Austin},
address = {Austin, TX, USA},
}
@article{355771,
author = {Blue,, James L.},
title = {A Portable Fortran Program to Find the Euclidean Norm of a Vector},
journal = {ACM Trans. Math. Softw.},
volume = {4},
number = {1},
year = {1978},
issn = {0098-3500},
pages = {15--23},
doi = {http://doi.acm.org/10.1145/355769.355771},
publisher = {ACM},
address = {New York, NY, USA},
}
@InCollection{Cody:1971:SEF,
author = "W. J. Cody",
title = "Software for the Elementary Functions",
crossref = "Rice:1971:MS",
pages = "171--186",
year = "1971",
bibdate = "Thu Sep 15 18:56:47 1994",
bibsource = "garbo.uwasa.fi:/pc/doc-soft/fpbiblio.txt",
acknowledgement = ack-nj,
}
@techreport{ieee754-1985,
author = {Stevenson, David },
booktitle = {ANSI/IEEE Std 754-1985},
citeulike-article-id = {1677576},
journal = {ANSI/IEEE Std 754-1985},
keywords = {floating-point, para08},
month = {August},
posted-at = {2008-04-21 12:47:40},
priority = {1},
title = {IEEE standard for binary floating-point arithmetic},
url = {http://ieeexplore.ieee.org/xpls/abs\_all.jsp?arnumber=30711},
year = {1985}
}
@Book{P754:2008:ISF,
author = "{IEEE Task P754}",
title = "{IEEE 754-2008, Standard for Floating-Point
Arithmetic}",
publisher = pub-IEEE-STD,
address = pub-IEEE-STD:adr,
pages = "58",
day = "29",
month = aug,
year = "2008",
DOI = "http://doi.ieeecomputersociety.org/10.1109/IEEESTD.2008.4610935",
ISBN = "0-7381-5753-8 (paper), 0-7381-5752-X (electronic)",
ISBN-13 = "978-0-7381-5753-5 (paper), 978-0-7381-5752-8
(electronic)",
bibdate = "Thu Sep 25 09:50:30 2008",
URL = "http://ieeexplore.ieee.org/servlet/opac?punumber=4610933;
http://en.wikipedia.org/wiki/IEEE_754-2008",
abstract = "This standard specifies interchange and arithmetic
formats and methods for binary and decimal
floating-point arithmetic in computer programming
environments. This standard specifies exception
conditions and their default handling. An
implementation of a floating-point system conforming to
this standard may be realized entirely in software,
entirely in hardware, or in any combination of software
and hardware. For operations specified in the normative
part of this standard, numerical results and exceptions
are uniquely determined by the values of the input
data, sequence of operations, and destination formats,
all under user control.",
acknowledgement = ack-nhfb,
}
@article{matlab-hypot,
note={\url{http://www.mathworks.com/access/helpdesk/help/techdoc/index.html?/access/helpdesk/help/techdoc/ref/hypot.html&http://www.google.fr/search?q=hypot&ie=utf-8&oe=utf-8&aq=t&rls=org.mozilla:fr:official&client=firefox-a}},
title={Matlab - hypot : square root of sum of squares},
}
@article{fdlibm,
title = {A Freely Distributable C Math Library},
author = {Sun Microsystems, Inc.},
year = {1993},
note = {\url{http://www.netlib.org/fdlibm}},
}
@book{261217,
author = {Muller,, Jean-Michel},
title = {Elementary functions: algorithms and implementation},
year = {1997},
isbn = {0-8176-3990-X},
publisher = {Birkhauser Boston, Inc.},
address = {Secaucus, NJ, USA},
}
@UNPUBLISHED{WWWMapleSoft,
title = {MapleSoft},
author = {Maple},
note = {\url{http://www.maplesoft.com}}
}
@UNPUBLISHED{WWWMathematica,
title = {Mathematica},
author = {Wolfram Research},
note = {\url{http://www.wolfram.com/products/mathematica}}
}
@UNPUBLISHED{WWWWolframAlpha,
title = {Wolfram Alpha},
author = {Wolfram Research},
note = {\url{http://www.wolframalpha.com}}
}
@UNPUBLISHED{WWWMaxima,
title = {Maxima},
note = {\url{http://maxima.sourceforge.net/}}
}
@UNPUBLISHED{WWWScilab,
title = {Scilab},
author = {The Scilab Consortium},
note = {\url{http://www.scilab.org}}
}
@UNPUBLISHED{WWWMatlab,
title = {Matlab},
author = {The MathWorks},
note = {\url{http://www.mathworks.fr}}
}
@UNPUBLISHED{WWWOctave,
title = {Octave},
note = {\url{http://www.gnu.org/software/octave}}
}
@Book{Higham:2002:ASN,
author = "Nicholas J. Higham",
title = "Accuracy and Stability of Numerical Algorithms",
publisher = "Society for Industrial and Applied Mathematics",
address = "Philadelphia, PA, USA",
edition = "Second",
year = "2002",
pages = "xxx+680",
ISBN = "0-89871-521-0"
}
@article{JenkinsTraub1970,
jstor_articletype = {primary_article},
title = {A Three-Stage Algorithm for Real Polynomials Using Quadratic Iteration},
author = {Jenkins, M. A. and Traub, J. F.},
journal = {SIAM Journal on Numerical Analysis},
jstor_issuetitle = {},
volume = {7},
number = {4},
jstor_formatteddate = {Dec., 1970},
pages = {545--566},
url = {http://www.jstor.org/stable/2949376},
ISSN = {00361429},
abstract = {},
language = {},
year = {1970},
publisher = {Society for Industrial and Applied Mathematics},
copyright = {Copyright © 1970 Society for Industrial and Applied Mathematics},
}
@book{KelleyNewtonMethod,
author = {Kelley, C. T.},
title = {Solving nonlinear equations with Newton's method},
year = {2003},
publisher= {SIAM}}
@article{DumontetVignes1977,
title = {D\'etermination du pas optimal dans le calcul des d\'eriv\'ees sur ordinateur},
author = {Dumontet, J. and Vignes, J.},
journal = {R.A.I.R.O Analyse num\'erique},
volume = {11},
number = {1},
pages = {13--25},
url = {http://www.numdam.org/item?id=M2AN_1977__11_1_13_0},
year = {1977},
}
@article{1979SteplemanWinarsky,
jstor_articletype = {primary_article},
title = {Adaptive Numerical Differentiation},
author = {Stepleman, R. S. and Winarsky, N. D.},
journal = {Mathematics of Computation},
jstor_issuetitle = {},
volume = {33},
number = {148},
jstor_formatteddate = {Oct., 1979},
pages = {1257--1264},
url = {http://www.jstor.org/stable/2006459},
ISSN = {00255718},
abstract = {It is well known that the calculation of an accurate approximate derivative $f'(x)$ of a nontabular function $f(x)$ on a finite-precision computer by the formula $d(h) = (f(x + h) - f(x - h))/2h$ is a delicate task. If $h$ is too large, truncation errors cause poor answers, while if $h$ is too small, cancellation and other "rounding" errors cause poor answers. We will show that by using simple results on the nature of the asymptotic convergence of $d(h)$ to $f'$, a reliable numerical method can be obtained which can yield efficiently the theoretical maximum number of accurate digits for the given machine precision.},
language = {},
year = {1979},
publisher = {American Mathematical Society},
copyright = {Copyright © 1979 American Mathematical Society},
}
@book{Gill81MurrayWright,
address = {London},
author = {Gill, P. E. and Murray, W. and Wright, M. H. },
citeulike-article-id = {1055426},
keywords = {bibtex-import},
posted-at = {2007-01-20 09:02:15},
priority = {2},
publisher = {Academic Press},
title = {Practical optimization},
year = {1981}
}
@Book{MullerEtAl2010,
title = {Handbook of Floating-Point Arithmetic},
author = {Muller, Jean-Michel and Brisebarre, Nicolas and de
Dinechin, Florent and Jeannerod, Claude-Pierre and
Lef{\`e}vre, Vincent and Melquiond, Guillaume and Revol,
Nathalie and Stehl{\'e}, Damien and Torres, Serge},
publisher = {{B}irkh\"auser {B}oston },
pages = {572 },
note = {{ACM} {G}.1.0; {G}.1.2; {G}.4; {B}.2.0; {B}.2.4; {F}.2.1.,
ISBN 978-0-8176-4704-9},
year = {2010},
}
@article{Monniaux2008,
author = {Monniaux, David},
title = {The pitfalls of verifying floating-point computations},
journal = {ACM Trans. Program. Lang. Syst.},
volume = {30},
number = {3},
year = {2008},
issn = {0164-0925},
pages = {1--41},
doi = {http://doi.acm.org/10.1145/1353445.1353446},
publisher = {ACM},
address = {New York, NY, USA},
}
@techreport{CordenKreitzerIntel2009,
title = {Consistency of Floating-Point Results using the Intel Compiler or Why doesn't my application always give the same answer?},
author = {Corden, Martyn J. and Kreitzer, David},
institution = {Intel Corporation, Software Solutions Group},
year = {2009}
}
@techreport{GCCManual2008,
title = {The GNU Compiler Collection},
author = {Free Software Foundation},
year = {2008}
}
@UNPUBLISHED{Intel64IA32Architecture,
title = {Intel 64 and IA-32 Architectures Software Developer's Manual. Volume 1: Basic Architecture},
author = {Intel Corporation},
year = {2009},
note = {\url{http://www.intel.com/products/processor/manuals}}
}
@UNPUBLISHED{MPFRWeb,
author = {MPFR},
title = {The MPFR Library},
note = {\url{http://www.mpfr.org}}
}
@Article{Fousse:2007:MMP,
author = "Laurent Fousse and Guillaume Hanrot and Vincent Lef\`evre and Patrick
P\'elissier and Paul Zimmermann",
title = "{MPFR}: A Multiple-Precision Binary Floating-Point Library with Correct Rounding",
journal = "{ACM} Transactions on Mathematical Software",
volume = "33",
number = "2",
month = jun,
year = "2007",
pages = "13:1--13:15",
URL = "http://doi.acm.org/10.1145/1236463.1236468",
abstract = "This paper presents a multiple-precision binary floating-point library,
written in the ISO C language, and based on the GNU MP library. Its
particularity is to extend to arbitrary-precision ideas from the IEEE 754
standard, by providing \emph{correct rounding} and \emph{exceptions}.
We demonstrate how these strong semantics are achieved --- with no
significant slowdown with respect to other arbitrary-precision tools
--- and discuss a few applications where such a library can be useful.",
}
@UNPUBLISHED{GMPWeb,
author = {GMP},
title = {GNU Multiple Precision Arithmetic Library},
note = {\url{http://gmplib.org}}
}
@book{GolubVanLoad1996,
address = {Baltimore, MD, USA},
author = {Golub, Gene H. and Van Loan, Charles F. },
isbn = {0801854148},
publisher = {Johns Hopkins University Press},
title = {Matrix computations (3rd ed.)},
url = {http://portal.acm.org/citation.cfm?id=248979},
year = {1996}
}
@techreport{ProgrammingC,
title = {Programming languages ? C, ISO/IEC 9899:TC3},
author = {ISO/IEC},
year = {2007},
note = {\url{http://www.open-std.org/jtc1/sc22/wg14/www/docs/n1256.pdf}}
}
@UNPUBLISHED{KahanMarketing2000,
author = {William Kahan},
title = {Marketing versus Mathematics},
year = {2000},
note = {\url{www.cs.berkeley.edu/~wkahan/MktgMath.ps}}
}
@UNPUBLISHED{Li2000,
author = {X. S. Li and J. W. Demmel and D. H. Bailey and G. Henry and Y. Hida and J. Iskandar and
W. Kahan and A. Kapur and M. C. Martin and T. Tung and D. J. Yoo},
title = {Design, Implementation and Testing of Extended and Mixed Precision BLAS},
year = {2000},
note = {\url{www.netlib.org/lapack/lawnspdf/lawn149.pdf}}
}
@Article{Stewart:1986:CNC,
author = "G. W. Stewart",
title = "Corrigendum: ``{A} Note on Complex Division''",
journal = j-TOMS,
volume = "12",
number = "3",
pages = "285--285",
month = sep,
year = "1986",
CODEN = "ACMSCU",
ISSN = "0098-3500",
bibdate = "Fri Sep 30 01:17:25 1994",
note = "See \cite{Stewart:1985:NCD}.",
acknowledgement = ack-nhfb,
}

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