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-rw-r--r--Master/texmf-dist/source/generic/xint/xint.dtx833
1 files changed, 489 insertions, 344 deletions
diff --git a/Master/texmf-dist/source/generic/xint/xint.dtx b/Master/texmf-dist/source/generic/xint/xint.dtx
index 96abc77b56a..7e8e6bb2c34 100644
--- a/Master/texmf-dist/source/generic/xint/xint.dtx
+++ b/Master/texmf-dist/source/generic/xint/xint.dtx
@@ -1,17 +1,20 @@
% -*- coding: iso-latin-1; -*-
-%
-% File: xint.dtx
-% Source of the xint bundle
-% Version: 1.01 of April 4, 2013
-% Copyright (C) 2013 by Jean-Francois Burnol
-%
-% Current style files in the bundle:
+% This file: xint.dtx (1.02, 2013/04/05)
+%%
+%%----------------------------------------------------------------
+%% The xint bundle (version 1.02 of April 5, 2013)
+%<xint>%% xint: Expandable operations on long numbers
+%<xintgcd>%% xintgcd: Euclidean algorithm with xint package
+%% Copyright (C) 2013 by Jean-Francois Burnol
+%%----------------------------------------------------------------
+%%
+% Style files in the bundle:
% (base) xint.sty Expandable operations on long numbers
% xintgcd.sty Euclidean algorithm with xint package
-%
-% This work consists of the source file xint.dtx and of its
-% derived files xint.sty, xintgcd.sty, xint.ins and the
-% documentation xint.pdf (or xint.dvi).
+%
+% This work consists of the source file xint.dtx and of its derived
+% files xint.sty, xintgcd.sty, xint.ins and the documentation xint.pdf
+% (or xint.dvi).
%
% This work may be distributed and/or modified under the
% conditions of the LaTeX Project Public License, either
@@ -55,9 +58,9 @@
% \input xintgcd.sty\relax
%
%<*none>
-\def\pkgversion{1.01}
-\def\pkgdate{2013/04/04}
-\def\lasttimestamp{Time-stamp: <04-04-2013 17:36:58 CEST jfb>}
+\def\pkgversion{1.02}
+\def\pkgdate{2013/04/05}
+\def\lasttimestamp{Time-stamp: <05-04-2013 16:49:16 CEST JF>}
\def\striptimestamp #1 <#2 #3 #4 #5>{#2 at #3 #4}
\def\getdocdate #1 <#2-#3-#4 #5>{#4/#3/#2}
\edef\docdate{\expandafter\getdocdate\lasttimestamp}
@@ -73,14 +76,6 @@
\endgroup
\iffalse
%</none>
-%%
-%%----------------------------------------------------------------
-%% Part of the xint bundle (version 1.01 of 2013/04/04)
-%<xint>%% xint: Expandable operations on long numbers
-%<xintgcd>%% xintgcd: Euclidean algorithm with xint package
-%% Copyright (C) 2013 by Jean-Francois Burnol
-%%----------------------------------------------------------------
-%%
%<*ins>
%----------- to .ins file ----------------------------------------
%%
@@ -366,40 +361,6 @@ numbers (not digits!!!) will work thousands of times faster (or
more, I don't know) than what can be achieved using \TeX{} to
manipulate strings of ASCII representations of digits!
-% \pdfresettimer
-% \edef\x{\xintPow{1325798301}{137}}
-% \the\pdfelapsedtime\
-% \xintLen{\x}
-
-% \pdfresettimer
-% \edef\x{\xintFac{1000}}
-% \edef\T{\the\pdfelapsedtime}\T=
-% \xintQuo\T{65536} secondes\par
-% \pdfresettimer
-% \edef\y{\xintSqr{\x}}
-% \edef\T{\the\pdfelapsedtime}\T=
-% \xintQuo\T{65536} secondes\par
-% \xintLen{\x}\par
-% \xintLen{\y}\par
-
-% Sur l'iMac c'est un peu plus rapide:
-% 55570 1250
-% 573033= 8 secondes
-% 3382960= 51 secondes
-% 2568
-% 5136
-
-% This warning being issued, \xintname computes &1325798301^{137}&
-% which has 1250 digits in less than 1 second (on my 2012 acquired
-% laptop). It checks a Bezout identity involving two multiplications
-% of 200 digits numbers (and a subtraction) in one 12th of a second.
-% It computes 1000! (which has 2568 digits) in less than 10 seconds
-% and its square in less than 60 seconds: of course this will be
-% dwarfed by any specialized software. Communicating such
-% computation times from runs on an unspecified machine is not very
-% precise, but I guess my laptop is representative of the models of
-% the last two years.
-
\section{Expansions}
Except otherwise stated all macros are completely expandable. For
@@ -433,7 +394,7 @@ of typesetting this documentation and not have big numbers extend
into the margin and go beyond the page physical limits, I use this
little macro (not provided by the package):
\begin{verbatim}
-\def\allownumbersplit #1 {\ifx #1\relax \else #1\hskip 0pt plus 1pt
+\def\allownumbersplit #1{\ifx #1\relax \else #1\hskip 0pt plus 1pt
\expandafter\allownumbersplit\fi}%
\end{verbatim}
To provoke the double expansion first, it is used as in:
@@ -776,6 +737,7 @@ strings not containing these control sequence names. Anything
within braces is treated as one unit: one level of exterior braces
is removed and the contents are not reverted.
+\clearpage
\section{Loading and usage}
\begin{verbatim}
@@ -1042,8 +1004,33 @@ argument. \centeredline{%
\subsection{\chb{xintMul}}
-\csa{xintMul\n\m} returns the product of the two numbers. It is
-more efficient to have the first one be the longer of the two.
+\csa{xintMul\n\m} returns the product of the two numbers. The order of the
+numbers influences the efficiency of the computation:
+\begin{enumerate}
+\item if the shortest number has at most 4 digits, then it is (always) more
+ efficient if it comes as the second argument,
+\item if both numbers have at most 50 digits, then it is generally a bit
+ more efficient to have the longest one be the first argument and the shortest
+ one be the second argument,
+\item if one of the number has more than 250 digits, it is always advantageous
+ if the shortest number is the first argument (except if it has only 4 digits
+ or less!). For example, 50 digits $\times$ 1000 digits is five times faster
+ than 1000 digits $\times$ 50 digits,
+\item if both numbers have less than 250 digits, then it is advantageous to have
+ the shortest one be the first argument, as long as it is not too short. The
+ limit depends on the size of the longer number; roughly, when the longer
+ number has 100 digits, this limit on the shorter one is already of 12 digits,
+ and the longer the long number, the lower the limit for the shorter.
+\end{enumerate}
+So when both numbers have at most 50 digits, put the longer one first and the
+shorter one second; when both numbers have at least 50 digits, put the shorter
+one first and the longer one second. The gain will be substantial if the long
+number is very long.
+
+When one of the number has at most 4 digits, always make it second. But if the
+shortest has 5 digits or more, it is advantageous to have it in first
+position when the longer number has 250 digits or more.
+
\subsection{\chb{xintSqr}}
@@ -1064,8 +1051,18 @@ and returns 1: |\xintPrd {}=|\texttt{\xintPrd {}}. A product
reduced to a single term returns this number: |\xintPrd
{{-1234}}=|\texttt{\xintPrd {{-1234}}}. Attention that |\xintPrd
{-1234}| is not legal input and will make the \TeX{} compilation
-fail. On the other hand |\xintPrd {1234}=|\texttt{\xintPrd {1234}}.
-
+fail. On the other hand |\xintPrd {1234}=|\texttt{\xintPrd
+ {1234}}.
+\centeredline{&2^{200}3^{100}7^{100}&}
+\centeredline{=|\xintPrd {{\xintPow {2}{200}}{\xintPow
+ {3}{100}}{\xintPow {7}{100}}}|}
+=\expandafter\expandafter\expandafter\allownumbersplit
+ \xintPrd {{\xintPow {2}{200}}{\xintPow {3}{100}}{\xintPow
+ {7}{100}}}\relax
+\centeredline{=|\xintPow {\xintMul {\xintPow {42}{9}}{43008}}{10}|}
+
+% \expandafter\expandafter\expandafter\allownumbersplit
+% \xintPow {\xintMul {\xintPow {42}{9}}{43008}}{10}\relax
\subsection{\chb{xintProductExpr}}
@@ -1080,7 +1077,15 @@ argument.
\csa{xintFac\n} returns the factorial. It is an error if the
argument is negative or at least &10^9&. It is not recommended to
launch the computation of things such as &100000!&, if you need
-your computer for other tasks.
+your computer for other tasks. On my laptop &1000!& (2568 digits)
+is computed in a little less than ten seconds, &2000!& (5736
+digits) is computed in a little less than one hundred seconds, and
+&3000!& (which has 9131 digits) needs close to seven minutes\dots
+I have no idea how much time &10000!& would need (do rather
+&9999!& if you can, the algorithm has some overhead at the
+transition from &N=9999& to &10000& and higher; &10000!& has 35660
+digits). Not to mention &100000!& which, from the Stirling formula,
+should have 456574 digits.
\subsection{\chb{xintPow}}
@@ -1519,7 +1524,7 @@ and modify it to what is needed.
\fi
\expandafter\x\csname ver@xint.sty\endcsname
\ProvidesPackage{xint}%
- [2013/04/04 v1.01 Expandable operations on long numbers (jfB)]%
+ [2013/04/05 v1.02 Expandable operations on long numbers (jfB)]%
% \end{macrocode}
% \subsection{Token management macros}
% \begin{macrocode}
@@ -3070,7 +3075,20 @@ and modify it to what is needed.
% \begin{verbatim}
% MULTIPLICATION
% Ici #1#2 = 2e input et #3#4 = 1er input
-% Algorithme plus efficace pour #3#4 plus long que #1#2
+% Quelques précisions apportées à l'occasion de 1.02 (qui modifie le
+% fonctionnement de \xintPrd pour tenir compte des règles ci-dessous, et aussi
+% améliore au passage très légèrement la vitesse de calcul de la factorielle):
+% La multiplication est plus rapide sous les conditions suivantes:
+% - si le nombre le plus court a au plus 4 chiffres, il DOIT être en 2e
+% - si les deux ont au plus 50 chiffres, le plus court en *second*
+% - si les deux ont au moins 50 chiffres, le plus court en *premier*
+% - si le plus long a au moins 250 chiffres, mettre ce plus long
+% toujours en *second* (sauf si l'autre a au plus 4 chiffres)
+% - si le plus long a au moins 100 chiffres, en second si le premier a au
+% moins 12 chiffres.
+% La règle générale est donc à peu près: pour les 'gros calculs' mettre le
+% plus court en premier (SAUF pour multiplication par un nombre < 10000), et
+% pour les 'petits calculs' le plus long en premier.
% \end{verbatim}
% \vspace*{-1.5\baselineskip}
% \begin{macrocode}
@@ -3124,8 +3142,7 @@ and modify it to what is needed.
% \end{macrocode}
% \vspace*{-.5\baselineskip}
% \begin{verbatim}
-% Ici #3#1 correspond au **2e input** celui censé être
-% pyschologiquement plus petit
+% Ici #3#1 correspond au **2e input**
% \end{verbatim}
% \vspace*{-1.5\baselineskip}
% \begin{macrocode}
@@ -3461,6 +3478,19 @@ and modify it to what is needed.
% \begin{verbatim}
% \xintPrd {{a}...{z}}
% \xintProductExpr {a}...{z}\relax
+% Release 1.02 modifies \XINT@posprod. The new version takes into account that
+% multiplication on long numbers is more efficient with the shorter one first;
+% the earlier version was on the premise that it was more efficient to give
+% the longer number first and used a special version of multiplication to
+% produce its output in reversed order to serve as next first number. This was
+% optimal for numbers of at most 50 digits, but the bad choice for long
+% numbers. As \xintPrd should be fast when it is used to produce long numbers
+% (producing short numbers means few factors means computation does not take
+% much time anyhow), I revert this choice here. On the other hand this was the
+% correct choice (and tested so) for use in the \xintPow recursion which each
+% times multiplies by something even bigger than what has been obtained so
+% far. So the original version is renamed and moved to serve for \xintPow.
+% The factorial receives separate special revision.
% \end{verbatim}
% \vspace*{-1.5\baselineskip}
% \begin{macrocode}
@@ -3471,13 +3501,12 @@ and modify it to what is needed.
\def\XINT@pprod@checkifempty #1%
{%
\xint@relax #1\XINT@pprod@emptyproduct\relax
- \XINT@pprod@RQfirst #1%
+ \XINT@pprod@first #1%
}%
\def\XINT@pprod@emptyproduct #1\Z { 1}%
-\def\XINT@pprod@RQfirst #1\Z
+\def\XINT@pprod@first #1\Z
{%
- \expandafter\XINT@pprod@getnext\expandafter
- {\romannumeral0\XINT@RQ {}#1\R\R\R\R\R\R\R\R\Z}%
+ \XINT@pprod@getnext {#1}%
}%
\def\XINT@pprod@getnext #1#2%
{%
@@ -3486,21 +3515,22 @@ and modify it to what is needed.
\def\XINT@pprod@checkiffinished #1%
{%
\xint@relax #1\XINT@pprod@end\relax
- \XINT@pprod@compute #1%
+ \XINT@pprod@RQnew #1%
}%
-\def\XINT@pprod@compute #1\Z #2%
+\def\XINT@pprod@RQnew #1\Z
{%
+ \expandafter\XINT@pprod@compute
\expandafter
- \XINT@pprod@getnext
- \expandafter
- {\romannumeral0\XINT@prod@mul@enter #2\W\X\Y\Z #1\W\X\Y\Z}%
+ {\romannumeral0\XINT@RQ {}#1\R\R\R\R\R\R\R\R\Z }%
}%
-\def\XINT@pprod@end\relax\XINT@pprod@compute #1\Z #2%
+\def\XINT@pprod@compute #1#2%
{%
\expandafter
- \xint@cleanupzeros@andstop
- \romannumeral0\XINT@rev {#2}%
+ \XINT@pprod@getnext
+ \expandafter
+ {\romannumeral0\XINT@mul@enter #1\W\X\Y\Z #2\W\X\Y\Z }%
}%
+\def\XINT@pprod@end\relax\XINT@pprod@RQnew #1\Z #2{ #2}%
\def\xintProductExpr {\romannumeral0\xintproductexpr }%
\def\xintPrd {\romannumeral0\xintprd }%
\def\xintprd #1%
@@ -3531,13 +3561,11 @@ and modify it to what is needed.
\def\XINT@prod@returnzero #1\relax { 0}%
\def\XINT@prod@firstisneg #1#2\Z
{%
- \expandafter\XINT@prod@xpxpnext\expandafter
- 0\expandafter{\romannumeral0\XINT@RQ {}#2\R\R\R\R\R\R\R\R\Z}%
+ \XINT@prod@xpxpnext 0{#2}%
}%
\def\XINT@prod@firstispos #1\Z
{%
- \expandafter\XINT@prod@xpxpnext\expandafter
- 1\expandafter{\romannumeral0\XINT@RQ {}#1\R\R\R\R\R\R\R\R\Z }%
+ \XINT@prod@xpxpnext 1{#1}%
}%
\def\XINT@prod@xpxpnext #1#2#3%
{%
@@ -3553,279 +3581,61 @@ and modify it to what is needed.
{%
\xint@zero #1\XINT@prod@returnzero0%
\xint@UDsignfork
- #1\dummy \XINT@prod@neg
- -\dummy \XINT@prod@pos
+ #1\dummy \XINT@prod@neg@RQnew
+ -\dummy \XINT@prod@pos@RQnew
\xint@UDkrof
#1%
}%
-\def\XINT@prod@pos #1\Z #2#3%
+\def\XINT@prod@pos@RQnew #1\Z
+{%
+ \expandafter
+ \XINT@prod@pos
+ \expandafter
+ {\romannumeral0\XINT@RQ {}#1\R\R\R\R\R\R\R\R\Z }%
+}%
+\def\XINT@prod@pos #1#2#3%
{%
\expandafter
\XINT@prod@xpxpnext
\expandafter
#3%
\expandafter
- {\romannumeral0\XINT@prod@mul@enter #2\W\X\Y\Z #1\W\X\Y\Z }%
+ {\romannumeral0\XINT@mul@enter #1\W\X\Y\Z #2\W\X\Y\Z }%
}%
-\def\XINT@prod@neg #1#2\Z #3#4%
+\def\XINT@prod@neg@RQnew #1#2\Z
+{%
+ \expandafter
+ \XINT@prod@neg
+ \expandafter
+ {\romannumeral0\XINT@RQ {}#2\R\R\R\R\R\R\R\R\Z }%
+}%
+\def\XINT@prod@neg #1#2#3%
{%
\expandafter
\XINT@prod@xpxpnext
\expandafter
- {\the\numexpr 1-#4\expandafter}%
+ {\the\numexpr 1-#3\expandafter}%
\expandafter
- {\romannumeral0\XINT@prod@mul@enter #3\W\X\Y\Z #2\W\X\Y\Z }%
+ {\romannumeral0\XINT@mul@enter #1\W\X\Y\Z #2\W\X\Y\Z }%
}%
\def\XINT@prod@end\relax\XINT@prod@checksign #1\Z #2#3%
{%
- \expandafter
- \xint@prod@cleanupzeros
- \romannumeral0\XINT@rev {#2#3}%
+ \xint@prod@cleanupzeros #3#2%
}%
\def\xint@prod@cleanupzeros #1#2#3#4#5%
{%
\expandafter\space\the\numexpr (2*#1-1)*#2#3#4#5\relax
}%
% \end{macrocode}
-% \vspace*{-.5\baselineskip}
-% \begin{verbatim}
-% MULTIPLICATION ET ADDITION POUR LES PRODUITS
-% \end{verbatim}
-% \vspace*{-1.5\baselineskip}
-% \begin{macrocode}
-\def\XINT@prod@add@A #1#2#3#4#5#6%
-{%
- \xint@w
- #3\xint@prod@add@az
- \W\XINT@prod@add@AB #1{#3#4#5#6}{#2}%
-}%
-\def\xint@prod@add@az\W\XINT@prod@add@AB #1#2%
-{%
- \XINT@prod@add@AC@checkcarry #1%
-}%
-\def\XINT@prod@add@AC@checkcarry #1%
-{%
- \xint@zero #1\xint@prod@add@AC@nocarry 0\XINT@prod@add@C
-}%
-\def\xint@prod@add@AC@nocarry 0\XINT@prod@add@C
-{%
- \XINT@prod@add@F
-}%
-\def\XINT@prod@add@AB #1#2#3#4\W\X\Y\Z #5#6#7#8%
-{%
- \XINT@prod@add@ABE #1#2{#8#7#6#5}{#3}#4\W\X\Y\Z
-}%
-\def\XINT@prod@add@ABE #1#2#3#4#5#6%
-{\expandafter
- \XINT@prod@add@ABEA\the\numexpr #1+10#5#4#3#2+#6\relax
-}%
-\def\XINT@prod@add@ABEA #1#2#3#4#5#6#7%
-{%
- \XINT@prod@add@A #2{#7#6#5#4#3}%<-- attention on met donc \`a droite
-}%
-\def\XINT@prod@add@C #1#2#3#4#5%
-{%
- \xint@w
- #5\xint@prod@add@cw
- #4\xint@prod@add@cx
- #3\xint@prod@add@cy
- #2\xint@prod@add@cz
- \W\XINT@prod@add@CD {#5#4#3#2}{#1}%
-}%
-\def\XINT@prod@add@CD #1%
-{\expandafter
- \XINT@prod@add@CC\the\numexpr 1+10#1\relax
-}%
-\def\XINT@prod@add@CC #1#2#3#4#5#6#7%
-{%
- \XINT@prod@add@AC@checkcarry #2{#7#6#5#4#3}%
-}%
-\def\xint@prod@add@cw
- #1\xint@prod@add@cx
- #2\xint@prod@add@cy
- #3\xint@prod@add@cz
- \W\XINT@prod@add@CD
-{\expandafter
- \XINT@prod@add@CDw\the\numexpr 1+10#1#2#3\relax
-}%
-\def\XINT@prod@add@CDw #1#2#3#4#5#6%
-{%
- \xint@quatrezeros #2#3#4#5\XINT@prod@add@endDw@zeros
- 0000\XINT@prod@add@endDw #2#3#4#5%
-}%
-\def\XINT@prod@add@endDw@zeros 0000\XINT@prod@add@endDw 0000#1\X\Y\Z{ #1}%
-\def\XINT@prod@add@endDw #1#2#3#4#5\X\Y\Z{ #5#4#3#2#1}%
-\def\xint@prod@add@cx
- #1\xint@prod@add@cy
- #2\xint@prod@add@cz
- \W\XINT@prod@add@CD
-{\expandafter
- \XINT@prod@add@CDx\the\numexpr 1+100#1#2\relax
-}%
-\def\XINT@prod@add@CDx #1#2#3#4#5#6%
-{%
- \xint@quatrezeros #2#3#4#5\XINT@prod@add@endDx@zeros
- 0000\XINT@prod@add@endDx #2#3#4#5%
-}%
-\def\XINT@prod@add@endDx@zeros 0000\XINT@prod@add@endDx 0000#1\Y\Z{ #1}%
-\def\XINT@prod@add@endDx #1#2#3#4#5\Y\Z{ #5#4#3#2#1}%
-\def\xint@prod@add@cy
- #1\xint@prod@add@cz
- \W\XINT@prod@add@CD
-{\expandafter
- \XINT@prod@add@CDy\the\numexpr 1+1000#1\relax
-}%
-\def\XINT@prod@add@CDy #1#2#3#4#5#6%
-{%
- \xint@quatrezeros #2#3#4#5\XINT@prod@add@endDy@zeros
- 0000\XINT@prod@add@endDy #2#3#4#5%
-}%
-\def\XINT@prod@add@endDy@zeros 0000\XINT@prod@add@endDy 0000#1\Z{ #1}%
-\def\XINT@prod@add@endDy #1#2#3#4#5\Z{ #5#4#3#2#1}%
-\def\xint@prod@add@cz\W\XINT@prod@add@CD #1#2{ #21000}%
-\def\XINT@prod@add@F #1#2#3#4#5%
-{%
- \xint@w
- #5\xint@prod@add@Gw
- #4\xint@prod@add@Gx
- #3\xint@prod@add@Gy
- #2\xint@prod@add@Gz
- \W\XINT@prod@add@G {#2#3#4#5}{#1}%
-}%
-\def\XINT@prod@add@G #1#2%
-{%
- \XINT@prod@add@F {#2#1}%
-}%
-\def\xint@prod@add@Gw
- #1\xint@prod@add@Gx
- #2\xint@prod@add@Gy
- #3\xint@prod@add@Gz
- \W\XINT@prod@add@G #4%
-{%
- \xint@quatrezeros #3#2#10\XINT@prod@add@endGw@zeros
- 0000\XINT@prod@add@endGw #3#2#10%
-}%
-\def\XINT@prod@add@endGw@zeros 0000\XINT@prod@add@endGw 0000#1\X\Y\Z{ #1}%
-\def\XINT@prod@add@endGw #1#2#3#4#5\X\Y\Z{ #5#1#2#3#4}%
-\def\xint@prod@add@Gx
- #1\xint@prod@add@Gy
- #2\xint@prod@add@Gz
- \W\XINT@prod@add@G #3%
-{%
- \xint@quatrezeros #2#100\XINT@prod@add@endGx@zeros
- 0000\XINT@prod@add@endGx #2#100%
-}%
-\def\XINT@prod@add@endGx@zeros 0000\XINT@prod@add@endGx 0000#1\Y\Z{ #1}%
-\def\XINT@prod@add@endGx #1#2#3#4#5\Y\Z{ #5#1#2#3#4}%
-\def\xint@prod@add@Gy
- #1\xint@prod@add@Gz
- \W\XINT@prod@add@G #2%
-{%
- \xint@quatrezeros #1000\XINT@prod@add@endGy@zeros
- 0000\XINT@prod@add@endGy #1000%
-}%
-\def\XINT@prod@add@endGy@zeros 0000\XINT@prod@add@endGy 0000#1\Z{ #1}%
-\def\XINT@prod@add@endGy #1#2#3#4#5\Z{ #5#1#2#3#4}%
-\def\xint@prod@add@Gz\W\XINT@prod@add@G #1#2{ #2}%
-% \end{macrocode}
-% \vspace*{-.5\baselineskip}
+% \subsection{\ch{xintFac}}
% \begin{verbatim}
-%--- multiplication spéciale
+% Modified a bit with 1.02, following changes to \XINT@posprod
+% I am tempted, here and elsewhere, to use \ifcase\XINT@Geq {#1}{1000000000}
+% rather than \ifnum\XINT@Length {#1}>9 but for the time being I leave things
+% as they stand.
% \end{verbatim}
% \vspace*{-1.5\baselineskip}
% \begin{macrocode}
-\def\XINT@prod@mul@enter #1\W\X\Y\Z #2#3#4#5%
-{%
- \xint@w
- #5\xint@prod@mul@enterw
- #4\xint@prod@mul@enterx
- #3\xint@prod@mul@entery
- #2\xint@prod@mul@enterz
- \W\XINT@prod@mul@start {#2#3#4#5}#1\W\X\Y\Z
-}%
-\def\xint@prod@mul@enterw
- #1\xint@prod@mul@enterx
- #2\xint@prod@mul@entery
- #3\xint@prod@mul@enterz
- \W\XINT@prod@mul@start #4#5\W\X\Y\Z \X\Y\Z
-{%
- \XINT@mul@Mr {#3#2#1}#5\Z\Z\Z\Z
-}%
-\def\xint@prod@mul@enterx
- #1\xint@prod@mul@entery
- #2\xint@prod@mul@enterz
- \W\XINT@prod@mul@start #3#4\W\X\Y\Z \Y\Z
-{%
- \XINT@mul@Mr {#2#1}#4\Z\Z\Z\Z
-}%
-\def\xint@prod@mul@entery
- #1\xint@prod@mul@enterz
- \W\XINT@prod@mul@start #2#3\W\X\Y\Z \Z
-{%
- \XINT@mul@Mr {#1}#3\Z\Z\Z\Z
-}%
-\def\XINT@prod@mul@start #1#2\W\X\Y\Z
-{\expandafter
- \XINT@prod@mul@main \expandafter
- {\romannumeral0%
- \XINT@mul@Mr {#1}#2\Z\Z\Z\Z
- }#2\W\X\Y\Z
-}%
-\def\XINT@prod@mul@main #1#2\W\X\Y\Z #3#4#5#6%
-{%
- \xint@w
- #6\xint@prod@mul@mainw
- #5\xint@prod@mul@mainx
- #4\xint@prod@mul@mainy
- #3\xint@prod@mul@mainz
- \W\XINT@prod@mul@compute {#1}{#3#4#5#6}#2\W\X\Y\Z
-}%
-\def\XINT@prod@mul@compute #1#2#3\W\X\Y\Z
-{\expandafter
- \XINT@prod@mul@main \expandafter
- {\romannumeral0\expandafter
- \XINT@mul@Ar \expandafter0\expandafter{\expandafter}%
- \romannumeral0\XINT@mul@Mr {#2}#3\Z\Z\Z\Z \W\X\Y\Z 0000#1\W\X\Y\Z
- }#3\W\X\Y\Z
-}%
-\def\xint@prod@mul@mainw
- #1\xint@prod@mul@mainx
- #2\xint@prod@mul@mainy
- #3\xint@prod@mul@mainz
- \W\XINT@prod@mul@compute #4#5#6\W\X\Y\Z \X\Y\Z
-{%
- \expandafter
- \XINT@prod@add@A \expandafter0\expandafter{\expandafter}%
- \romannumeral0%
- \XINT@mul@Mr {#3#2#1}#6\Z\Z\Z\Z
- \W\X\Y\Z 000#4\W\X\Y\Z
-}%
-\def\xint@prod@mul@mainx
- #1\xint@prod@mul@mainy
- #2\xint@prod@mul@mainz
- \W\XINT@prod@mul@compute #3#4#5\W\X\Y\Z \Y\Z
-{%
- \expandafter
- \XINT@prod@add@A \expandafter0\expandafter{\expandafter}%
- \romannumeral0%
- \XINT@mul@Mr {#2#1}#5\Z\Z\Z\Z
- \W\X\Y\Z 00#3\W\X\Y\Z
-}%
-\def\xint@prod@mul@mainy
- #1\xint@prod@mul@mainz
- \W\XINT@prod@mul@compute #2#3#4\W\X\Y\Z \Z
-{%
- \expandafter
- \XINT@prod@add@A \expandafter0\expandafter{\expandafter}%
- \romannumeral0%
- \XINT@mul@Mr {#1}#4\Z\Z\Z\Z
- \W\X\Y\Z 0#2\W\X\Y\Z
-}%
-\def\xint@prod@mul@mainz\W\XINT@prod@mul@compute #1#2#3\W\X\Y\Z
-{ #1}%
-% \end{macrocode}
-% \subsection{\ch{xintFac}}
-% \begin{macrocode}
\def\xintFac {\romannumeral0\xintfac }%
\def\xintfac #1%
{%
@@ -3849,19 +3659,44 @@ and modify it to what is needed.
}%
\def\XINT@fac@checklength #1%
{%
- \ifnum\xintLen {#1} > 9
- \xint@afterfi{\xintError:FactorialOfTooBigNumber
- \expandafter\space\expandafter 1\xint@gobble@three }%
+ \ifnum \XINT@Length {#1}> 9
+ \xint@afterfi{\xintError:FactorialOfTooBigNumber
+ \expandafter\space\expandafter 1\xint@gobble }%
+ \else
+ \xint@afterfi{\ifnum #1>9999
+ \expandafter\XINT@fac@big@loop
+ \else
+ \expandafter\XINT@fac@loop
+ \fi }%
+ \fi
+ {#1}%
+}%
+\def\XINT@fac@big@loop #1{\XINT@fac@big@loop@main {10000}{#1}{}}%
+\def\XINT@fac@big@loop@main #1#2#3%
+{%
+ \ifnum #1<#2
+ \expandafter
+ \XINT@fac@big@loop@main
+ \expandafter
+ {\the\numexpr #1+1\expandafter }%
\else
- \expandafter\XINT@fac@loop
+ \expandafter\XINT@fac@big@docomputation
\fi
- {1}{#1}{}%
+ {#2}{#3{#1}}%
+}%
+\def\XINT@fac@big@docomputation #1#2%
+{%
+ \expandafter
+ \XINT@pprod@getnext
+ \expandafter
+ {\romannumeral0\XINT@fac@loop {9999}}#2\relax
}%
-\def\XINT@fac@loop #1#2#3%
+\def\XINT@fac@loop #1{\XINT@fac@loop@main 1{#1}{}}%
+\def\XINT@fac@loop@main #1#2#3%
{%
\ifnum #1<#2
\expandafter
- \XINT@fac@loop
+ \XINT@fac@loop@main
\expandafter
{\the\numexpr #1+1\expandafter }%
\else
@@ -3871,10 +3706,39 @@ and modify it to what is needed.
}%
\def\XINT@fac@docomputation #1#2%
{%
- \XINT@posprod #2\relax
+ \XINT@fprod@getnext {1000}#2\relax
+}%
+\def\XINT@fprod@getnext #1#2%
+{%
+ \XINT@fprod@checkiffinished #2\Z {#1}%
+}%
+\def\XINT@fprod@checkiffinished #1%
+{%
+ \xint@relax #1\XINT@fprod@end\relax
+ \XINT@fprod@compute #1%
+}%
+\def\XINT@fprod@compute #1\Z #2%
+{%
+ \expandafter
+ \XINT@fprod@getnext
+ \expandafter
+ {\romannumeral0\XINT@mul@Mr {#1}#2\Z\Z\Z\Z }%
+}%
+\def\XINT@fprod@end\relax\XINT@fprod@compute #1\Z #2%
+{%
+ \expandafter
+ \xint@cleanupzeros@andstop
+ \romannumeral0\XINT@rev {#2}%
}%
% \end{macrocode}
% \subsection{\ch{xintPow}}
+% \begin{verbatim}
+% 1.02 this is the same as in earlier versions, but I had to move here the
+% special routine of Product as it was previously done, with its accompanying
+% special multiplication and addition (to maintain the intermediate results in a
+% special reversed form), now that I have modified the \XINT@posprod routine.
+% \end{verbatim}
+% \vspace*{-1.5\baselineskip}
% \begin{macrocode}
\def\xintPow {\romannumeral0\xintpow }%
\def\xintpow #1%
@@ -4000,7 +3864,7 @@ and modify it to what is needed.
\else
\expandafter\XINT@pow@loop
\fi
- {#1}{#2}\XINT@posprod
+ {#1}{#2}\XINT@pow@posprod
\xint@UNDEF
\xint@undef\xint@undef\xint@undef\xint@undef
\xint@undef\xint@undef\xint@undef\xint@undef
@@ -4014,10 +3878,8 @@ and modify it to what is needed.
\expandafter\XINT@pow@loop@end
\else
\xint@afterfi{\expandafter\XINT@pow@loop@a
- \expandafter{\the\numexpr 2*(#1/2)-#1\expandafter }%
- % b mod 2
- \expandafter{\the\numexpr #1-#1/2\expandafter }%
- % [b/2]
+ \expandafter{\the\numexpr 2*(#1/2)-#1\expandafter }% b mod 2
+ \expandafter{\the\numexpr #1-#1/2\expandafter }% [b/2]
\expandafter{\romannumeral0\xintsqr{#2}}}%
\fi
{{#2}}%
@@ -4036,6 +3898,289 @@ and modify it to what is needed.
\XINT@pow@loop {#1}{#2}%
}%
% \end{macrocode}
+% \vspace*{-.5\baselineskip}
+% \begin{verbatim}
+% Routine de produit servant pour le calcul des puissances.
+% \end{verbatim}
+% \vspace*{-1.5\baselineskip}
+% \begin{macrocode}
+\def\XINT@pow@posprod #1%
+{%
+ \XINT@pow@pprod@checkifempty #1\Z
+}%
+\def\XINT@pow@pprod@checkifempty #1%
+{%
+ \xint@relax #1\XINT@pow@pprod@emptyproduct\relax
+ \XINT@pow@pprod@RQfirst #1%
+}%
+\def\XINT@pow@pprod@emptyproduct #1\Z { 1}%
+\def\XINT@pow@pprod@RQfirst #1\Z
+{%
+ \expandafter\XINT@pow@pprod@getnext\expandafter
+ {\romannumeral0\XINT@RQ {}#1\R\R\R\R\R\R\R\R\Z}%
+}%
+\def\XINT@pow@pprod@getnext #1#2%
+{%
+ \XINT@pow@pprod@checkiffinished #2\Z {#1}%
+}%
+\def\XINT@pow@pprod@checkiffinished #1%
+{%
+ \xint@relax #1\XINT@pow@pprod@end\relax
+ \XINT@pow@pprod@compute #1%
+}%
+\def\XINT@pow@pprod@compute #1\Z #2%
+{%
+ \expandafter
+ \XINT@pow@pprod@getnext
+ \expandafter
+ {\romannumeral0\XINT@pow@mul@enter #2\W\X\Y\Z #1\W\X\Y\Z}%
+}%
+\def\XINT@pow@pprod@end\relax\XINT@pow@pprod@compute #1\Z #2%
+{%
+ \expandafter
+ \xint@cleanupzeros@andstop
+ \romannumeral0\XINT@rev {#2}%
+}%
+% \end{macrocode}
+% \vspace*{-.5\baselineskip}
+% \begin{verbatim}
+% Multiplication spéciale pour emploi par le Produit servant pour le calcul
+% des Puissances (sic)
+% \end{verbatim}
+% \vspace*{-1.5\baselineskip}
+% \begin{macrocode}
+\def\XINT@pow@mul@enter #1\W\X\Y\Z #2#3#4#5%
+{%
+ \xint@w
+ #5\xint@pow@mul@enterw
+ #4\xint@pow@mul@enterx
+ #3\xint@pow@mul@entery
+ #2\xint@pow@mul@enterz
+ \W\XINT@pow@mul@start {#2#3#4#5}#1\W\X\Y\Z
+}%
+\def\xint@pow@mul@enterw
+ #1\xint@pow@mul@enterx
+ #2\xint@pow@mul@entery
+ #3\xint@pow@mul@enterz
+ \W\XINT@pow@mul@start #4#5\W\X\Y\Z \X\Y\Z
+{%
+ \XINT@mul@Mr {#3#2#1}#5\Z\Z\Z\Z
+}%
+\def\xint@pow@mul@enterx
+ #1\xint@pow@mul@entery
+ #2\xint@pow@mul@enterz
+ \W\XINT@pow@mul@start #3#4\W\X\Y\Z \Y\Z
+{%
+ \XINT@mul@Mr {#2#1}#4\Z\Z\Z\Z
+}%
+\def\xint@pow@mul@entery
+ #1\xint@pow@mul@enterz
+ \W\XINT@pow@mul@start #2#3\W\X\Y\Z \Z
+{%
+ \XINT@mul@Mr {#1}#3\Z\Z\Z\Z
+}%
+\def\XINT@pow@mul@start #1#2\W\X\Y\Z
+{\expandafter
+ \XINT@pow@mul@main \expandafter
+ {\romannumeral0%
+ \XINT@mul@Mr {#1}#2\Z\Z\Z\Z
+ }#2\W\X\Y\Z
+}%
+\def\XINT@pow@mul@main #1#2\W\X\Y\Z #3#4#5#6%
+{%
+ \xint@w
+ #6\xint@pow@mul@mainw
+ #5\xint@pow@mul@mainx
+ #4\xint@pow@mul@mainy
+ #3\xint@pow@mul@mainz
+ \W\XINT@pow@mul@compute {#1}{#3#4#5#6}#2\W\X\Y\Z
+}%
+\def\XINT@pow@mul@compute #1#2#3\W\X\Y\Z
+{\expandafter
+ \XINT@pow@mul@main \expandafter
+ {\romannumeral0\expandafter
+ \XINT@mul@Ar \expandafter0\expandafter{\expandafter}%
+ \romannumeral0\XINT@mul@Mr {#2}#3\Z\Z\Z\Z \W\X\Y\Z 0000#1\W\X\Y\Z
+ }#3\W\X\Y\Z
+}%
+\def\xint@pow@mul@mainw
+ #1\xint@pow@mul@mainx
+ #2\xint@pow@mul@mainy
+ #3\xint@pow@mul@mainz
+ \W\XINT@pow@mul@compute #4#5#6\W\X\Y\Z \X\Y\Z
+{%
+ \expandafter
+ \XINT@pow@add@A \expandafter0\expandafter{\expandafter}%
+ \romannumeral0%
+ \XINT@mul@Mr {#3#2#1}#6\Z\Z\Z\Z
+ \W\X\Y\Z 000#4\W\X\Y\Z
+}%
+\def\xint@pow@mul@mainx
+ #1\xint@pow@mul@mainy
+ #2\xint@pow@mul@mainz
+ \W\XINT@pow@mul@compute #3#4#5\W\X\Y\Z \Y\Z
+{%
+ \expandafter
+ \XINT@pow@add@A \expandafter0\expandafter{\expandafter}%
+ \romannumeral0%
+ \XINT@mul@Mr {#2#1}#5\Z\Z\Z\Z
+ \W\X\Y\Z 00#3\W\X\Y\Z
+}%
+\def\xint@pow@mul@mainy
+ #1\xint@pow@mul@mainz
+ \W\XINT@pow@mul@compute #2#3#4\W\X\Y\Z \Z
+{%
+ \expandafter
+ \XINT@pow@add@A \expandafter0\expandafter{\expandafter}%
+ \romannumeral0%
+ \XINT@mul@Mr {#1}#4\Z\Z\Z\Z
+ \W\X\Y\Z 0#2\W\X\Y\Z
+}%
+\def\xint@pow@mul@mainz\W\XINT@pow@mul@compute #1#2#3\W\X\Y\Z
+{ #1}%
+% \end{macrocode}
+% \vspace*{-.5\baselineskip}
+% \begin{verbatim}
+% ADDITION spéciale pour emploi dans la routine de Multiplication utilisée
+% dans la routine de Produit servant pour le calcul des puissances (sic).
+% \end{verbatim}
+% \vspace*{-1.5\baselineskip}
+% \begin{macrocode}
+\def\XINT@pow@add@A #1#2#3#4#5#6%
+{%
+ \xint@w
+ #3\xint@pow@add@az
+ \W\XINT@pow@add@AB #1{#3#4#5#6}{#2}%
+}%
+\def\xint@pow@add@az\W\XINT@pow@add@AB #1#2%
+{%
+ \XINT@pow@add@AC@checkcarry #1%
+}%
+\def\XINT@pow@add@AC@checkcarry #1%
+{%
+ \xint@zero #1\xint@pow@add@AC@nocarry 0\XINT@pow@add@C
+}%
+\def\xint@pow@add@AC@nocarry 0\XINT@pow@add@C
+{%
+ \XINT@pow@add@F
+}%
+\def\XINT@pow@add@AB #1#2#3#4\W\X\Y\Z #5#6#7#8%
+{%
+ \XINT@pow@add@ABE #1#2{#8#7#6#5}{#3}#4\W\X\Y\Z
+}%
+\def\XINT@pow@add@ABE #1#2#3#4#5#6%
+{\expandafter
+ \XINT@pow@add@ABEA\the\numexpr #1+10#5#4#3#2+#6\relax
+}%
+\def\XINT@pow@add@ABEA #1#2#3#4#5#6#7%
+{%
+ \XINT@pow@add@A #2{#7#6#5#4#3}%<-- attention on met donc \`a droite
+}%
+\def\XINT@pow@add@C #1#2#3#4#5%
+{%
+ \xint@w
+ #5\xint@pow@add@cw
+ #4\xint@pow@add@cx
+ #3\xint@pow@add@cy
+ #2\xint@pow@add@cz
+ \W\XINT@pow@add@CD {#5#4#3#2}{#1}%
+}%
+\def\XINT@pow@add@CD #1%
+{\expandafter
+ \XINT@pow@add@CC\the\numexpr 1+10#1\relax
+}%
+\def\XINT@pow@add@CC #1#2#3#4#5#6#7%
+{%
+ \XINT@pow@add@AC@checkcarry #2{#7#6#5#4#3}%
+}%
+\def\xint@pow@add@cw
+ #1\xint@pow@add@cx
+ #2\xint@pow@add@cy
+ #3\xint@pow@add@cz
+ \W\XINT@pow@add@CD
+{\expandafter
+ \XINT@pow@add@CDw\the\numexpr 1+10#1#2#3\relax
+}%
+\def\XINT@pow@add@CDw #1#2#3#4#5#6%
+{%
+ \xint@quatrezeros #2#3#4#5\XINT@pow@add@endDw@zeros
+ 0000\XINT@pow@add@endDw #2#3#4#5%
+}%
+\def\XINT@pow@add@endDw@zeros 0000\XINT@pow@add@endDw 0000#1\X\Y\Z{ #1}%
+\def\XINT@pow@add@endDw #1#2#3#4#5\X\Y\Z{ #5#4#3#2#1}%
+\def\xint@pow@add@cx
+ #1\xint@pow@add@cy
+ #2\xint@pow@add@cz
+ \W\XINT@pow@add@CD
+{\expandafter
+ \XINT@pow@add@CDx\the\numexpr 1+100#1#2\relax
+}%
+\def\XINT@pow@add@CDx #1#2#3#4#5#6%
+{%
+ \xint@quatrezeros #2#3#4#5\XINT@pow@add@endDx@zeros
+ 0000\XINT@pow@add@endDx #2#3#4#5%
+}%
+\def\XINT@pow@add@endDx@zeros 0000\XINT@pow@add@endDx 0000#1\Y\Z{ #1}%
+\def\XINT@pow@add@endDx #1#2#3#4#5\Y\Z{ #5#4#3#2#1}%
+\def\xint@pow@add@cy
+ #1\xint@pow@add@cz
+ \W\XINT@pow@add@CD
+{\expandafter
+ \XINT@pow@add@CDy\the\numexpr 1+1000#1\relax
+}%
+\def\XINT@pow@add@CDy #1#2#3#4#5#6%
+{%
+ \xint@quatrezeros #2#3#4#5\XINT@pow@add@endDy@zeros
+ 0000\XINT@pow@add@endDy #2#3#4#5%
+}%
+\def\XINT@pow@add@endDy@zeros 0000\XINT@pow@add@endDy 0000#1\Z{ #1}%
+\def\XINT@pow@add@endDy #1#2#3#4#5\Z{ #5#4#3#2#1}%
+\def\xint@pow@add@cz\W\XINT@pow@add@CD #1#2{ #21000}%
+\def\XINT@pow@add@F #1#2#3#4#5%
+{%
+ \xint@w
+ #5\xint@pow@add@Gw
+ #4\xint@pow@add@Gx
+ #3\xint@pow@add@Gy
+ #2\xint@pow@add@Gz
+ \W\XINT@pow@add@G {#2#3#4#5}{#1}%
+}%
+\def\XINT@pow@add@G #1#2%
+{%
+ \XINT@pow@add@F {#2#1}%
+}%
+\def\xint@pow@add@Gw
+ #1\xint@pow@add@Gx
+ #2\xint@pow@add@Gy
+ #3\xint@pow@add@Gz
+ \W\XINT@pow@add@G #4%
+{%
+ \xint@quatrezeros #3#2#10\XINT@pow@add@endGw@zeros
+ 0000\XINT@pow@add@endGw #3#2#10%
+}%
+\def\XINT@pow@add@endGw@zeros 0000\XINT@pow@add@endGw 0000#1\X\Y\Z{ #1}%
+\def\XINT@pow@add@endGw #1#2#3#4#5\X\Y\Z{ #5#1#2#3#4}%
+\def\xint@pow@add@Gx
+ #1\xint@pow@add@Gy
+ #2\xint@pow@add@Gz
+ \W\XINT@pow@add@G #3%
+{%
+ \xint@quatrezeros #2#100\XINT@pow@add@endGx@zeros
+ 0000\XINT@pow@add@endGx #2#100%
+}%
+\def\XINT@pow@add@endGx@zeros 0000\XINT@pow@add@endGx 0000#1\Y\Z{ #1}%
+\def\XINT@pow@add@endGx #1#2#3#4#5\Y\Z{ #5#1#2#3#4}%
+\def\xint@pow@add@Gy
+ #1\xint@pow@add@Gz
+ \W\XINT@pow@add@G #2%
+{%
+ \xint@quatrezeros #1000\XINT@pow@add@endGy@zeros
+ 0000\XINT@pow@add@endGy #1000%
+}%
+\def\XINT@pow@add@endGy@zeros 0000\XINT@pow@add@endGy 0000#1\Z{ #1}%
+\def\XINT@pow@add@endGy #1#2#3#4#5\Z{ #5#1#2#3#4}%
+\def\xint@pow@add@Gz\W\XINT@pow@add@G #1#2{ #2}%
+% \end{macrocode}
% \subsection{\ch{xintDivision}, \ch{xintQuo}, \ch{xintRem}}
% \begin{macrocode}
\def\xintQuo {\romannumeral0\xintquo }%
@@ -4583,7 +4728,7 @@ and modify it to what is needed.
\expandafter
\XINT@div@body@h
\romannumeral0\XINT@div@sub@xpxp
- {\romannumeral0\XINT@prod@mul@enter #3\W\X\Y\Z #1\W\X\Y\Z }%
+ {\romannumeral0\XINT@pow@mul@enter #3\W\X\Y\Z #1\W\X\Y\Z }%
{#2}\Z
{#3}{#1}%
}%
@@ -4628,7 +4773,7 @@ and modify it to what is needed.
\expandafter
\XINT@div@body@l
\expandafter{\romannumeral0\XINT@div@sub@xpxp
- {\romannumeral0\XINT@prod@mul@enter #3\W\X\Y\Z #1\W\X\Y\Z }%
+ {\romannumeral0\XINT@pow@mul@enter #3\W\X\Y\Z #1\W\X\Y\Z }%
{\XINT@Rev{#2}}}%
{#4+#1}%
}%
@@ -5632,7 +5777,7 @@ and modify it to what is needed.
\fi
\expandafter\x\csname ver@xintgcd.sty\endcsname
\ProvidesPackage{xintgcd}%
- [2013/04/04 v1.01 Euclide algorithm with xint package (jfB)]%
+ [2013/04/05 v1.02 Euclide algorithm with xint package (jfB)]%
% \end{macrocode}
% \subsection{\ch{xintGCD}}
% \begin{macrocode}
@@ -6340,7 +6485,7 @@ and modify it to what is needed.
Grave accent \` Left brace \{ Vertical bar \|
Right brace \} Tilde \~}
-\CheckSum{6420}
+\CheckSum{6549}
\Finale
%%
generated by cgit v1.2.3 (git 2.39.1) at 2024-11-05 19:08:41 +0000