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diff --git a/Master/texmf-dist/source/generic/xint/xint.dtx b/Master/texmf-dist/source/generic/xint/xint.dtx new file mode 100644 index 00000000000..a652e6fa7c0 --- /dev/null +++ b/Master/texmf-dist/source/generic/xint/xint.dtx @@ -0,0 +1,6164 @@ +% -*- coding: iso-latin-1; -*- +% File: xint.dtx +%%---------------------------------------------------------------- +%% xint: Expandable operations on long numbers +%% xintgcd: Euclidean algorithm with xint package +%% Copyright (C) 2013 by Jean-Francois Burnol +%%---------------------------------------------------------------- +%<*ins> +\def\pkgname{xint} +\def\pkgdate{2013/03/28} +\def\pkgversion{v1.0} +\def\pkgdescription{Expandable operations on long numbers (jfB)} +%</ins> +% +% This work consists of the main source file xint.dtx and +% its derived files, among them the style files xint.sty, +% xintgcd.sty, and the documentation file xint.pdf. +% +% The author of this work is Jean-Francois Burnol +% <jfbu at free dot fr> +% +% This work may be distributed and/or modified under the +% conditions of the LaTeX Project Public License, either +% version 1.3c of this license or (at your option) any later +% version. This version of this license is in +% http://www.latex-project.org/lppl/lppl-1-3c.txt +% and the latest version of this license is in +% http://www.latex-project.org/lppl.txt +% and version 1.3 or later is part of all distributions of +% LaTeX version 2005/12/01 or later. +% +% Installation and Usage: +% ======================= +% +% Run tex or latex on xint.dtx. +% +% This will extract the style files xint.sty and xintgcd.sty (and +% xint.ins). Files with the same names and in the same repertory +% will be overwritten. The tex (not latex) run will stop with the +% complaint that it does not understand \NeedsTeXFormat, but the +% style files will already have been extracted by that time. +% +% Alternatively, run tex or latex on xint.ins if available. +% +% To get xint.pdf run pdflatex thrice on xint.dtx +% +% xint.sty, xintgcd.sty -> TDS:tex/generic/xint/ +% xint.dtx -> TDS:source/generic/xint/ +% xint.pdf -> TDS:doc/generic/xint/ +% +% It may well be necessary to then refresh the TeX installation +% filename database. +% +% Usage with LaTeX: \usepackage{xint} +% \usepackage{xintgcd} +% +% Usage with TeX: \input xint.sty\relax +% \input xintgcd.sty\relax +% +%<*none> +\def\lasttimestamp{Time-stamp: <31-03-2013 20:55:34 CEST jfb>} +\def\docdate{2013/03/31} +\def\striptimestamp#1 <#2 #3 #4 #5>{#2 at #3 #4} +\edef\dtxtimestamp{\expandafter\striptimestamp\lasttimestamp} +\begingroup +\input docstrip.tex +\askforoverwritefalse +\generate{\nopreamble +\file{\pkgname.ins}{\from{\pkgname.dtx}{ins}} +\usepreamble\defaultpreamble +\file{\pkgname.sty}{\from{\pkgname.dtx}{package}} +\file{\pkgname gcd.sty}{\from{\pkgname.dtx}{gcd}}} +\endgroup +\iffalse +%</none> +%<*ins> +%----------- -> .ins file ---------------------------------------- +%% +%% This is a generated file. Run tex or latex on this file to +%% extract xint.sty and xintgcd.sty from xint.dtx +%% +%% See xint.dtx for the statements of copyright and conditions of +%% distribution and/or modification of this work. +%% +\input docstrip.tex +\askforoverwritefalse +\generate{\usepreamble\defaultpreamble +\file{\pkgname.sty}{\from{\pkgname.dtx}{package}} +\file{\pkgname gcd.sty}{\from{\pkgname.dtx}{gcd}}} +\endbatchfile +%----------- end of .ins file ------------------------------------ +%</ins> +%<*none> +\fi +\NeedsTeXFormat{LaTeX2e} +\ProvidesFile{\pkgname.dtx} + [`\pkgname' source and documentation (\dtxtimestamp)] +\documentclass[a4paper,11pt,abstract]{scrdoc} +\pagestyle{headings} +\usepackage[latin1]{inputenc} +\usepackage[T1]{fontenc} +\usepackage[hscale=0.66,vscale=0.75]{geometry} + +\usepackage{xint} +\usepackage{xintgcd} + +\usepackage{txfonts} + +% malheureusement, comme j'utilise des diacritiques dans mes +% parties commentées, imprimées verbatim, je ne pourrai pas +% utiliser dvipdfmx qui a un problème avec txtt + +\DeclareFontFamily{T1}{txtt}{} +\DeclareFontShape{T1}{txtt}{m}{n}{ %medium + <->s*[.96] t1xtt% +}{} +\DeclareFontShape{T1}{txtt}{m}{sc}{ %cap & small cap + <->s*[.96] t1xttsc% +}{} +\DeclareFontShape{T1}{txtt}{m}{sl}{ %slanted + <->s*[.96] t1xttsl% +}{} +\DeclareFontShape{T1}{txtt}{m}{it}{ %italic + <->ssub * txtt/m/sl% +}{} +\DeclareFontShape{T1}{txtt}{m}{ui}{ %unslanted italic + <->ssub * txtt/m/sl% +}{} +\DeclareFontShape{T1}{txtt}{bx}{n}{ %bold extended + <->t1xbtt% +}{} +\DeclareFontShape{T1}{txtt}{bx}{sc}{ %bold extended cap & small cap + <->t1xbttsc% +}{} +\DeclareFontShape{T1}{txtt}{bx}{sl}{ %bold extended slanted + <->t1xbttsl% +}{} +\DeclareFontShape{T1}{txtt}{bx}{it}{ %bold extended italic + <->ssub * txtt/bx/sl% +}{} +\DeclareFontShape{T1}{txtt}{bx}{ui}{ %bold extended unslanted italic + <->ssub * txtt/bx/sl% +}{} +\DeclareFontShape{T1}{txtt}{b}{n}{ %bold + <->ssub * txtt/bx/n% +}{} +\DeclareFontShape{T1}{txtt}{b}{sc}{ %bold cap & small cap + <->ssub * txtt/bx/sc% +}{} +\DeclareFontShape{T1}{txtt}{b}{sl}{ %bold slanted + <->ssub * txtt/bx/sl% +}{} +\DeclareFontShape{T1}{txtt}{b}{it}{ %bold italic + <->ssub * txtt/bx/it% +}{} +\DeclareFontShape{T1}{txtt}{b}{ui}{ %bold unslanted italic + <->ssub * txtt/bx/ui% +}{} + +\usepackage{xspace} +\usepackage{color} + +\definecolor{joli}{RGB}{225,95,0} +\definecolor{JOLI}{RGB}{225,95,0} +\definecolor{BLUE}{RGB}{0,0,255} +\definecolor{niceone}{RGB}{38,128,192} + +\usepackage[english]{babel} + +\usepackage[% +pdfencoding=pdfdoc,bookmarks=true]{hyperref} + +\hypersetup{% +linktoc=all,% +breaklinks=true,% +hidelinks,% +pdfauthor={Jean-Fran\c cois Burnol},% +pdftitle={The xint and xintgcd packages},% +pdfsubject={Arithmetic with TeX},% +pdfkeywords={Expansion, arithmetic, TeX},% +pdfstartview=FitH,% +pdfpagemode=UseOutlines} + + +\makeatletter +% 7 mars 2013 +% This macro allows to conveniently center a line inside a paragraph and still +% use therein \verb or other commands changing catcodes. +% A proposito, the \LaTeX \centerline uses \hsize and not \linewidth ! +\def\@centeredline {\hbox to \linewidth + \bgroup \hss \bgroup + \aftergroup\centeredline@ } + \newcommand*\centeredline {% + \ifhmode + \\\relax + \def\centeredline@{\hss\egroup\hskip\z@skip}% + \else + \def\centeredline@{\hss\egroup}% + \fi + \afterassignment\@centeredline + \let\next=} +\makeatother + +\makeatletter +\let\original@check@percent\check@percent +\let\check@percent\relax +% le \verb de doc.sty est très chiant car il a retiré +% \verbatim@font pour mettre un \ttfamily hard-coded +% à la place. +% +% Par ailleurs j'en ai marre des erreurs dues au fait que mes +% paragraphes reformatés dans emacs passent à la ligne au milieu +% d'un \verb. Je décide donc d'annuler l'effet du \dospecials sur +% les espaces dans la source. Et donc je retire le +% \verb@eol@error et il n'y a donc plus lieu d'un comportement +% différent pour l'impression des blancs, donné par la version étoilée. +% +% Et il n'y avait donc pas de \obeylines puisque la fin de ligne +% devenait un message d'erreur dans \verb@eol@error +% +\def\verb {\relax \ifmmode \hbox \else \leavevmode \null \fi + \bgroup \let \do \do@noligs \verbatim@nolig@list \verbatim@font + \let \do \@makeother \dospecials \catcode 32 10 \@ifstar + {\@sverb }{\@sverb }} +% ça c'est pour mes petits morceaux de code: +\def\verbatim@font {\ttfamily } +\def\MacroFont{\ttfamily\baselineskip12pt\relax} +% Mais j'ai besoin d'un verbatim différent pour les nombres car je +% ne veux pas passer en mode mathématique et je ne veux pas les 0 +% du txtt pour cela. Comme je n'utilise pas de tabulation, je vais +% utiliser & +\catcode`\& 13 +\def&{\begingroup\let\do\@makeother\dospecials\catcode`\& 13 \@jfverb } +\def\@jfverb #1&{#1\endgroup } +\makeatother + +\DeclareRobustCommand\csa[1]{{\ttfamily\char`\\#1}} +\DeclareRobustCommand\csb[1]{{\color{blue}\ttfamily\char`\\#1}} + +\newcommand\ch[1]{\texorpdfstring{\csa{#1}}{\textbackslash #1}} +\newcommand\chb[1]{\texorpdfstring{\csb{#1}}{\textbackslash #1}} + + +\newcommand\xintname{% + \texorpdfstring{{\color{joli}\ttfamily\bfseries xint}} + {xint}\xspace} + +\newcommand\xintgcdname{% + \texorpdfstring{{\color{joli}\ttfamily\bfseries xintgcd}} + {xintgcd}\xspace} + +\frenchspacing + +\renewcommand\familydefault\sfdefault + +\usepackage{framed} + +\begin{document} +\thispagestyle{empty} +\rmfamily + +\begin{center} + {\normalfont\Large The \xintname and \xintgcdname packages}\\ + \textsc{Jean-François Burnol}\par + \footnotesize \ttfamily + jfbu (at) free (dot) fr\\ + Package version: \pkgversion\ (\pkgdate)\\ + Documentation generated from the source file\\ + with timestamp ``\dtxtimestamp'' +\end{center} + +\begin{abstract} + The \xintname package implements with expandable \TeX{} macros + the basic arithmetic operations of addition, subtraction, + multiplication and division, as applied to arbitrarily long + numbers represented as chains of digits with an optional minus + sign. + + The \xintgcdname package provides implementations of the + Euclidean algorithm and of its typesetting. + + The packages may be used with Plain and with \LaTeX. +\end{abstract} + + +% à cause des XX.YY, mais franchement tout ce qui concerne la +% table des matières est une catastrophe de conception avec LaTeX +% et scrartcl n'améliore pas les choses tant que ça ici. +\makeatletter +\def\l@subsection {\bprot@dottedtocline {2}{1.5em}{2.8em}} +\makeatother + + +\tableofcontents + +\section{Origins of this package} + +The package |bigintcalc| by \textsc{Heiko Oberdiek} already +provides expandable arithmetic operations on ``big numbers'', +exceeding the \TeX{} limits (of &2^{31}-1&), so why another +one? + +I got started on this in early March 2013, via a thread on the +|c.t.tex| usenet group, where \textsc{Ulrich D\,i\,e\,z} used the +previously cited package together with a macro (|\ReverseOrder|) +which I had contributed to another thread. \footnote{The + \csa{ReverseOrder} could be avoided in that circumstance, but it + does play a crucial r\^ole here.} What I had learned in this +other thread thanks to interaction with \textsc{Ulrich D\,i\,e\,z} and +\textsc{GL} on expandable manipulations of tokens motivated me to +try my hands at addition and multiplication. + +I wrote macros \csa{bigMul} and \csa{bigAdd} which I posted to the +newsgroup; they appeared to work comparatively fast. These first +versions did not use the \eTeX{} \csa{numexpr} macro, they worked +one digit at a time, having previously stored digit arithmetic in +(many) macros. + +I noticed that the |bigintcalc| package used the \csa{numexpr} +\eTeX{} primitive when available, but (as far as I could tell) not +to do computations many digits at a time. Using \csa{numexpr} for +one digit at a time for \csa{bigAdd} and \csa{bigMul} slowed them +a tiny bit but avoided cluttering \TeX{} memory with 1200 macros +storing pre-computed arithmetic with 2 or 3 digits. I wondered +if some speed could be gained by using +\csa{numexpr} to do four digits at a time for elementary +multiplications (as the maximal admissible number for +\csa{numexpr} has ten digits). + +The present package is the result of this initial questioning. + +\begin{framed}\centering + \xintname requires the \eTeX{} \csa{numexpr} primitive. +\end{framed} + +I have aimed at speed wherever I could, and to the extent that I +could guess what was more efficient for \TeX{}. After a while +though I did opt for more readable coding style in those parts of +the code which were not at the heart of repeatedly used loops. In +particular I started using \csa{ifnum} and \csa{ifcase} constructs +which I had completely avoided so far, working only with macro +expansions. + +This implementation is thus a \TeX nical thing, quite different +from what one would do in a structured programming language like +|C|, although the underlying algorithms are just the standard +steps applied to hand computations (nothing fancy like +Fast Fourier Transform...) + +By the way, yes \xintname enjoys working fast and efficiently with +200 digits numbers, but surely any program (even poorly written) +in |C| using the |CPU| for arithmetic operations on arrays of +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 +example, with the following code snippet within |myfile.tex| +\begin{verbatim} +\newwrite\outfile +\immediate\openout\outfile \jobname-out\relax +\immediate\write\outfile {\xintQuo{\xintPow{2}{1000}}{\xintFac{100}}} +% \immediate\closeout\outfile +\end{verbatim} +the tex run creates a file |myfile-out.tex| +containing the decimal representation of the integer quotient &2^{1000}/100!&. +Or, similar things can happen inside a |\csname...\endcsname|, and +of course in an |\edef|. + +\edef\x{\xintQuo{\xintPow {2}{1000}}{\xintFac{100}}} +\edef\y{\xintLen{\x}} +\def\allownumbersplit #1% + {\ifx #1\relax \else #1\hskip 0pt plus 1pt + \expandafter\allownumbersplit\fi}% + +Furthermore the package macros give their final results in two +expansion steps. They twice expand their arguments so that they +can be arbitrarily chained. Hence \centeredline{% + |\xintLen{\xintQuo{\xintPow{2}{1000}}{\xintFac{100}}}|} expands +in two steps and tells us that &[2^{1000}/100!]& has {\y} +digits. This is not so many and we could print it here: +{\expandafter\expandafter\expandafter\allownumbersplit + \xintQuo{\xintPow {2}{1000}}{\xintFac{100}}\relax}. For the sake +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 + \expandafter\allownumbersplit\fi}% +\expandafter\expandafter\expandafter\allownumbersplit + \xintQuo{\xintPow {2}{1000}}{\xintFac{100}}\relax +\end{verbatim} + +Remarks on the double expansion of arguments: +\begin{enumerate} +\item When I say that the macros expand twice their arguments, + this means that they expand the first token seen (for each + argument), then expand again the first token of the result of + the first expansion. For example + \centeredline{|\def\x{12}\def\y{34}|% + |\xintAdd {\x}{\x\y}|} is \emph{not} a legal construct. It works here + by sheer luck as the |\y| gets expanded inside a |\numexpr|. But + this would fail in general: if you need a more complete + (expandable...) expansion of your initial input, you should use + the \fbox{\csa{bigintcalcNum}} macro from the |bigintcalc| + package. Or, outside of an expandable-only context, just massage + your inputs through \csa{edef}'s. + +\item Unfortunately, after |\def\x {12}|, one can not use just + |-\x| as input to one of the package macros: the rules above + explain that the twice expansion will act only on the minus sign, + hence do nothing. The only way is to use the \csb{xintOpp} + macro, as in for example |\xintAdd + {\xintOpp\x}{\x}|\,=\,{\xintAdd {\xintOpp\x}{\x}}. + +\def\x {12}% +\item With the definition \centeredline{% + |\def\AplusBC #1#2#3{\xintAdd {#1}{\xintMul {#2}{#3}}}|} one + obtains an expandable macro producing the expected result, not + in two, but rather in three steps: a first expansion is consumed + by the macro expanding to its definition. As a result {|\xintAdd + {\AplusBC {1}{2}{3}}{4}|} would then miserably fail. The + solution is to use the \emph{lowercase} form of + \csa{xintAdd}: \centeredline {|\def\AplusBC + #1#2#3{\romannumeral0\xintadd {#1}{\xintMul {#2}{#3}}}|}% + and then \csa{AplusBC} will share the same properties as do the + other \xintname `primitive' macros. +% ENFIN DÉBARRASSÉ DES TRÈS TRÈS TRÈS CHIANTS EOL ERROR DE \verb !!! + + Don't leave any space after the zero, and use the lowercase form + \emph{only} for the external highest level of chained commands. + All \xintname provided public macros have such a lowercase form + for this purpose. +\end{enumerate} + +\section {Inputs} + +After a twice expansion of the arguments, the ensuing numbers have +to be strings of digits with one (and not more) optional minus +sign (and not a plus sign). The first digit is not +zero if there are more than one digit. And |-0| is not legal +input. Syntax such as +|\xintMul\A\B| is accepted and equivalent to |\xintMul {\A}{\B}|. +Or course |\xintAdd\xintMul\A\B\C| does not work, the product +operation must be put within braces: +|\xintAdd{\xintMul\A\B}\C|. + +It would be nice to have a functional form |\add(x,\mul(y,z))| but +this is not provided by the package. Arguments must be either +within braces or a single control sequence. + +For the division (but not for addition, subtraction, or +division), the two inputs must have at most +&2^{31}-9=&{\xintSub{\xintPow {2}{31}}{9}} digits. + +Anyhow I guess that even much smaller sizes exceed the \TeX{} +memory limits on any installation. But if the situation did arise +nevertheless of such a gigantic input, an arithmetic overflow +would occur (after some long time I guess) as \xintname first +computes the length of the inputs by using \csa{numexpr} with +successive additions of the number |8| to itself until the whole +input has been parsed (this initial step is only for the division +algorithm, the three other arithmetic operations remain unaware of +the sizes of their inputs, although they do experience them in a +sense, as they initially reverse the order of digits of at least +one of the input, which means they have to scan it entirely). + +Also: the factorial function \csa{xintFac} will refuse to +(start...) compute |N!| if |N| $\geq$ 1000000000, and the power function +|\xintPow {A}{B}|, when the absolute value \verb+|A|+ is at +least two, will refuse to start the computation if |B| $\geq$ 1000000000 +(the minimal outcome is &2^{1000000000}& which has 301029996 digits...). + +In those latter cases, no arithmetic overflow will happen, but rather, +copied from package |bigintcalc|, undefined control sequences with +names indicating the source of the problem are inserted in the +token stream and will appear in the log file in \TeX{} `undefined +macro' error messages. This will not stop the +computation, which (most of the time) will output a zero. + +No check is done on the format of the inputs after the initial +twice expansion. Often, but not always, something starting with a +|0| will be assumed to be zero (throwing the rest away, or +sometimes not which then will lead to errors). Plus signs are not +accepted and will cause errors. + +The sole exception is the macro \csb{xintNum} which accepts numbers +starting with an arbitrary long sequence of plus signs, minus +signs, followed by zeros and will remove all of them, keeping only +the correct sign: \centeredline{|\xintNum + {+-+-+----++-++----00000000009876543210}|\texttt{=\xintNum + {+-+-+----++-++----0000000009876543210}}} But don't insert +zeros within the initial signs. As with all other package macros, +\csa{xintNum} expands twice its argument, and obtains its final +result in two expansion steps. + +\begin{framed} + \TeX{}'s count registers cannot be directly used but must be + prefixed by |\the| or |\number|. The same for \csa{numexpr} + expressions. +\end{framed} + +\section{Outputs} + +The output, when it consists of a single number, is always in the +normalized form described in the previous section. Some macros +have an output consisting of more than one number, each one is +then within braces. For example \csb{xintDivision} gives first the +quotient and then the remainder, each of them within braces. This +is for programming purposes to avoid having to do twice the +division, once for the quotient, the other one for the remainder: but +of course macros \csb{xintQuo} and \csb{xintRem} are provided for easier +direct access. + +The macro \csb{xintDecSplit} cuts its second argument at a +location specified by its first argument |x|. When |x| is negative +the cut location is from the left end of the number, and if it +exceeds the right end (least significant digit), the second member +of the \csa{xintDecSplit} output will be an \emph{empty} pair of braces; +and if the cut is not too far to the right, the leading zeros of +the right half will not be removed. This is the only case where a +package macro may output something which would need to be input to +\csa{xintNum} before further processing by the other package +macros. + +When using things such as |\ifcase \xintSgn {\A}| one has to leave +a space after the closing brace for \TeX{} to +stop its scanning for a number: once \TeX{} has finished expanding +|\xintSgn{\A}| and has so far obtained either |1|, |0|, or |-1|, a +space (or something `unexpandable') must stop it looking for more digits. + +\section{Assignments} + +\xintAssign\xintBezout{357}{323}\to\tmpA\tmpB\tmpU\tmpV\tmpD + +The end user might not need to maintain at all times complete +expandability. For example why not allow oneself the two definitions +|\edef\A {\xintQuo{100}{3}}| and |\edef\B {\xintRem {100}{3}}|. A special + syntax is provided to make these things more efficient, as we + know that \csa{xintDivision} computes both the quotient and the + remainder at the same time: + \centeredline{\csb{xintAssign}\csa{xintDivision}|{100}{3}|\csb{to}|\A\B|} + \centeredline{\csb{xintAssign}\csa{xintDivision}% +|{\xintPow {2}{1000}}{\xintFac{100}}|\csb{to}|\A\B|} gives +\xintAssign\xintDivision{\xintPow {2}{1000}}{\xintFac{100}}\to\A\B +|\meaning\A|\texttt{: \expandafter\allownumbersplit\meaning\A\relax} and +|\meaning\B|\texttt{: \expandafter\allownumbersplit\meaning\B\relax}. + + + Another example (which uses a macro from the \xintgcdname + package): + \centeredline{\csb{xintAssign}\csa{xintBezout}|{357}{323}|% + \csb{to}|\A\B\U\V\D|} is equivalent to setting |\A| to + \texttt{\tmpA}, |\B| to \texttt{\tmpB}, |\U| to \texttt{\tmpU}, + |\V| to \texttt{\tmpV}, and |\D| to \texttt{\tmpD}. And indeed + (\tmpU)$\times$\tmpA-(\tmpV)$\times$\tmpB= + \xintSub{\xintMul\tmpU\tmpA}{\xintMul\tmpV\tmpB} + is a Bezout Identity. +\xintAssign\xintBezout{3570902836026}{200467139463}\to\tmpA\tmpB\tmpU\tmpV\tmpD +\centeredline{\csb{xintAssign}\csa{xintBezout}|{3570902836026}{200467139463}|% + \csb{to}|\A\B\U\V\D|} gives then |\U|\texttt{: + \expandafter\allownumbersplit\meaning\tmpU\relax} and +|\V|\texttt{: \expandafter\allownumbersplit\meaning\tmpV\relax}. + + + + When one does not know in advance the number of tokens, one can + use \csa{xintAssignArray} or its synonym \csa{xintDigitsOf}: + \centeredline{\csb{xintDigitsOf}\csa{xintPow}|{2}{100}|\csb{to}\csa{Out}} + This defines \csa{Out} to be macro with one parameter, + \csa{Out}|{0}| gives the size |N| of the array and + \csa{Out}|{n}|, for |n| from |1| to |N| then gives the |n|th + element of the array, here the |n|th digit of &2^{100}&, from + the most significant to the least significant. As usual, the + generated macro \csa{Out} is completely expandable and expands twice its + (unique) argument. Consider the following code snippet: +\begin{verbatim} +\newcount\cnta +\newcount\cntb +\begingroup +\xintDigitsOf\xintPow{2}{100}\to\Out +\cnta = 1 +\cntb = 0 +\loop +\advance \cntb \xintSqr{\Out{\the\cnta}} +\ifnum \cnta < \Out{0} +\advance\cnta 1 +\repeat + +|2^{100}| (=\xintPow {2}{100}) has \Out{0} digits and the sum of +their squares is \the\cntb. These digits are, from the least to +the most significant: \cnta = \Out{0} +\loop \Out{\the\cnta}\ifnum \cnta > 1 \advance\cnta -1 , \repeat. +\endgroup +\end{verbatim} + +\newcount\cnta +\newcount\cntb +\begingroup +\xintDigitsOf\xintPow{2}{100}\to\Out +\cnta = 1 +\cntb = 0 +\loop +\advance \cntb \xintSqr{\Out{\the\cnta}} +\ifnum \cnta < \Out{0} +\advance\cnta 1 +\repeat + +&2^{100}& (=\xintPow {2}{100}) has \Out{0} digits and the sum of +their squares is \the\cntb. These digits are, from the least to +the most significant: \cnta = \Out{0} +\loop \Out{\the\cnta}\ifnum \cnta > 1 \advance\cnta -1 , \repeat. +\endgroup + +We used a group in order to release the memory taken by the +\csa{Out} array: indeed internally, besides \csa{Out} itself, +additional macros are defined which are \csa{Out0}, \csa{Out00}, +\csa{Out1}, \csa{Out2}, ..., \csa{OutN}, where |N| is the size of +the array (which is the value returned by |\Out{0}|; the digits +are parts of the names not arguments). + +The command \csb{xintRelaxArray}\csa{Out} sets all these macros to +\csa{relax}, but it was simpler to put everything withing a group. + +Needless to say \csb{xintAssign}, \csb{xintAssignArray} and +\csb{xintDigitsOf} do not do any check on whether the macros they +define are already defined. + +In the example above, we deliberately broke all rules of complete +expandability, but had we wanted to compute the sum of the digits, +not the sum of the squares, we could just have written: +\centeredline{\csb{xintSum}|{\xintPow{2}{100}}=|\texttt{% + \xintSum{\xintPow{2}{100}}}} Indeed, \csa{xintSum} is usually +used as in \centeredline{% + \csb{xintSum}|{{123}{-345}{\xintFac{7}}{\xintOpp{\xintRem{3347}{591}}}}=|\texttt{% + \xintSum{{123}{-345}{\xintFac{7}}{\xintOpp{\xintRem{3347}{591}}}}}} +but in the example above each digit of &2^{100}& is treated as +would have been a summand enclosed within braces, due to the rules +of \TeX{} for parsing macro arguments. + +Note that |{-\xintRem{3347}{591}}| is not a valid input, because +the double expansion will apply only to the minus sign and leave +unaffected the |\xintRem|. So we used \csa{xintOpp} which replaces +a number with its opposite. + +As another use of \csa{xintAssignArray} let us extract one line +from the source code of the \xintgcdname macro +\csb{xintTypesetEuclideAlgorithm}. +\centeredline{|\xintAssignArray\xintEuclideAlgorithm + {#1}{#2}\to\U|} +This is done inside a group. After this command |\U{1}| contains +the number |N| of steps of the algorithm (not to be confused with +|\U{0}=2N+4| which is the number of elements in the |\U| array), +and the GCD is to be found in |\U{3}|, a convenient location +between |\U{2}| and |\U{4}| which are (absolute values of the +twice expansion of) the +initial inputs. Then follow |N| quotients and remainders +from the first to the last step of the algorithm. The +\csa{xintTypesetEuclideAlgorithm} macro organizes this data +for typesetting: this is just an example of one way to do it. + + +%% As an example: \xintTypesetEuclideAlgorithm {2362001530033}{981106461701} + +\section{Error messages} + +We employ the same method as in the |bigintcalc| package. But the +error is always thrown \emph{before} the end of the +|romannumeral0| expansion so as to not disturb further processing +of the token stream, if the operation was a secondary one whose +output is expected by a first one. Here is the list of possible +errors: +\begin{verbatim} +\xintError:ArrayIndexIsNegative +\xintError:ArrayIndexBeyondLimit +\xintError:FactorialOfNegativeNumber +\xintError:FactorialOfTooBigNumber +\xintError:DivisionByZero +\xintError:FractionRoundedToZero +\xintError:ExponentTooBig +\xintError:TooBigDecimalShift +\xintError:TooBigDecimalSplit +\xintError:NoBezoutForZeros +\end{verbatim} + +\section{Package namespace} + +Inner macros of the \xintname and \xintgcdname packages all begin +either with |\XINT@| or with |\xint@|. The package public commands +all start with |\xint|. The major forms have their initials +capitalized, and lowercase forms, prefixed with |\romannumeral0|, +allow definitions of further macros expanding in two steps to +their full expansion (and can thus be chained with the `primitive' +\xintname macros). Some other control sequence names are used +only as delimiters, and left undefined. + +The |\xintReverseOrder|\marg{tokens} macro uses |\XINT@UNDEF| and +|\XINT@undef| as dummy tokens and can be used on arbitrary token +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. + +\section{Loading and usage} + +\begin{verbatim} + Usage with LaTeX: \usepackage{xint} + \usepackage{xintgcd} + + Usage with TeX: \input xint.sty\relax + \input xintgcd.sty\relax +\end{verbatim} + +We have added, directly copied from packages by \textsc{Heiko + Oberdiek}, a mecanism of re-load and \eTeX{} detection, +especially for Plain \TeX{}. As \eTeX{} is required, the +executable |tex| can not be used, |etex| or |pdftex| (version +|1.40| or later) or ..., must +be invoked. + +Furthermore, the package \xintgcdname will check for previous +loading of \xintname, and will try to load it if this was not +already done. + +Also inspired from the \textsc{Heiko Oberdiek} packages we have +included a complete catcode protection mecanism. The packages may +be loaded in any catcode configuration satisfying these +requirements: the percent is comment character, the backslash is +escape character, digits have category code other and letters have +category code letter. Nothing else is assumed, and the previous +configuration is restored after the loading of the packages. + +This is for the loading of the packages. For the actual use of the +macros, note that when feeding them with negative numbers the +minus sign must have category code other, as is standard. + +\xintname presupposes that the usual \csa{space} and +\csa{empty} macros are pre-defined, which is the case in Plain +\TeX{} as well as in \LaTeX. + +Lastly, the macros \csa{xintRelaxArray} (of \xintname) and +\csa{xintTypesetEuclideAlgorithm} and +\csa{xintTypesetBezoutAlgorithm} (of \xintgcdname) use +\csa{loop}, both Plain and \LaTeX{} incarnations are +compatible. \csa{xintTypesetBezoutAlgorithm} also uses the +\csa{endgraf} macro. + + +\section{Installation} + +\begin{verbatim} + Run tex or latex on xint.dtx. + + This will extract the style files xint.sty and xintgcd.sty (and + xint.ins). Files with the same names and in the same repertory + will be overwritten. The tex (not latex) run will stop with the + complaint that it does not understand \NeedsTeXFormat, but the + style files will already have been extracted by that time. + + Alternatively, run tex or latex on xint.ins if available. + + To get xint.pdf run pdflatex thrice on xint.dtx + + xint.sty, xintgcd.sty -> TDS:tex/generic/xint/ + xint.dtx -> TDS:source/generic/xint/ + xint.pdf -> TDS:doc/generic/xint/ + + It may well be necessary to then refresh the TeX installation + filename database. +\end{verbatim} + + +\section{Commands of the \xintname package} + +\def\n{\string{N\string}} +\def\m{\string{M\string}} +\def\x{\string{x\string}} + +\n{} stands for a normalised number within braces as described in +the documentation, or for a control sequence expanding in at most +two steps to such a number (without the braces!), or for a control +sequence within braces expanding in at most two steps to such a +number, of for material within braces which expands in two +expansion of the first token to such a number. + +\subsection{\chb{xintRev}} + +\csa{xintRev\n} will revert the order of the digits of the number, +keeping the optional sign. Leading zeros +resulting from the operation are not removed (see the +\csa{xintNum} macro for this). +\centeredline{|\xintRev{-123000}|\texttt{=\xintRev{-123000}}} +\centeredline{|\xintNum{\xintRev{-123000}}|\texttt{=\xintNum{\xintRev{-123000}}}} + +\subsection{\chb{xintReverseOrder}} + +\csa{xintReverseOrder}\marg{token\_list} does not do any +expansion of its argument and just reverses the order of the +tokens. Brace pairs encountered are removed once and the enclosed +material does not get reverted. + +\subsection{\chb{xintNum}} + +\csa{xintNum\n} removes chains of plus or minus signs, followed by +zeros. +\centeredline{|\xintNum{+---++----+--000000000367941789479}|\texttt +{=\xintNum{+---++----+--000000000367941789479}}} + +\subsection{\chb{xintLen}} + +\csa{xintLen\n} returns the length of the number, not counting the +sign. +\centeredline{|\xintLen{-12345678901234567890123456789}|\texttt +{=\xintLen{-12345678901234567890123456789}}} + +\subsection{\chb{xintLength}} + +\csa{xintLength}\marg{token\_list} does not do any expansion of +its argument and just counts how many tokens there are. Things +enclosed in braces count as one, and there should be no such +brace group within the final eight slots. + +\subsection{\chb{xintAssign}} + +\csa{xintAssign}\meta{braced things}\csa{to}% +\meta{as many cs as they are things} defines (without checking if +something gets overwritten) the control sequences on the right of +\csa{to} to be the complete expansions of the successive things on +the left of \csa{to} enclosed within braces. + +Important: a double expansion is applied first to the material +extending up to \csa{to}. + +\xintAssign\xintPow {7}{13}\to\SevenToThePowerThirteen +\xintAssign\xintDivision{1000000000000}{133333333}\to\Q\R + +As a special exception, if after this initial double expansion a +brace does not immediately follows \csa{xintAssign}, it is assumed +that there is only one control sequence to define and it is then +defined to be the complete expansion of the material between +\csa{xintAssign} and \csa{to}. +\centeredline{|\xintAssign\xintDivision{1000000000000}{133333333}\to\Q\R|} +\centeredline{|\meaning\Q: |\texttt{\meaning\Q}, |\meaning\R: + |\texttt{\meaning\R}} \centeredline{|\xintAssign\xintPow + {7}{13}\to\SevenToThePowerThirteen|} +\centeredline{|\SevenToThePowerThirteen=|\texttt{\SevenToThePowerThirteen}} + +Of course this macro and its cousins completely break usage in +pure expansion contexts, as assignments are made via the +\csa{edef} primitive. + +\subsection{\chb{xintAssignArray}} + +\xintAssignArray\xintBezout {1000}{113}\to\Bez + +\csa{xintAssign}\meta{braced things}\csa{to}\csa{myArray} first +double expands the first token then defines \csa{myArray} to be a +macro with one parameter, such that \csa{myArray\n} expands in two +steps (which include the twice-expansion of \texttt{\n}) to give +the |N|th braced thing, itself completely expanded. +\csa{myArray}|{0}| returns the number |M| of elements of the array +so that the successive elements are \csa{myArray}|{1}|, \dots, +\csa{myArray}|{M}|. \centeredline{|\xintAssignArray\xintBezout + {1000}{113}\to\Bez|} will set |\Bez{0}| to \texttt{\Bez0}, +|\Bez{1}| to \texttt{\Bez1}, |\Bez{2}| to \texttt{\Bez2}, +|\Bez{3}| to \texttt{\Bez3}, |\Bez{4}| to \texttt{\Bez4}, and +|\Bez{5}| to \texttt{\Bez5}: +(\Bez3)${}\times{}$\Bez1${}-{}$(\Bez4)${}\times{}$\Bez2${}={}$\Bez5. + +\subsection{\chb{xintRelaxArray}} + +\csa{xintRelaxArray}\csa{myArray} sets to \csa{relax} all +macros which were defined by the previous \csa{xintAssignArray} +with \csa{myArray} as array name. + +\subsection{\chb{xintDigitsOf}} + +This is a synonym for \csa{xintAssignArray}, to be used to define +an array giving all the digits of a given number. + +\subsection{\chb{xintSgn}} + +\csa{xintSgn\n} returns 1 if the number is positive, 0 if it is +zero and -1 if it is negative. + +\subsection{\chb{xintOpp}} + +\csa{xintOpp\n} returns the opposite |-N| of the number |N|. + +\subsection{\chb{xintAbs}} + +\csa{xintAbs\n} returns the absolute value of the number. + +\subsection{\chb{xintAdd}} + +\csa{xintAdd\n\m} returns the sum of the two numbers. + +\subsection{\chb{xintSub}} + +\csa{xintSub\n\m} returns the difference |N-M|. + +\subsection{\chb{xintCmp}} + +\csa{xintCmp\n\m} returns 1 if |N>M|, 0 if |N=M|, and -1 if |N<M|. + +\subsection{\chb{xintGeq}} + +\csa{xintGeq\n\m} returns 1 if the absolute value of the first +number is at least equal to the absolute value of the second +number. If \verb+|N|<|M|+ it returns 0. + +\subsection{\chb{xintMax}} + +\csa{xintMax\n\m} returns the largest of the two in the sense of the order +structure on the relative integers (\emph{i.e.} the right-most +number if they are put on a line with positive numbers on the right). + +\subsection{\chb{xintMin}} + +\csa{xintMin\n\m} returns the smallest of the two in the sense of the order +structure on the relative integers (\emph{i.e.} the left-most +number if they are put on a line with positive numbers on the right). + +\subsection{\chb{xintSum}} + +\csa{xintSum}\marg{braced things} after expanding its argument +twice expects to find a sequence of tokens (or braced material). +Each is twice-expanded, and the sum of all these numbers is +returned. +\centeredline{% + \csa{xintSum}|{{123}{-98763450}{\xintFac{7}}{\xintMul{3347}{591}}}=|\texttt{% + \xintSum{{123}{-98763450}{\xintFac{7}}{\xintMul{3347}{591}}}}} +\centeredline{\csa{xintSum}|{1234567890}=|\texttt{% + \xintSum{1234567890}}} + +\subsection{\chb{xintSumExpr}} + +\csa{xintSum}\meta{braced things}\csa{relax} is to what +\csa{xintSum} reduces after its initial double expansion of its +argument. \centeredline{% + \csa{xintSumExpr}| {123}{-98763450}|% + |{\xintFac{7}}{\xintMul{3347}{591}}\relax=|\texttt{% + \xintSumExpr {123}{-98763450}{\xintFac{7}}{\xintMul{3347}{591}}\relax}} + +\subsection{\chb{xintMul}} + +\csa{xintMul\n\m} returns the product of the two numbers. + +\subsection{\chb{xintSqr}} + +\csa{xintSqr\n} returns the square. + +\subsection{\chb{xintPrd}} + +\csa{xintPrd}\marg{braced things} after expanding its argument +twice expects to find a sequence of tokens (or braced material). +Each is twice-expanded, and the product of all these numbers is +returned. +\centeredline{% + \csa{xintPrd}|{{-9876}{\xintFac{7}}{\xintMul{3347}{591}}}=|% +\texttt{% + \xintPrd{{-9876}{\xintFac{7}}{\xintMul{3347}{591}}}}} +\centeredline{\csa{xintPrd}|{123456789123456789}=|\texttt{% + \xintPrd{123456789123456789}}} + + +\subsection{\chb{xintProductExpr}} + +\csa{xintProductExpr}\meta{braced things}\csa{relax} is to what +\csa{xintPrd} reduces after its initial double expansion of its +argument. +\centeredline{\csa{xintProductExpr}| 123456789123456789\relax=|\texttt{% + \xintProductExpr 123456789123456789\relax}} + +\subsection{\chb{xintFac}} + +\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. + +\subsection{\chb{xintPow}} + +\csa{xintPow\n\m} returns |N^M|. When |M| is zero, this is 1. Some +cases (|N| zero and |M| negative, \verb+|N|>1+ and |M| negative, +\verb+|N|>1+ and |M| at least &10^9&) make \xintname throw errors. + +\subsection{\chb{xintDivision}} + +\csa{xintDivision\n\m} returns |{quotient Q}{remainder R}|. This +is euclidean division: |N = QM + R|, |0|${}\leq{}$\verb+R < |M|+. So the +remainder is always non-negative and the formula |N = QM + R| +always holds independently of the signs of |N| or |M|. Division by +zero is of course an error (even if |N| vanishes) and returns |{0}{0}|. + +\subsection{\chb{xintQuo}} + +\csa{xintQuo\n\m} returns the quotient from the euclidean division. + +\subsection{\chb{xintRem}} + +\csa{xintRem\n\m} returns the remainder from the euclidean division. + + +\subsection{\chb{xintFDg}} + +\csa{xintFDg\n} returns the first digit (most significant) of the +decimal expansion. + +\subsection{\chb{xintLDg}} + +\csa{xintLDg\n} returns the least significant digit. When the +number is positive, this is the same as the remainder in the +euclidean division by ten. + +\subsection{\chb{xintOdd}} + +\csa{xintOdd\n} is 1 if the number is odd and 0 otherwise. + +\subsection{\chb{xintDSL}} + +\csa{xintDSL\n} is decimal shift left, \emph{i.e.} multiplication +by ten. + +\subsection{\chb{xintDSR}} + +\csa{xintDSR\n} is decimal shift right, \emph{i.e.} it removes the +last digit (keeping the sign). For a positive number, this is the +same as the quotient from the +euclidean division by ten. + +\subsection{\chb{xintDSH}} + +\csa{xintDSH\x\n} is parametrized decimal shift. When |x| is +negative, it is like iterating \csa{xintDSL} \verb+|x|+ times +(\emph{i.e.} multiplication by &10^{-&|x|&}&). When |x| is +positive, it is like iterating \csa{DSR} |x| times. When |x| +exceeds the length of the number, the result is zero. + +\subsection{\chb{xintDecSplit}} + + +\csa{xintDecSplit\x\n} cuts the number into two pieces (each +within a pair of enclosing braces). First the +sign if present is \emph{removed}. Then, when |x| +is positive or vanishes, this is like the +euclidean division by &10^{&|x|&}&. When |x| is negative the +number is split into a first piece with the \verb+|x|+ most +significant digits and a second piece with the remaining digits. +Leading zeros in this second piece are not removed. In the case +where the absolute value of |x| is at least the length of the +number, the second piece is empty (not zero!). So the absolute +value of the original number is always the concatenation of the +first and second piece, in this case with a negative |x|. +\xintAssign\xintDecSplit {0}{-123004321}\to\L\R +\centeredline{|\xintAssign\xintDecSplit {0}{-123004321}\to\L\R|} +|\meaning\L: |\texttt{\meaning\L}, |\meaning\R: |\texttt{\meaning\R} +\xintAssign\xintDecSplit {6}{-123004321}\to\L\R +\centeredline{|\xintAssign\xintDecSplit {6}{-123004321}\to\L\R|} +|\meaning\L: |\texttt{\meaning\L}, |\meaning\R: |\texttt{\meaning\R} +\xintAssign\xintDecSplit {-5}{-12300004321}\to\L\R +\centeredline{|\xintAssign\xintDecSplit {-5}{-12300004321}\to\L\R|} +|\meaning\L: |\texttt{\meaning\L}, |\meaning\R: |\texttt{\meaning\R} +\xintAssign\xintDecSplit {-11}{-12300004321}\to\L\R +\centeredline{|\xintAssign\xintDecSplit {-11}{-12300004321}\to\L\R|} +|\meaning\L: |\texttt{\meaning\L}, |\meaning\R: |\texttt{\meaning\R} + +\subsection{\chb{xintDecSplitL}} + +\csa{xintDecSplitL\x\n} returns the first piece after the action +of \csa{xintDecSplit}. + +\subsection{\chb{xintDecSplitR}} + +\csa{xintDecSplitR\x\n} returns the second piece after the action +of \csa{xintDecSplit}. + + +\section{Commands of the \xintgcdname package} + + +\subsection{\chb{xintGCD}} + +\csa{xintGCD\n\m} computes the greatest common divisor. It is +positive, except when both |N| and |M| vanish, for which the macro +returns zero. +\centeredline{\csa{xintGCD}|{10000}{1113}=|\texttt{\xintGCD{10000}{1113}}} + +\subsection{\chb{xintBezout}} + +\xintAssign{{\xintBezout {10000}{1113}}}\to\X +\xintAssign {\xintBezout {10000}{1113}}\to\A\B\U\V\D + +\csa{xintBezout\n\m} returns five numbers |A|, |B|, |U|, |V|, |D| within +braces. |A| is the first (twice-expanded) input number, |B| the +second, |D| is the GCD, and \texttt{UA - VB = D}. +\centeredline{|\xintAssign {{\xintBezout {10000}{1113}}}\to\X|} +\centeredline{|\meaning\X: |\texttt{\meaning\X }.} +\centeredline{|\xintAssign {\xintBezout {10000}{1113}}\to\A\B\U\V\D|} +|\meaning\A: |\texttt{\meaning\A }, +|\meaning\B: |\texttt{\meaning\B }, +|\meaning\U: |\texttt{\meaning\U }, +|\meaning\V: |\texttt{\meaning\V }, +|\meaning\D: |\texttt{\meaning\D }. + +\subsection{\chb{xintEuclideAlgorithm}} + +\xintAssign {{\xintEuclideAlgorithm {10000}{1113}}}\to\X + +\def\restorebracecatcodes + {\catcode`\{=1 \catcode`\}=2 } + +\def\allowlistsplit + {\catcode`\{=12 \catcode`\}=12 \allowlistsplita } + +\def\allowlistsplitx {\futurelet\listnext\allowlistsplitxx } + +\def\allowlistsplitxx {\ifx\listnext\relax \restorebracecatcodes + \else \expandafter\allowlistsplitxxx \fi } +\begingroup +\catcode`\[=1 +\catcode`\]=2 +\catcode`\{=12 +\catcode`\}=12 +\gdef\allowlistsplita #1{[#1\allowlistsplitx {] +\gdef\allowlistsplitxxx {#1}% + [{#1}\hskip 0pt plus 1pt \allowlistsplitx ] +\endgroup + +\csa{xintEuclideAlgorithm\n\m} applies the Euclide algorithm and +keeps a copy of all quotients and remainders. +\centeredline{|\xintAssign {{\xintEuclideAlgorithm {10000}{1113}}}\to\X|} + +|\meaning\X: |\texttt{\expandafter\allowlistsplit\meaning\X + \relax }. +The first token is the number of steps, the second is |N|, the +third is the GCD, the fourth is |M| then the first quotient and +remainder, the second quotient and remainder, \dots until the +final quotient and last (zero) remainder. + +\subsection{\chb{xintBezoutAlgorithm}} + +\catcode`\& 4 + +\xintAssign {{\xintBezoutAlgorithm {10000}{1113}}}\to\X + +\csa{xintBezoutAlgorithm\n\m} applies the Euclide algorithm and +keeps a copy of all quotients and remainders. Furthermore it +computes the entries of the successive products of the 2 by 2 matrices +$\left(\vcenter{\halign {\,#&\,#\cr q & 1 \cr 1 & 0 \cr}}\right)$ +formed from the quotients arising in the algorithm. +\centeredline{|\xintAssign {{\xintEuclideAlgorithm {10000}{1113}}}\to\X|} + +|\meaning\X: |\texttt{\expandafter\allowlistsplit\meaning\X \relax}. + +The first token is the number of steps, the second is |N|, then +|0|, |1|, the GCD, |M|, |1|, |0|, the first quotient, the first +remainder, the top left entry of the first matrix, the bottom left +entry, and then these four things at each step until the end. + +\subsection{\chb{xintTypesetEuclideAlgorithm}} + +This macro is just an example of how to organize the data returned +by \csa{xintEuclideAlgorithm}. See the source code and modify it +to what is needed. +\centeredline{|\xintTypesetEuclideAlgorithm {10000}{1113}|} +\xintTypesetEuclideAlgorithm {10000}{1113} + + +\subsection{\chb{xintTypesetBezoutAlgorithm}} + +This macro is just an example of how to organize the data returned +by \csa{xintBezoutAlgorithm}. See the source code and modify it +to what is needed. +\centeredline{|\xintTypesetBezoutAlgorithm {10000}{1113}|} +\xintTypesetBezoutAlgorithm {10000}{1113} + + + + +\makeatletter +\let\check@percent\original@check@percent +\StopEventually{\check@checksum\end{document}\endinput} +\makeatother + +\def\MacroFont{\ttfamily\small\baselineskip12pt\relax} + +\MakePercentIgnore +% +% \catcode`\<=0 \catcode`\>=11 \catcode`\*=11 \catcode`\/=11 +% \let</none>\relax +% \def<*package>{\catcode`\<=12 \catcode`\>=12 \catcode`\*=12 \catcode`\/=12} +% +%</none> +%<*package> +% \section {Package \xintname implementation} +% +% The commenting of the macros is currently (\docdate) very +% sparse. Some comments may be left-overs from previous versions +% of the macro, with parameters in another order for example. +% +% \subsection{Catcodes, \eTeX{} detection, reload detection} +% +% The method for package identification and reload detection is +% copied verbatim from the packages by \textsc{Heiko Oberdiek}. +% +% The method for catcodes was also inspired by these packages, we +% proceed slightly differently. +% +% \begin{macrocode} +\begingroup\catcode61\catcode48\catcode32=10\relax% + \catcode13=5 % ^^M + \endlinechar=13 % + \catcode123=1 % { + \catcode125=2 % } + \catcode64=11 % @ + \catcode35=6 % # + \catcode44=12 % , + \catcode45=12 % - + \catcode46=12 % . + \catcode58=12 % : + \expandafter\let\expandafter\x\csname ver@xint.sty\endcsname + \expandafter + \ifx\csname PackageInfo\endcsname\relax + \def\y#1#2{\immediate\write-1{Package #1 Info: #2.}}% + \else + \def\y#1#2{\PackageInfo{#1}{#2}}% + \fi + \expandafter + \ifx\csname numexpr\endcsname\relax + \y{xint}{\numexpr not available, aborting input}% + \aftergroup\endinput + \else + \ifx\x\relax % plain-TeX, first loading + \else + \def\empty {}% + \ifx\x\empty % LaTeX, first loading, + % variable is initialized, but \ProvidesPackage not yet seen + \else + \y{xint}{I was already loaded, aborting input}% + \aftergroup\endinput + \fi + \fi + \fi + \def\ChangeCatcodesIfInputNotAborted + {% + \endgroup + \edef\XINT@restorecatcodes@endinput + {% + \catcode47=\the\catcode47 % / + \catcode41=\the\catcode41 % ) + \catcode40=\the\catcode40 % ( + \catcode42=\the\catcode42 % * + \catcode43=\the\catcode43 % + + \catcode62=\the\catcode62 % > + \catcode60=\the\catcode60 % < + \catcode58=\the\catcode58 % : + \catcode46=\the\catcode46 % . + \catcode45=\the\catcode45 % - + \catcode44=\the\catcode44 % , + \catcode35=\the\catcode35 % # + \catcode64=\the\catcode64 % @ + \catcode125=\the\catcode125 % } + \catcode123=\the\catcode123 % { + \endlinechar=\the\endlinechar + \catcode13=\the\catcode13 % ^^M + \catcode32=\the\catcode32 % + \catcode61=\the\catcode61 % = + \noexpand\endinput + }% + \def\XINT@setcatcodes + {% + \catcode61=12 % = + \catcode32=10 % space + \catcode13=5 % ^^M + \endlinechar=13 % + \catcode123=1 % { + \catcode125=2 % } + \catcode64=11 % @ + \catcode35=6 % # + \catcode44=12 % , + \catcode45=12 % - + \catcode46=12 % . + \catcode58=11 % : (made letter for error cs) + \catcode60=12 % < + \catcode62=12 % > + \catcode43=12 % + + \catcode42=12 % * + \catcode40=12 % ( + \catcode41=12 % ) + \catcode47=12 % / + }% + \XINT@setcatcodes + }% +\ChangeCatcodesIfInputNotAborted +% \end{macrocode} +% \subsection{Package identification} +% +% Copied verbatim from \textsc{Heiko Oberdiek}'s packages. +% +% \begin{macrocode} +\begingroup + \catcode91=12 % [ + \catcode93=12 % ] + \catcode58=12 % : (does not really matter, was letter) + \expandafter\ifx\csname ProvidesPackage\endcsname\relax + \def\x#1#2#3[#4]{\endgroup + \immediate\write-1{Package: #3 #4}% + \xdef#1{#4}% + }% + \else + \def\x#1#2[#3]{\endgroup + #2[{#3}]% + \ifx#1\@undefined + \xdef#1{#3}% + \fi + \ifx#1\relax + \xdef#1{#3}% + \fi + }% + \fi +\expandafter\x\csname ver@xint.sty\endcsname +\ProvidesPackage{xint}% + [2013/03/28 v1.0 Expandable operations on long numbers (jfB)]% +% \end{macrocode} +% \subsection{Token management macros} +% \begin{macrocode} +\def\xint@gobble #1{}% +\def\xint@gobble@one #1{}% +\def\xint@gobble@two #1#2{}% +\def\xint@gobble@three #1#2#3{}% +\def\xint@gobble@four #1#2#3#4{}% +\def\xint@gobble@five #1#2#3#4#5{}% +\def\xint@gobble@six #1#2#3#4#5#6{}% +\def\xint@gobble@seven #1#2#3#4#5#6#7{}% +\def\xint@gobble@eight #1#2#3#4#5#6#7#8{}% +\def\xint@secondoftwo #1#2{#2}% +\def\xint@firstoftwo@andstop #1#2{ #1}% +\def\xint@secondoftwo@andstop #1#2{ #2}% +\def\xint@exchangetwo@keepbraces #1#2{{#2}{#1}}% +\def\xint@exchangetwo@keepbraces@andstop #1#2{ {#2}{#1}}% +\def\xint@xpxp@andstop {\expandafter\expandafter\expandafter\space }% +\def\xint@r #1\R {}% +\def\xint@w #1\W {}% +\def\xint@z #1\Z {}% +\def\xint@zero #10{}% +\def\xint@one #11{}% +\def\xint@minus #1-{}% +\def\xint@relax #1\relax {}% +\def\xint@quatrezeros #10000{}% +\def\xint@bracedundef {\xint@undef }% +\def\xint@UDzerofork #10\dummy #2#3\xint@UDforkzero {#2}% +\def\xint@UDzerosfork #100\dummy #2#3\xint@UDforkzeros {#2}% +\def\xint@UDsignfork #1-\dummy #2#3\xint@UDforksign {#2}% +\def\xint@UDsignsfork #1--\dummy #2#3\xint@UDforksigns {#2}% +\def\xint@UDzerominusfork #10-\dummy #2#3\xint@UDforkminuszero {#2}% +\def\xint@afterfi #1#2\fi {\fi #1}% +% \end{macrocode} +% \subsection{\ch{xintRev}, \ch{xintReverseOrder}} +% \begin{verbatim} +% \xintRev: fait la double expansion, vérifie le signe +% \xintReverseOrder: ne fait PAS la double expansion, ne regarde +% PAS le signe. +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\xintRev {\romannumeral0\xintrev }% +\def\xintrev #1% +{% + \expandafter\expandafter\expandafter + \xint@rev + \expandafter\expandafter\expandafter + {#1}% +}% +\def\xint@rev #1% +{% + \XINT@rev@fork #1\Z +}% +\def\XINT@rev@fork #1#2% +{% + \xint@UDsignfork + #1\dummy \XINT@rev@negative + -\dummy \XINT@rev@nonnegative + \xint@UDforksign + #1#2% +}% +\def\XINT@rev@negative #1#2\Z +{% + \expandafter + \space + \expandafter + -% + \romannumeral0\XINT@rev {#2}% +}% +\def\XINT@rev@nonnegative #1\Z +{% + \XINT@rev {#1}% +}% +\def\XINT@Rev {\romannumeral0\XINT@rev }% +\let\xintReverseOrder \XINT@Rev +\def\XINT@rev #1% +{% + \XINT@rord@main {}#1% + \xint@UNDEF + \xint@undef\xint@undef\xint@undef\xint@undef + \xint@undef\xint@undef\xint@undef\xint@undef + \xint@UNDEF +}% +\def\XINT@rord@main #1#2#3#4#5#6#7#8#9% +{% + \XINT@strip@undef #9\XINT@rord@cleanup\xint@undef + \XINT@rord@main {#9#8#7#6#5#4#3#2#1}% +}% +\def\XINT@rord@cleanup\xint@undef\XINT@rord@main #1#2\xint@UNDEF +{% + \expandafter\space\XINT@strip@UNDEF #1% +}% +\def\XINT@strip@undef #1\xint@undef {}% +\def\XINT@strip@UNDEF #1\xint@UNDEF {}% +% \end{macrocode} +% \subsection{\ch{XINT@RQ}} +% \begin{verbatim} +% cette macro renverse et ajoute le nombre minimal de zéros à +% la fin pour que la longueur soit alors multiple de 4 +% \romannumeral0\XINT@RQ {}<le truc à renverser>\R\R\R\R\R\R\R\R\Z +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@RQ #1#2#3#4#5#6#7#8#9% +{% + \xint@r #9\XINT@RQ@end\R + \XINT@RQ {#9#8#7#6#5#4#3#2#1}% +}% +\def\XINT@RQ@end\R\XINT@RQ #1#2\Z +{% + \XINT@RQ@end@ #1\Z +}% +\def\XINT@RQ@end@ #1#2#3#4#5#6#7#8% +{% + \xint@r #8\XINT@RQ@end@viii + #7\XINT@RQ@end@vii + #6\XINT@RQ@end@vi + #5\XINT@RQ@end@v + #4\XINT@RQ@end@iv + #3\XINT@RQ@end@iii + #2\XINT@RQ@end@ii + \R\XINT@RQ@end@i + \Z #2#3#4#5#6#7#8% +}% +\def\XINT@RQ@end@viii #1\Z #2#3#4#5#6#7#8#9\Z { #9}% +\def\XINT@RQ@end@vii #1\Z #2#3#4#5#6#7#8#9\Z { #8#9000}% +\def\XINT@RQ@end@vi #1\Z #2#3#4#5#6#7#8#9\Z { #7#8#900}% +\def\XINT@RQ@end@v #1\Z #2#3#4#5#6#7#8#9\Z { #6#7#8#90}% +\def\XINT@RQ@end@iv #1\Z #2#3#4#5#6#7#8#9\Z { #5#6#7#8#9}% +\def\XINT@RQ@end@iii #1\Z #2#3#4#5#6#7#8#9\Z { #4#5#6#7#8#9000}% +\def\XINT@RQ@end@ii #1\Z #2#3#4#5#6#7#8#9\Z { #3#4#5#6#7#8#900}% +\def\XINT@RQ@end@i \Z #1#2#3#4#5#6#7#8\Z { #1#2#3#4#5#6#7#80}% +% \end{macrocode} +% \subsection{\ch{XINT@cuz}} +% \begin{macrocode} +\def\xint@cleanupzeros@andstop #1#2#3#4% +{\expandafter + \space + \the\numexpr #1#2#3#4\relax +}% +\def\xint@cleanupzeros@nospace #1#2#3#4% +{% + \the\numexpr #1#2#3#4\relax +}% +\def\XINT@Rev@andcleanupzeros #1% +{% + \romannumeral0\expandafter + \xint@cleanupzeros@andstop + \romannumeral0\XINT@rord@main {}#1% + \xint@UNDEF + \xint@undef\xint@undef\xint@undef\xint@undef + \xint@undef\xint@undef\xint@undef\xint@undef + \xint@UNDEF +}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% routine CleanUpZeros. Utilisée en particulier par la +% soustraction. +% INPUT: longueur **multiple de 4** (<-- ATTENTION) +% OUTPUT: on a retiré tous les leading zéros, on n'est **plus* +% nécessairement de longueur 4n +% Délimiteur pour @main: \W\W\W\W\W\W\W\Z avec SEPT \W +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@cuz #1% +{% + \XINT@cuz@loop #1\W\W\W\W\W\W\W\Z% +}% +\def\XINT@cuz@loop #1#2#3#4#5#6#7#8% +{% + \xint@w #8\xint@cuz@enda\W + \xint@z #8\xint@cuz@endb\Z + \XINT@cuz@checka {#1#2#3#4#5#6#7#8}% +}% +\def\xint@cuz@enda #1\XINT@cuz@checka #2% +{% + \xint@cuz@endaa #2% +}% +\def\xint@cuz@endaa #1#2#3#4#5\Z +{% + \expandafter\space\the\numexpr #1#2#3#4\relax +}% +\def\xint@cuz@endb\Z\XINT@cuz@checka #1{ 0}% +\def\XINT@cuz@checka #1% +{% + \expandafter \XINT@cuz@checkb \the\numexpr #1\relax +}% +\def\XINT@cuz@checkb #1% +{% + \xint@zero #1\xint@cuz@backtoloop 0\XINT@cuz@Stop #1% +}% +\def\XINT@cuz@Stop #1\W #2\Z{ #1}% +\def\xint@cuz@backtoloop 0\XINT@cuz@Stop 0{\XINT@cuz@loop }% +% \end{macrocode} +% \subsection{\ch{xintNum}} +% \begin{verbatim} +% For example \xintNum {----+-+++---+----000000000000003} +% \end{verbatim} +% \begin{macrocode} +\def\xintNum {\romannumeral0\xintnum }% +\def\xintnum #1% +{% + \expandafter\expandafter\expandafter + \XINT@num + \expandafter\expandafter\expandafter + {#1}% +}% +\def\XINT@Num {\romannumeral0\XINT@num }% +\def\XINT@num #1{\XINT@num@loop #1\R\R\R\R\R\R\R\R\Z }% +\def\XINT@num@loop #1#2#3#4#5#6#7#8% +{% + \xint@r #8\XINT@num@end\R\XINT@num@NumEight #1#2#3#4#5#6#7#8% +}% +\def\XINT@num@end\R\XINT@num@NumEight #1\R #2\Z +{% + \expandafter\space\the\numexpr #1+0\relax +}% +\def\XINT@num@NumEight #1#2#3#4#5#6#7#8% +{% + \ifnum \numexpr #1#2#3#4#5#6#7#8+0\relax = 0 + \xint@afterfi {\expandafter\XINT@num@keepsign@a + \the\numexpr #1#2#3#4#5#6#7#81\relax}% + \else + \xint@afterfi {\expandafter\XINT@num@finish + \the\numexpr #1#2#3#4#5#6#7#8\relax}% + \fi +}% +\def\XINT@num@keepsign@a #1% +{% + \xint@one#1\XINT@num@gobacktoloop 1\XINT@num@keepsign@b +}% +\def\XINT@num@gobacktoloop 1\XINT@num@keepsign@b {\XINT@num@loop }% +\def\XINT@num@keepsign@b #1{\XINT@num@loop -}% +\def\XINT@num@finish #1\R #2\Z { #1}% +% \end{macrocode} +% \subsection{\ch{xintLen}, \ch{xintLength}} +% \begin{verbatim} +% \xintLen -> fait la double expansion, ne compte PAS le signe +% \xintLength -> ne fait PAS la double expansion, compte le signe +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\xintLen {\romannumeral0\xintlen }% +\def\xintlen #1% +{% + \expandafter\expandafter\expandafter + \XINT@length@fork #1\R\R\R\R\R\R\R\R\Z +}% +\def\XINT@Len #1{\romannumeral0\XINT@length@fork #1\R\R\R\R\R\R\R\R\Z }% +\def\XINT@length@fork #1% +{% + \expandafter\XINT@length@loop + \xint@UDsignfork + #1\dummy {{0}}% + -\dummy {{0}#1}% + \xint@UDforksign +}% +\def\XINT@Length #1{\romannumeral0\XINT@length@loop {0}#1\R\R\R\R\R\R\R\R\Z }% +\def\XINT@length #1{\XINT@length@loop {0}#1\R\R\R\R\R\R\R\R\Z }% +\let\xintLength\XINT@Length +\def\XINT@length@loop #1#2#3#4#5#6#7#8#9% +{% + \xint@r #9\XINT@length@end {#2#3#4#5#6#7#8#9}\R + \expandafter\XINT@length@loop\expandafter {\the\numexpr #1+8\relax}% +}% +\def\XINT@length@end #1\R\expandafter\XINT@length@loop\expandafter #2#3\Z +{% + \XINT@length@end@ #1\W\W\W\W\W\W\W\W\Z {#2}% +}% +\def\XINT@length@end@ #1\R #2#3#4#5#6#7#8#9\Z +{% + \xint@w #2\XINT@length@end@i + #3\XINT@length@end@ii + #4\XINT@length@end@iii + #5\XINT@length@end@iv + #6\XINT@length@end@v + #7\XINT@length@end@vi + #8\XINT@length@end@vii + \W\XINT@length@end@viii +}% +\def\XINT@length@end@viii #1% + {\expandafter\space\the\numexpr #1-8\relax}% +\def\XINT@length@end@vii #1\XINT@length@end@viii #2% + {\expandafter\space\the\numexpr #2-7\relax}% +\def\XINT@length@end@vi #1\XINT@length@end@viii #2% + {\expandafter\space\the\numexpr #2-6\relax}% +\def\XINT@length@end@v #1\XINT@length@end@viii #2% + {\expandafter\space\the\numexpr #2-5\relax}% +\def\XINT@length@end@iv #1\XINT@length@end@viii #2% + {\expandafter\space\the\numexpr #2-4\relax}% +\def\XINT@length@end@iii #1\XINT@length@end@viii #2% + {\expandafter\space\the\numexpr #2-3\relax}% +\def\XINT@length@end@ii #1\XINT@length@end@viii #2% + {\expandafter\space\the\numexpr #2-2\relax}% +\def\XINT@length@end@i #1\XINT@length@end@viii #2% + {\expandafter\space\the\numexpr #2-1\relax}% +% \end{macrocode} +% \subsection{\ch{xintAssign}, \ch{xintAssignArray}, \ch{xintDigitsOf}} +% \begin{verbatim} +% \xintAssign {a}{b}..{z}\to\A\B...\Z, +% \xintAssignArray {a}{b}..{z}\to\U +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\xintAssign #1\to +{% + \expandafter\expandafter\expandafter + \XINT@assign@a #1{}\to +}% +\def\XINT@assign@a #1% attention to the # at the beginning of next line +#{% + \def\xint@temp {#1}% + \ifx\empty\xint@temp + \expandafter\XINT@assign@b + \else + \expandafter\XINT@assign@B + \fi +}% +\def\XINT@assign@b #1#2\to #3% +{% + \edef #3{#1}\def\xint@temp {#2}% + \ifx\empty\xint@temp + \else + \xint@afterfi{\XINT@assign@a #2\to }% + \fi +}% +\def\XINT@assign@B #1\to #2% +{% + \edef #2{\xint@temp}% +}% +\def\xintRelaxArray #1% +{% + \edef\xint@arrayname {\expandafter\xint@gobble\string #1}% + \expandafter\let\expandafter\xint@temp + \csname\xint@arrayname 0\endcsname + \count 255 0 + \loop + \global\expandafter\let + \csname\xint@arrayname\the\count255\endcsname\relax + \ifnum \count 255 < \xint@temp + \advance\count 255 1 + \repeat + \global\expandafter\let\csname\xint@arrayname 00\endcsname\relax + \global\let #1\relax +}% +\def\xintAssignArray #1\to #2% +{% + \edef\xint@arrayname {\expandafter\xint@gobble\string #2}% + \count 255 0 + \expandafter\expandafter\expandafter + \XINT@assignarray@loop #1\xint@undef + \csname\xint@arrayname 00\endcsname + \csname\xint@arrayname 0\endcsname + {\xint@arrayname}% + #2% +}% +\def\XINT@assignarray@loop #1% +{% + \def\xint@temp {#1}% + \ifx\xint@bracedundef\xint@temp + \edef\xint@temp{\the\count 255 }% + \expandafter\let\csname\xint@arrayname0\endcsname\xint@temp + \expandafter\XINT@assignarray@end + \else + \advance\count 255 1 + \expandafter\edef + \csname\xint@arrayname\the\count 255\endcsname{\xint@temp}% + \expandafter\XINT@assignarray@loop + \fi +}% +\def\XINT@assignarray@end {\expandafter\XINT@assignarray@@end }% +\def\XINT@assignarray@@end #1% +{% + \expandafter\XINT@assignarray@@@end\expandafter #1% +}% +\def\XINT@assignarray@@@end #1#2#3% +{% + \expandafter\XINT@assignarray@@@@end + \expandafter #1\expandafter #2\expandafter{#3}% +}% +\def\XINT@assignarray@@@@end #1#2#3#4% +{% + \def #4##1% + {\romannumeral0% + \expandafter\expandafter\expandafter + #1% + \expandafter\expandafter\expandafter + {##1}% + }% + \def #1##1% + {% + \ifnum ##1< 0 + \xint@afterfi {\xintError:ArrayIndexIsNegative + \expandafter\space 0}% + \else + \xint@afterfi {% + \ifnum ##1> #2 + \xint@afterfi {\xintError:ArrayIndexBeyondLimit + \expandafter\space 0}% + \else + \xint@afterfi + {\expandafter\expandafter\expandafter + \space\csname #3##1\endcsname}% + \fi}% + \fi + }% +}% +\let\xintDigitsOf\xintAssignArray +% \end{macrocode} +% \subsection{\ch{xintSgn}} +% \begin{macrocode} +\def\xintSgn {\romannumeral0\xintsgn }% +\def\xintsgn #1% +{% + \expandafter\expandafter\expandafter + \XINT@sgn #1\Z% +}% +\def\XINT@Sgn #1{\romannumeral0\XINT@sgn #1\Z }% +\def\XINT@sgn #1% +{% + \xint@xpxp@andstop + \xint@UDzerominusfork + #1-\dummy {\expandafter0}% zero + 0#1\dummy {\expandafter-\expandafter1}% n\'egatif + 0-\dummy {\expandafter1}% positif + \xint@UDforkminuszero + \xint@z +}% +% \end{macrocode} +% \subsection{\ch{xintOpp}} +% \begin{macrocode} +\def\xintOpp {\romannumeral0\xintopp }% +\def\xintopp #1% +{% + \expandafter\expandafter\expandafter + \XINT@opp #1% +}% +\def\XINT@Opp #1{\romannumeral0\XINT@opp #1}% +\def\XINT@opp #1% +{% + \expandafter\space + \xint@UDzerominusfork + #1-\dummy 0% zero + 0#1\dummy {}% negative + 0-\dummy {-#1}% positive + \xint@UDforkminuszero +}% +% \end{macrocode} +% \subsection{\ch{xintAbs}} +% \begin{macrocode} +\def\xintAbs {\romannumeral0\xintabs }% +\def\xintabs #1% +{% + \expandafter\expandafter\expandafter + \XINT@abs #1% +}% +\def\XINT@Abs {\romannumeral0\XINT@abs }% +\def\XINT@abs #1% +{% + \xint@UDsignfork + #1\dummy \XINT@abs@isnegative + -\dummy \XINT@abs@isnonnegative + \xint@UDforksign + #1% +}% +\def\XINT@abs@isnegative #1{ }% +\def\XINT@abs@isnonnegative #1{ #1}% +% \end{macrocode} +% \begin{verbatim} +%----------------------------------------------------------------- +%----------------------------------------------------------------- +% ARITHMETIC OPERATIONS: ADDITION, SUBTRACTION, SUMS, +% MULTIPLICATION, PRODUCTS, FACTORIAL, POWERS, EUCLIDEAN DIVISION. +% \end{verbatim} +% \vspace*{-2\baselineskip} +% \subsection{\ch{xintAdd}} +% \begin{macrocode} +\def\xintAdd {\romannumeral0\xintadd }% +\def\xintadd #1% +{% + \expandafter\expandafter\expandafter + \xint@add + \expandafter\expandafter\expandafter + {#1}% +}% +\def\xint@add #1#2% +{% + \expandafter\expandafter\expandafter + \XINT@add@fork #2\Z #1\Z +}% +\def\XINT@Add #1#2{\romannumeral0\XINT@add@fork #2\Z #1\Z }% +\def\XINT@add #1#2{\XINT@add@fork #2\Z #1\Z }% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% ADDITION +% Ici #1#2 vient du *deuxième* argument de \xintAdd +% et #3#4 donc du *premier* [algo plus efficace lorsque +% le premier est plus long que le second] +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@add@fork #1#2\Z #3#4\Z +{% + \xint@UDzerofork + #1\dummy \XINT@add@secondiszero + #3\dummy \XINT@add@firstiszero + 0\dummy + {\xint@UDsignsfork + #1#3\dummy \XINT@add@minusminus % #1 = #3 = - + #1-\dummy \XINT@add@minusplus % #1 = - + #3-\dummy \XINT@add@plusminus % #3 = - + --\dummy \XINT@add@plusplus + \xint@UDforksigns}% + \xint@UDforkzero + {#2}{#4}#1#3% +}% +\def\XINT@add@secondiszero #1#2#3#4{ #4#2}% +\def\XINT@add@firstiszero #1#2#3#4{ #3#1}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% #1 vient du *deuxième* et #2 vient du *premier* +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@add@minusminus #1#2#3#4% +{% + \expandafter\space\expandafter-% + \romannumeral0\XINT@add@pre {#2}{#1}% +}% +\def\XINT@add@minusplus #1#2#3#4% +{% + \XINT@sub@pre {#4#2}{#1}% +}% +\def\XINT@add@plusminus #1#2#3#4% +{% + \XINT@sub@pre {#3#1}{#2}% +}% +\def\XINT@add@plusplus #1#2#3#4% +{% + \XINT@add@pre {#4#2}{#3#1}% +}% +\def\XINT@add@pre #1% +{% + \expandafter\XINT@add@@pre\expandafter{% + \romannumeral0\XINT@RQ {}#1\R\R\R\R\R\R\R\R\Z + }% +}% +\def\XINT@add@@pre #1#2% +{% + \expandafter\XINT@add@A + \expandafter0\expandafter{\expandafter}% + \romannumeral0\XINT@RQ {}#2\R\R\R\R\R\R\R\R\Z + \W\X\Y\Z #1\W\X\Y\Z +}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% ADDITION \XINT@add@A +% INPUT: +% \romannumeral0\XINT@add@A <N1>\W\X\Y\Z <N2>\W\X\Y\Z +% avec: N1 et N2 sur **4n**, et **renversés**, et le plus long ne +% doit pas se terminer par 0000. [Donc on peut avoir 0000 comme +% input si l'autre est >0 et ne se termine pas en 0000 bien sûr]. +% OUTPUT: +% La somme N1+N2, *PAS* sur 4n, dans l'ordre *normal*, et *sans +% leading zeros* +% La procédure est plus rapide lorsque la longueur de N2 est +% supérieure à celle de N1 +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@add@A #1#2#3#4#5#6% +{% + \xint@w + #3\xint@add@az + \W\XINT@add@AB #1{#3#4#5#6}{#2}% +}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% 1er nombre fini. +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\xint@add@az\W\XINT@add@AB #1#2% +{% + \XINT@add@AC@checkcarry #1% +}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% ici #2 est prévu pour l'addition, mais attention il devra être renversé pour +% \numexpr. #3 = résultat partiel. #4 = chiffres qui restent +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@add@AB #1#2#3#4\W\X\Y\Z #5#6#7#8% +{% + \xint@w + #5\xint@add@bz + \W\XINT@add@ABE #1#2{#8#7#6#5}{#3}#4\W\X\Y\Z +}% +\def\XINT@add@ABE #1#2#3#4#5#6% +{\expandafter + \XINT@add@ABEA\the\numexpr #1+10#5#4#3#2+#6\relax.% +}% +\def\XINT@add@ABEA #1#2#3.#4% +{% + \XINT@add@A #2{#3#4}% +}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% ici le deuxième nombre est fini +% #6 part à la poubelle, #2#3#4#5 est le #2 dans \XINT@add@AB +% on ne vérifie pas la retenue cette fois, mais les fois suivantes +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\xint@add@bz\W\XINT@add@ABE #1#2#3#4#5#6% +{\expandafter + \XINT@add@CC\the\numexpr #1+10#5#4#3#2\relax.% +}% +\def\XINT@add@CC #1#2#3.#4% +{% + \XINT@add@AC@checkcarry #2{#3#4}% on va examiner et \'eliminer #2 +}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% retenue plus chiffres qui restent de l'un des deux nombres. +% #2 = résultat partiel +% #3#4#5#6 = summand, avec plus significatif à droite +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@add@AC@checkcarry #1% +{% + \xint@zero #1\xint@add@AC@nocarry 0\XINT@add@C +}% +\def\xint@add@AC@nocarry 0\XINT@add@C #1#2\W\X\Y\Z +{% + \expandafter + \xint@cleanupzeros@andstop + \romannumeral0% + \XINT@rord@main {}#2% + \xint@UNDEF + \xint@undef\xint@undef\xint@undef\xint@undef + \xint@undef\xint@undef\xint@undef\xint@undef + \xint@UNDEF + #1% +}% +\def\XINT@add@C #1#2#3#4#5% +{% + \xint@w + #2\xint@add@cz + \W\XINT@add@CD {#5#4#3#2}{#1}% +}% +\def\XINT@add@CD #1% +{\expandafter + \XINT@add@CC\the\numexpr 1+10#1\relax.% +}% +\def\xint@add@cz\W\XINT@add@CD #1#2{ 1#2}% +% \end{macrocode} +% \subsection{\ch{xintSub}} +% \begin{macrocode} +\def\xintSub {\romannumeral0\xintsub }% +\def\xintsub #1% +{% + \expandafter\expandafter\expandafter + \xint@sub + \expandafter\expandafter\expandafter + {#1}% +}% +\def\xint@sub #1#2% +{% + \expandafter\expandafter\expandafter + \XINT@sub@fork #2\Z #1\Z +}% +\def\XINT@Sub #1#2{\romannumeral0\XINT@sub@fork #2\Z #1\Z }% +\def\XINT@sub #1#2{\XINT@sub@fork #2\Z #1\Z }% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% SOUSTRACTION +% #3#4-#1#2 +% #3#4 vient du *premier* +% #1#2 vient du *second* +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@sub@fork #1#2\Z #3#4\Z +{% + \xint@UDsignsfork + #1#3\dummy \XINT@sub@minusminus + #1-\dummy \XINT@sub@minusplus % attention, #3=0 possible + #3-\dummy \XINT@sub@plusminus % attention, #1=0 possible + --\dummy {\xint@UDzerofork + #1\dummy \XINT@sub@secondiszero + #3\dummy \XINT@sub@firstiszero + 0\dummy \XINT@sub@plusplus + \xint@UDforkzero}% + \xint@UDforksigns + {#2}{#4}#1#3% +}% +\def\XINT@sub@secondiszero #1#2#3#4{ #4#2}% +\def\XINT@sub@firstiszero #1#2#3#4{ -#3#1}% +\def\XINT@sub@plusplus #1#2#3#4% +{% + \XINT@sub@pre {#4#2}{#3#1}% +}% +\def\XINT@sub@minusminus #1#2#3#4% +{% + \XINT@sub@pre {#1}{#2}% +}% +\def\XINT@sub@minusplus #1#2#3#4% +{% + \xint@zero #4\xint@sub@mp0\XINT@add@pre {#4#2}{#1}% +}% +\def\xint@sub@mp0\XINT@add@pre #1#2{ #2}% +\def\XINT@sub@plusminus #1#2#3#4% +{% + \xint@zero #3\xint@sub@pm0\expandafter\space\expandafter-% + \romannumeral0\XINT@add@pre {#2}{#3#1}% +}% +\def\xint@sub@pm #1\XINT@add@pre #2#3{ -#2}% +\def\XINT@sub@pre #1% +{% + \expandafter\XINT@sub@@pre\expandafter{% + \romannumeral0\XINT@RQ {}#1\R\R\R\R\R\R\R\R\Z + }% +}% +\def\XINT@sub@@pre #1#2% +{% + \expandafter\XINT@sub@A + \expandafter1\expandafter{\expandafter}% + \romannumeral0\XINT@RQ {}#2\R\R\R\R\R\R\R\R\Z + \W\X\Y\Z #1 \W\X\Y\Z +}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% \romannumeral0\XINT@subA 1{}<N1>\W\X\Y\Z<N2>\W\X\Y\Z +% N1 et N2 sont présentés à l'envers ET ON A RAJOUTÉ DES ZÉROS +% POUR QUE LEURS LONGUEURS À CHACUN SOIENT MULTIPLES DE 4, MAIS +% AUCUN NE SE TERMINE EN 0000 +% output: N2 - N1 +% Elle donne le résultat dans le **bon ordre**, avec le bon signe, +% et sans zéros superflus. +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@sub@A #1#2#3\W\X\Y\Z #4#5#6#7% +{% + \xint@w + #4\xint@sub@az + \W\XINT@sub@B #1{#4#5#6#7}{#2}#3\W\X\Y\Z +}% +\def\XINT@sub@B #1#2#3#4#5#6#7% +{% + \xint@w + #4\xint@sub@bz + \W\XINT@sub@onestep #1#2{#7#6#5#4}{#3}% +}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% d'abord la branche principale +% #6 = 4 chiffres de N1, plus significatif en *premier*, +% #2#3#4#5 chiffres de N2, plus significatif en *dernier* +% On veut N2 - N1. +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@sub@onestep #1#2#3#4#5#6% +{\expandafter + \XINT@sub@backtoA\the\numexpr 11#5#4#3#2-#6+#1-1\relax.% +}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% ON PRODUIT LE RÉSULTAT DANS LE BON ORDRE +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@sub@backtoA #1#2#3.#4% +{% + \XINT@sub@A #2{#3#4}% +}% +\def\xint@sub@bz + \W\XINT@sub@onestep #1#2#3#4#5#6#7% +{% + \xint@UDzerofork + #1\dummy \XINT@sub@C % une retenue + 0\dummy \XINT@sub@D % pas de retenue + \xint@UDforkzero + {#7}#2#3#4#5% +}% +\def\XINT@sub@D #1#2\W\X\Y\Z +{% + \expandafter + \xint@cleanupzeros@andstop + \romannumeral0% + \XINT@rord@main {}#2% + \xint@UNDEF + \xint@undef\xint@undef\xint@undef\xint@undef + \xint@undef\xint@undef\xint@undef\xint@undef + \xint@UNDEF + #1% +}% +\def\XINT@sub@C #1#2#3#4#5% +{% + \xint@w + #2\xint@sub@cz + \W\XINT@sub@AC@onestep {#5#4#3#2}{#1}% +}% +\def\XINT@sub@AC@onestep #1% +{\expandafter + \XINT@sub@backtoC\the\numexpr 11#1-1\relax.% +}% +\def\XINT@sub@backtoC #1#2#3.#4% +{% + \XINT@sub@AC@checkcarry #2{#3#4}% la retenue va \^etre examin\'ee +}% +\def\XINT@sub@AC@checkcarry #1% +{% + \xint@one #1\xint@sub@AC@nocarry 1\XINT@sub@C +}% +\def\xint@sub@AC@nocarry 1\XINT@sub@C #1#2\W\X\Y\Z +{% + \expandafter + \XINT@cuz@loop + \romannumeral0% + \XINT@rord@main {}#2% + \xint@UNDEF + \xint@undef\xint@undef\xint@undef\xint@undef + \xint@undef\xint@undef\xint@undef\xint@undef + \xint@UNDEF + #1\W\W\W\W\W\W\W\Z +}% +\def\xint@sub@cz\W\XINT@sub@AC@onestep #1% +{% + \XINT@cuz +}% +\def\xint@sub@az\W\XINT@sub@B #1#2#3#4#5#6#7% +{% + \xint@w + #4\xint@sub@ez + \W\XINT@sub@Eenter #1{#3}#4#5#6#7% +}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% le premier nombre continue, le résultat sera < 0. +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@sub@Eenter #1#2% +{% + \expandafter + \XINT@sub@E\expandafter1\expandafter{\expandafter}% + \romannumeral0% + \XINT@rord@main {}#2% + \xint@UNDEF + \xint@undef\xint@undef\xint@undef\xint@undef + \xint@undef\xint@undef\xint@undef\xint@undef + \xint@UNDEF + \W\X\Y\Z #1% +}% +\def\XINT@sub@E #1#2#3#4#5#6% +{% + \xint@w #3\xint@sub@F\W\XINT@sub@Eonestep + #1{#6#5#4#3}{#2}% +}% +\def\XINT@sub@Eonestep #1#2% +{\expandafter + \XINT@sub@backtoE\the\numexpr 110000-#2+#1-1\relax.% +}% +\def\XINT@sub@backtoE #1#2#3.#4% +{% + \XINT@sub@E #2{#3#4}% +}% +\def\xint@sub@F\W\XINT@sub@Eonestep #1#2#3#4% +{% + \xint@sub@Fthreewayfork + #4#1\dummy {\XINT@sub@Fdec 0}% soustraire 1. Et faire signe - + #1#4\dummy {\XINT@sub@Finc 1}% additionner 1. Et faire signe - + 10\dummy \XINT@sub@DD % terminer. Mais avec signe - + \xint@sub@Fforkthreeway + {#3}% +}% +\def\xint@sub@Fthreewayfork #110\dummy #2#3\xint@sub@Fforkthreeway {#2}% +\def\XINT@sub@DD +{\expandafter\space\expandafter-\romannumeral0\XINT@sub@D }% +\def\XINT@sub@Fdec #1#2#3#4#5#6% +{% + \xint@w + #3\xint@sub@Fdec@finish\W\XINT@sub@Fdec@onestep + #1{#6#5#4#3}{#2}% +}% +\def\XINT@sub@Fdec@onestep #1#2% +{\expandafter + \XINT@sub@backtoFdec\the\numexpr 11#2+#1-1\relax.% +}% +\def\XINT@sub@backtoFdec #1#2#3.#4% +{% + \XINT@sub@Fdec #2{#3#4}% +}% +\def\xint@sub@Fdec@finish\W\XINT@sub@Fdec@onestep #1#2% +{% + \expandafter\space\expandafter-\romannumeral0\XINT@cuz +}% +\def\XINT@sub@Finc #1#2#3#4#5#6% +{% + \xint@w + #3\xint@sub@Finc@finish\W\XINT@sub@Finc@onestep + #1{#6#5#4#3}{#2}% +}% +\def\XINT@sub@Finc@onestep #1#2% +{\expandafter + \XINT@sub@backtoFinc\the\numexpr 10#2+#1\relax.% +}% +\def\XINT@sub@backtoFinc #1#2#3.#4% +{% + \XINT@sub@Finc #2{#3#4}% +}% +\def\xint@sub@Finc@finish\W\XINT@sub@Finc@onestep #1#2#3% +{% + \xint@UDzerofork + #1\dummy {\expandafter\space\expandafter-% + \xint@cleanupzeros@nospace}% + 0\dummy { -1}% + \xint@UDforkzero + #3% +}% +\def\xint@sub@ez\W\XINT@sub@Eenter #1% +{% + \xint@UDzerofork + #1\dummy \XINT@sub@K % il y a une retenue + 0\dummy \XINT@sub@L % pas de retenue + \xint@UDforkzero +}% +\def\XINT@sub@L #1\W\X\Y\Z + {\XINT@cuz@loop #1\W\W\W\W\W\W\W\Z }% +\def\XINT@sub@K #1% +{% + \expandafter + \XINT@sub@KK\expandafter1\expandafter{\expandafter}% + \romannumeral0% + \XINT@rord@main {}#1% + \xint@UNDEF + \xint@undef\xint@undef\xint@undef\xint@undef + \xint@undef\xint@undef\xint@undef\xint@undef + \xint@UNDEF +}% +\def\XINT@sub@KK #1#2#3#4#5#6% +{% + \xint@w + #3\xint@sub@KK@finish\W\XINT@sub@KK@onestep + #1{#6#5#4#3}{#2}% +}% +\def\XINT@sub@KK@onestep #1#2% +{\expandafter + \XINT@sub@backtoKK\the\numexpr 110000-#2+#1-1\relax.% +}% +\def\XINT@sub@backtoKK #1#2#3.#4% +{% + \XINT@sub@KK #2{#3#4}% +}% +\def\xint@sub@KK@finish\W\XINT@sub@KK@onestep #1#2#3% +{% + \expandafter\space\expandafter-\romannumeral + 0\XINT@cuz@loop #3\W\W\W\W\W\W\W\Z +}% +% \end{macrocode} +% \subsection{\ch{xintCmp}} +% \begin{macrocode} +\def\xintCmp {\romannumeral0\xintcmp }% +\def\xintcmp #1% +{% + \expandafter\expandafter\expandafter + \xint@cmp + \expandafter\expandafter\expandafter + {#1}% +}% +\def\xint@cmp #1#2% +{% + \expandafter\expandafter\expandafter + \XINT@cmp@fork #2\Z #1\Z +}% +\def\XINT@Cmp #1#2{\romannumeral0\XINT@cmp@fork #2\Z #1\Z }% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% COMPARAISON +% 1 si #3#4>#1#2, 0 si #3#4=#1#2, -1 si #3#4<#1#2 +% #3#4 vient du *premier* +% #1#2 vient du *second* +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@cmp@fork #1#2\Z #3#4\Z +{% + \xint@UDsignsfork + #1#3\dummy \XINT@cmp@minusminus + #1-\dummy \XINT@cmp@minusplus + #3-\dummy \XINT@cmp@plusminus + --\dummy {\xint@UDzerosfork + #1#3\dummy \XINT@cmp@zerozero + #10\dummy \XINT@cmp@zeroplus + #30\dummy \XINT@cmp@pluszero + 00\dummy \XINT@cmp@plusplus + \xint@UDforkzeros}% + \xint@UDforksigns + {#2}{#4}#1#3% +}% +\def\XINT@cmp@minusplus #1#2#3#4{ 1}% +\def\XINT@cmp@plusminus #1#2#3#4{ -1}% +\def\XINT@cmp@zerozero #1#2#3#4{ 0}% +\def\XINT@cmp@zeroplus #1#2#3#4{ 1}% +\def\XINT@cmp@pluszero #1#2#3#4{ -1}% +\def\XINT@cmp@plusplus #1#2#3#4% +{% + \XINT@cmp@pre {#4#2}{#3#1}% +}% +\def\XINT@cmp@minusminus #1#2#3#4% +{% + \XINT@cmp@pre {#1}{#2}% +}% +\def\XINT@cmp@pre #1% +{% + \expandafter\XINT@cmp@@pre\expandafter{% + \romannumeral0\XINT@RQ {}#1\R\R\R\R\R\R\R\R\Z + }% +}% +\def\XINT@cmp@@pre #1#2% +{% + \expandafter\XINT@cmp@A + \expandafter1\expandafter{\expandafter}% + \romannumeral0\XINT@RQ {}#2\R\R\R\R\R\R\R\R\Z + \W\X\Y\Z #1\W\X\Y\Z +}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% COMPARAISON +% N1 et N2 sont présentés à l'envers ET ON A RAJOUTÉ DES ZÉROS +% POUR QUE LEURS LONGUEURS À CHACUN SOIENT MULTIPLES DE 4, MAIS +% AUCUN NE SE TERMINE EN 0000 +% routine appelée via \XINT@cmp@A 1{}<N1>\W\X\Y\Z<N2>\W\X\Y\Z +% ATTENTION RENVOIE 1 SI N1 < N2, 0 si N1 = N2, -1 si N1 > N2 +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@cmp@A #1#2#3\W\X\Y\Z #4#5#6#7% +{% + \xint@w + #4\xint@cmp@az + \W\XINT@cmp@B #1{#4#5#6#7}{#2}#3\W\X\Y\Z +}% +\def\XINT@cmp@B #1#2#3#4#5#6#7% +{% + \xint@w + #4\xint@cmp@bz + \W\XINT@cmp@onestep #1#2{#7#6#5#4}{#3}% +}% +\def\XINT@cmp@onestep #1#2#3#4#5#6% +{\expandafter + \XINT@cmp@backtoA\the\numexpr 11#5#4#3#2-#6+#1-1\relax.% +}% +\def\XINT@cmp@backtoA #1#2#3.#4% +{% + \XINT@cmp@A #2{#3#4}% +}% +\def\xint@cmp@bz + \W\XINT@cmp@onestep #1\Z { 1}% +\def\xint@cmp@az\W\XINT@cmp@B #1#2#3#4#5#6#7% +{% + \xint@w + #4\xint@cmp@ez + \W\XINT@cmp@Eenter #1{#3}#4#5#6#7% +}% +\def\XINT@cmp@Eenter #1\Z { -1}% +\def\xint@cmp@ez\W\XINT@cmp@Eenter #1% +{% + \xint@UDzerofork + #1\dummy \XINT@cmp@K % il y a une retenue + 0\dummy \XINT@cmp@L % pas de retenue + \xint@UDforkzero +}% +\def\XINT@cmp@K #1\Z { -1}% +\def\XINT@cmp@L #1{\XINT@OneIfPositive@main #1}% +\def\XINT@OneIfPositive #1% +{% + \XINT@OneIfPositive@main #1\W\X\Y\Z% +}% +\def\XINT@OneIfPositive@main #1#2#3#4% +{% + \xint@z #4\xint@OneIfPositive@terminated\Z\XINT@OneIfPositive@onestep + #1#2#3#4% +}% +\def\xint@OneIfPositive@terminated\Z\XINT@OneIfPositive@onestep\W\X\Y\Z { 0}% +\def\XINT@OneIfPositive@onestep #1#2#3#4% +{% + \expandafter + \XINT@OneIfPositive@check + \the\numexpr #1#2#3#4\relax +}% +\def\XINT@OneIfPositive@check #1% +{% + \xint@zero + #1\xint@OneIfPositive@backtomain 0\XINT@OneIfPositive@finish #1% +}% +\def\XINT@OneIfPositive@finish #1\W\X\Y\Z{ 1}% +\def\xint@OneIfPositive@backtomain 0\XINT@OneIfPositive@finish 0% + {\XINT@OneIfPositive@main }% +% \end{macrocode} +% \subsection{\ch{xintGeq}} +% \begin{verbatim} +% PLUS GRAND OU ÉGAL +% attention compare les **valeurs absolues** +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\xintGeq {\romannumeral0\xintgeq }% +\def\xintgeq #1% +{% + \expandafter\expandafter\expandafter + \xint@geq + \expandafter\expandafter\expandafter + {#1}% +}% +\def\xint@geq #1#2% +{\expandafter\expandafter\expandafter + \XINT@geq@fork #2\Z #1\Z +}% +\def\XINT@Geq #1#2{\romannumeral0\XINT@geq@fork #2\Z #1\Z }% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% PLUS GRAND OU ÉGAL +% ATTENTION, TESTE les VALEURS ABSOLUES +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@geq@fork #1#2\Z #3#4\Z +{% + \xint@UDzerofork + #1\dummy \XINT@geq@secondiszero % |#1#2|=0 + #3\dummy \XINT@geq@firstiszero % |#1#2|>0 + 0\dummy {\xint@UDsignsfork + #1#3\dummy \XINT@geq@minusminus + #1-\dummy \XINT@geq@minusplus + #3-\dummy \XINT@geq@plusminus + --\dummy \XINT@geq@plusplus + \xint@UDforksigns}% + \xint@UDforkzero + {#2}{#4}#1#3% +}% +\def\XINT@geq@secondiszero #1#2#3#4{ 1}% +\def\XINT@geq@firstiszero #1#2#3#4{ 0}% +\def\XINT@geq@plusplus #1#2#3#4% + {\XINT@geq@pre {#4#2}{#3#1}}% +\def\XINT@geq@minusminus #1#2#3#4% + {\XINT@geq@pre {#2}{#1}}% +\def\XINT@geq@minusplus #1#2#3#4% + {\XINT@geq@pre {#4#2}{#1}}% +\def\XINT@geq@plusminus #1#2#3#4% + {\XINT@geq@pre {#2}{#3#1}}% +\def\XINT@geq@pre #1% +{% + \expandafter\XINT@geq@@pre\expandafter{% + \romannumeral0\XINT@RQ {}#1\R\R\R\R\R\R\R\R\Z + }% +}% +\def\XINT@geq@@pre #1#2% +{% + \expandafter\XINT@geq@A + \expandafter1\expandafter{\expandafter}% + \romannumeral0\XINT@RQ {}#2\R\R\R\R\R\R\R\R\Z + \W\X\Y\Z #1 \W\X\Y\Z +}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% PLUS GRAND OU ÉGAL +% N1 et N2 sont présentés à l'envers ET ON A RAJOUTÉ DES ZÉROS +% POUR QUE LEURS LONGUEURS À CHACUN SOIENT MULTIPLES DE 4, MAIS +% AUCUN NE SE TERMINE EN 0000 +% routine appelée via +% \romannumeral0\XINT@geq@A 1{}<N1>\W\X\Y\Z<N2>\W\X\Y\Z +% ATTENTION RENVOIE 1 SI N1 < N2 ou N1 = N2 et 0 si N1 > N2 +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@geq@A #1#2#3\W\X\Y\Z #4#5#6#7% +{% + \xint@w + #4\xint@geq@az + \W\XINT@geq@B #1{#4#5#6#7}{#2}#3\W\X\Y\Z +}% +\def\XINT@geq@B #1#2#3#4#5#6#7% +{% + \xint@w + #4\xint@geq@bz + \W\XINT@geq@onestep #1#2{#7#6#5#4}{#3}% +}% +\def\XINT@geq@onestep #1#2#3#4#5#6% +{\expandafter + \XINT@geq@backtoA\the\numexpr 11#5#4#3#2-#6+#1-1\relax.% +}% +\def\XINT@geq@backtoA #1#2#3.#4% +{% + \XINT@geq@A #2{#3#4}% +}% +\def\xint@geq@bz\W\XINT@geq@onestep #1\W\X\Y\Z { 1}% +\def\xint@geq@az\W\XINT@geq@B #1#2#3#4#5#6#7% +{% + \xint@w + #4\xint@geq@ez + \W\XINT@geq@Eenter #1% +}% +\def\XINT@geq@Eenter #1\W\X\Y\Z { 0}% +\def\xint@geq@ez\W\XINT@geq@Eenter #1% +{% + \xint@UDzerofork + #1\dummy { 0} % il y a une retenue + 0\dummy { 1} % pas de retenue + \xint@UDforkzero +}% +% \end{macrocode} +% \subsection{\ch{xintMax}} +% \begin{macrocode} +\def\xintMax {\romannumeral0\xintmax }% +\def\xintmax #1% +{% + \expandafter\expandafter\expandafter + \xint@max + \expandafter\expandafter\expandafter + {#1}% +}% +\def\xint@max #1#2% +{% + \expandafter\expandafter\expandafter + \XINT@max@fork #2\Z #1\Z +}% +\def\XINT@Max #1#2{\romannumeral0\XINT@max@fork #2\Z #1\Z }% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% #3#4 vient du *premier* +% #1#2 vient du *second* +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@max@fork #1#2\Z #3#4\Z +{% + \xint@UDsignsfork + #1#3\dummy \XINT@max@minusminus % A < 0, B < 0 + #1-\dummy \XINT@max@minusplus % B < 0, A >= 0 + #3-\dummy \XINT@max@plusminus % A < 0, B >= 0 + --\dummy {\xint@UDzerosfork + #1#3\dummy \XINT@max@zerozero % A = B = 0 + #10\dummy \XINT@max@zeroplus % B = 0, A > 0 + #30\dummy \XINT@max@pluszero % A = 0, B > 0 + 00\dummy \XINT@max@plusplus % A, B > 0 + \xint@UDforkzeros}% + \xint@UDforksigns + {#2}{#4}#1#3% +}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% A = #4#2, B = #3#1 +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@max@zerozero #1#2#3#4{ 0}% +\def\XINT@max@zeroplus #1#2#3#4{ #4#2}% +\def\XINT@max@pluszero #1#2#3#4{ #3#1}% +\def\XINT@max@minusplus #1#2#3#4{ #4#2}% +\def\XINT@max@plusminus #1#2#3#4{ #3#1}% +\def\XINT@max@plusplus #1#2#3#4% +{% + \ifodd\XINT@Geq {#4#2}{#3#1} + \xint@afterfi { #4#2}% + \else + \xint@afterfi { #3#1}% + \fi +}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% #3=-, #4=-, #1 = |B| = -B, #2 = |A| = -A +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@max@minusminus #1#2#3#4% +{% + \ifodd\XINT@Geq {#1}{#2} + \xint@afterfi { -#2}% + \else + \xint@afterfi { -#1}% + \fi +}% +% \end{macrocode} +% \subsection{\ch{xintMin}} +% \begin{macrocode} +\def\xintMin {\romannumeral0\xintmin }% +\def\xintmin #1% +{% + \expandafter\expandafter\expandafter + \xint@min + \expandafter\expandafter\expandafter + {#1}% +}% +\def\xint@min #1#2% +{% + \expandafter\expandafter\expandafter + \XINT@min@fork #2\Z #1\Z +}% +\def\XINT@Min #1#2{\romannumeral0\XINT@min@fork #2\Z #1\Z }% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% #3#4 vient du *premier* +% #1#2 vient du *second* +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@min@fork #1#2\Z #3#4\Z +{% + \xint@UDsignsfork + #1#3\dummy \XINT@min@minusminus % A < 0, B < 0 + #1-\dummy \XINT@min@minusplus % B < 0, A >= 0 + #3-\dummy \XINT@min@plusminus % A < 0, B >= 0 + --\dummy {\xint@UDzerosfork + #1#3\dummy \XINT@min@zerozero % A = B = 0 + #10\dummy \XINT@min@zeroplus % B = 0, A > 0 + #30\dummy \XINT@min@pluszero % A = 0, B > 0 + 00\dummy \XINT@min@plusplus % A, B > 0 + \xint@UDforkzeros}% + \xint@UDforksigns + {#2}{#4}#1#3% +}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% A = #4#2, B = #3#1 +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@min@zerozero #1#2#3#4{ 0}% +\def\XINT@min@zeroplus #1#2#3#4{ 0}% +\def\XINT@min@pluszero #1#2#3#4{ 0}% +\def\XINT@min@minusplus #1#2#3#4{ #3#1}% +\def\XINT@min@plusminus #1#2#3#4{ #4#2}% +\def\XINT@min@plusplus #1#2#3#4% +{% + \ifodd\XINT@Geq {#4#2}{#3#1} + \xint@afterfi { #3#1}% + \else + \xint@afterfi { #4#2}% + \fi +}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% #3=-, #4=-, #1 = |B| = -B, #2 = |A| = -A +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@min@minusminus #1#2#3#4% +{% + \ifodd\XINT@Geq {#1}{#2} + \xint@afterfi { -#1}% + \else + \xint@afterfi { -#2}% + \fi +}% +% \end{macrocode} +% \subsection{\ch{xintSum}, \ch{xintSumExpr}} +% \begin{verbatim} +% \xintSum {{a}{b}...{z}} +% \xintSumExpr {a}{b}...{z}\relax +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@psum #1% +{% + \romannumeral0\XINT@psum@checkifemptysum #1\Z +}% +\def\XINT@psum@checkifemptysum #1% +{% + \xint@relax #1\XINT@psum@returnzero\relax \XINT@psum@RQfirst #1% +}% +\def\XINT@psum@returnzero #1\Z { 0}% +\def\XINT@psum@RQfirst #1\Z +{% + \expandafter\XINT@psum@loop\expandafter + {\romannumeral0\XINT@RQ {}#1\R\R\R\R\R\R\R\R\Z}% avant: #1\Z +}% +\def\XINT@psum@loop #1#2% +{% + \xint@relax #2\XINT@psum@end\relax + \expandafter + \XINT@psum@loop\expandafter + {\romannumeral0\expandafter\XINT@sum@A + \expandafter0\expandafter{\expandafter}% + \romannumeral0\XINT@RQ {}#2\R\R\R\R\R\R\R\R\Z + \W\X\Y\Z #1\W\X\Y\Z }% +}% +\def\XINT@psum@end\relax\expandafter + \XINT@psum@loop\expandafter #1% + {\XINT@psum@end@ #1}% +\def\XINT@psum@end@ #1\W\X\Y\Z #2\W\X\Y\Z +{% + \expandafter + \xint@cleanupzeros@andstop\romannumeral0\XINT@rev {#2}% +}% +\def\xintSumExpr {\romannumeral0\xintsumexpr }% +\def\xintSum {\romannumeral0\xintsum }% +\def\xintsum #1% +{% + \expandafter\expandafter\expandafter + \xintsumexpr #1\relax +}% +\def\xintsumexpr #1% +{% + \expandafter\expandafter\expandafter + \XINT@sum@checkifempty #1\Z {\XINT@psum }{\XINT@psum }% +}% +\def\XINT@sum@checkifempty #1% +{% + \xint@relax #1\XINT@sum@returnzero\relax + \XINT@sum@checksign #1% +}% +\def\XINT@sum@returnzero #1\Z #2#3{ 0}% +\def\XINT@sum@checksign #1% +{% + \xint@zero #1\XINT@sum@skipzeroinput0% + \xint@UDsignfork + #1\dummy \XINT@sum@pushneg + -\dummy \XINT@sum@pushpos + \xint@UDforksign + #1% +}% +\def\XINT@sum@skipzeroinput #1\xint@UDforksign #2\Z #3#4% +{% + \XINT@sum@xpxpnext {#3}{#4}% +}% +\def\XINT@sum@pushpos #1#2\Z #3#4% +{% + \XINT@sum@xpxpnext {#3{#1#2}}{#4}% +}% +\def\XINT@sum@pushneg #1#2\Z #3#4% +{% + \XINT@sum@xpxpnext {#3}{#4{#2}}% +}% +\def\XINT@sum@xpxpnext #1#2#3% +{% + \expandafter\expandafter\expandafter + \XINT@sum@checkiffinished #3\Z {#1}{#2}% +}% +\def\XINT@sum@checkiffinished #1% +{% + \xint@relax #1\XINT@sum@end\relax + \XINT@sum@checksign #1% +}% +\def\XINT@sum@end\relax\XINT@sum@checksign\relax #1\Z #2#3% + {\xintsub{#2\relax}{#3\relax}}% +\def\XINT@sum@A #1#2#3#4#5#6% +{% + \xint@w + #3\xint@sum@az + \W\XINT@sum@B #1{#3#4#5#6}{#2}% +}% +\def\xint@sum@az\W\XINT@sum@B #1#2% +{% + \XINT@sum@AC@checkcarry #1% +}% +\def\XINT@sum@B #1#2#3#4\W\X\Y\Z #5#6#7#8% +{% + \xint@w + #5\xint@sum@bz + \W\XINT@sum@E #1#2{#8#7#6#5}{#3}#4\W\X\Y\Z +}% +\def\XINT@sum@E #1#2#3#4#5#6% +{\expandafter + \XINT@sum@ABEA\the\numexpr #1+10#5#4#3#2+#6\relax +}% +\def\XINT@sum@ABEA #1#2#3#4#5#6#7% +{% + \XINT@sum@A #2{#7#6#5#4#3}% +}% +\def\xint@sum@bz\W\XINT@sum@E #1#2#3#4#5#6% +{\expandafter + \XINT@sum@CC\the\numexpr #1+10#5#4#3#2\relax +}% +\def\XINT@sum@CC #1#2#3#4#5#6#7% +{% + \XINT@sum@AC@checkcarry #2{#7#6#5#4#3}% +}% +\def\XINT@sum@AC@checkcarry #1% +{% + \xint@zero #1\xint@sum@AC@nocarry 0\XINT@sum@C +}% +\def\xint@sum@AC@nocarry 0\XINT@sum@C #1#2\W\X\Y\Z { #1#2}% +\def\XINT@sum@C #1#2#3#4#5% +{% + \xint@w + #2\xint@sum@cz + \W\XINT@sum@D {#5#4#3#2}{#1}% +}% +\def\XINT@sum@D #1% +{\expandafter + \XINT@sum@CC\the\numexpr 1+10#1\relax +}% +\def\xint@sum@cz\W\XINT@sum@D #1#2{ #21000}% +% \end{macrocode} +% \subsection{\ch{xintMul}} +% \begin{macrocode} +\def\xintMul {\romannumeral0\xintmul }% +\def\xintmul #1% +{% + \expandafter\expandafter\expandafter + \xint@mul + \expandafter\expandafter\expandafter + {#1}% +}% +\def\xint@mul #1#2% +{\expandafter\expandafter\expandafter + \XINT@mul@fork #2\Z #1\Z +}% +\def\XINT@Mul #1#2{\romannumeral0\XINT@mul@fork #2\Z #1\Z }% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% MULTIPLICATION +% Ici #1#2 = 2e input et #3#4 = 1er input +% Algorithme plus efficace pour #3#4 plus long que #1#2 +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@mul@fork #1#2\Z #3#4\Z +{% + \xint@UDzerofork + #1\dummy \XINT@mul@zero + #3\dummy \XINT@mul@zero + 0\dummy + {\xint@UDsignsfork + #1#3\dummy \XINT@mul@minusminus % #1 = #3 = - + #1-\dummy \XINT@mul@minusplus % #1 = - + #3-\dummy \XINT@mul@plusminus % #3 = - + --\dummy \XINT@mul@plusplus + \xint@UDforksigns}% + \xint@UDforkzero + {#2}{#4}#1#3% +}% +\def\XINT@mul@zero #1#2#3#4{ 0}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% Dans ce qui suit #3#1 vient du #1#2 initial correspondant au +% ** 2e ** input. +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@mul@minusminus #1#2#3#4% +{% + \expandafter + \XINT@mul@enter\romannumeral0% + \XINT@RQ {}#2\R\R\R\R\R\R\R\R\Z + \W\X\Y\Z #1\W\X\Y\Z +}% +\def\XINT@mul@minusplus #1#2#3#4% +{% + \expandafter\space\expandafter-% + \romannumeral0\expandafter + \XINT@mul@enter\romannumeral0% + \XINT@RQ {}#4#2\R\R\R\R\R\R\R\R\Z + \W\X\Y\Z #1\W\X\Y\Z +}% +\def\XINT@mul@plusminus #1#2#3#4% +{% + \expandafter\space\expandafter-% + \romannumeral0\expandafter + \XINT@mul@enter\romannumeral0% + \XINT@RQ {}#2\R\R\R\R\R\R\R\R\Z + \W\X\Y\Z #3#1\W\X\Y\Z +}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% Ici #3#1 correspond au **2e input** celui censé être +% pyschologiquement plus petit +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@mul@plusplus #1#2#3#4% +{% + \expandafter + \XINT@mul@enter\romannumeral0% + \XINT@RQ {}#4#2\R\R\R\R\R\R\R\R\Z + \W\X\Y\Z #3#1\W\X\Y\Z +}% +\def\XINT@mul@add@A #1#2#3#4#5#6% +{% + \xint@w + #3\xint@mul@add@az + \W\XINT@mul@add@AB #1{#3#4#5#6}{#2}% +}% +\def\xint@mul@add@az\W\XINT@mul@add@AB #1#2% +{% + \XINT@mul@add@AC@checkcarry #1% +}% +\def\XINT@mul@add@AB #1#2#3#4\W\X\Y\Z #5#6#7#8% +{% + \XINT@mul@add@ABE #1#2{#8#7#6#5}{#3}#4\W\X\Y\Z +}% +\def\XINT@mul@add@ABE #1#2#3#4#5#6% +{\expandafter + \XINT@mul@add@ABEA\the\numexpr #1+10#5#4#3#2+#6\relax.% +}% +\def\XINT@mul@add@ABEA #1#2#3.#4% +{% + \XINT@mul@add@A #2{#3#4}% +}% +\def\XINT@mul@add@AC@checkcarry #1% +{% + \xint@zero #1\xint@mul@add@AC@nocarry 0\XINT@mul@add@C +}% +\def\xint@mul@add@AC@nocarry 0\XINT@mul@add@C #1#2\W\X\Y\Z +{% + \expandafter + \xint@cleanupzeros@andstop + \romannumeral0% + \XINT@rord@main {}#2% + \xint@UNDEF + \xint@undef\xint@undef\xint@undef\xint@undef + \xint@undef\xint@undef\xint@undef\xint@undef + \xint@UNDEF + #1% +}% +\def\XINT@mul@add@C #1#2#3#4#5% +{% + \xint@w + #5\xint@mul@add@cw + #4\xint@mul@add@cx + #3\xint@mul@add@cy + #2\xint@mul@add@cz + \W\XINT@mul@add@CD {#5#4#3#2}{#1}% +}% +\def\XINT@mul@add@CD #1% +{\expandafter + \XINT@mul@add@CC\the\numexpr 1+10#1\relax.% +}% +\def\XINT@mul@add@CC #1#2#3.#4% +{% + \XINT@mul@add@AC@checkcarry #2{#3#4}% +}% +\def\xint@mul@add@cw + #1\xint@mul@add@cx + #2\xint@mul@add@cy + #3\xint@mul@add@cz + \W\XINT@mul@add@CD +{\expandafter + \XINT@mul@add@CDw\the\numexpr 1+#1#2#3\relax.% +}% +\def\XINT@mul@add@CDw #1.#2#3\X\Y\Z +{% + \XINT@mul@add@end #1#3% +}% +\def\xint@mul@add@cx + #1\xint@mul@add@cy + #2\xint@mul@add@cz + \W\XINT@mul@add@CD +{\expandafter + \XINT@mul@add@CDx\the\numexpr 1+#1#2\relax.% +}% +\def\XINT@mul@add@CDx #1.#2#3\Y\Z +{% + \XINT@mul@add@end #1#3% +}% +\def\xint@mul@add@cy + #1\xint@mul@add@cz + \W\XINT@mul@add@CD +{\expandafter + \XINT@mul@add@CDy\the\numexpr 1+#1\relax.% +}% +\def\XINT@mul@add@CDy #1.#2#3\Z +{% + \XINT@mul@add@end #1#3% +}% +\def\xint@mul@add@cz\W\XINT@mul@add@CD #1#2#3{\XINT@mul@add@end #1#3}% +\def\XINT@mul@add@end #1#2#3#4#5% +{\expandafter\space + \the\numexpr #1#2#3#4#5\relax +}% +\def\XINT@mul@Ar #1#2#3#4#5#6% +{% + \xint@z #6\xint@mul@br\Z\XINT@mul@Br #1{#6#5#4#3}{#2}% +}% +\def\xint@mul@br\Z\XINT@mul@Br #1#2% +{% + \XINT@sum@AC@checkcarry #1% +}% +\def\XINT@mul@Br #1#2#3#4\W\X\Y\Z #5#6#7#8% +{\expandafter + \XINT@mul@ABEAr\the\numexpr #1+10#2+#8#7#6#5\relax.{#3}#4\W\X\Y\Z +}% +\def\XINT@mul@ABEAr #1#2#3#4#5#6.#7% +{% + \XINT@mul@Ar #2{#7#6#5#4#3}% +}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% Mr renvoie le résultat ***à l'envers***, sur ***4n chiffres*** +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@mul@Mr #1% +{% + \expandafter + \XINT@mul@Mr@checkifzeroorone + \expandafter{\the\numexpr #1\relax}% +}% +\def\XINT@mul@Mr@checkifzeroorone #1% +{% + \ifcase #1 + \expandafter\XINT@mul@Mr@zero + \or + \expandafter\XINT@mul@Mr@one + \else + \expandafter\XINT@mul@Nr + \fi + {0000}{}{#1}% +}% +\def\XINT@mul@Mr@zero #1\Z\Z\Z\Z { 0000}% +\def\XINT@mul@Mr@one #1#2#3#4\Z\Z\Z\Z { #4}% +\def\XINT@mul@Nr #1#2#3#4#5#6#7% +{% + \xint@z #4\xint@mul@pr\Z\XINT@mul@Pr {#1}{#3}{#7#6#5#4}{#2}{#3}% +}% +\def\XINT@mul@Pr #1#2#3% +{\expandafter + \XINT@mul@Lr\the\numexpr 10000#1+#2*#3\relax +}% +\def\XINT@mul@Lr 1#1#2#3#4#5#6#7#8#9% +{% + \XINT@mul@Nr {#1#2#3#4}{#9#8#7#6#5}% +}% +\def\xint@mul@pr\Z\XINT@mul@Pr #1#2#3#4#5% +{% + \xint@quatrezeros #1\XINT@mul@Mr@end@nocarry 0000\XINT@mul@Mr@end@carry + #1{#4}% +}% +\def\XINT@mul@Mr@end@nocarry 0000\XINT@mul@Mr@end@carry 0000#1{ #1}% +\def\XINT@mul@Mr@end@carry #1#2#3#4#5{ #5#4#3#2#1}% +\def\XINT@mul@M #1% +{\expandafter + \XINT@mul@M@checkifzeroorone + \expandafter{\the\numexpr #1\relax}% +}% +\def\XINT@mul@M@checkifzeroorone #1% +{% + \ifcase #1 + \expandafter\XINT@mul@M@zero + \or + \expandafter\XINT@mul@M@one + \else + \expandafter\XINT@mul@N + \fi + {0000}{}{#1}% +}% +\def\XINT@mul@M@zero #1\Z\Z\Z\Z { 0}% +\def\XINT@mul@M@one #1#2#3#4\Z\Z\Z\Z {% + \expandafter + \xint@cleanupzeros@andstop + \romannumeral0\XINT@rev{#4}% +}% +\def\XINT@mul@N #1#2#3#4#5#6#7% +{% + \xint@z #4\xint@mul@p\Z\XINT@mul@P {#1}{#3}{#7#6#5#4}{#2}{#3}% +}% +\def\XINT@mul@P #1#2#3% +{\expandafter + \XINT@mul@L\the\numexpr 10000#1+#2*#3\relax +}% +\def\XINT@mul@L 1#1#2#3#4#5#6#7#8#9% +{% + \XINT@mul@N {#1#2#3#4}{#5#6#7#8#9}% +}% +\def\xint@mul@p\Z\XINT@mul@P #1#2#3#4#5% +{% + \XINT@mul@M@end #1#4% +}% +\def\XINT@mul@M@end #1#2#3#4#5#6#7#8% +{\expandafter\space + \the\numexpr #1#2#3#4#5#6#7#8\relax +}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% Routine de multiplication principale +% délimiteur \W\X\Y\Z +% Le résultat partiel est toujours maintenu avec significatif à +% droite et il a un nombre multiple de 4 de chiffres +% \romannumeral0\XINT@mul@enter <N1>\W\X\Y\Z <N2>\W\X\Y\Z +% avec N1: *renversé*, *longueur 4n* (zéros éventuellement ajoutés +% au-delà du chiffre le plus significatif) +% et N2 = dans l'ordre *normal*, et pas forcément longueur 4n, +% et N2 est *non nul*. +% pas de signes +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@mul@enter #1\W\X\Y\Z #2#3#4#5% +{% + \xint@w + #5\xint@mul@enterw + #4\xint@mul@enterx + #3\xint@mul@entery + #2\xint@mul@enterz + \W\XINT@mul@start {#2#3#4#5}#1\W\X\Y\Z +}% +\def\xint@mul@enterw + #1\xint@mul@enterx + #2\xint@mul@entery + #3\xint@mul@enterz + \W\XINT@mul@start #4#5\W\X\Y\Z \X\Y\Z +{% + \XINT@mul@M {#3#2#1}#5\Z\Z\Z\Z +}% +\def\xint@mul@enterx + #1\xint@mul@entery + #2\xint@mul@enterz + \W\XINT@mul@start #3#4\W\X\Y\Z \Y\Z +{% + \XINT@mul@M {#2#1}#4\Z\Z\Z\Z +}% +\def\xint@mul@entery + #1\xint@mul@enterz + \W\XINT@mul@start #2#3\W\X\Y\Z \Z +{% + \XINT@mul@M {#1}#3\Z\Z\Z\Z +}% +\def\XINT@mul@start #1#2\W\X\Y\Z +{\expandafter + \XINT@mul@main \expandafter + {\romannumeral0\XINT@mul@Mr {#1}#2\Z\Z\Z\Z}#2\W\X\Y\Z +}% +\def\XINT@mul@main #1#2\W\X\Y\Z #3#4#5#6% +{% + \xint@w + #6\xint@mul@mainw + #5\xint@mul@mainx + #4\xint@mul@mainy + #3\xint@mul@mainz + \W\XINT@mul@compute {#1}{#3#4#5#6}#2\W\X\Y\Z +}% +\def\XINT@mul@compute #1#2#3\W\X\Y\Z +{\expandafter + \XINT@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@mul@mainw + #1\xint@mul@mainx + #2\xint@mul@mainy + #3\xint@mul@mainz + \W\XINT@mul@compute #4#5#6\W\X\Y\Z \X\Y\Z +{% + \expandafter + \XINT@mul@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@mul@mainx + #1\xint@mul@mainy + #2\xint@mul@mainz + \W\XINT@mul@compute #3#4#5\W\X\Y\Z \Y\Z +{% + \expandafter + \XINT@mul@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@mul@mainy + #1\xint@mul@mainz + \W\XINT@mul@compute #2#3#4\W\X\Y\Z \Z +{% + \expandafter + \XINT@mul@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@mul@mainz\W\XINT@mul@compute #1#2#3\W\X\Y\Z +{% + \expandafter + \xint@cleanupzeros@andstop\romannumeral0\XINT@rev{#1}% +}% +% \end{macrocode} +% \subsection{\ch{xintSqr}} +% \begin{macrocode} +\def\xintSqr {\romannumeral0\xintsqr }% +\def\xintsqr #1% +{% + \expandafter\expandafter\expandafter + \XINT@sqr + \expandafter\expandafter\expandafter + {\xintAbs{#1}}% fait l'expansion de #1 et se d\'ebarrasse du signe +}% +\def\XINT@sqr #1% +{\expandafter + \XINT@mul@enter + \romannumeral0% + \XINT@RQ {}#1\R\R\R\R\R\R\R\R\Z + \W\X\Y\Z #1\W\X\Y\Z +}% +% \end{macrocode} +% \subsection{\ch{xintPrd}, \ch{xintProductExpr}} +% \begin{verbatim} +% \xintPrd {{a}...{z}} +% \xintProductExpr {a}...{z}\relax +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@posprod #1% +{% + \XINT@pprod@checkifempty #1\Z +}% +\def\XINT@pprod@checkifempty #1% +{% + \xint@relax #1\XINT@pprod@emptyproduct\relax + \XINT@pprod@RQfirst #1% +}% +\def\XINT@pprod@emptyproduct #1\Z { 1}% +\def\XINT@pprod@RQfirst #1\Z +{% + \expandafter\XINT@pprod@getnext\expandafter + {\romannumeral0\XINT@RQ {}#1\R\R\R\R\R\R\R\R\Z}% +}% +\def\XINT@pprod@getnext #1#2% +{% + \XINT@pprod@checkiffinished #2\Z {#1}% +}% +\def\XINT@pprod@checkiffinished #1% +{% + \xint@relax #1\XINT@pprod@end\relax + \XINT@pprod@compute #1% +}% +\def\XINT@pprod@compute #1\Z #2% +{% + \expandafter + \XINT@pprod@getnext + \expandafter + {\romannumeral0\XINT@prod@mul@enter #2\W\X\Y\Z #1\W\X\Y\Z}% +}% +\def\XINT@pprod@end\relax\XINT@pprod@compute #1\Z #2% +{% + \expandafter + \xint@cleanupzeros@andstop + \romannumeral0\XINT@rev {#2}% +}% +\def\xintProductExpr {\romannumeral0\xintproductexpr }% +\def\xintPrd {\romannumeral0\xintprd }% +\def\xintprd #1% +{% + \expandafter\expandafter\expandafter + \xintproductexpr #1\relax +}% +\def\xintproductexpr #1% +{% + \expandafter\expandafter\expandafter + \XINT@prod@checkifempty #1\Z +}% +\def\XINT@prod@checkifempty #1% +{% + \xint@relax #1\XINT@prod@emptyproduct\relax + \XINT@prod@checkfirstsign #1% +}% +\def\XINT@prod@emptyproduct #1\Z { 1}% +\def\XINT@prod@checkfirstsign #1% +{% + \xint@zero #1\XINT@prod@returnzero0% + \xint@UDsignfork + #1\dummy \XINT@prod@firstisneg + -\dummy \XINT@prod@firstispos + \xint@UDforksign + #1% +}% +\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}% +}% +\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 }% +}% +\def\XINT@prod@xpxpnext #1#2#3% +{% + \expandafter\expandafter\expandafter + \XINT@prod@checkiffinished #3\Z {#2}#1% +}% +\def\XINT@prod@checkiffinished #1% +{% + \xint@relax #1\XINT@prod@end\relax + \XINT@prod@checksign #1% +}% +\def\XINT@prod@checksign #1% +{% + \xint@zero #1\XINT@prod@returnzero0% + \xint@UDsignfork + #1\dummy \XINT@prod@neg + -\dummy \XINT@prod@pos + \xint@UDforksign + #1% +}% +\def\XINT@prod@pos #1\Z #2#3% +{% + \expandafter + \XINT@prod@xpxpnext + \expandafter + #3% + \expandafter + {\romannumeral0\XINT@prod@mul@enter #2\W\X\Y\Z #1\W\X\Y\Z }% +}% +\def\XINT@prod@neg #1#2\Z #3#4% +{% + \expandafter + \XINT@prod@xpxpnext + \expandafter + {\the\numexpr 1-#4\expandafter}% + \expandafter + {\romannumeral0\XINT@prod@mul@enter #3\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}% +}% +\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} +% \begin{verbatim} +%--- multiplication spéciale +% \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% +{% + \expandafter\expandafter\expandafter + \XINT@fac@fork + \expandafter\expandafter\expandafter + {#1}% +}% +\def\XINT@Fac {\romannumeral0\XINT@fac@fork }% +\def\XINT@fac@fork #1% +{% + \ifcase\xintSgn {#1} + \xint@afterfi{\expandafter\space\expandafter 1\xint@gobble }% + \or + \expandafter\XINT@fac@checklength + \else + \xint@afterfi{\xintError:FactorialOfNegativeNumber + \expandafter\space\expandafter 1\xint@gobble }% + \fi + {#1}% +}% +\def\XINT@fac@checklength #1% +{% + \ifnum\xintLen {#1} > 9 + \xint@afterfi{\xintError:FactorialOfTooBigNumber + \expandafter\space\expandafter 1\xint@gobble@three }% + \else + \expandafter\XINT@fac@loop + \fi + {1}{#1}{}% +}% +\def\XINT@fac@loop #1#2#3% +{% + \ifnum #1<#2 + \expandafter + \XINT@fac@loop + \expandafter + {\the\numexpr #1+1\expandafter }% + \else + \expandafter\XINT@fac@docomputation + \fi + {#2}{#3{#1}}% +}% +\def\XINT@fac@docomputation #1#2% +{% + \XINT@posprod #2\relax +}% +% \end{macrocode} +% \subsection{\ch{xintPow}} +% \begin{macrocode} +\def\xintPow {\romannumeral0\xintpow }% +\def\xintpow #1% +{% + \expandafter\expandafter\expandafter + \xint@pow + #1\Z% +}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% #1#2 = A +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\xint@pow #1#2\Z +{% + \xint@UDsignfork + #1\dummy \XINT@pow@Aneg + -\dummy \XINT@pow@Anonneg + \xint@UDforksign + #1{#2}% +}% +\def\XINT@pow@Aneg #1#2#3% +{% + \expandafter\expandafter\expandafter + \XINT@pow@Aneg@ + \expandafter\expandafter\expandafter + {#3}{#2}% +}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% B = #1, xpxp déjà fait +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@pow@Aneg@ #1% +{% + \ifcase\XINT@Odd{#1} + \or \expandafter\XINT@pow@Aneg@Bodd + \fi + \XINT@pow@Anonneg@ {#1}% +}% +\def\XINT@pow@Aneg@Bodd #1% +{% + \expandafter\XINT@opp\romannumeral0\XINT@pow@Anonneg@ +}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% B = #3, faire le xpxp +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@pow@Anonneg #1#2#3% +{% + \expandafter\expandafter\expandafter + \XINT@pow@Anonneg@ + \expandafter\expandafter\expandafter + {#3}{#1#2}% +}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% #1 = B, #2 = |A| +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@pow@Anonneg@ #1#2% +{% + \ifcase\XINT@Cmp {#2}{1} + \expandafter\XINT@pow@AisOne + \or + \expandafter\XINT@pow@AatleastTwo + \else + \expandafter\XINT@pow@AisZero + \fi + {#1}{#2}% +}% +\def\XINT@pow@AisOne #1#2{ 1}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% #1 = B +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@pow@AisZero #1#2% +{% + \ifcase\XINT@Sgn {#1} + \xint@afterfi { 1}% + \or + \xint@afterfi { 0}% + \else + \xint@afterfi {\xintError:DivisionByZero\space 0}% + \fi +}% +\def\XINT@pow@AatleastTwo #1% +{% + \ifcase\XINT@Sgn {#1} + \expandafter\XINT@pow@BisZero + \or + \expandafter\XINT@pow@checkBlength + \else + \expandafter\XINT@pow@BisNegative + \fi + {#1}% +}% +\def\XINT@pow@BisNegative #1#2{\xintError:FractionRoundedToZero\space 0}% +\def\XINT@pow@BisZero #1#2{ 1}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% B = #1 > 0, A = #2 > 1 +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@pow@checkBlength #1#2% +{% + \ifnum\xintLen{#1} >9 + \expandafter\XINT@pow@BtooBig + \else + \expandafter\XINT@pow@loop + \fi + {#1}{#2}\XINT@posprod + \xint@UNDEF + \xint@undef\xint@undef\xint@undef\xint@undef + \xint@undef\xint@undef\xint@undef\xint@undef + \xint@UNDEF +}% +\def\XINT@pow@BtooBig #1\xint@UNDEF #2\xint@UNDEF + {\xintError:ExponentTooBig\space 0}% +\def\XINT@pow@loop #1#2% +{% + \ifnum #1 = 1 + \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{\romannumeral0\xintsqr{#2}}}% + \fi + {{#2}}% +}% +\def\XINT@pow@loop@end {\romannumeral0\XINT@rord@main {}\relax }% +\def\XINT@pow@loop@a #1% +{% + \ifnum #1 = 1 + \expandafter\XINT@pow@loop + \else + \expandafter\XINT@pow@loop@throwaway + \fi +}% +\def\XINT@pow@loop@throwaway #1#2#3% +{% + \XINT@pow@loop {#1}{#2}% +}% +% \end{macrocode} +% \subsection{\ch{xintDivision}, \ch{xintQuo}, \ch{xintRem}} +% \begin{macrocode} +\def\xintQuo {\romannumeral0\xintquo }% +\def\xintRem {\romannumeral0\xintrem }% +\def\xintquo {% + \expandafter + \xint@firstoftwo@andstop + \romannumeral0\xintdivision }% +\def\xintrem {% + \expandafter + \xint@secondoftwo@andstop + \romannumeral0\xintdivision }% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% #1 = A, #2 = B. On calcule le quotient de A par B +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\xintDivision {\romannumeral0\xintdivision }% +\def\xintdivision #1% +{% + \expandafter\expandafter\expandafter + \xint@division + \expandafter\expandafter\expandafter + {#1}% +}% +\def\xint@division #1#2% +{% + \expandafter\expandafter\expandafter + \XINT@div@fork #2\Z #1\Z +}% +\def\XINT@Division #1#2{\romannumeral0\XINT@div@fork #2\Z #1\Z }% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% #1#2 = 2e input = diviseur = B +% #3#4 = 1er input = divisé = A +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@div@fork #1#2\Z #3#4\Z +{% + \xint@UDzerofork + #1\dummy \XINT@div@BisZero + #3\dummy \XINT@div@AisZero + 0\dummy + {\xint@UDsignfork + #1\dummy \XINT@div@BisNegative % B < 0 + #3\dummy \XINT@div@AisNegative % A < 0, B > 0 + -\dummy \XINT@div@plusplus % B > 0, A > 0 + \xint@UDforksign }% + \xint@UDforkzero + {#2}{#4}#1#3% #1#2=B, #3#4=A +}% +\def\XINT@div@BisZero #1#2#3#4% + {\xintError:DivisionByZero\space {0}{0}}% +\def\XINT@div@AisZero #1#2#3#4{ {0}{0}}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% jusqu'à présent c'est facile. +% minusplus signifie B < 0, A > 0 +% plusminus signifie B > 0, A < 0 +% Ici #3#1 correspond au diviseur B et #4#2 au divisé A +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@div@plusplus #1#2#3#4% +{% + \XINT@div@prepare {#3#1}{#4#2}% +}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% B = #3#1 < 0, A non nul positif ou négatif +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@div@BisNegative #1#2#3#4% +{% + \expandafter\XINT@div@BisNegative@post + \romannumeral0\XINT@div@fork #1\Z #4#2\Z +}% +\def\XINT@div@BisNegative@post #1#2% +{% + \expandafter\space\expandafter + {\romannumeral0\XINT@opp #1}{#2}% +}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% B = #3#1 > 0, A =-#2< 0 +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@div@AisNegative #1#2#3#4% +{% + \expandafter\XINT@div@AisNegative@post + \romannumeral0\XINT@div@prepare {#3#1}{#2}{#3#1}% +}% +\def\XINT@div@AisNegative@post #1#2% +{% + \ifcase\xintSgn {#2} + \expandafter \XINT@div@AisNegative@zerorem + \or + \expandafter \XINT@div@AisNegative@posrem + \fi + {#1}{#2}% +}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% en #3 on a une copie de B (à l'endroit) +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@div@AisNegative@zerorem #1#2#3% +{% + \expandafter\space\expandafter + {\romannumeral0\XINT@opp #1}{0}% +}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% #1 = quotient, #2 = reste, #3 = diviseur initial (à l'endroit) +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@div@AisNegative@posrem #1% +{% + \expandafter + \XINT@div@AisNegative@posrem@b + \expandafter + {\romannumeral0\xintopp {\XINT@Add{#1}{1}}}% +}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% remplace Reste par B - Reste, après avoir remplacé Q par -(Q+1) +% de sorte que la formule a = qb + r, 0<= r < |b| est valable +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@div@AisNegative@posrem@b #1#2#3% +{% + \expandafter + \xint@exchangetwo@keepbraces@andstop + \expandafter + {\romannumeral0\XINT@sub {#3}{#2}}{#1}% +}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% par la suite A et B sont > 0. +% #1 = B. Pour le moment à l'endroit. +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@div@prepare #1% +{% + \expandafter + \XINT@div@prepareB@a + \expandafter + {\romannumeral0\XINT@length {#1}}{#1}% B > 0 ici +}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% Calcul du plus petit K = 4n >= longueur de B +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@div@prepareB@a #1% +{% + \expandafter\XINT@div@prepareB@b\expandafter + {\the\numexpr 4*((#1+1)/4)\relax}{#1}% +}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% #1 = K +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@div@prepareB@b #1#2% +{% + \expandafter\XINT@div@prepareB@c \expandafter + {\the\numexpr #1-#2\relax}{#1}% +}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% #1 = c +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@div@prepareB@c #1% +{% + \ifcase #1 + \expandafter\XINT@div@prepareB@di + \or \expandafter\XINT@div@prepareB@dii + \or \expandafter\XINT@div@prepareB@diii + \else \expandafter\XINT@div@prepareB@div + \fi +}% +\def\XINT@div@prepareB@di {\XINT@div@prepareB@e {}{0}}% +\def\XINT@div@prepareB@dii {\XINT@div@prepareB@e {0}{1}}% +\def\XINT@div@prepareB@diii {\XINT@div@prepareB@e {00}{2}}% +\def\XINT@div@prepareB@div {\XINT@div@prepareB@e {000}{3}}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% #1 = zéros à rajouter à B, #2=c, #3=K, #4 = B +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@div@prepareB@e #1#2#3#4% +{% + \XINT@div@prepareB@f #4#1\Z {#3}{#2}{#1}% +}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% x = #1#2#3#4 = 4 premiers chiffres de B. #1 est non nul. +% Ensuite on renverse B pour calculs plus rapides par la suite. +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@div@prepareB@f #1#2#3#4#5\Z +{% + \expandafter + \XINT@div@prepareB@g + \expandafter + {\romannumeral0\XINT@rev {#1#2#3#4#5}}{#1#2#3#4}% +}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% #3= K, #4 = c, #5= {} ou {0} ou {00} ou {000}, #6 = A initial +% #1 = B préparé et renversé, #2 = x = quatre premiers chiffres +% On multiplie aussi A par 10^c. +% B, x, K, c, {} ou {0} ou {00} ou {000}, A initial +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@div@prepareB@g #1#2#3#4#5#6% +{% + \XINT@div@prepareA@a {#6#5}{#2}{#3}{#1}{#4}% +}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% A, x, K, B, c, +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@div@prepareA@a #1% +{% + \expandafter + \XINT@div@prepareA@b + \expandafter + {\romannumeral0\XINT@length {#1}}{#1}% A >0 ici +}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% L0, A, x, K, B, ... +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@div@prepareA@b #1% +{% + \expandafter\XINT@div@prepareA@c\expandafter + {\the\numexpr 4*((#1+1)/4)\relax}{#1}% +}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% L, L0, A, x, K, B,... +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@div@prepareA@c #1#2% +{% + \expandafter\XINT@div@prepareA@d \expandafter + {\the\numexpr #1-#2\relax}{#1}% +}% +\def\XINT@div@prepareA@d #1% +{% + \ifcase #1 + \expandafter\XINT@div@prepareA@di + \or \expandafter\XINT@div@prepareA@dii + \or \expandafter\XINT@div@prepareA@diii + \else \expandafter\XINT@div@prepareA@div + \fi +}% +\def\XINT@div@prepareA@di {\XINT@div@prepareA@e {}}% +\def\XINT@div@prepareA@dii {\XINT@div@prepareA@e {0}}% +\def\XINT@div@prepareA@diii {\XINT@div@prepareA@e {00}}% +\def\XINT@div@prepareA@div {\XINT@div@prepareA@e {000}}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% #1#3 = A préparé, #2 = longueur de ce A préparé, +% +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@div@prepareA@e #1#2#3% +{% + \XINT@div@startswitch {#1#3}{#2}% +}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% A, L, x, K, B, ... +% A, L, x, K, B, c +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@div@startswitch #1#2#3#4% +{% + \ifnum #2 > #4 + \expandafter\XINT@div@body@a + \else + \ifnum #2 = #4 + \expandafter\expandafter\expandafter + \XINT@div@final@a + \else + \expandafter\expandafter\expandafter + \XINT@div@finished@a + \fi\fi {#1}{#4}{#3}{0000}{#2}% +}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% A, K, x, Q, L, B, c +% ---- "Finished" +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@div@finished@a #1#2#3% +{% + \expandafter + \XINT@div@finished@b + \expandafter + {\romannumeral0\XINT@cuz {#1}}% +}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% A, Q, L, B, c +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@div@finished@b #1#2#3#4#5% +{% + \ifcase \XINT@Sgn {#1} + \xint@afterfi {\XINT@div@finished@c {0}}% + \or + \xint@afterfi {\expandafter\XINT@div@finished@c + \expandafter + {\romannumeral0\XINT@dsh@preparegobble {#1}{#5}}}% + \fi + {#2}% +}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% Reste Final, Q à renverser +% #2 = Quotient, #1 = Reste. +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@div@finished@c #1#2% +{% + \expandafter + \space + \expandafter + {\romannumeral0\expandafter\xint@cleanupzeros@andstop + \romannumeral0\XINT@rev {#2}}{#1}% +}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% ---- "Final" +% A, K, x, Q, L, B, c +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@div@final@a #1% +{% + \XINT@div@final@b #1\Z +}% +\def\XINT@div@final@b #1#2#3#4#5\Z +{% + \xint@quatrezeros #1#2#3#4\xint@div@final@c0000% + \XINT@div@final@c {#1#2#3#4}{#1#2#3#4#5}% +}% +\def\xint@div@final@c0000\XINT@div@final@c #1% + {\XINT@div@finished@a }% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% a, A, K, x, Q, L, B ,c +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@div@final@c #1#2#3#4% +{% + \expandafter + \XINT@div@final@d + \expandafter + {\the\numexpr #1/#4\relax}{#2}% +}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% q, A, Q, L, B à l'envers sur 4n, c +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@div@final@d #1#2#3#4#5% +{% + \expandafter + \XINT@div@final@e + \expandafter + {\romannumeral0\xintsub {\xint@cleanupzeros@nospace #2}% + {\romannumeral0\XINT@mul@M {#1}#5\Z\Z\Z\Z }}% + {#1}{#2}{#3}{#4}{#5}% +}% +\def\XINT@div@final@e #1#2% +{% + \ifnum\xintSgn{#1} < 0 + \expandafter\XINT@div@final@d % en arri\`ere toute + \expandafter{\the\numexpr #2-1\expandafter + \expandafter\expandafter }% + \expandafter\xint@gobble@two + \else + \expandafter\XINT@div@final@f + \fi + {#1}{#2}% +}% +\def\XINT@div@final@f #1#2#3#4#5#6#7% +{% + \ifcase \XINT@Sgn {#1} + \xint@afterfi {\XINT@div@final@end {0}}% + \or + \xint@afterfi {\expandafter\XINT@div@final@end + \expandafter + {\romannumeral0\XINT@dsh@preparegobble {#1}{#7}}}% + \fi + {\romannumeral0\xintadd {\XINT@Rev@andcleanupzeros{#4}}{#2}}% +}% +\def\XINT@div@final@end #1#2% +{% + \expandafter\space\expandafter + {#2}{#1}% +}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% Boucle Principale +% A, K, x, Q, L, B, c +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@div@body@a #1% +{% + \XINT@div@body@b #1\Z +}% +\def\XINT@div@body@b #1#2#3#4#5#6#7#8#9\Z +{% + \XINT@div@body@c + {#1#2#3#4#5#6#7#8#9}% + {#1#2#3#4#5#6#7#8}% +}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% A, a, K, x, Q, L, B, c +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@div@body@c #1#2#3% +{% + \XINT@div@body@d {#3}{}#1\Z {#2}{#3}% +}% +\def\XINT@div@body@d #1#2#3#4#5#6% +{% + \ifnum #1 > 0 + \expandafter + \XINT@div@body@d + \expandafter + {\the\numexpr #1-4\expandafter }% + \else + \expandafter + \XINT@div@body@e + \fi + {#6#5#4#3#2}% +}% +\def\XINT@div@body@e #1#2\Z #3% +{% + \XINT@div@body@f {#3}{#1}{#2}% +}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% a, alpha, alpha', K, x, Q, L, B, c +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@div@body@f #1#2#3#4#5#6#7#8% +{% + \expandafter\XINT@div@body@g + \expandafter + {\the\numexpr (#1+(#5+1)/2)/(#5+1)-1\relax }% + {#2}{#8}{#4}{#5}{#3}{#6}{#7}{#8}% +}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% q1, alpha, B, K, x, alpha', Q, L, B, c +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@div@body@g #1#2#3% +{% + \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 }% + {#2}\Z + {#3}{#1}% +}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% alpha1 = alpha-q1 B, \Z, B, q1, K, x, alpha', Q, L, B, c +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@div@body@h #1#2#3#4#5#6#7#8#9\Z +{% + \ifnum #1#2#3#4>0 + \xint@afterfi{\XINT@div@body@i {#1#2#3#4#5#6#7#8}}% + \else + \expandafter\XINT@div@body@k + \fi + {#1#2#3#4#5#6#7#8#9}% +}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% a1, alpha1, B, q1, K, x, alpha', Q, L, B, c +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@div@body@i #1#2#3#4#5#6% +{% + \expandafter\XINT@div@body@j + \expandafter{\the\numexpr (#1+(#6+1)/2)/(#6+1)-1\relax }% + {#2}{#3}{#4}{#5}{#6}% +}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% q2, alpha1, B, q1, K, x, alpha', Q, L, B, c +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@div@body@j #1#2#3#4% +{% + \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 }% + {\XINT@Rev{#2}}}% + {#4+#1}% +}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% alpha2, q1+q2, K, x, alpha', Q, L, B, c +% attention body@j -> body@l +% alpha1, B, q=q1, K, x, alpha', Q, L, B, c +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@div@body@k #1#2% +{% + \XINT@div@body@l {#1}% +}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% alpha2, q= q1+q2, K, x, alpha', Q, L, B, c +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@div@body@l #1#2#3#4#5#6#7% +{% + \expandafter + \XINT@div@body@m + \the\numexpr 100000000+#2\relax + {#6}{#3}{#7}{#1#5}{#4}% +}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% chiffres de q, Q, K, L, A', x, B, c +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@div@body@m #1#2#3#4#5#6#7#8#9% +{% + \ifnum #2#3#4#5>0 + \xint@afterfi {\XINT@div@body@n {#9#8#7#6#5#4#3#2}}% + \else + \xint@afterfi {\XINT@div@body@n {#9#8#7#6}}% + \fi +}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% q renversé, Q, K, L, A', x, B, c +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@div@body@n #1#2% +{% + \expandafter\XINT@div@body@o\expandafter + {\romannumeral0\XINT@sum@A 0{}#1\W\X\Y\Z #2\W\X\Y\Z }% +}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% q+Q, K, L, A', x, B, c +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@div@body@o #1#2#3#4% +{% + \XINT@div@body@p {#3}{#2}{}#4\Z {#1}% +}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% L, K, {}, A'\Z, q+Q, x, B, c +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@div@body@p #1#2#3#4#5#6#7% +{% + \ifnum #1 > #2 + \xint@afterfi + {\ifnum #4#5#6#7 > 0 + \expandafter\XINT@div@body@q + \else + \expandafter\XINT@div@body@repeatp + \fi }% + \else + \expandafter\XINT@div@gotofinal@a + \fi + {#1}{#2}{#3}#4#5#6#7% +}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% L, K, zeros, A' avec moins de zéros\Z, q+Q, x, B, c +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@div@body@repeatp #1#2#3#4#5#6#7% +{% + \expandafter + \XINT@div@body@p + \expandafter + {\the\numexpr #1-4\relax}{#2}{0000#3}% +}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% L -> L-4, zeros->zeros+0000, répéter jusqu'à ce que soit L=K +% soit on ne trouve plus 0000 +% nouveau L, K, zeros, nouveau A=#4, Q+q, x, B, c +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@div@body@q #1#2#3#4\Z #5#6% +{% + \XINT@div@body@a {#4}{#2}{#6}{#3#5}{#1}% +}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% A, K, x, Q, L, B, c --> iterate +% ---- +% Boucle Principale achevée +% ATTENTION IL FAUT AJOUTER 4 ZEROS DE MOINS QUE CEUX +% QUI ONT ÉTÉ PRÉPARÉS DANS #3!! +% L, K (L=K), zeros, A\Z, Q, x, B, c +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@div@gotofinal@a #1#2#3#4\Z % +{% + \XINT@div@gotofinal@b #3\Z {#4}{#1}% +}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% zeros\Z, A, L=K, Q, x, B, c +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@div@gotofinal@b 0000#1\Z #2#3#4#5% +{% + \XINT@div@final@a {#2}{#3}{#5}{#1#4}{#3}% +}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% A, L=K, x, Q avec zéros, L, B, c +% La soustraction spéciale. Étendre deux fois les arguments +% pour \XINT@div@sub@enter longueur multiple de 4 on sait que #2>#1, +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@div@sub@xpxp #1% +{% + \expandafter + \XINT@div@sub@xpxp@ + \expandafter + {#1}% +}% +\def\XINT@div@sub@xpxp@ #1#2% +{% + \expandafter\expandafter\expandafter + \XINT@div@sub@xpxp@@ + #2\W\X\Y\Z #1\W\X\Y\Z +}% +\def\XINT@div@sub@xpxp@@ +{% + \XINT@div@sub@A 1{}% +}% +\def\XINT@div@sub@A #1#2#3#4#5#6% +{% + \xint@w + #3\xint@div@sub@az + \W\XINT@div@sub@B #1{#3#4#5#6}{#2}% +}% +\def\XINT@div@sub@B #1#2#3#4\W\X\Y\Z #5#6#7#8% +{% + \xint@w + #5\xint@div@sub@bz + \W\XINT@div@sub@onestep #1#2{#8#7#6#5}{#3}#4\W\X\Y\Z +}% +\def\XINT@div@sub@onestep #1#2#3#4#5#6% +{\expandafter + \XINT@div@sub@backtoA\the\numexpr 11#5#4#3#2-#6+#1-1\relax.% +}% +\def\XINT@div@sub@backtoA #1#2#3.#4% +{% + \XINT@div@sub@A #2{#3#4}% +}% +\def\xint@div@sub@bz + \W\XINT@div@sub@onestep #1#2#3#4#5#6#7% +{% + \xint@UDzerofork + #1\dummy \XINT@div@sub@C % + 0\dummy \XINT@div@sub@D % pas de retenue + \xint@UDforkzero + {#7}#2#3#4#5% +}% +\def\XINT@div@sub@D #1#2\W\X\Y\Z +{% + \expandafter\space + \romannumeral0% + \XINT@rord@main {}#2% + \xint@UNDEF + \xint@undef\xint@undef\xint@undef\xint@undef + \xint@undef\xint@undef\xint@undef\xint@undef + \xint@UNDEF + #1% +}% +\def\XINT@div@sub@C #1#2#3#4#5% +{% + \xint@w + #2\xint@div@sub@cz + \W\XINT@div@sub@AC@onestep {#5#4#3#2}{#1}% +}% +\def\XINT@div@sub@AC@onestep #1% +{\expandafter + \XINT@div@sub@backtoC\the\numexpr 11#1-1\relax.% +}% +\def\XINT@div@sub@backtoC #1#2#3.#4% +{% + \XINT@div@sub@AC@checkcarry #2{#3#4}% la retenue va \^etre examin\'ee +}% +\def\XINT@div@sub@AC@checkcarry #1% +{% + \xint@one #1\xint@div@sub@AC@nocarry 1\XINT@div@sub@C +}% +\def\xint@div@sub@AC@nocarry 1\XINT@div@sub@C #1#2\W\X\Y\Z +{% + \expandafter\space + \romannumeral0% + \XINT@rord@main {}#2% + \xint@UNDEF + \xint@undef\xint@undef\xint@undef\xint@undef + \xint@undef\xint@undef\xint@undef\xint@undef + \xint@UNDEF + #1% +}% +\def\xint@div@sub@cz\W\XINT@div@sub@AC@onestep #1#2{ #2}% +\def\xint@div@sub@az\W\XINT@div@sub@B #1#2#3#4\Z { #3}% +% \end{macrocode} +% \begin{verbatim} +%----------------------------------------------------------------- +%----------------------------------------------------------------- +% DECIMAL OPERATIONS: FIRST DIGIT, LASTDIGIT, ODDNESS, +% MULTIPLICATION BY TEN, QUOTIENT BY TEN, QUOTIENT OR +% MULTIPLICATION BY POWER OF TEN, SPLIT OPERATION. +% \end{verbatim} +% \vspace*{-2\baselineskip} +% \subsection{\ch{xintFDg}} +% \begin{verbatim} +% FIRST DIGIT +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\xintFDg {\romannumeral0\xintfdg }% +\def\xintfdg #1% +{% + \expandafter\expandafter\expandafter + \XINT@fdg #1\W\Z +}% +\def\XINT@FDg #1{\romannumeral0\XINT@fdg #1\W\Z }% +\def\XINT@fdg #1#2% +{% + \xint@xpxp@andstop + \xint@UDzerominusfork + #1-\dummy {\expandafter 0}% zero + 0#1\dummy {\expandafter #2}% negative + 0-\dummy {\expandafter #1}% positive + \xint@UDforkminuszero + \xint@z +}% +% \end{macrocode} +% \subsection{\ch{xintLDg}} +% \begin{verbatim} +% LAST DIGIT +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\xintLDg {\romannumeral0\xintldg }% +\def\xintldg #1% +{% + \expandafter\expandafter\expandafter + \XINT@ldg + \expandafter\expandafter\expandafter + {#1}% +}% +\def\XINT@LDg #1{\romannumeral0\XINT@ldg {#1}}% +\def\XINT@ldg #1% +{% + \expandafter + \XINT@ldg@ + \romannumeral0\XINT@rev {#1}\Z +}% +\def\XINT@ldg@ #1% +{% + \expandafter\space\expandafter #1\xint@z +}% +% \end{macrocode} +% \subsection{\ch{xintOdd}} +% \begin{verbatim} +% ODDNESS +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\xintOdd {\romannumeral0\xintodd }% +\def\xintodd #1% +{% + \ifodd\xintLDg{#1} + \xint@afterfi{ 1}% + \else + \xint@afterfi{ 0}% + \fi +}% +\def\XINT@Odd #1% +{\romannumeral0% + \ifodd\XINT@LDg{#1} + \xint@afterfi{ 1}% + \else + \xint@afterfi{ 0}% + \fi +}% +% \end{macrocode} +% \subsection{\ch{xintDSL}} +% \begin{verbatim} +% DECIMAL SHIFT LEFT (=MULTIPLICATION PAR 10) +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\xintDSL {\romannumeral0\xintdsl }% +\def\xintdsl #1% +{% + \expandafter\expandafter\expandafter + \XINT@dsl #1\Z +}% +\def\XINT@DSL #1{\romannumeral0\XINT@dsl #1\Z }% +\def\XINT@dsl #1% +{% + \xint@zero #1\xint@dsl@zero 0\XINT@dsl@ #1% +}% +\def\xint@dsl@zero 0\XINT@dsl@ 0#1\Z { 0}% +\def\XINT@dsl@ #1\Z { #10}% +% \end{macrocode} +% \subsection{\ch{xintDSR}} +% \begin{verbatim} +% DECIMAL SHIFT RIGHT (=DIVISION PAR 10) +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\xintDSR {\romannumeral0\xintdsr }% +\def\xintdsr #1% +{% + \expandafter\expandafter\expandafter + \XINT@dsr@a + \expandafter\expandafter\expandafter + {#1}\W\Z +}% +\def\XINT@DSR #1{\romannumeral0\XINT@dsr@a {#1}\W\Z }% +\def\XINT@dsr@a +{% + \expandafter + \XINT@dsr@b + \romannumeral0\XINT@rev +}% +\def\XINT@dsr@b #1#2#3\Z +{% + \xint@w #2\xint@dsr@onedigit\W + \xint@minus #2\xint@dsr@onedigit-% + \expandafter + \XINT@dsr@removew + \romannumeral0\XINT@rev {#2#3}% +}% +\def\xint@dsr@onedigit #1\XINT@rev #2{ 0}% +\def\XINT@dsr@removew #1\W { }% +% \end{macrocode} +% \subsection{\ch{xintDSH}} +% \begin{verbatim} +% DECIMAL SHIFTS +% \xintDSH {x}{A} +% si x <= 0, fait A -> A.10^(|x|) +% si x > 0, et A >=0, fait A -> quo(A,10^(x)) +% si x > 0, et A < 0, fait A -> -quo(-A,10^(x)) +% (donc pour x > 0 c'est comme DSR itéré x fois) +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\xintDSH {\romannumeral0\xintdsh }% +\def\xintdsh #1% +{% + \expandafter\expandafter\expandafter + \xint@dsh + \expandafter\expandafter\expandafter + {#1}% +}% +\def\xint@dsh #1#2% +{% + \expandafter\expandafter\expandafter\expandafter + \expandafter\expandafter\expandafter + \XINT@dsh@checkxsize + \expandafter\expandafter\expandafter + \xint@exchangetwo@keepbraces + \expandafter\expandafter\expandafter + {#2}{#1}% +}% +\def\XINT@DSH {\romannumeral0\XINT@dsh@checkxsize }% +\def\XINT@dsh@checkxsize #1% +{% + \ifnum\XINT@Len {#1} > 9 + \expandafter\XINT@dsh@bigx + \else + \expandafter\XINT@dsh + \fi + {#1}% +}% +\def\XINT@dsh@bigx #1#2% +{% + \ifcase\XINT@Sgn {#1} + \or \xint@afterfi { 0}% + \else \xint@afterfi + {% + \ifodd\XINT@Sgn {#2} + \xint@afterfi{\xintError:TooBigDecimalShift + \space0}% + \else \xint@afterfi { 0}% + \fi + }% + \fi +}% +\def\XINT@dsh #1#2{\XINT@dsh@checkAsign #2\Z {#1}}% +\def\XINT@dsh@checkAsign #1% +{% + \xint@UDzerominusfork + #1-\dummy \XINT@dsh@AisZero + 0#1\dummy \XINT@dsh@AisNeg + 0-\dummy {\XINT@dsh@directionfork #1}% + \xint@UDforkminuszero +}% +\def\XINT@dsh@AisZero #1\Z #2{ 0}% +\def\XINT@dsh@AisNeg {\expandafter\XINT@dsh@neg@checkifreturnedzero + \romannumeral0\XINT@dsh@directionfork }% +\def\XINT@dsh@neg@checkifreturnedzero #1% +{% + \expandafter\space + \xint@UDzerofork + #1\dummy {0}% + 0\dummy {-#1}% + \xint@UDforkzero +}% +\def\XINT@dsh@directionfork #1\Z #2% +{% + \XINT@dsh@checkxsign #2\Z {#1}% +}% +\def\XINT@dsh@checkxsign #1% +{% + \xint@UDzerominusfork + #1-\dummy \XINT@dsh@donothing + 0#1\dummy \XINT@dsh@shiftleft + 0-\dummy {\XINT@dsh@shiftright #1}% + \xint@UDforkminuszero +}% +\def\XINT@dsh@donothing #1\Z #2{ #2}% +\def\XINT@dsh@shiftright #1\Z #2% +{% + \ifnum \XINT@Length {#2} > #1 + \expandafter\XINT@dsh@preparegobble + \else + \expandafter\XINT@dsh@returnzero + \fi + {#2}{#1}% +}% +\def\XINT@dsh@returnzero #1#2{ 0}% +\def\XINT@dsh@preparegobble #1% +{% + \expandafter + \XINT@dsh@preparegobble@ + \expandafter + {\romannumeral0\XINT@rev{#1}}% +}% +\def\XINT@dsh@preparegobble@ #1#2{\XINT@dsh@gobbleloop {#2}#1\Z }% +\def\XINT@dsh@gobbleloop #1% +{% + \ifcase #1 + \expandafter\XINT@dsh@endgobble + \or + \expandafter\XINT@dsh@gobble@one@andend + \or + \expandafter\XINT@dsh@gobble@two@andend + \or + \expandafter\XINT@dsh@gobble@three@andend + \or + \expandafter\XINT@dsh@gobble@four@andend + \or + \expandafter\XINT@dsh@gobble@five@andend + \or + \expandafter\XINT@dsh@gobble@six@andend + \or + \expandafter\XINT@dsh@gobble@seven@andend + \else + \expandafter \XINT@dsh@gobbleloop + \expandafter + {\the\numexpr + #1-8\expandafter\expandafter\expandafter }% + \expandafter + \xint@gobble@eight + \fi +}% +\def\XINT@dsh@gobble@one@andend + {\expandafter\XINT@dsh@endgobble\xint@gobble@one }% +\def\XINT@dsh@gobble@two@andend + {\expandafter\XINT@dsh@endgobble\xint@gobble@two }% +\def\XINT@dsh@gobble@three@andend + {\expandafter\XINT@dsh@endgobble\xint@gobble@three }% +\def\XINT@dsh@gobble@four@andend + {\expandafter\XINT@dsh@endgobble\xint@gobble@four }% +\def\XINT@dsh@gobble@five@andend + {\expandafter\XINT@dsh@endgobble\xint@gobble@five }% +\def\XINT@dsh@gobble@six@andend + {\expandafter\XINT@dsh@endgobble\xint@gobble@six }% +\def\XINT@dsh@gobble@seven@andend + {\expandafter\XINT@dsh@endgobble\xint@gobble@seven }% +\def\XINT@dsh@endgobble #1\Z +{% + \XINT@rev{#1}% +}% +\def\XINT@dsh@shiftleft #1\Z +{% + \XINT@dsh@zeroloop {#1}\Z +}% +\def\XINT@dsh@zeroloop #1% +{% + \ifcase #1 + \expandafter \XINT@dsh@exit + \or + \expandafter \XINT@dsh@exiti + \or + \expandafter \XINT@dsh@exitii + \or + \expandafter \XINT@dsh@exitiii + \or + \expandafter \XINT@dsh@exitiv + \or + \expandafter \XINT@dsh@exitv + \or + \expandafter \XINT@dsh@exitvi + \or + \expandafter \XINT@dsh@exitvii + \else + \expandafter \XINT@dsh@zeroloop + \expandafter + {\the\numexpr + #1-8\expandafter\expandafter\expandafter }% + \expandafter + \XINT@dsh@addeightzeros + \fi +}% +\def\XINT@dsh@addeightzeros {00000000}% +\def\XINT@dsh@exit #1\Z + {\XINT@dsh@addzeros {#1}}% +\def\XINT@dsh@exiti #1\Z + {\XINT@dsh@addzeros {0#1}}% +\def\XINT@dsh@exitii #1\Z + {\XINT@dsh@addzeros {00#1}}% +\def\XINT@dsh@exitiii #1\Z + {\XINT@dsh@addzeros {000#1}}% +\def\XINT@dsh@exitiv #1\Z + {\XINT@dsh@addzeros {0000#1}}% +\def\XINT@dsh@exitv #1\Z + {\XINT@dsh@addzeros {00000#1}}% +\def\XINT@dsh@exitvi #1\Z + {\XINT@dsh@addzeros {000000#1}}% +\def\XINT@dsh@exitvii #1\Z + {\XINT@dsh@addzeros {0000000#1}}% +\def\XINT@dsh@addzeros #1#2{ #2#1}% +% \end{macrocode} +% \subsection{\ch{xintDecSplit}, \ch{xintDecSplitL}, \ch{xintDecSplitR}} +% \begin{verbatim} +% DECIMAL SPLIT +% Elle commence par remplacer A par |A| +% si x = 0 elle renvoie {A}{0} +% si x > 0, elle fait A -> [A/10^x], R est le reste SANS leading zeros. +% et si x = ou > longueur de A ça donne {0}{A} +% si x < 0, on part de la gauche. On découpe en deux. si |x| = ou > +% longueur de A tout A est mis dans Q et R est **vide** (pas 0 !!) +% R PEUT AVOIR DES LEADING ZEROS DANS CE CAS x <0. +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\xintDecSplitL {\romannumeral0\xintdecsplitl }% +\def\xintDecSplitR {\romannumeral0\xintdecsplitr }% +\def\xintdecsplitl +{% + \expandafter + \xint@firstoftwo@andstop + \romannumeral0\xintdecsplit +}% +\def\xintdecsplitr +{% + \expandafter + \xint@secondoftwo@andstop + \romannumeral0\xintdecsplit +}% +\def\xintDecSplit {\romannumeral0\xintdecsplit }% +\def\xintdecsplit #1% +{% + \expandafter\expandafter\expandafter + \xint@split + \expandafter\expandafter\expandafter + {#1}% +}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% ON REMPLACE A PAR |A| !! +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\xint@split #1#2% +{% + \expandafter + \XINT@split@checkifAzero + \expandafter + {\romannumeral0\xintabs {#2}}{#1}% fait expansion de A +}% +\def\XINT@split@checkifAzero #1#2% +{% + \ifcase \XINT@Sgn {#1} + \expandafter\XINT@split@AisZero + \fi + \XINT@split@checkxsize {#2}{#1}% +}% +\def\XINT@split@AisZero\XINT@split@checkxsize #1#2{ {0}{0}}% +\def\XINT@split@checkxsize #1% +{% + \ifnum\XINT@Len {#1} > 9 + \expandafter\XINT@split@bigx + \else + \expandafter\XINT@split@xfork + \fi + #1\Z +}% +\def\XINT@split@bigx #1\Z #2% +{% + \ifcase\XINT@Sgn {#1} + \or \xint@afterfi { {0}{#2}}% + \else + \xint@afterfi + {\expandafter\xintError:TooBigDecimalSplit + \space{0}{0}}% + \fi +}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% si x > 0 division par 10^x +% si x < 0 division par 10^{longueur(A)-|x|} +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@split@xfork #1% +{% + \xint@UDzerominusfork + #1-\dummy \XINT@split@donothing + 0#1\dummy \XINT@split@fromleft + 0-\dummy {\XINT@split@splitright #1}% + \xint@UDforkminuszero +}% +\def\XINT@split@donothing #1\Z #2{ {#2}{0}}% +\def\XINT@split@fromleft #1\Z #2% +{% + \XINT@split@fromleft@loop {#1}{}#2\W\W\W\W\W\W\W\W\Z % +}% +\def\XINT@split@fromleft@loop #1% +{% + \ifcase #1 + \expandafter\XINT@split@fromleft@endsplit + \or + \expandafter\XINT@split@fromleft@one@andend + \or + \expandafter\XINT@split@fromleft@two@andend + \or + \expandafter\XINT@split@fromleft@three@andend + \or + \expandafter\XINT@split@fromleft@four@andend + \or + \expandafter\XINT@split@fromleft@five@andend + \or + \expandafter\XINT@split@fromleft@six@andend + \or + \expandafter\XINT@split@fromleft@seven@andend + \else + \expandafter \XINT@split@fromleft@loop@perhaps + \expandafter + {\the\numexpr + #1-8\expandafter\expandafter\expandafter }% + \expandafter + \XINT@split@fromleft@eight + \fi +}% +\def\XINT@split@fromleft@endsplit #1#2\W #3\Z + { {#1}{#2}}% +\def\XINT@split@fromleft@eight #1#2#3#4#5#6#7#8#9% +{% + #9{#1#2#3#4#5#6#7#8#9}% +}% +\def\XINT@split@fromleft@loop@perhaps #1#2% +{% + \xint@w #2\XINT@split@fromleft@toofar\W \XINT@split@fromleft@loop + {#1}% +}% +\def\XINT@split@fromleft@toofar\W \XINT@split@fromleft@loop #1#2#3\Z +{% + \XINT@split@fromleft@toofar@b #2\Z +}% +\def\XINT@split@fromleft@toofar@b #1\W #2\Z +{% + \space {#1}{}% +}% +\def\XINT@split@fromleft@one@andend + {\expandafter\XINT@split@fromleft@checkiftoofar\XINT@split@fromleft@one }% +\def\XINT@split@fromleft@one #1#2{#2{#1#2}}% +\def\XINT@split@fromleft@two@andend + {\expandafter\XINT@split@fromleft@checkiftoofar\XINT@split@fromleft@two }% +\def\XINT@split@fromleft@two #1#2#3{#3{#1#2#3}}% +\def\XINT@split@fromleft@three@andend + {\expandafter\XINT@split@fromleft@checkiftoofar\XINT@split@fromleft@three }% +\def\XINT@split@fromleft@three #1#2#3#4{#4{#1#2#3#4}}% +\def\XINT@split@fromleft@four@andend + {\expandafter\XINT@split@fromleft@checkiftoofar\XINT@split@fromleft@four }% +\def\XINT@split@fromleft@four #1#2#3#4#5{#5{#1#2#3#4#5}}% +\def\XINT@split@fromleft@five@andend + {\expandafter\XINT@split@fromleft@checkiftoofar\XINT@split@fromleft@five }% +\def\XINT@split@fromleft@five #1#2#3#4#5#6{#6{#1#2#3#4#5#6}}% +\def\XINT@split@fromleft@six@andend + {\expandafter\XINT@split@fromleft@checkiftoofar\XINT@split@fromleft@six }% +\def\XINT@split@fromleft@six #1#2#3#4#5#6#7{#7{#1#2#3#4#5#6#7}}% +\def\XINT@split@fromleft@seven@andend + {\expandafter\XINT@split@fromleft@checkiftoofar\XINT@split@fromleft@seven }% +\def\XINT@split@fromleft@seven #1#2#3#4#5#6#7#8{#8{#1#2#3#4#5#6#7#8}}% +\def\XINT@split@fromleft@checkiftoofar #1#2#3\W #4\Z +{% + \xint@w #1\XINT@split@fromleft@wenttoofar\W + \space {#2}{#3}% +}% +\def\XINT@split@fromleft@wenttoofar\W\space #1% +{% + \XINT@split@fromleft@wenttoofar@b #1\Z +}% +\def\XINT@split@fromleft@wenttoofar@b #1\W #2\Z +{% + \space {#1}% +}% +\def\XINT@split@splitright #1\Z #2% +{% + \ifnum \XINT@Length {#2} > #1 + \expandafter\XINT@split@pre + \else + \expandafter\XINT@split@quotientiszero + \fi + {#2}{#1}% +}% +\def\XINT@split@quotientiszero #1#2{ {0}{#1}}% +\def\XINT@split@pre #1% +{% + \expandafter + \XINT@split@@pre + \expandafter + {\romannumeral0\XINT@rev{#1}}% +}% +\def\XINT@split@@pre #1#2% +{% + \XINT@split@loop {#2}{}#1\Z +}% +\def\XINT@split@loop #1% +{% + \ifcase #1 + \expandafter\XINT@split@endsplit + \or + \expandafter\XINT@split@one@andend + \or + \expandafter\XINT@split@two@andend + \or + \expandafter\XINT@split@three@andend + \or + \expandafter\XINT@split@four@andend + \or + \expandafter\XINT@split@five@andend + \or + \expandafter\XINT@split@six@andend + \or + \expandafter\XINT@split@seven@andend + \else + \expandafter \XINT@split@loop + \expandafter + {\the\numexpr + #1-8\expandafter\expandafter\expandafter }% + \expandafter + \XINT@split@eight + \fi +}% +\def\XINT@split@eight #1#2#3#4#5#6#7#8#9{{#1#2#3#4#5#6#7#8#9}}% +\def\XINT@split@one@andend + {\expandafter\XINT@split@endsplit\XINT@split@one }% +\def\XINT@split@one #1#2{{#1#2000}}% +\def\XINT@split@two@andend + {\expandafter\XINT@split@endsplit\XINT@split@two }% +\def\XINT@split@two #1#2#3{{#1#2#300}}% +\def\XINT@split@three@andend + {\expandafter\XINT@split@endsplit\XINT@split@three }% +\def\XINT@split@three #1#2#3#4{{#1#2#3#40}}% +\def\XINT@split@four@andend + {\expandafter\XINT@split@endsplit\XINT@split@four }% +\def\XINT@split@four #1#2#3#4#5{{#1#2#3#4#5}}% +\def\XINT@split@five@andend + {\expandafter\XINT@split@endsplit\XINT@split@five }% +\def\XINT@split@five #1#2#3#4#5#6{{#1#2#3#4#5#6000}}% +\def\XINT@split@six@andend + {\expandafter\XINT@split@endsplit\XINT@split@six }% +\def\XINT@split@six #1#2#3#4#5#6#7{{#1#2#3#4#5#6#700}}% +\def\XINT@split@seven@andend + {\expandafter\XINT@split@endsplit\XINT@split@seven }% +\def\XINT@split@seven #1#2#3#4#5#6#7#8{{#1#2#3#4#5#6#7#80}}% +\def\XINT@split@endsplit #1#2\Z +{% + \expandafter\expandafter\expandafter\XINT@split@endsplit@ + \expandafter\expandafter\expandafter + {\romannumeral0\XINT@rev + {\Z\W\W\W\W\W\W\W #1\XINT@cuz@loop0\romannumeral}}% + {\romannumeral0\XINT@rev{#2}}% +}% +\def\XINT@split@endsplit@ #1#2% +{% + \expandafter\space\expandafter {#2}{#1}% +}% +\XINT@restorecatcodes@endinput% +% \end{macrocode} +%</package> +%<*gcd> +% \section{Package \xintgcdname implementation} +% +% The commenting is currently (\docdate) very sparse. +% +% \subsection{Catcodes, \eTeX{} detection, reload detection} +% +% The code for reload detection is copied from \textsc{Heiko +% Oberdiek}'s packages, and adapted here to check for previous +% loading of the master \xintname package. +% +% The method for catcodes is slightly different, but still +% directly inspired by these packages. +% +% \begin{macrocode} +\begingroup\catcode61\catcode48\catcode32=10\relax% + \catcode13=5 % ^^M + \endlinechar=13 % + \catcode123=1 % { + \catcode125=2 % } + \catcode64=11 % @ + \catcode35=6 % # + \catcode44=12 % , + \catcode45=12 % - + \catcode46=12 % . + \catcode58=12 % : + \def\space { }% + \let\z\endgroup + \expandafter\let\expandafter\x\csname ver@xintgcd.sty\endcsname + \expandafter\let\expandafter\w\csname ver@xint.sty\endcsname + \expandafter + \ifx\csname PackageInfo\endcsname\relax + \def\y#1#2{\immediate\write-1{Package #1 Info: #2.}}% + \else + \def\y#1#2{\PackageInfo{#1}{#2}}% + \fi + \expandafter + \ifx\csname numexpr\endcsname\relax + \y{xintgcd}{\numexpr not available, aborting input}% + \aftergroup\endinput + \else + \ifx\x\relax % plain-TeX, first loading of xintgcd.sty + \ifx\w\relax % but xint.sty not yet loaded. + \y{xintgcd}{Package xint is required}% + \y{xintgcd}{Will try \string\input\space xint.sty}% + \def\z{\endgroup\input xint.sty\relax}% + \fi + \else + \def\empty {}% + \ifx\x\empty % LaTeX, first loading, + % variable is initialized, but \ProvidesPackage not yet seen + \ifx\w\relax % xint.sty not yet loaded. + \y{xintgcd}{Package xint is required}% + \y{xintgcd}{Will try \string\RequirePackage{xint}}% + \def\z{\endgroup\RequirePackage{xint}}% + \fi + \else + \y{xintgcd}{I was already loaded, aborting input}% + \aftergroup\endinput + \fi + \fi + \fi +\z% +% \end{macrocode} +% \subsection{Validation of \xintname loading} +% \begin{macrocode} +\begingroup\catcode61\catcode48\catcode32=10\relax% + \catcode13=5 % ^^M + \endlinechar=13 % + \catcode123=1 % { + \catcode125=2 % } + \catcode64=11 % @ + \catcode35=6 % # + \catcode44=12 % , + \catcode45=12 % - + \catcode46=12 % . + \catcode58=12 % : + \expandafter + \ifx\csname PackageInfo\endcsname\relax + \def\y#1#2{\immediate\write-1{Package #1 Info: #2.}}% + \else + \def\y#1#2{\PackageInfo{#1}{#2}}% + \fi + \def\empty {}% + \expandafter\let\expandafter\w\csname ver@xint.sty\endcsname + \ifx\w\relax % Plain TeX, user gave a file name at the prompt + \y{xintgcd}{Loading of package xint failed, aborting input}% + \aftergroup\endinput + \fi + \ifx\w\empty % LaTeX, user gave a file name at the prompt + \y{xintgcd}{Loading of package xint failed, aborting input}% + \aftergroup\endinput + \fi +\endgroup% +% \end{macrocode} +% \subsection{Catcodes} +% +% Perhaps catcodes have changed after the loading of \xintname +% and prior to the current loading of \xintgcdname, so we can not employ +% the |\XINT@restorecatcodes@endinput| in this style file. But +% there is no problem using |\XINT@setcatcodes|. +% +% \begin{macrocode} +\begingroup\catcode61\catcode48\catcode32=10\relax% + \catcode13=5 % ^^M + \endlinechar=13 % + \catcode123=1 % { + \catcode125=2 % } + \catcode64=11 % @ + \def\x + {% + \endgroup + \edef\XINT@gcd@restorecatcodes@endinput + {% + \catcode36=\the\catcode36 % $ + \catcode47=\the\catcode47 % / + \catcode41=\the\catcode41 % ) + \catcode40=\the\catcode40 % ( + \catcode42=\the\catcode42 % * + \catcode43=\the\catcode43 % + + \catcode62=\the\catcode62 % > + \catcode60=\the\catcode60 % < + \catcode58=\the\catcode58 % : + \catcode46=\the\catcode46 % . + \catcode45=\the\catcode45 % - + \catcode44=\the\catcode44 % , + \catcode35=\the\catcode35 % # + \catcode64=\the\catcode64 % @ + \catcode125=\the\catcode125 % } + \catcode123=\the\catcode123 % { + \endlinechar=\the\endlinechar + \catcode13=\the\catcode13 % ^^M + \catcode32=\the\catcode32 % + \catcode61=\the\catcode61 % = + \noexpand\endinput + }% + \XINT@setcatcodes + \catcode36=3 % $ + }% +\x +% \end{macrocode} +% \subsection{Package identification} +% \begin{macrocode} +\begingroup + \catcode91=12 % [ + \catcode93=12 % ] + \catcode58=12 % : + \expandafter\ifx\csname ProvidesPackage\endcsname\relax + \def\x#1#2#3[#4]{\endgroup + \immediate\write-1{Package: #3 #4}% + \xdef#1{#4}% + }% + \else + \def\x#1#2[#3]{\endgroup + #2[{#3}]% + \ifx#1\@undefined + \xdef#1{#3}% + \fi + \ifx#1\relax + \xdef#1{#3}% + \fi + }% + \fi +\expandafter\x\csname ver@xintgcd.sty\endcsname +\ProvidesPackage{xintgcd}% + [2013/03/28 v1.0 Euclide algorithm with xint package (jfB)]% +% \end{macrocode} +% \subsection{\ch{xintGCD}} +% \begin{macrocode} +\def\xintGCD {\romannumeral0\xintgcd }% +\def\xintgcd #1% +{% + \expandafter + \XINT@gcd + \expandafter + {\romannumeral0\xintabs {#1}}% +}% +\def\XINT@gcd #1#2% +{% + \expandafter + \XINT@gcd@fork + \romannumeral0\xintabs {#2}\Z #1\Z +}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% Ici #3#4=A, #1#2=B +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@gcd@fork #1#2\Z #3#4\Z +{% + \xint@UDzerofork + #1\dummy \XINT@gcd@BisZero + #3\dummy \XINT@gcd@AisZero + 0\dummy \XINT@gcd@loop + \xint@UDforkzero + {#1#2}{#3#4}% +}% +\def\XINT@gcd@AisZero #1#2{ #1}% +\def\XINT@gcd@BisZero #1#2{ #2}% +\def\XINT@gcd@CheckRem #1#2\Z +{% + \xint@zero #1\xint@gcd@end0\XINT@gcd@loop {#1#2}% +}% +\def\xint@gcd@end0\XINT@gcd@loop #1#2{ #2}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% #1=B, #2=A +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@gcd@loop #1#2% +{% + \expandafter\expandafter\expandafter + \XINT@gcd@CheckRem + \expandafter\xint@secondoftwo + \romannumeral0\XINT@div@prepare {#1}{#2}\Z + {#1}% +}% +% \end{macrocode} +% \subsection{\ch{xintBezout}} +% \begin{macrocode} +\def\xintBezout {\romannumeral0\xintbezout }% +\def\xintbezout #1% +{% + \expandafter\expandafter\expandafter + \xint@bezout + \expandafter\expandafter\expandafter + {#1}% +}% +\def\xint@bezout #1#2% +{\expandafter\expandafter\expandafter + \XINT@bezout@fork #2\Z #1\Z +}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% #3#4 = A, #1#2=B +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@bezout@fork #1#2\Z #3#4\Z +{% + \xint@UDzerosfork + #1#3\dummy \XINT@bezout@botharezero + #10\dummy \XINT@bezout@secondiszero + #30\dummy \XINT@bezout@firstiszero + 00\dummy + {\xint@UDsignsfork + #1#3\dummy \XINT@bezout@minusminus % A < 0, B < 0 + #1-\dummy \XINT@bezout@minusplus % A > 0, B < 0 + #3-\dummy \XINT@bezout@plusminus % A < 0, B > 0 + --\dummy \XINT@bezout@plusplus % A > 0, B > 0 + \xint@UDforksigns }% + \xint@UDforkzeros + {#2}{#4}#1#3{#3#4}{#1#2}% #1#2=B, #3#4=A +}% +\def\XINT@bezout@botharezero #1#2#3#4#5#6% +{% + \xintError:NoBezoutForZeros + \space {0}{0}{0}{0}{0}% +}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% attention première entrée doit être ici (-1)^n donc 1 +% #4#2=0 = A, B = #3#1 +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@bezout@firstiszero #1#2#3#4#5#6% +{% + \xint@UDsignfork + #3\dummy { {0}{#3#1}{0}{1}{#1}}% + -\dummy { {0}{#3#1}{0}{-1}{#1}}% + \xint@UDforksign +}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% #4#2= A, B = #3#1 = 0 +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@bezout@secondiszero #1#2#3#4#5#6% +{% + \xint@UDsignfork + #4\dummy{ {#4#2}{0}{-1}{0}{#2}}% + -\dummy{ {#4#2}{0}{1}{0}{#2}}% + \xint@UDforksign +}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% #4#2= A < 0, #3#1 = B < 0 +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@bezout@minusminus #1#2#3#4% +{% + \expandafter\XINT@bezout@mm@post + \romannumeral0\XINT@bezout@loop@a 1{#1}{#2}1001% +}% +\def\XINT@bezout@mm@post #1#2% +{% + \expandafter + \XINT@bezout@mm@postb + \expandafter + {\romannumeral0\xintopp{#2}}{\romannumeral0\xintopp{#1}}% +}% +\def\XINT@bezout@mm@postb #1#2% +{% + \expandafter + \XINT@bezout@mm@postc + \expandafter {#2}{#1}% +}% +\def\XINT@bezout@mm@postc #1#2#3#4#5% +{% + \space {#4}{#5}{#1}{#2}{#3}% +}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% minusplus #4#2= A > 0, B < 0 +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@bezout@minusplus #1#2#3#4% +{% + \expandafter\XINT@bezout@mp@post + \romannumeral0\XINT@bezout@loop@a 1{#1}{#4#2}1001% +}% +\def\XINT@bezout@mp@post #1#2% +{% + \expandafter + \XINT@bezout@mp@postb + \expandafter + {\romannumeral0\xintopp {#2}}{#1}% +}% +\def\XINT@bezout@mp@postb #1#2#3#4#5% +{% + \space {#4}{#5}{#2}{#1}{#3}% +}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% plusminus A < 0, B > 0 +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@bezout@plusminus #1#2#3#4% +{% + \expandafter\XINT@bezout@pm@post + \romannumeral0\XINT@bezout@loop@a 1{#3#1}{#2}1001% +}% +\def\XINT@bezout@pm@post #1% +{% + \expandafter + \XINT@bezout@pm@postb + \expandafter + {\romannumeral0\xintopp{#1}}% +}% +\def\XINT@bezout@pm@postb #1#2#3#4#5% +{% + \space {#4}{#5}{#1}{#2}{#3}% +}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% plusplus +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@bezout@plusplus #1#2#3#4% +{% + \expandafter\XINT@bezout@pp@post + \romannumeral0\XINT@bezout@loop@a 1{#3#1}{#4#2}1001% +}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% la parité (-1)^N est en #1, et on la jette ici. +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@bezout@pp@post #1#2#3#4#5% +{% + \space {#4}{#5}{#1}{#2}{#3}% +}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% n = 0: 1BAalpha(0)beta(0)alpha(-1)beta(-1) +% n général: +% {(-1)^n}{r(n-1)}{r(n-2)}{alpha(n-1)}{beta(n-1)}{alpha(n-2)}{beta(n-2)} +% #2 = B, #3 = A +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@bezout@loop@a #1#2#3% +{% + \expandafter\XINT@bezout@loop@b + \expandafter{\the\numexpr -#1\expandafter }% + \romannumeral0\XINT@div@prepare {#2}{#3}{#2}% +}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% Le q(n) a ici une existence éphémère, dans le version Bezout Algorithm +% il faudra le conserver. On voudra à la fin +% {{q(n)}{r(n)}{alpha(n)}{beta(n)}} +% De plus ce n'est plus (-1)^n que l'on veut mais n. (ou dans un autre ordre) +% {-(-1)^n}{q(n)}{r(n)}{r(n-1)}{alpha(n-1)}{beta(n-1)}{alpha(n-2)}{beta(n-2)} +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@bezout@loop@b #1#2#3#4#5#6#7#8% +{% + \expandafter + \XINT@bezout@loop@c + \expandafter + {\romannumeral0\xintadd{\XINT@Mul{#5}{#2}}{#7}}% + {\romannumeral0\xintadd{\XINT@Mul{#6}{#2}}{#8}}% + {#1}{#3}{#4}{#5}{#6}% +}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% {alpha(n)}{->beta(n)}{-(-1)^n}{r(n)}{r(n-1)}{alpha(n-1)}{beta(n-1)} +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@bezout@loop@c #1#2% +{% + \expandafter + \XINT@bezout@loop@d + \expandafter + {#2}{#1}% +}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% {beta(n)}{alpha(n)}{(-1)^(n+1)}{r(n)}{r(n-1)}{alpha(n-1)}{beta(n-1)} +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@bezout@loop@d #1#2#3#4#5% +{% + \XINT@bezout@loop@e #4\Z {#3}{#5}{#2}{#1}% +}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% r(n)\Z {(-1)^(n+1)}{r(n-1)}{alpha(n)}{beta(n)}{alpha(n-1)}{beta(n-1)} +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@bezout@loop@e #1#2\Z +{% + \xint@zero #1\xint@bezout@loop@exit0\XINT@bezout@loop@f + {#1#2}% +}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% {r(n)}{(-1)^(n+1)}{r(n-1)}{alpha(n)}{beta(n)}{alpha(n-1)}{beta(n-1)} +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@bezout@loop@f #1#2% +{% + \XINT@bezout@loop@a {#2}{#1}% +}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% {(-1)^(n+1)}{r(n)}{r(n-1)}{alpha(n)}{beta(n)}{alpha(n-1)}{beta(n-1)} +% et itération +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\xint@bezout@loop@exit0\XINT@bezout@loop@f #1#2% +{% + \ifcase #2 + \or \expandafter\XINT@bezout@exiteven + \else\expandafter\XINT@bezout@exitodd + \fi +}% +\def\XINT@bezout@exiteven #1#2#3#4#5% +{% + \space {#5}{#4}{#1}% +}% +\def\XINT@bezout@exitodd #1#2#3#4#5% +{% + \space {-#5}{-#4}{#1}% +}% +% \end{macrocode} +% \subsection{\ch{xintEuclideAlgorithm}} +% \begin{verbatim} +% Pour Euclide: +% {N}{A}{D=r(n)}{B}{q1}{r1}{q2}{r2}{q3}{r3}....{qN}{rN=0} +% u<2n> = u<2n+3>u<2n+2> + u<2n+4> à la n ième étape +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\xintEuclideAlgorithm {\romannumeral0\xinteuclidealgorithm }% +\def\xinteuclidealgorithm #1% +{% + \expandafter + \XINT@euc + \expandafter + {\romannumeral0\xintabs {#1}}% +}% +\def\XINT@euc #1#2% +{% + \expandafter + \XINT@euc@fork + \romannumeral0\xintabs {#2}\Z #1\Z +}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% Ici #3#4=A, #1#2=B +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@euc@fork #1#2\Z #3#4\Z +{% + \xint@UDzerofork + #1\dummy \XINT@euc@BisZero + #3\dummy \XINT@euc@AisZero + 0\dummy \XINT@euc@a + \xint@UDforkzero + {0}{#1#2}{#3#4}{{#3#4}{#1#2}}{}\Z +}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% Le {} pour protéger {{A}{B}} si on s'arrête après une étape (B divise A) +% On va renvoyer: +% {N}{A}{D=r(n)}{B}{q1}{r1}{q2}{r2}{q3}{r3}....{qN}{rN=0} +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@euc@AisZero #1#2#3#4#5#6{ {1}{0}{#2}{#2}{0}{0}}% +\def\XINT@euc@BisZero #1#2#3#4#5#6{ {1}{0}{#3}{#3}{0}{0}}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% {n}{rn}{an}{{qn}{rn}}...{{A}{B}}{}\Z +% an = r(n-1) +% Pour n=0 on a juste {0}{B}{A}{{A}{B}}{}\Z +% \XINT@div@prepare {u}{v} divise v par u +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@euc@a #1#2#3% +{% + \expandafter + \XINT@euc@b + \expandafter {\the\numexpr #1+1\expandafter }% + \romannumeral0\XINT@div@prepare {#2}{#3}{#2}% +}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% {n+1}{q(n+1)}{r(n+1)}{rn}{{qn}{rn}}... +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@euc@b #1#2#3#4% +{% + \XINT@euc@c #3\Z {#1}{#3}{#4}{{#2}{#3}}% +}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% r(n+1)\Z {n+1}{r(n+1)}{r(n)}{{q(n+1)}{r(n+1)}}{{qn}{rn}}... +% Test si r(n+1) est nul. +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@euc@c #1#2\Z +{% + \xint@zero #1\xint@euc@end0\XINT@euc@a +}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% {n+1}{r(n+1)}{r(n)}{{q(n+1)}{r(n+1)}}...{}\Z +% Ici r(n+1) = 0. On arrête on se prépare à inverser. +% {n+1}{0}{r(n)}{{q(n+1)}{r(n+1)}}.....{{q1}{r1}}{{A}{B}}{}\Z +% On veut renvoyer: +% {N=n+1}{A}{D=r(n)}{B}{q1}{r1}{q2}{r2}{q3}{r3}....{qN}{rN=0} +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\xint@euc@end0\XINT@euc@a #1#2#3#4\Z% +{% + \expandafter\xint@euc@end@ + \romannumeral0% + \XINT@rord@main {}#4{{#1}{#3}}% + \xint@UNDEF + \xint@undef\xint@undef\xint@undef\xint@undef + \xint@undef\xint@undef\xint@undef\xint@undef + \xint@UNDEF +}% +\def\xint@euc@end@ #1#2#3% +{% + \space {#1}{#3}{#2}% +}% +% \end{macrocode} +% \subsection{\ch{xintBezoutAlgorithm}} +% \begin{verbatim} +% Pour Bezout: objectif, renvoyer +% alpha0=1, beta0=0 +% alpha(-1)=0, beta(-1)=1 +% {N}{A}{0}{1}{D=r(n)}{B}{1}{0}{q1}{r1}{alpha1=q1}{beta1=1} +% {q2}{r2}{alpha2}{beta2}....{qN}{rN=0}{alphaN=B/D}{betaN=B/D} +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\xintBezoutAlgorithm {\romannumeral0\xintbezoutalgorithm }% +\def\xintbezoutalgorithm #1% +{% + \expandafter + \XINT@bezalg + \expandafter + {\romannumeral0\xintabs {#1}}% +}% +\def\XINT@bezalg #1#2% +{% + \expandafter + \XINT@bezalg@fork + \romannumeral0\xintabs {#2}\Z #1\Z +}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% Ici #3#4=A, #1#2=B +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@bezalg@fork #1#2\Z #3#4\Z +{% + \xint@UDzerofork + #1\dummy \XINT@bezalg@BisZero + #3\dummy \XINT@bezalg@AisZero + 0\dummy \XINT@bezalg@a + \xint@UDforkzero + 0{#1#2}{#3#4}1001{{#3#4}{#1#2}}{}\Z +}% +\def\XINT@bezalg@AisZero #1#2#3\Z{ {1}{0}{0}{1}{#2}{#2}{1}{0}{0}{0}{0}{1}}% +\def\XINT@bezalg@BisZero #1#2#3#4\Z{ {1}{0}{0}{1}{#3}{#3}{1}{0}{0}{0}{0}{1}}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% pour préparer l'étape n+1 il faut +% {n}{r(n)}{r(n-1)}{alpha(n)}{beta(n)}{alpha(n-1)}{beta(n-1)} +% {{q(n)}{r(n)}{alpha(n)}{beta(n)}}... +% division de #3 par #2 +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@bezalg@a #1#2#3% +{% + \expandafter + \XINT@bezalg@b + \expandafter {\the\numexpr #1+1\expandafter }% + \romannumeral0\XINT@div@prepare {#2}{#3}{#2}% +}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% {n+1}{q(n+1)}{r(n+1)}{r(n)}{alpha(n)}{beta(n)}{alpha(n-1)}{beta(n-1)}... +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@bezalg@b #1#2#3#4#5#6#7#8% +{% + \expandafter\XINT@bezalg@c\expandafter + {\romannumeral0\xintadd {\xintMul {#6}{#2}}{#8}}% + {\romannumeral0\xintadd {\xintMul {#5}{#2}}{#7}}% + {#1}{#2}{#3}{#4}{#5}{#6}% +}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% {beta(n+1)}{alpha(n+1)}{n+1}{q(n+1)}{r(n+1)}{r(n)}{alpha(n)}{beta(n}}% +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@bezalg@c #1#2#3#4#5#6% +{% + \expandafter\XINT@bezalg@d\expandafter + {#2}{#3}{#4}{#5}{#6}{#1}% +}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% {alpha(n+1)}{n+1}{q(n+1)}{r(n+1)}{r(n)}{beta(n+1)} +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@bezalg@d #1#2#3#4#5#6#7#8% +{% + \XINT@bezalg@e #4\Z {#2}{#4}{#5}{#1}{#6}{#7}{#8}{{#3}{#4}{#1}{#6}}% +}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% r(n+1)\Z {n+1}{r(n+1)}{r(n)}{alpha(n+1)}{beta(n+1)} +% {alpha(n)}{beta(n)}{q,r,alpha,beta(n+1)} +% Test si r(n+1) est nul. +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@bezalg@e #1#2\Z +{% + \xint@zero #1\xint@bezalg@end0\XINT@bezalg@a +}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% Ici r(n+1) = 0. On arrête on se prépare à inverser. +% {n+1}{r(n+1)}{r(n)}{alpha(n+1)}{beta(n+1)}% +% {alpha(n)}{beta(n)}% +% {q,r,alpha,beta(n+1)}...{{A}{B}}{}\Z +% On veut renvoyer +% {N}{A}{0}{1}{D=r(n)}{B}{1}{0}{q1}{r1}{alpha1=q1}{beta1=1} +% {q2}{r2}{alpha2}{beta2}....{qN}{rN=0}{alphaN=B/D}{betaN=B/D} +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\xint@bezalg@end0\XINT@bezalg@a #1#2#3#4#5#6#7#8\Z +{% + \expandafter\xint@bezalg@end@ + \romannumeral0% + \XINT@rord@main {}#8{{#1}{#3}}% + \xint@UNDEF + \xint@undef\xint@undef\xint@undef\xint@undef + \xint@undef\xint@undef\xint@undef\xint@undef + \xint@UNDEF +}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% {N}{D}{A}{B}{q1}{r1}{alpha1=q1}{beta1=1}{q2}{r2}{alpha2}{beta2} +% ....{qN}{rN=0}{alphaN=B/D}{betaN=B/D} +% On veut renvoyer +% {N}{A}{0}{1}{D=r(n)}{B}{1}{0}{q1}{r1}{alpha1=q1}{beta1=1} +% {q2}{r2}{alpha2}{beta2}....{qN}{rN=0}{alphaN=B/D}{betaN=B/D} +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\xint@bezalg@end@ #1#2#3#4% +{% + \space {#1}{#3}{0}{1}{#2}{#4}{1}{0}% +}% +% \end{macrocode} +% \subsection{\ch{xintTypesetEuclideAlgorithm}} +% \begin{verbatim} +% TYPESETTING +% Organisation: +% {N}{A}{D}{B}{q1}{r1}{q2}{r2}{q3}{r3}....{qN}{rN=0} +% \U1 = N = nombre d'étapes, \U3 = PGCD, \U2 = A, \U4=B +% q1 = \U5, q2 = \U7 --> qn = \U<2n+3>, rn = \U<2n+4> +% bn = rn. B = r0. A=r(-1) +% r(n-2) = q(n)r(n-1)+r(n) (n e étape) (n au moins 1) +% \U{2n} = \U{2n+3} \times \U{2n+2} + \U{2n+4}, n e étape. +% avec n entre 1 et N. +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\xintTypesetEuclideAlgorithm #1#2% +{% l'algo remplace #1 et #2 par |#1| et |#2| + \par + \begingroup + \xintAssignArray\xintEuclideAlgorithm {#1}{#2}\to\U + \edef\A{\U2}\edef\B{\U4}\edef\N{\U1}% + \setbox 0 \vbox{\halign {$##$\cr \A\cr \B \cr}}% + \noindent + \count 255 1 + \loop + \hbox to \wd 0 {\hfil$\U{\the\numexpr 2*\count 255\relax}$}% + ${} = \U{\the\numexpr 2*\count 255 + 3\relax} + \times \U{\the\numexpr 2*\count 255 + 2\relax} + + \U{\the\numexpr 2*\count 255 + 4\relax}$% + \ifnum \count 255 < \N + \hfill\break + \advance \count 255 1 + \repeat + \par + \endgroup +}% +% \end{macrocode} +% \subsection{\ch{xintTypesetBezoutAlgorithm}} +% \begin{verbatim} +% Pour Bezout on a: +% {N}{A}{0}{1}{D=r(n)}{B}{1}{0}{q1}{r1}{alpha1=q1}{beta1=1} +% {q2}{r2}{alpha2}{beta2}....{qN}{rN=0}{alphaN=B/D}{betaN=B/D}% +% Donc 4N+8 termes +% U1 = N, U2= A, U5=D, U6=B, +% q1 = U9, qn = U{4n+5}, n au moins 1 +% rn = U{4n+6} , n au moins -1 +% alpha(n) = U{4n+7}, n au moins -1 +% beta(n) = U{4n+8}, n au moins -1 +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\xintTypesetBezoutAlgorithm #1#2% +{% + \par + \begingroup + \parindent0pt + \xintAssignArray\xintBezoutAlgorithm {#1}{#2}\to\BEZ + \edef\A{\BEZ2}\edef\B{\BEZ6}\edef\N{\BEZ1}% A = |#1|, B = |#2| + \setbox 0 \vbox{\halign {$##$\cr \A\cr \B \cr}}% + \count 255 1 + \loop + \noindent + \hbox to \wd 0 {\hfil$\BEZ{\the\numexpr 4*\count 255 - 2\relax}$}% + ${} = \BEZ{\the\numexpr 4*\count 255 + 5\relax} + \times \BEZ{\the\numexpr 4*\count 255 + 2\relax} + + \BEZ{\the\numexpr 4*\count 255 + 6\relax}$\hfill\break + \hbox to \wd 0 {\hfil$\BEZ{\the\numexpr 4*\count 255 +7\relax}$}% + ${} = \BEZ{\the\numexpr 4*\count 255 + 5\relax} + \times \BEZ{\the\numexpr 4*\count 255 + 3\relax} + + \BEZ{\the\numexpr 4*\count 255 - 1\relax}$\hfill\break + \hbox to \wd 0 {\hfil$\BEZ{\the\numexpr 4*\count 255 +8\relax}$}% + ${} = \BEZ{\the\numexpr 4*\count 255 + 5\relax} + \times \BEZ{\the\numexpr 4*\count 255 + 4\relax} + + \BEZ{\the\numexpr 4*\count 255 \relax}$ + \endgraf + \ifnum \count 255 < \N + \advance \count 255 1 + \repeat + \par + \edef\U{\BEZ{\the\numexpr 4*\N + 4\relax}}% + \edef\V{\BEZ{\the\numexpr 4*\N + 3\relax}}% + \edef\D{\BEZ5}% + \ifodd\N\relax + $\U\times\A - \V\times \B = -\D$% + \else + $\U\times\A - \V\times\B = \D$% + \fi + \par + \endgroup +}% +\XINT@gcd@restorecatcodes@endinput% +% \end{macrocode} +% \DeleteShortVerb{\|} +%</gcd> +%<*none> +% \MakePercentComment +\CharacterTable + {Upper-case \A\B\C\D\E\F\G\H\I\J\K\L\M\N\O\P\Q\R\S\T\U\V\W\X\Y\Z + Lower-case \a\b\c\d\e\f\g\h\i\j\k\l\m\n\o\p\q\r\s\t\u\v\w\x\y\z + Digits \0\1\2\3\4\5\6\7\8\9 + Exclamation \! Double quote \" Hash (number) \# + Dollar \$ Percent \% Ampersand \& + Acute accent \' Left paren \( Right paren \) + Asterisk \* Plus \+ Comma \, + Minus \- Point \. Solidus \/ + Colon \: Semicolon \; Less than \< + Equals \= Greater than \> Question mark \? + Commercial at \@ Left bracket \[ Backslash \\ + Right bracket \] Circumflex \^ Underscore \_ + Grave accent \` Left brace \{ Vertical bar \| + Right brace \} Tilde \~} + +\CheckSum{6418} + +\Finale +%% +%% End of file `xint.dtx'. + |