texlive[46377] Master/texmf-dist: polexpr (18jan18)
commits+karl at tug.org
commits+karl at tug.org
Fri Jan 19 00:25:25 CET 2018
Revision: 46377
http://tug.org/svn/texlive?view=revision&revision=46377
Author: karl
Date: 2018-01-19 00:25:24 +0100 (Fri, 19 Jan 2018)
Log Message:
-----------
polexpr (18jan18)
Modified Paths:
--------------
trunk/Master/texmf-dist/doc/latex/polexpr/README.md
trunk/Master/texmf-dist/doc/latex/polexpr/polexpr.html
trunk/Master/texmf-dist/doc/latex/polexpr/polexpr.txt
trunk/Master/texmf-dist/tex/latex/polexpr/polexpr.sty
Modified: trunk/Master/texmf-dist/doc/latex/polexpr/README.md
===================================================================
--- trunk/Master/texmf-dist/doc/latex/polexpr/README.md 2018-01-18 23:25:12 UTC (rev 46376)
+++ trunk/Master/texmf-dist/doc/latex/polexpr/README.md 2018-01-18 23:25:24 UTC (rev 46377)
@@ -22,8 +22,8 @@
The Author of this Work is Jean-François Burnol.
-This Work consists of the package file polexpr.sty, this README.md and the
-documentation file polexpr.txt.
+This Work consists of the package file polexpr.sty, this README.md and
+the documentation file polexpr.txt.
Abstract
--------
@@ -37,10 +37,10 @@
functions or other constructs as recognized by the `\xintexpr` numerical
parser.
-The polynomials are then not only genuine `\xintexpr` numerical
-functions but additionally are also known to the package via their
-coefficients. This allows dedicated macros to implement polynomial
-algorithmics.
+The polynomials are then not only genuine `\xintexpr` (and
+`\xintfloatexpr`) numerical functions but additionally are known to the
+package via their coefficients. This allows dedicated macros to
+implement polynomial algorithmics.
CHANGE LOG
----------
@@ -52,7 +52,6 @@
- Various utilities such as `\PolFromCSV`, `\PolMapCoeffs`,
`\PolToCSV`, `\PolToExpr`, ...
Only one-variable polynomials so far.
-
- v0.2 (2018/01/14)
- Fix: `"README thinks \numexpr recognizes ^ operator"`.
- Convert README to reStructuredText markup.
@@ -64,4 +63,45 @@
- Convert README to (CTAN compatible) Markdown markup.
Due to lack of available time the test suite might not be extensive
enough. Bug reports are very welcome!
+- v0.3 (2018/01/17)
+ - bug fixes:
+ - the `0.1` `\PolEval` accepted expressions for its second
+ argument, but this was removed by mistake at `0.2`.
+ Restored.
+ - incompatible or breaking changes:
+ - `\PolToExpr` now by default uses *descending* powers (it
+ also treats differently coefficients equal to 1 or -1.) Use
+ `\PolToExpr*` for *ascending* powers.
+ - `\PolEval` reduced the output to smallest terms, but as this is
+ costly with big fractions and not needed if e.g. wrapped in
+ an `\xintRound` or `\xintFloat`, this step has been removed;
+ the former meaning is available as `\PolEvalReduced`.
+ - new (or newly documented) macros:
+ - `\PolTypesetCmd`
+ - `\PolTypesetCmdPrefix`,
+ - `\PolTypesetMonomialCmd`,
+ - `\PolEvalReduced`,
+ - `\PolFloatEval`,
+ - `\PolToFloatExpr`,
+ - `\PolToExprOneTerm`,
+ - `\PolToFloatExprOneTerm`,
+ - `\PolToExprCmd`,
+ - `\PolToFloatExprCmd`,
+ - `\PolToExprTermPrefix`,
+ - `\PolToExprVar`,
+ - `\PolToExprTimes`.
+ - improvements:
+ - documentation has a table of contents, internal hyperlinks,
+ standardized signature notations and added explanations.
+ - one can do `\PolLet{g}={f}` or `\PolLet{g}{f}`.
+ - `\PolToExpr{f}` is highly customizable.
+ - `\poldef` and other defining macros prepare the polynomial
+ functions for usage within `\xintthefloatexpr` (or
+ `\xintdeffloatvar`). Coefficients are pre-rounded to the
+ floating point precision. Indispensible for numerical
+ algorithms, as exact fractions, even reduced, quickly become
+ very big. See the documentation about how to use the exact
+ polynomials also in floating point context.
+- v0.3.1 (2018/01/18)
+ Fixes two typos in example code included in the documentation.
Modified: trunk/Master/texmf-dist/doc/latex/polexpr/polexpr.html
===================================================================
--- trunk/Master/texmf-dist/doc/latex/polexpr/polexpr.html 2018-01-18 23:25:12 UTC (rev 46376)
+++ trunk/Master/texmf-dist/doc/latex/polexpr/polexpr.html 2018-01-18 23:25:24 UTC (rev 46377)
@@ -6,7 +6,7 @@
<meta name="generator" content="Docutils 0.14: http://docutils.sourceforge.net/" />
<title>Package polexpr documentation</title>
<style type="text/css">
-body{font-size: 14pt;}
+body{font-size: 13pt;}
/*
:Author: David Goodger (goodger at python.org)
:Id: $Id: html4css1.css 7952 2016-07-26 18:15:59Z milde $
@@ -362,35 +362,101 @@
<body>
<div class="document" id="package-polexpr-documentation">
<h1 class="title">Package polexpr documentation</h1>
+<h2 class="subtitle" id="id1">0.3.1 (2018/01/18)</h2>
<!-- comment: -*- fill-column: 72; mode: rst; -*- -->
+<div class="contents topic" id="contents">
+<p class="topic-title first">Contents</p>
+<ul class="simple">
+<li><a class="reference internal" href="#first-examples" id="id15">First Examples</a></li>
+<li><a class="reference internal" href="#non-expandable-macros" id="id16">Non-expandable macros</a><ul>
+<li><a class="reference internal" href="#poldef-polname-letter-expression-in-letter" id="id17"><tt class="docutils literal">\poldef <span class="pre">polname(letter):=</span> expression in letter;</tt></a></li>
+<li><a class="reference internal" href="#poldef-letter-polname-expression-in-letter" id="id18"><tt class="docutils literal"><span class="pre">\PolDef[letter]{polname}{expression</span> in letter}</tt></a></li>
+<li><a class="reference internal" href="#pollet-polname-2-polname-1" id="id19"><tt class="docutils literal"><span class="pre">\PolLet{polname_2}={polname_1}</span></tt></a></li>
+<li><a class="reference internal" href="#polassign-polname-toarray-macro" id="id20"><tt class="docutils literal"><span class="pre">\PolAssign{polname}\toarray\macro</span></tt></a></li>
+<li><a class="reference internal" href="#polget-polname-fromarray-macro" id="id21"><tt class="docutils literal"><span class="pre">\PolGet{polname}\fromarray\macro</span></tt></a></li>
+<li><a class="reference internal" href="#polfromcsv-polname-csv" id="id22"><tt class="docutils literal"><span class="pre">\PolFromCSV{polname}{<csv>}</span></tt></a></li>
+<li><a class="reference internal" href="#poltypeset-polname" id="id23"><tt class="docutils literal">\PolTypeset{polname}</tt></a><ul>
+<li><a class="reference internal" href="#poltypesetcmd-raw-coeff" id="id24"><tt class="docutils literal">\PolTypesetCmd{raw_coeff}</tt></a></li>
+<li><a class="reference internal" href="#poltypesetcmdprefix-raw-coeff" id="id25"><tt class="docutils literal">\PolTypesetCmdPrefix{raw_coeff}</tt></a></li>
+<li><a class="reference internal" href="#poltypesetmonomialcmd" id="id26"><tt class="docutils literal">\PolTypesetMonomialCmd</tt></a></li>
+</ul>
+</li>
+<li><a class="reference internal" href="#id5" id="id27"><tt class="docutils literal"><span class="pre">\PolTypeset*{polname}</span></tt></a></li>
+<li><a class="reference internal" href="#poldiff-polname-1-polname-2" id="id28"><tt class="docutils literal"><span class="pre">\PolDiff{polname_1}{polname_2}</span></tt></a></li>
+<li><a class="reference internal" href="#poldiff-n-polname-1-polname-2" id="id29"><tt class="docutils literal"><span class="pre">\PolDiff[N]{polname_1}{polname_2}</span></tt></a></li>
+<li><a class="reference internal" href="#polantidiff-polname-1-polname-2" id="id30"><tt class="docutils literal"><span class="pre">\PolAntiDiff{polname_1}{polname_2}</span></tt></a></li>
+<li><a class="reference internal" href="#polantidiff-n-polname-1-polname-2" id="id31"><tt class="docutils literal"><span class="pre">\PolAntiDiff[N]{polname_1}{polname_2}</span></tt></a></li>
+<li><a class="reference internal" href="#poldivide-polname-1-polname-2-polname-q-polname-r" id="id32"><tt class="docutils literal"><span class="pre">\PolDivide{polname_1}{polname_2}{polname_Q}{polname_R}</span></tt></a></li>
+<li><a class="reference internal" href="#polgcd-polname-1-polname-2-polname-gcd" id="id33"><tt class="docutils literal"><span class="pre">\PolGCD{polname_1}{polname_2}{polname_GCD}</span></tt></a></li>
+<li><a class="reference internal" href="#polmapcoeffs-macro-polname" id="id34"><tt class="docutils literal"><span class="pre">\PolMapCoeffs{\macro}{polname}</span></tt></a></li>
+<li><a class="reference internal" href="#polreducecoeffs-polname" id="id35"><tt class="docutils literal">\PolReduceCoeffs{polname}</tt></a></li>
+</ul>
+</li>
+<li><a class="reference internal" href="#expandable-macros" id="id36">Expandable macros</a><ul>
+<li><a class="reference internal" href="#poleval-polname-at-numerical-expression" id="id37"><tt class="docutils literal"><span class="pre">\PolEval{polname}\At{numerical</span> expression}</tt></a></li>
+<li><a class="reference internal" href="#polevalreduced-polname-at-numerical-expression" id="id38"><tt class="docutils literal"><span class="pre">\PolEvalReduced{polname}\At{numerical</span> expression}</tt></a></li>
+<li><a class="reference internal" href="#polfloateval-polname-at-numerical-expression" id="id39"><tt class="docutils literal"><span class="pre">\PolFloatEval{polname}\At{numerical</span> expression}</tt></a></li>
+<li><a class="reference internal" href="#polnthcoeff-polname-number" id="id40"><tt class="docutils literal"><span class="pre">\PolNthCoeff{polname}{number}</span></tt></a></li>
+<li><a class="reference internal" href="#poldegree-polname" id="id41"><tt class="docutils literal">\PolDegree{polname}</tt></a></li>
+<li><a class="reference internal" href="#poltoexpr-polname" id="id42"><tt class="docutils literal">\PolToExpr{polname}</tt></a><ul>
+<li><a class="reference internal" href="#poltoexproneterm-raw-coeff-number" id="id43"><tt class="docutils literal"><span class="pre">\PolToExprOneTerm{raw_coeff}{number}</span></tt></a></li>
+<li><a class="reference internal" href="#poltoexpronetermstyleb-raw-coeff-number" id="id44"><tt class="docutils literal"><span class="pre">\PolToExprOneTermStyleB{raw_coeff}{number}</span></tt></a></li>
+<li><a class="reference internal" href="#poltoexprcmd-raw-coeff" id="id45"><tt class="docutils literal">\PolToExprCmd{raw_coeff}</tt></a></li>
+<li><a class="reference internal" href="#poltoexprtermprefix-raw-coeff" id="id46"><tt class="docutils literal">\PolToExprTermPrefix{raw_coeff}</tt></a></li>
+<li><a class="reference internal" href="#poltoexprvar" id="id47"><tt class="docutils literal">\PolToExprVar</tt></a></li>
+<li><a class="reference internal" href="#poltoexprtimes" id="id48"><tt class="docutils literal">\PolToExprTimes</tt></a></li>
+</ul>
+</li>
+<li><a class="reference internal" href="#id12" id="id49"><tt class="docutils literal"><span class="pre">\PolToExpr*{polname}</span></tt></a></li>
+<li><a class="reference internal" href="#poltofloatexpr-polname" id="id50"><tt class="docutils literal">\PolToFloatExpr{polname}</tt></a><ul>
+<li><a class="reference internal" href="#poltofloatexproneterm-raw-coeff-number" id="id51"><tt class="docutils literal"><span class="pre">\PolToFloatExprOneTerm{raw_coeff}{number}</span></tt></a></li>
+<li><a class="reference internal" href="#poltofloatexprcmd-raw-coeff" id="id52"><tt class="docutils literal">\PolToFloatExprCmd{raw_coeff}</tt></a></li>
+</ul>
+</li>
+<li><a class="reference internal" href="#id13" id="id53"><tt class="docutils literal"><span class="pre">\PolToFloatExpr*{polname}</span></tt></a></li>
+<li><a class="reference internal" href="#poltolist-polname" id="id54"><tt class="docutils literal">\PolToList{polname}</tt></a></li>
+<li><a class="reference internal" href="#poltocsv-polname" id="id55"><tt class="docutils literal">\PolToCSV{polname}</tt></a></li>
+</ul>
+</li>
+<li><a class="reference internal" href="#booleans-with-default-setting-as-indicated" id="id56">Booleans (with default setting as indicated)</a><ul>
+<li><a class="reference internal" href="#xintverbosefalse" id="id57"><tt class="docutils literal">\xintverbosefalse</tt></a></li>
+<li><a class="reference internal" href="#poltypesetallfalse" id="id58"><tt class="docutils literal">\poltypesetallfalse</tt></a></li>
+<li><a class="reference internal" href="#poltoexprallfalse" id="id59"><tt class="docutils literal">\poltoexprallfalse</tt></a></li>
+</ul>
+</li>
+<li><a class="reference internal" href="#technicalities" id="id60">Technicalities</a></li>
+<li><a class="reference internal" href="#releases" id="id61">RELEASES</a></li>
+<li><a class="reference internal" href="#acknowledgments" id="id62">Acknowledgments</a></li>
+</ul>
+</div>
<div class="section" id="first-examples">
-<h1>First Examples</h1>
+<h1><a class="toc-backref" href="#id15">First Examples</a></h1>
<p>The syntax is:</p>
<pre class="literal-block">
-\poldef <name>(x):=<expression in variable x>;
+\poldef polname(x):= expression in variable x;
</pre>
<p>where in place of <tt class="docutils literal">x</tt> an arbitrary <em>dummy variable</em> is authorized
-(i.e. per default any of <tt class="docutils literal"><span class="pre">[a..z|A..Z]</span></tt>; more letters can be declared
+(i.e. per default any of <tt class="docutils literal"><span class="pre">[a-z|A-Z]</span></tt>; more letters can be declared
under Unicode engines.) One can also issue:</p>
<pre class="literal-block">
-\PolDef{name}{expression in variable x}
+\PolDef{polname}{expression in variable x}
</pre>
<p>which admits an optional first argument to modify the variable letter
from its default <tt class="docutils literal">x</tt>.</p>
<dl class="docutils">
<dt><tt class="docutils literal">\poldef <span class="pre">f(x):=</span> <span class="pre">1-x+x^2;</span></tt></dt>
-<dd>defines polynomial <tt class="docutils literal">f</tt>. Polynomial names must start with a letter
-and may contain letters, digits, and underscores. The variable must
-be a single letter. The colon character is optional. The semi-colon
-at end of expression is mandatory.</dd>
+<dd>defines polynomial <tt class="docutils literal">f</tt>. Polynomial names must start with a
+letter and may contain letters, digits, and underscores. The
+variable must be a single letter. The colon character is optional.
+The semi-colon at end of expression is mandatory.</dd>
<dt><tt class="docutils literal"><span class="pre">\PolDef{f}{1-x+x^2}</span></tt></dt>
<dd>does the same as <tt class="docutils literal">\poldef <span class="pre">f(x):=</span> <span class="pre">1-x+x^2;</span></tt> To use another letter
than <tt class="docutils literal">x</tt> in the expression, one must pass it as an extra optional
argument to <tt class="docutils literal">\PolDef</tt>. Useful if the semi-colon has been assigned
some non-standard catcode by some package.</dd>
-<dt><tt class="docutils literal"><span class="pre">\PolLet{g}{f}</span></tt></dt>
-<dd>saves a copy of <tt class="docutils literal">f</tt> under name <tt class="docutils literal">g</tt>.</dd>
+<dt><tt class="docutils literal"><span class="pre">\PolLet{g}={f}</span></tt></dt>
+<dd>saves a copy of <tt class="docutils literal">f</tt> under name <tt class="docutils literal">g</tt>. Also usable without <tt class="docutils literal">=</tt>.</dd>
<dt><tt class="docutils literal">\poldef <span class="pre">f(z):=</span> <span class="pre">f(z)^2;</span></tt></dt>
<dd>redefines <tt class="docutils literal">f</tt> in terms of itself.</dd>
<dt><tt class="docutils literal">\poldef <span class="pre">f(T):=</span> <span class="pre">f(f(T));</span></tt></dt>
@@ -429,68 +495,127 @@
\poldef k(x):= (x-1)(x-2)(x-3)(x-4)/(x^2-5x+4);
</pre>
</div>
-<div class="admonition attention">
+<div class="admonition attention" id="warningtacit">
<p class="first admonition-title">Attention!</p>
-<p class="last"><tt class="docutils literal">1/2 x</tt> skips the space and is treated like <tt class="docutils literal"><span class="pre">1/(2x)</span></tt> because of
-the tacit multiplication rules of xintexpr. But this means it gives
-zero! Thus one must use <tt class="docutils literal">(1/2)x</tt> or <tt class="docutils literal">1/2*x</tt> or <tt class="docutils literal"><span class="pre">(1/2)*x</span></tt> for
-disambiguation: <tt class="docutils literal"><span class="pre">x-1/2*x^2+1/3*x^3...</span></tt></p>
+<p><tt class="docutils literal">1/2 x^2</tt> skips the space and is treated like <tt class="docutils literal"><span class="pre">1/(2*x^2)</span></tt> because
+of the tacit multiplication rules of xintexpr. But this means it
+gives zero! Thus one must use <tt class="docutils literal">(1/2)x^2</tt> or <tt class="docutils literal">1/2*x^2</tt> or
+<tt class="docutils literal"><span class="pre">(1/2)*x^2</span></tt> for disambiguation: <tt class="docutils literal">x - 1/2*x^2 + <span class="pre">1/3*x^3...</span></tt>. It is
+even simpler to move the denominator to the right: <tt class="docutils literal">x - x^2/2 +
+x^3/3 - ...</tt>.</p>
+<p class="last">It is worth noting that <tt class="docutils literal"><span class="pre">1/2(x-1)(x-2)</span></tt> suffers the same issue:
+<a class="reference external" href="http://www.ctan.org/pkg/xint">xint</a> tacit multiplication always "ties more", hence this gets
+interpreted as <tt class="docutils literal"><span class="pre">1/(2*(x-1)*(x-2))</span></tt> which gives zero by polynomial
+division. Thus, use one of <tt class="docutils literal"><span class="pre">(1/2)(x-1)(x-2)</span></tt>, <tt class="docutils literal"><span class="pre">1/2*(x-1)(x-2)</span></tt> or
+<tt class="docutils literal"><span class="pre">(x-1)(x-2)/2</span></tt>.</p>
</div>
<p>After:</p>
<pre class="literal-block">
-\poldef f1(x):= 25(x-1)(x^2-2)(x-3)(x-4)(x-5);%
-\poldef f2(x):= 37(x-1)(x^2-2)(x-6)(x-7)(x-8);%
+\poldef f_1(x):= 25(x-1)(x^2-2)(x-3)(x-4)(x-5);%
+\poldef f_2(x):= 37(x-1)(x^2-2)(x-6)(x-7)(x-8);%
</pre>
-<p>the macro call <tt class="docutils literal"><span class="pre">\PolGCD{f1}{f2}{k}</span></tt> sets <tt class="docutils literal">k</tt> to the (unitary) GCD of
-<tt class="docutils literal">f1</tt> and <tt class="docutils literal">f2</tt>.</p>
+<p>the macro call <tt class="docutils literal"><span class="pre">\PolGCD{f_1}{f_2}{k}</span></tt> sets <tt class="docutils literal">k</tt> to the (unitary) GCD of
+<tt class="docutils literal">f_1</tt> and <tt class="docutils literal">f_2</tt> (hence to the expansion of <tt class="docutils literal"><span class="pre">(x-1)(x^2-2)</span></tt>.)</p>
<dl class="docutils">
<dt><tt class="docutils literal">\PolToExpr{k}</tt></dt>
-<dd>will thus (expandably) give in this case <tt class="docutils literal"><span class="pre">2-2*x^1-1*x^2+1*x^3</span></tt>.
-This is useful for console or file output (the syntax is Maple- and
-PSTricks-compatible; currently the letter <tt class="docutils literal">x</tt> in output is not
-customizable, but this can easily be added if requested from author.)</dd>
+<dd>will (expandably) give in this case <tt class="docutils literal"><span class="pre">x^3-x^2-2*x+2</span></tt>. This is
+useful for console or file output (the syntax is Maple- and
+PSTricks-compatible; the letter used in output can be
+(non-expandably) changed via a redefinition of <a class="reference internal" href="#poltoexprvar">\PolToExprVar</a>.)</dd>
+<dt><tt class="docutils literal"><span class="pre">\PolToExpr*{k}</span></tt></dt>
+<dd>gives ascending powers: <tt class="docutils literal"><span class="pre">2-2*x-x^2+x^3</span></tt>.</dd>
</dl>
</div>
<div class="section" id="non-expandable-macros">
-<h1>Non-expandable macros</h1>
-<dl class="docutils">
-<dt><tt class="docutils literal">\poldef <span class="pre">name(letter):=</span> polynomial expression using letter;</tt></dt>
-<dd><p class="first">This evaluates the polynomial expression and stores the coefficients
+<h1><a class="toc-backref" href="#id16">Non-expandable macros</a></h1>
+<div class="section" id="poldef-polname-letter-expression-in-letter">
+<span id="poldef"></span><h2><a class="toc-backref" href="#id17"><tt class="docutils literal">\poldef <span class="pre">polname(letter):=</span> expression in letter;</tt></a></h2>
+<blockquote>
+<p>This evaluates the <em>polynomial expression</em> and stores the coefficients
in a private structure accessible later via other package macros,
-under the user-chosen <tt class="docutils literal">name</tt>. Of course previously defined
-polynomials are allowed in a new expression. Names must start with a
+under the user-chosen <tt class="docutils literal">polname</tt>. Of course the <em>expression</em> can
+use other previously defined polynomials. Names must start with a
letter and are constituted of letters, digits and underscore
-characters. See Examples above.</p>
-<p>As a side effect the function <tt class="docutils literal">name()</tt> is recognized as a genuine
-<tt class="docutils literal"><span class="pre">\xintexpr...\relax</span></tt> function for (exact) numerical evaluation. It
-computes values not according to the original expression but via
-the Horner scheme corresponding to the polynomial coefficients.</p>
-<p class="last">The original expression is lost after parsing, and in particular
+characters. The whole <a class="reference external" href="http://www.ctan.org/pkg/xint">xintexpr</a> syntax is authorized:</p>
+<pre class="literal-block">
+\poldef sin(z) := add((-1)^i z^(2i+1)/(2i+1)!, i = 0..10);
+</pre>
+<p>With fractional coefficients, beware the <a class="reference internal" href="#warningtacit">tacit multiplication issue</a>.</p>
+<p>As a side effect the function <tt class="docutils literal">polname()</tt> is recognized as a
+genuine <tt class="docutils literal"><span class="pre">\xintexpr...\relax</span></tt> function for (exact) numerical
+evaluation (or within an <tt class="docutils literal">\xintdefvar</tt> assignment.) It computes
+values not according to the original expression but via the Horner
+scheme corresponding to the polynomial coefficients.</p>
+<p>Also, a function with the same name is created for use within
+<tt class="docutils literal">\xintfloatexpr</tt> (or <tt class="docutils literal">\xintdeffloatvar</tt>.) This is indispensible
+for numerical algorithms as exact computations very quickly lead to
+very big fractions. Addition and multiplication steps of the Horner
+scheme will be executed as floating-point operations. The
+coefficients have already been rounded at time of definition,
+according to the then prevailing <tt class="docutils literal">\xinttheDigits</tt> value.</p>
+<div class="admonition important">
+<p class="first admonition-title">Important</p>
+<p>Package macros (such as derivatives or Euclidean division)
+operate with the "exact" polynomials; "floating point"
+polynomials are always obtained in a second step.</p>
+<p>To modifiy "in-place" the original coefficients of a polynomial
+and round them to float precision:</p>
+<pre class="literal-block">
+\PolMapCoeffs{\xintFloat}{polname}
+% or \xintFloat[P] for precision P digits
+</pre>
+<p class="last">See <a class="reference internal" href="#polmapcoeffs-macro-polname">\PolMapCoeffs{\macro}{polname}</a>.</p>
+</div>
+<p>The original expression is lost after parsing, and in particular
the package provides no way to typeset it. This has to be done
manually, if needed.</p>
-</dd>
-<dt><tt class="docutils literal"><span class="pre">\PolDef{name}{P(x)}</span></tt></dt>
-<dd>Does the same but the variable is assumed to be <tt class="docutils literal">x</tt>. To use
-another letter, pass it as first optional argument:
-<tt class="docutils literal"><span class="pre">\PolDef[X]{name}{P(X)}</span></tt>.</dd>
-<dt><tt class="docutils literal"><span class="pre">\PolLet{g}{f}</span></tt></dt>
-<dd>Makes a copy of already defined polynomial f to new one g. Same
-effect as <tt class="docutils literal"><span class="pre">\PolDef{g}{f(x)}</span></tt> but faster.</dd>
-<dt><tt class="docutils literal"><span class="pre">\PolAssign{f}\toarray\Array</span></tt></dt>
-<dd><p class="first">Defines a one-argument expandable macro <tt class="docutils literal"><span class="pre">\Array{#1}</span></tt> which expands
+</blockquote>
+</div>
+<div class="section" id="poldef-letter-polname-expression-in-letter">
+<span id="id2"></span><h2><a class="toc-backref" href="#id18"><tt class="docutils literal"><span class="pre">\PolDef[letter]{polname}{expression</span> in letter}</tt></a></h2>
+<blockquote>
+Does the same in an undelimited macro format (thus avoiding
+potential problems with the catcode of the semi-colon in presence of
+some packages.) In absence of the <tt class="docutils literal">[letter]</tt> optional argument,
+the variable is assumed to be <tt class="docutils literal">x</tt>.</blockquote>
+</div>
+<div class="section" id="pollet-polname-2-polname-1">
+<h2><a class="toc-backref" href="#id19"><tt class="docutils literal"><span class="pre">\PolLet{polname_2}={polname_1}</span></tt></a></h2>
+<blockquote>
+Makes a copy of the already defined polynomial <tt class="docutils literal">polname_1</tt> to a
+new one <tt class="docutils literal">polname_2</tt>. Same effect as
+<tt class="docutils literal"><span class="pre">\PolDef{polname_2}{polname_1(x)}</span></tt> but with less overhead. The
+<tt class="docutils literal">=</tt> is optional.</blockquote>
+</div>
+<div class="section" id="polassign-polname-toarray-macro">
+<h2><a class="toc-backref" href="#id20"><tt class="docutils literal"><span class="pre">\PolAssign{polname}\toarray\macro</span></tt></a></h2>
+<blockquote>
+<p>Defines a one-argument expandable macro <tt class="docutils literal"><span class="pre">\macro{#1}</span></tt> which expands
to the (raw) #1th polynomial coefficient.</p>
-<ul class="last simple">
+<ul class="simple">
<li>Attention, coefficients here are indexed starting at 1.</li>
-<li>With #1=-1, -2, ..., <tt class="docutils literal"><span class="pre">\Array{#1}</span></tt> returns leading coefficients.</li>
+<li>With #1=-1, -2, ..., <tt class="docutils literal"><span class="pre">\macro{#1}</span></tt> returns leading coefficients.</li>
<li>With #1=0, returns the number of coefficients, i.e. <tt class="docutils literal">1 + deg f</tt>
for non-zero polynomials.</li>
<li>Out-of-range #1's return <tt class="docutils literal">0/1[0]</tt>.</li>
</ul>
-</dd>
-<dt><tt class="docutils literal"><span class="pre">\PolGet{f}\fromarray\Array</span></tt></dt>
-<dd><p class="first">Does the converse operation to <tt class="docutils literal"><span class="pre">\PolAssign{f}\toarray\Array</span></tt>. No
+<p>See also <a class="reference internal" href="#polnthcoeff-polname-number">\PolNthCoeff{polname}{number}</a>. The main difference is that
+with <tt class="docutils literal">\PolAssign</tt>, <tt class="docutils literal">\macro</tt> is made a prefix to <tt class="docutils literal">1 + deg f</tt>
+already defined (hidden to user) macros holding individually the
+coefficients but <a class="reference internal" href="#polnthcoeff-polname-number">\PolNthCoeff{polname}{number}</a> does each time the job
+to expandably recover the <tt class="docutils literal">Nth</tt> coefficient, and due to
+expandability can not store it in a macro for future usage (of course,
+it can be an argument in an <tt class="docutils literal">\edef</tt>.) The other difference
+is the shift by one in indexing, mentioned above (negative
+indices act the same in both.)</p>
+</blockquote>
+</div>
+<div class="section" id="polget-polname-fromarray-macro">
+<h2><a class="toc-backref" href="#id21"><tt class="docutils literal"><span class="pre">\PolGet{polname}\fromarray\macro</span></tt></a></h2>
+<blockquote>
+<p>Does the converse operation to <tt class="docutils literal"><span class="pre">\PolAssign{polname}\toarray\macro</span></tt>. No
error checks on validity of coefficients as numbers. Each
-<tt class="docutils literal">\Array{index}</tt> is expanded in an <tt class="docutils literal">\edef</tt> before being assigned
+<tt class="docutils literal">\macro{number}</tt> is expanded in an <tt class="docutils literal">\edef</tt> before being assigned
to a coefficient. Leading zero coefficients are removed from the
polynomial.</p>
<p>(contrived) Example:</p>
@@ -498,159 +623,441 @@
\xintAssignArray{1}{-2}{5}{-3}\to\foo
\PolGet{f}\fromarray\foo
</pre>
-<p class="last">This will define <tt class="docutils literal">f</tt> as would have <tt class="docutils literal">\poldef <span class="pre">f(x):=1-2x+5x^2-3x^3;</span></tt>.
+<p>This will define <tt class="docutils literal">f</tt> as would have <tt class="docutils literal">\poldef <span class="pre">f(x):=1-2x+5x^2-3x^3;</span></tt>.
However the coefficients are still in their original form (i.e.
they were not subjected to <tt class="docutils literal">\xintRaw</tt> or similar xintfrac macro.)</p>
-</dd>
-<dt><tt class="docutils literal"><span class="pre">\PolFromCSV{f}{comma</span> separated coefficients}</tt></dt>
-<dd><p class="first">Defines a polynomial directly from the comma separated list (or a
-macro expanding to such a list) of its coefficients, the constant
-term being the first item. No validity checks. Spaces from the list
-argument are trimmed. List items are expanded in an <tt class="docutils literal">\edef</tt>, but
-currently left in their original form like e.g. <tt class="docutils literal">1.5e3</tt> which is
-not converted to <tt class="docutils literal">15/1[2]</tt> <em>raw</em> xintfrac format (this may
-change).</p>
+</blockquote>
+</div>
+<div class="section" id="polfromcsv-polname-csv">
+<h2><a class="toc-backref" href="#id22"><tt class="docutils literal"><span class="pre">\PolFromCSV{polname}{<csv>}</span></tt></a></h2>
+<blockquote>
+<p>Defines a polynomial directly from the comma separated list of
+values (or a macro expanding to such a list) of its coefficients,
+the constant term being the first item. No validity checks. Spaces
+from the list argument are trimmed. List items are each expanded in
+an <tt class="docutils literal">\edef</tt>, but currently left in their original form like e.g.
+<tt class="docutils literal">1.5e3</tt> which is not converted to <tt class="docutils literal">15/1[2]</tt> <em>raw</em> xintfrac
+format (this may change).</p>
<p>Leading zero coefficients are removed:</p>
<pre class="literal-block">
-\PolFromCSV{J}{0, 0, 0, 0, 0, 0, 0, 0, 0, 0}
+\PolFromCSV{f}{0, 0, 0, 0, 0, 0, 0, 0, 0, 0}
</pre>
<p>defines the zero polynomial, which has only one (zero) coefficient.</p>
-<p class="last">See also expandable macro <tt class="docutils literal">\PolToCSV</tt>.</p>
-</dd>
-<dt><tt class="docutils literal">\PolTypeset{name}</tt></dt>
-<dd><p class="first">Typesets in descending powers in math mode. It uses letter <tt class="docutils literal">x</tt> but
+<p>See also expandable macro <a class="reference internal" href="#poltocsv-polname">\PolToCSV</a>.</p>
+</blockquote>
+</div>
+<div class="section" id="poltypeset-polname">
+<h2><a class="toc-backref" href="#id23"><tt class="docutils literal">\PolTypeset{polname}</tt></a></h2>
+<blockquote>
+<p>Typesets in descending powers in math mode. It uses letter <tt class="docutils literal">x</tt> but
this can be changed via an optional argument:</p>
<pre class="literal-block">
-\PolTypeset[z]{name}
+\PolTypeset[z]{polname}
</pre>
<p>By default zero coefficients are skipped (issue <tt class="docutils literal">\poltypesetalltrue</tt>
to get all of them in output).</p>
-<p class="last">Macros <tt class="docutils literal">\PolTypesetCmd</tt>, <tt class="docutils literal">\PolTypesetPlus</tt>, <tt class="docutils literal">\PolTypesetMonomial</tt>
-can help configure the output. See the package code.</p>
-</dd>
-<dt><tt class="docutils literal"><span class="pre">\PolTypeset*{name}</span></tt></dt>
-<dd>Typesets in ascending powers. Change the letter from its default
-<tt class="docutils literal">x</tt> by optional argument.</dd>
-<dt><tt class="docutils literal"><span class="pre">\PolDiff{f1}{f2}</span></tt></dt>
-<dd><p class="first">This sets f2 to the first derivative of <tt class="docutils literal">f1</tt>. It is allowed to
-issue <tt class="docutils literal"><span class="pre">\PolDiff{f}{f}</span></tt>, effectively replacing <tt class="docutils literal">f</tt> by <tt class="docutils literal">f'</tt>.</p>
-<p class="last">Coefficients of the result <tt class="docutils literal">f2</tt> are irreducible fractions
+<p>These commands (whose meanings will be found in the package code)
+can be re-defined for customization. Their default definitions are
+expandable, but this is not a requirement.</p>
+</blockquote>
+<div class="section" id="poltypesetcmd-raw-coeff">
+<h3><a class="toc-backref" href="#id24"><tt class="docutils literal">\PolTypesetCmd{raw_coeff}</tt></a></h3>
+<blockquote>
+<p>Basically will use <tt class="docutils literal">\xintSignedFrac</tt> from <a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a>, but checks if
+the coefficient is <tt class="docutils literal">1</tt> or <tt class="docutils literal"><span class="pre">-1</span></tt> and then skips printing the
+<tt class="docutils literal">1</tt>, except for the constant term...</p>
+<p>One can do things such as for example: <a class="footnote-reference" href="#id4" id="id3">[1]</a></p>
+<pre class="literal-block">
+\renewcommand\PolTypesetCmd[1]{\num{\xintPFloat[5]{#1}}}
+\renewcommand\PolTypesetCmd[1]{\num{\xintRound{4}{#1}}}
+</pre>
+<p>where e.g. we used the <tt class="docutils literal">\num</tt> macro of <tt class="docutils literal">siunitx</tt> as it
+understands floating point notation.</p>
+<table class="docutils footnote" frame="void" id="id4" rules="none">
+<colgroup><col class="label" /><col /></colgroup>
+<tbody valign="top">
+<tr><td class="label"><a class="fn-backref" href="#id3">[1]</a></td><td>the difference in the syntaxes of <tt class="docutils literal">\xintPFloat</tt> and
+<tt class="docutils literal">\xintRound</tt> is explained from the fact that
+<tt class="docutils literal">\xintPFloat</tt> by default uses the prevailing precision
+hence the extra argument like here <tt class="docutils literal">5</tt> is an optional one.</td></tr>
+</tbody>
+</table>
+</blockquote>
+</div>
+<div class="section" id="poltypesetcmdprefix-raw-coeff">
+<h3><a class="toc-backref" href="#id25"><tt class="docutils literal">\PolTypesetCmdPrefix{raw_coeff}</tt></a></h3>
+<blockquote>
+Expands to a <tt class="docutils literal">+</tt> if the <tt class="docutils literal">raw_coeff</tt> is zero or positive, and to
+nothing if <tt class="docutils literal">raw_coeff</tt> is negative, as in latter case the
+<tt class="docutils literal">\xintSignedFrac</tt> used by <a class="reference internal" href="#poltypesetcmd-raw-coeff">\PolTypesetCmd{raw_coeff}</a> will put
+the <tt class="docutils literal">-</tt> sign in front of the fraction (if it is a fraction) and
+this will thus serve as separator in the typeset formula. Not used
+for the first term.</blockquote>
+</div>
+<div class="section" id="poltypesetmonomialcmd">
+<h3><a class="toc-backref" href="#id26"><tt class="docutils literal">\PolTypesetMonomialCmd</tt></a></h3>
+<blockquote>
+This decides how a monomial (in variable <tt class="docutils literal">\PolVar</tt> and with
+exponent <tt class="docutils literal">\PolIndex</tt>) is to be printed. The default does nothing
+for the constant term, <tt class="docutils literal">\PolVar</tt> for the first degree and
+<tt class="docutils literal"><span class="pre">\PolVar^{\PolIndex}</span></tt> for higher degrees monomials. Beware that
+<tt class="docutils literal">\PolIndex</tt> expands to digit tokens and needs termination in
+<tt class="docutils literal">\ifnum</tt> tests.</blockquote>
+</div>
+</div>
+<div class="section" id="id5">
+<h2><a class="toc-backref" href="#id27"><tt class="docutils literal"><span class="pre">\PolTypeset*{polname}</span></tt></a></h2>
+<blockquote>
+Typesets in ascending powers. Use e.g. <tt class="docutils literal">[h]</tt> optional argument
+(after the <tt class="docutils literal">*</tt>) to use letter <tt class="docutils literal">h</tt> rather than <tt class="docutils literal">x</tt>.</blockquote>
+</div>
+<div class="section" id="poldiff-polname-1-polname-2">
+<h2><a class="toc-backref" href="#id28"><tt class="docutils literal"><span class="pre">\PolDiff{polname_1}{polname_2}</span></tt></a></h2>
+<blockquote>
+<p>This sets <tt class="docutils literal">polname_2</tt> to the first derivative of <tt class="docutils literal">polname_1</tt>. It
+is allowed to issue <tt class="docutils literal"><span class="pre">\PolDiff{f}{f}</span></tt>, effectively replacing <tt class="docutils literal">f</tt>
+by <tt class="docutils literal">f'</tt>.</p>
+<p>Coefficients of the result <tt class="docutils literal">polname_2</tt> are irreducible fractions
(see <a class="reference internal" href="#technicalities">Technicalities</a> for the whole story.)</p>
-</dd>
-<dt><tt class="docutils literal"><span class="pre">\PolDiff[N]{f1}{f2}</span></tt></dt>
-<dd>This sets <tt class="docutils literal">f2</tt> to the <tt class="docutils literal">N</tt>-th derivative of <tt class="docutils literal">f1</tt>. Identical
-arguments is allowed. With <tt class="docutils literal">N=0</tt>, same effect as
-<tt class="docutils literal"><span class="pre">\PolLet{f2}{f1}</span></tt>. With negative <tt class="docutils literal">N</tt>, switches to using
-<tt class="docutils literal">\PolAntiDiff</tt>.</dd>
-<dt><tt class="docutils literal"><span class="pre">\PolAntiDiff{f1}{f2}</span></tt></dt>
-<dd><p class="first">This sets <tt class="docutils literal">f2</tt> to the primitive of <tt class="docutils literal">f1</tt> vanishing at zero.</p>
-<p class="last">Coefficients of the result <tt class="docutils literal">f2</tt> are irreducible fractions
+</blockquote>
+</div>
+<div class="section" id="poldiff-n-polname-1-polname-2">
+<h2><a class="toc-backref" href="#id29"><tt class="docutils literal"><span class="pre">\PolDiff[N]{polname_1}{polname_2}</span></tt></a></h2>
+<blockquote>
+This sets <tt class="docutils literal">polname_2</tt> to the <tt class="docutils literal">N</tt>-th derivative of <tt class="docutils literal">polname_1</tt>.
+Identical arguments is allowed. With <tt class="docutils literal">N=0</tt>, same effect as
+<tt class="docutils literal"><span class="pre">\PolLet{polname_2}={polname_1}</span></tt>. With negative <tt class="docutils literal">N</tt>, switches to
+using <tt class="docutils literal">\PolAntiDiff</tt>.</blockquote>
+</div>
+<div class="section" id="polantidiff-polname-1-polname-2">
+<h2><a class="toc-backref" href="#id30"><tt class="docutils literal"><span class="pre">\PolAntiDiff{polname_1}{polname_2}</span></tt></a></h2>
+<blockquote>
+<p>This sets <tt class="docutils literal">polname_2</tt> to the primitive of <tt class="docutils literal">polname_1</tt> vanishing
+at zero.</p>
+<p>Coefficients of the result <tt class="docutils literal">polname_2</tt> are irreducible fractions
(see <a class="reference internal" href="#technicalities">Technicalities</a> for the whole story.)</p>
-</dd>
-<dt><tt class="docutils literal"><span class="pre">\PolAntiDiff[N]{f1}{f2}</span></tt></dt>
-<dd>This sets <tt class="docutils literal">f2</tt> to the result of <tt class="docutils literal">N</tt> successive integrations on
-<tt class="docutils literal">f1</tt>. With negative <tt class="docutils literal">N</tt>, it switches to using <tt class="docutils literal">\PolDiff</tt>.</dd>
-<dt><tt class="docutils literal"><span class="pre">\PolDivide{f1}{f2}{Q}{R}</span></tt></dt>
-<dd>This sets <tt class="docutils literal">Q</tt> and <tt class="docutils literal">R</tt> to be the quotient and remainder in the
-Euclidean division of <tt class="docutils literal">f1</tt> by <tt class="docutils literal">f2</tt>.</dd>
-<dt><tt class="docutils literal"><span class="pre">\PolGCD{f}{g}{k}</span></tt></dt>
-<dd>This sets <tt class="docutils literal">k</tt> to be the G.C.D. It is a unitary polynomial except
-if both <tt class="docutils literal">f</tt> and <tt class="docutils literal">g</tt> vanish, then <tt class="docutils literal">k</tt> is the zero polynomial.</dd>
-<dt><tt class="docutils literal"><span class="pre">\PolMapCoeffs{\macro}{name}</span></tt></dt>
-<dd><p class="first">It modifies each coefficient of the defined polynomial via the
-<em>expandable</em> macro <tt class="docutils literal">\macro</tt>. The degree is adjusted as necessary
-if some leading coefficients vanish after the operation. In
-replacement text of <tt class="docutils literal">\macro</tt>, <tt class="docutils literal">\index</tt> expands to the
-coefficient index (which is defined to be zero for the constant
-term).</p>
+</blockquote>
+</div>
+<div class="section" id="polantidiff-n-polname-1-polname-2">
+<h2><a class="toc-backref" href="#id31"><tt class="docutils literal"><span class="pre">\PolAntiDiff[N]{polname_1}{polname_2}</span></tt></a></h2>
+<blockquote>
+This sets <tt class="docutils literal">polname_2</tt> to the result of <tt class="docutils literal">N</tt> successive integrations on
+<tt class="docutils literal">polname_1</tt>. With negative <tt class="docutils literal">N</tt>, it switches to using <tt class="docutils literal">\PolDiff</tt>.</blockquote>
+</div>
+<div class="section" id="poldivide-polname-1-polname-2-polname-q-polname-r">
+<h2><a class="toc-backref" href="#id32"><tt class="docutils literal"><span class="pre">\PolDivide{polname_1}{polname_2}{polname_Q}{polname_R}</span></tt></a></h2>
+<blockquote>
+This sets <tt class="docutils literal">polname_Q</tt> and <tt class="docutils literal">polname_R</tt> to be the quotient and
+remainder in the Euclidean division of <tt class="docutils literal">polname_1</tt> by
+<tt class="docutils literal">polname_2</tt>.</blockquote>
+</div>
+<div class="section" id="polgcd-polname-1-polname-2-polname-gcd">
+<h2><a class="toc-backref" href="#id33"><tt class="docutils literal"><span class="pre">\PolGCD{polname_1}{polname_2}{polname_GCD}</span></tt></a></h2>
+<blockquote>
+This sets <tt class="docutils literal">polname_GCD</tt> to be the G.C.D. It is a unitary
+polynomial except if both <tt class="docutils literal">polname_1</tt> and <tt class="docutils literal">polname_2</tt> vanish,
+then <tt class="docutils literal">polname_GCD</tt> is the zero polynomial.</blockquote>
+</div>
+<div class="section" id="polmapcoeffs-macro-polname">
+<h2><a class="toc-backref" href="#id34"><tt class="docutils literal"><span class="pre">\PolMapCoeffs{\macro}{polname}</span></tt></a></h2>
+<blockquote>
+<p>It modifies ('in-place': original coefficients get lost) each
+coefficient of the defined polynomial via the <em>expandable</em> macro
+<tt class="docutils literal">\macro</tt>. The degree is adjusted as necessary if some leading
+coefficients vanish after the operation. In replacement text of
+<tt class="docutils literal">\macro</tt>, <tt class="docutils literal">\index</tt> expands to the coefficient index (which is
+defined to be zero for the constant term).</p>
<p>Notice that <tt class="docutils literal">\macro</tt> will have to handle inputs of the shape
-<tt class="docutils literal">A/B[N]</tt> (xintfrac internal notation). This means that it probably
+<tt class="docutils literal">A/B[N]</tt> (<a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a> internal notation). This means that it probably
will have to be expressed in terms of macros from xintfrac package.</p>
<p>Example:</p>
<pre class="literal-block">
\def\foo#1{\xintMul{#1}{\the\numexpr\index*\index\relax}}
</pre>
-<p class="last">(or with <tt class="docutils literal"><span class="pre">\xintSqr{\xindex}</span></tt>) to replace <tt class="docutils literal">n</tt>-th coefficient
+<p>(or with <tt class="docutils literal"><span class="pre">\xintSqr{\index}</span></tt>) to replace <tt class="docutils literal">n</tt>-th coefficient
<tt class="docutils literal">f_n</tt> by <tt class="docutils literal">f_n*n^2</tt>.</p>
-</dd>
-<dt><tt class="docutils literal">\PolReduceCoeffs{name}</tt></dt>
-<dd>About the same as <tt class="docutils literal"><span class="pre">\PolMapCoeffs{\xintIrr}{name}</span></tt> (but adds
-<tt class="docutils literal">[0]</tt> postfix which speeds up xintfrac operations when
-evaluating.)</dd>
-</dl>
+</blockquote>
</div>
+<div class="section" id="polreducecoeffs-polname">
+<h2><a class="toc-backref" href="#id35"><tt class="docutils literal">\PolReduceCoeffs{polname}</tt></a></h2>
+<blockquote>
+About the same as <tt class="docutils literal"><span class="pre">\PolMapCoeffs{\xintIrr}{polname}</span></tt> (but
+maintaining a <tt class="docutils literal">[0]</tt> postfix for speedier <a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a> parsing when
+polynomial function is used for computations.) This is a
+one-argument macro, working 'in-place'.</blockquote>
+</div>
+</div>
<div class="section" id="expandable-macros">
-<h1>Expandable macros</h1>
+<h1><a class="toc-backref" href="#id36">Expandable macros</a></h1>
<p>All these macros expand completely in two steps except <tt class="docutils literal">\PolToExpr</tt>
-which needs a <tt class="docutils literal">\write</tt>, <tt class="docutils literal">\edef</tt> or a <tt class="docutils literal"><span class="pre">\csname...\endcsname</span></tt> context.</p>
-<dl class="docutils">
-<dt><tt class="docutils literal"><span class="pre">\PolEval{name}\At{value}</span></tt></dt>
-<dd>It boils down to <tt class="docutils literal">\xinttheexpr <span class="pre">reduce(name(value))\relax</span></tt>.</dd>
-<dt><tt class="docutils literal"><span class="pre">\PolNthCoeff{name}{N}</span></tt></dt>
-<dd>It expands to the raw <tt class="docutils literal">N</tt>-th coefficient (<tt class="docutils literal">0/1[0]</tt> if index is
-out of range). With <tt class="docutils literal"><span class="pre">N=-1</span></tt>, <tt class="docutils literal"><span class="pre">-2</span></tt>, ... expands to the leading
-coefficients.</dd>
-<dt><tt class="docutils literal">\PolDegree{name}</tt></dt>
-<dd>It expands to the degree. This is <tt class="docutils literal"><span class="pre">-1</span></tt> if zero polynomial but this
-may change in future. Should it then expand to <tt class="docutils literal"><span class="pre">-\infty</span></tt> ?</dd>
-<dt><tt class="docutils literal">\PolToExpr{f}</tt></dt>
-<dd><p class="first">Expands to <tt class="docutils literal">f_0 + f_1*x + f_2*x^2 + ...</tt> (ascending powers). <a class="footnote-reference" href="#id3" id="id1">[1]</a>,
-<a class="footnote-reference" href="#id4" id="id2">[2]</a></p>
-<table class="docutils footnote" frame="void" id="id3" rules="none">
+and <tt class="docutils literal">\PolToFloatExpr</tt> (and their auxiliaries) which need a
+<tt class="docutils literal">\write</tt>, <tt class="docutils literal">\edef</tt> or a <tt class="docutils literal"><span class="pre">\csname...\endcsname</span></tt> context.</p>
+<div class="section" id="poleval-polname-at-numerical-expression">
+<h2><a class="toc-backref" href="#id37"><tt class="docutils literal"><span class="pre">\PolEval{polname}\At{numerical</span> expression}</tt></a></h2>
+<blockquote>
+<p>It boils down to <tt class="docutils literal">\xinttheexpr polname(numerical <span class="pre">expression)\relax</span></tt>.</p>
+<div class="admonition note">
+<p class="first admonition-title">Note</p>
+<p>The <tt class="docutils literal">0.2</tt> version stupidly tried to be clever and as a result
+of a misguided optimization choked if <tt class="docutils literal">value</tt> was not a number
+but a numerical expression (a sum e.g.), but the more powerful
+behaviour has been reinstored at <tt class="docutils literal">0.3</tt> release.</p>
+<p class="last">The <tt class="docutils literal">0.1</tt> and <tt class="docutils literal">0.2</tt> version did a <tt class="docutils literal">reduce</tt> which however is
+costly on big fractions and irrelevant if the output is served as
+argument of <tt class="docutils literal">\xintRound</tt> or <tt class="docutils literal">\xintFloat</tt>. Thus <tt class="docutils literal">reduce</tt> was
+removed, and former meaning is now available as
+<a class="reference internal" href="#polevalreduced-polname-at-numerical-expression">\PolEvalReduced{polname}\At{numerical expression}</a></p>
+</div>
+</blockquote>
+</div>
+<div class="section" id="polevalreduced-polname-at-numerical-expression">
+<h2><a class="toc-backref" href="#id38"><tt class="docutils literal"><span class="pre">\PolEvalReduced{polname}\At{numerical</span> expression}</tt></a></h2>
+<blockquote>
+Boils down to <tt class="docutils literal">\xinttheexpr reduce(polname(numerical <span class="pre">expression))\relax</span></tt>.</blockquote>
+</div>
+<div class="section" id="polfloateval-polname-at-numerical-expression">
+<h2><a class="toc-backref" href="#id39"><tt class="docutils literal"><span class="pre">\PolFloatEval{polname}\At{numerical</span> expression}</tt></a></h2>
+<blockquote>
+<p>Boils down to <tt class="docutils literal">\xintthefloatexpr polname(numerical <span class="pre">expression)\relax</span></tt>.</p>
+<p>This is done via a Horner Scheme (see <a class="reference internal" href="#poldef">\poldef</a>), with
+already rounded coefficients. <a class="footnote-reference" href="#id8" id="id6">[2]</a> To use the <em>exact coefficients</em>
+(and <em>exact</em> additions and multiplications), just insert it in the
+float expression as in this example: <a class="footnote-reference" href="#id9" id="id7">[3]</a></p>
+<pre class="literal-block">
+\xintthefloatexpr 3.27*\xintexpr f(2.53)\relax^2\relax
+</pre>
+<p>The <tt class="docutils literal">f(2.53)</tt> is exactly computed then rounded at the time of
+getting raised to the power <tt class="docutils literal">2</tt>. Moving the <tt class="docutils literal">^2</tt> inside, that
+operation would also be treated exactly.</p>
+<table class="docutils footnote" frame="void" id="id8" rules="none">
<colgroup><col class="label" /><col /></colgroup>
<tbody valign="top">
-<tr><td class="label"><a class="fn-backref" href="#id1">[1]</a></td><td>in a <tt class="docutils literal">\write</tt>, <tt class="docutils literal">\edef</tt>, or <tt class="docutils literal"><span class="pre">\csname...\endcsname</span></tt>, but
-not under <tt class="docutils literal"><span class="pre">\romannumeral-`0</span></tt>.</td></tr>
+<tr><td class="label"><a class="fn-backref" href="#id6">[2]</a></td><td>Anyway each floating point operation starts by rounding its
+operands to the floating point precision.</td></tr>
</tbody>
</table>
-<table class="docutils footnote" frame="void" id="id4" rules="none">
+<table class="docutils footnote" frame="void" id="id9" rules="none">
<colgroup><col class="label" /><col /></colgroup>
<tbody valign="top">
-<tr><td class="label"><a class="fn-backref" href="#id2">[2]</a></td><td>the letter <tt class="docutils literal">x</tt> is (in this release) not customizable.</td></tr>
+<tr><td class="label"><a class="fn-backref" href="#id7">[3]</a></td><td>The <tt class="docutils literal">\xintexpr</tt> could be <tt class="docutils literal">\xinttheexpr</tt> but that would be
+less efficient. Cf. <a class="reference external" href="http://www.ctan.org/pkg/xint">xintexpr</a> documentation about nested
+expressions.</td></tr>
</tbody>
</table>
-<p>By default zero coefficients are skipped (issue <tt class="docutils literal">\poltoexprtrue</tt> to
+</blockquote>
+</div>
+<div class="section" id="polnthcoeff-polname-number">
+<h2><a class="toc-backref" href="#id40"><tt class="docutils literal"><span class="pre">\PolNthCoeff{polname}{number}</span></tt></a></h2>
+<blockquote>
+It expands to the raw <tt class="docutils literal">N</tt>-th coefficient (<tt class="docutils literal">0/1[0]</tt> if the index
+number is out of range). With <tt class="docutils literal"><span class="pre">N=-1</span></tt>, <tt class="docutils literal"><span class="pre">-2</span></tt>, ... expands to the
+leading coefficients.</blockquote>
+</div>
+<div class="section" id="poldegree-polname">
+<h2><a class="toc-backref" href="#id41"><tt class="docutils literal">\PolDegree{polname}</tt></a></h2>
+<blockquote>
+It expands to the degree. This is <tt class="docutils literal"><span class="pre">-1</span></tt> if zero polynomial but this
+may change in future. Should it then expand to <tt class="docutils literal"><span class="pre">-\infty</span></tt> ?</blockquote>
+</div>
+<div class="section" id="poltoexpr-polname">
+<h2><a class="toc-backref" href="#id42"><tt class="docutils literal">\PolToExpr{polname}</tt></a></h2>
+<blockquote>
+<p>Expands <a class="footnote-reference" href="#id11" id="id10">[4]</a> to <tt class="docutils literal"><span class="pre">coeff_N*x^N+...</span></tt> (descending powers.)</p>
+<table class="docutils footnote" frame="void" id="id11" rules="none">
+<colgroup><col class="label" /><col /></colgroup>
+<tbody valign="top">
+<tr><td class="label"><a class="fn-backref" href="#id10">[4]</a></td><td>in a <tt class="docutils literal">\write</tt>, <tt class="docutils literal">\edef</tt>, or <tt class="docutils literal"><span class="pre">\csname...\endcsname</span></tt>, but
+not under <tt class="docutils literal"><span class="pre">\romannumeral-`0</span></tt>.</td></tr>
+</tbody>
+</table>
+<p>By default zero coefficients are skipped (issue <tt class="docutils literal">\poltoexpralltrue</tt> to
get all of them in output).</p>
-<p>No <tt class="docutils literal">+</tt> sign before negative coefficients, for compliance with Maple
-input format. This means though that parsing the result back via
-naive delimited macros is difficult, see <tt class="docutils literal">\PolToList</tt> and <tt class="docutils literal">\PolToCSV</tt>
-for more low-level formats making it easier to get expandably some
-output of one's choice, which may possibly be parsed later on by
-other macros of one's design, or from other packages.</p>
-<p>Of course <tt class="docutils literal">\PolToExpr{f}</tt> can be inserted in a poldef, as the
+<p>By default, no <tt class="docutils literal">+</tt> sign before negative coefficients, for
+compliance with Maple input format (but see
+<a class="reference internal" href="#poltoexprtermprefix-raw-coeff">\PolToExprTermPrefix{raw_coeff}</a>.) Also, like the default
+behaviour of <a class="reference internal" href="#poltypeset-polname">\PolTypeset{polname}</a>, does not print (for the non
+constant terms) coefficients equal to plus or minus one. The degree
+one monomial is output as <tt class="docutils literal">x</tt>, not <tt class="docutils literal">x^1</tt>. Complete customization is
+possible, see next macros.</p>
+<p>Of course <tt class="docutils literal">\PolToExpr{f}</tt> can be inserted in a <tt class="docutils literal">\poldef</tt>, as the
latter expands token by token, hence will force complete expansion
-of <tt class="docutils literal">\PolToExpr{f}</tt>, but simply <tt class="docutils literal">f(x)</tt> will be more efficient for the
-identical result.</p>
-<p class="last"><tt class="docutils literal">\PolToExprCmd</tt> is the one-argument macro used by <tt class="docutils literal">\PolToExpr</tt> for the
-coefficients, it defaults to <tt class="docutils literal"><span class="pre">\xintPRaw{\xintRawWithZeros{#1}}</span></tt>. One
-will have to redefine it to use <tt class="docutils literal"><span class="pre">\xintIrr{#1}</span></tt> in place of
-<tt class="docutils literal"><span class="pre">\xintRawWithZeros{#1}</span></tt> to get in output reduced coefficients.</p>
-</dd>
-<dt><tt class="docutils literal">\PolToList{f}</tt></dt>
-<dd>Expands to <tt class="docutils literal"><span class="pre">{f_0}{f_1}...{f_N}</span></tt> with <tt class="docutils literal">N</tt> = degree of f (except
-zero polynomial which does give <tt class="docutils literal">{0/1[0]}</tt> and not an empty
-output.)</dd>
-<dt><tt class="docutils literal">\PolToCSV{f}</tt></dt>
-<dd>Expands to <tt class="docutils literal">f_0, f_1, f_2, <span class="pre">.....,</span> f_N</tt>. Converse to
-<tt class="docutils literal">\PolFromCSV</tt>.</dd>
-</dl>
+of <tt class="docutils literal">\PolToExpr{f}</tt>, but a simple <tt class="docutils literal">f(x)</tt> is more efficient for
+the identical result.</p>
+</blockquote>
+<div class="section" id="poltoexproneterm-raw-coeff-number">
+<h3><a class="toc-backref" href="#id43"><tt class="docutils literal"><span class="pre">\PolToExprOneTerm{raw_coeff}{number}</span></tt></a></h3>
+<blockquote>
+<p>This two argument expandable command takes care of the monomial and
+its coefficient. The default definition is done in order for
+coefficients of absolute value <tt class="docutils literal">1</tt> not be printed explicitely
+(except of course for the constant term). Also by default, the
+monomial of degree one is <tt class="docutils literal">x</tt> not <tt class="docutils literal">x^1</tt>, and <tt class="docutils literal">x^0</tt> is skipped.</p>
+<p>For compatibility with Maple input requirements, by default a <tt class="docutils literal">*</tt>
+always precedes the <tt class="docutils literal">x^number</tt>, except if the coefficient is a one
+or a minus one. See <a class="reference internal" href="#poltoexprtimes">\PolToExprTimes</a>.</p>
+</blockquote>
</div>
+<div class="section" id="poltoexpronetermstyleb-raw-coeff-number">
+<h3><a class="toc-backref" href="#id44"><tt class="docutils literal"><span class="pre">\PolToExprOneTermStyleB{raw_coeff}{number}</span></tt></a></h3>
+<blockquote>
+<p>For output in this style:</p>
+<pre class="literal-block">
+2*x^11/3+3*x^8/7-x^5−x^4/4−x^3−x^2/2−2*x+1
+</pre>
+<p>issue <tt class="docutils literal">\let\PolToExprOneTerm\PolToExprOneTermStyleB</tt> before using
+<tt class="docutils literal">\PolToExpr</tt>. Note that then <tt class="docutils literal">\PolToExprCmd</tt> isn't used at all.</p>
+<p>To suppress the <tt class="docutils literal">*</tt>'s, cf. <a class="reference internal" href="#poltoexprtimes">\PolToExprTimes</a>.</p>
+</blockquote>
+</div>
+<div class="section" id="poltoexprcmd-raw-coeff">
+<h3><a class="toc-backref" href="#id45"><tt class="docutils literal">\PolToExprCmd{raw_coeff}</tt></a></h3>
+<blockquote>
+It is the one-argument macro used by the package definition of
+<tt class="docutils literal">\PolToExprOneTerm</tt> for the coefficients themselves (when not
+equal to plus or minus one), and it defaults to
+<tt class="docutils literal"><span class="pre">\xintPRaw{\xintRawWithZeros{#1}}</span></tt>. One will have to redefine it
+to <tt class="docutils literal"><span class="pre">\xintIrr{#1}</span></tt> or to <tt class="docutils literal"><span class="pre">\xintPRaw{\xintIrr{#1}}</span></tt> to obtain in the
+output forcefully reduced coefficients.</blockquote>
+</div>
+<div class="section" id="poltoexprtermprefix-raw-coeff">
+<h3><a class="toc-backref" href="#id46"><tt class="docutils literal">\PolToExprTermPrefix{raw_coeff}</tt></a></h3>
+<blockquote>
+Defined identically as <a class="reference internal" href="#poltypesetcmdprefix-raw-coeff">\PolTypesetCmdPrefix{raw_coeff}</a>. It
+prefixes with a plus sign for non-negative coefficients, because
+they don't carry one by themselves.</blockquote>
+</div>
+<div class="section" id="poltoexprvar">
+<h3><a class="toc-backref" href="#id47"><tt class="docutils literal">\PolToExprVar</tt></a></h3>
+<blockquote>
+This expands to the variable to use in output (it does not have to
+be a single letter, may be an expandable macro.) Initial definition
+is <tt class="docutils literal">x</tt>.</blockquote>
+</div>
+<div class="section" id="poltoexprtimes">
+<h3><a class="toc-backref" href="#id48"><tt class="docutils literal">\PolToExprTimes</tt></a></h3>
+<blockquote>
+This expands to the symbol used for multiplication of an
+<tt class="docutils literal"><span class="pre">x^{number}</span></tt> by the corresponding coefficient. The default is
+<tt class="docutils literal">*</tt>. Redefine the macro to expand to nothing to get rid of it (but
+this will give output incompatible with some professional computer
+algebra software).</blockquote>
+</div>
+</div>
+<div class="section" id="id12">
+<h2><a class="toc-backref" href="#id49"><tt class="docutils literal"><span class="pre">\PolToExpr*{polname}</span></tt></a></h2>
+<blockquote>
+Expands to <tt class="docutils literal"><span class="pre">coeff_0+coeff_1*x+coeff_2*x^2+...</span></tt> (ascending powers).
+Customizable like <a class="reference internal" href="#poltoexpr-polname">\PolToExpr{polname}</a> via the same macros.</blockquote>
+</div>
+<div class="section" id="poltofloatexpr-polname">
+<h2><a class="toc-backref" href="#id50"><tt class="docutils literal">\PolToFloatExpr{polname}</tt></a></h2>
+<blockquote>
+<p>Similar to <a class="reference internal" href="#poltoexpr-polname">\PolToExpr{polname}</a> but uses <a class="reference external" href="\PolToFloatExprCmd{raw_coeff}">\PolToFloatExprCmd</a>
+which by default rounds and converts the coefficients to floating
+point format.</p>
+<div class="admonition note">
+<p class="first admonition-title">Note</p>
+<p>The polynomial function for usage in <tt class="docutils literal">\xintfloatexpr</tt> is
+already prepared with the rounded coefficients, but the latter
+are not easily recoverable (and especially not expandably) from
+this. Thus <tt class="docutils literal">\PolToFloatExprCmd</tt> operates from the <em>exact</em>
+coefficients anew. This means though that if the prevailing float
+precision was changed with <tt class="docutils literal"><span class="pre">\xintDigits:=P;</span></tt> syntax, the output
+will obey this precision <tt class="docutils literal">P</tt>, but the polynomial function was
+defined earlier and operates on floating point numbers with
+coefficients which were rounded at time of definition.</p>
+<p class="last">This may change in future, if the pre-rounded coefficients are
+stored in a more easily accessible data structure.</p>
+</div>
+</blockquote>
+<div class="section" id="poltofloatexproneterm-raw-coeff-number">
+<h3><a class="toc-backref" href="#id51"><tt class="docutils literal"><span class="pre">\PolToFloatExprOneTerm{raw_coeff}{number}</span></tt></a></h3>
+<blockquote>
+Similar to <a class="reference external" href="\PolToExprOneTerm{raw_coeff}{number}">\PolToExprOneTerm</a>. But does not treat
+especially coefficients equal to plus or minus one.</blockquote>
+</div>
+<div class="section" id="poltofloatexprcmd-raw-coeff">
+<h3><a class="toc-backref" href="#id52"><tt class="docutils literal">\PolToFloatExprCmd{raw_coeff}</tt></a></h3>
+<blockquote>
+<p>It is the one-argument macro used by <tt class="docutils literal">\PolToFloatExprOneTerm</tt>.
+Its package definition is <tt class="docutils literal"><span class="pre">\xintFloat{#1}</span></tt>.</p>
+<div class="admonition caution">
+<p class="first admonition-title">Caution!</p>
+<p>Currently (<a class="reference external" href="http://www.ctan.org/pkg/xint">xint</a> <tt class="docutils literal">1.2p</tt>) <tt class="docutils literal">\xintFloat{0}</tt> outputs <tt class="docutils literal">0.e0</tt>
+which is perfectly acceptable input for Python, but not for
+Maple. Thus, one should better leave the <a class="reference internal" href="#poltoexprallfalse">\poltoexprallfalse</a>
+toggle to its default <tt class="docutils literal">\iffalse</tt> state, if one intends to use
+the output in a Maple worksheet.</p>
+<p>But even then the zero polynomial will cause a problem. Workaround:</p>
+<pre class="literal-block">
+\renewcommand\PolToFloatExprCmd[1]{\xintiiifZero{#1}{0.0}{\xintFloat{#1}}}
+</pre>
+<p class="last">Usage of <tt class="docutils literal">\xintiiifZero</tt> and not <tt class="docutils literal">\xintifZero</tt> is only for
+optimization (I can't help it) because <tt class="docutils literal">#1</tt> is known to be
+in <tt class="docutils literal">xintfrac</tt> raw format.</p>
+</div>
+</blockquote>
+</div>
+</div>
+<div class="section" id="id13">
+<h2><a class="toc-backref" href="#id53"><tt class="docutils literal"><span class="pre">\PolToFloatExpr*{polname}</span></tt></a></h2>
+<blockquote>
+Typesets in ascending powers.</blockquote>
+</div>
+<div class="section" id="poltolist-polname">
+<h2><a class="toc-backref" href="#id54"><tt class="docutils literal">\PolToList{polname}</tt></a></h2>
+<blockquote>
+Expands to <tt class="docutils literal"><span class="pre">{coeff_0}{coeff_1}...{coeff_N}</span></tt> with <tt class="docutils literal">N</tt> = degree
+(except zero polynomial which does give <tt class="docutils literal">{0/1[0]}</tt> and not an
+empty output.)</blockquote>
+</div>
+<div class="section" id="poltocsv-polname">
+<h2><a class="toc-backref" href="#id55"><tt class="docutils literal">\PolToCSV{polname}</tt></a></h2>
+<blockquote>
+Expands to <tt class="docutils literal">coeff_0, coeff_1, coeff_2, <span class="pre">.....,</span> coeff_N</tt>. Converse
+to <a class="reference internal" href="#polfromcsv-polname-csv">\PolFromCSV</a>.</blockquote>
+</div>
+</div>
+<div class="section" id="booleans-with-default-setting-as-indicated">
+<h1><a class="toc-backref" href="#id56">Booleans (with default setting as indicated)</a></h1>
+<div class="section" id="xintverbosefalse">
+<h2><a class="toc-backref" href="#id57"><tt class="docutils literal">\xintverbosefalse</tt></a></h2>
+<blockquote>
+<p>This is actually an <a class="reference external" href="http://www.ctan.org/pkg/xint">xintexpr</a> configuration. Setting it to
+<tt class="docutils literal">true</tt> triggers the writing of information to the log when new
+polynomials are defined.</p>
+<div class="admonition caution">
+<p class="first admonition-title">Caution!</p>
+<p class="last">The macro meanings as written to the log are to be considered
+unstable and undocumented internal structures.</p>
+</div>
+</blockquote>
+</div>
+<div class="section" id="poltypesetallfalse">
+<h2><a class="toc-backref" href="#id58"><tt class="docutils literal">\poltypesetallfalse</tt></a></h2>
+<blockquote>
+If <tt class="docutils literal">true</tt>, <a class="reference internal" href="#poltypeset-polname">\PolTypeset{polname}</a> will also typeset the vanishing
+coefficients.</blockquote>
+</div>
+<div class="section" id="poltoexprallfalse">
+<h2><a class="toc-backref" href="#id59"><tt class="docutils literal">\poltoexprallfalse</tt></a></h2>
+<blockquote>
+If <tt class="docutils literal">true</tt>, <a class="reference internal" href="#poltoexpr-polname">\PolToExpr{polname}</a> and <a class="reference internal" href="#poltofloatexpr-polname">\PolToFloatExpr{polname}</a> will
+also include the vanishing coefficients in their outputs.</blockquote>
+</div>
+</div>
<div class="section" id="technicalities">
-<h1>Technicalities</h1>
+<h1><a class="toc-backref" href="#id60">Technicalities</a></h1>
<ul>
-<li><p class="first">The catcode of the semi-colon is reset temporarily by <tt class="docutils literal">\poldef</tt> macro in
-case some other package (for example the French babel module) may have
-made it active. This will fail though if the whole thing was already
-part of a macro argument, in such cases one can use <tt class="docutils literal">\PolDef</tt> rather.
-The colon in <tt class="docutils literal">:=</tt> may be active with no consequences.</p>
+<li><p class="first">The catcode of the semi-colon is reset temporarily by <a class="reference internal" href="#poldef">\poldef</a> macro in case some other package (for example the French
+babel module) may have made it active. This will fail though if the
+whole thing was already part of a macro argument, in such cases one
+can use <a class="reference internal" href="#id2">\PolDef{f}{P(x)}</a>
+rather. The colon in <tt class="docutils literal">:=</tt> may be active with no consequences.</p>
</li>
-<li><p class="first">Beware the <tt class="docutils literal">1/2 x</tt> problem: as mentioned above, it will be give zero due
-to the tacit multiplication rules of <tt class="docutils literal">\xintexpr</tt> and to the fact that
-the package will do the Euclidean division of <tt class="docutils literal">1</tt> by polynomial <tt class="docutils literal">2x</tt>.</p>
-</li>
<li><p class="first">During execution of polynomial operations by <tt class="docutils literal">\poldef</tt> (but not
during the initial purely numerical parsing of the expression), the
-xintfrac macro <tt class="docutils literal">\xintAdd</tt> is temporarily patched to always express
+<a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a> macro <tt class="docutils literal">\xintAdd</tt> is temporarily patched to always express
<tt class="docutils literal">a/b + c/d</tt> with <tt class="docutils literal"><span class="pre">l.c.m.(b,d)</span></tt> as denominator. Indeed the current
(xint 1.2p) <tt class="docutils literal">\xintAdd</tt> uses <tt class="docutils literal"><span class="pre">(ad+bc)/bd</span></tt> formula except if <tt class="docutils literal">b</tt>
divides <tt class="docutils literal">d</tt> or <tt class="docutils literal">d</tt> divides <tt class="docutils literal">b</tt>, which quickly leads in real life
@@ -677,34 +1084,42 @@
<p>where all coefficients have the same denominator 6 (which in this
example is the <tt class="docutils literal">l.c.m</tt> of the denominators of the reduced coefficients.)</p>
</li>
-<li><p class="first"><tt class="docutils literal">\PolDiff</tt> always applies <tt class="docutils literal">\xintIrr</tt> to the resulting coefficients, except
-that the <em>power of ten</em> part <tt class="docutils literal">[N]</tt> (for example an input in scientific
-notation such as <tt class="docutils literal">1.23e5</tt> gives <tt class="docutils literal">123/1[3]</tt> internally in xintfrac) is not
-taken into account in the reduction of the fraction. This is tentative
-and may change.</p>
-<p>Same remark for <tt class="docutils literal">\PolAntiDiff</tt>.</p>
+<li><p class="first"><a class="reference internal" href="#poldiff-polname-1-polname-2">\PolDiff{polname_1}{polname_2}</a> always applies <tt class="docutils literal">\xintIrr</tt> to the
+resulting coefficients, except that the <em>power of ten</em> part <tt class="docutils literal">[N]</tt>
+(for example an input in scientific notation such as <tt class="docutils literal">1.23e5</tt> gives
+<tt class="docutils literal">123/1[3]</tt> internally in xintfrac) is not taken into account in the
+reduction of the fraction. This is tentative and may change.</p>
+<p>Same remark for <a class="reference internal" href="#polantidiff-polname-1-polname-2">\PolAntiDiff{polname_1}{polname_2}</a>.</p>
</li>
-<li><p class="first">If <tt class="docutils literal">f</tt> was created from comma separated values by macro PolFromCSV,
-then the coefficients will be in the output of <tt class="docutils literal">\PolToList</tt> and
-<tt class="docutils literal">\PolToCSV</tt> in the same format as originally in input: a <tt class="docutils literal">1.3e2</tt>
-will again be a <tt class="docutils literal">1.3e2</tt>.</p>
+<li><p class="first">If <tt class="docutils literal">f</tt> was created from comma separated values by macro
+<a class="reference internal" href="#polfromcsv-polname-csv">\PolFromCSV{polname}{<csv>}</a>, then the coefficients will be in
+the output of <a class="reference internal" href="#poltolist-polname">\PolToList{polname}</a> and <a class="reference internal" href="#poltocsv-polname">\PolToCSV{polname}</a> in
+the same format as originally in input: a <tt class="docutils literal">1.3e2</tt> will again be a
+<tt class="docutils literal">1.3e2</tt>.</p>
<p>In contrast when such coefficients are used in a <tt class="docutils literal">\poldef</tt> (or
<tt class="docutils literal">\PolDef</tt>) expression, they get transformed during the parsing to
-the xintfrac <em>raw</em> format. This is an unavoidable consequence of usage
-by poldef of <tt class="docutils literal">\xintdeffunc</tt> which itself is based on <tt class="docutils literal">\xintexpr.</tt>
+the xintfrac <em>raw</em> format.
This <em>raw</em> format speeds up expansion of xintfrac macros for numerical
evaluations.</p>
</li>
-<li><p class="first">Currently, the package does not as a result of <tt class="docutils literal">\poldef</tt> add to the TeX
-memory an already pre-computed <em>array</em> structure for the polynomial
-coefficients, as would be constructed by <tt class="docutils literal"><span class="pre">\PolAssign{f}\toarray\macro</span></tt>.
-Such structures are used, but for internal calculations in temporarily
-restricted scopes. Apart from the function <tt class="docutils literal">f()</tt> known to the
-(numerical) <tt class="docutils literal">\xintexpr</tt> parser (whose meaning can be found in the log
-file after xintverbosetrue), the data is (currently) stored in a
-single other macro encapsulating the degree, and the coefficients as a
-list. This may evolve in future.</p>
+<li><p class="first">Currently, the package stores all coefficients from index <tt class="docutils literal">0</tt> to
+index equal to the polynomial degree inside a single macro, as a list.
+This data structure is obviously very inefficient for polynomials of
+high degree and few coefficients (as an example with <tt class="docutils literal">\poldef
+<span class="pre">f(x):=x^1000</span> + x^500;</tt> the subsequent definition <tt class="docutils literal">\poldef <span class="pre">g(x):=</span>
+<span class="pre">f(x)^2;</span></tt> will do of the order of 1,000,000 multiplications and
+additions involvings only zeroes... which does take time). This
+may change in the future.</p>
</li>
+<li><p class="first">Tests have been made with Newton's iteration (for which computing
+exactly the derivative is precisely what this package is made for) or
+Regula Falsi method for locating roots: using exact computations leads
+quickly to gigantic fractions (but dichotomy method much less so). It
+is thus recommended to use <tt class="docutils literal">\xintdeffloatvar</tt> or
+<tt class="docutils literal">\xintthefloatexpr</tt> contexts for any kind of numerical mathematics.
+Of course, exact computations are invaluable for number theory or
+combinatorics...</p>
+</li>
<li><p class="first">As is to be expected internal structures of the package are barely
documented and unstable. Don't use them.</p>
</li>
@@ -711,12 +1126,23 @@
</ul>
</div>
<div class="section" id="releases">
-<h1>RELEASES</h1>
-<ul class="simple">
-<li>0.1 (2018/01/11): initial release (files README, polexpr.sty).</li>
-<li>0.2 (2018/01/14): documentation moved to polexpr.{txt,html}.</li>
+<h1><a class="toc-backref" href="#id61">RELEASES</a></h1>
+<ul>
+<li><p class="first">0.1 (2018/01/11)</p>
+<p>Initial release (files README, polexpr.sty).</p>
+</li>
+<li><p class="first">0.2 (2018/01/14)</p>
+<p>Documentation moved to polexpr.{txt,html}.</p>
+</li>
+<li><p class="first">0.3 (2018/01/17)</p>
+<p>Make polynomials known to <tt class="docutils literal">\xintfloatexpr</tt> and improve
+documentation.</p>
+</li>
+<li><p class="first">0.3.1 (2018/01/18)</p>
+<p>Fix two typos in documentation.</p>
+</li>
</ul>
-<p>Files of 0.2 release:</p>
+<p>Files of 0.3.1 release:</p>
<ul class="simple">
<li>README.md,</li>
<li>polexpr.sty (package file),</li>
@@ -723,8 +1149,16 @@
<li>polexpr.txt (documentation),</li>
<li>polexpr.html (conversion via <a class="reference external" href="http://docutils.sourceforge.net/docs/index.html">DocUtils</a> rst2html.py)</li>
</ul>
-<p>See README.md for the License and the change log.</p>
+<p>See README.md for the License and the change log (there were
+some breaking changes from 0.2 to 0.3).</p>
</div>
+<div class="section" id="acknowledgments">
+<h1><a class="toc-backref" href="#id62">Acknowledgments</a></h1>
+<p>Thanks to Jürgen Gilg whose question about <a class="reference external" href="http://www.ctan.org/pkg/xint">xint</a> usage for
+differentiating polynomials was the initial trigger leading to this
+package, and to Jürgen Gilg and Thomas Söll for testing it on some
+concrete problems.</p>
</div>
+</div>
</body>
</html>
Modified: trunk/Master/texmf-dist/doc/latex/polexpr/polexpr.txt
===================================================================
--- trunk/Master/texmf-dist/doc/latex/polexpr/polexpr.txt 2018-01-18 23:25:12 UTC (rev 46376)
+++ trunk/Master/texmf-dist/doc/latex/polexpr/polexpr.txt 2018-01-18 23:25:24 UTC (rev 46377)
@@ -1,29 +1,35 @@
.. comment: -*- fill-column: 72; mode: rst; -*-
-Package polexpr documentation
-=============================
+===============================
+ Package polexpr documentation
+===============================
+0.3.1 (2018/01/18)
+==================
+
+.. contents::
+
First Examples
--------------
The syntax is::
- \poldef <name>(x):=<expression in variable x>;
+ \poldef polname(x):= expression in variable x;
where in place of ``x`` an arbitrary *dummy variable* is authorized
-(i.e. per default any of ``[a..z|A..Z]``; more letters can be declared
+(i.e. per default any of ``[a-z|A-Z]``; more letters can be declared
under Unicode engines.) One can also issue::
- \PolDef{name}{expression in variable x}
+ \PolDef{polname}{expression in variable x}
which admits an optional first argument to modify the variable letter
from its default ``x``.
``\poldef f(x):= 1-x+x^2;``
- defines polynomial ``f``. Polynomial names must start with a letter
- and may contain letters, digits, and underscores. The variable must
- be a single letter. The colon character is optional. The semi-colon
- at end of expression is mandatory.
+ defines polynomial ``f``. Polynomial names must start with a
+ letter and may contain letters, digits, and underscores. The
+ variable must be a single letter. The colon character is optional.
+ The semi-colon at end of expression is mandatory.
``\PolDef{f}{1-x+x^2}``
does the same as ``\poldef f(x):= 1-x+x^2;`` To use another letter
@@ -31,9 +37,9 @@
argument to ``\PolDef``. Useful if the semi-colon has been assigned
some non-standard catcode by some package.
-``\PolLet{g}{f}``
- saves a copy of ``f`` under name ``g``.
-
+``\PolLet{g}={f}``
+ saves a copy of ``f`` under name ``g``. Also usable without ``=``.
+
``\poldef f(z):= f(z)^2;``
redefines ``f`` in terms of itself.
@@ -46,7 +52,7 @@
``\PolDiff{f}{df_dx}``
sets ``df_dx`` to the derivative of ``f``.
-
+
``\PolDiff{df_dx}{f_xx}``
obtains second derivative.
@@ -77,63 +83,119 @@
\poldef k(x):= (x-1)(x-2)(x-3)(x-4)/(x^2-5x+4);
+.. _warningtacit:
+
.. attention::
- ``1/2 x`` skips the space and is treated like ``1/(2x)`` because of
- the tacit multiplication rules of \xintexpr. But this means it gives
- zero! Thus one must use ``(1/2)x`` or ``1/2*x`` or ``(1/2)*x`` for
- disambiguation: ``x-1/2*x^2+1/3*x^3...``
+ ``1/2 x^2`` skips the space and is treated like ``1/(2*x^2)`` because
+ of the tacit multiplication rules of \xintexpr. But this means it
+ gives zero! Thus one must use ``(1/2)x^2`` or ``1/2*x^2`` or
+ ``(1/2)*x^2`` for disambiguation: ``x - 1/2*x^2 + 1/3*x^3...``. It is
+ even simpler to move the denominator to the right: ``x - x^2/2 +
+ x^3/3 - ...``.
+ It is worth noting that ``1/2(x-1)(x-2)`` suffers the same issue:
+ xint_ tacit multiplication always "ties more", hence this gets
+ interpreted as ``1/(2*(x-1)*(x-2))`` which gives zero by polynomial
+ division. Thus, use one of ``(1/2)(x-1)(x-2)``, ``1/2*(x-1)(x-2)`` or
+ ``(x-1)(x-2)/2``.
+
After::
- \poldef f1(x):= 25(x-1)(x^2-2)(x-3)(x-4)(x-5);%
- \poldef f2(x):= 37(x-1)(x^2-2)(x-6)(x-7)(x-8);%
+ \poldef f_1(x):= 25(x-1)(x^2-2)(x-3)(x-4)(x-5);%
+ \poldef f_2(x):= 37(x-1)(x^2-2)(x-6)(x-7)(x-8);%
-the macro call ``\PolGCD{f1}{f2}{k}`` sets ``k`` to the (unitary) GCD of
-``f1`` and ``f2``.
+the macro call ``\PolGCD{f_1}{f_2}{k}`` sets ``k`` to the (unitary) GCD of
+``f_1`` and ``f_2`` (hence to the expansion of ``(x-1)(x^2-2)``.)
``\PolToExpr{k}``
- will thus (expandably) give in this case ``2-2*x^1-1*x^2+1*x^3``.
- This is useful for console or file output (the syntax is Maple- and
- PSTricks-compatible; currently the letter ``x`` in output is not
- customizable, but this can easily be added if requested from author.)
+ will (expandably) give in this case ``x^3-x^2-2*x+2``. This is
+ useful for console or file output (the syntax is Maple- and
+ PSTricks-compatible; the letter used in output can be
+ (non-expandably) changed via a redefinition of `\\PolToExprVar`_.)
+``\PolToExpr*{k}``
+ gives ascending powers: ``2-2*x-x^2+x^3``.
+
Non-expandable macros
---------------------
-``\poldef name(letter):= polynomial expression using letter;``
- This evaluates the polynomial expression and stores the coefficients
+.. _poldef;:
+
+``\poldef polname(letter):= expression in letter;``
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+ This evaluates the *polynomial expression* and stores the coefficients
in a private structure accessible later via other package macros,
- under the user-chosen ``name``. Of course previously defined
- polynomials are allowed in a new expression. Names must start with a
+ under the user-chosen ``polname``. Of course the *expression* can
+ use other previously defined polynomials. Names must start with a
letter and are constituted of letters, digits and underscore
- characters. See Examples above.
+ characters. The whole xintexpr_ syntax is authorized::
- As a side effect the function ``name()`` is recognized as a genuine
- ``\xintexpr...\relax`` function for (exact) numerical evaluation. It
- computes values not according to the original expression but via
- the Horner scheme corresponding to the polynomial coefficients.
+ \poldef sin(z) := add((-1)^i z^(2i+1)/(2i+1)!, i = 0..10);
+ With fractional coefficients, beware the `tacit multiplication issue
+ <warningtacit_>`_.
+
+ As a side effect the function ``polname()`` is recognized as a
+ genuine ``\xintexpr...\relax`` function for (exact) numerical
+ evaluation (or within an ``\xintdefvar`` assignment.) It computes
+ values not according to the original expression but via the Horner
+ scheme corresponding to the polynomial coefficients.
+
+ Also, a function with the same name is created for use within
+ ``\xintfloatexpr`` (or ``\xintdeffloatvar``.) This is indispensible
+ for numerical algorithms as exact computations very quickly lead to
+ very big fractions. Addition and multiplication steps of the Horner
+ scheme will be executed as floating-point operations. The
+ coefficients have already been rounded at time of definition,
+ according to the then prevailing ``\xinttheDigits`` value.
+
+ .. important::
+
+ Package macros (such as derivatives or Euclidean division)
+ operate with the "exact" polynomials; "floating point"
+ polynomials are always obtained in a second step.
+
+ To modifiy "in-place" the original coefficients of a polynomial
+ and round them to float precision::
+
+ \PolMapCoeffs{\xintFloat}{polname}
+ % or \xintFloat[P] for precision P digits
+
+ See `\\PolMapCoeffs{\\macro}{polname}`_.
+
The original expression is lost after parsing, and in particular
the package provides no way to typeset it. This has to be done
manually, if needed.
-``\PolDef{name}{P(x)}``
- Does the same but the variable is assumed to be ``x``. To use
- another letter, pass it as first optional argument:
- ``\PolDef[X]{name}{P(X)}``.
+.. _PolDef:
-``\PolLet{g}{f}``
- Makes a copy of already defined polynomial f to new one g. Same
- effect as ``\PolDef{g}{f(x)}`` but faster.
+``\PolDef[letter]{polname}{expression in letter}``
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
-``\PolAssign{f}\toarray\Array``
- Defines a one-argument expandable macro ``\Array{#1}`` which expands
+ Does the same in an undelimited macro format (thus avoiding
+ potential problems with the catcode of the semi-colon in presence of
+ some packages.) In absence of the ``[letter]`` optional argument,
+ the variable is assumed to be ``x``.
+
+``\PolLet{polname_2}={polname_1}``
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+ Makes a copy of the already defined polynomial ``polname_1`` to a
+ new one ``polname_2``. Same effect as
+ ``\PolDef{polname_2}{polname_1(x)}`` but with less overhead. The
+ ``=`` is optional.
+
+``\PolAssign{polname}\toarray\macro``
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+ Defines a one-argument expandable macro ``\macro{#1}`` which expands
to the (raw) #1th polynomial coefficient.
- Attention, coefficients here are indexed starting at 1.
- - With #1=-1, -2, ..., ``\Array{#1}`` returns leading coefficients.
+ - With #1=-1, -2, ..., ``\macro{#1}`` returns leading coefficients.
- With #1=0, returns the number of coefficients, i.e. ``1 + deg f``
for non-zero polynomials.
@@ -140,10 +202,22 @@
- Out-of-range #1's return ``0/1[0]``.
-``\PolGet{f}\fromarray\Array``
- Does the converse operation to ``\PolAssign{f}\toarray\Array``. No
+ See also `\\PolNthCoeff{polname}{number}`_. The main difference is that
+ with ``\PolAssign``, ``\macro`` is made a prefix to ``1 + deg f``
+ already defined (hidden to user) macros holding individually the
+ coefficients but `\\PolNthCoeff{polname}{number}`_ does each time the job
+ to expandably recover the ``Nth`` coefficient, and due to
+ expandability can not store it in a macro for future usage (of course,
+ it can be an argument in an ``\edef``.) The other difference
+ is the shift by one in indexing, mentioned above (negative
+ indices act the same in both.)
+
+``\PolGet{polname}\fromarray\macro``
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+ Does the converse operation to ``\PolAssign{polname}\toarray\macro``. No
error checks on validity of coefficients as numbers. Each
- ``\Array{index}`` is expanded in an ``\edef`` before being assigned
+ ``\macro{number}`` is expanded in an ``\edef`` before being assigned
to a coefficient. Leading zero coefficients are removed from the
polynomial.
@@ -156,80 +230,147 @@
However the coefficients are still in their original form (i.e.
they were not subjected to ``\xintRaw`` or similar xintfrac macro.)
-``\PolFromCSV{f}{comma separated coefficients}``
- Defines a polynomial directly from the comma separated list (or a
- macro expanding to such a list) of its coefficients, the constant
- term being the first item. No validity checks. Spaces from the list
- argument are trimmed. List items are expanded in an ``\edef``, but
- currently left in their original form like e.g. ``1.5e3`` which is
- not converted to ``15/1[2]`` *raw* xintfrac format (this may
- change).
+``\PolFromCSV{polname}{<csv>}``
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+ Defines a polynomial directly from the comma separated list of
+ values (or a macro expanding to such a list) of its coefficients,
+ the constant term being the first item. No validity checks. Spaces
+ from the list argument are trimmed. List items are each expanded in
+ an ``\edef``, but currently left in their original form like e.g.
+ ``1.5e3`` which is not converted to ``15/1[2]`` *raw* xintfrac
+ format (this may change).
+
Leading zero coefficients are removed::
- \PolFromCSV{J}{0, 0, 0, 0, 0, 0, 0, 0, 0, 0}
+ \PolFromCSV{f}{0, 0, 0, 0, 0, 0, 0, 0, 0, 0}
defines the zero polynomial, which has only one (zero) coefficient.
- See also expandable macro ``\PolToCSV``.
+ See also expandable macro `\\PolToCSV <\\PolToCSV{polname}_>`_.
-``\PolTypeset{name}``
+``\PolTypeset{polname}``
+~~~~~~~~~~~~~~~~~~~~~~~~
+
Typesets in descending powers in math mode. It uses letter ``x`` but
this can be changed via an optional argument::
- \PolTypeset[z]{name}
+ \PolTypeset[z]{polname}
By default zero coefficients are skipped (issue ``\poltypesetalltrue``
to get all of them in output).
- Macros ``\PolTypesetCmd``, ``\PolTypesetPlus``, ``\PolTypesetMonomial``
- can help configure the output. See the package code.
+ These commands (whose meanings will be found in the package code)
+ can be re-defined for customization. Their default definitions are
+ expandable, but this is not a requirement.
-``\PolTypeset*{name}``
- Typesets in ascending powers. Change the letter from its default
- ``x`` by optional argument.
+``\PolTypesetCmd{raw_coeff}``
+^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-``\PolDiff{f1}{f2}``
- This sets f2 to the first derivative of ``f1``. It is allowed to
- issue ``\PolDiff{f}{f}``, effectively replacing ``f`` by ``f'``.
+ Basically will use ``\xintSignedFrac`` from xintfrac_, but checks if
+ the coefficient is ``1`` or ``-1`` and then skips printing the
+ ``1``, except for the constant term...
- Coefficients of the result ``f2`` are irreducible fractions
+ One can do things such as for example: [#]_
+
+ ::
+
+ \renewcommand\PolTypesetCmd[1]{\num{\xintPFloat[5]{#1}}}
+ \renewcommand\PolTypesetCmd[1]{\num{\xintRound{4}{#1}}}
+
+ where e.g. we used the ``\num`` macro of ``siunitx`` as it
+ understands floating point notation.
+
+ .. [#] the difference in the syntaxes of ``\xintPFloat`` and
+ ``\xintRound`` is explained from the fact that
+ ``\xintPFloat`` by default uses the prevailing precision
+ hence the extra argument like here ``5`` is an optional one.
+
+``\PolTypesetCmdPrefix{raw_coeff}``
+^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+
+ Expands to a ``+`` if the ``raw_coeff`` is zero or positive, and to
+ nothing if ``raw_coeff`` is negative, as in latter case the
+ ``\xintSignedFrac`` used by `\\PolTypesetCmd{raw_coeff}`_ will put
+ the ``-`` sign in front of the fraction (if it is a fraction) and
+ this will thus serve as separator in the typeset formula. Not used
+ for the first term.
+
+``\PolTypesetMonomialCmd``
+^^^^^^^^^^^^^^^^^^^^^^^^^^
+
+ This decides how a monomial (in variable ``\PolVar`` and with
+ exponent ``\PolIndex``) is to be printed. The default does nothing
+ for the constant term, ``\PolVar`` for the first degree and
+ ``\PolVar^{\PolIndex}`` for higher degrees monomials. Beware that
+ ``\PolIndex`` expands to digit tokens and needs termination in
+ ``\ifnum`` tests.
+
+``\PolTypeset*{polname}``
+~~~~~~~~~~~~~~~~~~~~~~~~~
+
+ Typesets in ascending powers. Use e.g. ``[h]`` optional argument
+ (after the ``*``) to use letter ``h`` rather than ``x``.
+
+``\PolDiff{polname_1}{polname_2}``
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+ This sets ``polname_2`` to the first derivative of ``polname_1``. It
+ is allowed to issue ``\PolDiff{f}{f}``, effectively replacing ``f``
+ by ``f'``.
+
+ Coefficients of the result ``polname_2`` are irreducible fractions
(see `Technicalities`_ for the whole story.)
-``\PolDiff[N]{f1}{f2}``
- This sets ``f2`` to the ``N``-th derivative of ``f1``. Identical
- arguments is allowed. With ``N=0``, same effect as
- ``\PolLet{f2}{f1}``. With negative ``N``, switches to using
- ``\PolAntiDiff``.
+``\PolDiff[N]{polname_1}{polname_2}``
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
-``\PolAntiDiff{f1}{f2}``
- This sets ``f2`` to the primitive of ``f1`` vanishing at zero.
+ This sets ``polname_2`` to the ``N``-th derivative of ``polname_1``.
+ Identical arguments is allowed. With ``N=0``, same effect as
+ ``\PolLet{polname_2}={polname_1}``. With negative ``N``, switches to
+ using ``\PolAntiDiff``.
- Coefficients of the result ``f2`` are irreducible fractions
+``\PolAntiDiff{polname_1}{polname_2}``
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+ This sets ``polname_2`` to the primitive of ``polname_1`` vanishing
+ at zero.
+
+ Coefficients of the result ``polname_2`` are irreducible fractions
(see `Technicalities`_ for the whole story.)
-``\PolAntiDiff[N]{f1}{f2}``
- This sets ``f2`` to the result of ``N`` successive integrations on
- ``f1``. With negative ``N``, it switches to using ``\PolDiff``.
+``\PolAntiDiff[N]{polname_1}{polname_2}``
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
-``\PolDivide{f1}{f2}{Q}{R}``
- This sets ``Q`` and ``R`` to be the quotient and remainder in the
- Euclidean division of ``f1`` by ``f2``.
+ This sets ``polname_2`` to the result of ``N`` successive integrations on
+ ``polname_1``. With negative ``N``, it switches to using ``\PolDiff``.
-``\PolGCD{f}{g}{k}``
- This sets ``k`` to be the G.C.D. It is a unitary polynomial except
- if both ``f`` and ``g`` vanish, then ``k`` is the zero polynomial.
+``\PolDivide{polname_1}{polname_2}{polname_Q}{polname_R}``
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
-``\PolMapCoeffs{\macro}{name}``
- It modifies each coefficient of the defined polynomial via the
- *expandable* macro ``\macro``. The degree is adjusted as necessary
- if some leading coefficients vanish after the operation. In
- replacement text of ``\macro``, ``\index`` expands to the
- coefficient index (which is defined to be zero for the constant
- term).
+ This sets ``polname_Q`` and ``polname_R`` to be the quotient and
+ remainder in the Euclidean division of ``polname_1`` by
+ ``polname_2``.
+``\PolGCD{polname_1}{polname_2}{polname_GCD}``
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+ This sets ``polname_GCD`` to be the G.C.D. It is a unitary
+ polynomial except if both ``polname_1`` and ``polname_2`` vanish,
+ then ``polname_GCD`` is the zero polynomial.
+
+``\PolMapCoeffs{\macro}{polname}``
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+ It modifies ('in-place': original coefficients get lost) each
+ coefficient of the defined polynomial via the *expandable* macro
+ ``\macro``. The degree is adjusted as necessary if some leading
+ coefficients vanish after the operation. In replacement text of
+ ``\macro``, ``\index`` expands to the coefficient index (which is
+ defined to be zero for the constant term).
+
Notice that ``\macro`` will have to handle inputs of the shape
- ``A/B[N]`` (xintfrac internal notation). This means that it probably
+ ``A/B[N]`` (xintfrac_ internal notation). This means that it probably
will have to be expressed in terms of macros from xintfrac package.
Example::
@@ -236,86 +377,285 @@
\def\foo#1{\xintMul{#1}{\the\numexpr\index*\index\relax}}
- (or with ``\xintSqr{\xindex}``) to replace ``n``-th coefficient
+ (or with ``\xintSqr{\index}``) to replace ``n``-th coefficient
``f_n`` by ``f_n*n^2``.
-``\PolReduceCoeffs{name}``
- About the same as ``\PolMapCoeffs{\xintIrr}{name}`` (but adds
- ``[0]`` postfix which speeds up xintfrac operations when
- evaluating.)
+``\PolReduceCoeffs{polname}``
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+ About the same as ``\PolMapCoeffs{\xintIrr}{polname}`` (but
+ maintaining a ``[0]`` postfix for speedier xintfrac_ parsing when
+ polynomial function is used for computations.) This is a
+ one-argument macro, working 'in-place'.
+
Expandable macros
-----------------
All these macros expand completely in two steps except ``\PolToExpr``
-which needs a ``\write``, ``\edef`` or a ``\csname...\endcsname`` context.
+and ``\PolToFloatExpr`` (and their auxiliaries) which need a
+``\write``, ``\edef`` or a ``\csname...\endcsname`` context.
-``\PolEval{name}\At{value}``
- It boils down to ``\xinttheexpr reduce(name(value))\relax``.
+``\PolEval{polname}\At{numerical expression}``
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
-``\PolNthCoeff{name}{N}``
- It expands to the raw ``N``-th coefficient (``0/1[0]`` if index is
- out of range). With ``N=-1``, ``-2``, ... expands to the leading
- coefficients.
+ It boils down to ``\xinttheexpr polname(numerical expression)\relax``.
-``\PolDegree{name}``
+ .. note::
+
+ The ``0.2`` version stupidly tried to be clever and as a result
+ of a misguided optimization choked if ``value`` was not a number
+ but a numerical expression (a sum e.g.), but the more powerful
+ behaviour has been reinstored at ``0.3`` release.
+
+ The ``0.1`` and ``0.2`` version did a ``reduce`` which however is
+ costly on big fractions and irrelevant if the output is served as
+ argument of ``\xintRound`` or ``\xintFloat``. Thus ``reduce`` was
+ removed, and former meaning is now available as
+ `\\PolEvalReduced{polname}\\At{numerical expression}`_
+
+``\PolEvalReduced{polname}\At{numerical expression}``
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+ Boils down to ``\xinttheexpr reduce(polname(numerical expression))\relax``.
+
+``\PolFloatEval{polname}\At{numerical expression}``
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+ Boils down to ``\xintthefloatexpr polname(numerical expression)\relax``.
+
+ This is done via a Horner Scheme (see `\\poldef <poldef;_>`_), with
+ already rounded coefficients. [#]_ To use the *exact coefficients*
+ (and *exact* additions and multiplications), just insert it in the
+ float expression as in this example: [#]_
+
+ ::
+
+ \xintthefloatexpr 3.27*\xintexpr f(2.53)\relax^2\relax
+
+ The ``f(2.53)`` is exactly computed then rounded at the time of
+ getting raised to the power ``2``. Moving the ``^2`` inside, that
+ operation would also be treated exactly.
+
+ .. [#] Anyway each floating point operation starts by rounding its
+ operands to the floating point precision.
+
+ .. [#] The ``\xintexpr`` could be ``\xinttheexpr`` but that would be
+ less efficient. Cf. xintexpr_ documentation about nested
+ expressions.
+
+``\PolNthCoeff{polname}{number}``
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+ It expands to the raw ``N``-th coefficient (``0/1[0]`` if the index
+ number is out of range). With ``N=-1``, ``-2``, ... expands to the
+ leading coefficients.
+
+``\PolDegree{polname}``
+~~~~~~~~~~~~~~~~~~~~~~~
+
It expands to the degree. This is ``-1`` if zero polynomial but this
may change in future. Should it then expand to ``-\infty`` ?
-``\PolToExpr{f}``
- Expands to ``f_0 + f_1*x + f_2*x^2 + ...`` (ascending powers). [1]_,
- [2]_
+``\PolToExpr{polname}``
+~~~~~~~~~~~~~~~~~~~~~~~
- .. [1] in a ``\write``, ``\edef``, or ``\csname...\endcsname``, but
+ Expands [#]_ to ``coeff_N*x^N+...`` (descending powers.)
+
+ .. [#] in a ``\write``, ``\edef``, or ``\csname...\endcsname``, but
not under ``\romannumeral-`0``.
- .. [2] the letter ``x`` is (in this release) not customizable.
-
- By default zero coefficients are skipped (issue ``\poltoexprtrue`` to
+ By default zero coefficients are skipped (issue ``\poltoexpralltrue`` to
get all of them in output).
- No ``+`` sign before negative coefficients, for compliance with Maple
- input format. This means though that parsing the result back via
- naive delimited macros is difficult, see ``\PolToList`` and ``\PolToCSV``
- for more low-level formats making it easier to get expandably some
- output of one's choice, which may possibly be parsed later on by
- other macros of one's design, or from other packages.
+ By default, no ``+`` sign before negative coefficients, for
+ compliance with Maple input format (but see
+ `\\PolToExprTermPrefix{raw_coeff}`_.) Also, like the default
+ behaviour of `\\PolTypeset{polname}`_, does not print (for the non
+ constant terms) coefficients equal to plus or minus one. The degree
+ one monomial is output as ``x``, not ``x^1``. Complete customization is
+ possible, see next macros.
- Of course ``\PolToExpr{f}`` can be inserted in a \poldef, as the
+ Of course ``\PolToExpr{f}`` can be inserted in a ``\poldef``, as the
latter expands token by token, hence will force complete expansion
- of ``\PolToExpr{f}``, but simply ``f(x)`` will be more efficient for the
- identical result.
+ of ``\PolToExpr{f}``, but a simple ``f(x)`` is more efficient for
+ the identical result.
- ``\PolToExprCmd`` is the one-argument macro used by ``\PolToExpr`` for the
- coefficients, it defaults to ``\xintPRaw{\xintRawWithZeros{#1}}``. One
- will have to redefine it to use ``\xintIrr{#1}`` in place of
- ``\xintRawWithZeros{#1}`` to get in output reduced coefficients.
+``\PolToExprOneTerm{raw_coeff}{number}``
+^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-``\PolToList{f}``
- Expands to ``{f_0}{f_1}...{f_N}`` with ``N`` = degree of f (except
- zero polynomial which does give ``{0/1[0]}`` and not an empty
- output.)
+ This two argument expandable command takes care of the monomial and
+ its coefficient. The default definition is done in order for
+ coefficients of absolute value ``1`` not be printed explicitely
+ (except of course for the constant term). Also by default, the
+ monomial of degree one is ``x`` not ``x^1``, and ``x^0`` is skipped.
-``\PolToCSV{f}``
- Expands to ``f_0, f_1, f_2, ....., f_N``. Converse to
- ``\PolFromCSV``.
+ For compatibility with Maple input requirements, by default a ``*``
+ always precedes the ``x^number``, except if the coefficient is a one
+ or a minus one. See `\\PolToExprTimes`_.
+``\PolToExprOneTermStyleB{raw_coeff}{number}``
+^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+
+ For output in this style::
+
+ 2*x^11/3+3*x^8/7-x^5−x^4/4−x^3−x^2/2−2*x+1
+
+ issue ``\let\PolToExprOneTerm\PolToExprOneTermStyleB`` before using
+ ``\PolToExpr``. Note that then ``\PolToExprCmd`` isn't used at all.
+
+ To suppress the ``*``'s, cf. `\\PolToExprTimes`_.
+
+``\PolToExprCmd{raw_coeff}``
+^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+
+ It is the one-argument macro used by the package definition of
+ ``\PolToExprOneTerm`` for the coefficients themselves (when not
+ equal to plus or minus one), and it defaults to
+ ``\xintPRaw{\xintRawWithZeros{#1}}``. One will have to redefine it
+ to ``\xintIrr{#1}`` or to ``\xintPRaw{\xintIrr{#1}}`` to obtain in the
+ output forcefully reduced coefficients.
+
+``\PolToExprTermPrefix{raw_coeff}``
+^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+
+ Defined identically as `\\PolTypesetCmdPrefix{raw_coeff}`_. It
+ prefixes with a plus sign for non-negative coefficients, because
+ they don't carry one by themselves.
+
+``\PolToExprVar``
+^^^^^^^^^^^^^^^^^
+
+ This expands to the variable to use in output (it does not have to
+ be a single letter, may be an expandable macro.) Initial definition
+ is ``x``.
+
+``\PolToExprTimes``
+^^^^^^^^^^^^^^^^^^^
+
+ This expands to the symbol used for multiplication of an
+ ``x^{number}`` by the corresponding coefficient. The default is
+ ``*``. Redefine the macro to expand to nothing to get rid of it (but
+ this will give output incompatible with some professional computer
+ algebra software).
+
+``\PolToExpr*{polname}``
+~~~~~~~~~~~~~~~~~~~~~~~~
+
+ Expands to ``coeff_0+coeff_1*x+coeff_2*x^2+...`` (ascending powers).
+ Customizable like `\\PolToExpr{polname}`_ via the same macros.
+
+``\PolToFloatExpr{polname}``
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+ Similar to `\\PolToExpr{polname}`_ but uses `\\PolToFloatExprCmd
+ <\\PolToFloatExprCmd{raw_coeff}>`_
+ which by default rounds and converts the coefficients to floating
+ point format.
+
+ .. note::
+
+ The polynomial function for usage in ``\xintfloatexpr`` is
+ already prepared with the rounded coefficients, but the latter
+ are not easily recoverable (and especially not expandably) from
+ this. Thus ``\PolToFloatExprCmd`` operates from the *exact*
+ coefficients anew. This means though that if the prevailing float
+ precision was changed with ``\xintDigits:=P;`` syntax, the output
+ will obey this precision ``P``, but the polynomial function was
+ defined earlier and operates on floating point numbers with
+ coefficients which were rounded at time of definition.
+
+ This may change in future, if the pre-rounded coefficients are
+ stored in a more easily accessible data structure.
+
+``\PolToFloatExprOneTerm{raw_coeff}{number}``
+^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+
+ Similar to `\\PolToExprOneTerm
+ <\\PolToExprOneTerm{raw_coeff}{number}>`_. But does not treat
+ especially coefficients equal to plus or minus one.
+
+``\PolToFloatExprCmd{raw_coeff}``
+^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+
+ It is the one-argument macro used by ``\PolToFloatExprOneTerm``.
+ Its package definition is ``\xintFloat{#1}``.
+
+ .. caution::
+
+ Currently (xint_ ``1.2p``) ``\xintFloat{0}`` outputs ``0.e0``
+ which is perfectly acceptable input for Python, but not for
+ Maple. Thus, one should better leave the `\\poltoexprallfalse`_
+ toggle to its default ``\iffalse`` state, if one intends to use
+ the output in a Maple worksheet.
+
+ But even then the zero polynomial will cause a problem. Workaround::
+
+ \renewcommand\PolToFloatExprCmd[1]{\xintiiifZero{#1}{0.0}{\xintFloat{#1}}}
+
+ Usage of ``\xintiiifZero`` and not ``\xintifZero`` is only for
+ optimization (I can't help it) because ``#1`` is known to be
+ in ``xintfrac`` raw format.
+
+``\PolToFloatExpr*{polname}``
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+ Typesets in ascending powers.
+
+``\PolToList{polname}``
+~~~~~~~~~~~~~~~~~~~~~~~
+
+ Expands to ``{coeff_0}{coeff_1}...{coeff_N}`` with ``N`` = degree
+ (except zero polynomial which does give ``{0/1[0]}`` and not an
+ empty output.)
+
+``\PolToCSV{polname}``
+~~~~~~~~~~~~~~~~~~~~~~
+
+ Expands to ``coeff_0, coeff_1, coeff_2, ....., coeff_N``. Converse
+ to `\\PolFromCSV <\\PolFromCSV{polname}{\<csv\>}_>`_.
+
+Booleans (with default setting as indicated)
+--------------------------------------------
+
+``\xintverbosefalse``
+~~~~~~~~~~~~~~~~~~~~~
+
+ This is actually an xintexpr_ configuration. Setting it to
+ ``true`` triggers the writing of information to the log when new
+ polynomials are defined.
+
+ .. caution::
+
+ The macro meanings as written to the log are to be considered
+ unstable and undocumented internal structures.
+
+``\poltypesetallfalse``
+~~~~~~~~~~~~~~~~~~~~~~~
+
+ If ``true``, `\\PolTypeset{polname}`_ will also typeset the vanishing
+ coefficients.
+
+
+``\poltoexprallfalse``
+~~~~~~~~~~~~~~~~~~~~~~
+
+ If ``true``, `\\PolToExpr{polname}`_ and `\\PolToFloatExpr{polname}`_ will
+ also include the vanishing coefficients in their outputs.
+
+
Technicalities
--------------
-- The catcode of the semi-colon is reset temporarily by ``\poldef`` macro in
- case some other package (for example the French babel module) may have
- made it active. This will fail though if the whole thing was already
- part of a macro argument, in such cases one can use ``\PolDef`` rather.
- The colon in ``:=`` may be active with no consequences.
+- The catcode of the semi-colon is reset temporarily by `\\poldef
+ <poldef;_>`_ macro in case some other package (for example the French
+ babel module) may have made it active. This will fail though if the
+ whole thing was already part of a macro argument, in such cases one
+ can use `\\PolDef{f}{P(x)} <PolDef_>`_
+ rather. The colon in ``:=`` may be active with no consequences.
-- Beware the ``1/2 x`` problem: as mentioned above, it will be give zero due
- to the tacit multiplication rules of ``\xintexpr`` and to the fact that
- the package will do the Euclidean division of ``1`` by polynomial ``2x``.
-
- During execution of polynomial operations by ``\poldef`` (but not
during the initial purely numerical parsing of the expression), the
- xintfrac macro ``\xintAdd`` is temporarily patched to always express
+ xintfrac_ macro ``\xintAdd`` is temporarily patched to always express
``a/b + c/d`` with ``l.c.m.(b,d)`` as denominator. Indeed the current
(xint 1.2p) ``\xintAdd`` uses ``(ad+bc)/bd`` formula except if ``b``
divides ``d`` or ``d`` divides ``b``, which quickly leads in real life
@@ -343,36 +683,44 @@
where all coefficients have the same denominator 6 (which in this
example is the ``l.c.m`` of the denominators of the reduced coefficients.)
-- ``\PolDiff`` always applies ``\xintIrr`` to the resulting coefficients, except
- that the *power of ten* part ``[N]`` (for example an input in scientific
- notation such as ``1.23e5`` gives ``123/1[3]`` internally in xintfrac) is not
- taken into account in the reduction of the fraction. This is tentative
- and may change.
+- `\\PolDiff{polname_1}{polname_2}`_ always applies ``\xintIrr`` to the
+ resulting coefficients, except that the *power of ten* part ``[N]``
+ (for example an input in scientific notation such as ``1.23e5`` gives
+ ``123/1[3]`` internally in xintfrac) is not taken into account in the
+ reduction of the fraction. This is tentative and may change.
- Same remark for ``\PolAntiDiff``.
+ Same remark for `\\PolAntiDiff{polname_1}{polname_2}`_.
-- If ``f`` was created from comma separated values by macro \PolFromCSV,
- then the coefficients will be in the output of ``\PolToList`` and
- ``\PolToCSV`` in the same format as originally in input: a ``1.3e2``
- will again be a ``1.3e2``.
-
+- If ``f`` was created from comma separated values by macro
+ `\\PolFromCSV{polname}{\<csv\>}`_, then the coefficients will be in
+ the output of `\\PolToList{polname}`_ and `\\PolToCSV{polname}`_ in
+ the same format as originally in input: a ``1.3e2`` will again be a
+ ``1.3e2``.
+
In contrast when such coefficients are used in a ``\poldef`` (or
``\PolDef``) expression, they get transformed during the parsing to
- the xintfrac *raw* format. This is an unavoidable consequence of usage
- by \poldef of ``\xintdeffunc`` which itself is based on ``\xintexpr.``
+ the xintfrac *raw* format.
This *raw* format speeds up expansion of xintfrac macros for numerical
evaluations.
-- Currently, the package does not as a result of ``\poldef`` add to the TeX
- memory an already pre-computed *array* structure for the polynomial
- coefficients, as would be constructed by ``\PolAssign{f}\toarray\macro``.
- Such structures are used, but for internal calculations in temporarily
- restricted scopes. Apart from the function ``f()`` known to the
- (numerical) ``\xintexpr`` parser (whose meaning can be found in the log
- file after \xintverbosetrue), the data is (currently) stored in a
- single other macro encapsulating the degree, and the coefficients as a
- list. This may evolve in future.
+- Currently, the package stores all coefficients from index ``0`` to
+ index equal to the polynomial degree inside a single macro, as a list.
+ This data structure is obviously very inefficient for polynomials of
+ high degree and few coefficients (as an example with ``\poldef
+ f(x):=x^1000 + x^500;`` the subsequent definition ``\poldef g(x):=
+ f(x)^2;`` will do of the order of 1,000,000 multiplications and
+ additions involvings only zeroes... which does take time). This
+ may change in the future.
+- Tests have been made with Newton's iteration (for which computing
+ exactly the derivative is precisely what this package is made for) or
+ Regula Falsi method for locating roots: using exact computations leads
+ quickly to gigantic fractions (but dichotomy method much less so). It
+ is thus recommended to use ``\xintdeffloatvar`` or
+ ``\xintthefloatexpr`` contexts for any kind of numerical mathematics.
+ Of course, exact computations are invaluable for number theory or
+ combinatorics...
+
- As is to be expected internal structures of the package are barely
documented and unstable. Don't use them.
@@ -379,11 +727,25 @@
RELEASES
--------
-- 0.1 (2018/01/11): initial release (files README, polexpr.sty).
-- 0.2 (2018/01/14): documentation moved to polexpr.{txt,html}.
+- 0.1 (2018/01/11)
-Files of 0.2 release:
+ Initial release (files README, polexpr.sty).
+- 0.2 (2018/01/14)
+
+ Documentation moved to polexpr.{txt,html}.
+
+- 0.3 (2018/01/17)
+
+ Make polynomials known to ``\xintfloatexpr`` and improve
+ documentation.
+
+- 0.3.1 (2018/01/18)
+
+ Fix two typos in documentation.
+
+Files of 0.3.1 release:
+
- README.md,
- polexpr.sty (package file),
- polexpr.txt (documentation),
@@ -391,4 +753,17 @@
__ http://docutils.sourceforge.net/docs/index.html
-See README.md for the License and the change log.
+See README.md for the License and the change log (there were
+some breaking changes from 0.2 to 0.3).
+
+Acknowledgments
+---------------
+
+Thanks to Jürgen Gilg whose question about xint_ usage for
+differentiating polynomials was the initial trigger leading to this
+package, and to Jürgen Gilg and Thomas Söll for testing it on some
+concrete problems.
+
+.. _xintfrac:
+.. _xintexpr:
+.. _xint: http://www.ctan.org/pkg/xint
Modified: trunk/Master/texmf-dist/tex/latex/polexpr/polexpr.sty
===================================================================
--- trunk/Master/texmf-dist/tex/latex/polexpr/polexpr.sty 2018-01-18 23:25:12 UTC (rev 46376)
+++ trunk/Master/texmf-dist/tex/latex/polexpr/polexpr.sty 2018-01-18 23:25:24 UTC (rev 46377)
@@ -1,7 +1,7 @@
% author: Jean-François Burnol
% License: LPPL 1.3c (author-maintained)
\ProvidesPackage{polexpr}%
- [2018/01/14 v0.2 Polynomial expressions with rational coefficients (JFB)]%
+ [2018/01/18 v0.3.1 Polynomial expressions with rational coefficients (JFB)]%
\RequirePackage{xintexpr}[2016/03/19]% xint 1.2g (or 1.2c 2015/11/16 at least)
\edef\POL at restorecatcodes
{\catcode`\noexpand\_ \the\catcode`\_ \catcode0 \the\catcode0\relax}%
@@ -76,12 +76,16 @@
%%
\def\POL at newpol#1{%
\expandafter\POL at ifZero\csname POLuserpol@#1\endcsname
- {\@namedef{XINT_expr_userfunc_#1}##1,{0/1[0]}}%
+ {\@namedef{XINT_expr_userfunc_#1}##1,{0/1[0]}%
+ \@namedef{XINT_flexpr_userfunc_#1}##1,{0[0]}}%
{\POL at newpolhorner{#1}}%
\unless\ifcsname XINT_expr_userfuncNE:#1\endcsname\POL at addtoextras{#1}\fi
\expandafter\XINT_expr_defuserfunc
\csname XINT_expr_func_#1\expandafter\endcsname
\csname XINT_expr_userfunc_#1\endcsname
+ \expandafter\XINT_expr_defuserfunc
+ \csname XINT_flexpr_func_#1\expandafter\endcsname
+ \csname XINT_flexpr_userfunc_#1\endcsname
\ifxintverbose\POL at info{#1}\fi
}%
\def\POL at info #1{%
@@ -90,7 +94,11 @@
associated to \string\XINT_expr_userfunc_#1\space
whose meaning uses Horner scheme:
\expandafter\meaning
- \csname XINT_expr_userfunc_#1\endcsname}%
+ \csname XINT_expr_userfunc_#1\endcsname^^J%
+ \@spaces And \string\XINT_flexpr_userfunc_#1\space for
+ the \string\xintfloatexpr\space parser has meaning
+ \expandafter\meaning
+ \csname XINT_flexpr_userfunc_#1\endcsname}%
}%
%
\def\POL at newpolhorner#1{%
@@ -102,17 +110,37 @@
\edef\POL at var@coeffs{\xintRevWithBraces{\POL at var@coeffs}}%
\begingroup
\expandafter\POL at newpol@horner\POL at var@coeffs\relax
+ \expandafter
\endgroup
- \expandafter\let\csname XINT_expr_userfunc_#1\endcsname\POL at tmp
+ \expandafter\def\csname XINT_expr_userfunc_#1\expandafter\endcsname
+ \expandafter##\expandafter1\expandafter,\expandafter{\POL at tmp{##1}}%
+ \begingroup
+ \expandafter\POL at newpol@floathorner\POL at var@coeffs\relax
+ \expandafter
+ \endgroup
+ \expandafter\def\csname XINT_flexpr_userfunc_#1\expandafter\endcsname
+ \expandafter##\expandafter1\expandafter,\expandafter{\POL at tmp{##1}}%
}%
\def\POL at newpol@horner#1{\let\xintAdd\relax\let\xintMul\relax
- \gdef\POL at tmp##1,{#1}\POL at newpol@horner at loop.}%
+ \def\POL at tmp##1{#1}\POL at newpol@horner at loop.}%
\def\POL at newpol@horner at loop.#1{%
\if\relax#1\expandafter\xint_gob_til_dot\fi
- \xdef\POL at tmp##1,{\xintiiifZero{#1}{\@firstofone}{\xintAdd{#1}}%
- {\xintMul{##1}{\POL at tmp##1,}}}%
+ \edef\POL at tmp##1{\xintiiifZero{#1}
+ {\@firstofone}{\xintAdd{#1}}%
+ {\xintMul{##1}{\POL at tmp{##1}}}}%
\POL at newpol@horner at loop.%
}%
+\def\POL at newpol@floathorner#1{\let\XINTinFloatAdd\relax\let\XINTinFloatMul\relax
+ \def\xintAdd{\XINTinFloatAdd}\def\xintMul{\XINTinFloatMul}%
+ \edef\POL at tmp##1{\XINTinFloatdigits{#1}}%
+ \POL at newpol@floathorner at loop.}%
+\def\POL at newpol@floathorner at loop.#1{%
+ \if\relax#1\expandafter\xint_gob_til_dot\fi
+ \edef\POL at tmp##1{\xintiiifZero{#1}
+ {\@firstofone}{\xintAdd{\XINTinFloatdigits{#1}}}%
+ {\xintMul{##1}{\POL at tmp{##1}}}}%
+ \POL at newpol@floathorner at loop.%
+}%
%% Customizes xintexpr.sty's \XINT_NewExpr (\POL at addtoextras{name})
\begingroup
\catcode`~ 12
@@ -150,15 +178,24 @@
\def\POL at redefinemacros{\POL at original_redefinemacros\POL at redefineextras}%
\let\POL at redefineextras\@empty
%
-\newcommand\PolLet[2]{%
+\newcommand\PolLet[2]{\if=\noexpand#2\expandafter\xint_firstoftwo
+ \else\expandafter\xint_secondoftwo\fi
+ \POL@@let\POL at let {#1}{#2}}%
+\def\POL@@let#1#2#3{\POL at let{#1}{#3}}%
+\def\POL at let#1#2{%
\expandafter\let\csname POLuserpol@#1\expandafter\endcsname
\csname POLuserpol@#2\endcsname
\unless\ifcsname XINT_expr_userfuncNE:#1\endcsname\POL at addtoextras{#1}\fi
\expandafter\let\csname XINT_expr_userfunc_#1\expandafter\endcsname
\csname XINT_expr_userfunc_#2\endcsname
+ \expandafter\let\csname XINT_flexpr_userfunc_#1\expandafter\endcsname
+ \csname XINT_flexpr_userfunc_#2\endcsname
\expandafter\XINT_expr_defuserfunc
\csname XINT_expr_func_#1\expandafter\endcsname
\csname XINT_expr_userfunc_#1\endcsname
+ \expandafter\XINT_expr_defuserfunc
+ \csname XINT_flexpr_func_#1\expandafter\endcsname
+ \csname XINT_flexpr_userfunc_#1\endcsname
\ifxintverbose\POL at info{#1}\fi
}%
@@ -230,12 +267,12 @@
\newif\ifpoltypesetall
-\newcommand\PolTypesetPlus[1]{\xintiiifSgn{#1}{}{+}{+}}%
+\newcommand\PolTypesetCmdPrefix[1]{\xintiiifSgn{#1}{}{+}{+}}%
\newcommand\PolTypesetCmd[1]{\xintifOne{\xintiiAbs{#1}}%
{\ifnum\PolIndex=\z@\xintiiSgn{#1}\else
\xintiiifSgn{#1}{-}{}{}\fi}%
{\xintSignedFrac{#1}}}%
-\newcommand\PolTypesetMonomial{%
+\newcommand\PolTypesetMonomialCmd{%
\ifcase\PolIndex\space
%
\or\PolVar
@@ -262,12 +299,12 @@
\fi
\def\PolVar{#1}%
\ifnum\POL at var@deg<\z@
- \PolTypesetCmd{0/1[0]}%
+ \PolTypesetCmd{0/1[0]}\PolTypesetMonomialCmd
\else
\ifnum\POL at var@deg=\z@
- \expandafter\PolTypesetCmd\POL at var@coeffs
+ \expandafter\PolTypesetCmd\POL at var@coeffs\PolTypesetMonomialCmd
\else
- \def\POL at ts@plus##1{\let\POL at ts@plus\PolTypesetPlus}%
+ \def\POL at ts@prefix##1{\let\POL at ts@prefix\PolTypesetCmdPrefix}%
\expandafter\POL at ts@loop
\romannumeral-`0\POL at ts@reverse{\POL at var@coeffs}\relax
\fi
@@ -282,12 +319,12 @@
\if\relax#1\expandafter\xint_gob_til_dot\fi
\xintiiifZero{#1}%
{}%
- {\POL at ts@plus{#1}\PolTypesetCmd{#1}\PolTypesetMonomial}%
+ {\POL at ts@prefix{#1}\PolTypesetCmd{#1}\PolTypesetMonomialCmd}%
\edef\PolIndex{\the\numexpr\PolIndex+\POL@@ne at or@m at ne}\POL at ts@check.%
}%
\def\POL at ts@nocheck.#1{%
\if\relax#1\expandafter\xint_gob_til_dot\fi
- \POL at ts@plus{#1}\PolTypesetCmd{#1}\PolTypesetMonomial
+ \POL at ts@prefix{#1}\PolTypesetCmd{#1}\PolTypesetMonomialCmd
\edef\PolIndex{\the\numexpr\PolIndex+\POL@@ne at or@m at ne}\POL at ts@nocheck.%
}%
@@ -495,7 +532,7 @@
\else\xint_afterfi{\POL at Iterate\POL at Diff@one}%
\fi
}%
-\def\POL at Diff@no #1#2{\PolLet{#2}{#1}}%
+\def\POL at Diff@no #1#2{\POL at let{#2}{#1}}%
\def\POL at Diff@one #1#2{%
\expandafter\expandafter\expandafter\POL at split
\csname POLuserpol@#1\endcsname;\POL at var@deg\POL at var@coeffs
@@ -920,11 +957,17 @@
%% EXPANDABLE MACROS
\newcommand\PolEval{}%
-\def\PolEval#1#2\At#3{%
+\def\PolEval#1#2\At#3{\romannumeral`^^@\xinttheexpr #1(#3)\relax}%
+%
+\newcommand\PolEvalReduced{}%
+\def\PolEvalReduced#1#2\At#3{%
\romannumeral0\xintpraw % only serves to not print denominator if = 1
- {\xintIrr{\csname XINT_expr_userfunc_#1\endcsname {#3},}[0]}%
+ {\xintIrr{\romannumeral`^^@\xintthebareeval#1(#3)\relax}[0]}%
}%
%
+\newcommand\PolFloatEval{}%
+\def\PolFloatEval#1#2\At#3{\romannumeral`^^@\xintthefloatexpr #1(#3)\relax}%
+%
\newcommand\PolNthCoeff[2]{\romannumeral`^^@%
\expandafter\POL at nthcoeff
\romannumeral0\xintnthelt{\ifnum\numexpr#2<\z@#2\else(#2)+1\fi}%
@@ -949,34 +992,119 @@
\newif\ifpoltoexprall
\newcommand\PolToExprCmd[1]{\xintPRaw{\xintRawWithZeros{#1}}}%
-\newcommand\PolToExprCmdWithSep[1]{\xintiiifSgn{#1}{}{+}{+}\PolToExprCmd{#1}}%
-\newcommand\PolToExpr[1]{\expandafter\expandafter\expandafter
- \POL at toexpr\csname POLuserpol@#1\endcsname\relax}%
+\newcommand\PolToFloatExprCmd[1]{\xintFloat{#1}}%
+\let\PolToExprTermPrefix\PolTypesetCmdPrefix
+\newcommand\PolToExprOneTerm[2]{%
+ \ifnum#2=\z@
+ \PolToExprCmd{#1}%
+ \else
+ \xintifOne{\xintiiAbs{#1}}
+ {\xintiiifSgn{#1}{-}{}{}}% + from \PolToExprTermPrefix
+ {\PolToExprCmd{#1}\PolToExprTimes}%
+ \fi
+ \ifcase\xintiiAbs{#2} %<-- space here mandatory
+ \or\PolToExprVar
+ \else\PolToExprVar^\xintiiAbs{#2}%
+ \fi
+}%
+\newcommand\PolToExprOneTermStyleB[2]{%
+ \ifnum#2=\z@
+ \xintNumerator{#1}%
+ \else
+ \xintifOne{\xintiiAbs{\xintNumerator{#1}}}
+ {\xintiiifSgn{#1}{-}{}{}}% + from \PolToExprTermPrefix
+ {\xintNumerator{#1}\PolToExprTimes}%
+ \fi
+ \ifcase\xintiiAbs{#2} %<-- space here mandatory
+ \or\PolToExprVar
+ \else\PolToExprVar^\xintiiAbs{#2}%
+ \fi
+ \xintiiifOne{\xintDenominator{#1}}{}{/\xintDenominator{#1}}%
+}%
+\newcommand\PolToFloatExprOneTerm[2]{%
+ \ifnum#2=\z@
+ \PolToFloatExprCmd{#1}%
+ \else
+ \PolToFloatExprCmd{#1}\PolToExprTimes
+ \fi
+ \ifcase\xintiiAbs{#2} %<-- space here mandatory
+ \or\PolToExprVar
+ \else\PolToExprVar^\xintiiAbs{#2}%
+ \fi
+}%
+\newcommand\PolToExprTimes{*}%
+\newcommand\PolToExprVar{x}%
+\newcommand\PolToExpr[1]{%
+ \if*\noexpand#1\expandafter\xint_firstoftwo\else
+ \expandafter\xint_secondoftwo\fi
+ \PolToExprAscending\PolToExprDescending{#1}}%
+\newcommand\PolToFloatExpr[1]{%
+ \if*\noexpand#1\expandafter\xint_firstoftwo\else
+ \expandafter\xint_secondoftwo\fi
+ \PolToFloatExprAscending\PolToFloatExprDescending{#1}}%
+\newcommand\PolToExprAscending[2]{%
+ \expandafter\POL at toexpr\csname POLuserpol@#2\endcsname
+ \PolToExprOneTerm\POL at toexprA}%
+\newcommand\PolToFloatExprAscending[2]{%
+ \expandafter\POL at toexpr\csname POLuserpol@#2\endcsname
+ \PolToFloatExprOneTerm\POL at toexprA}%
+\newcommand\PolToExprDescending[1]{%
+ \expandafter\POL at toexpr\csname POLuserpol@#1\endcsname
+ \PolToExprOneTerm\POL at toexprD}%
+\newcommand\PolToFloatExprDescending[1]{%
+ \expandafter\POL at toexpr\csname POLuserpol@#1\endcsname
+ \PolToFloatExprOneTerm\POL at toexprD}%
%
-\def\POL at toexpr#1.\empty{%
- \ifnum#1<\z@
- \PolToExprCmd{0/1[0]}\expandafter\@gobbletwo
+\def\POL at toexpr#1#2#3{\expandafter\POL at toexpr@
+ \expandafter#3\expandafter#2#1\relax}%
+\def\POL at toexpr@#1#2#3.{%
+ \ifnum#3<\z@
+ #2{0/1[0]}{0}\expandafter\xint_gobble_v
\else
- \expandafter\POL at toexpr@a
- \fi}%
+ \expandafter#1%
+ \fi {#3}#2}%
%
-\def\POL at toexpr@a #1{\ifpoltoexprall\expandafter\POL at toexprall@b
- \else\expandafter\POL at toexpr@b
- \fi{#1}{}\PolToExprCmd0.}%
-%
+\def\POL at toexprA #1#2\empty#3{%
+ \ifpoltoexprall\expandafter\POL at toexprall@b
+ \else\expandafter\POL at toexpr@b
+ \fi {#3}#2{0}1.%
+}%
+\def\POL at toexprD #1#2#3\relax{% #3 has \empty to prevent brace removal
+ \expandafter\POL at toexprD@a\expandafter#2%
+ \the\numexpr #1\expandafter.\romannumeral0\xintrevwithbraces{#3}\relax
+}%
+\def\POL at toexprD@a #1#2.#3{%
+ \ifpoltoexprall\expandafter\POL at toexprall@b
+ \else\expandafter\POL at toexpr@b
+ \fi{#3}#1{-#2}\the\numexpr\@ne+-#2.%
+}%
\def\POL at toexpr@b #1#2#3{%
\xintiiifZero{#1}%
- {\expandafter\POL at toexpr@loop\expandafter#3}%
- {#3{#1}#2\expandafter\POL at toexpr@loop\expandafter\PolToExprCmdWithSep}%
- \expandafter\POL at toexpr@b\the\numexpr\@ne+%
+ {\expandafter\POL at toexpr@loop\expandafter\POL at toexpr@b}%
+ {#2{#1}{#3}%
+ \expandafter\POL at toexpr@loop\expandafter\POL at toexpr@c}%
+ \expandafter#2%
}%
+\def\POL at toexpr@c #1#2#3{%
+ \xintiiifZero{#1}%
+ {}%
+ {\PolToExprTermPrefix{#1}#2{#1}{#3}}%
+ \expandafter\POL at toexpr@loop\expandafter\POL at toexpr@c
+ \expandafter#2%
+}%
\def\POL at toexprall@b #1#2#3{%
- #3{#1}#2%
- \expandafter\POL at toexpr@loop\expandafter\PolToExprCmdWithSep
- \expandafter\POL at toexprall@b\the\numexpr\@ne+%
+ #2{#1}{#3}%
+ \expandafter\POL at toexpr@loop\expandafter\POL at toexprall@c
+ \expandafter#2%
}%
+\def\POL at toexprall@c #1#2#3{%
+ \PolToExprTermPrefix{#1}#2{#1}{#3}%
+ \expandafter\POL at toexpr@loop\expandafter\POL at toexprall@c
+ \expandafter#2%
+}%
\def\POL at toexpr@loop#1#2#3.#4{%
- \if\relax#4\expandafter\xint_gob_til_dot\fi #2{#4}{*x^#3}#1#3.%
+ \if\relax#4\expandafter\xint_gob_til_dot\fi
+ #1{#4}#2{#3}\the\numexpr\@ne+#3.%
}%
More information about the tex-live-commits
mailing list