texlive[46653] Master/texmf-dist: polexpr (16feb18)
commits+karl at tug.org
commits+karl at tug.org
Fri Feb 16 23:30:10 CET 2018
Revision: 46653
http://tug.org/svn/texlive?view=revision&revision=46653
Author: karl
Date: 2018-02-16 23:30:09 +0100 (Fri, 16 Feb 2018)
Log Message:
-----------
polexpr (16feb18)
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 2018-02-16 01:23:20 UTC (rev 46652)
+++ trunk/Master/texmf-dist/doc/latex/polexpr/README.md 2018-02-16 22:30:09 UTC (rev 46653)
@@ -42,66 +42,41 @@
package via their coefficients. This allows dedicated macros to
implement polynomial algorithmics.
-CHANGE LOG
-----------
+Releases
+--------
-- v0.1 (2018/01/11): initial release. Features:
- - The `\poldef` parser itself,
- - Differentiation and anti-differentiation,
- - Euclidean division and GCDs,
- - 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.
- - Move main documentation from README to separate `polexpr.txt`
- file.
- - Provide `polexpr.html` as obtained via
- [DocUtils](http://docutils.sourceforge.net/docs/index.html)
- `rst2html.py`.
- - 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.
+- 0.1 (2018/01/11)
+ 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.
+- 0.4 (2018/02/16)
+ - Revert 0.3 automatic generation of floating point variants.
+ - Move CHANGE LOG from README.md to HTML documentation.
+ - A few bug fixes and breaking changes. Please refer to
+ `polexpr.html`.
+ - Main new feature: root localization via [Sturm
+ Theorem](https://en.wikipedia.org/wiki/Sturm%27s_theorem).
+Files of 0.4 release:
+
+- README.md,
+- polexpr.sty (package file),
+- polexpr.txt (documentation),
+- polexpr.html (conversion via
+ [DocUtils](http://docutils.sourceforge.net/docs/index.html)
+ rst2html.py)
+
+Acknowledgments
+---------------
+
+Thanks to Jürgen Gilg whose question about
+[xint](http://www.ctan.org/pkg/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.
+
+Renewed thanks on occasion of `0.4` release!
Modified: trunk/Master/texmf-dist/doc/latex/polexpr/polexpr.html
===================================================================
--- trunk/Master/texmf-dist/doc/latex/polexpr/polexpr.html 2018-02-16 01:23:20 UTC (rev 46652)
+++ trunk/Master/texmf-dist/doc/latex/polexpr/polexpr.html 2018-02-16 22:30:09 UTC (rev 46653)
@@ -362,76 +362,128 @@
<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>
+<h2 class="subtitle" id="id1">0.4 (2018/02/16)</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>
+<li><a class="reference internal" href="#basic-examples" id="id33">Basic Examples</a></li>
+<li><a class="reference internal" href="#examples-of-localization-of-roots" id="id34">Examples of localization of roots</a><ul>
+<li><a class="reference internal" href="#a-typical-example" id="id35">A typical example</a></li>
+<li><a class="reference internal" href="#a-degree-four-polynomial-with-nearby-roots" id="id36">A degree four polynomial with nearby roots</a></li>
+<li><a class="reference internal" href="#the-degree-nine-polynomial-with-0-99-0-999-0-9999-as-triple-roots" id="id37">The degree nine polynomial with 0.99, 0.999, 0.9999 as triple roots</a></li>
+<li><a class="reference internal" href="#a-mignotte-type-polynomial" id="id38">A Mignotte type polynomial</a></li>
+<li><a class="reference internal" href="#the-degree-41-polynomial-with-2-1-9-1-8-0-0-1-1-9-2-as-roots" id="id39">The degree 41 polynomial with -2, -1.9, -1.8, ..., 0, 0.1, ..., 1.9, 2 as roots</a></li>
+<li><a class="reference internal" href="#roots-of-chebyshev-polynomials" id="id40">Roots of Chebyshev polynomials</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>
+<li><a class="reference internal" href="#non-expandable-macros" id="id41">Non-expandable macros</a><ul>
+<li><a class="reference internal" href="#poldef-polname-letter-expression-in-letter" id="id42"><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="id43"><tt class="docutils literal"><span class="pre">\PolDef[letter]{polname}{expression</span> in letter}</tt></a></li>
+<li><a class="reference internal" href="#polgenfloatvariant-polname" id="id44"><tt class="docutils literal">\PolGenFloatVariant{polname}</tt></a></li>
+<li><a class="reference internal" href="#pollet-polname-2-polname-1" id="id45"><tt class="docutils literal"><span class="pre">\PolLet{polname_2}={polname_1}</span></tt></a></li>
+<li><a class="reference internal" href="#polgloballet-polname-2-polname-1" id="id46"><tt class="docutils literal"><span class="pre">\PolGlobalLet{polname_2}={polname_1}</span></tt></a></li>
+<li><a class="reference internal" href="#polassign-polname-toarray-macro" id="id47"><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="id48"><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="id49"><tt class="docutils literal"><span class="pre">\PolFromCSV{polname}{<csv>}</span></tt></a></li>
+<li><a class="reference internal" href="#poltypeset-polname" id="id50"><tt class="docutils literal">\PolTypeset{polname}</tt></a><ul>
+<li><a class="reference internal" href="#poltypesetcmd-raw-coeff" id="id51"><tt class="docutils literal">\PolTypesetCmd{raw_coeff}</tt></a></li>
+<li><a class="reference internal" href="#poltypesetone-raw-coeff" id="id52"><tt class="docutils literal">\PolTypesetOne{raw_coeff}</tt></a></li>
+<li><a class="reference internal" href="#id6" id="id53"><tt class="docutils literal">\PolTypesetMonomialCmd</tt></a></li>
+<li><a class="reference internal" href="#poltypesetcmdprefix-raw-coeff" id="id54"><tt class="docutils literal">\PolTypesetCmdPrefix{raw_coeff}</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>
+<li><a class="reference internal" href="#id8" id="id55"><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="id56"><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="id57"><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="id58"><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="id59"><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="id60"><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="#polquo-polname-1-polname-2-polname-q" id="id61"><tt class="docutils literal"><span class="pre">\PolQuo{polname_1}{polname_2}{polname_Q}</span></tt></a></li>
+<li><a class="reference internal" href="#polrem-polname-1-polname-2-polname-r" id="id62"><tt class="docutils literal"><span class="pre">\PolRem{polname_1}{polname_2}{polname_R}</span></tt></a></li>
+<li><a class="reference internal" href="#polgcd-polname-1-polname-2-polname-gcd" id="id63"><tt class="docutils literal"><span class="pre">\PolGCD{polname_1}{polname_2}{polname_GCD}</span></tt></a></li>
+<li><a class="reference internal" href="#poltosturm-polname-sturmname" id="id64"><tt class="docutils literal"><span class="pre">\PolToSturm{polname}{sturmname}</span></tt></a></li>
+<li><a class="reference internal" href="#id10" id="id65"><tt class="docutils literal"><span class="pre">\PolToSturm*{polname}{sturmname}</span></tt></a></li>
+<li><a class="reference internal" href="#polsettosturmchainsignchangesat-macro-sturmname-fraction" id="id66"><tt class="docutils literal"><span class="pre">\PolSetToSturmChainSignChangesAt{\macro}{sturmname}{fraction}</span></tt></a></li>
+<li><a class="reference internal" href="#polsettonbofzeroswithin-macro-sturmname-value-a-value-b" id="id67"><tt class="docutils literal"><span class="pre">\PolSetToNbOfZerosWithin{\macro}{sturmname}{value_a}{value_b}</span></tt></a></li>
+<li><a class="reference internal" href="#polsturmisolatezeros-sturmname" id="id68"><tt class="docutils literal">\PolSturmIsolateZeros{sturmname}</tt></a></li>
+<li><a class="reference internal" href="#polrefineinterval-sturmname-index" id="id69"><tt class="docutils literal"><span class="pre">\PolRefineInterval*{sturmname}{index}</span></tt></a></li>
+<li><a class="reference internal" href="#polrefineinterval-n-sturmname-index" id="id70"><tt class="docutils literal"><span class="pre">\PolRefineInterval[N]{sturmname}{index}</span></tt></a></li>
+<li><a class="reference internal" href="#polensureintervallength-sturmname-index-e" id="id71"><tt class="docutils literal"><span class="pre">\PolEnsureIntervalLength{sturmname}{index}{E}</span></tt></a></li>
+<li><a class="reference internal" href="#polensureintervallengths-sturmname-e" id="id72"><tt class="docutils literal"><span class="pre">\PolEnsureIntervalLengths{sturmname}{E}</span></tt></a></li>
+<li><a class="reference internal" href="#polprintintervals-varname-sturmname" id="id73"><tt class="docutils literal"><span class="pre">\PolPrintIntervals[varname]{sturmname}</span></tt></a><ul>
+<li><a class="reference internal" href="#id11" id="id74"><tt class="docutils literal">\PolPrintIntervalsPrintExactZero</tt></a></li>
+<li><a class="reference internal" href="#id12" id="id75"><tt class="docutils literal">\PolPrintIntervalsPrintLeftEndPoint</tt></a></li>
+<li><a class="reference internal" href="#id13" id="id76"><tt class="docutils literal">\PolPrintIntervalsPrintRightEndPoint</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>
+<li><a class="reference internal" href="#polmapcoeffs-macro-polname" id="id77"><tt class="docutils literal"><span class="pre">\PolMapCoeffs{\macro}{polname}</span></tt></a></li>
+<li><a class="reference internal" href="#polreducecoeffs-polname" id="id78"><tt class="docutils literal">\PolReduceCoeffs{polname}</tt></a></li>
+<li><a class="reference internal" href="#id15" id="id79"><tt class="docutils literal"><span class="pre">\PolReduceCoeffs*{polname}</span></tt></a></li>
+<li><a class="reference internal" href="#polmakemonic-polname" id="id80"><tt class="docutils literal">\PolMakeMonic{polname}</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>
+<li><a class="reference internal" href="#expandable-macros" id="id81">Expandable macros</a><ul>
+<li><a class="reference internal" href="#poleval-polname-atexpr-numerical-expression" id="id82"><tt class="docutils literal"><span class="pre">\PolEval{polname}\AtExpr{numerical</span> expression}</tt></a></li>
+<li><a class="reference internal" href="#poleval-polname-at-fraction" id="id83"><tt class="docutils literal"><span class="pre">\PolEval{polname}\At{fraction}</span></tt></a></li>
+<li><a class="reference internal" href="#polevalreduced-polname-atexpr-numerical-expression" id="id84"><tt class="docutils literal"><span class="pre">\PolEvalReduced{polname}\AtExpr{numerical</span> expression}</tt></a></li>
+<li><a class="reference internal" href="#polevalreduced-polname-at-fraction" id="id85"><tt class="docutils literal"><span class="pre">\PolEvalReduced{polname}\At{fraction}</span></tt></a></li>
+<li><a class="reference internal" href="#polfloateval-polname-atexpr-numerical-expression" id="id86"><tt class="docutils literal"><span class="pre">\PolFloatEval{polname}\AtExpr{numerical</span> expression}</tt></a></li>
+<li><a class="reference internal" href="#polfloateval-polname-at-fraction" id="id87"><tt class="docutils literal"><span class="pre">\PolFloatEval{polname}\At{fraction}</span></tt></a></li>
+<li><a class="reference internal" href="#polifcoeffisplusorminusone-a-b" id="id88"><tt class="docutils literal"><span class="pre">\PolIfCoeffIsPlusOrMinusOne{A}{B}</span></tt></a></li>
+<li><a class="reference internal" href="#polleadingcoeff-polname" id="id89"><tt class="docutils literal">\PolLeadingCoeff{polname}</tt></a></li>
+<li><a class="reference internal" href="#polnthcoeff-polname-number" id="id90"><tt class="docutils literal"><span class="pre">\PolNthCoeff{polname}{number}</span></tt></a></li>
+<li><a class="reference internal" href="#poldegree-polname" id="id91"><tt class="docutils literal">\PolDegree{polname}</tt></a></li>
+<li><a class="reference internal" href="#poltoexpr-polname" id="id92"><tt class="docutils literal">\PolToExpr{polname}</tt></a><ul>
+<li><a class="reference internal" href="#poltoexproneterm-raw-coeff-number" id="id93"><tt class="docutils literal"><span class="pre">\PolToExprOneTerm{raw_coeff}{number}</span></tt></a></li>
+<li><a class="reference internal" href="#poltoexpronetermstylea-raw-coeff-number" id="id94"><tt class="docutils literal"><span class="pre">\PolToExprOneTermStyleA{raw_coeff}{number}</span></tt></a></li>
+<li><a class="reference internal" href="#poltoexpronetermstyleb-raw-coeff-number" id="id95"><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="id96"><tt class="docutils literal">\PolToExprCmd{raw_coeff}</tt></a></li>
+<li><a class="reference internal" href="#poltoexprtermprefix-raw-coeff" id="id97"><tt class="docutils literal">\PolToExprTermPrefix{raw_coeff}</tt></a></li>
+<li><a class="reference internal" href="#id23" id="id98"><tt class="docutils literal">\PolToExprVar</tt></a></li>
+<li><a class="reference internal" href="#id24" id="id99"><tt class="docutils literal">\PolToExprTimes</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>
+<li><a class="reference internal" href="#id26" id="id100"><tt class="docutils literal"><span class="pre">\PolToExpr*{polname}</span></tt></a></li>
+<li><a class="reference internal" href="#poltofloatexpr-polname" id="id101"><tt class="docutils literal">\PolToFloatExpr{polname}</tt></a><ul>
+<li><a class="reference internal" href="#poltofloatexproneterm-raw-coeff-number" id="id102"><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="id103"><tt class="docutils literal">\PolToFloatExprCmd{raw_coeff}</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>
+<li><a class="reference internal" href="#id30" id="id104"><tt class="docutils literal"><span class="pre">\PolToFloatExpr*{polname}</span></tt></a></li>
+<li><a class="reference internal" href="#poltolist-polname" id="id105"><tt class="docutils literal">\PolToList{polname}</tt></a></li>
+<li><a class="reference internal" href="#poltocsv-polname" id="id106"><tt class="docutils literal">\PolToCSV{polname}</tt></a></li>
+<li><a class="reference internal" href="#polsturmchainlength-sturmname" id="id107"><tt class="docutils literal">\PolSturmChainLength{sturmname}</tt></a></li>
+<li><a class="reference internal" href="#polsturmifzeroexactlyknown-sturmname-index-a-b" id="id108"><tt class="docutils literal"><span class="pre">\PolSturmIfZeroExactlyKnown{sturmname}{index}{A}{B}</span></tt></a></li>
+<li><a class="reference internal" href="#polsturmisolatedzeroleft-sturmname-index" id="id109"><tt class="docutils literal"><span class="pre">\PolSturmIsolatedZeroLeft{sturmname}{index}</span></tt></a></li>
+<li><a class="reference internal" href="#polsturmisolatedzeroright-sturmname-index" id="id110"><tt class="docutils literal"><span class="pre">\PolSturmIsolatedZeroRight{sturmname}{index}</span></tt></a></li>
+<li><a class="reference internal" href="#polsturmnbofisolatedzeros-sturmname" id="id111"><tt class="docutils literal">\PolSturmNbOfIsolatedZeros{sturmname}</tt></a></li>
+<li><a class="reference internal" href="#polintervalwidth-sturmname-index" id="id112"><tt class="docutils literal"><span class="pre">\PolIntervalWidth{sturmname}{index}</span></tt></a></li>
+<li><a class="reference internal" href="#macros-for-use-within-execution-of-polprintintervals" id="id113">Macros for use within execution of <tt class="docutils literal">\PolPrintIntervals</tt></a><ul>
+<li><a class="reference internal" href="#id31" id="id114"><tt class="docutils literal">\PolPrintIntervalsTheEndPoint</tt></a></li>
+<li><a class="reference internal" href="#id32" id="id115"><tt class="docutils literal">\PolPrintIntervalsTheIndex</tt></a></li>
+<li><a class="reference internal" href="#polifendpointispositive-a-b" id="id116"><tt class="docutils literal"><span class="pre">\PolIfEndPointIsPositive{A}{B}</span></tt></a></li>
+<li><a class="reference internal" href="#polifendpointisnegative-a-b" id="id117"><tt class="docutils literal"><span class="pre">\PolIfEndPointIsNegative{A}{B}</span></tt></a></li>
+<li><a class="reference internal" href="#polifendpointiszero-a-b" id="id118"><tt class="docutils literal"><span class="pre">\PolIfEndPointIsZero{A}{B}</span></tt></a></li>
</ul>
+</li>
+<li><a class="reference internal" href="#poldectostring-decimal-number" id="id119"><tt class="docutils literal">\PolDecToString{decimal number}</tt></a></li>
+</ul>
+</li>
+<li><a class="reference internal" href="#booleans-with-default-setting-as-indicated" id="id120">Booleans (with default setting as indicated)</a><ul>
+<li><a class="reference internal" href="#xintverbosefalse" id="id121"><tt class="docutils literal">\xintverbosefalse</tt></a></li>
+<li><a class="reference internal" href="#poltypesetallfalse" id="id122"><tt class="docutils literal">\poltypesetallfalse</tt></a></li>
+<li><a class="reference internal" href="#poltoexprallfalse" id="id123"><tt class="docutils literal">\poltoexprallfalse</tt></a></li>
+</ul>
+</li>
+<li><a class="reference internal" href="#technicalities" id="id124">Technicalities</a></li>
+<li><a class="reference internal" href="#change-log" id="id125">CHANGE LOG</a></li>
+<li><a class="reference internal" href="#acknowledgments" id="id126">Acknowledgments</a></li>
+</ul>
</div>
-<div class="section" id="first-examples">
-<h1><a class="toc-backref" href="#id15">First Examples</a></h1>
+<div class="section" id="basic-examples">
+<h1><a class="toc-backref" href="#id33">Basic Examples</a></h1>
<p>The syntax is:</p>
<pre class="literal-block">
\poldef polname(x):= expression in variable x;
@@ -526,10 +578,200 @@
<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="examples-of-localization-of-roots">
+<h1><a class="toc-backref" href="#id34">Examples of localization of roots</a></h1>
+<p>First some remarks about auxiliaries.</p>
+<ul>
+<li><p class="first">To make printed decimal numbers more enjoyable than via
+<tt class="docutils literal">\xintSignedFrac</tt>:</p>
+<pre class="literal-block">
+\renewcommand\PolTypesetOne[1]{\PolDecToString{\xintREZ{#1}}}%
+</pre>
+<p><tt class="docutils literal">\PolDecToString</tt> will use decimal notation to incorporate the power
+of ten part; and the <tt class="docutils literal">\xintREZ</tt> will have the effect to suppress
+trailing zeros if present in raw numerator (if those digits end up
+after decimal mark.) Notice that the above are expandable macros and
+that one can also do:</p>
+<pre class="literal-block">
+\renewcommand\PolToExprCmd[1]{\PolDecToString{\xintREZ{#1}}}%
+</pre>
+<p>to modify output of <a class="reference internal" href="#poltoexpr-polname">\PolToExpr{polname}</a>.</p>
+</li>
+<li><p class="first">for extra info in log file <tt class="docutils literal">\xintverbosetrue</tt>. In fact one can also
+work from command line then (I recommend <tt class="docutils literal">rlwrap</tt> for encapsulating
+<tt class="docutils literal">latex</tt>).</p>
+</li>
+</ul>
+<div class="section" id="a-typical-example">
+<h2><a class="toc-backref" href="#id35">A typical example</a></h2>
+<pre class="literal-block">
+\poldef f(x) := x^7 - x^6 - 2x + 1;
+
+\PolToSturm{f}{f}
+\PolSturmIsolateZeros{f}
+The \PolTypeset{f} polynomial has \PolSturmNbOfIsolatedZeros{f} distinct real
+roots which are located in the following intervals:
+\PolPrintIntervals{f}
+Here is the second root with ten more decimal digits:
+\PolRefineInterval[10]{f}{2}
+\[\PolSturmIsolatedZeroLeft{f}{2}<Z_2<\PolSturmIsolatedZeroRight{f}{2}\]
+And here is the first root with twenty digits after decimal mark:
+\PolEnsureIntervalLength{f}{1}{-20}
+\[\PolSturmIsolatedZeroLeft{f}{1}<Z_1<\PolSturmIsolatedZeroRight{f}{1}\]
+The derivative polynomial is \PolTypeset{f_1}.
+\PolToSturm{f_1}{f_1}\PolSturmIsolateZeros{f_1}%
+It has \PolSturmNbOfIsolatedZeros{f_1} distinct real
+roots:
+\PolPrintIntervals[W]{f_1}
+\PolEnsureIntervalLengths{f_1}{-10}%
+Here they are with ten digits after decimal mark:
+\PolPrintIntervals[W]{f_1}
+\PolDiff{f_1}{f_xx}
+\PolToSturm{f_xx}{f_xx}
+\PolSturmIsolateZeros{f_xx}
+The second derivative is \PolTypeset{f_xx}.
+It has \PolSturmNbOfIsolatedZeros{f_xx} distinct real
+roots:
+\PolPrintIntervals[X]{f_xx}
+Here is the positive one with 20 digits after decimal mark:
+\PolEnsureIntervalLength{f_xx}{2}{-20}%
+\[X_2 = \PolSturmIsolatedZeroLeft{f_xx}{2}\dots\]
+The more mathematically advanced among our dear readers will be able
+to give the exact value for $X_2$!
+</pre>
+</div>
+<div class="section" id="a-degree-four-polynomial-with-nearby-roots">
+<h2><a class="toc-backref" href="#id36">A degree four polynomial with nearby roots</a></h2>
+<pre class="literal-block">
+\PolDef{Q}{(x-1.050001)(x-1.105001)(x-1.110501)(x-1.111051)}
+\PolTypeset{Q}
+\PolToSturm{Q}{Q} % it is allowed to use same prefix for Sturm chain
+\PolSturmIsolateZeros{Q}
+\PolPrintIntervals{Q}
+% reports 1.0 < Z_1 < 1.1, 1.10 < Z_2 < 1.11, 1.110 < Z_3 < 1.111, and 1.111 < Z_4 < 1.112
+% but the above bounds do not allow minimizing separation between roots
+% so we refine:
+\PolRefineInterval*{Q}{1}
+\PolRefineInterval*{Q}{2}
+\PolRefineInterval*{Q}{3}
+\PolRefineInterval*{Q}{4}
+\PolPrintIntervals{Q}
+% reports 1.05 < Z_1 < 1.06, 1.105 < Z_2 < 1.106, 1.1105 < Z_3 < 1.1106,
+% and 1.11105 < Z_4 < 1.11106.
+\PolEnsureIntervalLengths{Q}{-6}
+\PolPrintIntervals{Q}
+% of course finds here all roots exactly
+</pre>
+</div>
+<div class="section" id="the-degree-nine-polynomial-with-0-99-0-999-0-9999-as-triple-roots">
+<h2><a class="toc-backref" href="#id37">The degree nine polynomial with 0.99, 0.999, 0.9999 as triple roots</a></h2>
+<pre class="literal-block">
+\PolDef{P}{(x-0.99)^3(x-0.999)^3(x-0.9999)^3}
+\PolTypeset{P}\par
+\PolToSturm{P}{P}%
+\PolLet{Psqfree}{P_0}\PolMakeMonic{Psqfree}\PolReduceCoeffs*{Psqfree}
+\par
+The monic square-free radical is \PolTypeset{Psqfree}.
+\PolSturmIsolateZeros{P}
+\par
+It has \PolSturmNbOfIsolatedZeros{P} real roots.
+\PolPrintIntervals{P}% all three roots found exactly
+</pre>
+</div>
+<div class="section" id="a-mignotte-type-polynomial">
+<h2><a class="toc-backref" href="#id38">A Mignotte type polynomial</a></h2>
+<pre class="literal-block">
+\PolDef{P}{x^10 - (10x-1)^2}%
+\PolTypeset{P} % prints it in expanded form
+\PolToSturm{P}{P} % we can use same prefix for Sturm chain
+\PolSturmIsolateZeros{P} % finds 4 real roots
+\PolPrintIntervals{P}%
+% reports -2 < Z_1 < -1, 0 < Z_2 < 0.1, 0.1 < Z_3 < 0.2, 1 < Z_4 < 2
+\PolRefineInterval*{P}{2}% will refine to 0.0999990 < Z_2 < 0.0999991
+\PolRefineInterval*{P}{3}% will refine to 0.100001 < Z_3 < 0.100002
+\PolPrintIntervals{P}%
+\PolEnsureIntervalLengths{P}{-10}%
+\PolPrintIntervals{P}% now all roots are known 10 decimal digits after mark
+\PolEnsureIntervalLength{P}{2}{-20}% makes Z_2 known with 20 digits after mark
+There are $\PolSturmNbOfIsolatedZeros{P}$ distinct real roots and there holds
+$\PolSturmIsolatedZeroLeft{P}{2}<Z_2<\PolSturmIsolatedZeroRight{P}{2}$.
+</pre>
+<p>The last line produces:</p>
+<pre class="literal-block">
+0.09999900004999650028 < Z_2 < 0.09999900004999650029
+</pre>
+</div>
+<div class="section" id="the-degree-41-polynomial-with-2-1-9-1-8-0-0-1-1-9-2-as-roots">
+<h2><a class="toc-backref" href="#id39">The degree 41 polynomial with -2, -1.9, -1.8, ..., 0, 0.1, ..., 1.9, 2 as roots</a></h2>
+<pre class="literal-block">
+\PolDef{P}{mul((x-i*1e-1), i=-20..20)}% i/10 is same but less efficient
+</pre>
+<p>In the defining expression we could have used <tt class="docutils literal">i/10</tt> but this gives
+less efficient internal form for the coefficients (the <tt class="docutils literal">10</tt>'s end up
+in denominators). Using <tt class="docutils literal">\PolToExpr{P}</tt> after having done</p>
+<pre class="literal-block">
+\renewcommand\PolToExprCmd[1]{\PolDecToString{\xintREZ{#1}}}
+</pre>
+<p>we get this expanded form:</p>
+<pre class="literal-block">
+x^41
+-28.7*x^39
++375.7117*x^37
+-2975.11006*x^35
++15935.28150578*x^33
+-61167.527674162*x^31
++173944.259366417394*x^29
+-373686.963560544648*x^27
++613012.0665016658846445*x^25
+-771182.31133138163125495*x^23
++743263.86672885754888959569*x^21
+-545609.076599482896371978698*x^19
++301748.325708943677229642930528*x^17
+-123655.8987669450434698869844544*x^15
++36666.1782054884005855608205864192*x^13
+-7607.85821367459445649518380016128*x^11
++1053.15135918687298508885950223794176*x^9
+-90.6380005918141132650786081964032*x^7
++4.33701563847327366842552218288128*x^5
+-0.0944770968420804735498178265088*x^3
++0.00059190121813899276854174416896*x
+</pre>
+<p>which shows coefficients with up to 36 significant digits...</p>
+<p>Stress test: not a hard challenge to <tt class="docutils literal">xint + polexpr</tt>, but be a bit patient!</p>
+<pre class="literal-block">
+\PolDef{P}{mul((x-i*1e-1), i=-20..20)}%
+\PolToSturm{P}{S} % dutifully computes S_0, ..., S_{41}
+\PolSturmIsolateZeros{S} % finds *exactly* (but a bit slowly) all 41 roots!
+\PolPrintIntervals{S} % nice, isn't it?
+</pre>
+</div>
+<div class="section" id="roots-of-chebyshev-polynomials">
+<h2><a class="toc-backref" href="#id40">Roots of Chebyshev polynomials</a></h2>
+<pre class="literal-block">
+\newcount\mycount
+\poldef T_0(x) := 1;
+\poldef T_1(x) := x;
+\mycount 2
+\xintloop
+ \poldef T_\the\mycount(x) :=
+ 2x*T_\the\numexpr\mycount-1(x)
+ - T_\the\numexpr\mycount-2(x);
+\ifnum\mycount<15
+\advance\mycount 1
+\repeat
+
+\[T_{15} = \PolTypeset[X]{T_15}\]
+\PolToSturm{T_15}{T_15}
+\PolSturmIsolateZeros{T_15}
+\PolEnsureIntervalLengths{T_15}{-10}
+\PolPrintIntervals{T_15}
+</pre>
+</div>
+</div>
<div class="section" id="non-expandable-macros">
-<h1><a class="toc-backref" href="#id16">Non-expandable macros</a></h1>
+<h1><a class="toc-backref" href="#id41">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>
+<span id="poldef"></span><h2><a class="toc-backref" href="#id42"><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,
@@ -546,25 +788,18 @@
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 class="admonition attention">
+<p class="first admonition-title">Attention!</p>
+<p>Release <tt class="docutils literal">0.3</tt> also did the necessary set-up to let the
+polynomial be known to the <tt class="docutils literal">\xintfloatexpr</tt> (or
+<tt class="docutils literal">\xintdeffloatvar</tt>) parser.</p>
+<p>Since <tt class="docutils literal">0.4</tt> this isn't done automatically. Even more, a
+previously existing floating point variant of the same name will
+be let undefined again, to avoid hard to debug mismatches between
+exact and floating point polynomials. This also applies when the
+polynomial is produced not via <tt class="docutils literal">\poldef</tt> or <tt class="docutils literal">\PolDef</tt> but as
+a product of the other package macros.</p>
+<p class="last">See <a class="reference internal" href="#polgenfloatvariant-polname">\PolGenFloatVariant{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
@@ -572,15 +807,34 @@
</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>
+<span id="id2"></span><h2><a class="toc-backref" href="#id43"><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>
+Does the same as <a class="reference external" href="poldef;">\poldef</a> 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="polgenfloatvariant-polname">
+<span id="polgenfloatvariant"></span><h2><a class="toc-backref" href="#id44"><tt class="docutils literal">\PolGenFloatVariant{polname}</tt></a></h2>
+<blockquote>
+<p>Makes the polynomial also usable in the <tt class="docutils literal">\xintfloatexpr</tt> parser.
+It will therein evaluates via an Horner scheme with coefficients
+already pre-rounded to the float precision.</p>
+<p>See also <a class="reference internal" href="#poltofloatexpr-polname">\PolToFloatExpr{polname}</a>.</p>
+<div class="admonition attention">
+<p class="first admonition-title">Attention!</p>
+<p>Release <tt class="docutils literal">0.3</tt> did this automatically on <tt class="docutils literal">\PolDef</tt> and
+<tt class="docutils literal">\poldef</tt> but this was removed at <tt class="docutils literal">0.4</tt> for optimization.</p>
+<p class="last">Any operation, for example generating the derivative polynomial,
+or dividing two polynomials or using the <tt class="docutils literal">\PolLet</tt>, <strong>must</strong> be
+followed by explicit usage of <tt class="docutils literal">\PolGenFloatVariant{polname}</tt> if
+the new polynomial is to be used in <tt class="docutils literal">\xintfloatexpr</tt> or alike
+context.</p>
+</div>
+</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>
+<span id="pollet"></span><h2><a class="toc-backref" href="#id45"><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
@@ -587,8 +841,13 @@
<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="polgloballet-polname-2-polname-1">
+<span id="polgloballet"></span><h2><a class="toc-backref" href="#id46"><tt class="docutils literal"><span class="pre">\PolGlobalLet{polname_2}={polname_1}</span></tt></a></h2>
+<blockquote>
+Acts globally.</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>
+<span id="polassign"></span><h2><a class="toc-backref" href="#id47"><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>
@@ -611,7 +870,7 @@
</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>
+<span id="polget"></span><h2><a class="toc-backref" href="#id48"><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
@@ -625,11 +884,11 @@
</pre>
<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>
+they were not subjected to <tt class="docutils literal">\xintRaw</tt> or similar <a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a> macro.)</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>
+<span id="polfromcsv"></span><h2><a class="toc-backref" href="#id49"><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,
@@ -636,7 +895,7 @@
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
+<tt class="docutils literal">1.5e3</tt> which is not converted to <tt class="docutils literal">15/1[2]</tt> <em>raw</em> <a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a>
format (this may change).</p>
<p>Leading zero coefficients are removed:</p>
<pre class="literal-block">
@@ -647,7 +906,7 @@
</blockquote>
</div>
<div class="section" id="poltypeset-polname">
-<h2><a class="toc-backref" href="#id23"><tt class="docutils literal">\PolTypeset{polname}</tt></a></h2>
+<span id="poltypeset"></span><h2><a class="toc-backref" href="#id50"><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>
@@ -661,31 +920,52 @@
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>
+<span id="poltypesetcmd"></span><h3><a class="toc-backref" href="#id51"><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>
+<p>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. Also it sets conditional
+<a class="reference internal" href="#polifcoeffisplusorminusone-a-b">\PolIfCoeffIsPlusOrMinusOne{A}{B}</a>.</p>
+<p>The actual printing of the coefficients, when not equal to plus or
+minus one is handled by <a class="reference internal" href="#poltypesetone-raw-coeff">\PolTypesetOne{raw_coeff}</a>.</p>
+</blockquote>
+</div>
+<div class="section" id="poltypesetone-raw-coeff">
+<span id="poltypesetone"></span><h3><a class="toc-backref" href="#id52"><tt class="docutils literal">\PolTypesetOne{raw_coeff}</tt></a></h3>
+<blockquote>
+<p>The default is <tt class="docutils literal">\xintSignedFrac</tt> but this macro is annoying as it
+insists to use a power of ten, and not decimal notation.</p>
+<p>One can do things such as for example: <a class="footnote-reference" href="#id5" id="id4">[1]</a></p>
<pre class="literal-block">
-\renewcommand\PolTypesetCmd[1]{\num{\xintPFloat[5]{#1}}}
-\renewcommand\PolTypesetCmd[1]{\num{\xintRound{4}{#1}}}
+\renewcommand\PolTypesetOne[1]{\num{\xintPFloat[5]{#1}}}
+\renewcommand\PolTypesetOne[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">
+<table class="docutils footnote" frame="void" id="id5" 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
+<tr><td class="label"><a class="fn-backref" href="#id4">[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>
+<p>One can also give a try to using <a class="reference internal" href="#poldectostring-decimal-number">\PolDecToString{decimal number}</a>
+which uses decimal notation (at least for the numerator part).</p>
</blockquote>
</div>
+<div class="section" id="id6">
+<span id="poltypesetmonomialcmd"></span><h3><a class="toc-backref" href="#id53"><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 class="section" id="poltypesetcmdprefix-raw-coeff">
-<h3><a class="toc-backref" href="#id25"><tt class="docutils literal">\PolTypesetCmdPrefix{raw_coeff}</tt></a></h3>
+<span id="poltypesetcmdprefix"></span><h3><a class="toc-backref" href="#id54"><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
@@ -694,25 +974,15 @@
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>
+<div class="section" id="id8">
+<span id="id7"></span><h2><a class="toc-backref" href="#id55"><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>
+<span id="poldiff"></span><h2><a class="toc-backref" href="#id56"><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>
@@ -722,7 +992,7 @@
</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>
+<span id="poldiff-n"></span><h2><a class="toc-backref" href="#id57"><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
@@ -730,7 +1000,7 @@
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>
+<span id="polantidiff"></span><h2><a class="toc-backref" href="#id58"><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>
@@ -739,27 +1009,287 @@
</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>
+<span id="polantidiff-n"></span><h2><a class="toc-backref" href="#id59"><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>
+<span id="poldivide"></span><h2><a class="toc-backref" href="#id60"><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="polquo-polname-1-polname-2-polname-q">
+<span id="polquo"></span><h2><a class="toc-backref" href="#id61"><tt class="docutils literal"><span class="pre">\PolQuo{polname_1}{polname_2}{polname_Q}</span></tt></a></h2>
+<blockquote>
+This sets <tt class="docutils literal">polname_Q</tt> to be the quotient 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="polrem-polname-1-polname-2-polname-r">
+<span id="polrem"></span><h2><a class="toc-backref" href="#id62"><tt class="docutils literal"><span class="pre">\PolRem{polname_1}{polname_2}{polname_R}</span></tt></a></h2>
+<blockquote>
+This sets <tt class="docutils literal">polname_R</tt> to be the 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>
+<span id="polgcd"></span><h2><a class="toc-backref" href="#id63"><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>
+This sets <tt class="docutils literal">polname_GCD</tt> to be the (monic) GCD of the two first
+polynomials. 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>
+<!-- ``\PolIGCD{polname_1}{polname_2}{polname_iGCD}``
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+ **NOT YET**
+
+ This **assumes** that the two polynomials have integer coefficients.
+ It then computes the greatest common divisor in the integer
+ polynomial ring, normalized to have a positive leading coefficient
+ (if the inputs are not both zero).
+
+``\PolIContent{polname}``
+~~~~~~~~~~~~~~~~~~~~~~~~~
+
+ **NOT YET**
+
+ This computes a positive rational number such that dividing the
+ polynomial with it returns an integer coefficients polynomial with
+ no common factor among the coefficients. -->
</div>
+<div class="section" id="poltosturm-polname-sturmname">
+<span id="poltosturm"></span><h2><a class="toc-backref" href="#id64"><tt class="docutils literal"><span class="pre">\PolToSturm{polname}{sturmname}</span></tt></a></h2>
+<blockquote>
+<p>With, for example, <tt class="docutils literal">polname</tt> being <tt class="docutils literal">P</tt> and <tt class="docutils literal">sturmname</tt> being
+<tt class="docutils literal">S</tt>, the macro starts by computing polynomials <tt class="docutils literal">S_0 = P</tt>, <tt class="docutils literal">S_1
+= P'</tt>, ..., with <tt class="docutils literal">S_{n+1}</tt> the opposite of the remainder of
+euclidean division of <tt class="docutils literal"><span class="pre">S_{n-1}</span></tt> by <tt class="docutils literal">S_{n}</tt>. The last non-zero
+remainder <tt class="docutils literal">S_N</tt> is up to a factor the GCD of <tt class="docutils literal">P</tt> and <tt class="docutils literal">P'</tt>
+hence a constant if and only if <tt class="docutils literal">P</tt> is square-free.</p>
+<p>In case <tt class="docutils literal">S_N</tt> is not a constant, the macro then goes on with
+dividing all <tt class="docutils literal">S_k</tt>'s with <tt class="docutils literal">S_N</tt> (which becomes <tt class="docutils literal">1</tt>).</p>
+<p>Thus <tt class="docutils literal">S_0</tt> now has exactly the same real and complex
+roots as polynomial <tt class="docutils literal">polname</tt>, but each with multiplicity one.</p>
+</blockquote>
+</div>
+<div class="section" id="id10">
+<span id="id9"></span><h2><a class="toc-backref" href="#id65"><tt class="docutils literal"><span class="pre">\PolToSturm*{polname}{sturmname}</span></tt></a></h2>
+<blockquote>
+Does not divide the Sturm chain by its last element.</blockquote>
+</div>
+<div class="section" id="polsettosturmchainsignchangesat-macro-sturmname-fraction">
+<span id="polsettosturmchainsignchangesat"></span><h2><a class="toc-backref" href="#id66"><tt class="docutils literal"><span class="pre">\PolSetToSturmChainSignChangesAt{\macro}{sturmname}{fraction}</span></tt></a></h2>
+<blockquote>
+<p>Sets macro <tt class="docutils literal">\macro</tt> to the number of sign changes in the Sturm
+chain with name prefix <tt class="docutils literal">sturmname</tt>, at location <tt class="docutils literal">fraction</tt>
+(which must be in format as acceptable by the <a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a> macros.)</p>
+<div class="admonition note">
+<p class="first admonition-title">Note</p>
+<p>The author was lazy and did not provide rather an expandable
+variant, where one would do <tt class="docutils literal"><span class="pre">\edef\macro{\PolNbOf...}</span></tt>.</p>
+<p>This will presumably get added in a future release.</p>
+<p class="last">After some hesitation it was decided the macro would by default
+act globally. To make the scope of its macro definition local,
+use <tt class="docutils literal">[\empty]</tt> as extra optional argument.</p>
+</div>
+</blockquote>
+</div>
+<div class="section" id="polsettonbofzeroswithin-macro-sturmname-value-a-value-b">
+<span id="polsettonbofzeroswithin"></span><h2><a class="toc-backref" href="#id67"><tt class="docutils literal"><span class="pre">\PolSetToNbOfZerosWithin{\macro}{sturmname}{value_a}{value_b}</span></tt></a></h2>
+<blockquote>
+<p>Applies the <a class="reference external" href="https://en.wikipedia.org/wiki/Sturm%27s_theorem">Sturm Theorem</a> to set <tt class="docutils literal">\macro</tt> to the exact number
+of distinct roots of <tt class="docutils literal">sturmname_0</tt> in the interval <tt class="docutils literal">(value_a,
+value_b]</tt> (the macro first re-orders the value for <tt class="docutils literal">value_a <=
+value_b</tt> to hold).</p>
+<div class="admonition note">
+<p class="first admonition-title">Note</p>
+<p>The author was lazy and did not provide rather an expandable
+variant, where one would do <tt class="docutils literal"><span class="pre">\edef\macro{\PolNbOf...}</span></tt>.</p>
+<p>This will presumably get added in future.</p>
+<p class="last">After some hesitation it was decided the macro would by default
+act globally. To make the scope of its macro definition local,
+use <tt class="docutils literal">[\empty]</tt> as extra optional argument.</p>
+</div>
+</blockquote>
+</div>
+<div class="section" id="polsturmisolatezeros-sturmname">
+<span id="polsturmisolatezeros"></span><h2><a class="toc-backref" href="#id68"><tt class="docutils literal">\PolSturmIsolateZeros{sturmname}</tt></a></h2>
+<blockquote>
+<p>First, it evaluates using <a class="reference external" href="https://en.wikipedia.org/wiki/Sturm%27s_theorem">Sturm theorem</a> the number of distinct
+real roots of <tt class="docutils literal">sturmname_0</tt>.</p>
+<div class="admonition important">
+<p class="first admonition-title">Important</p>
+<p class="last">The Sturm chain <strong>must</strong> be of the reduced type, i.e.
+as constructed via <a class="reference internal" href="#poltosturm-polname-sturmname">\PolToSturm{polname}{sturmname}</a>.</p>
+</div>
+<p>Then it locates, again using <a class="reference external" href="https://en.wikipedia.org/wiki/Sturm%27s_theorem">Sturm theorem</a>, as many disjoint
+intervals as there are roots. Some intervals reduce to singleton
+which are roots. Non-singleton intervals get refined to make sure
+none of their two limit points is a root: they contain each a single
+root, in their respective interiors.</p>
+<!-- This procedure is covariant
+with the independent variable ``x`` becoming ``-x``.
+Hmm, pas sûr et trop fatigué -->
+<p>The interval boundaries are decimal numbers, originating
+in iterated decimal subdivision from initial intervals
+<tt class="docutils literal"><span class="pre">(-10^E,</span> 0)</tt> and <tt class="docutils literal">(0, 10^E)</tt>; if zero is a root it is always
+identified individually. The non-singleton intervals are of the
+type <tt class="docutils literal">(a/10^f, <span class="pre">(a+1)/10^f)</span></tt> with <tt class="docutils literal">a</tt> an integer, which is
+neither <tt class="docutils literal">0</tt> nor <tt class="docutils literal"><span class="pre">-1</span></tt>. Hence <tt class="docutils literal">a</tt> and <tt class="docutils literal">a+1</tt> are both positive
+or both negative.</p>
+<p>The interval boundaries (and exactly found roots) are made available
+for future computations in <tt class="docutils literal">\xintexpr</tt>-essions or polynomial
+definitions as variables <tt class="docutils literal"><sturmname>L_1</tt>,
+<tt class="docutils literal"><sturmname>L_2</tt>, etc..., for the left end-points and
+<tt class="docutils literal"><sturmname>R_1</tt>, <tt class="docutils literal"><sturmname>R_2</tt>, ..., for the right
+end-points.</p>
+<p>Also two macro arrays (in the sense of
+<a class="reference external" href="http://www.ctan.org/pkg/xint">xinttools</a>'s <tt class="docutils literal">\xintAssignArray</tt>) are created for holding the
+interval end-points written out in standard decimal notation
+(see <a class="reference internal" href="#poldectostring-decimal-number">\PolDecToString{decimal number}</a>).
+To access these values, macros
+<a class="reference internal" href="#polsturmisolatedzeroleft-sturmname-index">\PolSturmIsolatedZeroLeft{sturmname}{index}</a> and
+<a class="reference internal" href="#polsturmisolatedzeroright-sturmname-index">\PolSturmIsolatedZeroRight{sturmname}{index}</a> are provided.</p>
+<div class="admonition important">
+<p class="first admonition-title">Important</p>
+<p class="last">Trailing zeroes in these stored decimal numbers are significant:
+they are also present in the decimal expansion of the exact root.</p>
+</div>
+<div class="admonition note">
+<p class="first admonition-title">Note</p>
+<p class="last">The actual array macros are <tt class="docutils literal">\POL_ZeroInt<sturmname>L</tt> and
+<tt class="docutils literal">\POL_ZeroInt<sturmname>R</tt> but as these names use the
+non-letter character <tt class="docutils literal">_</tt> and possibly also digits from
+<tt class="docutils literal">sturmname</tt>, the accessor macros above have been made part of
+the package.</p>
+</div>
+<p>The start of decimal expansion of a positive <tt class="docutils literal">k</tt>-th root is given
+by <tt class="docutils literal"><span class="pre">\PolSturmIsolatedZeroLeft{sturmname}{k}</span></tt>, and for a negative
+root it is given by <tt class="docutils literal"><span class="pre">\PolSturmIsolatedZeroRight{sturmname}{k}</span></tt>.
+These two decimal numbers are either both zero or both of the same
+sign.</p>
+<p>The number of distinct roots is obtainable as
+<tt class="docutils literal">\PolSturmNbOfIsolatedZeros{sturmname}</tt>.</p>
+<div class="admonition note">
+<p class="first admonition-title">Note</p>
+<p class="last">In the current implementation the <tt class="docutils literal"><span class="pre"><sturmname>...</span></tt> variables
+and the <tt class="docutils literal"><span class="pre">\POL_ZeroInt...</span></tt> arrays are globally defined. On the
+other hand the Sturm sequence polynomials obey the current scope.</p>
+</div>
+<div class="admonition note">
+<p class="first admonition-title">Note</p>
+<p class="last">When two successive roots are located in adjacent intervals, the
+separation between them is not lower bounded. See
+<a class="reference internal" href="#polrefineinterval-sturmname-index">\PolRefineInterval*{sturmname}{index}</a>.</p>
+</div>
+<div class="admonition note">
+<p class="first admonition-title">Note</p>
+<p>As all computations are done <em>exactly</em> there can be no errors...
+apart those due to bad coding by author. The results are exact
+bounds for the mathematically exact real roots.</p>
+<p class="last">Future releases will perhaps also provide macros based on Newton
+or Regula Falsi methods. Exact computations with such methods
+lead however quickly to very big fractions, and this forces usage
+of some rounding scheme for the abscissas if computation times
+are to remain reasonable. This raises issues of its own, which
+are studied in numerical mathematics.</p>
+</div>
+</blockquote>
+</div>
+<div class="section" id="polrefineinterval-sturmname-index">
+<span id="polrefineinterval"></span><h2><a class="toc-backref" href="#id69"><tt class="docutils literal"><span class="pre">\PolRefineInterval*{sturmname}{index}</span></tt></a></h2>
+<blockquote>
+The <tt class="docutils literal">index</tt>-th interval (starting indexing at one) is further
+subdivided as many times as is necessary in order for the newer
+interval to have both its end-points distinct from the end-points of
+the original interval. This means that the <tt class="docutils literal">k</tt>th root is then
+strictly separated from the other roots.</blockquote>
+</div>
+<div class="section" id="polrefineinterval-n-sturmname-index">
+<span id="polrefineinterval-n"></span><h2><a class="toc-backref" href="#id70"><tt class="docutils literal"><span class="pre">\PolRefineInterval[N]{sturmname}{index}</span></tt></a></h2>
+<blockquote>
+The <tt class="docutils literal">index</tt>-th interval (starting count at one) is further
+subdivided once, reducing its length by a factor of 10. This is done
+<tt class="docutils literal">N</tt> times if the optional argument <tt class="docutils literal">[N]</tt> is present.</blockquote>
+</div>
+<div class="section" id="polensureintervallength-sturmname-index-e">
+<span id="polensureintervallength"></span><h2><a class="toc-backref" href="#id71"><tt class="docutils literal"><span class="pre">\PolEnsureIntervalLength{sturmname}{index}{E}</span></tt></a></h2>
+<blockquote>
+The <tt class="docutils literal">index</tt>-th interval is subdivided until its length becomes at
+most <tt class="docutils literal">10^E</tt>. This means (for <tt class="docutils literal">E<0</tt>) that the first <tt class="docutils literal"><span class="pre">-E</span></tt> digits
+after decimal mark of the <tt class="docutils literal">k</tt>th root will then be known exactly.</blockquote>
+</div>
+<div class="section" id="polensureintervallengths-sturmname-e">
+<span id="polensureintervallengths"></span><h2><a class="toc-backref" href="#id72"><tt class="docutils literal"><span class="pre">\PolEnsureIntervalLengths{sturmname}{E}</span></tt></a></h2>
+<blockquote>
+<p>The intervals as obtained from <tt class="docutils literal">\PolSturmIsolateZeros</tt> are (if
+necessary) subdivided further by (base 10) dichotomy in order for
+each of them to have length at most <tt class="docutils literal">10^E</tt> (length will be shorter
+than <tt class="docutils literal">10^E</tt> in output only if it did not change or became zero.)</p>
+<p>This means that decimal expansions of all roots will be known with
+<tt class="docutils literal"><span class="pre">-E</span></tt> digits (for <tt class="docutils literal">E<0</tt>) after decimal mark.</p>
+</blockquote>
+</div>
+<div class="section" id="polprintintervals-varname-sturmname">
+<span id="polprintintervals"></span><h2><a class="toc-backref" href="#id73"><tt class="docutils literal"><span class="pre">\PolPrintIntervals[varname]{sturmname}</span></tt></a></h2>
+<blockquote>
+<p>This is a convenience macro which prints the bounds for the roots
+<tt class="docutils literal">Z_1</tt>, <tt class="docutils literal">Z_2</tt>, ... (the optional argument <tt class="docutils literal">varname</tt> allows to
+specify a replacement for the default <tt class="docutils literal">Z</tt>). This will be done in a
+math mode <tt class="docutils literal">array</tt>, one interval per row, and pattern <tt class="docutils literal">rcccl</tt>,
+where the second and fourth column hold the <tt class="docutils literal"><</tt> sign, except when
+the interval reduces to a singleton, which means the root is known
+exactly. The user is invited to renewcommand the macro if some other
+type of tabular environment for example is wanted.</p>
+<p>In each array cell the corresponding interval end-point (which may
+be an exactly known root) is available as macro
+<a class="reference internal" href="#polprintintervalstheendpoint">\PolPrintIntervalsTheEndPoint</a> (in decimal notation). And the
+corresponding interval index is available as
+<a class="reference internal" href="#polprintintervalstheindex">\PolPrintIntervalsTheIndex</a>.</p>
+<p>These values may be tested to decide some on-the-fly customization
+(color for example), via the following auxiliaries which can be
+modified by user. Furthermore these auxiliaries can also use the
+following conditionals: <a class="reference internal" href="#polifendpointispositive-a-b">\PolIfEndPointIsPositive{A}{B}</a>,
+<a class="reference internal" href="#polifendpointisnegative-a-b">\PolIfEndPointIsNegative{A}{B}</a>, <a class="reference internal" href="#polifendpointiszero-a-b">\PolIfEndPointIsZero{A}{B}</a>.</p>
+</blockquote>
+<div class="section" id="id11">
+<span id="polprintintervalsprintexactzero"></span><h3><a class="toc-backref" href="#id74"><tt class="docutils literal">\PolPrintIntervalsPrintExactZero</tt></a></h3>
+<blockquote>
+<p>This is provided to help customize how an exactly known root is
+printed in the right most column of the array. The package
+definition is:</p>
+<pre class="literal-block">
+\newcommand\PolPrintIntervalsPrintExactZero{\PolPrintIntervalsTheEndPoint}%
+</pre>
+<p>Recall that this is expanded in an array cell.</p>
+<p>If for example you want to print in red the third root, known
+exactly, the macro could make a test for the value of
+<a class="reference internal" href="#polprintintervalstheindex">\PolPrintIntervalsTheIndex</a> and act accordingly.</p>
+</blockquote>
+</div>
+<div class="section" id="id12">
+<span id="polprintintervalsprintleftendpoint"></span><h3><a class="toc-backref" href="#id75"><tt class="docutils literal">\PolPrintIntervalsPrintLeftEndPoint</tt></a></h3>
+<blockquote>
+<p>Package definition is:</p>
+<pre class="literal-block">
+\newcommand\PolPrintIntervalsPrintLeftEndPoint{\PolPrintIntervalsTheEndPoint}%
+</pre>
+</blockquote>
+</div>
+<div class="section" id="id13">
+<span id="polprintintervalsprintrightendpoint"></span><h3><a class="toc-backref" href="#id76"><tt class="docutils literal">\PolPrintIntervalsPrintRightEndPoint</tt></a></h3>
+<blockquote>
+<p>Package definition is:</p>
+<pre class="literal-block">
+\newcommand\PolPrintIntervalsPrintRightEndPoint{\PolPrintIntervalsTheEndPoint}%
+</pre>
+</blockquote>
+</div>
+</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>
+<span id="polmapcoeffs"></span><h2><a class="toc-backref" href="#id77"><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
@@ -769,7 +1299,7 @@
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> (<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>
+will have to be expressed in terms of macros from <a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a> package.</p>
<p>Example:</p>
<pre class="literal-block">
\def\foo#1{\xintMul{#1}{\the\numexpr\index*\index\relax}}
@@ -779,7 +1309,7 @@
</blockquote>
</div>
<div class="section" id="polreducecoeffs-polname">
-<h2><a class="toc-backref" href="#id35"><tt class="docutils literal">\PolReduceCoeffs{polname}</tt></a></h2>
+<span id="polreducecoeffs"></span><h2><a class="toc-backref" href="#id78"><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
@@ -786,43 +1316,89 @@
polynomial function is used for computations.) This is a
one-argument macro, working 'in-place'.</blockquote>
</div>
+<div class="section" id="id15">
+<span id="id14"></span><h2><a class="toc-backref" href="#id79"><tt class="docutils literal"><span class="pre">\PolReduceCoeffs*{polname}</span></tt></a></h2>
+<blockquote>
+<p>This starred variant leaves un-touched the decimal exponent in the
+internal representation of the fractional coefficients, i.e. if a
+coefficient is internally <tt class="docutils literal">A/B[N]</tt>, then <tt class="docutils literal">A/B</tt> is reduced to
+smallest terms, but the <tt class="docutils literal">10^N</tt> part is kept as is. Note: if the
+polynomial is freshly defined directly via <a class="reference internal" href="#polfromcsv">\PolFromCSV</a> its coefficients might still be internally in some
+format like <tt class="docutils literal">1.5e7</tt>; the macro will anyhow always first do the
+needed conversion to strict format <tt class="docutils literal">A/B[N]</tt>.</p>
+<p>Evaluations with polynomials treated by this can be much faster than
+with those handled by the non-starred variant
+<a class="reference internal" href="#polreducecoeffs-polname">\PolReduceCoeffs{polname}</a>: as the numerators and denominators
+remain smaller, this proves very beneficial in favorable cases
+(especially when the coefficients are decimal numbers) to the
+expansion speed of the <a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a> macros used internally by
+<a class="reference internal" href="#polevalat">\PolEval</a>.</p>
+</blockquote>
</div>
+<div class="section" id="polmakemonic-polname">
+<span id="polmakemonic"></span><h2><a class="toc-backref" href="#id80"><tt class="docutils literal">\PolMakeMonic{polname}</tt></a></h2>
+<blockquote>
+Divides by the leading coefficient. It is recommended to execute
+<a class="reference internal" href="#id15">\PolReduceCoeffs*{polname}</a> immediately afterwards. This is not
+done automatically, due to the case the original polynomial had integer
+coefficients and we want to keep the leading one as common
+denominator.</blockquote>
+</div>
+</div>
<div class="section" id="expandable-macros">
-<h1><a class="toc-backref" href="#id36">Expandable macros</a></h1>
+<h1><a class="toc-backref" href="#id81">Expandable macros</a></h1>
<p>All these macros expand completely in two steps except <tt class="docutils literal">\PolToExpr</tt>
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>
+<div class="section" id="poleval-polname-atexpr-numerical-expression">
+<span id="polevalatexpr"></span><h2><a class="toc-backref" href="#id82"><tt class="docutils literal"><span class="pre">\PolEval{polname}\AtExpr{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>
+It boils down to
+<tt class="docutils literal">\xinttheexpr polname(numerical <span class="pre">expression)\relax</span></tt>.</blockquote>
</div>
+<div class="section" id="poleval-polname-at-fraction">
+<span id="polevalat"></span><h2><a class="toc-backref" href="#id83"><tt class="docutils literal"><span class="pre">\PolEval{polname}\At{fraction}</span></tt></a></h2>
+<blockquote>
+<p>Evaluates the polynomial at value <tt class="docutils literal">fraction</tt> which must be in (or
+expand to) a format acceptable to the <a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a> macros.</p>
+<div class="admonition attention">
+<p class="first admonition-title">Attention!</p>
+<p>Meaning was changed at <tt class="docutils literal">0.4</tt>. Formerly <tt class="docutils literal"><span class="pre">\PolEval{P}\At{foo}</span></tt>
+accepted for <tt class="docutils literal">foo</tt> an expression which was handled by
+<tt class="docutils literal">\xintexpr</tt>. See <a class="reference internal" href="#poleval-polname-atexpr-numerical-expression">\PolEval{polname}\AtExpr{numerical
+expression}</a>.</p>
+<p class="last">In particular, to use an <tt class="docutils literal">\xintexpr</tt> user-declared variable (or
+e.g. the variables as defined by <a class="reference external" href="PolSturmIsolateZeros">\PolSturmIsolateZeros</a>) one <strong>must</strong> use the <tt class="docutils literal">\AtExpr</tt> syntax.</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>
+<div class="section" id="polevalreduced-polname-atexpr-numerical-expression">
+<span id="polevalreducedatexpr"></span><h2><a class="toc-backref" href="#id84"><tt class="docutils literal"><span class="pre">\PolEvalReduced{polname}\AtExpr{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>
+<div class="section" id="polevalreduced-polname-at-fraction">
+<span id="polevalreducedat"></span><h2><a class="toc-backref" href="#id85"><tt class="docutils literal"><span class="pre">\PolEvalReduced{polname}\At{fraction}</span></tt></a></h2>
<blockquote>
+<p>Evaluates the polynomial at value <tt class="docutils literal">fraction</tt> which must be in (or
+expand to) a format acceptable to the <a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a> macros, and produce
+an irreducible fraction.</p>
+<div class="admonition attention">
+<p class="first admonition-title">Attention!</p>
+<p class="last">Meaning was changed at <tt class="docutils literal">0.4</tt>. Formerly the evaluation point
+could be given as an expression.</p>
+</div>
+</blockquote>
+</div>
+<div class="section" id="polfloateval-polname-atexpr-numerical-expression">
+<span id="polfloatevalatexpr"></span><h2><a class="toc-backref" href="#id86"><tt class="docutils literal"><span class="pre">\PolFloatEval{polname}\AtExpr{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>
+<p>This is done via a Horner Scheme (see <a class="reference internal" href="#poldef">\poldef</a> and
+<a class="reference internal" href="#polgenfloatvariant-polname">\PolGenFloatVariant{polname}</a>), with already rounded
+coefficients. <a class="footnote-reference" href="#id19" id="id17">[2]</a> To use the <em>exact coefficients</em> with <em>exactly
+executed</em> additions and multiplications, just insert it in the float
+expression as in this example: <a class="footnote-reference" href="#id20" id="id18">[3]</a></p>
<pre class="literal-block">
\xintthefloatexpr 3.27*\xintexpr f(2.53)\relax^2\relax
</pre>
@@ -829,25 +1405,61 @@
<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">
+<div class="admonition attention">
+<p class="first admonition-title">Attention!</p>
+<p class="last">At <tt class="docutils literal">polexpr 0.3</tt>, polynoms were automatically also prepared for
+use in floating point contexts. This got dropped at <tt class="docutils literal">0.4</tt> for
+optimization purposes. See <a class="reference internal" href="#polgenfloatvariant-polname">\PolGenFloatVariant{polname}</a>.</p>
+</div>
+<table class="docutils footnote" frame="void" id="id19" rules="none">
<colgroup><col class="label" /><col /></colgroup>
<tbody valign="top">
-<tr><td class="label"><a class="fn-backref" href="#id6">[2]</a></td><td>Anyway each floating point operation starts by rounding its
+<tr><td class="label"><a class="fn-backref" href="#id17">[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="id9" rules="none">
+<table class="docutils footnote" frame="void" id="id20" rules="none">
<colgroup><col class="label" /><col /></colgroup>
<tbody valign="top">
-<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>
+<tr><td class="label"><a class="fn-backref" href="#id18">[3]</a></td><td>The <tt class="docutils literal">\xintexpr</tt> here 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>
</blockquote>
</div>
+<div class="section" id="polfloateval-polname-at-fraction">
+<span id="polfloatevalat"></span><h2><a class="toc-backref" href="#id87"><tt class="docutils literal"><span class="pre">\PolFloatEval{polname}\At{fraction}</span></tt></a></h2>
+<blockquote>
+<p>Evaluates the polynomial at value <tt class="docutils literal">fraction</tt> which must be in (or
+expand to) a format acceptable to the <a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a> macros, and produces
+a floating point number.</p>
+<div class="admonition attention">
+<p class="first admonition-title">Attention!</p>
+<p class="last">Meaning was changed at <tt class="docutils literal">0.4</tt>. Formerly the evaluation point
+could be given as an expression.</p>
+</div>
+</blockquote>
+</div>
+<div class="section" id="polifcoeffisplusorminusone-a-b">
+<span id="polifcoeffisplusorminusone"></span><h2><a class="toc-backref" href="#id88"><tt class="docutils literal"><span class="pre">\PolIfCoeffIsPlusOrMinusOne{A}{B}</span></tt></a></h2>
+<blockquote>
+<p>This macro is a priori undefined.</p>
+<p>It is defined via the default <a class="reference internal" href="#poltypesetcmd-raw-coeff">\PolTypesetCmd{raw_coeff}</a> to be
+used if needed in the execution of <a class="reference internal" href="#poltypesetmonomialcmd">\PolTypesetMonomialCmd</a>,
+e.g. to insert a <tt class="docutils literal">\cdot</tt> in front of <tt class="docutils literal"><span class="pre">\PolVar^{\PolIndex}</span></tt> if
+the coefficient is not plus or minus one.</p>
+<p>The macro will execute <tt class="docutils literal">A</tt> if the coefficient has been found to be
+plus or minus one, and <tt class="docutils literal">B</tt> if not.</p>
+</blockquote>
+</div>
+<div class="section" id="polleadingcoeff-polname">
+<span id="polleadingcoeff"></span><h2><a class="toc-backref" href="#id89"><tt class="docutils literal">\PolLeadingCoeff{polname}</tt></a></h2>
+<blockquote>
+Expands to the leading coefficient.</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>
+<span id="polnthcoeff"></span><h2><a class="toc-backref" href="#id90"><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
@@ -854,19 +1466,19 @@
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>
+<span id="poldegree"></span><h2><a class="toc-backref" href="#id91"><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>
+<span id="poltoexpr"></span><h2><a class="toc-backref" href="#id92"><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">
+<p>Expands <a class="footnote-reference" href="#id22" id="id21">[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="id22" 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
+<tr><td class="label"><a class="fn-backref" href="#id21">[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>
@@ -885,7 +1497,7 @@
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>
+<span id="poltoexproneterm"></span><h3><a class="toc-backref" href="#id93"><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
@@ -897,20 +1509,28 @@
or a minus one. See <a class="reference internal" href="#poltoexprtimes">\PolToExprTimes</a>.</p>
</blockquote>
</div>
+<div class="section" id="poltoexpronetermstylea-raw-coeff-number">
+<span id="poltoexpronetermstylea"></span><h3><a class="toc-backref" href="#id94"><tt class="docutils literal"><span class="pre">\PolToExprOneTermStyleA{raw_coeff}{number}</span></tt></a></h3>
+<blockquote>
+Holds the default package meaning of
+<a class="reference internal" href="#poltoexproneterm-raw-coeff-number">\PolToExprOneTerm{raw_coeff}{number}</a>.</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>
+<span id="poltoexpronetermstyleb"></span><h3><a class="toc-backref" href="#id95"><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>issue <tt class="docutils literal">\let\PolToExprOneTerm\PolToExprOneTermStyleB</tt> before usage of
+<tt class="docutils literal">\PolToExpr</tt>. Note that then <tt class="docutils literal">\PolToExprCmd</tt> isn't used at all.
+To revert to package default, issue
+<tt class="docutils literal">\let\PolToExprOneTerm\PolToExprOneTermStyleA</tt>.</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>
+<span id="poltoexprcmd"></span><h3><a class="toc-backref" href="#id96"><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
@@ -920,21 +1540,21 @@
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>
+<span id="poltoexprtermprefix"></span><h3><a class="toc-backref" href="#id97"><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>
+<div class="section" id="id23">
+<span id="poltoexprvar"></span><h3><a class="toc-backref" href="#id98"><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>
+<div class="section" id="id24">
+<span id="poltoexprtimes"></span><h3><a class="toc-backref" href="#id99"><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
@@ -943,14 +1563,14 @@
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>
+<div class="section" id="id26">
+<span id="id25"></span><h2><a class="toc-backref" href="#id100"><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>
+<span id="poltofloatexpr"></span><h2><a class="toc-backref" href="#id101"><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
@@ -957,27 +1577,25 @@
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>
+<p>It is not necessary to have issued
+<a class="reference internal" href="#polgenfloatvariant-polname">\PolGenFloatVariant{polname}</a>. The rounded coefficients are
+not easily recoverable from the <tt class="docutils literal">\xintfloatexpr</tt> polynomial
+function hence <tt class="docutils literal">\PolToFloatExprCmd</tt> operates from the <em>exact</em>
+coefficients anew.</p>
+<p class="last">Attention that both macros obey the prevailing float precision.
+If it is changed between those macro calls, then a mismatch
+exists between the coefficients as used in <tt class="docutils literal">\xintfloatexpr</tt> and
+those output by <tt class="docutils literal">\PolToFloatExpr{polname}</tt>.</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>
+<span id="poltofloatexproneterm"></span><h3><a class="toc-backref" href="#id102"><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>
+<span id="id28"></span><h3><a class="toc-backref" href="#id103"><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>
@@ -999,13 +1617,13 @@
</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>
+<div class="section" id="id30">
+<span id="id29"></span><h2><a class="toc-backref" href="#id104"><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>
+<span id="poltolist"></span><h2><a class="toc-backref" href="#id105"><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
@@ -1012,16 +1630,138 @@
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>
+<span id="poltocsv"></span><h2><a class="toc-backref" href="#id106"><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 class="section" id="polsturmchainlength-sturmname">
+<span id="polsturmchainlength"></span><h2><a class="toc-backref" href="#id107"><tt class="docutils literal">\PolSturmChainLength{sturmname}</tt></a></h2>
+<blockquote>
+<p>Returns the integer <tt class="docutils literal">N</tt> such that <tt class="docutils literal">sturmname_N</tt> is the last one
+in the Sturm chain <tt class="docutils literal">sturmname_0</tt>, <tt class="docutils literal">sturmname_1</tt>, ...</p>
+<p>See <a class="reference internal" href="#poltosturm-polname-sturmname">\PolToSturm{polname}{sturmname}</a>.</p>
+</blockquote>
</div>
+<div class="section" id="polsturmifzeroexactlyknown-sturmname-index-a-b">
+<span id="polsturmifzeroexactlyknown"></span><h2><a class="toc-backref" href="#id108"><tt class="docutils literal"><span class="pre">\PolSturmIfZeroExactlyKnown{sturmname}{index}{A}{B}</span></tt></a></h2>
+<blockquote>
+<p>Executes <tt class="docutils literal">A</tt> if the <tt class="docutils literal">index</tt>th interval reduces to a singleton,
+i.e. the root is known exactly, else <tt class="docutils literal">B</tt>.</p>
+<div class="admonition note">
+<p class="first admonition-title">Note</p>
+<p><tt class="docutils literal">index</tt> may be a TeX count, or a <tt class="docutils literal">\value{latexcounter}</tt>, or a
+numerical expression as parsable by <tt class="docutils literal">\numexpr</tt>: it does not
+have to be given via explicit digits.</p>
+<p class="last">This remark applies also to the other package macros with
+<tt class="docutils literal">index</tt> being the name of the argument in this documentation.
+There is also an out-of-range check done for some reasonable
+error message (right before everything goes haywire).</p>
+</div>
+</blockquote>
+</div>
+<div class="section" id="polsturmisolatedzeroleft-sturmname-index">
+<span id="polsturmisolatedzeroleft"></span><h2><a class="toc-backref" href="#id109"><tt class="docutils literal"><span class="pre">\PolSturmIsolatedZeroLeft{sturmname}{index}</span></tt></a></h2>
+<blockquote>
+Expands to the left end-point for the <tt class="docutils literal">index</tt>th interval
+obtained via <a class="reference internal" href="#polsturmisolatezeros-sturmname">\PolSturmIsolateZeros{sturmname}</a> and possibly
+refined afterwards.</blockquote>
+</div>
+<div class="section" id="polsturmisolatedzeroright-sturmname-index">
+<span id="polsturmisolatedzeroright"></span><h2><a class="toc-backref" href="#id110"><tt class="docutils literal"><span class="pre">\PolSturmIsolatedZeroRight{sturmname}{index}</span></tt></a></h2>
+<blockquote>
+Expands to the right end-point for the <tt class="docutils literal">index</tt>th interval
+obtained via <a class="reference internal" href="#polsturmisolatezeros-sturmname">\PolSturmIsolateZeros{sturmname}</a> and possibly
+refined afterwards.</blockquote>
+</div>
+<div class="section" id="polsturmnbofisolatedzeros-sturmname">
+<span id="polsturmnbofisolatedzeros"></span><h2><a class="toc-backref" href="#id111"><tt class="docutils literal">\PolSturmNbOfIsolatedZeros{sturmname}</tt></a></h2>
+<blockquote>
+Expands to the number of real roots of the polynomial
+<tt class="docutils literal"><sturmname>_0</tt> (which is the number of distinct real roots of the
+polynomial used to create the Sturm chain via
+<a class="reference internal" href="#poltosturm-polname-sturmname">\PolToSturm{polname}{sturmname}</a>.</blockquote>
+</div>
+<div class="section" id="polintervalwidth-sturmname-index">
+<span id="polintervalwidth"></span><h2><a class="toc-backref" href="#id112"><tt class="docutils literal"><span class="pre">\PolIntervalWidth{sturmname}{index}</span></tt></a></h2>
+<blockquote>
+The <tt class="docutils literal">10^E</tt> width of the current <tt class="docutils literal">index</tt>th root localization
+interval. Output is in <a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a> raw <tt class="docutils literal">1/1[E]</tt> format (if not zero).</blockquote>
+</div>
+<div class="section" id="macros-for-use-within-execution-of-polprintintervals">
+<h2><a class="toc-backref" href="#id113">Macros for use within execution of <tt class="docutils literal">\PolPrintIntervals</tt></a></h2>
+<p>More precisely, they can be used within the replacement texts of the
+<a class="reference internal" href="#polprintintervalsprintleftendpoint">\PolPrintIntervalsPrintLeftEndPoint</a>, etc, macros.</p>
+<div class="section" id="id31">
+<span id="polprintintervalstheendpoint"></span><h3><a class="toc-backref" href="#id114"><tt class="docutils literal">\PolPrintIntervalsTheEndPoint</tt></a></h3>
+<blockquote>
+Within a custom <a class="reference internal" href="#polprintintervalsprintleftendpoint">\PolPrintIntervalsPrintLeftEndPoint</a>, custom
+<a class="reference internal" href="#polprintintervalsprintrightendpoint">\PolPrintIntervalsPrintRightEndPoint</a>, or custom
+<a class="reference internal" href="#polprintintervalsprintexactzero">\PolPrintIntervalsPrintExactZero</a> this macro expands to the left
+or right end point of the considered interval. Serves as default
+replacement for <a class="reference internal" href="#polprintintervalsprintleftendpoint">\PolPrintIntervalsPrintLeftEndPoint</a> , etc...</blockquote>
+</div>
+<div class="section" id="id32">
+<span id="polprintintervalstheindex"></span><h3><a class="toc-backref" href="#id115"><tt class="docutils literal">\PolPrintIntervalsTheIndex</tt></a></h3>
+<blockquote>
+Within a custom <a class="reference internal" href="#polprintintervalsprintleftendpoint">\PolPrintIntervalsPrintLeftEndPoint</a>, custom
+<a class="reference internal" href="#polprintintervalsprintrightendpoint">\PolPrintIntervalsPrintRightEndPoint</a>, or custom
+<a class="reference internal" href="#polprintintervalsprintexactzero">\PolPrintIntervalsPrintExactZero</a> this macro expands to the index
+of the considered interval. For example if user wants to print the
+corresponding end points in red, the index value can thus be tested
+in the replacement text of <a class="reference internal" href="#polprintintervalsprintleftendpoint">\PolPrintIntervalsPrintLeftEndPoint</a> and
+the other two similar macros.</blockquote>
+</div>
+<div class="section" id="polifendpointispositive-a-b">
+<span id="polifendpointispositive"></span><h3><a class="toc-backref" href="#id116"><tt class="docutils literal"><span class="pre">\PolIfEndPointIsPositive{A}{B}</span></tt></a></h3>
+<blockquote>
+Within a custom <a class="reference internal" href="#polprintintervalsprintleftendpoint">\PolPrintIntervalsPrintLeftEndPoint</a>, custom
+<a class="reference internal" href="#polprintintervalsprintrightendpoint">\PolPrintIntervalsPrintRightEndPoint</a>, or custom
+<a class="reference internal" href="#polprintintervalsprintexactzero">\PolPrintIntervalsPrintExactZero</a> this macro executes <tt class="docutils literal">A</tt> if
+the considered interval end-point is positive, else <tt class="docutils literal">B</tt>.</blockquote>
+</div>
+<div class="section" id="polifendpointisnegative-a-b">
+<span id="polifendpointisnegative"></span><h3><a class="toc-backref" href="#id117"><tt class="docutils literal"><span class="pre">\PolIfEndPointIsNegative{A}{B}</span></tt></a></h3>
+<blockquote>
+Within a custom <a class="reference internal" href="#polprintintervalsprintleftendpoint">\PolPrintIntervalsPrintLeftEndPoint</a>, custom
+<a class="reference internal" href="#polprintintervalsprintrightendpoint">\PolPrintIntervalsPrintRightEndPoint</a>, or custom
+<a class="reference internal" href="#polprintintervalsprintexactzero">\PolPrintIntervalsPrintExactZero</a> this macro executes <tt class="docutils literal">A</tt> if
+the considered interval end-point is negative, else <tt class="docutils literal">B</tt>.</blockquote>
+</div>
+<div class="section" id="polifendpointiszero-a-b">
+<span id="polifendpointiszero"></span><h3><a class="toc-backref" href="#id118"><tt class="docutils literal"><span class="pre">\PolIfEndPointIsZero{A}{B}</span></tt></a></h3>
+<blockquote>
+Within a custom <a class="reference internal" href="#polprintintervalsprintleftendpoint">\PolPrintIntervalsPrintLeftEndPoint</a>, custom
+<a class="reference internal" href="#polprintintervalsprintrightendpoint">\PolPrintIntervalsPrintRightEndPoint</a>, or custom
+<a class="reference internal" href="#polprintintervalsprintexactzero">\PolPrintIntervalsPrintExactZero</a> this macro executes <tt class="docutils literal">A</tt> if
+the considered interval end-point is zero, else <tt class="docutils literal">B</tt>.</blockquote>
+</div>
+</div>
+<div class="section" id="poldectostring-decimal-number">
+<span id="poldectostring"></span><h2><a class="toc-backref" href="#id119"><tt class="docutils literal">\PolDecToString{decimal number}</tt></a></h2>
+<blockquote>
+<p>This is a utility macro to print decimal numbers. Indeed for legacy
+reasons, <a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a> does not yet have user-level ready-to-use macros
+handling specifically the printing of decimal numbers from their
+internal representations such as <tt class="docutils literal">A/1[N]</tt>.</p>
+<p>For example
+<tt class="docutils literal"><span class="pre">\PolDecToString{123.456e-8}</span></tt> will expand to <tt class="docutils literal">0.00000123456</tt>
+and <tt class="docutils literal"><span class="pre">\PolDecToString{123.450e-8}</span></tt> to <tt class="docutils literal">0.00000123450</tt>. This
+illustrates that trailing zeros are not trimmed (to achieve that one
+can use <tt class="docutils literal"><span class="pre">\PolDecToString{\xintREZ{#1}}</span></tt>.)</p>
+<p>The macro does not try to identify if the fraction has a denominator
+consisting only of two's and five's; such a denominator will be left
+at right-end of output.</p>
+<p>This utility macro will presumably be incorporated (possibly in a
+more powerful form) to <a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a> (or rather to a decimal module) in
+a future <a class="reference external" href="http://www.ctan.org/pkg/xint">xint</a> release.</p>
+</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>
+<h1><a class="toc-backref" href="#id120">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>
+<h2><a class="toc-backref" href="#id121"><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
@@ -1034,13 +1774,13 @@
</blockquote>
</div>
<div class="section" id="poltypesetallfalse">
-<h2><a class="toc-backref" href="#id58"><tt class="docutils literal">\poltypesetallfalse</tt></a></h2>
+<h2><a class="toc-backref" href="#id122"><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>
+<h2><a class="toc-backref" href="#id123"><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>
@@ -1047,7 +1787,7 @@
</div>
</div>
<div class="section" id="technicalities">
-<h1><a class="toc-backref" href="#id60">Technicalities</a></h1>
+<h1><a class="toc-backref" href="#id124">Technicalities</a></h1>
<ul>
<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
@@ -1058,7 +1798,7 @@
<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
<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
+<tt class="docutils literal">a/b + c/d</tt> with <tt class="docutils literal">lcm(b,d)</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
to big denominators.</p>
@@ -1082,7 +1822,8 @@
1/6+4/6*x^1+4/6*x^2+6/6*x^3+20/6*x^4+16/6*x^5+9/6*x^6+24/6*x^7+16/6*x^8
</pre>
<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>
+example is the least common multiple of the denominators of the
+reduced coefficients.)</p>
</li>
<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>
@@ -1111,53 +1852,193 @@
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>
</ul>
</div>
-<div class="section" id="releases">
-<h1><a class="toc-backref" href="#id61">RELEASES</a></h1>
+<div class="section" id="change-log">
+<h1><a class="toc-backref" href="#id125">CHANGE LOG</a></h1>
<ul>
-<li><p class="first">0.1 (2018/01/11)</p>
-<p>Initial release (files README, polexpr.sty).</p>
+<li><p class="first">v0.1 (2018/01/11): initial release. Features:</p>
+<ul class="simple">
+<li>The <a class="reference internal" href="#poldef">\poldef</a> parser itself,</li>
+<li>Differentiation and anti-differentiation,</li>
+<li>Euclidean division and GCDs,</li>
+<li>Various utilities such as <a class="reference internal" href="#polfromcsv">\PolFromCSV</a>,
+<a class="reference internal" href="#polmapcoeffs">\PolMapCoeffs</a>,
+<a class="reference internal" href="#poltocsv">\PolToCSV</a>, <a class="reference internal" href="#poltoexpr">\PolToExpr</a>, ...</li>
+</ul>
+<p>Only one-variable polynomials so far.</p>
</li>
-<li><p class="first">0.2 (2018/01/14)</p>
-<p>Documentation moved to polexpr.{txt,html}.</p>
+<li><p class="first">v0.2 (2018/01/14)</p>
+<ul class="simple">
+<li>Fix: <tt class="docutils literal">"README thinks \numexpr recognizes ^ operator"</tt>.</li>
+<li>Convert README to reStructuredText markup.</li>
+<li>Move main documentation from README to separate <tt class="docutils literal">polexpr.txt</tt> file.</li>
+<li>Provide <tt class="docutils literal">polexpr.html</tt> as obtained via <a class="reference external" href="http://docutils.sourceforge.net/docs/index.html">DocUtils</a> <tt class="docutils literal">rst2html.py</tt>.</li>
+<li>Convert README to (CTAN compatible) Markdown markup.</li>
+</ul>
+<p>Due to lack of available time the test suite might not be extensive
+enough. Bug reports are very welcome!</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><p class="first">v0.3 (2018/01/17)</p>
+<ul>
+<li><p class="first">bug fixes:</p>
+<ul>
+<li><p class="first">the <tt class="docutils literal">0.1</tt> <a class="reference internal" href="#polevalat">\PolEval</a> accepted expressions for its second
+argument, but this was removed by mistake at <tt class="docutils literal">0.2</tt>. Restored.</p>
+<p><strong>Attention</strong>: at <tt class="docutils literal">0.4</tt> this has been reverted again, and
+<a class="reference internal" href="#polevalatexpr">\PolEval{P}\AtExpr{foo}</a> syntax is needed for
+using expressions in the second argument.</p>
</li>
-<li><p class="first">0.3.1 (2018/01/18)</p>
-<p>Fix two typos in documentation.</p>
+</ul>
</li>
+<li><p class="first">incompatible or breaking changes:</p>
+<ul class="simple">
+<li><a class="reference internal" href="#poltoexpr">\PolToExpr</a> now by default uses <em>descending</em>
+powers (it also treats differently coefficients equal to 1 or -1.)
+Use <a class="reference internal" href="#id25">\PolToExpr*</a> for <em>ascending</em> powers.</li>
+<li><a class="reference internal" href="#polevalat">\PolEval</a> reduced the output to smallest terms,
+but as this is costly with big fractions and not needed if e.g.
+wrapped in an <tt class="docutils literal">\xintRound</tt> or <tt class="docutils literal">\xintFloat</tt>, this step has been
+removed; the former meaning is available as <a class="reference internal" href="#polevalreducedat">\PolEvalReduced</a>.</li>
</ul>
-<p>Files of 0.3.1 release:</p>
+</li>
+<li><p class="first">new (or newly documented) macros:</p>
<ul class="simple">
-<li>README.md,</li>
-<li>polexpr.sty (package file),</li>
-<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>
+<li><a class="reference internal" href="#poltypesetcmd">\PolTypesetCmd</a></li>
+<li><a class="reference internal" href="#poltypesetcmdprefix">\PolTypesetCmdPrefix</a></li>
+<li><a class="reference internal" href="#poltypesetmonomialcmd">\PolTypesetMonomialCmd</a></li>
+<li><a class="reference internal" href="#polevalreducedat">\PolEvalReducedAt</a></li>
+<li><a class="reference internal" href="#poltofloatexpr">\PolToFloatExpr</a></li>
+<li><a class="reference internal" href="#poltoexproneterm">\PolToExprOneTerm</a></li>
+<li><a class="reference internal" href="#poltofloatexproneterm">\PolToFloatExprOneTerm</a></li>
+<li><a class="reference internal" href="#poltoexprcmd">\PolToExprCmd</a></li>
+<li><a class="reference internal" href="#id28">\PolToFloatExprCmd</a></li>
+<li><a class="reference internal" href="#poltoexprtermprefix">\PolToExprTermPrefix</a></li>
+<li><a class="reference internal" href="#poltoexprvar">\PolToExprVar</a></li>
+<li><a class="reference internal" href="#poltoexprtimes">\PolToExprTimes</a></li>
</ul>
-<p>See README.md for the License and the change log (there were
-some breaking changes from 0.2 to 0.3).</p>
+</li>
+<li><p class="first">improvements:</p>
+<ul>
+<li><p class="first">documentation has a table of contents, internal hyperlinks,
+standardized signature notations and added explanations.</p>
+</li>
+<li><p class="first">one can do <tt class="docutils literal"><span class="pre">\PolLet{g}={f}</span></tt> or <tt class="docutils literal"><span class="pre">\PolLet{g}{f}</span></tt>.</p>
+</li>
+<li><p class="first"><tt class="docutils literal">\PolToExpr{f}</tt> is highly customizable.</p>
+</li>
+<li><p class="first"><a class="reference internal" href="#poldef">\poldef</a> and other defining macros prepare the polynomial
+functions for usage within <tt class="docutils literal">\xintthefloatexpr</tt> (or
+<tt class="docutils literal">\xintdeffloatvar</tt>). 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.</p>
+<p><strong>Attention</strong>: this has been reverted at <tt class="docutils literal">0.4</tt>. The macro
+<a class="reference internal" href="#polgenfloatvariant">\PolGenFloatVariant</a> must be used for
+generation floating point polynomial functions.</p>
+</li>
+</ul>
+</li>
+</ul>
+</li>
+<li><p class="first">v0.3.1 (2018/01/18)</p>
+<p>Fixes two typos in example code included in the documentation.</p>
+</li>
+<li><p class="first">v0.4 (2018/02/16)</p>
+<ul>
+<li><p class="first">bug fixes:</p>
+<ul class="simple">
+<li>when Euclidean division gave a zero remainder, the internal
+representation of this zero polynomial could be faulty; this
+could cause mysterious bugs in conjunction with other package
+macros such as <a class="reference internal" href="#polmapcoeffs">\PolMapCoeffs</a>.</li>
+<li><a class="reference internal" href="#polgcd">\PolGCD</a> was buggy in case of first polynomial being
+of lesser degree than the second one.</li>
+</ul>
+</li>
+<li><p class="first">breaking changes:</p>
+<ul>
+<li><p class="first">formerly <a class="reference internal" href="#polevalat">\PolEval{P}\At{foo}</a> allowed <tt class="docutils literal">foo</tt> to
+be an expression, which was transparently handled via
+<tt class="docutils literal">\xinttheexpr</tt>. Now, <tt class="docutils literal">foo</tt> must be a fraction (or a macro
+expanding to such) in the format acceptable by <tt class="docutils literal">xintfrac.sty</tt>
+macros. Use <a class="reference internal" href="#polevalatexpr">\PolEval{P}\AtExpr{foo}</a> for more
+general arguments using expression syntax. E.g., if <tt class="docutils literal">foo</tt> is the
+name of a variable known to <tt class="docutils literal">\xintexpr</tt>.</p>
+<p>The same holds for <a class="reference internal" href="#polevalreducedat">\PolEvalReduced</a>
+and <a class="reference internal" href="#polfloatevalat">\PolFloatEval</a>.</p>
+</li>
+<li><p class="first">the <tt class="docutils literal">3.0</tt> automatic generation of floating point variants has
+been reverted. Not only do <em>not</em> the package macros automatically
+generate floating point variants of newly created polynomials,
+they actually make pre-existing such variant undefined.</p>
+<p>See <a class="reference internal" href="#polgenfloatvariant">\PolGenFloatVariant</a>.</p>
+</li>
+</ul>
+</li>
+<li><p class="first">new non-expandable macros:</p>
+<ul class="simple">
+<li><a class="reference internal" href="#polgenfloatvariant">\PolGenFloatVariant</a></li>
+<li><a class="reference internal" href="#polgloballet">\PolGlobalLet</a></li>
+<li><a class="reference internal" href="#poltypesetone">\PolTypesetOne</a></li>
+<li><a class="reference internal" href="#polquo">\PolQuo</a></li>
+<li><a class="reference internal" href="#polrem">\PolRem</a></li>
+<li><a class="reference internal" href="#poltosturm">\PolToSturm</a></li>
+<li><a class="reference internal" href="#id9">\PolToSturm*</a></li>
+<li><a class="reference internal" href="#polsettosturmchainsignchangesat">\PolSetToSturmChainSignChangesAt</a></li>
+<li><a class="reference internal" href="#polsettonbofzeroswithin">\PolSetToNbOfZerosWithin</a></li>
+<li><a class="reference internal" href="#polsturmisolatezeros">\PolSturmIsolateZeros</a></li>
+<li><a class="reference internal" href="#polrefineinterval">\PolRefineInterval*</a></li>
+<li><a class="reference internal" href="#polrefineinterval-n">\PolRefineInterval[N]</a></li>
+<li><a class="reference internal" href="#polensureintervallength">\PolEnsureIntervalLength</a></li>
+<li><a class="reference internal" href="#polensureintervallengths">\PolEnsureIntervalLengths</a></li>
+<li><a class="reference internal" href="#polprintintervals">\PolPrintIntervals</a></li>
+<li><a class="reference internal" href="#polprintintervalsprintexactzero">\PolPrintIntervalsPrintExactZero</a></li>
+<li><a class="reference internal" href="#polprintintervalsprintleftendpoint">\PolPrintIntervalsPrintLeftEndPoint</a></li>
+<li><a class="reference internal" href="#polprintintervalsprintrightendpoint">\PolPrintIntervalsPrintRightEndPoint</a></li>
+<li><a class="reference internal" href="#id14">\PolReduceCoeffs*</a></li>
+<li><a class="reference internal" href="#polmakemonic">\PolMakeMonic</a></li>
+</ul>
+</li>
+<li><p class="first">new expandable macros:</p>
+<ul class="simple">
+<li><a class="reference internal" href="#poltoexpronetermstylea">\PolToExprOneTermStyleA</a></li>
+<li><a class="reference internal" href="#polifcoeffisplusorminusone">\PolIfCoeffIsPlusOrMinusOne</a></li>
+<li><a class="reference internal" href="#polleadingcoeff">\PolLeadingCoeff</a></li>
+<li><a class="reference internal" href="#polsturmchainlength">\PolSturmChainLength</a></li>
+<li><a class="reference internal" href="#polsturmnbofisolatedzeros">\PolSturmNbOfIsolatedZeros</a></li>
+<li><a class="reference internal" href="#polsturmifzeroexactlyknown">\PolSturmIfZeroExactlyKnown</a></li>
+<li><a class="reference internal" href="#polsturmisolatedzeroleft">\PolSturmIsolatedZeroLeft</a></li>
+<li><a class="reference internal" href="#polsturmisolatedzeroright">\PolSturmIsolatedZeroRight</a></li>
+<li><a class="reference internal" href="#polprintintervalstheendpoint">\PolPrintIntervalsTheEndPoint</a></li>
+<li><a class="reference internal" href="#polprintintervalstheindex">\PolPrintIntervalsTheIndex</a></li>
+<li><a class="reference internal" href="#polifendpointispositive">\PolIfEndPointIsPositive</a></li>
+<li><a class="reference internal" href="#polifendpointisnegative">\PolIfEndPointIsNegative</a></li>
+<li><a class="reference internal" href="#polifendpointiszero">\PolIfEndPointIsZero</a></li>
+<li><a class="reference internal" href="#polintervalwidth">\PolIntervalWidth</a></li>
+<li><a class="reference internal" href="#poldectostring">\PolDecToString</a></li>
+</ul>
+</li>
+<li><p class="first">improvements:</p>
+<p>The main new feature is implementation of the <a class="reference external" href="https://en.wikipedia.org/wiki/Sturm%27s_theorem">Sturm algorithm</a>
+for localization of the real roots of polynomials.</p>
+</li>
+</ul>
+</li>
+</ul>
</div>
<div class="section" id="acknowledgments">
-<h1><a class="toc-backref" href="#id62">Acknowledgments</a></h1>
+<h1><a class="toc-backref" href="#id126">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>
+<p>Renewed thanks on occasion of <tt class="docutils literal">0.4</tt> release!</p>
+<p>See README.md for the License.</p>
</div>
</div>
</body>
Modified: trunk/Master/texmf-dist/doc/latex/polexpr/polexpr.txt
===================================================================
--- trunk/Master/texmf-dist/doc/latex/polexpr/polexpr.txt 2018-02-16 01:23:20 UTC (rev 46652)
+++ trunk/Master/texmf-dist/doc/latex/polexpr/polexpr.txt 2018-02-16 22:30:09 UTC (rev 46653)
@@ -4,12 +4,12 @@
Package polexpr documentation
===============================
-0.3.1 (2018/01/18)
-==================
+0.4 (2018/02/16)
+================
.. contents::
-First Examples
+Basic Examples
--------------
The syntax is::
@@ -68,7 +68,7 @@
$f''(z) = \PolTypeset[z]{f_xx}$\newline
$f'''(z)= \PolTypeset[z]{d3f_dx3}$\par
-.. important::
+.. important::
The package does not currently know rational functions: ``/`` in
a parsed polynomial expression does the Euclidean quotient::
@@ -85,7 +85,7 @@
.. _warningtacit:
-.. attention::
+.. attention::
``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
@@ -117,6 +117,209 @@
``\PolToExpr*{k}``
gives ascending powers: ``2-2*x-x^2+x^3``.
+Examples of localization of roots
+---------------------------------
+
+First some remarks about auxiliaries.
+
+- To make printed decimal numbers more enjoyable than via
+ ``\xintSignedFrac``::
+
+ \renewcommand\PolTypesetOne[1]{\PolDecToString{\xintREZ{#1}}}%
+
+ ``\PolDecToString`` will use decimal notation to incorporate the power
+ of ten part; and the ``\xintREZ`` will have the effect to suppress
+ trailing zeros if present in raw numerator (if those digits end up
+ after decimal mark.) Notice that the above are expandable macros and
+ that one can also do::
+
+ \renewcommand\PolToExprCmd[1]{\PolDecToString{\xintREZ{#1}}}%
+
+ to modify output of `\\PolToExpr{polname}`_.
+
+- for extra info in log file ``\xintverbosetrue``. In fact one can also
+ work from command line then (I recommend ``rlwrap`` for encapsulating
+ ``latex``).
+
+
+A typical example
+~~~~~~~~~~~~~~~~~
+
+::
+
+ \poldef f(x) := x^7 - x^6 - 2x + 1;
+
+ \PolToSturm{f}{f}
+ \PolSturmIsolateZeros{f}
+ The \PolTypeset{f} polynomial has \PolSturmNbOfIsolatedZeros{f} distinct real
+ roots which are located in the following intervals:
+ \PolPrintIntervals{f}
+ Here is the second root with ten more decimal digits:
+ \PolRefineInterval[10]{f}{2}
+ \[\PolSturmIsolatedZeroLeft{f}{2}<Z_2<\PolSturmIsolatedZeroRight{f}{2}\]
+ And here is the first root with twenty digits after decimal mark:
+ \PolEnsureIntervalLength{f}{1}{-20}
+ \[\PolSturmIsolatedZeroLeft{f}{1}<Z_1<\PolSturmIsolatedZeroRight{f}{1}\]
+ The derivative polynomial is \PolTypeset{f_1}.
+ \PolToSturm{f_1}{f_1}\PolSturmIsolateZeros{f_1}%
+ It has \PolSturmNbOfIsolatedZeros{f_1} distinct real
+ roots:
+ \PolPrintIntervals[W]{f_1}
+ \PolEnsureIntervalLengths{f_1}{-10}%
+ Here they are with ten digits after decimal mark:
+ \PolPrintIntervals[W]{f_1}
+ \PolDiff{f_1}{f_xx}
+ \PolToSturm{f_xx}{f_xx}
+ \PolSturmIsolateZeros{f_xx}
+ The second derivative is \PolTypeset{f_xx}.
+ It has \PolSturmNbOfIsolatedZeros{f_xx} distinct real
+ roots:
+ \PolPrintIntervals[X]{f_xx}
+ Here is the positive one with 20 digits after decimal mark:
+ \PolEnsureIntervalLength{f_xx}{2}{-20}%
+ \[X_2 = \PolSturmIsolatedZeroLeft{f_xx}{2}\dots\]
+ The more mathematically advanced among our dear readers will be able
+ to give the exact value for $X_2$!
+
+A degree four polynomial with nearby roots
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+::
+
+ \PolDef{Q}{(x-1.050001)(x-1.105001)(x-1.110501)(x-1.111051)}
+ \PolTypeset{Q}
+ \PolToSturm{Q}{Q} % it is allowed to use same prefix for Sturm chain
+ \PolSturmIsolateZeros{Q}
+ \PolPrintIntervals{Q}
+ % reports 1.0 < Z_1 < 1.1, 1.10 < Z_2 < 1.11, 1.110 < Z_3 < 1.111, and 1.111 < Z_4 < 1.112
+ % but the above bounds do not allow minimizing separation between roots
+ % so we refine:
+ \PolRefineInterval*{Q}{1}
+ \PolRefineInterval*{Q}{2}
+ \PolRefineInterval*{Q}{3}
+ \PolRefineInterval*{Q}{4}
+ \PolPrintIntervals{Q}
+ % reports 1.05 < Z_1 < 1.06, 1.105 < Z_2 < 1.106, 1.1105 < Z_3 < 1.1106,
+ % and 1.11105 < Z_4 < 1.11106.
+ \PolEnsureIntervalLengths{Q}{-6}
+ \PolPrintIntervals{Q}
+ % of course finds here all roots exactly
+
+
+The degree nine polynomial with 0.99, 0.999, 0.9999 as triple roots
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+::
+
+ \PolDef{P}{(x-0.99)^3(x-0.999)^3(x-0.9999)^3}
+ \PolTypeset{P}\par
+ \PolToSturm{P}{P}%
+ \PolLet{Psqfree}{P_0}\PolMakeMonic{Psqfree}\PolReduceCoeffs*{Psqfree}
+ \par
+ The monic square-free radical is \PolTypeset{Psqfree}.
+ \PolSturmIsolateZeros{P}
+ \par
+ It has \PolSturmNbOfIsolatedZeros{P} real roots.
+ \PolPrintIntervals{P}% all three roots found exactly
+
+A Mignotte type polynomial
+~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+::
+
+ \PolDef{P}{x^10 - (10x-1)^2}%
+ \PolTypeset{P} % prints it in expanded form
+ \PolToSturm{P}{P} % we can use same prefix for Sturm chain
+ \PolSturmIsolateZeros{P} % finds 4 real roots
+ \PolPrintIntervals{P}%
+ % reports -2 < Z_1 < -1, 0 < Z_2 < 0.1, 0.1 < Z_3 < 0.2, 1 < Z_4 < 2
+ \PolRefineInterval*{P}{2}% will refine to 0.0999990 < Z_2 < 0.0999991
+ \PolRefineInterval*{P}{3}% will refine to 0.100001 < Z_3 < 0.100002
+ \PolPrintIntervals{P}%
+ \PolEnsureIntervalLengths{P}{-10}%
+ \PolPrintIntervals{P}% now all roots are known 10 decimal digits after mark
+ \PolEnsureIntervalLength{P}{2}{-20}% makes Z_2 known with 20 digits after mark
+ There are $\PolSturmNbOfIsolatedZeros{P}$ distinct real roots and there holds
+ $\PolSturmIsolatedZeroLeft{P}{2}<Z_2<\PolSturmIsolatedZeroRight{P}{2}$.
+
+The last line produces::
+
+ 0.09999900004999650028 < Z_2 < 0.09999900004999650029
+
+The degree 41 polynomial with -2, -1.9, -1.8, ..., 0, 0.1, ..., 1.9, 2 as roots
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+::
+
+ \PolDef{P}{mul((x-i*1e-1), i=-20..20)}% i/10 is same but less efficient
+
+In the defining expression we could have used ``i/10`` but this gives
+less efficient internal form for the coefficients (the ``10``'s end up
+in denominators). Using ``\PolToExpr{P}`` after having done
+
+::
+
+ \renewcommand\PolToExprCmd[1]{\PolDecToString{\xintREZ{#1}}}
+
+we get this expanded form::
+
+ x^41
+ -28.7*x^39
+ +375.7117*x^37
+ -2975.11006*x^35
+ +15935.28150578*x^33
+ -61167.527674162*x^31
+ +173944.259366417394*x^29
+ -373686.963560544648*x^27
+ +613012.0665016658846445*x^25
+ -771182.31133138163125495*x^23
+ +743263.86672885754888959569*x^21
+ -545609.076599482896371978698*x^19
+ +301748.325708943677229642930528*x^17
+ -123655.8987669450434698869844544*x^15
+ +36666.1782054884005855608205864192*x^13
+ -7607.85821367459445649518380016128*x^11
+ +1053.15135918687298508885950223794176*x^9
+ -90.6380005918141132650786081964032*x^7
+ +4.33701563847327366842552218288128*x^5
+ -0.0944770968420804735498178265088*x^3
+ +0.00059190121813899276854174416896*x
+
+which shows coefficients with up to 36 significant digits...
+
+Stress test: not a hard challenge to ``xint + polexpr``, but be a bit patient!
+
+::
+
+ \PolDef{P}{mul((x-i*1e-1), i=-20..20)}%
+ \PolToSturm{P}{S} % dutifully computes S_0, ..., S_{41}
+ \PolSturmIsolateZeros{S} % finds *exactly* (but a bit slowly) all 41 roots!
+ \PolPrintIntervals{S} % nice, isn't it?
+
+Roots of Chebyshev polynomials
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+::
+
+ \newcount\mycount
+ \poldef T_0(x) := 1;
+ \poldef T_1(x) := x;
+ \mycount 2
+ \xintloop
+ \poldef T_\the\mycount(x) :=
+ 2x*T_\the\numexpr\mycount-1(x)
+ - T_\the\numexpr\mycount-2(x);
+ \ifnum\mycount<15
+ \advance\mycount 1
+ \repeat
+
+ \[T_{15} = \PolTypeset[X]{T_15}\]
+ \PolToSturm{T_15}{T_15}
+ \PolSturmIsolateZeros{T_15}
+ \PolEnsureIntervalLengths{T_15}{-10}
+ \PolPrintIntervals{T_15}
+
+
Non-expandable macros
---------------------
@@ -143,28 +346,21 @@
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.
+ .. attention::
- .. important::
+ Release ``0.3`` also did the necessary set-up to let the
+ polynomial be known to the ``\xintfloatexpr`` (or
+ ``\xintdeffloatvar``) parser.
- Package macros (such as derivatives or Euclidean division)
- operate with the "exact" polynomials; "floating point"
- polynomials are always obtained in a second step.
+ Since ``0.4`` this isn't done automatically. Even more, a
+ previously existing floating point variant of the same name will
+ be let undefined again, to avoid hard to debug mismatches between
+ exact and floating point polynomials. This also applies when the
+ polynomial is produced not via ``\poldef`` or ``\PolDef`` but as
+ a product of the other package macros.
- To modifiy "in-place" the original coefficients of a polynomial
- and round them to float precision::
+ See `\\PolGenFloatVariant{polname}`_.
- \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.
@@ -174,11 +370,35 @@
``\PolDef[letter]{polname}{expression in letter}``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
- 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``.
+ Does the same as `\\poldef <poldef;>`_ 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``.
+.. _PolGenFloatVariant:
+
+``\PolGenFloatVariant{polname}``
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+ Makes the polynomial also usable in the ``\xintfloatexpr`` parser.
+ It will therein evaluates via an Horner scheme with coefficients
+ already pre-rounded to the float precision.
+
+ See also `\\PolToFloatExpr{polname}`_.
+
+ .. attention::
+
+ Release ``0.3`` did this automatically on ``\PolDef`` and
+ ``\poldef`` but this was removed at ``0.4`` for optimization.
+
+ Any operation, for example generating the derivative polynomial,
+ or dividing two polynomials or using the ``\PolLet``, **must** be
+ followed by explicit usage of ``\PolGenFloatVariant{polname}`` if
+ the new polynomial is to be used in ``\xintfloatexpr`` or alike
+ context.
+
+.. _PolLet:
+
``\PolLet{polname_2}={polname_1}``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
@@ -187,6 +407,15 @@
``\PolDef{polname_2}{polname_1(x)}`` but with less overhead. The
``=`` is optional.
+.. _PolGlobalLet:
+
+``\PolGlobalLet{polname_2}={polname_1}``
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+ Acts globally.
+
+.. _PolAssign:
+
``\PolAssign{polname}\toarray\macro``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
@@ -212,6 +441,8 @@
is the shift by one in indexing, mentioned above (negative
indices act the same in both.)
+.. _PolGet:
+
``\PolGet{polname}\fromarray\macro``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
@@ -228,8 +459,10 @@
This will define ``f`` as would have ``\poldef f(x):=1-2x+5x^2-3x^3;``.
However the coefficients are still in their original form (i.e.
- they were not subjected to ``\xintRaw`` or similar xintfrac macro.)
+ they were not subjected to ``\xintRaw`` or similar xintfrac_ macro.)
+.. _PolFromCSV:
+
``\PolFromCSV{polname}{<csv>}``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
@@ -238,7 +471,7 @@
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
+ ``1.5e3`` which is not converted to ``15/1[2]`` *raw* xintfrac_
format (this may change).
Leading zero coefficients are removed::
@@ -249,6 +482,8 @@
See also expandable macro `\\PolToCSV <\\PolToCSV{polname}_>`_.
+.. _PolTypeset:
+
``\PolTypeset{polname}``
~~~~~~~~~~~~~~~~~~~~~~~~
@@ -264,19 +499,32 @@
can be re-defined for customization. Their default definitions are
expandable, but this is not a requirement.
+.. _PolTypesetCmd:
+
``\PolTypesetCmd{raw_coeff}``
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
- 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...
+ Checks if the coefficient is ``1`` or ``-1`` and then skips printing
+ the ``1``, except for the constant term. Also it sets conditional
+ `\\PolIfCoeffIsPlusOrMinusOne{A}{B}`_.
+ The actual printing of the coefficients, when not equal to plus or
+ minus one is handled by `\\PolTypesetOne{raw_coeff}`_.
+
+.. _PolTypesetOne:
+
+``\PolTypesetOne{raw_coeff}``
+^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+
+ The default is ``\xintSignedFrac`` but this macro is annoying as it
+ insists to use a power of ten, and not decimal notation.
+
One can do things such as for example: [#]_
::
- \renewcommand\PolTypesetCmd[1]{\num{\xintPFloat[5]{#1}}}
- \renewcommand\PolTypesetCmd[1]{\num{\xintRound{4}{#1}}}
+ \renewcommand\PolTypesetOne[1]{\num{\xintPFloat[5]{#1}}}
+ \renewcommand\PolTypesetOne[1]{\num{\xintRound{4}{#1}}}
where e.g. we used the ``\num`` macro of ``siunitx`` as it
understands floating point notation.
@@ -286,15 +534,10 @@
``\xintPFloat`` by default uses the prevailing precision
hence the extra argument like here ``5`` is an optional one.
-``\PolTypesetCmdPrefix{raw_coeff}``
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+ One can also give a try to using `\\PolDecToString{decimal number}`_
+ which uses decimal notation (at least for the numerator part).
- 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:
``\PolTypesetMonomialCmd``
^^^^^^^^^^^^^^^^^^^^^^^^^^
@@ -306,6 +549,20 @@
``\PolIndex`` expands to digit tokens and needs termination in
``\ifnum`` tests.
+.. _PolTypesetCmdPrefix:
+
+``\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.
+
+.. _PolTypeset*:
+
``\PolTypeset*{polname}``
~~~~~~~~~~~~~~~~~~~~~~~~~
@@ -312,6 +569,8 @@
Typesets in ascending powers. Use e.g. ``[h]`` optional argument
(after the ``*``) to use letter ``h`` rather than ``x``.
+.. _PolDiff:
+
``\PolDiff{polname_1}{polname_2}``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
@@ -322,6 +581,8 @@
Coefficients of the result ``polname_2`` are irreducible fractions
(see `Technicalities`_ for the whole story.)
+.. _PolDiff[N]:
+
``\PolDiff[N]{polname_1}{polname_2}``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
@@ -330,6 +591,8 @@
``\PolLet{polname_2}={polname_1}``. With negative ``N``, switches to
using ``\PolAntiDiff``.
+.. _PolAntiDiff:
+
``\PolAntiDiff{polname_1}{polname_2}``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
@@ -339,6 +602,8 @@
Coefficients of the result ``polname_2`` are irreducible fractions
(see `Technicalities`_ for the whole story.)
+.. _PolAntiDiff[N]:
+
``\PolAntiDiff[N]{polname_1}{polname_2}``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
@@ -345,6 +610,8 @@
This sets ``polname_2`` to the result of ``N`` successive integrations on
``polname_1``. With negative ``N``, it switches to using ``\PolDiff``.
+.. _PolDivide:
+
``\PolDivide{polname_1}{polname_2}{polname_Q}{polname_R}``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
@@ -352,13 +619,315 @@
remainder in the Euclidean division of ``polname_1`` by
``polname_2``.
+.. _PolQuo:
+
+``\PolQuo{polname_1}{polname_2}{polname_Q}``
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+ This sets ``polname_Q`` to be the quotient in the Euclidean division
+ of ``polname_1`` by ``polname_2``.
+
+.. _PolRem:
+
+``\PolRem{polname_1}{polname_2}{polname_R}``
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+ This sets ``polname_R`` to be the remainder in the Euclidean division
+ of ``polname_1`` by ``polname_2``.
+
+.. _PolGCD:
+
``\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.
+ This sets ``polname_GCD`` to be the (monic) GCD of the two first
+ polynomials. It is a unitary polynomial except if both ``polname_1``
+ and ``polname_2`` vanish, then ``polname_GCD`` is the zero
+ polynomial.
+.. ``\PolIGCD{polname_1}{polname_2}{polname_iGCD}``
+ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+ **NOT YET**
+
+ This **assumes** that the two polynomials have integer coefficients.
+ It then computes the greatest common divisor in the integer
+ polynomial ring, normalized to have a positive leading coefficient
+ (if the inputs are not both zero).
+
+ ``\PolIContent{polname}``
+ ~~~~~~~~~~~~~~~~~~~~~~~~~
+
+ **NOT YET**
+
+ This computes a positive rational number such that dividing the
+ polynomial with it returns an integer coefficients polynomial with
+ no common factor among the coefficients.
+
+.. _PolToSturm:
+
+``\PolToSturm{polname}{sturmname}``
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+ With, for example, ``polname`` being ``P`` and ``sturmname`` being
+ ``S``, the macro starts by computing polynomials ``S_0 = P``, ``S_1
+ = P'``, ..., with ``S_{n+1}`` the opposite of the remainder of
+ euclidean division of ``S_{n-1}`` by ``S_{n}``. The last non-zero
+ remainder ``S_N`` is up to a factor the GCD of ``P`` and ``P'``
+ hence a constant if and only if ``P`` is square-free.
+
+ In case ``S_N`` is not a constant, the macro then goes on with
+ dividing all ``S_k``'s with ``S_N`` (which becomes ``1``).
+
+ Thus ``S_0`` now has exactly the same real and complex
+ roots as polynomial ``polname``, but each with multiplicity one.
+
+.. _PolToSturm*:
+
+``\PolToSturm*{polname}{sturmname}``
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+ Does not divide the Sturm chain by its last element.
+
+.. _PolSetToSturmChainSignChangesAt:
+
+``\PolSetToSturmChainSignChangesAt{\macro}{sturmname}{fraction}``
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+ Sets macro ``\macro`` to the number of sign changes in the Sturm
+ chain with name prefix ``sturmname``, at location ``fraction``
+ (which must be in format as acceptable by the xintfrac_ macros.)
+
+ .. note::
+
+ The author was lazy and did not provide rather an expandable
+ variant, where one would do ``\edef\macro{\PolNbOf...}``.
+
+ This will presumably get added in a future release.
+
+ After some hesitation it was decided the macro would by default
+ act globally. To make the scope of its macro definition local,
+ use ``[\empty]`` as extra optional argument.
+
+.. _PolSetToNbOfZerosWithin:
+
+``\PolSetToNbOfZerosWithin{\macro}{sturmname}{value_a}{value_b}``
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+ Applies the `Sturm Theorem`_ to set ``\macro`` to the exact number
+ of distinct roots of ``sturmname_0`` in the interval ``(value_a,
+ value_b]`` (the macro first re-orders the value for ``value_a <=
+ value_b`` to hold).
+
+ .. note::
+
+ The author was lazy and did not provide rather an expandable
+ variant, where one would do ``\edef\macro{\PolNbOf...}``.
+
+ This will presumably get added in future.
+
+ After some hesitation it was decided the macro would by default
+ act globally. To make the scope of its macro definition local,
+ use ``[\empty]`` as extra optional argument.
+
+.. _PolSturmIsolateZeros:
+
+``\PolSturmIsolateZeros{sturmname}``
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+ First, it evaluates using `Sturm theorem`_ the number of distinct
+ real roots of ``sturmname_0``.
+
+ .. important::
+
+ The Sturm chain **must** be of the reduced type, i.e.
+ as constructed via `\\PolToSturm{polname}{sturmname}`_.
+
+ Then it locates, again using `Sturm theorem`_, as many disjoint
+ intervals as there are roots. Some intervals reduce to singleton
+ which are roots. Non-singleton intervals get refined to make sure
+ none of their two limit points is a root: they contain each a single
+ root, in their respective interiors.
+
+ .. This procedure is covariant
+ with the independent variable ``x`` becoming ``-x``.
+ Hmm, pas sûr et trop fatigué
+
+ The interval boundaries are decimal numbers, originating
+ in iterated decimal subdivision from initial intervals
+ ``(-10^E, 0)`` and ``(0, 10^E)``; if zero is a root it is always
+ identified individually. The non-singleton intervals are of the
+ type ``(a/10^f, (a+1)/10^f)`` with ``a`` an integer, which is
+ neither ``0`` nor ``-1``. Hence ``a`` and ``a+1`` are both positive
+ or both negative.
+
+ The interval boundaries (and exactly found roots) are made available
+ for future computations in ``\xintexpr``-essions or polynomial
+ definitions as variables ``<sturmname>L_1``,
+ ``<sturmname>L_2``, etc..., for the left end-points and
+ ``<sturmname>R_1``, ``<sturmname>R_2``, ..., for the right
+ end-points.
+
+ Also two macro arrays (in the sense of
+ xinttools_'s ``\xintAssignArray``) are created for holding the
+ interval end-points written out in standard decimal notation
+ (see `\\PolDecToString{decimal number}`_).
+ To access these values, macros
+ `\\PolSturmIsolatedZeroLeft{sturmname}{index}`_ and
+ `\\PolSturmIsolatedZeroRight{sturmname}{index}`_ are provided.
+
+ .. important::
+
+ Trailing zeroes in these stored decimal numbers are significant:
+ they are also present in the decimal expansion of the exact root.
+
+ .. note::
+
+ The actual array macros are ``\POL_ZeroInt<sturmname>L`` and
+ ``\POL_ZeroInt<sturmname>R`` but as these names use the
+ non-letter character ``_`` and possibly also digits from
+ ``sturmname``, the accessor macros above have been made part of
+ the package.
+
+ The start of decimal expansion of a positive ``k``-th root is given
+ by ``\PolSturmIsolatedZeroLeft{sturmname}{k}``, and for a negative
+ root it is given by ``\PolSturmIsolatedZeroRight{sturmname}{k}``.
+ These two decimal numbers are either both zero or both of the same
+ sign.
+
+ The number of distinct roots is obtainable as
+ ``\PolSturmNbOfIsolatedZeros{sturmname}``.
+
+ .. note::
+
+ In the current implementation the ``<sturmname>...`` variables
+ and the ``\POL_ZeroInt...`` arrays are globally defined. On the
+ other hand the Sturm sequence polynomials obey the current scope.
+
+ .. note::
+
+ When two successive roots are located in adjacent intervals, the
+ separation between them is not lower bounded. See
+ `\\PolRefineInterval*{sturmname}{index}`_.
+
+ .. note::
+
+ As all computations are done *exactly* there can be no errors...
+ apart those due to bad coding by author. The results are exact
+ bounds for the mathematically exact real roots.
+
+ Future releases will perhaps also provide macros based on Newton
+ or Regula Falsi methods. Exact computations with such methods
+ lead however quickly to very big fractions, and this forces usage
+ of some rounding scheme for the abscissas if computation times
+ are to remain reasonable. This raises issues of its own, which
+ are studied in numerical mathematics.
+
+.. _PolRefineInterval*:
+
+``\PolRefineInterval*{sturmname}{index}``
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+ The ``index``\ -th interval (starting indexing at one) is further
+ subdivided as many times as is necessary in order for the newer
+ interval to have both its end-points distinct from the end-points of
+ the original interval. This means that the ``k``\ th root is then
+ strictly separated from the other roots.
+
+.. _PolRefineInterval[N]:
+
+``\PolRefineInterval[N]{sturmname}{index}``
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+ The ``index``\ -th interval (starting count at one) is further
+ subdivided once, reducing its length by a factor of 10. This is done
+ ``N`` times if the optional argument ``[N]`` is present.
+
+.. _PolEnsureIntervalLength:
+
+``\PolEnsureIntervalLength{sturmname}{index}{E}``
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+ The ``index``\ -th interval is subdivided until its length becomes at
+ most ``10^E``. This means (for ``E<0``) that the first ``-E`` digits
+ after decimal mark of the ``k``\ th root will then be known exactly.
+
+.. _PolEnsureIntervalLengths:
+
+``\PolEnsureIntervalLengths{sturmname}{E}``
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+ The intervals as obtained from ``\PolSturmIsolateZeros`` are (if
+ necessary) subdivided further by (base 10) dichotomy in order for
+ each of them to have length at most ``10^E`` (length will be shorter
+ than ``10^E`` in output only if it did not change or became zero.)
+
+ This means that decimal expansions of all roots will be known with
+ ``-E`` digits (for ``E<0``) after decimal mark.
+
+.. _PolPrintIntervals:
+
+``\PolPrintIntervals[varname]{sturmname}``
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+ This is a convenience macro which prints the bounds for the roots
+ ``Z_1``, ``Z_2``, ... (the optional argument ``varname`` allows to
+ specify a replacement for the default ``Z``). This will be done in a
+ math mode ``array``, one interval per row, and pattern ``rcccl``,
+ where the second and fourth column hold the ``<`` sign, except when
+ the interval reduces to a singleton, which means the root is known
+ exactly. The user is invited to renewcommand the macro if some other
+ type of tabular environment for example is wanted.
+
+ In each array cell the corresponding interval end-point (which may
+ be an exactly known root) is available as macro
+ `\\PolPrintIntervalsTheEndPoint`_ (in decimal notation). And the
+ corresponding interval index is available as
+ `\\PolPrintIntervalsTheIndex`_.
+
+ These values may be tested to decide some on-the-fly customization
+ (color for example), via the following auxiliaries which can be
+ modified by user. Furthermore these auxiliaries can also use the
+ following conditionals: `\\PolIfEndPointIsPositive{A}{B}`_,
+ `\\PolIfEndPointIsNegative{A}{B}`_, `\\PolIfEndPointIsZero{A}{B}`_.
+
+.. _PolPrintIntervalsPrintExactZero:
+
+``\PolPrintIntervalsPrintExactZero``
+^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+ This is provided to help customize how an exactly known root is
+ printed in the right most column of the array. The package
+ definition is::
+
+ \newcommand\PolPrintIntervalsPrintExactZero{\PolPrintIntervalsTheEndPoint}%
+
+ Recall that this is expanded in an array cell.
+
+ If for example you want to print in red the third root, known
+ exactly, the macro could make a test for the value of
+ `\\PolPrintIntervalsTheIndex`_ and act accordingly.
+
+
+.. _PolPrintIntervalsPrintLeftEndPoint:
+
+``\PolPrintIntervalsPrintLeftEndPoint``
+^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+
+ Package definition is::
+
+ \newcommand\PolPrintIntervalsPrintLeftEndPoint{\PolPrintIntervalsTheEndPoint}%
+
+.. _PolPrintIntervalsPrintRightEndPoint:
+
+``\PolPrintIntervalsPrintRightEndPoint``
+^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+
+ Package definition is::
+
+ \newcommand\PolPrintIntervalsPrintRightEndPoint{\PolPrintIntervalsTheEndPoint}%
+
+.. _PolMapCoeffs:
+
``\PolMapCoeffs{\macro}{polname}``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
@@ -371,7 +940,7 @@
Notice that ``\macro`` will have to handle inputs of the shape
``A/B[N]`` (xintfrac_ internal notation). This means that it probably
- will have to be expressed in terms of macros from xintfrac package.
+ will have to be expressed in terms of macros from xintfrac_ package.
Example::
@@ -380,6 +949,8 @@
(or with ``\xintSqr{\index}``) to replace ``n``-th coefficient
``f_n`` by ``f_n*n^2``.
+.. _PolReduceCoeffs:
+
``\PolReduceCoeffs{polname}``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
@@ -388,6 +959,39 @@
polynomial function is used for computations.) This is a
one-argument macro, working 'in-place'.
+.. _PolReduceCoeffs*:
+
+``\PolReduceCoeffs*{polname}``
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+ This starred variant leaves un-touched the decimal exponent in the
+ internal representation of the fractional coefficients, i.e. if a
+ coefficient is internally ``A/B[N]``, then ``A/B`` is reduced to
+ smallest terms, but the ``10^N`` part is kept as is. Note: if the
+ polynomial is freshly defined directly via `\\PolFromCSV
+ <PolFromCSV_>`_ its coefficients might still be internally in some
+ format like ``1.5e7``; the macro will anyhow always first do the
+ needed conversion to strict format ``A/B[N]``.
+
+ Evaluations with polynomials treated by this can be much faster than
+ with those handled by the non-starred variant
+ `\\PolReduceCoeffs{polname}`_: as the numerators and denominators
+ remain smaller, this proves very beneficial in favorable cases
+ (especially when the coefficients are decimal numbers) to the
+ expansion speed of the xintfrac_ macros used internally by
+ `\\PolEval <PolEvalAt_>`_.
+
+.. _PolMakeMonic:
+
+``\PolMakeMonic{polname}``
+~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+ Divides by the leading coefficient. It is recommended to execute
+ `\\PolReduceCoeffs*{polname}`_ immediately afterwards. This is not
+ done automatically, due to the case the original polynomial had integer
+ coefficients and we want to keep the leading one as common
+ denominator.
+
Expandable macros
-----------------
@@ -395,38 +999,67 @@
and ``\PolToFloatExpr`` (and their auxiliaries) which need a
``\write``, ``\edef`` or a ``\csname...\endcsname`` context.
-``\PolEval{polname}\At{numerical expression}``
-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+.. _PolEvalAtExpr:
- It boils down to ``\xinttheexpr polname(numerical expression)\relax``.
+``\PolEval{polname}\AtExpr{numerical expression}``
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
- .. note::
+ It boils down to
+ ``\xinttheexpr polname(numerical expression)\relax``.
- 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}`_
+.. _PolEvalAt:
-``\PolEvalReduced{polname}\At{numerical expression}``
-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+``\PolEval{polname}\At{fraction}``
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+ Evaluates the polynomial at value ``fraction`` which must be in (or
+ expand to) a format acceptable to the xintfrac_ macros.
+
+ .. attention::
+
+ Meaning was changed at ``0.4``. Formerly ``\PolEval{P}\At{foo}``
+ accepted for ``foo`` an expression which was handled by
+ ``\xintexpr``. See `\\PolEval{polname}\\AtExpr{numerical
+ expression}`_.
+
+ In particular, to use an ``\xintexpr`` user-declared variable (or
+ e.g. the variables as defined by `\\PolSturmIsolateZeros
+ <PolSturmIsolateZeros>`_) one **must** use the ``\AtExpr`` syntax.
+
+.. _PolEvalReducedAtExpr:
+
+``\PolEvalReduced{polname}\AtExpr{numerical expression}``
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
Boils down to ``\xinttheexpr reduce(polname(numerical expression))\relax``.
-``\PolFloatEval{polname}\At{numerical expression}``
-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+.. _PolEvalReducedAt:
+``\PolEvalReduced{polname}\At{fraction}``
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+ Evaluates the polynomial at value ``fraction`` which must be in (or
+ expand to) a format acceptable to the xintfrac_ macros, and produce
+ an irreducible fraction.
+
+ .. attention::
+
+ Meaning was changed at ``0.4``. Formerly the evaluation point
+ could be given as an expression.
+
+.. _PolFloatEvalAtExpr:
+
+``\PolFloatEval{polname}\AtExpr{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: [#]_
+ This is done via a Horner Scheme (see `\\poldef <poldef;_>`_ and
+ `\\PolGenFloatVariant{polname}`_), with already rounded
+ coefficients. [#]_ To use the *exact coefficients* with *exactly
+ executed* additions and multiplications, just insert it in the float
+ expression as in this example: [#]_
::
@@ -436,13 +1069,57 @@
getting raised to the power ``2``. Moving the ``^2`` inside, that
operation would also be treated exactly.
+ .. attention::
+
+ At ``polexpr 0.3``, polynoms were automatically also prepared for
+ use in floating point contexts. This got dropped at ``0.4`` for
+ optimization purposes. See `\\PolGenFloatVariant{polname}`_.
+
.. [#] 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.
+ .. [#] The ``\xintexpr`` here could be ``\xinttheexpr`` but that
+ would be less efficient. Cf. xintexpr_ documentation about
+ nested expressions.
+.. _PolFloatEvalAt:
+
+``\PolFloatEval{polname}\At{fraction}``
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+ Evaluates the polynomial at value ``fraction`` which must be in (or
+ expand to) a format acceptable to the xintfrac_ macros, and produces
+ a floating point number.
+
+ .. attention::
+
+ Meaning was changed at ``0.4``. Formerly the evaluation point
+ could be given as an expression.
+
+.. _PolIfCoeffIsPlusOrMinusOne:
+
+``\PolIfCoeffIsPlusOrMinusOne{A}{B}``
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+ This macro is a priori undefined.
+
+ It is defined via the default `\\PolTypesetCmd{raw_coeff}`_ to be
+ used if needed in the execution of `\\PolTypesetMonomialCmd`_,
+ e.g. to insert a ``\cdot`` in front of ``\PolVar^{\PolIndex}`` if
+ the coefficient is not plus or minus one.
+
+ The macro will execute ``A`` if the coefficient has been found to be
+ plus or minus one, and ``B`` if not.
+
+.. _PolLeadingCoeff:
+
+``\PolLeadingCoeff{polname}``
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+ Expands to the leading coefficient.
+
+.. _PolNthCoeff:
+
``\PolNthCoeff{polname}{number}``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
@@ -450,6 +1127,8 @@
number is out of range). With ``N=-1``, ``-2``, ... expands to the
leading coefficients.
+.. _PolDegree:
+
``\PolDegree{polname}``
~~~~~~~~~~~~~~~~~~~~~~~
@@ -456,6 +1135,8 @@
It expands to the degree. This is ``-1`` if zero polynomial but this
may change in future. Should it then expand to ``-\infty`` ?
+.. _PolToExpr:
+
``\PolToExpr{polname}``
~~~~~~~~~~~~~~~~~~~~~~~
@@ -480,6 +1161,8 @@
of ``\PolToExpr{f}``, but a simple ``f(x)`` is more efficient for
the identical result.
+.. _PolToExprOneTerm:
+
``\PolToExprOneTerm{raw_coeff}{number}``
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
@@ -493,6 +1176,16 @@
always precedes the ``x^number``, except if the coefficient is a one
or a minus one. See `\\PolToExprTimes`_.
+.. _PolToExprOneTermStyleA:
+
+``\PolToExprOneTermStyleA{raw_coeff}{number}``
+^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+
+ Holds the default package meaning of
+ `\\PolToExprOneTerm{raw_coeff}{number}`_.
+
+.. _PolToExprOneTermStyleB:
+
``\PolToExprOneTermStyleB{raw_coeff}{number}``
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
@@ -500,11 +1193,15 @@
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
+ issue ``\let\PolToExprOneTerm\PolToExprOneTermStyleB`` before usage of
``\PolToExpr``. Note that then ``\PolToExprCmd`` isn't used at all.
+ To revert to package default, issue
+ ``\let\PolToExprOneTerm\PolToExprOneTermStyleA``.
To suppress the ``*``'s, cf. `\\PolToExprTimes`_.
+.. _PolToExprCmd:
+
``\PolToExprCmd{raw_coeff}``
^^^^^^^^^^^^^^^^^^^^^^^^^^^^
@@ -515,6 +1212,8 @@
to ``\xintIrr{#1}`` or to ``\xintPRaw{\xintIrr{#1}}`` to obtain in the
output forcefully reduced coefficients.
+.. _PolToExprTermPrefix:
+
``\PolToExprTermPrefix{raw_coeff}``
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
@@ -522,6 +1221,8 @@
prefixes with a plus sign for non-negative coefficients, because
they don't carry one by themselves.
+.. _PolToExprVar:
+
``\PolToExprVar``
^^^^^^^^^^^^^^^^^
@@ -529,6 +1230,8 @@
be a single letter, may be an expandable macro.) Initial definition
is ``x``.
+.. _PolToExprTimes:
+
``\PolToExprTimes``
^^^^^^^^^^^^^^^^^^^
@@ -538,6 +1241,8 @@
this will give output incompatible with some professional computer
algebra software).
+.. _PolToExpr*:
+
``\PolToExpr*{polname}``
~~~~~~~~~~~~~~~~~~~~~~~~
@@ -544,6 +1249,8 @@
Expands to ``coeff_0+coeff_1*x+coeff_2*x^2+...`` (ascending powers).
Customizable like `\\PolToExpr{polname}`_ via the same macros.
+.. _PolToFloatExpr:
+
``\PolToFloatExpr{polname}``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~
@@ -554,19 +1261,19 @@
.. 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.
+ It is not necessary to have issued
+ `\\PolGenFloatVariant{polname}`_. The rounded coefficients are
+ not easily recoverable from the ``\xintfloatexpr`` polynomial
+ function hence ``\PolToFloatExprCmd`` operates from the *exact*
+ coefficients anew.
- This may change in future, if the pre-rounded coefficients are
- stored in a more easily accessible data structure.
+ Attention that both macros obey the prevailing float precision.
+ If it is changed between those macro calls, then a mismatch
+ exists between the coefficients as used in ``\xintfloatexpr`` and
+ those output by ``\PolToFloatExpr{polname}``.
+.. _PolToFloatExprOneTerm:
+
``\PolToFloatExprOneTerm{raw_coeff}{number}``
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
@@ -574,6 +1281,8 @@
<\\PolToExprOneTerm{raw_coeff}{number}>`_. But does not treat
especially coefficients equal to plus or minus one.
+.. _PolToFloatExprCmd:
+
``\PolToFloatExprCmd{raw_coeff}``
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
@@ -596,11 +1305,15 @@
optimization (I can't help it) because ``#1`` is known to be
in ``xintfrac`` raw format.
+.. _PolToFloatExpr*:
+
``\PolToFloatExpr*{polname}``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Typesets in ascending powers.
+.. _PolToList:
+
``\PolToList{polname}``
~~~~~~~~~~~~~~~~~~~~~~~
@@ -608,6 +1321,8 @@
(except zero polynomial which does give ``{0/1[0]}`` and not an
empty output.)
+.. _PolToCSV:
+
``\PolToCSV{polname}``
~~~~~~~~~~~~~~~~~~~~~~
@@ -614,6 +1329,157 @@
Expands to ``coeff_0, coeff_1, coeff_2, ....., coeff_N``. Converse
to `\\PolFromCSV <\\PolFromCSV{polname}{\<csv\>}_>`_.
+.. _PolSturmChainLength:
+
+``\PolSturmChainLength{sturmname}``
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+ Returns the integer ``N`` such that ``sturmname_N`` is the last one
+ in the Sturm chain ``sturmname_0``, ``sturmname_1``, ...
+
+ See `\\PolToSturm{polname}{sturmname}`_.
+
+.. _PolSturmIfZeroExactlyKnown:
+
+``\PolSturmIfZeroExactlyKnown{sturmname}{index}{A}{B}``
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+ Executes ``A`` if the ``index``\ th interval reduces to a singleton,
+ i.e. the root is known exactly, else ``B``.
+
+ .. note::
+
+ ``index`` may be a TeX count, or a ``\value{latexcounter}``, or a
+ numerical expression as parsable by ``\numexpr``: it does not
+ have to be given via explicit digits.
+
+ This remark applies also to the other package macros with
+ ``index`` being the name of the argument in this documentation.
+ There is also an out-of-range check done for some reasonable
+ error message (right before everything goes haywire).
+
+
+.. _PolSturmIsolatedZeroLeft:
+
+``\PolSturmIsolatedZeroLeft{sturmname}{index}``
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+ Expands to the left end-point for the ``index``\ th interval
+ obtained via `\\PolSturmIsolateZeros{sturmname}`_ and possibly
+ refined afterwards.
+
+.. _PolSturmIsolatedZeroRight:
+
+``\PolSturmIsolatedZeroRight{sturmname}{index}``
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+ Expands to the right end-point for the ``index``\ th interval
+ obtained via `\\PolSturmIsolateZeros{sturmname}`_ and possibly
+ refined afterwards.
+
+.. _PolSturmNbOfIsolatedZeros:
+
+``\PolSturmNbOfIsolatedZeros{sturmname}``
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+ Expands to the number of real roots of the polynomial
+ ``<sturmname>_0`` (which is the number of distinct real roots of the
+ polynomial used to create the Sturm chain via
+ `\\PolToSturm{polname}{sturmname}`_.
+
+.. _PolIntervalWidth:
+
+``\PolIntervalWidth{sturmname}{index}``
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+ The ``10^E`` width of the current ``index``\ th root localization
+ interval. Output is in xintfrac_ raw ``1/1[E]`` format (if not zero).
+
+Macros for use within execution of ``\PolPrintIntervals``
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+More precisely, they can be used within the replacement texts of the
+`\\PolPrintIntervalsPrintLeftEndPoint`_, etc, macros.
+
+
+.. _PolPrintIntervalsTheEndPoint:
+
+``\PolPrintIntervalsTheEndPoint``
+^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+
+ Within a custom `\\PolPrintIntervalsPrintLeftEndPoint`_, custom
+ `\\PolPrintIntervalsPrintRightEndPoint`_, or custom
+ `\\PolPrintIntervalsPrintExactZero`_ this macro expands to the left
+ or right end point of the considered interval. Serves as default
+ replacement for `\\PolPrintIntervalsPrintLeftEndPoint`_ , etc...
+
+.. _PolPrintIntervalsTheIndex:
+
+``\PolPrintIntervalsTheIndex``
+^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+
+ Within a custom `\\PolPrintIntervalsPrintLeftEndPoint`_, custom
+ `\\PolPrintIntervalsPrintRightEndPoint`_, or custom
+ `\\PolPrintIntervalsPrintExactZero`_ this macro expands to the index
+ of the considered interval. For example if user wants to print the
+ corresponding end points in red, the index value can thus be tested
+ in the replacement text of `\\PolPrintIntervalsPrintLeftEndPoint`_ and
+ the other two similar macros.
+
+.. _PolIfEndPointIsPositive:
+
+``\PolIfEndPointIsPositive{A}{B}``
+^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+
+ Within a custom `\\PolPrintIntervalsPrintLeftEndPoint`_, custom
+ `\\PolPrintIntervalsPrintRightEndPoint`_, or custom
+ `\\PolPrintIntervalsPrintExactZero`_ this macro executes ``A`` if
+ the considered interval end-point is positive, else ``B``.
+
+.. _PolIfEndPointIsNegative:
+
+``\PolIfEndPointIsNegative{A}{B}``
+^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+
+ Within a custom `\\PolPrintIntervalsPrintLeftEndPoint`_, custom
+ `\\PolPrintIntervalsPrintRightEndPoint`_, or custom
+ `\\PolPrintIntervalsPrintExactZero`_ this macro executes ``A`` if
+ the considered interval end-point is negative, else ``B``.
+
+.. _PolIfEndPointIsZero:
+
+``\PolIfEndPointIsZero{A}{B}``
+^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+
+ Within a custom `\\PolPrintIntervalsPrintLeftEndPoint`_, custom
+ `\\PolPrintIntervalsPrintRightEndPoint`_, or custom
+ `\\PolPrintIntervalsPrintExactZero`_ this macro executes ``A`` if
+ the considered interval end-point is zero, else ``B``.
+
+.. _PolDecToString:
+
+``\PolDecToString{decimal number}``
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+ This is a utility macro to print decimal numbers. Indeed for legacy
+ reasons, xintfrac_ does not yet have user-level ready-to-use macros
+ handling specifically the printing of decimal numbers from their
+ internal representations such as ``A/1[N]``.
+
+ For example
+ ``\PolDecToString{123.456e-8}`` will expand to ``0.00000123456``
+ and ``\PolDecToString{123.450e-8}`` to ``0.00000123450``. This
+ illustrates that trailing zeros are not trimmed (to achieve that one
+ can use ``\PolDecToString{\xintREZ{#1}}``.)
+
+ The macro does not try to identify if the fraction has a denominator
+ consisting only of two's and five's; such a denominator will be left
+ at right-end of output.
+
+ This utility macro will presumably be incorporated (possibly in a
+ more powerful form) to xintfrac_ (or rather to a decimal module) in
+ a future xint_ release.
+
Booleans (with default setting as indicated)
--------------------------------------------
@@ -656,7 +1522,7 @@
- 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
- ``a/b + c/d`` with ``l.c.m.(b,d)`` as denominator. Indeed the current
+ ``a/b + c/d`` with ``lcm(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
to big denominators.
@@ -681,7 +1547,8 @@
1/6+4/6*x^1+4/6*x^2+6/6*x^3+20/6*x^4+16/6*x^5+9/6*x^6+24/6*x^7+16/6*x^8
where all coefficients have the same denominator 6 (which in this
- example is the ``l.c.m`` of the denominators of the reduced coefficients.)
+ example is the least common multiple of the denominators of the
+ reduced coefficients.)
- `\\PolDiff{polname_1}{polname_2}`_ always applies ``\xintIrr`` to the
resulting coefficients, except that the *power of ten* part ``[N]``
@@ -712,50 +1579,164 @@
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.
-RELEASES
---------
-- 0.1 (2018/01/11)
+CHANGE LOG
+----------
- Initial release (files README, polexpr.sty).
+- v0.1 (2018/01/11): initial release. Features:
-- 0.2 (2018/01/14)
+ * The `\\poldef <poldef;_>`_ parser itself,
+ * Differentiation and anti-differentiation,
+ * Euclidean division and GCDs,
+ * Various utilities such as `\\PolFromCSV <PolFromCSV_>`_,
+ `\\PolMapCoeffs <PolMapCoeffs_>`_,
+ `\\PolToCSV <PolToCSV_>`_, `\\PolToExpr <PolToExpr_>`_, ...
- Documentation moved to polexpr.{txt,html}.
+ Only one-variable polynomials so far.
-- 0.3 (2018/01/17)
-
- Make polynomials known to ``\xintfloatexpr`` and improve
- documentation.
+- v0.2 (2018/01/14)
-- 0.3.1 (2018/01/18)
+ * Fix: ``"README thinks \numexpr recognizes ^ operator"``.
+ * Convert README to reStructuredText markup.
+ * Move main documentation from README to separate ``polexpr.txt`` file.
+ * Provide ``polexpr.html`` as obtained via DocUtils_ ``rst2html.py``.
+ * Convert README to (CTAN compatible) Markdown markup.
- Fix two typos in documentation.
+ Due to lack of available time the test suite might not be extensive
+ enough. Bug reports are very welcome!
-Files of 0.3.1 release:
+- v0.3 (2018/01/17)
-- README.md,
-- polexpr.sty (package file),
-- polexpr.txt (documentation),
-- polexpr.html (conversion via `DocUtils`__ rst2html.py)
+ * bug fixes:
- __ http://docutils.sourceforge.net/docs/index.html
+ - the ``0.1`` `\\PolEval <PolEvalAt_>`_ accepted expressions for its second
+ argument, but this was removed by mistake at ``0.2``. Restored.
-See README.md for the License and the change log (there were
-some breaking changes from 0.2 to 0.3).
+ **Attention**: at ``0.4`` this has been reverted again, and
+ `\\PolEval{P}\\AtExpr{foo} <PolEvalAtExpr_>`_ syntax is needed for
+ using expressions in the second argument.
+ * incompatible or breaking changes:
+ - `\\PolToExpr <PolToExpr_>`_ now by default uses *descending*
+ powers (it also treats differently coefficients equal to 1 or -1.)
+ Use `\\PolToExpr* <PolToExpr*_>`_ for *ascending* powers.
+ - `\\PolEval <PolEvalAt_>`_ 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
+ <PolEvalReducedAt_>`_.
+ * new (or newly documented) macros:
+
+ - `\\PolTypesetCmd <PolTypesetCmd_>`_
+ - `\\PolTypesetCmdPrefix <PolTypesetCmdPrefix_>`_
+ - `\\PolTypesetMonomialCmd <PolTypesetMonomialCmd_>`_
+ - `\\PolEvalReducedAt <PolEvalReducedAt_>`_
+ - `\\PolToFloatExpr <PolToFloatExpr_>`_
+ - `\\PolToExprOneTerm <PolToExprOneTerm_>`_
+ - `\\PolToFloatExprOneTerm <PolToFloatExprOneTerm_>`_
+ - `\\PolToExprCmd <PolToExprCmd_>`_
+ - `\\PolToFloatExprCmd <PolToFloatExprCmd_>`_
+ - `\\PolToExprTermPrefix <PolToExprTermPrefix_>`_
+ - `\\PolToExprVar <PolToExprVar_>`_
+ - `\\PolToExprTimes <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 <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.
+
+ **Attention**: this has been reverted at ``0.4``. The macro
+ `\\PolGenFloatVariant <PolGenFloatVariant_>`_ must be used for
+ generation floating point polynomial functions.
+
+- v0.3.1 (2018/01/18)
+
+ Fixes two typos in example code included in the documentation.
+
+- v0.4 (2018/02/16)
+
+ * bug fixes:
+
+ - when Euclidean division gave a zero remainder, the internal
+ representation of this zero polynomial could be faulty; this
+ could cause mysterious bugs in conjunction with other package
+ macros such as `\\PolMapCoeffs <PolMapCoeffs_>`_.
+ - `\\PolGCD <PolGCD_>`_ was buggy in case of first polynomial being
+ of lesser degree than the second one.
+ * breaking changes:
+
+ - formerly `\\PolEval{P}\\At{foo} <PolEvalAt_>`_ allowed ``foo`` to
+ be an expression, which was transparently handled via
+ ``\xinttheexpr``. Now, ``foo`` must be a fraction (or a macro
+ expanding to such) in the format acceptable by ``xintfrac.sty``
+ macros. Use `\\PolEval{P}\\AtExpr{foo} <PolEvalAtExpr_>`_ for more
+ general arguments using expression syntax. E.g., if ``foo`` is the
+ name of a variable known to ``\xintexpr``.
+
+ The same holds for `\\PolEvalReduced <PolEvalReducedAt_>`_
+ and `\\PolFloatEval <PolFloatEvalAt_>`_.
+ - the ``3.0`` automatic generation of floating point variants has
+ been reverted. Not only do *not* the package macros automatically
+ generate floating point variants of newly created polynomials,
+ they actually make pre-existing such variant undefined.
+
+ See `\\PolGenFloatVariant <PolGenFloatVariant_>`_.
+ * new non-expandable macros:
+
+ - `\\PolGenFloatVariant <PolGenFloatVariant_>`_
+ - `\\PolGlobalLet <PolGlobalLet_>`_
+ - `\\PolTypesetOne <PolTypesetOne_>`_
+ - `\\PolQuo <PolQuo_>`_
+ - `\\PolRem <PolRem_>`_
+ - `\\PolToSturm <PolToSturm_>`_
+ - `\\PolToSturm\* <PolToSturm*_>`_
+ - `\\PolSetToSturmChainSignChangesAt <PolSetToSturmChainSignChangesAt_>`_
+ - `\\PolSetToNbOfZerosWithin <PolSetToNbOfZerosWithin_>`_
+ - `\\PolSturmIsolateZeros <PolSturmIsolateZeros_>`_
+ - `\\PolRefineInterval* <PolRefineInterval*_>`_
+ - `\\PolRefineInterval[N] <PolRefineInterval[N]_>`_
+ - `\\PolEnsureIntervalLength <PolEnsureIntervalLength_>`_
+ - `\\PolEnsureIntervalLengths <PolEnsureIntervalLengths_>`_
+ - `\\PolPrintIntervals <PolPrintIntervals_>`_
+ - `\\PolPrintIntervalsPrintExactZero <PolPrintIntervalsPrintExactZero_>`_
+ - `\\PolPrintIntervalsPrintLeftEndPoint <PolPrintIntervalsPrintLeftEndPoint_>`_
+ - `\\PolPrintIntervalsPrintRightEndPoint <PolPrintIntervalsPrintRightEndPoint_>`_
+ - `\\PolReduceCoeffs* <PolReduceCoeffs*_>`_
+ - `\\PolMakeMonic <PolMakeMonic_>`_
+ * new expandable macros:
+
+ - `\\PolToExprOneTermStyleA <PolToExprOneTermStyleA_>`_
+ - `\\PolIfCoeffIsPlusOrMinusOne <PolIfCoeffIsPlusOrMinusOne_>`_
+ - `\\PolLeadingCoeff <PolLeadingCoeff_>`_
+ - `\\PolSturmChainLength <PolSturmChainLength_>`_
+ - `\\PolSturmNbOfIsolatedZeros <PolSturmNbOfIsolatedZeros_>`_
+ - `\\PolSturmIfZeroExactlyKnown <PolSturmIfZeroExactlyKnown_>`_
+ - `\\PolSturmIsolatedZeroLeft <PolSturmIsolatedZeroLeft_>`_
+ - `\\PolSturmIsolatedZeroRight <PolSturmIsolatedZeroRight_>`_
+ - `\\PolPrintIntervalsTheEndPoint <PolPrintIntervalsTheEndPoint_>`_
+ - `\\PolPrintIntervalsTheIndex <PolPrintIntervalsTheIndex_>`_
+ - `\\PolIfEndPointIsPositive <PolIfEndPointIsPositive_>`_
+ - `\\PolIfEndPointIsNegative <PolIfEndPointIsNegative_>`_
+ - `\\PolIfEndPointIsZero <PolIfEndPointIsZero_>`_
+ - `\\PolIntervalWidth <PolIntervalWidth_>`_
+ - `\\PolDecToString <PolDecToString_>`_
+ * improvements:
+
+ The main new feature is implementation of the `Sturm algorithm`_
+ for localization of the real roots of polynomials.
+
+
Acknowledgments
---------------
@@ -764,6 +1745,16 @@
package, and to Jürgen Gilg and Thomas Söll for testing it on some
concrete problems.
+Renewed thanks on occasion of ``0.4`` release!
+
+See README.md for the License.
+
+.. _xinttools:
.. _xintfrac:
.. _xintexpr:
.. _xint: http://www.ctan.org/pkg/xint
+
+.. _Sturm algorithm:
+.. _Sturm Theorem: https://en.wikipedia.org/wiki/Sturm%27s_theorem
+
+.. _DocUtils: http://docutils.sourceforge.net/docs/index.html
Modified: trunk/Master/texmf-dist/tex/latex/polexpr/polexpr.sty
===================================================================
--- trunk/Master/texmf-dist/tex/latex/polexpr/polexpr.sty 2018-02-16 01:23:20 UTC (rev 46652)
+++ trunk/Master/texmf-dist/tex/latex/polexpr/polexpr.sty 2018-02-16 22:30:09 UTC (rev 46653)
@@ -1,7 +1,7 @@
% author: Jean-François Burnol
% License: LPPL 1.3c (author-maintained)
\ProvidesPackage{polexpr}%
- [2018/01/18 v0.3.1 Polynomial expressions with rational coefficients (JFB)]%
+ [2018/02/16 v0.4 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}%
@@ -8,7 +8,13 @@
\catcode`\_ 11 \catcode0 12
%% AUXILIARIES
-\newif\ifPOL at pol % (cf core algebra macros)
+\newcount\POL at count
+\newif\ifPOL at pol
+\newif\ifxintveryverbose
+\newif\ifpoltypesetall
+\newif\ifPOL at sturm@normalize
+\newif\ifPOL at isolz@nextwillneedrefine
+\newif\ifpoltoexprall
%% the main exchange structure (stored in macros \POLuserpol@<name>)
%% is: degree.\empty{coeff0}{coeff1}....{coeffN}
%% (degree=N except zero polynomial recognized from degree set to -1
@@ -22,19 +28,18 @@
{\def#3{#1}\expandafter\def\expandafter#4\expandafter{#2}}%
%
\def\POL at resultfromarray #1{% ATTENTION, **MUST** be executed with
-% \count@ set to 1 + degree (thus \count@ = 0 for zero polynomial)
- \edef\POL at result{%
+% \count@ set to 1 + degree (\count@ = 0 for zero polynomial)
+ \edef\POL at result{\ifnum\count@>\z@
\the\numexpr\count at -\@ne.\noexpand\empty
\xintiloop [1+1]%
-% always done at least once with index 1, hence ok for zero polynomial
\expandafter\POL at braceit\csname POL at array#1\xintiloopindex\endcsname
\ifnum\xintiloopindex<\count@
- \repeat}%
+ \repeat
+ \else-1.\noexpand\empty{0/1[0]}\fi}%
}%
\def\POL at braceit#1{{#1}}% needed as \xintiloopindex can not "see" through braces
-\newif\ifxintveryverbose
\newcommand\PolDef[3][x]{\poldef #2(#1):=#3;}%
\def\poldef{\edef\POL at restoresemicolon{\catcode59=\the\catcode59\relax}%
\catcode59 12 \POL at defpol}%
@@ -76,17 +81,23 @@
%%
\def\POL at newpol#1{%
\expandafter\POL at ifZero\csname POLuserpol@#1\endcsname
- {\@namedef{XINT_expr_userfunc_#1}##1,{0/1[0]}%
- \@namedef{XINT_flexpr_userfunc_#1}##1,{0[0]}}%
+ {\@namedef{XINT_expr_userfunc_#1}##1,{0/1[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\let\csname XINT_flexpr_func_#1\endcsname\@undefined
+ \ifxintverbose\POL at info{#1}\fi
+}%
+\def\POL at newfloatpol#1{%
+ \expandafter\POL at ifZero\csname POLuserpol@#1\endcsname
+ {\@namedef{XINT_flexpr_userfunc_#1}##1,{0[0]}}%
+ {\POL at newfloatpolhorner{#1}}%
\expandafter\XINT_expr_defuserfunc
\csname XINT_flexpr_func_#1\expandafter\endcsname
\csname XINT_flexpr_userfunc_#1\endcsname
- \ifxintverbose\POL at info{#1}\fi
+ \ifxintverbose\POL at floatinfo{#1}\fi
}%
\def\POL at info #1{%
\xintMessage {polexpr}{Info}%
@@ -94,9 +105,13 @@
associated to \string\XINT_expr_userfunc_#1\space
whose meaning uses Horner scheme:
\expandafter\meaning
- \csname XINT_expr_userfunc_#1\endcsname^^J%
- \@spaces And \string\XINT_flexpr_userfunc_#1\space for
- the \string\xintfloatexpr\space parser has meaning
+ \csname XINT_expr_userfunc_#1\endcsname}%
+}%
+\def\POL at floatinfo #1{%
+ \xintMessage {polexpr}{Info}%
+ {Function #1 for the \string\xintfloatexpr\space parser is
+ associated to \string\XINT_flexpr_userfunc_#1\space
+ whose meaning uses Horner scheme:
\expandafter\meaning
\csname XINT_flexpr_userfunc_#1\endcsname}%
}%
@@ -114,6 +129,14 @@
\endgroup
\expandafter\def\csname XINT_expr_userfunc_#1\expandafter\endcsname
\expandafter##\expandafter1\expandafter,\expandafter{\POL at tmp{##1}}%
+}%
+\def\POL at newfloatpolhorner#1{%
+ %% redefine function to expand by Horner scheme. Is this useful?
+ %% perhaps bad idea for numerical evaluation of thing such as (1+x)^10?
+% note: I added {0/1[0]} item to zero polynomial also to facilitate this
+ \expandafter\expandafter\expandafter\POL at split
+ \csname POLuserpol@#1\endcsname;\POL at var@deg\POL at var@coeffs
+ \edef\POL at var@coeffs{\xintRevWithBraces{\POL at var@coeffs}}%
\begingroup
\expandafter\POL at newpol@floathorner\POL at var@coeffs\relax
\expandafter
@@ -177,10 +200,14 @@
%\let\POL at original@redefinemacros\XINT_expr_redefinemacros % do locally
\def\POL at redefinemacros{\POL at original_redefinemacros\POL at redefineextras}%
\let\POL at redefineextras\@empty
-%
+
+
+\newcommand\PolGenFloatVariant[1]{\POL at newfloatpol{#1}}%
+
+
\newcommand\PolLet[2]{\if=\noexpand#2\expandafter\xint_firstoftwo
\else\expandafter\xint_secondoftwo\fi
- \POL@@let\POL at let {#1}{#2}}%
+ \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
@@ -188,18 +215,21 @@
\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
}%
+\newcommand\PolGlobalLet[2]{\begingroup
+ \globaldefs\@ne
+ \if=\noexpand#2\expandafter\xint_firstoftwo
+ \else\expandafter\xint_secondoftwo\fi
+% There is a potential problem related to \POL at addtoextras, (the local set-up
+% will become the global one) but I will reconsider that another day
+ \POL@@globallet\POL at globallet {#1}{#2}}%
+\def\POL@@globallet#1#2#3{\POL at globallet{#1}{#3}}%
+\def\POL at globallet#1#2{\POL at let{#1}{#2}\endgroup}%
-
\newcommand\PolAssign[1]{\def\POL at polname{#1}\POL at assign}% zap spaces in #1?
\def\POL at assign#1\toarray#2{%
\expandafter\expandafter\expandafter\POL at split
@@ -224,13 +254,15 @@
}%
-\newcommand\PolGet[1]{\def\POL at polname{#1}% zap spaces in #1?
- \begingroup % closed in \POL at getfrom
- \POL at getfrom}%
-% attention au name clash proche avec \POL at get auxiliaire de \POL at add etc..
-\def\POL at getfrom#1\fromarray#2{%
- \count@#2{0} % must be > 0, else could create infinite loop
- % \ifnum\count@>\z@\else\InvalidArrayError_PolGet\fi
+\newcommand\PolGet{}
+\def\PolGet#1#2\fromarray#3{%
+ \begingroup % closed in \POL at getfromarray
+ \POL at getfromarray{#1}{#3}%
+ \POL at newpol{#1}%
+}%
+\def\POL at getfromarray#1#2{%
+ \count@=#2{0} %<- intentional space,
+ % must be > 0, else could create infinite loop
\xintloop
\edef\POL at tmp{#2{\count@}}%
\xintiiifZero{\POL at tmp}%
@@ -251,27 +283,30 @@
\expandafter
\endgroup
\expandafter
- \def\csname POLuserpol@\POL at polname\expandafter\endcsname
+ \def\csname POLuserpol@#1\expandafter\endcsname
\expandafter{\POL at result}%
- \expandafter\POL at newpol\expandafter{\POL at polname}%
}%
-\newcommand\PolFromCSV[2]{\def\POL at polname{#1}%
- \begingroup % closed in \POL at getfrom
+\newcommand\PolFromCSV[2]{%
+ \begingroup % closed in \POL at getfromarray
\xintAssignArray\xintCSVtoList{#2}\to\POL at arrayA
- \POL at getfrom\fromarray\POL at arrayA
+ \POL at getfromarray{#1}\POL at arrayA
+ \POL at newpol{#1}%
% semble un peu indirect et sous-optimal
% mais je veux élaguer les coefficients nuls. Peut-être à revoir.
}%
-\newif\ifpoltypesetall
\newcommand\PolTypesetCmdPrefix[1]{\xintiiifSgn{#1}{}{+}{+}}%
\newcommand\PolTypesetCmd[1]{\xintifOne{\xintiiAbs{#1}}%
{\ifnum\PolIndex=\z@\xintiiSgn{#1}\else
- \xintiiifSgn{#1}{-}{}{}\fi}%
- {\xintSignedFrac{#1}}}%
+ \xintiiifSgn{#1}{-}{}{}\fi
+ \let\PolIfCoeffIsPlusOrMinusOne\@firstoftwo}%
+ {\PolTypesetOne{#1}%
+ \let\PolIfCoeffIsPlusOrMinusOne\@secondoftwo}%
+ }%
+\newcommand\PolTypesetOne{\xintSignedFrac}%
\newcommand\PolTypesetMonomialCmd{%
\ifcase\PolIndex\space
%
@@ -330,6 +365,10 @@
\newcommand\PolMapCoeffs[2]{% #1 = macro, #2 = name
+ \POL at mapcoeffs{#1}{#2}%
+ \POL at newpol{#2}%
+}%
+\def\POL at mapcoeffs#1#2{%
\begingroup
\def\POL at map@macro{#1}%
\expandafter\expandafter\expandafter\POL at split
@@ -352,7 +391,6 @@
\endgroup
\expandafter
\def\csname POLuserpol@#2\expandafter\endcsname\expandafter{\POL at result}%
- \POL at newpol{#2}%
}%
\def\POL at map@loop.#1{\if\relax#1\expandafter\xint_gob_til_dot\fi
\advance\count@\@ne
@@ -362,11 +400,34 @@
\edef\index{\the\numexpr\index+\@ne}%
\POL at map@loop.}%
\def\POL at xintIrr#1{\xintIrr{#1}[0]}%
-\newcommand\PolReduceCoeffs[1]{\PolMapCoeffs{\POL at xintIrr}{#1}}%
+\def\POL at special@xintIrr#1{%
+ \expandafter\POL at special@xintIrr at i\romannumeral0\xintraw{#1}}%
+\def\POL at special@xintIrr at i#1/#2[#3]{\xintIrr{#1/#2[0]}[#3]}%
+\newcommand\PolReduceCoeffs{\@ifstar\POL at sreducecoeffs\POL at reducecoeffs}%
+\def\POL at reducecoeffs#1{\PolMapCoeffs{\POL at xintIrr}{#1}}%
+\def\POL at sreducecoeffs#1{\PolMapCoeffs{\POL at special@xintIrr}{#1}}%
+\def\POL at special@xintIrr at skipraw#1{\POL at special@xintIrr at i#1}% used by ToSturm
%% EUCLIDEAN DIVISION
\newcommand\PolDivide[4]{% #3=quotient, #4=remainder of #1 by #2
+ \POL at divide{#1}{#2}%
+ \expandafter\let\csname POLuserpol@#3\endcsname\POL at Q
+ \POL at newpol{#3}%
+ \expandafter\let\csname POLuserpol@#4\endcsname\POL at R
+ \POL at newpol{#4}%
+}%
+\newcommand\PolQuo[3]{% #3=quotient of #1 by #2
+ \POL at divide{#1}{#2}%
+ \expandafter\let\csname POLuserpol@#3\endcsname\POL at Q
+ \POL at newpol{#3}%
+}%
+\newcommand\PolRem[3]{% #3=remainder of #1 by #2
+ \POL at divide{#1}{#2}%
+ \expandafter\let\csname POLuserpol@#3\endcsname\POL at R
+ \POL at newpol{#3}%
+}%
+\newcommand\POL at divide[2]{%
\begingroup
\let\xintScalarSub\xintSub
\let\XINT_fadd_C\POL_fadd_C
@@ -386,17 +447,19 @@
\expandafter
\endgroup
\expandafter
- \def\csname POLuserpol@#3\expandafter\expandafter\expandafter\endcsname
+ \def\csname POL at Q\expandafter\expandafter\expandafter\endcsname
\expandafter\expandafter\expandafter{\expandafter\POL at Q\expandafter}%
\expandafter
- \def\csname POLuserpol@#4\expandafter\endcsname\expandafter{\POL at R}%
- \POL at newpol{#3}%
- \POL at newpol{#4}%
+ \def\csname POL at R\expandafter\endcsname\expandafter{\POL at R}%
}%
%% GCD
\newcommand\PolGCD[3]{% sets #3 to the (unitary) G.C.D. of #1 and #2
+ \POL at GCD{#1}{#2}{#3}%
+ \POL at newpol{#3}%
+}%
+\def\POL at GCD #1#2#3{%
\begingroup
\let\xintScalarSub\xintSub
\let\XINT_fadd_C\POL_fadd_C
@@ -424,7 +487,8 @@
\POL at gcd@exit AB}%
{\ifnum\POL at degA<\POL at degB\space
\let\POL at tmp\POL at B\let\POL at B\POL at A\let\POL at A\POL at tmp
- \let\POL at tmp\POL at degB\let\POL at degB\POL at degA\let\POL at degA\POL at degB
+ \let\POL at tmp\POL at degB\let\POL at degB\POL at degA\let\POL at degA\POL at tmp
+ \let\POL at tmp\POL at polB\let\POL at polB\POL at polA\let\POL at polA\POL at tmp
\fi
\xintAssignArray\POL at polA\to\POL at arrayA
\xintAssignArray\POL at polB\to\POL at arrayB
@@ -434,7 +498,6 @@
\endgroup
\expandafter\def\csname POLuserpol@#3\expandafter\endcsname
\expandafter{\POL at result}%
- \POL at newpol{#3}%
}%
\def\POL at normalize#1{%
\expandafter\def\expandafter\POL at tmp\expandafter
@@ -450,7 +513,7 @@
{\POL at normalize@leading}}[0]}%
\advance\count@\m at ne
\repeat
-}%
+}%
\def\POL at gcd#1#2{%
\POL at normalize{#2}%
\edef\POL at degQ{\the\numexpr\csname POL at deg#1\endcsname
@@ -498,8 +561,8 @@
\count@\numexpr\csname POL at deg#1\endcsname+\@ne\relax
\POL at resultfromarray #1%
}%
-
+
%% TODO: BEZOUT
@@ -533,7 +596,8 @@
\fi
}%
\def\POL at Diff@no #1#2{\POL at let{#2}{#1}}%
-\def\POL at Diff@one #1#2{%
+\def\POL at Diff@one #1#2{\POL at Diff@@one {#1}{#2}\POL at newpol{#2}}%
+\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
\ifnum\POL at var@deg<\@ne
@@ -543,7 +607,6 @@
\expandafter\edef\csname POLuserpol@#2\endcsname
{\the\numexpr\POL at var@deg-\@ne.\noexpand\empty\POL at var@coeffs}%
\fi
- \POL at newpol{#2}%
}%
% lazy way but allows to share with AntiDiff
\def\POL at Iterate#1#2#3{%
@@ -593,7 +656,8 @@
\fi
}%
\let\POL at AntiDiff@no\POL at Diff@no
-\def\POL at AntiDiff@one #1#2{%
+\def\POL at AntiDiff@one #1#2{\POL at AntiDiff@@one{#1}{#2}\POL at newpol{#2}}%
+\def\POL at AntiDiff@@one#1#2{%
\expandafter\expandafter\expandafter\POL at split
\csname POLuserpol@#1\endcsname;\POL at var@deg\POL at var@coeffs
\ifnum\POL at var@deg<\z@
@@ -603,10 +667,801 @@
\expandafter\edef\csname POLuserpol@#2\endcsname
{\the\numexpr\POL at var@deg+\@ne.\noexpand\empty{0/1[0]}\POL at var@coeffs}%
\fi
- \POL at newpol{#2}%
}%
+%% Sturm Algorithm (polexpr 0.4)
+\newcommand\PolToSturm{%
+ \@ifstar{\POL at sturm@normalizefalse}{\POL at sturm@normalizetrue}%
+ \POL at ToSturm
+}%
+\newcommand\POL at ToSturm[2]{%
+ \edef\POL at sturmname{#2}%
+ \POL at let{\POL at sturmname _0}{#1}%
+ \POL at mapcoeffs{\POL at special@xintIrr}{\POL at sturmname _0}%
+ \POL at Diff@@one{\POL at sturmname _0}{\POL at sturmname _1}%
+ \POL at count\@ne
+ \xintloop
+ \POL at divide{\POL at sturmname _\the\numexpr\POL at count-\@ne}%
+ {\POL at sturmname _\the\POL at count}%
+ \expandafter\POL at split\POL at R;\POL at degR\POL at polR
+ \unless\ifnum\POL at degR=\m at ne
+ \edef\POL at polR{\xintApply{\POL at dooppandirr}{\POL at polR}}%
+ \advance\POL at count\@ne
+ \expandafter\edef\csname POLuserpol@\POL at sturmname _\the\POL at count\endcsname
+ {\POL at degR.\noexpand\empty\POL at polR}%
+ \repeat
+ \edef\POL at sturm@N{\the\POL at count}%
+ \ifPOL at sturm@normalize
+ \ifnum\PolDegree{\POL at sturmname _\POL at sturm@N}>\z@
+ \xintloop
+ \advance\POL at count\m at ne
+ \POL at divide{\POL at sturmname _\the\POL at count}%
+ {\POL at sturmname _\POL at sturm@N}%
+ \expandafter\POL at split\POL at Q;\POL at degQ\POL at polQ
+ \edef\POL at polQ{\xintApply{\POL at special@xintIrr at skipraw}{\POL at polQ}}%
+ \edef\POL at Q{\POL at degQ.\noexpand\empty\POL at polQ}%
+ \expandafter\edef\csname POLuserpol@\POL at sturmname _\the\POL at count\endcsname
+ {\POL at degQ.\noexpand\empty\POL at polQ}%
+ \ifnum\POL at count>\z@
+ \repeat
+ \@namedef{POLuserpol@\POL at sturmname _\POL at sturm@N}{0.\empty{1/1[0]}}%
+ \fi
+ \fi
+ \POL at count\z@
+ \xintloop
+ \POL at newpol{\POL at sturmname _\the\POL at count}%
+ \unless\ifnum\POL at sturm@N=\POL at count
+ \advance\POL at count\@ne
+ \repeat
+ \expandafter\let\csname PolSturmChainLength_\POL at sturmname \endcsname\POL at sturm@N
+}%
+\def\POL at dooppandirr#1{\xintiiOpp{\POL at special@xintIrr at i#1}}%
+\newcommand\PolSturmChainLength[1]
+ {\romannumeral`^^@\csname PolSturmChainLength_#1\endcsname}%
+\newcommand\PolSetToSturmChainSignChangesAt[4][\global]{%
+ \edef\POL at sturmchain@X{\xintREZ{#4}}%
+ \edef\POL at sturmname{#3}%
+ \edef\POL at sturmlength{\PolSturmChainLength{\POL at sturmname}}%
+ \POL at sturmchain@getSV at at\POL at sturmchain@X
+ #1\let#2\POL at sturmchain@SV
+}%
+\def\POL at sturmchain@getSV at at#1{% ATTENTION USES \POL at count
+ \def\POL at sturmchain@SV{0}%
+ \edef\POL at isolz@lastsign{\xintiiSgn{\PolEval{\POL at sturmname _0}\At{#1}}}%
+ \let\POL at IsoRightSign\POL at isolz@lastsign % needed only for SturmIsolate etc...
+ \POL at count \z@
+ \ifnum\POL at isolz@lastsign=\z@
+ \edef\POL at isolz@lastsign
+ {\xintiiSgn{\PolEval{\POL at sturmname _1}\At{#1}}}%
+ \POL at count \@ne
+ \fi
+ \xintloop
+ \unless\ifnum\POL at sturmlength=\POL at count
+ \advance\POL at count \@ne
+ \edef\POL at isolz@newsign
+ {\xintiiSgn{\PolEval{\POL at sturmname _\the\POL at count}\At{#1}}}%
+ \ifnum\POL at isolz@newsign=\numexpr-\POL at isolz@lastsign\relax
+ \edef\POL at sturmchain@SV{\the\numexpr\POL at sturmchain@SV+\@ne}%
+ \let\POL at isolz@lastsign=\POL at isolz@newsign
+ \fi
+ \repeat
+}%
+\newcommand\PolSetToNbOfZerosWithin[5][\global]{%
+ \edef\POL at tmpA{\xintREZ{#4}}%
+ \edef\POL at tmpB{\xintREZ{#5}}%
+ \edef\POL at sturmname{#3}%
+ \edef\POL at sturmlength{\PolSturmChainLength{\POL at sturmname}}%
+ \POL at sturmchain@getSV at at\POL at tmpA
+ \let\POL at SVA\POL at sturmchain@SV
+ \POL at sturmchain@getSV at at\POL at tmpB
+ \let\POL at SVB\POL at sturmchain@SV
+ \ifnum\POL at SVA<\POL at SVB\space
+ #1\edef#2{\the\numexpr\POL at SVB-\POL at SVA}%
+ \else
+ #1\edef#2{\the\numexpr\POL at SVA-\POL at SVB}%
+ \fi
+}%
+
+
+\newcommand\PolSturmIsolateZeros[1]{%
+ % #1 name of Sturm chain (already pre-computed from a given polynomial)
+ \edef\POL at sturmname{#1}%
+ \edef\POL at sturmlength{\PolSturmChainLength{#1}}%
+ % Count number of sign changes at plus infinity in Sturm sequence
+ \def\POL at isolz@plusinf at SV{0}%
+ \edef\POL at isolz@lastsign{\xintiiSgn{\PolLeadingCoeff{#1_0}}}%
+ \let\POL at isolz@plusinf at sign\POL at isolz@lastsign
+ \POL at count\@ne
+ \xintloop
+ \edef\POL at isolz@newsign{\xintiiSgn{\PolLeadingCoeff{#1_\the\POL at count}}}%
+ \unless\ifnum\POL at isolz@newsign=\POL at isolz@lastsign
+ \edef\POL at isolz@plusinf at SV{\the\numexpr\POL at isolz@plusinf at SV+\@ne}%
+ \fi
+ \let\POL at isolz@lastsign=\POL at isolz@newsign
+ \ifnum\POL at sturmlength>\POL at count
+ \advance\POL at count\@ne
+ \repeat
+ % Count number of sign changes at minus infinity in Sturm sequence
+ \def\POL at isolz@minusinf at SV{0}%
+ \edef\POL at isolz@lastsign{\xintiiSgn{\PolLeadingCoeff{#1_0}}}%
+ \ifodd\PolDegree{#1_0}
+ \edef\POL at isolz@lastsign{\xintiiOpp{\POL at isolz@lastsign}}%
+ \fi
+ \let\POL at isolz@minusinf at sign\POL at isolz@lastsign
+ \POL at count\@ne
+ \xintloop
+ \edef\POL at isolz@newsign{\xintiiSgn{\PolLeadingCoeff{#1_\the\POL at count}}}%
+ \ifodd\PolDegree{#1_\the\POL at count}
+ \edef\POL at isolz@newsign{\xintiiOpp{\POL at isolz@newsign}}%
+ \fi
+ \unless\ifnum\POL at isolz@newsign=\POL at isolz@lastsign
+ \edef\POL at isolz@minusinf at SV{\the\numexpr\POL at isolz@minusinf at SV+\@ne}%
+ \fi
+ \let\POL at isolz@lastsign=\POL at isolz@newsign
+ \ifnum\POL at sturmlength>\POL at count
+ \advance\POL at count\@ne
+ \repeat
+ \edef\POL at isolz@NbOfRoots
+ {\the\numexpr\POL at isolz@minusinf at SV-\POL at isolz@plusinf at SV}%
+ \ifnum\POL at isolz@NbOfRoots=\z@
+ \begingroup\globaldefs\@ne
+ \expandafter\xintAssignArray\expandafter\to\csname POL_ZeroInt#1L\endcsname
+ \expandafter\xintAssignArray\expandafter\to\csname POL_ZeroInt#1R\endcsname
+ \endgroup
+ \else
+ \begingroup\globaldefs\@ne
+ \expandafter\POL at isolz@initarray\csname POL_ZeroInt#1L\endcsname
+ \expandafter\POL at isolz@initarray\csname POL_ZeroInt#1R\endcsname
+ \endgroup
+ \expandafter\POL at isolz@getaprioribound
+ \fi
+}%
+\def\POL at isolz@initarray#1{%
+ \expandafter\xintAssignArray
+ \romannumeral\xintreplicate{\POL at isolz@NbOfRoots}{{0}}\to#1%
+}%
+% utility macro for a priori bound on root decimal exponent, via Float Rounding
+\def\POL at isolz@updateE #1e#2;%
+{\unless\ifnum#2<\POL at isolz@E\space\edef\POL at isolz@E{\the\numexpr#2+\@ne}\fi}%
+\def\POL at isolz@getaprioribound{%
+ \PolAssign{\POL at sturmname _0}\toarray\POL at arrayA
+ \edef\POL at isolz@leading{\POL at arrayA{\POL at arrayA{0}}}%
+ \POL at count\z@
+ \xintloop
+ \advance\POL at count\@ne
+ \ifnum\POL at arrayA{0}>\POL at count
+ \expandafter\edef\csname POL at arrayA\the\POL at count\endcsname
+ {\xintDiv{\POL at arrayA\POL at count}\POL at isolz@leading}%
+ \repeat
+ \def\POL at isolz@E{1}% WE SEEK SMALLEST E SUCH HAT -10^E < roots < +10^E
+ \advance\POL at count\m at ne
+ \xintloop
+ \ifnum\POL at count>\z@
+ \expandafter\POL at isolz@updateE
+ % use floating point to get decimal exponent
+ \romannumeral0\xintfloat[4]% should I use with [2] rather? (should work)
+ {\xintAdd{1/1[0]}{\xintAbs{\POL at arrayA\POL at count}}};%
+ \advance\POL at count\m at ne
+ \repeat
+ % \ifxintverbose\xintMessage{polexpr}{Info}%
+ % {Roots a priori bounded in absolute value by 10 to the \POL at isolz@E.}%
+ % \fi
+ \POL at isolz@main
+}%
+\def\POL at IsoRight@raw{\POL at IsoRight@Int/1[\POL at isolz@E]}%
+\def\POL at IsoLeft@raw {\POL at IsoLeft@Int/1[\POL at isolz@E]}%
+\def\POL at IsoRight@rawout{%
+ \ifnum\POL at IsoRightSign=\z@\expandafter\xintREZ\fi\POL at IsoRight@raw}%
+\def\POL at IsoLeft@rawout{%
+ \ifnum\POL at IsoRightSign=\z@
+ \expandafter\xint_firstoftwo\else\expandafter\xint_secondoftwo
+ \fi{\xintREZ\POL at IsoRight@raw}%
+ {\POL at IsoLeft@Int/1[\POL at isolz@E]}}%
+\def\POL at isolz@main {%
+% NOTE 2018/02/16. THIS WILL PRESUMABLY BE RE-ORGANIZED IN FUTURE TO DO
+% FIRST POSITIVE ROOTS THEN NEGATIVE ROOTS VIA CHANGE OF VARIABLE TO OPPOSITE.
+ \global\POL at isolz@nextwillneedrefinefalse
+ \def\POL at IsoRight@Int{0}%
+ \POL at sturmchain@getSV at at\POL at IsoRight@raw
+ \let\POL at IsoRightSV\POL at sturmchain@SV
+ \let\POL at IsoAtZeroSV\POL at IsoRightSV
+ \let\POL at IsoAtZeroSign\POL at IsoRightSign
+ \ifnum\POL at IsoAtZeroSign=\z@
+ \xdef\POL at isolz@IntervalIndex
+ {\the\numexpr\POL at isolz@minusinf at SV-\POL at IsoRightSV}%
+ \POL at refine@storeleftandright % store zero root
+ \edef\POL at IsoRightSV{\the\numexpr\POL at IsoRightSV+\@ne}%
+% subtlety here if original polynomial had multiplicities, but ok. I checked!
+ \edef\POL at IsoRightSign % evaluated twice, but that's not so bad
+ {\xintiiOpp{\xintiiSgn{\PolEval{\POL at sturmname _1}\At{0/1[0]}}}}%
+ \fi
+ \def\POL at IsoLeft@Int{-1}% -10^E isn't a root!
+ \let\POL at IsoLeftSV\POL at isolz@minusinf at SV
+ \let\POL at IsoLeftSign\POL at isolz@minusinf at sign
+ \edef\POL at isolz@NbOfNegRoots{\the\numexpr\POL at IsoLeftSV-\POL at IsoRightSV}%
+ \gdef\POL at isolz@IntervalIndex{0}%
+ \begingroup
+ \let\POL at IsoAtZeroSV\POL at IsoRightSV % locally shifted if root at zero
+ \let\POL at IsoAtZeroSign\POL at IsoRightSign
+ \ifnum\POL at isolz@NbOfNegRoots>\z@
+ \def\POL at IsoRight@Int{-1}%
+ \xintloop
+ \edef\POL at isolz@E{\the\numexpr\POL at isolz@E-\@ne}%
+ \POL at sturmchain@getSV at at\POL at IsoRight@raw
+ \let\POL at IsoRightSV\POL at sturmchain@SV
+ % would an \ifx test be quicker? (to be checked)
+ \ifnum\POL at IsoRightSV=\POL at IsoLeftSV\space
+ % no roots in-between, sign and SV kept
+ \repeat
+ \def\POL at IsoLeft@Int{-10}%
+ \let\POL@@IsoRightSign\POL at IsoRightSign % zero possible
+ \let\POL@@IsoRightSV\POL at IsoRightSV
+ \xintloop
+ \edef\POL at IsoRight@Int{\the\numexpr\POL at IsoLeft@Int+\@ne}%
+% we could arguably do a more efficient dichotomy here
+ \POL at sturmchain@getSV at at\POL at IsoRight@raw
+ \let\POL at IsoRightSV\POL at sturmchain@SV
+ \POL at isolz@check
+ \ifnum\POL at isolz@IntervalIndex=\POL at isolz@NbOfNegRoots\space
+ \expandafter\xintbreakloop
+ \fi
+ \let\POL at IsoLeft@Int\POL at IsoRight@Int
+ \let\POL at IsoLeftSign\POL at IsoRightSign
+ \let\POL at IsoLeftSV\POL at IsoRightSV
+ \ifnum\POL at IsoRight@Int < -\tw@
+ \repeat
+ \ifnum\POL at isolz@IntervalIndex<\POL at isolz@NbOfNegRoots\space
+ \def\POL at IsoRight@Int{-1}%
+ \let\POL at IsoRightSign\POL@@IsoRightSign
+ \let\POL at IsoRightSV\POL@@IsoRightSV
+ \POL at isolz@check
+ \ifnum\POL at isolz@IntervalIndex<\POL at isolz@NbOfNegRoots\space
+ \def\POL at IsoLeft@Int{-1}%
+ \let\POL at IsoLeftSign\POL at IsoRightSign
+ \let\POL at IsoLeftSV\POL at IsoRightSV
+ \def\POL at IsoRight@Int{0}%
+ \let\POL at IsoRightSV\POL at IsoAtZeroSV % altered if 0 was a root
+ \let\POL at IsoRightSign\POL at IsoAtZeroSign% id.
+% this will recurse to locate roots with smaller decimal exponents
+ \POL at isolz@check % attention that this should not re-evaluate at 0
+ \fi
+ \fi
+ \fi
+ \endgroup
+ \def\POL at IsoLeft@Int{0}%
+ \let\POL at IsoLeftSV\POL at IsoAtZeroSV
+ \let\POL at IsoLeftSign\POL at IsoAtZeroSign
+ \ifnum\POL at IsoLeftSign=\z@
+ \xdef\POL at isolz@IntervalIndex{\the\numexpr\POL at isolz@IntervalIndex+\@ne}%
+ \global\POL at isolz@nextwillneedrefinetrue
+ \else
+ \global\POL at isolz@nextwillneedrefinefalse
+ \fi
+ \let\POL@@IsoRightSV=\POL at isolz@plusinf at SV
+ \let\POL@@IsoRightSign=\POL at isolz@plusinf at sign % 10^E not a root!
+ \edef\POL at isolz@NbOfPosRoots
+ {\the\numexpr\POL at IsoLeftSV-\POL@@IsoRightSV}% attention @@
+ \ifnum\POL at isolz@NbOfPosRoots>\z@
+ \def\POL at IsoRight@Int{1}%
+ \xintloop
+ \edef\POL at isolz@E{\the\numexpr\POL at isolz@E-\@ne}%
+ \POL at sturmchain@getSV at at\POL at IsoRight@raw
+ \let\POL at IsoRightSV\POL at sturmchain@SV
+ \ifnum\POL at IsoRightSV=\POL@@IsoRightSV\space
+ \let\POL@@IsoRightSign\POL at IsoRightSign % root here possible!
+ \repeat
+ \unless\ifnum\POL at IsoRightSV=\POL at IsoLeftSV\space
+ \POL at isolz@check % will recurse inside groups if needed
+ \fi
+ \def\POL at IsoLeft@Int{1}%
+ \let\POL at IsoLeftSV\POL at IsoRightSV
+ \let\POL at IsoLeftSign\POL at IsoRightSign
+ \xintloop
+% we could arguably do a more efficient dichotomy here
+ \edef\POL at IsoRight@Int{\the\numexpr\POL at IsoLeft@Int+\@ne}%
+ \POL at sturmchain@getSV at at\POL at IsoRight@raw
+ \let\POL at IsoRightSV\POL at sturmchain@SV
+ \POL at isolz@check
+ \let\POL at IsoLeft@Int\POL at IsoRight@Int
+ \let\POL at IsoLeftSign\POL at IsoRightSign
+ \let\POL at IsoLeftSV\POL at IsoRightSV
+ \ifnum\POL at isolz@IntervalIndex=\POL at isolz@NbOfRoots\space
+ \expandafter\xintbreakloop
+ \fi
+ \ifnum\POL at IsoLeft@Int < \xint_c_ix
+ \repeat
+ \ifnum\POL at isolz@IntervalIndex<\POL at isolz@NbOfRoots\space
+ % get now the last, right most, root (or roots)
+ \def\POL at IsoRight@Int{10}%
+ \let\POL at IsoRightSign\POL@@IsoRightSign
+ \let\POL at IsoRightSV\POL@@IsoRightSV
+ \POL at isolz@check
+ \fi
+ \fi
+}%
+\def\POL at isolz@check{% \POL at IsoRightSign must be ready for use here
+% \ifxintverbose
+% \xintMessage{polexpr}{Info}%
+% {\the\numexpr\POL at IsoLeftSV-\POL at IsoRightSV\relax\space roots
+% in (\POL at IsoLeft@raw,\POL at IsoRight@raw] (E = \POL at isolz@E)}%
+% \fi
+ \ifcase\numexpr\POL at IsoLeftSV-\POL at IsoRightSV\relax
+ % no root in ]left, right]
+ \global\POL at isolz@nextwillneedrefinefalse
+ \or
+ % exactly one root in ]left, right]
+ \xdef\POL at isolz@IntervalIndex{\the\numexpr\POL at isolz@IntervalIndex+\@ne}%
+ \ifnum\POL at IsoRightSign=\z@
+ % if right boundary is a root, ignore previous flag
+ \global\POL at isolz@nextwillneedrefinefalse
+ \fi
+ % if left boundary is known to have been a root we refine interval
+ \ifPOL at isolz@nextwillneedrefine
+ \expandafter\expandafter\expandafter\POL at isolz@refine
+ \else
+ \POL at refine@storeleftandright
+ \ifnum\POL at IsoRightSign=\z@
+ \global\POL at isolz@nextwillneedrefinetrue
+ \fi
+ \fi
+ \else
+ % more than one root, we need to recurse
+ \expandafter\POL at isolz@recursedeeper
+ \fi
+}%
+\def\POL at isolz@recursedeeper{%
+% NOTE 2018/02/16. I SHOULD DO A REAL BINARY DICHOTOMY HERE WHICH ON AVERAGE
+% SHOULD BRING SOME GAIN (LIKE WHAT IS ALREADY DONE FOR THE "refine" MACROS.
+% THUS IN FUTURE THIS MIGHT BE REFACTORED.
+\begingroup
+ \edef\POL at isolz@E{\the\numexpr\POL at isolz@E-\@ne}%
+ \edef\POL@@IsoRight at Int{\xintDSL{\POL at IsoRight@Int}}%
+ \let\POL@@IsoRightSign \POL at IsoRightSign
+ \let\POL@@IsoRightSV \POL at IsoRightSV
+ \edef\POL at IsoLeft@Int {\xintDSL{\POL at IsoLeft@Int}}%
+ \xintiloop[1+1]
+ \edef\POL at IsoRight@Int{\xintInc{\POL at IsoLeft@Int}}%
+ \POL at sturmchain@getSV at at\POL at IsoRight@raw
+ \let\POL at IsoRightSV\POL at sturmchain@SV
+ \POL at isolz@check
+ \let\POL at IsoLeft@Int\POL at IsoRight@Int
+ \let\POL at IsoLeftSV\POL at IsoRightSV
+ \let\POL at IsoLeftSign\POL at IsoRightSign% not used, actually
+ \ifnum\POL at IsoLeftSV=\POL@@IsoRightSV\space
+ \expandafter\xintbreakiloop
+ \fi
+ \ifnum\xintiloopindex < \xint_c_ix
+ \repeat
+ \let\POL at IsoRight@Int\POL@@IsoRight at Int
+ \let\POL at IsoRightSign\POL@@IsoRightSign
+ \let\POL at IsoRightSV \POL@@IsoRightSV
+ % if we exited the loop via breakiloop this is superfluous
+ % but it only costs one \ifnum
+ \POL at isolz@check
+\endgroup
+}%
+\def\POL at isolz@refine{%
+ % starting point is first root = left < unique second root < right
+ % even if we hit exactly via refinement second root, we set flag false as
+ % processing will continue with original right end-point, which isn't a root
+ \global\POL at isolz@nextwillneedrefinefalse
+\begingroup
+ \let\POL@@IsoRightSign\POL at IsoRightSign % already evaluated
+ \xintloop
+ \edef\POL at isolz@E{\the\numexpr\POL at isolz@E-\@ne}%
+ \edef\POL at IsoLeft@Int {\xintDSL{\POL at IsoLeft@Int}}%
+ \edef\POL at IsoRight@Int{\xintInc{\POL at IsoLeft@Int}}%
+ \edef\POL at IsoRightSign
+ {\xintiiSgn{\PolEval{\POL at sturmname _0}\At{\POL at IsoRight@raw}}}%
+ \ifnum\POL at IsoRightSign=\POL@@IsoRightSign\space
+ \repeat
+ % now second root has been separated from the one at left end point
+% we update the storage of the root at left for it to have the same number
+% of digits in mantissa. No, I decided not to do that to avoid complications.
+ % \begingroup
+ % \let\POL at IsoRight@Int\POL at IsoLeft@Int
+ % \def\POL at IsoRightSign{0}%
+ % \edef\POL at isolz@IntervalIndex{\the\numexpr\POL at isolz@IntervalIndex-\@ne}%
+ % \POL at refine@storeleftandright
+ % \endgroup
+ \edef\POL@@IsoRight at Int{\xintDSL{\xintInc{\xintDSR{\POL at IsoLeft@Int}}}}%
+ \let\POL at IsoLeft@Int\POL at IsoRight@Int
+ \let\POL at IsoLeftSign\POL at IsoRightSign
+ \ifnum\POL at IsoRightSign=\z@ % check if new Left is actually a root
+ \else
+ \edef\POL at IsoRight@Int{\xintDec{\POL@@IsoRight at Int}}%
+ \edef\POL at IsoRightSign
+ {\xintiiSgn{\PolEval{\POL at sturmname _0}\At{\POL at IsoRight@raw}}}%
+ \ifnum\POL at IsoRightSign=\POL@@IsoRightSign\space
+ \POL at refine@doonce % we need to locate in interval (1, 9) in local scale
+ \else
+ \let\POL at IsoLeft@Int\POL at IsoRight@Int
+ \ifnum\POL at IsoRightSign=\z@
+ \def\POL at IsoLeftSign{0}%
+ \else
+ \let\POL at IsoRight@Int\POL@@IsoRight at Int
+ % the IsoRightSign is now wrong but here we don't care
+ \fi\fi
+ \fi
+ % on exit, exact root found iff \POL at IsoRightSign is zero
+ \POL at refine@storeleftandright
+\endgroup
+}%
+\def\POL at refine@doonce{% if exact root is found, always in IsoRight on exit
+% NOTE: FUTURE REFACTORING WILL GET RID OF \xintiiAdd WHICH ARE A BIT COSTLY
+% BUT BASICALLY NEEDED TO HANDLE BOTH NEGATIVE AND POSITIVE HERE.
+% I WILL RE-ORGANIZE THE WHOLE THING IN FUTURE TO GET ROOTS STARTING FROM
+% THE ORIGIN AND SIMPLY RE-LABEL THE NEGATIVE ONE AT THE END. 2018/02/16.
+ \let\POL@@IsoRight at Int\POL at IsoRight@Int % 9
+ \let\POL@@IsoRightSign\POL at IsoRightSign
+ \edef\POL at IsoRight@Int{\xintiiAdd{4}{\POL at IsoLeft@Int}}% 5
+ \edef\POL at IsoRightSign
+ {\xintiiSgn{\PolEval{\POL at sturmname _0}\At{\POL at IsoRight@raw}}}%
+ \ifnum\POL at IsoRightSign=\POL at IsoLeftSign\space
+ \let\POL at IsoLeft@Int\POL at IsoRight@Int % 5
+ \edef\POL at IsoRight@Int{\xintiiAdd{2}{\POL at IsoLeft@Int}}%
+ \edef\POL at IsoRightSign
+ {\xintiiSgn{\PolEval{\POL at sturmname _0}\At{\POL at IsoRight@raw}}}%
+ \ifnum\POL at IsoRightSign=\POL at IsoLeftSign\space
+ \let\POL at IsoLeft@Int\POL at IsoRight@Int % 7
+ \edef\POL at IsoRight@Int{\xintInc{\POL at IsoLeft@Int}}%
+ \edef\POL at IsoRightSign
+ {\xintiiSgn{\PolEval{\POL at sturmname _0}\At{\POL at IsoRight@raw}}}%
+ \ifnum\POL at IsoRightSign=\POL at IsoLeftSign\space
+ \let\POL at IsoLeft@Int\POL at IsoRight@Int % 8
+ \let\POL at IsoRight@Int\POL@@IsoRight at Int % 9
+ \let\POL at IsoRightSign\POL@@IsoRightSign % opposite of one at left
+ \fi % else 7, 8 with possible root at 8
+ \else
+ \ifnum\POL at IsoRightSign=\z@
+ \let\POL at IsoLeft@Int\POL at IsoRight@Int % root at 7
+ \def\POL at IsoLeftSign{0}%
+ \else
+ \let\POL@@IsoRight at Int\POL at IsoRight@Int % 7
+ \edef\POL at IsoRight@Int{\xintInc{\POL at IsoLeft@Int}}% 6
+ \edef\POL at IsoRightSign
+ {\xintiiSgn{\PolEval{\POL at sturmname _0}\At{\POL at IsoRight@raw}}}%
+ \ifnum\POL at IsoRightSign=\POL at IsoLeftSign\space
+ \let\POL at IsoLeft@Int\POL at IsoRight@Int % 6
+ \let\POL at IsoRight@Int\POL@@IsoRight at Int % 7
+ \let\POL at IsoRightSign\POL@@IsoRightSign
+ \fi % else 5, 6 with possible root at 6
+ \fi\fi
+ \else
+ \ifnum\POL at IsoRightSign=\z@
+ \let\POL at IsoLeft@Int\POL at IsoRight@Int % root at 5
+ \def\POL at IsoLeftSign{0}%
+ \else
+ \let\POL@@IsoRight at Int\POL at IsoRight@Int % 5
+ \edef\POL at IsoRight@Int{\xintiiAdd{2}{\POL at IsoLeft@Int}}%
+ \edef\POL at IsoRightSign
+ {\xintiiSgn{\PolEval{\POL at sturmname _0}\At{\POL at IsoRight@raw}}}%
+ \ifnum\POL at IsoRightSign=\POL at IsoLeftSign\space
+ \let\POL at IsoLeft@Int\POL at IsoRight@Int % 3
+ \edef\POL at IsoRight@Int{\xintInc{\POL at IsoLeft@Int}}% 4
+ \edef\POL at IsoRightSign
+ {\xintiiSgn{\PolEval{\POL at sturmname _0}\At{\POL at IsoRight@raw}}}%
+ \ifnum\POL at IsoRightSign=\POL at IsoLeftSign\space
+ \let\POL at IsoLeft@Int\POL at IsoRight@Int % 4
+ \let\POL at IsoRight@Int\POL@@IsoRight at Int % 5
+ \let\POL at IsoRightSign\POL@@IsoRightSign
+ \fi % else 3, 4 with possible root at 4
+ \else
+ \ifnum\POL at IsoRightSign=\z@
+ \let\POL at IsoLeft@Int\POL at IsoRight@Int % root at 3
+ \def\POL at IsoLeftSign{0}%
+ \else
+ \let\POL@@IsoRight at Int\POL at IsoRight@Int % 3
+ \edef\POL at IsoRight@Int{\xintInc{\POL at IsoLeft@Int}}% 2
+ \edef\POL at IsoRightSign
+ {\xintiiSgn{\PolEval{\POL at sturmname _0}\At{\POL at IsoRight@raw}}}%
+ \ifnum\POL at IsoRightSign=\POL at IsoLeftSign\space
+ \let\POL at IsoLeft@Int\POL at IsoRight@Int % 2
+ \let\POL at IsoRight@Int\POL@@IsoRight at Int % 3
+ \let\POL at IsoRightSign\POL@@IsoRightSign
+ \fi % else 1, 2 with possible root at 2
+ \fi\fi
+ \fi\fi
+}%
+\def\POL at refine@storeleftandright{%
+ \expandafter
+ \xdef\csname POL_ZeroInt\POL at sturmname
+ L\POL at isolz@IntervalIndex\endcsname
+ {\PolDecToString{\POL at IsoLeft@rawout}}%
+ \expandafter
+ \xdef\csname POL_ZeroInt\POL at sturmname
+ R\POL at isolz@IntervalIndex\endcsname
+ {\PolDecToString{\POL at IsoRight@rawout}}%
+ \begingroup\globaldefs\@ne
+ \xintdefvar\POL at sturmname
+ L_\POL at isolz@IntervalIndex:=qfrac(\POL at IsoLeft@rawout);%
+ \xintdefvar\POL at sturmname
+ R_\POL at isolz@IntervalIndex:=qfrac(\POL at IsoRight@rawout);%
+ \endgroup
+}%
+
+
+%% \PolRefineInterval
+\def\POL at xintexprGetVar#1{\expandafter\expandafter\expandafter
+ \XINT_expr_unlock\csname XINT_expr_var_#1\endcsname}%
+\def\POL at set@IsoLeft at rawin{%
+ \edef\POL at IsoLeft@rawin
+ {\POL at xintexprGetVar{\POL at sturmname L_\POL at isolz@IntervalIndex}}%
+}%
+\def\POL at set@IsoRight at rawin{%
+ \edef\POL at IsoRight@rawin
+ {\POL at xintexprGetVar{\POL at sturmname R_\POL at isolz@IntervalIndex}}%
+}%
+\def\POL at set@IsoLeft at Int #1/1[#2]{%
+ \edef\POL at IsoLeft@Int{\xintDSH{\POL at isolz@E-#2}{#1}}%
+}%
+\newcommand\PolRefineInterval{\@ifstar\POL at srefine@start\POL at refine@start}%
+\newcommand\POL at refine@start[3][1]{%
+ \edef\POL at isolz@IntervalIndex{\the\numexpr#3}%
+ \edef\POL at sturmname{#2}%
+ \expandafter\POL at refine@sharedbody\expandafter
+ {\expandafter\POL at refine@loop\expandafter{\the\numexpr#1}}%
+}%
+\def\POL at srefine@start#1#2{%
+ \edef\POL at isolz@IntervalIndex{\the\numexpr#2}%
+ \edef\POL at sturmname{#1}%
+ \POL at refine@sharedbody
+ {\let\POL at refine@left at next\POL at refine@main % we want to recurse if needed
+ \let\POL at refine@right at next\POL at refine@main % we want to recurse if needed
+ \POL at refine@main}%
+}%
+\def\POL at refine@sharedbody#1{%
+ \POL at set@IsoLeft at rawin
+ \edef\POL at IsoLeftSign
+ {\xintiiSgn{\PolEval{\POL at sturmname _0}\At{\POL at IsoLeft@rawin}}}%
+ \ifnum\POL at IsoLeftSign=\z@
+ % do nothing if that interval was already a singleton
+ \else
+ % else both end-points are not roots and there is a single one in-between
+ \POL at set@IsoRight at rawin
+ \edef\POL at IsoRightSign{\the\numexpr-\POL at IsoLeftSign}%
+ \edef\POL at isolz@E{\expandafter\POL at refine@getE
+ % je pense que le xintrez ici est superflu
+ \romannumeral0\xintrez{\xintSub{\POL at IsoRight@rawin}{\POL at IsoLeft@rawin}}}%
+ \expandafter\POL at set@IsoLeft at Int\POL at IsoLeft@rawin
+ \edef\POL at IsoRight@Int{\xintInc{\POL at IsoLeft@Int}}%
+ #1%
+ \POL at refine@storeleftandright
+ \fi
+}%
+\def\POL at refine@loop#1{%
+ \let\POL at refine@left at next \@empty % no recursion at end sub-intervals
+ \let\POL at refine@right at next\@empty
+ \xintiloop[1+1]
+ \POL at refine@main
+ \ifnum\POL at IsoRightSign=\z@
+ \expandafter\xintbreakiloop
+ \fi
+ \ifnum\xintiloopindex<#1
+ \repeat
+}%
+\def\POL at refine@main{%
+ \edef\POL at isolz@E{\the\numexpr\POL at isolz@E-\@ne}%
+ \edef\POL at IsoLeft@Int{\xintDSL{\POL at IsoLeft@Int}}%
+ \edef\POL at IsoRight@Int{\xintDSL{\POL at IsoRight@Int}}%
+ \let\POL@@IsoRight at Int\POL at IsoRight@Int
+ \let\POL@@IsoRightSign\POL at IsoRightSign
+ \edef\POL at IsoRight@Int{\xintInc{\POL at IsoLeft@Int}}%
+ \edef\POL at IsoRightSign
+ {\xintiiSgn{\PolEval{\POL at sturmname _0}\At{\POL at IsoRight@raw}}}%
+ \ifnum\POL at IsoRightSign=\z@
+ \let\POL at IsoLeft@Int\POL at IsoRight@Int % root at 1
+ \def\POL at IsoLeftSign{0}%
+ \let\POL at next\@empty
+ \else
+ \ifnum\POL at IsoRightSign=\POL@@IsoRightSign\space
+ \let\POL at next\POL at refine@left at next % may be \@empty or \POL at refine@main for recursion
+ \let\POL at refine@right at next\@empty
+ \else
+ \let\POL at IsoLeft@Int\POL at IsoRight@Int
+ \edef\POL at IsoRight@Int{\xintDec{\POL@@IsoRight at Int}}%
+ \edef\POL at IsoRightSign
+ {\xintiiSgn{\PolEval{\POL at sturmname _0}\At{\POL at IsoRight@raw}}}%
+ \ifnum\POL at IsoRightSign=\z@
+ \let\POL at IsoLeft@Int\POL at IsoRight@Int % root at 9
+ \def\POL at IsoLeftSign{0}%
+ \let\POL at next\@empty
+ \else
+ \ifnum\POL at IsoRightSign=\POL@@IsoRightSign\space
+ \let\POL at next\POL at refine@doonce
+ \else
+ \let\POL at IsoLeft@Int\POL at IsoRight@Int
+ \let\POL at IsoRight@Int\POL@@IsoRight at Int
+ \let\POL at IsoRightSign\POL@@IsoRightSign
+ \let\POL at next\POL at refine@right at next
+ \let\POL at refine@left at next\@empty
+ \fi
+ \fi
+ \fi\fi
+ \POL at next
+}%
+% lacking pre-defined xintfrac macro here (such as an \xintRawExponent)
+\def\POL at refine@getE#1[#2]{#2}% \xintREZ already applied, for safety
+
+
+\newcommand\PolIntervalWidth[2]{%
+% le \xintRez est à cause des E positifs, car trailing zéros explicites
+% si je travaillais à partir des variables xintexpr directement ne devrait
+% pas être nécessaire, mais trop fragile par rapport à chgt internes possibles
+ \romannumeral0\xintrez{\xintSub{\@nameuse{POL_ZeroInt#1R}{#2}}%
+ {\@nameuse{POL_ZeroInt#1L}{#2}}}
+}%
+
+
+\newcommand\PolEnsureIntervalLengths[2]{% #1 = Sturm chain name,
+ % localize roots in intervals of length at most 10^{#2}
+ \POL at count\z@
+ % \POL at count used by \POL at sturmchain@getSV at at but latter not used
+ \edef\POL at sturmname{#1}%
+ \edef\POL at ensure@targetE{\the\numexpr#2}%
+ \edef\POL at nbofroots{\csname POL_ZeroInt\POL at sturmname L\endcsname 0}%
+ \xintloop
+ \advance\POL at count\@ne
+ \edef\POL at isolz@IntervalIndex{\the\POL at count}%
+ \POL at ensure@one
+ \ifnum\POL at nbofroots>\POL at count
+ \repeat
+}%
+\newcommand\PolEnsureIntervalLength[3]{% #1 = Sturm chain name,
+ % #2 = index of interval
+ % localize roots in intervals of length at most 10^{#3}
+ \edef\POL at sturmname{#1}%
+ \edef\POL at ensure@targetE{\the\numexpr#3}%
+ \edef\POL at isolz@IntervalIndex{\the\numexpr#2}%
+ \POL at ensure@one
+}%
+\def\POL at ensure@one{%
+ \POL at set@IsoLeft at rawin
+ \POL at set@IsoRight at rawin
+ \edef\POL at ensure@delta{\xintREZ{\xintSub{\POL at IsoRight@rawin}{\POL at IsoLeft@rawin}}}%
+ \xintiiifZero{\POL at ensure@delta}
+ {}
+ {\edef\POL at isolz@E{\expandafter\POL at refine@getE\POL at ensure@delta}%
+ \expandafter\POL at set@IsoLeft at Int\POL at IsoLeft@rawin
+ \edef\POL at IsoRight@Int{\xintInc{\POL at IsoLeft@Int}}%
+ \ifnum\POL at isolz@E>\POL at ensure@targetE\space
+ \edef\POL at IsoLeftSign
+ {\xintiiSgn{\PolEval{\POL at sturmname _0}\At{\POL at IsoLeft@raw}}}%
+ % at start left and right are not roots, and values of opposite signs
+ % \edef\POL at IsoRightSign{\the\numexpr-\POL at IsoLeftSign}%
+ \xintloop
+ \POL at ensure@Eloopbody % decreases E by one at each iteration
+ % if separation level is still too coarse we recurse at deeper level
+ \ifnum\POL at isolz@E>\POL at ensure@targetE\space
+ \repeat
+ % will check if right is at a zero, needs \POL at IsoRightSign set up
+ \POL at refine@storeleftandright
+ \fi
+ }%
+}%
+\def\POL at ensure@Eloopbody {%
+ \edef\POL at isolz@E{\the\numexpr\POL at isolz@E-\@ne}%
+ \edef\POL at IsoLeft@Int{\xintDSL{\POL at IsoLeft@Int}}%
+ % this will loop at most ten times
+ \xintloop
+ \edef\POL at IsoRight@Int{\xintInc{\POL at IsoLeft@Int}}%
+ \edef\POL at IsoRightSign
+ {\xintiiSgn{\PolEval{\POL at sturmname _0}\At{\POL at IsoRight@raw}}}%
+ % if we have found a zero at right boundary the \ifnum test will fail
+ % and we exit the loop
+ % else we exit the loop if sign at right boundary is opposite of
+ % sign at left boundary (the latter is +1 or -1, never 0)
+ % this is a bit wasteful if we go ten times to the right, because
+ % we know that there the sign will be opposite, evaluation was superfluous
+ \ifnum\POL at IsoLeftSign=\POL at IsoRightSign\space
+ \let\POL at IsoLeft@Int\POL at IsoRight@Int
+ \repeat
+ % check for case when we exited the inner loop because we actually
+ % found a zero, then we force exit from the main (E decreasing) loop
+ \ifnum\POL at IsoRightSign=\z@
+ \expandafter\xintbreakloop
+ \fi
+}%
+
+
+\catcode`_ 8
+\newcommand\PolPrintIntervals[2][Z]{%
+ \POL at count \@nameuse{POL_ZeroInt#2L}{0}
+ \ifnum\POL at count=\z@
+% No real roots.\par
+ \else
+% There are \the\POL at count\space distinct real roots:\par
+ \[\count@\POL at count
+ \global\POL at count\@ne
+ \begin{array}{rcccl}
+ \xintloop
+ \POL at SturmIfZeroExactlyKnown{#2}\POL at count
+ {% exact root
+ &&
+ #1_{\the\POL at count}&=&
+ \POL at printintervals@prepare{#2R}%
+ \PolPrintIntervalsPrintExactZero
+ }%
+ {% interval with root in its strict interior
+ \POL at printintervals@prepare{#2L}%
+ \PolPrintIntervalsPrintLeftEndPoint&<&
+ #1_{\the\POL at count}&<&
+ \POL at printintervals@prepare{#2R}%
+ \PolPrintIntervalsPrintRightEndPoint
+ }%
+ \global\advance\POL at count\@ne
+ \unless\ifnum\POL at count>\count@
+ \\%
+ \repeat
+ \end{array}\]
+ \fi
+}%
+\catcode`_ 11
+\newcommand\PolPrintIntervalsPrintExactZero {\PolPrintIntervalsTheEndPoint}%
+\newcommand\PolPrintIntervalsPrintLeftEndPoint {\PolPrintIntervalsTheEndPoint}%
+\newcommand\PolPrintIntervalsPrintRightEndPoint{\PolPrintIntervalsTheEndPoint}%
+\def\POL at printintervals@prepare#1{%
+ \edef\PolPrintIntervalsTheIndex{\the\POL at count}%
+ \edef\PolPrintIntervalsTheEndPoint{\@nameuse{POL_ZeroInt#1}\POL at count}%
+ \xintiiifSgn{\POL at xintexprGetVar{#1_\PolPrintIntervalsTheIndex}}
+ {\let\PolIfEndPointIsPositive\xint_secondoftwo
+ \let\PolIfEndPointIsNegative\xint_firstoftwo
+ \let\PolIfEndPointIsZero\xint_secondoftwo}
+ {\let\PolIfEndPointIsPositive\xint_secondoftwo
+ \let\PolIfEndPointIsNegative\xint_secondoftwo
+ \let\PolIfEndPointIsZero\xint_firstoftwo}
+ {\let\PolIfEndPointIsPositive\xint_firstoftwo
+ \let\PolIfEndPointIsNegative\xint_secondoftwo
+ \let\PolIfEndPointIsZero\xint_secondoftwo}%
+}%
+\newcommand\POL at SturmIfZeroExactlyKnown[2]{% faster than public one,
+ % but does not check if #2 is in range
+ \romannumeral0\xintifeq{\POL at xintexprGetVar{#1L_\the\numexpr#2\relax}}%
+ {\POL at xintexprGetVar{#1R_\the\numexpr#2\relax}}%
+}%
+
+
+\newcommand\PolSturmIfZeroExactlyKnown[2]{%
+ \romannumeral0\xintifeq{\PolSturmIsolatedZeroLeft{#1}{#2}}%
+ {\PolSturmIsolatedZeroRight{#1}{#2}}%
+}%
+\newcommand\PolSturmIsolatedZeroLeft[2]{%
+ \romannumeral`^^@\csname POL_ZeroInt#1L\endcsname{#2}}%
+\newcommand\PolSturmIsolatedZeroRight[2]{%
+ \romannumeral`^^@\csname POL_ZeroInt#1R\endcsname{#2}}%
+\newcommand\PolSturmNbOfIsolatedZeros[1]{%
+ \romannumeral`^^@\csname POL_ZeroInt#1L0\endcsname
+}%
+
+%% \PolDecToString (should become an xintfrac macro at some point)
+\newcommand\PolDecToString[1]{\romannumeral0\expandafter
+ \POL at dectostring\romannumeral0\xintraw{#1}}%
+\def\POL at dectostring #1/#2[#3]{\xintiiifZero {#1}%
+ \POL at dectostring@z
+ {\if1\XINT_isOne{#2}\expandafter\POL at dectostring@a
+ \else\expandafter\POL at dectostring@b
+ \fi}%
+ #1/#2[#3]%
+}%
+\def\POL at dectostring@z#1[#2]{ 0}%
+\def\POL at dectostring@a#1/#2[#3]{%
+ \ifnum#3<\z@\xint_dothis{\xinttrunc{-#3}{#1[#3]}}\fi
+ \xint_orthat{\xintiie{#1}{#3}}%
+}%
+\def\POL at dectostring@b#1/#2[#3]{% just to handle this somehow
+ \ifnum#3<\z@\xint_dothis{\xinttrunc{-#3}{#1[#3]}/#2}\fi
+ \xint_orthat{\xintiie{#1}{#3}/#2}%
+}%
+
+
+\newcommand\PolMakeMonic[1]{%
+ \edef\POL at leadingcoeff{\PolLeadingCoeff{#1}}%
+ \edef\POL at leadingcoeff@inverse{\xintDiv{1/1[0]}{\POL at leadingcoeff}}%
+ \PolMapCoeffs{\xintMul{\POL at leadingcoeff@inverse}}{#1}%
+}%
+
+
%% CORE ALGEBRA MACROS
%% We do this non-expandably, but in a nestable way... this is the whole
%% point because \xintdeffunc as used by \poldef creates a big nested macro.
@@ -635,7 +1490,7 @@
\def\POL at add@b{\POL at get\POL at B\POL at add@c}%
\def\POL at add@c{%
\global\POL at poltrue
- \POL at ifZero\POL at A
+ \POL at ifZero\POL at A
{\let\POL at result\POL at B}%
{\POL at ifZero\POL at B
{\let\POL at result\POL at A}%
@@ -678,7 +1533,7 @@
\def\POL at mul@b{\POL at get\POL at B\POL at mul@c}%
\def\POL at mul@c{%
\global\POL at poltrue
- \POL at ifZero\POL at A
+ \POL at ifZero\POL at A
{\def\POL at result{-1.\empty{0/1[0]}}}%
{\POL at ifZero\POL at B
{\def\POL at result{-1.\empty{0/1[0]}}}%
@@ -731,7 +1586,7 @@
\ifnum\count\tw@<\count@
\repeat
\expandafter\let\csname POL at arrayC\the\count@\endcsname\POL at tmp
-}%
+}%
\def\POL@@mul at phaseIIloopbody{%
\advance\count@\@ne
\def\POL at tmp{0[0]}%
@@ -748,7 +1603,7 @@
{\@nameuse{POL at arrayB\the\numexpr\count at +\@ne-\count\tw@}}%
}%
}%
- \repeat
+ \repeat
\expandafter\let\csname POL at arrayC\the\count@\endcsname\POL at tmp
}%
\def\POL@@mul at phaseIIIloopbody{%
@@ -766,7 +1621,7 @@
}%
}%
\ifnum\@nameuse{POL at arrayA0}>\count\tw@
- \repeat
+ \repeat
\expandafter\let\csname POL at arrayC\the\count@\endcsname\POL at tmp
}%
@@ -848,7 +1703,7 @@
\ifodd\POL at pow@exp\space
\expandafter\POL@@pow at odd
\the\numexpr(\POL at pow@exp+\@ne)/\tw at -\@ne\expandafter.%
- \else
+ \else
\expandafter\POL@@pow at even
\the\numexpr(\POL at pow@exp+\@ne)/\tw at -\@ne\expandafter.%
\fi
@@ -915,7 +1770,7 @@
\xintScalarDiv{\csname POL at arrayR\the\count@\endcsname}%
{\POL at B@leading}}%
\expandafter\let\csname POL at arrayQ\the\count\tw@\endcsname
- \POL@@div at ratio
+ \POL@@div at ratio
\advance\count@\m at ne
\advance\count\tw@\m at ne
\count4 \count@
@@ -956,18 +1811,42 @@
%% EXPANDABLE MACROS
-\newcommand\PolEval{}%
-\def\PolEval#1#2\At#3{\romannumeral`^^@\xinttheexpr #1(#3)\relax}%
+\def\Pol at Eval@fork#1\At#2#3\krof{#2}%
+\newcommand\PolEval[3]{\romannumeral`^^@\Pol at Eval@fork
+ #2\PolEvalAt
+ \At\PolEvalAtExpr\krof {#1}{#3}%
+}%
+\newcommand\PolEvalAt[2]
+ {\xintpraw{\csname XINT_expr_userfunc_#1\endcsname{#2},}}%
+\newcommand\PolEvalAtExpr[2]{\xinttheexpr #1(#2)\relax}%
%
-\newcommand\PolEvalReduced{}%
-\def\PolEvalReduced#1#2\At#3{%
- \romannumeral0\xintpraw % only serves to not print denominator if = 1
- {\xintIrr{\romannumeral`^^@\xintthebareeval#1(#3)\relax}[0]}%
+\newcommand\PolEvalReduced[3]{\romannumeral`^^@\Pol at Eval@fork
+ #2\PolEvalReducedAt
+ \At\PolEvalReducedAtExpr\krof {#1}{#3}%
}%
+\newcommand\PolEvalReducedAt[2]{%
+ \xintpraw % in order not to print denominator if the latter equals 1
+ {\xintIrr{\csname XINT_expr_userfunc_#1\endcsname{#2},}[0]}%
+}%
+\newcommand\PolEvalReducedAtExpr[2]{%
+ \xintpraw
+ {\xintIrr{\romannumeral`^^@\xintthebareeval#1(#2)\relax}[0]}%
+}%
%
-\newcommand\PolFloatEval{}%
-\def\PolFloatEval#1#2\At#3{\romannumeral`^^@\xintthefloatexpr #1(#3)\relax}%
+\newcommand\PolFloatEval[3]{\romannumeral`^^@\Pol at Eval@fork
+ #2\PolFloatEvalAt
+ \At\PolFloatEvalAtExpr\krof {#1}{#3}%
+}%
+\newcommand\PolFloatEvalAt[2]
+ {\xintpfloat{\csname XINT_flexpr_userfunc_#1\endcsname{#2},}}%
+\newcommand\PolFloatEvalAtExpr[2]{\xintthefloatexpr #1(#2)\relax}%
%
+\newcommand\PolLeadingCoeff[1]{%
+ \romannumeral`^^@\expandafter\expandafter\expandafter\xintlastitem
+ \expandafter\expandafter\expandafter
+ {\csname POLuserpol@#1\endcsname}%
+}%
+%
\newcommand\PolNthCoeff[2]{\romannumeral`^^@%
\expandafter\POL at nthcoeff
\romannumeral0\xintnthelt{\ifnum\numexpr#2<\z@#2\else(#2)+1\fi}%
@@ -990,11 +1869,10 @@
\newcommand\PolToCSV[1]{\romannumeral0\xintlistwithsep{, }{\PolToList{#1}}}%
-\newif\ifpoltoexprall
\newcommand\PolToExprCmd[1]{\xintPRaw{\xintRawWithZeros{#1}}}%
\newcommand\PolToFloatExprCmd[1]{\xintFloat{#1}}%
\let\PolToExprTermPrefix\PolTypesetCmdPrefix
-\newcommand\PolToExprOneTerm[2]{%
+\newcommand\PolToExprOneTermStyleA[2]{%
\ifnum#2=\z@
\PolToExprCmd{#1}%
\else
@@ -1007,6 +1885,7 @@
\else\PolToExprVar^\xintiiAbs{#2}%
\fi
}%
+\let\PolToExprOneTerm\PolToExprOneTermStyleA
\newcommand\PolToExprOneTermStyleB[2]{%
\ifnum#2=\z@
\xintNumerator{#1}%
More information about the tex-live-commits
mailing list