texlive[49378] Master/texmf-dist: polexpr (10dec18)
commits+karl at tug.org
commits+karl at tug.org
Mon Dec 10 23:58:44 CET 2018
Revision: 49378
http://tug.org/svn/texlive?view=revision&revision=49378
Author: karl
Date: 2018-12-10 23:58:44 +0100 (Mon, 10 Dec 2018)
Log Message:
-----------
polexpr (10dec18)
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 2018-12-10 22:58:31 UTC (rev 49377)
+++ trunk/Master/texmf-dist/doc/latex/polexpr/README.md 2018-12-10 22:58:44 UTC (rev 49378)
@@ -75,8 +75,10 @@
New feature: multiplicity of roots.
- 0.7 (2018/12/08)
New feature: finding all rational roots.
+- 0.7.1, 0.7.2 (2018/12/09)
+ Emergency bugfix.
-Files of 0.7 release:
+Files of 0.7.2 release:
- README.md,
- polexpr.sty (package file),
Modified: trunk/Master/texmf-dist/doc/latex/polexpr/polexpr.html
===================================================================
--- trunk/Master/texmf-dist/doc/latex/polexpr/polexpr.html 2018-12-10 22:58:31 UTC (rev 49377)
+++ trunk/Master/texmf-dist/doc/latex/polexpr/polexpr.html 2018-12-10 22:58:44 UTC (rev 49378)
@@ -362,7 +362,7 @@
<body>
<div class="document" id="package-polexpr-documentation">
<h1 class="title">Package polexpr documentation</h1>
-<h2 class="subtitle" id="id1">0.7 (2018/12/08)</h2>
+<h2 class="subtitle" id="id1">0.7.2 (2018/12/09)</h2>
<!-- comment: -*- fill-column: 72; mode: rst; -*- -->
<div class="contents topic" id="contents">
@@ -376,142 +376,143 @@
<li><a class="reference internal" href="#a-degree-five-polynomial-with-three-rational-roots" id="id43">A degree five polynomial with three rational roots</a></li>
<li><a class="reference internal" href="#a-mignotte-type-polynomial" id="id44">A Mignotte type polynomial</a></li>
<li><a class="reference internal" href="#the-wilkinson-polynomial" id="id45">The Wilkinson 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="id46">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="id47">Roots of Chebyshev polynomials</a></li>
+<li><a class="reference internal" href="#the-second-wilkinson-polynomial" id="id46">The second Wilkinson 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="id47">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="id48">Roots of Chebyshev polynomials</a></li>
</ul>
</li>
-<li><a class="reference internal" href="#non-expandable-macros" id="id48">Non-expandable macros</a><ul>
-<li><a class="reference internal" href="#poldef-polname-letter-expression-in-letter" id="id49"><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="id50"><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="id51"><tt class="docutils literal">\PolGenFloatVariant{polname}</tt></a></li>
-<li><a class="reference internal" href="#pollet-polname-2-polname-1" id="id52"><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="id53"><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="id54"><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="id55"><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="id56"><tt class="docutils literal"><span class="pre">\PolFromCSV{polname}{<csv>}</span></tt></a></li>
-<li><a class="reference internal" href="#poltypeset-polname" id="id57"><tt class="docutils literal">\PolTypeset{polname}</tt></a><ul>
-<li><a class="reference internal" href="#poltypesetcmd-raw-coeff" id="id58"><tt class="docutils literal">\PolTypesetCmd{raw_coeff}</tt></a></li>
-<li><a class="reference internal" href="#poltypesetone-raw-coeff" id="id59"><tt class="docutils literal">\PolTypesetOne{raw_coeff}</tt></a></li>
-<li><a class="reference internal" href="#id6" id="id60"><tt class="docutils literal">\PolTypesetMonomialCmd</tt></a></li>
-<li><a class="reference internal" href="#poltypesetcmdprefix-raw-coeff" id="id61"><tt class="docutils literal">\PolTypesetCmdPrefix{raw_coeff}</tt></a></li>
+<li><a class="reference internal" href="#non-expandable-macros" id="id49">Non-expandable macros</a><ul>
+<li><a class="reference internal" href="#poldef-polname-letter-expression-in-letter" id="id50"><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="id51"><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="id52"><tt class="docutils literal">\PolGenFloatVariant{polname}</tt></a></li>
+<li><a class="reference internal" href="#pollet-polname-2-polname-1" id="id53"><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="id54"><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="id55"><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="id56"><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="id57"><tt class="docutils literal"><span class="pre">\PolFromCSV{polname}{<csv>}</span></tt></a></li>
+<li><a class="reference internal" href="#poltypeset-polname" id="id58"><tt class="docutils literal">\PolTypeset{polname}</tt></a><ul>
+<li><a class="reference internal" href="#poltypesetcmd-raw-coeff" id="id59"><tt class="docutils literal">\PolTypesetCmd{raw_coeff}</tt></a></li>
+<li><a class="reference internal" href="#poltypesetone-raw-coeff" id="id60"><tt class="docutils literal">\PolTypesetOne{raw_coeff}</tt></a></li>
+<li><a class="reference internal" href="#id6" id="id61"><tt class="docutils literal">\PolTypesetMonomialCmd</tt></a></li>
+<li><a class="reference internal" href="#poltypesetcmdprefix-raw-coeff" id="id62"><tt class="docutils literal">\PolTypesetCmdPrefix{raw_coeff}</tt></a></li>
</ul>
</li>
-<li><a class="reference internal" href="#id8" id="id62"><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="id63"><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="id64"><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="id65"><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="id66"><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="id67"><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="id68"><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="id69"><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="id70"><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="id71"><tt class="docutils literal"><span class="pre">\PolToSturm{polname}{sturmname}</span></tt></a></li>
-<li><a class="reference internal" href="#id10" id="id72"><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="id73"><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="id74"><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="id75"><tt class="docutils literal">\PolSturmIsolateZeros{sturmname}</tt></a></li>
-<li><a class="reference internal" href="#id12" id="id76"><tt class="docutils literal"><span class="pre">\PolSturmIsolateZeros*{sturmname}</span></tt></a></li>
-<li><a class="reference internal" href="#id14" id="id77"><tt class="docutils literal"><span class="pre">\PolSturmIsolateZeros**{sturmname}</span></tt></a></li>
-<li><a class="reference internal" href="#polsturmisolatezerosandgetmultiplicities-sturmname" id="id78"><tt class="docutils literal">\PolSturmIsolateZerosAndGetMultiplicities{sturmname}</tt></a></li>
-<li><a class="reference internal" href="#polsturmisolatezerosgetmultiplicitiesandrationalroots-sturmname" id="id79"><tt class="docutils literal">\PolSturmIsolateZerosGetMultiplicitiesAndRationalRoots{sturmname}</tt></a></li>
-<li><a class="reference internal" href="#polsturmisolatezerosandfindrationalroots-sturmname" id="id80"><tt class="docutils literal">\PolSturmIsolateZerosAndFindRationalRoots{sturmname}</tt></a></li>
-<li><a class="reference internal" href="#polrefineinterval-sturmname-index" id="id81"><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="id82"><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="id83"><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="id84"><tt class="docutils literal"><span class="pre">\PolEnsureIntervalLengths{sturmname}{E}</span></tt></a></li>
-<li><a class="reference internal" href="#polprintintervals-varname-sturmname" id="id85"><tt class="docutils literal"><span class="pre">\PolPrintIntervals[varname]{sturmname}</span></tt></a><ul>
-<li><a class="reference internal" href="#polprintintervalsnorealroots" id="id86"><tt class="docutils literal">\PolPrintIntervalsNoRealRoots</tt></a></li>
-<li><a class="reference internal" href="#polprintintervalsbeginenv" id="id87"><tt class="docutils literal">\PolPrintIntervalsBeginEnv</tt></a></li>
-<li><a class="reference internal" href="#polprintintervalsendenv" id="id88"><tt class="docutils literal">\PolPrintIntervalsEndEnv</tt></a></li>
-<li><a class="reference internal" href="#polprintintervalsknownroot" id="id89"><tt class="docutils literal">\PolPrintIntervalsKnownRoot</tt></a></li>
-<li><a class="reference internal" href="#polprintintervalsunknownroot" id="id90"><tt class="docutils literal">\PolPrintIntervalsUnknownRoot</tt></a></li>
-<li><a class="reference internal" href="#id15" id="id91"><tt class="docutils literal">\PolPrintIntervalsPrintExactZero</tt></a></li>
-<li><a class="reference internal" href="#id16" id="id92"><tt class="docutils literal">\PolPrintIntervalsPrintLeftEndPoint</tt></a></li>
-<li><a class="reference internal" href="#id17" id="id93"><tt class="docutils literal">\PolPrintIntervalsPrintRightEndPoint</tt></a></li>
+<li><a class="reference internal" href="#id8" id="id63"><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="id64"><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="id65"><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="id66"><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="id67"><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="id68"><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="id69"><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="id70"><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="id71"><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="id72"><tt class="docutils literal"><span class="pre">\PolToSturm{polname}{sturmname}</span></tt></a></li>
+<li><a class="reference internal" href="#id10" id="id73"><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="id74"><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="id75"><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="id76"><tt class="docutils literal">\PolSturmIsolateZeros{sturmname}</tt></a></li>
+<li><a class="reference internal" href="#id12" id="id77"><tt class="docutils literal"><span class="pre">\PolSturmIsolateZeros*{sturmname}</span></tt></a></li>
+<li><a class="reference internal" href="#id14" id="id78"><tt class="docutils literal"><span class="pre">\PolSturmIsolateZeros**{sturmname}</span></tt></a></li>
+<li><a class="reference internal" href="#polsturmisolatezerosandgetmultiplicities-sturmname" id="id79"><tt class="docutils literal">\PolSturmIsolateZerosAndGetMultiplicities{sturmname}</tt></a></li>
+<li><a class="reference internal" href="#polsturmisolatezerosgetmultiplicitiesandrationalroots-sturmname" id="id80"><tt class="docutils literal">\PolSturmIsolateZerosGetMultiplicitiesAndRationalRoots{sturmname}</tt></a></li>
+<li><a class="reference internal" href="#polsturmisolatezerosandfindrationalroots-sturmname" id="id81"><tt class="docutils literal">\PolSturmIsolateZerosAndFindRationalRoots{sturmname}</tt></a></li>
+<li><a class="reference internal" href="#polrefineinterval-sturmname-index" id="id82"><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="id83"><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="id84"><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="id85"><tt class="docutils literal"><span class="pre">\PolEnsureIntervalLengths{sturmname}{E}</span></tt></a></li>
+<li><a class="reference internal" href="#polprintintervals-varname-sturmname" id="id86"><tt class="docutils literal"><span class="pre">\PolPrintIntervals[varname]{sturmname}</span></tt></a><ul>
+<li><a class="reference internal" href="#polprintintervalsnorealroots" id="id87"><tt class="docutils literal">\PolPrintIntervalsNoRealRoots</tt></a></li>
+<li><a class="reference internal" href="#polprintintervalsbeginenv" id="id88"><tt class="docutils literal">\PolPrintIntervalsBeginEnv</tt></a></li>
+<li><a class="reference internal" href="#polprintintervalsendenv" id="id89"><tt class="docutils literal">\PolPrintIntervalsEndEnv</tt></a></li>
+<li><a class="reference internal" href="#polprintintervalsknownroot" id="id90"><tt class="docutils literal">\PolPrintIntervalsKnownRoot</tt></a></li>
+<li><a class="reference internal" href="#polprintintervalsunknownroot" id="id91"><tt class="docutils literal">\PolPrintIntervalsUnknownRoot</tt></a></li>
+<li><a class="reference internal" href="#id15" id="id92"><tt class="docutils literal">\PolPrintIntervalsPrintExactZero</tt></a></li>
+<li><a class="reference internal" href="#id16" id="id93"><tt class="docutils literal">\PolPrintIntervalsPrintLeftEndPoint</tt></a></li>
+<li><a class="reference internal" href="#id17" id="id94"><tt class="docutils literal">\PolPrintIntervalsPrintRightEndPoint</tt></a></li>
</ul>
</li>
-<li><a class="reference internal" href="#id19" id="id94"><tt class="docutils literal"><span class="pre">\PolPrintIntervals*[varname]{sturmname}</span></tt></a><ul>
-<li><a class="reference internal" href="#polprintintervalsprintmultiplicity" id="id95"><tt class="docutils literal">\PolPrintIntervalsPrintMultiplicity</tt></a></li>
+<li><a class="reference internal" href="#id19" id="id95"><tt class="docutils literal"><span class="pre">\PolPrintIntervals*[varname]{sturmname}</span></tt></a><ul>
+<li><a class="reference internal" href="#polprintintervalsprintmultiplicity" id="id96"><tt class="docutils literal">\PolPrintIntervalsPrintMultiplicity</tt></a></li>
</ul>
</li>
-<li><a class="reference internal" href="#polmapcoeffs-macro-polname" id="id96"><tt class="docutils literal"><span class="pre">\PolMapCoeffs{\macro}{polname}</span></tt></a></li>
-<li><a class="reference internal" href="#polreducecoeffs-polname" id="id97"><tt class="docutils literal">\PolReduceCoeffs{polname}</tt></a></li>
-<li><a class="reference internal" href="#id21" id="id98"><tt class="docutils literal"><span class="pre">\PolReduceCoeffs*{polname}</span></tt></a></li>
-<li><a class="reference internal" href="#polmakemonic-polname" id="id99"><tt class="docutils literal">\PolMakeMonic{polname}</tt></a></li>
-<li><a class="reference internal" href="#polmakeprimitive-polname" id="id100"><tt class="docutils literal">\PolMakePrimitive{polname}</tt></a></li>
+<li><a class="reference internal" href="#polmapcoeffs-macro-polname" id="id97"><tt class="docutils literal"><span class="pre">\PolMapCoeffs{\macro}{polname}</span></tt></a></li>
+<li><a class="reference internal" href="#polreducecoeffs-polname" id="id98"><tt class="docutils literal">\PolReduceCoeffs{polname}</tt></a></li>
+<li><a class="reference internal" href="#id21" id="id99"><tt class="docutils literal"><span class="pre">\PolReduceCoeffs*{polname}</span></tt></a></li>
+<li><a class="reference internal" href="#polmakemonic-polname" id="id100"><tt class="docutils literal">\PolMakeMonic{polname}</tt></a></li>
+<li><a class="reference internal" href="#polmakeprimitive-polname" id="id101"><tt class="docutils literal">\PolMakePrimitive{polname}</tt></a></li>
</ul>
</li>
-<li><a class="reference internal" href="#expandable-macros" id="id101">Expandable macros</a><ul>
-<li><a class="reference internal" href="#poleval-polname-atexpr-numerical-expression" id="id102"><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="id103"><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="id104"><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="id105"><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="id106"><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="id107"><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="id108"><tt class="docutils literal"><span class="pre">\PolIfCoeffIsPlusOrMinusOne{A}{B}</span></tt></a></li>
-<li><a class="reference internal" href="#polleadingcoeff-polname" id="id109"><tt class="docutils literal">\PolLeadingCoeff{polname}</tt></a></li>
-<li><a class="reference internal" href="#polnthcoeff-polname-number" id="id110"><tt class="docutils literal"><span class="pre">\PolNthCoeff{polname}{number}</span></tt></a></li>
-<li><a class="reference internal" href="#poldegree-polname" id="id111"><tt class="docutils literal">\PolDegree{polname}</tt></a></li>
-<li><a class="reference internal" href="#policontent-polname" id="id112"><tt class="docutils literal">\PolIContent{polname}</tt></a></li>
-<li><a class="reference internal" href="#poltoexpr-polname" id="id113"><tt class="docutils literal">\PolToExpr{polname}</tt></a><ul>
-<li><a class="reference internal" href="#poltoexproneterm-raw-coeff-number" id="id114"><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="id115"><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="id116"><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="id117"><tt class="docutils literal">\PolToExprCmd{raw_coeff}</tt></a></li>
-<li><a class="reference internal" href="#poltoexprtermprefix-raw-coeff" id="id118"><tt class="docutils literal">\PolToExprTermPrefix{raw_coeff}</tt></a></li>
-<li><a class="reference internal" href="#id28" id="id119"><tt class="docutils literal">\PolToExprVar</tt></a></li>
-<li><a class="reference internal" href="#id29" id="id120"><tt class="docutils literal">\PolToExprTimes</tt></a></li>
+<li><a class="reference internal" href="#expandable-macros" id="id102">Expandable macros</a><ul>
+<li><a class="reference internal" href="#poleval-polname-atexpr-numerical-expression" id="id103"><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="id104"><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="id105"><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="id106"><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="id107"><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="id108"><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="id109"><tt class="docutils literal"><span class="pre">\PolIfCoeffIsPlusOrMinusOne{A}{B}</span></tt></a></li>
+<li><a class="reference internal" href="#polleadingcoeff-polname" id="id110"><tt class="docutils literal">\PolLeadingCoeff{polname}</tt></a></li>
+<li><a class="reference internal" href="#polnthcoeff-polname-number" id="id111"><tt class="docutils literal"><span class="pre">\PolNthCoeff{polname}{number}</span></tt></a></li>
+<li><a class="reference internal" href="#poldegree-polname" id="id112"><tt class="docutils literal">\PolDegree{polname}</tt></a></li>
+<li><a class="reference internal" href="#policontent-polname" id="id113"><tt class="docutils literal">\PolIContent{polname}</tt></a></li>
+<li><a class="reference internal" href="#poltoexpr-polname" id="id114"><tt class="docutils literal">\PolToExpr{polname}</tt></a><ul>
+<li><a class="reference internal" href="#poltoexproneterm-raw-coeff-number" id="id115"><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="id116"><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="id117"><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="id118"><tt class="docutils literal">\PolToExprCmd{raw_coeff}</tt></a></li>
+<li><a class="reference internal" href="#poltoexprtermprefix-raw-coeff" id="id119"><tt class="docutils literal">\PolToExprTermPrefix{raw_coeff}</tt></a></li>
+<li><a class="reference internal" href="#id28" id="id120"><tt class="docutils literal">\PolToExprVar</tt></a></li>
+<li><a class="reference internal" href="#id29" id="id121"><tt class="docutils literal">\PolToExprTimes</tt></a></li>
</ul>
</li>
-<li><a class="reference internal" href="#id31" id="id121"><tt class="docutils literal"><span class="pre">\PolToExpr*{polname}</span></tt></a></li>
-<li><a class="reference internal" href="#poltofloatexpr-polname" id="id122"><tt class="docutils literal">\PolToFloatExpr{polname}</tt></a><ul>
-<li><a class="reference internal" href="#poltofloatexproneterm-raw-coeff-number" id="id123"><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="id124"><tt class="docutils literal">\PolToFloatExprCmd{raw_coeff}</tt></a></li>
+<li><a class="reference internal" href="#id31" id="id122"><tt class="docutils literal"><span class="pre">\PolToExpr*{polname}</span></tt></a></li>
+<li><a class="reference internal" href="#poltofloatexpr-polname" id="id123"><tt class="docutils literal">\PolToFloatExpr{polname}</tt></a><ul>
+<li><a class="reference internal" href="#poltofloatexproneterm-raw-coeff-number" id="id124"><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="id125"><tt class="docutils literal">\PolToFloatExprCmd{raw_coeff}</tt></a></li>
</ul>
</li>
-<li><a class="reference internal" href="#id35" id="id125"><tt class="docutils literal"><span class="pre">\PolToFloatExpr*{polname}</span></tt></a></li>
-<li><a class="reference internal" href="#poltolist-polname" id="id126"><tt class="docutils literal">\PolToList{polname}</tt></a></li>
-<li><a class="reference internal" href="#poltocsv-polname" id="id127"><tt class="docutils literal">\PolToCSV{polname}</tt></a></li>
-<li><a class="reference internal" href="#polsturmchainlength-sturmname" id="id128"><tt class="docutils literal">\PolSturmChainLength{sturmname}</tt></a></li>
-<li><a class="reference internal" href="#polsturmifzeroexactlyknown-sturmname-index-a-b" id="id129"><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="id130"><tt class="docutils literal"><span class="pre">\PolSturmIsolatedZeroLeft{sturmname}{index}</span></tt></a></li>
-<li><a class="reference internal" href="#polsturmisolatedzeroright-sturmname-index" id="id131"><tt class="docutils literal"><span class="pre">\PolSturmIsolatedZeroRight{sturmname}{index}</span></tt></a></li>
-<li><a class="reference internal" href="#polsturmisolatedzeromultiplicity-sturmname-index" id="id132"><tt class="docutils literal"><span class="pre">\PolSturmIsolatedZeroMultiplicity{sturmname}{index}</span></tt></a></li>
-<li><a class="reference internal" href="#polsturmnbofisolatedzeros-sturmname" id="id133"><tt class="docutils literal">\PolSturmNbOfIsolatedZeros{sturmname}</tt></a><ul>
-<li><a class="reference internal" href="#polsturmnbofrootsof-sturmname-lessthanorequalto-value" id="id134"><tt class="docutils literal"><span class="pre">\PolSturmNbOfRootsOf{sturmname}\LessThanOrEqualTo{value}</span></tt></a></li>
-<li><a class="reference internal" href="#polsturmnbofrootsof-sturmname-lessthanorequaltoexpr-expression" id="id135"><tt class="docutils literal"><span class="pre">\PolSturmNbOfRootsOf{sturmname}\LessThanOrEqualToExpr{expression}</span></tt></a></li>
-<li><a class="reference internal" href="#polsturmnbwithmultofrootsof-sturmname-lessthanorequalto-value" id="id136"><tt class="docutils literal"><span class="pre">\PolSturmNbWithMultOfRootsOf{sturmname}\LessThanOrEqualTo{value}</span></tt></a></li>
-<li><a class="reference internal" href="#polsturmnbwithmultofrootsof-sturmname-lessthanorequaltoexpr-expression" id="id137"><tt class="docutils literal"><span class="pre">\PolSturmNbWithMultOfRootsOf{sturmname}\LessThanOrEqualToExpr{expression}</span></tt></a></li>
+<li><a class="reference internal" href="#id35" id="id126"><tt class="docutils literal"><span class="pre">\PolToFloatExpr*{polname}</span></tt></a></li>
+<li><a class="reference internal" href="#poltolist-polname" id="id127"><tt class="docutils literal">\PolToList{polname}</tt></a></li>
+<li><a class="reference internal" href="#poltocsv-polname" id="id128"><tt class="docutils literal">\PolToCSV{polname}</tt></a></li>
+<li><a class="reference internal" href="#polsturmchainlength-sturmname" id="id129"><tt class="docutils literal">\PolSturmChainLength{sturmname}</tt></a></li>
+<li><a class="reference internal" href="#polsturmifzeroexactlyknown-sturmname-index-a-b" id="id130"><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="id131"><tt class="docutils literal"><span class="pre">\PolSturmIsolatedZeroLeft{sturmname}{index}</span></tt></a></li>
+<li><a class="reference internal" href="#polsturmisolatedzeroright-sturmname-index" id="id132"><tt class="docutils literal"><span class="pre">\PolSturmIsolatedZeroRight{sturmname}{index}</span></tt></a></li>
+<li><a class="reference internal" href="#polsturmisolatedzeromultiplicity-sturmname-index" id="id133"><tt class="docutils literal"><span class="pre">\PolSturmIsolatedZeroMultiplicity{sturmname}{index}</span></tt></a></li>
+<li><a class="reference internal" href="#polsturmnbofisolatedzeros-sturmname" id="id134"><tt class="docutils literal">\PolSturmNbOfIsolatedZeros{sturmname}</tt></a><ul>
+<li><a class="reference internal" href="#polsturmnbofrootsof-sturmname-lessthanorequalto-value" id="id135"><tt class="docutils literal"><span class="pre">\PolSturmNbOfRootsOf{sturmname}\LessThanOrEqualTo{value}</span></tt></a></li>
+<li><a class="reference internal" href="#polsturmnbofrootsof-sturmname-lessthanorequaltoexpr-expression" id="id136"><tt class="docutils literal"><span class="pre">\PolSturmNbOfRootsOf{sturmname}\LessThanOrEqualToExpr{expression}</span></tt></a></li>
+<li><a class="reference internal" href="#polsturmnbwithmultofrootsof-sturmname-lessthanorequalto-value" id="id137"><tt class="docutils literal"><span class="pre">\PolSturmNbWithMultOfRootsOf{sturmname}\LessThanOrEqualTo{value}</span></tt></a></li>
+<li><a class="reference internal" href="#polsturmnbwithmultofrootsof-sturmname-lessthanorequaltoexpr-expression" id="id138"><tt class="docutils literal"><span class="pre">\PolSturmNbWithMultOfRootsOf{sturmname}\LessThanOrEqualToExpr{expression}</span></tt></a></li>
</ul>
</li>
-<li><a class="reference internal" href="#polsturmnbofrationalroots-sturmname" id="id138"><tt class="docutils literal">\PolSturmNbOfRationalRoots{sturmname}</tt></a></li>
-<li><a class="reference internal" href="#polsturmnbofrationalrootswithmultiplicities-sturmname" id="id139"><tt class="docutils literal">\PolSturmNbOfRationalRootsWithMultiplicities{sturmname}</tt></a></li>
-<li><a class="reference internal" href="#polsturmrationalroot-sturmname-k" id="id140"><tt class="docutils literal"><span class="pre">\PolSturmRationalRoot{sturmname}{k}</span></tt></a></li>
-<li><a class="reference internal" href="#polsturmrationalrootindex-sturmname-k" id="id141"><tt class="docutils literal"><span class="pre">\PolSturmRationalRootIndex{sturmname}{k}</span></tt></a></li>
-<li><a class="reference internal" href="#polsturmrationalrootmultiplicity-sturmname-k" id="id142"><tt class="docutils literal"><span class="pre">\PolSturmRationalRootMultiplicity{sturmname}{k}</span></tt></a></li>
-<li><a class="reference internal" href="#polintervalwidth-sturmname-index" id="id143"><tt class="docutils literal"><span class="pre">\PolIntervalWidth{sturmname}{index}</span></tt></a></li>
-<li><a class="reference internal" href="#expandable-macros-for-use-within-execution-of-polprintintervals" id="id144">Expandable macros for use within execution of <tt class="docutils literal">\PolPrintIntervals</tt></a><ul>
-<li><a class="reference internal" href="#polprintintervalsthevar" id="id145"><tt class="docutils literal">\PolPrintIntervalsTheVar</tt></a></li>
-<li><a class="reference internal" href="#polprintintervalstheindex" id="id146"><tt class="docutils literal">\PolPrintIntervalsTheIndex</tt></a></li>
-<li><a class="reference internal" href="#polprintintervalsthesturmname" id="id147"><tt class="docutils literal">\PolPrintIntervalsTheSturmName</tt></a></li>
-<li><a class="reference internal" href="#polprintintervalstheleftendpoint" id="id148"><tt class="docutils literal">\PolPrintIntervalsTheLeftEndPoint</tt></a></li>
-<li><a class="reference internal" href="#polprintintervalstherightendpoint" id="id149"><tt class="docutils literal">\PolPrintIntervalsTheRightEndPoint</tt></a></li>
-<li><a class="reference internal" href="#polprintintervalsthemultiplicity" id="id150"><tt class="docutils literal">\PolPrintIntervalsTheMultiplicity</tt></a></li>
+<li><a class="reference internal" href="#polsturmnbofrationalroots-sturmname" id="id139"><tt class="docutils literal">\PolSturmNbOfRationalRoots{sturmname}</tt></a></li>
+<li><a class="reference internal" href="#polsturmnbofrationalrootswithmultiplicities-sturmname" id="id140"><tt class="docutils literal">\PolSturmNbOfRationalRootsWithMultiplicities{sturmname}</tt></a></li>
+<li><a class="reference internal" href="#polsturmrationalroot-sturmname-k" id="id141"><tt class="docutils literal"><span class="pre">\PolSturmRationalRoot{sturmname}{k}</span></tt></a></li>
+<li><a class="reference internal" href="#polsturmrationalrootindex-sturmname-k" id="id142"><tt class="docutils literal"><span class="pre">\PolSturmRationalRootIndex{sturmname}{k}</span></tt></a></li>
+<li><a class="reference internal" href="#polsturmrationalrootmultiplicity-sturmname-k" id="id143"><tt class="docutils literal"><span class="pre">\PolSturmRationalRootMultiplicity{sturmname}{k}</span></tt></a></li>
+<li><a class="reference internal" href="#polintervalwidth-sturmname-index" id="id144"><tt class="docutils literal"><span class="pre">\PolIntervalWidth{sturmname}{index}</span></tt></a></li>
+<li><a class="reference internal" href="#expandable-macros-for-use-within-execution-of-polprintintervals" id="id145">Expandable macros for use within execution of <tt class="docutils literal">\PolPrintIntervals</tt></a><ul>
+<li><a class="reference internal" href="#polprintintervalsthevar" id="id146"><tt class="docutils literal">\PolPrintIntervalsTheVar</tt></a></li>
+<li><a class="reference internal" href="#polprintintervalstheindex" id="id147"><tt class="docutils literal">\PolPrintIntervalsTheIndex</tt></a></li>
+<li><a class="reference internal" href="#polprintintervalsthesturmname" id="id148"><tt class="docutils literal">\PolPrintIntervalsTheSturmName</tt></a></li>
+<li><a class="reference internal" href="#polprintintervalstheleftendpoint" id="id149"><tt class="docutils literal">\PolPrintIntervalsTheLeftEndPoint</tt></a></li>
+<li><a class="reference internal" href="#polprintintervalstherightendpoint" id="id150"><tt class="docutils literal">\PolPrintIntervalsTheRightEndPoint</tt></a></li>
+<li><a class="reference internal" href="#polprintintervalsthemultiplicity" id="id151"><tt class="docutils literal">\PolPrintIntervalsTheMultiplicity</tt></a></li>
</ul>
</li>
-<li><a class="reference internal" href="#poldectostring-decimal-number" id="id151"><tt class="docutils literal">\PolDecToString{decimal number}</tt></a></li>
+<li><a class="reference internal" href="#poldectostring-decimal-number" id="id152"><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="id152">Booleans (with default setting as indicated)</a><ul>
-<li><a class="reference internal" href="#xintverbosefalse" id="id153"><tt class="docutils literal">\xintverbosefalse</tt></a></li>
-<li><a class="reference internal" href="#poltypesetallfalse" id="id154"><tt class="docutils literal">\poltypesetallfalse</tt></a></li>
-<li><a class="reference internal" href="#poltoexprallfalse" id="id155"><tt class="docutils literal">\poltoexprallfalse</tt></a></li>
+<li><a class="reference internal" href="#booleans-with-default-setting-as-indicated" id="id153">Booleans (with default setting as indicated)</a><ul>
+<li><a class="reference internal" href="#xintverbosefalse" id="id154"><tt class="docutils literal">\xintverbosefalse</tt></a></li>
+<li><a class="reference internal" href="#poltypesetallfalse" id="id155"><tt class="docutils literal">\poltypesetallfalse</tt></a></li>
+<li><a class="reference internal" href="#poltoexprallfalse" id="id156"><tt class="docutils literal">\poltoexprallfalse</tt></a></li>
</ul>
</li>
-<li><a class="reference internal" href="#polexprsetup" id="id156"><tt class="docutils literal">\polexprsetup</tt></a></li>
-<li><a class="reference internal" href="#technicalities" id="id157">Technicalities</a></li>
-<li><a class="reference internal" href="#change-log" id="id158">CHANGE LOG</a></li>
-<li><a class="reference internal" href="#acknowledgments" id="id159">Acknowledgments</a></li>
+<li><a class="reference internal" href="#polexprsetup" id="id157"><tt class="docutils literal">\polexprsetup</tt></a></li>
+<li><a class="reference internal" href="#technicalities" id="id158">Technicalities</a></li>
+<li><a class="reference internal" href="#change-log" id="id159">CHANGE LOG</a></li>
+<li><a class="reference internal" href="#acknowledgments" id="id160">Acknowledgments</a></li>
</ul>
</div>
<div class="section" id="basic-syntax">
@@ -818,7 +819,7 @@
\PolSturmIsolateZeros**{Q}
\PolPrintIntervals{Q}
-$Q_norr(x) = \PolTypeset{Q_norr}$
+$Q_{norr}(x) = \PolTypeset{Q_norr}$
</pre>
<p>Here, all real roots are rational:</p>
<pre class="literal-block">
@@ -849,7 +850,7 @@
\PolSturmIsolateZeros{P} % finds 4 real roots
This polynomial has \PolSturmNbOfIsolatedZeros{P} distinct real roots:
\PolPrintIntervals{P}%
-% reports -2 < Z_1 < -1, 0 < Z_2 < 0.1, 0.1 < Z_3 < 0.2, 1 < Z_4 < 2
+% reports -2 < Z_1 < -1, 0.09 < Z_2 < 0.10, 0.1 < Z_3 < 0.2, 1 < Z_4 < 2
Let us refine the second and third intervals to separate the corresponding
roots:
\PolRefineInterval*{P}{2}% will refine to 0.0999990 < Z_2 < 0.0999991
@@ -948,8 +949,55 @@
Z_10 = 20.8469081014...
</pre>
</div>
+<div class="section" id="the-second-wilkinson-polynomial">
+<h2><a class="toc-backref" href="#id46">The second Wilkinson polynomial</a></h2>
+<pre class="literal-block">
+\documentclass{article}
+\usepackage{polexpr}
+\begin{document}
+\poldef f(x) := mul(x - 2^-i, i = 1..20);
+
+%\PolTypeset{f}
+
+\PolToSturm{f}{f}
+\PolSturmIsolateZeros**{f}
+\PolPrintIntervals{f}
+\end{document}
+</pre>
+<p>This takes more time than the polynomial with 1, 2, .., 20 as roots but
+less than the latter modified by the <tt class="docutils literal"><span class="pre">2**-23</span></tt> change in one
+coefficient.</p>
+<p>Here is the output (with release 0.7.2):</p>
+<pre class="literal-block">
+Z_1 = 0.00000095367431640625
+Z_2 = 0.0000019073486328125
+Z_3 = 0.000003814697265625
+Z_4 = 0.00000762939453125
+Z_5 = 0.0000152587890625
+Z_6 = 0.000030517578125
+Z_7 = 0.00006103515625
+Z_8 = 0.0001220703125
+Z_9 = 1/4096
+Z_10 = 1/2048
+Z_11 = 1/1024
+Z_12 = 1/512
+Z_13 = 1/256
+Z_14 = 1/128
+Z_15 = 0.015625
+Z_16 = 0.03125
+Z_17 = 0.0625
+Z_18 = 0.125
+Z_19 = 0.25
+Z_20 = 0.5
+</pre>
+<p>There is some incoherence in output format which has its source in the
+fact that some roots are found in branches which can only find decimal
+roots, whereas some are found in branches which could find general
+fractions and they use <tt class="docutils literal">\xintIrr</tt> before storage of the found root.
+This may evolve in future.</p>
+</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="#id46">The degree 41 polynomial with -2, -1.9, -1.8, ..., 0, 0.1, ..., 1.9, 2 as roots</a></h2>
+<h2><a class="toc-backref" href="#id47">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>
@@ -1007,7 +1055,7 @@
</div>
</div>
<div class="section" id="roots-of-chebyshev-polynomials">
-<h2><a class="toc-backref" href="#id47">Roots of Chebyshev polynomials</a></h2>
+<h2><a class="toc-backref" href="#id48">Roots of Chebyshev polynomials</a></h2>
<pre class="literal-block">
\newcount\mycount
\poldef T_0(x) := 1;
@@ -1030,9 +1078,9 @@
</div>
</div>
<div class="section" id="non-expandable-macros">
-<h1><a class="toc-backref" href="#id48">Non-expandable macros</a></h1>
+<h1><a class="toc-backref" href="#id49">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="#id49"><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="#id50"><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,
@@ -1069,7 +1117,7 @@
</blockquote>
</div>
<div class="section" id="poldef-letter-polname-expression-in-letter">
-<span id="id2"></span><h2><a class="toc-backref" href="#id50"><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="#id51"><tt class="docutils literal"><span class="pre">\PolDef[letter]{polname}{expression</span> in letter}</tt></a></h2>
<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
@@ -1077,7 +1125,7 @@
<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="#id51"><tt class="docutils literal">\PolGenFloatVariant{polname}</tt></a></h2>
+<span id="polgenfloatvariant"></span><h2><a class="toc-backref" href="#id52"><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
@@ -1096,7 +1144,7 @@
</blockquote>
</div>
<div class="section" id="pollet-polname-2-polname-1">
-<span id="pollet"></span><h2><a class="toc-backref" href="#id52"><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="#id53"><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
@@ -1104,12 +1152,12 @@
<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="#id53"><tt class="docutils literal"><span class="pre">\PolGlobalLet{polname_2}={polname_1}</span></tt></a></h2>
+<span id="polgloballet"></span><h2><a class="toc-backref" href="#id54"><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">
-<span id="polassign"></span><h2><a class="toc-backref" href="#id54"><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="#id55"><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>
@@ -1132,7 +1180,7 @@
</blockquote>
</div>
<div class="section" id="polget-polname-fromarray-macro">
-<span id="polget"></span><h2><a class="toc-backref" href="#id55"><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="#id56"><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>. Each individual
@@ -1153,7 +1201,7 @@
</blockquote>
</div>
<div class="section" id="polfromcsv-polname-csv">
-<span id="polfromcsv"></span><h2><a class="toc-backref" href="#id56"><tt class="docutils literal"><span class="pre">\PolFromCSV{polname}{<csv>}</span></tt></a></h2>
+<span id="polfromcsv"></span><h2><a class="toc-backref" href="#id57"><tt class="docutils literal"><span class="pre">\PolFromCSV{polname}{<csv>}</span></tt></a></h2>
<blockquote>
<p>Defines a polynomial directly from the comma separated list of values
(or a macro expanding to such a list) of its coefficients, the <em>first
@@ -1175,7 +1223,7 @@
</blockquote>
</div>
<div class="section" id="poltypeset-polname">
-<span id="poltypeset"></span><h2><a class="toc-backref" href="#id57"><tt class="docutils literal">\PolTypeset{polname}</tt></a></h2>
+<span id="poltypeset"></span><h2><a class="toc-backref" href="#id58"><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>
@@ -1189,7 +1237,7 @@
expandable, but this is not a requirement.</p>
</blockquote>
<div class="section" id="poltypesetcmd-raw-coeff">
-<span id="poltypesetcmd"></span><h3><a class="toc-backref" href="#id58"><tt class="docutils literal">\PolTypesetCmd{raw_coeff}</tt></a></h3>
+<span id="poltypesetcmd"></span><h3><a class="toc-backref" href="#id59"><tt class="docutils literal">\PolTypesetCmd{raw_coeff}</tt></a></h3>
<blockquote>
<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
@@ -1199,7 +1247,7 @@
</blockquote>
</div>
<div class="section" id="poltypesetone-raw-coeff">
-<span id="poltypesetone"></span><h3><a class="toc-backref" href="#id59"><tt class="docutils literal">\PolTypesetOne{raw_coeff}</tt></a></h3>
+<span id="poltypesetone"></span><h3><a class="toc-backref" href="#id60"><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>
@@ -1224,7 +1272,7 @@
</blockquote>
</div>
<div class="section" id="id6">
-<span id="poltypesetmonomialcmd"></span><h3><a class="toc-backref" href="#id60"><tt class="docutils literal">\PolTypesetMonomialCmd</tt></a></h3>
+<span id="poltypesetmonomialcmd"></span><h3><a class="toc-backref" href="#id61"><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
@@ -1234,7 +1282,7 @@
<tt class="docutils literal">\ifnum</tt> tests.</blockquote>
</div>
<div class="section" id="poltypesetcmdprefix-raw-coeff">
-<span id="poltypesetcmdprefix"></span><h3><a class="toc-backref" href="#id61"><tt class="docutils literal">\PolTypesetCmdPrefix{raw_coeff}</tt></a></h3>
+<span id="poltypesetcmdprefix"></span><h3><a class="toc-backref" href="#id62"><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
@@ -1245,13 +1293,13 @@
</div>
</div>
<div class="section" id="id8">
-<span id="id7"></span><h2><a class="toc-backref" href="#id62"><tt class="docutils literal"><span class="pre">\PolTypeset*{polname}</span></tt></a></h2>
+<span id="id7"></span><h2><a class="toc-backref" href="#id63"><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">
-<span id="poldiff"></span><h2><a class="toc-backref" href="#id63"><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="#id64"><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>
@@ -1261,7 +1309,7 @@
</blockquote>
</div>
<div class="section" id="poldiff-n-polname-1-polname-2">
-<span id="poldiff-n"></span><h2><a class="toc-backref" href="#id64"><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="#id65"><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
@@ -1269,7 +1317,7 @@
using <tt class="docutils literal">\PolAntiDiff</tt>.</blockquote>
</div>
<div class="section" id="polantidiff-polname-1-polname-2">
-<span id="polantidiff"></span><h2><a class="toc-backref" href="#id65"><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="#id66"><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>
@@ -1278,13 +1326,13 @@
</blockquote>
</div>
<div class="section" id="polantidiff-n-polname-1-polname-2">
-<span id="polantidiff-n"></span><h2><a class="toc-backref" href="#id66"><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="#id67"><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">
-<span id="poldivide"></span><h2><a class="toc-backref" href="#id67"><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="#id68"><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
@@ -1291,19 +1339,19 @@
<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="#id68"><tt class="docutils literal"><span class="pre">\PolQuo{polname_1}{polname_2}{polname_Q}</span></tt></a></h2>
+<span id="polquo"></span><h2><a class="toc-backref" href="#id69"><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="#id69"><tt class="docutils literal"><span class="pre">\PolRem{polname_1}{polname_2}{polname_R}</span></tt></a></h2>
+<span id="polrem"></span><h2><a class="toc-backref" href="#id70"><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">
-<span id="polgcd"></span><h2><a class="toc-backref" href="#id70"><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="#id71"><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 (monic) GCD of the two first
polynomials. It is a unitary polynomial except if both <tt class="docutils literal">polname_1</tt>
@@ -1329,7 +1377,7 @@
no common factor among the coefficients. -->
</div>
<div class="section" id="poltosturm-polname-sturmname">
-<span id="poltosturm"></span><h2><a class="toc-backref" href="#id71"><tt class="docutils literal"><span class="pre">\PolToSturm{polname}{sturmname}</span></tt></a></h2>
+<span id="poltosturm"></span><h2><a class="toc-backref" href="#id72"><tt class="docutils literal"><span class="pre">\PolToSturm{polname}{sturmname}</span></tt></a></h2>
<blockquote>
<p>With <tt class="docutils literal">polname</tt> being for example <tt class="docutils literal">P</tt>, the macro starts by
computing polynomials <tt class="docutils literal">P</tt> and <tt class="docutils literal">P'</tt>, then computes the (opposite
@@ -1369,8 +1417,8 @@
<p>The polynomials <tt class="docutils literal">sturmname_k</tt> main utility is for the execution of
<a class="reference internal" href="#polsturmisolatezeros-sturmname">\PolSturmIsolateZeros{sturmname}</a>. Be careful not to use these
names <tt class="docutils literal">sturmname_0</tt>, <tt class="docutils literal">sturmname_1</tt>, etc... for defining other
-polynomials after having done <tt class="docutils literal"><span class="pre">\PolToSturm{polname}{sturmname}</span> and
-before executing <span class="pre">``\PolSturmIsolateZeros{sturmname}</span></tt> else the
+polynomials after having done <tt class="docutils literal"><span class="pre">\PolToSturm{polname}{sturmname}</span></tt> and
+before executing <tt class="docutils literal">\PolSturmIsolateZeros{sturmname}</tt> else the
latter will behave erroneously.</p>
<p><a class="reference internal" href="#polsturmchainlength-sturmname">\PolSturmChainLength{sturmname}</a> gives the index of the last
element of the Sturm chain.</p>
@@ -1377,7 +1425,7 @@
</blockquote>
</div>
<div class="section" id="id10">
-<span id="id9"></span><h2><a class="toc-backref" href="#id72"><tt class="docutils literal"><span class="pre">\PolToSturm*{polname}{sturmname}</span></tt></a></h2>
+<span id="id9"></span><h2><a class="toc-backref" href="#id73"><tt class="docutils literal"><span class="pre">\PolToSturm*{polname}{sturmname}</span></tt></a></h2>
<blockquote>
<p>Does the same as <a class="reference internal" href="#poltosturm">un-starred version</a> and additionally it
keeps for user usage the memory of the <em>un-normalized</em> Sturm chain
@@ -1401,7 +1449,7 @@
</blockquote>
</div>
<div class="section" id="polsettosturmchainsignchangesat-macro-sturmname-fraction">
-<span id="polsettosturmchainsignchangesat"></span><h2><a class="toc-backref" href="#id73"><tt class="docutils literal"><span class="pre">\PolSetToSturmChainSignChangesAt{\macro}{sturmname}{fraction}</span></tt></a></h2>
+<span id="polsettosturmchainsignchangesat"></span><h2><a class="toc-backref" href="#id74"><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>
@@ -1418,7 +1466,7 @@
</blockquote>
</div>
<div class="section" id="polsettonbofzeroswithin-macro-sturmname-value-a-value-b">
-<span id="polsettonbofzeroswithin"></span><h2><a class="toc-backref" href="#id74"><tt class="docutils literal"><span class="pre">\PolSetToNbOfZerosWithin{\macro}{sturmname}{value_a}{value_b}</span></tt></a></h2>
+<span id="polsettonbofzeroswithin"></span><h2><a class="toc-backref" href="#id75"><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 <strong>distinct</strong> roots of <tt class="docutils literal">sturmname_0</tt> in the interval <tt class="docutils literal">(value_a,
@@ -1447,7 +1495,7 @@
</blockquote>
</div>
<div class="section" id="polsturmisolatezeros-sturmname">
-<span id="polsturmisolatezeros"></span><h2><a class="toc-backref" href="#id75"><tt class="docutils literal">\PolSturmIsolateZeros{sturmname}</tt></a></h2>
+<span id="polsturmisolatezeros"></span><h2><a class="toc-backref" href="#id76"><tt class="docutils literal">\PolSturmIsolateZeros{sturmname}</tt></a></h2>
<blockquote>
<p>The macros locates, using <a class="reference external" href="https://en.wikipedia.org/wiki/Sturm%27s_theorem">Sturm theorem</a>, as many disjoint
intervals as there are (real) roots.</p>
@@ -1557,7 +1605,7 @@
</blockquote>
</div>
<div class="section" id="id12">
-<span id="id11"></span><h2><a class="toc-backref" href="#id76"><tt class="docutils literal"><span class="pre">\PolSturmIsolateZeros*{sturmname}</span></tt></a></h2>
+<span id="id11"></span><h2><a class="toc-backref" href="#id77"><tt class="docutils literal"><span class="pre">\PolSturmIsolateZeros*{sturmname}</span></tt></a></h2>
<blockquote>
<p>The macro does the same as <a class="reference internal" href="#polsturmisolatezeros-sturmname">\PolSturmIsolateZeros{sturmname}</a> and
then in addition it does the extra work to determine all
@@ -1583,7 +1631,7 @@
</blockquote>
</div>
<div class="section" id="id14">
-<span id="id13"></span><h2><a class="toc-backref" href="#id77"><tt class="docutils literal"><span class="pre">\PolSturmIsolateZeros**{sturmname}</span></tt></a></h2>
+<span id="id13"></span><h2><a class="toc-backref" href="#id78"><tt class="docutils literal"><span class="pre">\PolSturmIsolateZeros**{sturmname}</span></tt></a></h2>
<blockquote>
<p>The macro does the same as <a class="reference internal" href="#id12">\PolSturmIsolateZeros*{sturmname}</a> and
in addition it does the extra work to determine all the <em>rational</em>
@@ -1608,17 +1656,17 @@
</blockquote>
</div>
<div class="section" id="polsturmisolatezerosandgetmultiplicities-sturmname">
-<span id="polsturmisolatezerosandgetmultiplicities"></span><h2><a class="toc-backref" href="#id78"><tt class="docutils literal">\PolSturmIsolateZerosAndGetMultiplicities{sturmname}</tt></a></h2>
+<span id="polsturmisolatezerosandgetmultiplicities"></span><h2><a class="toc-backref" href="#id79"><tt class="docutils literal">\PolSturmIsolateZerosAndGetMultiplicities{sturmname}</tt></a></h2>
<blockquote>
This is another name for <a class="reference internal" href="#id12">\PolSturmIsolateZeros*{sturmname}</a>.</blockquote>
</div>
<div class="section" id="polsturmisolatezerosgetmultiplicitiesandrationalroots-sturmname">
-<span id="polsturmisolatezerosgetmultiplicitiesandrationalroots"></span><h2><a class="toc-backref" href="#id79"><tt class="docutils literal">\PolSturmIsolateZerosGetMultiplicitiesAndRationalRoots{sturmname}</tt></a></h2>
+<span id="polsturmisolatezerosgetmultiplicitiesandrationalroots"></span><h2><a class="toc-backref" href="#id80"><tt class="docutils literal">\PolSturmIsolateZerosGetMultiplicitiesAndRationalRoots{sturmname}</tt></a></h2>
<blockquote>
This is another name for <a class="reference internal" href="#id14">\PolSturmIsolateZeros**{sturmname}</a>.</blockquote>
</div>
<div class="section" id="polsturmisolatezerosandfindrationalroots-sturmname">
-<h2><a class="toc-backref" href="#id80"><tt class="docutils literal">\PolSturmIsolateZerosAndFindRationalRoots{sturmname}</tt></a></h2>
+<h2><a class="toc-backref" href="#id81"><tt class="docutils literal">\PolSturmIsolateZerosAndFindRationalRoots{sturmname}</tt></a></h2>
<blockquote>
<p>This works exactly like <a class="reference internal" href="#id14">\PolSturmIsolateZeros**{sturmname}</a>
(inclusive of declaring the polynomials <tt class="docutils literal">sturmname_sqf_norr</tt> and
@@ -1648,7 +1696,7 @@
</blockquote>
</div>
<div class="section" id="polrefineinterval-sturmname-index">
-<span id="polrefineinterval"></span><h2><a class="toc-backref" href="#id81"><tt class="docutils literal"><span class="pre">\PolRefineInterval*{sturmname}{index}</span></tt></a></h2>
+<span id="polrefineinterval"></span><h2><a class="toc-backref" href="#id82"><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
@@ -1657,7 +1705,7 @@
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="#id82"><tt class="docutils literal"><span class="pre">\PolRefineInterval[N]{sturmname}{index}</span></tt></a></h2>
+<span id="polrefineinterval-n"></span><h2><a class="toc-backref" href="#id83"><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
@@ -1664,7 +1712,7 @@
<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="#id83"><tt class="docutils literal"><span class="pre">\PolEnsureIntervalLength{sturmname}{index}{E}</span></tt></a></h2>
+<span id="polensureintervallength"></span><h2><a class="toc-backref" href="#id84"><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
@@ -1671,7 +1719,7 @@
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="#id84"><tt class="docutils literal"><span class="pre">\PolEnsureIntervalLengths{sturmname}{E}</span></tt></a></h2>
+<span id="polensureintervallengths"></span><h2><a class="toc-backref" href="#id85"><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
@@ -1682,7 +1730,7 @@
</blockquote>
</div>
<div class="section" id="polprintintervals-varname-sturmname">
-<span id="polprintintervals"></span><h2><a class="toc-backref" href="#id85"><tt class="docutils literal"><span class="pre">\PolPrintIntervals[varname]{sturmname}</span></tt></a></h2>
+<span id="polprintintervals"></span><h2><a class="toc-backref" href="#id86"><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
@@ -1701,7 +1749,7 @@
<p>See next macros which govern its output.</p>
</blockquote>
<div class="section" id="polprintintervalsnorealroots">
-<h3><a class="toc-backref" href="#id86"><tt class="docutils literal">\PolPrintIntervalsNoRealRoots</tt></a></h3>
+<h3><a class="toc-backref" href="#id87"><tt class="docutils literal">\PolPrintIntervalsNoRealRoots</tt></a></h3>
<blockquote>
<p>Executed in place of an <tt class="docutils literal">array</tt> environment, when there are no
real roots. Default definition:</p>
@@ -1711,7 +1759,7 @@
</blockquote>
</div>
<div class="section" id="polprintintervalsbeginenv">
-<h3><a class="toc-backref" href="#id87"><tt class="docutils literal">\PolPrintIntervalsBeginEnv</tt></a></h3>
+<h3><a class="toc-backref" href="#id88"><tt class="docutils literal">\PolPrintIntervalsBeginEnv</tt></a></h3>
<blockquote>
<p>Default definition:</p>
<pre class="literal-block">
@@ -1720,7 +1768,7 @@
</blockquote>
</div>
<div class="section" id="polprintintervalsendenv">
-<h3><a class="toc-backref" href="#id88"><tt class="docutils literal">\PolPrintIntervalsEndEnv</tt></a></h3>
+<h3><a class="toc-backref" href="#id89"><tt class="docutils literal">\PolPrintIntervalsEndEnv</tt></a></h3>
<blockquote>
<p>Default definition:</p>
<pre class="literal-block">
@@ -1729,7 +1777,7 @@
</blockquote>
</div>
<div class="section" id="polprintintervalsknownroot">
-<h3><a class="toc-backref" href="#id89"><tt class="docutils literal">\PolPrintIntervalsKnownRoot</tt></a></h3>
+<h3><a class="toc-backref" href="#id90"><tt class="docutils literal">\PolPrintIntervalsKnownRoot</tt></a></h3>
<blockquote>
<p>Default definition:</p>
<pre class="literal-block">
@@ -1741,7 +1789,7 @@
</blockquote>
</div>
<div class="section" id="polprintintervalsunknownroot">
-<h3><a class="toc-backref" href="#id90"><tt class="docutils literal">\PolPrintIntervalsUnknownRoot</tt></a></h3>
+<h3><a class="toc-backref" href="#id91"><tt class="docutils literal">\PolPrintIntervalsUnknownRoot</tt></a></h3>
<blockquote>
<p>Default definition:</p>
<pre class="literal-block">
@@ -1754,7 +1802,7 @@
</blockquote>
</div>
<div class="section" id="id15">
-<span id="polprintintervalsprintexactzero"></span><h3><a class="toc-backref" href="#id91"><tt class="docutils literal">\PolPrintIntervalsPrintExactZero</tt></a></h3>
+<span id="polprintintervalsprintexactzero"></span><h3><a class="toc-backref" href="#id92"><tt class="docutils literal">\PolPrintIntervalsPrintExactZero</tt></a></h3>
<blockquote>
<p>Default definition:</p>
<pre class="literal-block">
@@ -1763,7 +1811,7 @@
</blockquote>
</div>
<div class="section" id="id16">
-<span id="polprintintervalsprintleftendpoint"></span><h3><a class="toc-backref" href="#id92"><tt class="docutils literal">\PolPrintIntervalsPrintLeftEndPoint</tt></a></h3>
+<span id="polprintintervalsprintleftendpoint"></span><h3><a class="toc-backref" href="#id93"><tt class="docutils literal">\PolPrintIntervalsPrintLeftEndPoint</tt></a></h3>
<blockquote>
<p>Default definition:</p>
<pre class="literal-block">
@@ -1772,7 +1820,7 @@
</blockquote>
</div>
<div class="section" id="id17">
-<span id="polprintintervalsprintrightendpoint"></span><h3><a class="toc-backref" href="#id93"><tt class="docutils literal">\PolPrintIntervalsPrintRightEndPoint</tt></a></h3>
+<span id="polprintintervalsprintrightendpoint"></span><h3><a class="toc-backref" href="#id94"><tt class="docutils literal">\PolPrintIntervalsPrintRightEndPoint</tt></a></h3>
<blockquote>
<p>Default definition is:</p>
<pre class="literal-block">
@@ -1782,7 +1830,7 @@
</div>
</div>
<div class="section" id="id19">
-<span id="id18"></span><h2><a class="toc-backref" href="#id94"><tt class="docutils literal"><span class="pre">\PolPrintIntervals*[varname]{sturmname}</span></tt></a></h2>
+<span id="id18"></span><h2><a class="toc-backref" href="#id95"><tt class="docutils literal"><span class="pre">\PolPrintIntervals*[varname]{sturmname}</span></tt></a></h2>
<blockquote>
<p>This starred variant produces an alternative output (which
displays the root multiplicity), and is provided as an
@@ -1801,7 +1849,7 @@
<p>Multiplicities are printed using this auxiliary macro:</p>
</blockquote>
<div class="section" id="polprintintervalsprintmultiplicity">
-<h3><a class="toc-backref" href="#id95"><tt class="docutils literal">\PolPrintIntervalsPrintMultiplicity</tt></a></h3>
+<h3><a class="toc-backref" href="#id96"><tt class="docutils literal">\PolPrintIntervalsPrintMultiplicity</tt></a></h3>
<blockquote>
<p>whose default definition is:</p>
<pre class="literal-block">
@@ -1811,7 +1859,7 @@
</div>
</div>
<div class="section" id="polmapcoeffs-macro-polname">
-<span id="polmapcoeffs"></span><h2><a class="toc-backref" href="#id96"><tt class="docutils literal"><span class="pre">\PolMapCoeffs{\macro}{polname}</span></tt></a></h2>
+<span id="polmapcoeffs"></span><h2><a class="toc-backref" href="#id97"><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
@@ -1831,7 +1879,7 @@
</blockquote>
</div>
<div class="section" id="polreducecoeffs-polname">
-<span id="polreducecoeffs"></span><h2><a class="toc-backref" href="#id97"><tt class="docutils literal">\PolReduceCoeffs{polname}</tt></a></h2>
+<span id="polreducecoeffs"></span><h2><a class="toc-backref" href="#id98"><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
@@ -1839,7 +1887,7 @@
one-argument macro, working 'in-place'.</blockquote>
</div>
<div class="section" id="id21">
-<span id="id20"></span><h2><a class="toc-backref" href="#id98"><tt class="docutils literal"><span class="pre">\PolReduceCoeffs*{polname}</span></tt></a></h2>
+<span id="id20"></span><h2><a class="toc-backref" href="#id99"><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
@@ -1858,7 +1906,7 @@
</blockquote>
</div>
<div class="section" id="polmakemonic-polname">
-<span id="polmakemonic"></span><h2><a class="toc-backref" href="#id99"><tt class="docutils literal">\PolMakeMonic{polname}</tt></a></h2>
+<span id="polmakemonic"></span><h2><a class="toc-backref" href="#id100"><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="#id21">\PolReduceCoeffs*{polname}</a> immediately afterwards. This is not
@@ -1867,7 +1915,7 @@
denominator.</blockquote>
</div>
<div class="section" id="polmakeprimitive-polname">
-<span id="polmakeprimitive"></span><h2><a class="toc-backref" href="#id100"><tt class="docutils literal">\PolMakePrimitive{polname}</tt></a></h2>
+<span id="polmakeprimitive"></span><h2><a class="toc-backref" href="#id101"><tt class="docutils literal">\PolMakePrimitive{polname}</tt></a></h2>
<blockquote>
Divides by the integer content see (<a class="reference internal" href="#policontent">\PolIContent</a>). This thus produces a polynomial with integer
coefficients having no common factor. The sign of the leading
@@ -1875,29 +1923,29 @@
</div>
</div>
<div class="section" id="expandable-macros">
-<h1><a class="toc-backref" href="#id101">Expandable macros</a></h1>
+<h1><a class="toc-backref" href="#id102">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-atexpr-numerical-expression">
-<span id="polevalatexpr"></span><h2><a class="toc-backref" href="#id102"><tt class="docutils literal"><span class="pre">\PolEval{polname}\AtExpr{numerical</span> expression}</tt></a></h2>
+<span id="polevalatexpr"></span><h2><a class="toc-backref" href="#id103"><tt class="docutils literal"><span class="pre">\PolEval{polname}\AtExpr{numerical</span> expression}</tt></a></h2>
<blockquote>
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="#id103"><tt class="docutils literal"><span class="pre">\PolEval{polname}\At{fraction}</span></tt></a></h2>
+<span id="polevalat"></span><h2><a class="toc-backref" href="#id104"><tt class="docutils literal"><span class="pre">\PolEval{polname}\At{fraction}</span></tt></a></h2>
<blockquote>
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.</blockquote>
</div>
<div class="section" id="polevalreduced-polname-atexpr-numerical-expression">
-<span id="polevalreducedatexpr"></span><h2><a class="toc-backref" href="#id104"><tt class="docutils literal"><span class="pre">\PolEvalReduced{polname}\AtExpr{numerical</span> expression}</tt></a></h2>
+<span id="polevalreducedatexpr"></span><h2><a class="toc-backref" href="#id105"><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="polevalreduced-polname-at-fraction">
-<span id="polevalreducedat"></span><h2><a class="toc-backref" href="#id105"><tt class="docutils literal"><span class="pre">\PolEvalReduced{polname}\At{fraction}</span></tt></a></h2>
+<span id="polevalreducedat"></span><h2><a class="toc-backref" href="#id106"><tt class="docutils literal"><span class="pre">\PolEvalReduced{polname}\At{fraction}</span></tt></a></h2>
<blockquote>
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
@@ -1904,7 +1952,7 @@
an irreducible fraction.</blockquote>
</div>
<div class="section" id="polfloateval-polname-atexpr-numerical-expression">
-<span id="polfloatevalatexpr"></span><h2><a class="toc-backref" href="#id106"><tt class="docutils literal"><span class="pre">\PolFloatEval{polname}\AtExpr{numerical</span> expression}</tt></a></h2>
+<span id="polfloatevalatexpr"></span><h2><a class="toc-backref" href="#id107"><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> and
@@ -1936,7 +1984,7 @@
</blockquote>
</div>
<div class="section" id="polfloateval-polname-at-fraction">
-<span id="polfloatevalat"></span><h2><a class="toc-backref" href="#id107"><tt class="docutils literal"><span class="pre">\PolFloatEval{polname}\At{fraction}</span></tt></a></h2>
+<span id="polfloatevalat"></span><h2><a class="toc-backref" href="#id108"><tt class="docutils literal"><span class="pre">\PolFloatEval{polname}\At{fraction}</span></tt></a></h2>
<blockquote>
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
@@ -1943,7 +1991,7 @@
a floating point number.</blockquote>
</div>
<div class="section" id="polifcoeffisplusorminusone-a-b">
-<span id="polifcoeffisplusorminusone"></span><h2><a class="toc-backref" href="#id108"><tt class="docutils literal"><span class="pre">\PolIfCoeffIsPlusOrMinusOne{A}{B}</span></tt></a></h2>
+<span id="polifcoeffisplusorminusone"></span><h2><a class="toc-backref" href="#id109"><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
@@ -1955,12 +2003,12 @@
</blockquote>
</div>
<div class="section" id="polleadingcoeff-polname">
-<span id="polleadingcoeff"></span><h2><a class="toc-backref" href="#id109"><tt class="docutils literal">\PolLeadingCoeff{polname}</tt></a></h2>
+<span id="polleadingcoeff"></span><h2><a class="toc-backref" href="#id110"><tt class="docutils literal">\PolLeadingCoeff{polname}</tt></a></h2>
<blockquote>
Expands to the leading coefficient.</blockquote>
</div>
<div class="section" id="polnthcoeff-polname-number">
-<span id="polnthcoeff"></span><h2><a class="toc-backref" href="#id110"><tt class="docutils literal"><span class="pre">\PolNthCoeff{polname}{number}</span></tt></a></h2>
+<span id="polnthcoeff"></span><h2><a class="toc-backref" href="#id111"><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
@@ -1967,13 +2015,13 @@
leading coefficients.</blockquote>
</div>
<div class="section" id="poldegree-polname">
-<span id="poldegree"></span><h2><a class="toc-backref" href="#id111"><tt class="docutils literal">\PolDegree{polname}</tt></a></h2>
+<span id="poldegree"></span><h2><a class="toc-backref" href="#id112"><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="policontent-polname">
-<span id="policontent"></span><h2><a class="toc-backref" href="#id112"><tt class="docutils literal">\PolIContent{polname}</tt></a></h2>
+<span id="policontent"></span><h2><a class="toc-backref" href="#id113"><tt class="docutils literal">\PolIContent{polname}</tt></a></h2>
<blockquote>
<p>It expands to the contents of the polynomial, i.e. to the positive
fraction such that dividing by this fraction produces a polynomial
@@ -1982,7 +2030,7 @@
</blockquote>
</div>
<div class="section" id="poltoexpr-polname">
-<span id="poltoexpr"></span><h2><a class="toc-backref" href="#id113"><tt class="docutils literal">\PolToExpr{polname}</tt></a></h2>
+<span id="poltoexpr"></span><h2><a class="toc-backref" href="#id114"><tt class="docutils literal">\PolToExpr{polname}</tt></a></h2>
<blockquote>
<p>Expands <a class="footnote-reference" href="#id27" id="id26">[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="id27" rules="none">
@@ -2007,7 +2055,7 @@
the identical result.</p>
</blockquote>
<div class="section" id="poltoexproneterm-raw-coeff-number">
-<span id="poltoexproneterm"></span><h3><a class="toc-backref" href="#id114"><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="#id115"><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
@@ -2020,13 +2068,13 @@
</blockquote>
</div>
<div class="section" id="poltoexpronetermstylea-raw-coeff-number">
-<span id="poltoexpronetermstylea"></span><h3><a class="toc-backref" href="#id115"><tt class="docutils literal"><span class="pre">\PolToExprOneTermStyleA{raw_coeff}{number}</span></tt></a></h3>
+<span id="poltoexpronetermstylea"></span><h3><a class="toc-backref" href="#id116"><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">
-<span id="poltoexpronetermstyleb"></span><h3><a class="toc-backref" href="#id116"><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="#id117"><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">
@@ -2040,7 +2088,7 @@
</blockquote>
</div>
<div class="section" id="poltoexprcmd-raw-coeff">
-<span id="poltoexprcmd"></span><h3><a class="toc-backref" href="#id117"><tt class="docutils literal">\PolToExprCmd{raw_coeff}</tt></a></h3>
+<span id="poltoexprcmd"></span><h3><a class="toc-backref" href="#id118"><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
@@ -2050,7 +2098,7 @@
output forcefully reduced coefficients.</blockquote>
</div>
<div class="section" id="poltoexprtermprefix-raw-coeff">
-<span id="poltoexprtermprefix"></span><h3><a class="toc-backref" href="#id118"><tt class="docutils literal">\PolToExprTermPrefix{raw_coeff}</tt></a></h3>
+<span id="poltoexprtermprefix"></span><h3><a class="toc-backref" href="#id119"><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
@@ -2057,7 +2105,7 @@
they don't carry one by themselves.</blockquote>
</div>
<div class="section" id="id28">
-<span id="poltoexprvar"></span><h3><a class="toc-backref" href="#id119"><tt class="docutils literal">\PolToExprVar</tt></a></h3>
+<span id="poltoexprvar"></span><h3><a class="toc-backref" href="#id120"><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
@@ -2064,7 +2112,7 @@
is <tt class="docutils literal">x</tt>.</blockquote>
</div>
<div class="section" id="id29">
-<span id="poltoexprtimes"></span><h3><a class="toc-backref" href="#id120"><tt class="docutils literal">\PolToExprTimes</tt></a></h3>
+<span id="poltoexprtimes"></span><h3><a class="toc-backref" href="#id121"><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
@@ -2074,13 +2122,13 @@
</div>
</div>
<div class="section" id="id31">
-<span id="id30"></span><h2><a class="toc-backref" href="#id121"><tt class="docutils literal"><span class="pre">\PolToExpr*{polname}</span></tt></a></h2>
+<span id="id30"></span><h2><a class="toc-backref" href="#id122"><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">
-<span id="poltofloatexpr"></span><h2><a class="toc-backref" href="#id122"><tt class="docutils literal">\PolToFloatExpr{polname}</tt></a></h2>
+<span id="poltofloatexpr"></span><h2><a class="toc-backref" href="#id123"><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
@@ -2099,13 +2147,13 @@
</div>
</blockquote>
<div class="section" id="poltofloatexproneterm-raw-coeff-number">
-<span id="poltofloatexproneterm"></span><h3><a class="toc-backref" href="#id123"><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="#id124"><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">
-<span id="id33"></span><h3><a class="toc-backref" href="#id124"><tt class="docutils literal">\PolToFloatExprCmd{raw_coeff}</tt></a></h3>
+<span id="id33"></span><h3><a class="toc-backref" href="#id125"><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>
@@ -2128,12 +2176,12 @@
</div>
</div>
<div class="section" id="id35">
-<span id="id34"></span><h2><a class="toc-backref" href="#id125"><tt class="docutils literal"><span class="pre">\PolToFloatExpr*{polname}</span></tt></a></h2>
+<span id="id34"></span><h2><a class="toc-backref" href="#id126"><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">
-<span id="poltolist"></span><h2><a class="toc-backref" href="#id126"><tt class="docutils literal">\PolToList{polname}</tt></a></h2>
+<span id="poltolist"></span><h2><a class="toc-backref" href="#id127"><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, and
<tt class="docutils literal">coeff_N</tt> the leading coefficient
@@ -2141,7 +2189,7 @@
empty output.)</blockquote>
</div>
<div class="section" id="poltocsv-polname">
-<span id="poltocsv"></span><h2><a class="toc-backref" href="#id127"><tt class="docutils literal">\PolToCSV{polname}</tt></a></h2>
+<span id="poltocsv"></span><h2><a class="toc-backref" href="#id128"><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>, starting
with constant term and ending with leading coefficient. Converse
@@ -2148,7 +2196,7 @@
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="#id128"><tt class="docutils literal">\PolSturmChainLength{sturmname}</tt></a></h2>
+<span id="polsturmchainlength"></span><h2><a class="toc-backref" href="#id129"><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>
@@ -2156,7 +2204,7 @@
</blockquote>
</div>
<div class="section" id="polsturmifzeroexactlyknown-sturmname-index-a-b">
-<span id="polsturmifzeroexactlyknown"></span><h2><a class="toc-backref" href="#id129"><tt class="docutils literal"><span class="pre">\PolSturmIfZeroExactlyKnown{sturmname}{index}{A}{B}</span></tt></a></h2>
+<span id="polsturmifzeroexactlyknown"></span><h2><a class="toc-backref" href="#id130"><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>
@@ -2168,7 +2216,7 @@
</blockquote>
</div>
<div class="section" id="polsturmisolatedzeroleft-sturmname-index">
-<span id="polsturmisolatedzeroleft"></span><h2><a class="toc-backref" href="#id130"><tt class="docutils literal"><span class="pre">\PolSturmIsolatedZeroLeft{sturmname}{index}</span></tt></a></h2>
+<span id="polsturmisolatedzeroleft"></span><h2><a class="toc-backref" href="#id131"><tt class="docutils literal"><span class="pre">\PolSturmIsolatedZeroLeft{sturmname}{index}</span></tt></a></h2>
<blockquote>
<p>Expands to the left end-point for the <tt class="docutils literal">index</tt>-th interval, as
computed by some earlier <a class="reference internal" href="#polsturmisolatezeros-sturmname">\PolSturmIsolateZeros{sturmname}</a>.</p>
@@ -2181,7 +2229,7 @@
</blockquote>
</div>
<div class="section" id="polsturmisolatedzeroright-sturmname-index">
-<span id="polsturmisolatedzeroright"></span><h2><a class="toc-backref" href="#id131"><tt class="docutils literal"><span class="pre">\PolSturmIsolatedZeroRight{sturmname}{index}</span></tt></a></h2>
+<span id="polsturmisolatedzeroright"></span><h2><a class="toc-backref" href="#id132"><tt class="docutils literal"><span class="pre">\PolSturmIsolatedZeroRight{sturmname}{index}</span></tt></a></h2>
<blockquote>
<p>Expands to the right end-point for the <tt class="docutils literal">index</tt>-th interval as
computed by some earlier <a class="reference internal" href="#polsturmisolatezeros-sturmname">\PolSturmIsolateZeros{sturmname}</a> and
@@ -2190,7 +2238,7 @@
</blockquote>
</div>
<div class="section" id="polsturmisolatedzeromultiplicity-sturmname-index">
-<span id="polsturmisolatedzeromultiplicity"></span><h2><a class="toc-backref" href="#id132"><tt class="docutils literal"><span class="pre">\PolSturmIsolatedZeroMultiplicity{sturmname}{index}</span></tt></a></h2>
+<span id="polsturmisolatedzeromultiplicity"></span><h2><a class="toc-backref" href="#id133"><tt class="docutils literal"><span class="pre">\PolSturmIsolatedZeroMultiplicity{sturmname}{index}</span></tt></a></h2>
<blockquote>
<p>Expands to the multiplicity of the unique root contained in the
<tt class="docutils literal">index</tt>-th interval.</p>
@@ -2203,7 +2251,7 @@
</blockquote>
</div>
<div class="section" id="polsturmnbofisolatedzeros-sturmname">
-<span id="polsturmnbofisolatedzeros"></span><h2><a class="toc-backref" href="#id133"><tt class="docutils literal">\PolSturmNbOfIsolatedZeros{sturmname}</tt></a></h2>
+<span id="polsturmnbofisolatedzeros"></span><h2><a class="toc-backref" href="#id134"><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>, i.e. the number of distinct real roots of the
@@ -2218,7 +2266,7 @@
documented package macros anyway.</p>
</div>
<div class="section" id="polsturmnbofrootsof-sturmname-lessthanorequalto-value">
-<h3><a class="toc-backref" href="#id134"><tt class="docutils literal"><span class="pre">\PolSturmNbOfRootsOf{sturmname}\LessThanOrEqualTo{value}</span></tt></a></h3>
+<h3><a class="toc-backref" href="#id135"><tt class="docutils literal"><span class="pre">\PolSturmNbOfRootsOf{sturmname}\LessThanOrEqualTo{value}</span></tt></a></h3>
<blockquote>
<p>Expands to the number of distinct roots (of the polynomial used to
create the Sturm chain) less than or equal to the <tt class="docutils literal">value</tt> (i.e. a
@@ -2234,7 +2282,7 @@
</blockquote>
</div>
<div class="section" id="polsturmnbofrootsof-sturmname-lessthanorequaltoexpr-expression">
-<h3><a class="toc-backref" href="#id135"><tt class="docutils literal"><span class="pre">\PolSturmNbOfRootsOf{sturmname}\LessThanOrEqualToExpr{expression}</span></tt></a></h3>
+<h3><a class="toc-backref" href="#id136"><tt class="docutils literal"><span class="pre">\PolSturmNbOfRootsOf{sturmname}\LessThanOrEqualToExpr{expression}</span></tt></a></h3>
<blockquote>
<p>Expands to the number of distinct roots (of the polynomial
used to create the Sturm chain) which are less than or equal to the
@@ -2247,7 +2295,7 @@
</blockquote>
</div>
<div class="section" id="polsturmnbwithmultofrootsof-sturmname-lessthanorequalto-value">
-<h3><a class="toc-backref" href="#id136"><tt class="docutils literal"><span class="pre">\PolSturmNbWithMultOfRootsOf{sturmname}\LessThanOrEqualTo{value}</span></tt></a></h3>
+<h3><a class="toc-backref" href="#id137"><tt class="docutils literal"><span class="pre">\PolSturmNbWithMultOfRootsOf{sturmname}\LessThanOrEqualTo{value}</span></tt></a></h3>
<blockquote>
<p>Expands to the number counted with multiplicities of the roots (of
the polynomial used to create the Sturm chain) which are less than
@@ -2260,7 +2308,7 @@
</blockquote>
</div>
<div class="section" id="polsturmnbwithmultofrootsof-sturmname-lessthanorequaltoexpr-expression">
-<h3><a class="toc-backref" href="#id137"><tt class="docutils literal"><span class="pre">\PolSturmNbWithMultOfRootsOf{sturmname}\LessThanOrEqualToExpr{expression}</span></tt></a></h3>
+<h3><a class="toc-backref" href="#id138"><tt class="docutils literal"><span class="pre">\PolSturmNbWithMultOfRootsOf{sturmname}\LessThanOrEqualToExpr{expression}</span></tt></a></h3>
<blockquote>
<p>Expands to the total number of roots (counted with multiplicities)
which are less than or equal to the given <tt class="docutils literal">expression</tt>.</p>
@@ -2273,7 +2321,7 @@
</div>
</div>
<div class="section" id="polsturmnbofrationalroots-sturmname">
-<h2><a class="toc-backref" href="#id138"><tt class="docutils literal">\PolSturmNbOfRationalRoots{sturmname}</tt></a></h2>
+<h2><a class="toc-backref" href="#id139"><tt class="docutils literal">\PolSturmNbOfRationalRoots{sturmname}</tt></a></h2>
<blockquote>
<p>Expands to the number of rational roots (without multiplicities).</p>
<div class="admonition attention">
@@ -2284,7 +2332,7 @@
</blockquote>
</div>
<div class="section" id="polsturmnbofrationalrootswithmultiplicities-sturmname">
-<h2><a class="toc-backref" href="#id139"><tt class="docutils literal">\PolSturmNbOfRationalRootsWithMultiplicities{sturmname}</tt></a></h2>
+<h2><a class="toc-backref" href="#id140"><tt class="docutils literal">\PolSturmNbOfRationalRootsWithMultiplicities{sturmname}</tt></a></h2>
<blockquote>
<p>Expands to the number of rational roots (counted with multiplicities).</p>
<div class="admonition attention">
@@ -2295,7 +2343,7 @@
</blockquote>
</div>
<div class="section" id="polsturmrationalroot-sturmname-k">
-<h2><a class="toc-backref" href="#id140"><tt class="docutils literal"><span class="pre">\PolSturmRationalRoot{sturmname}{k}</span></tt></a></h2>
+<h2><a class="toc-backref" href="#id141"><tt class="docutils literal"><span class="pre">\PolSturmRationalRoot{sturmname}{k}</span></tt></a></h2>
<blockquote>
<p>Expands to the <tt class="docutils literal">k</tt>th rational root (they are ordered and indexed
starting at 1 for the most negative).</p>
@@ -2307,7 +2355,7 @@
</blockquote>
</div>
<div class="section" id="polsturmrationalrootindex-sturmname-k">
-<h2><a class="toc-backref" href="#id141"><tt class="docutils literal"><span class="pre">\PolSturmRationalRootIndex{sturmname}{k}</span></tt></a></h2>
+<h2><a class="toc-backref" href="#id142"><tt class="docutils literal"><span class="pre">\PolSturmRationalRootIndex{sturmname}{k}</span></tt></a></h2>
<blockquote>
<p>Expands to <tt class="docutils literal">index</tt> of the <tt class="docutils literal">k</tt>th rational root as part of the
ordered real roots (without multiplicities). I.e., above macro
@@ -2324,7 +2372,7 @@
</blockquote>
</div>
<div class="section" id="polsturmrationalrootmultiplicity-sturmname-k">
-<h2><a class="toc-backref" href="#id142"><tt class="docutils literal"><span class="pre">\PolSturmRationalRootMultiplicity{sturmname}{k}</span></tt></a></h2>
+<h2><a class="toc-backref" href="#id143"><tt class="docutils literal"><span class="pre">\PolSturmRationalRootMultiplicity{sturmname}{k}</span></tt></a></h2>
<blockquote>
<p>Expands to the multiplicity of the <tt class="docutils literal">k</tt>th rational root.</p>
<div class="admonition attention">
@@ -2335,13 +2383,13 @@
</blockquote>
</div>
<div class="section" id="polintervalwidth-sturmname-index">
-<span id="polintervalwidth"></span><h2><a class="toc-backref" href="#id143"><tt class="docutils literal"><span class="pre">\PolIntervalWidth{sturmname}{index}</span></tt></a></h2>
+<span id="polintervalwidth"></span><h2><a class="toc-backref" href="#id144"><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="expandable-macros-for-use-within-execution-of-polprintintervals">
-<h2><a class="toc-backref" href="#id144">Expandable macros for use within execution of <tt class="docutils literal">\PolPrintIntervals</tt></a></h2>
+<h2><a class="toc-backref" href="#id145">Expandable macros for use within execution of <tt class="docutils literal">\PolPrintIntervals</tt></a></h2>
<p>These macros are for usage within custom user redefinitions of
<a class="reference internal" href="#polprintintervalsknownroot">\PolPrintIntervalsKnownRoot</a>, <a class="reference internal" href="#polprintintervalsunknownroot">\PolPrintIntervalsUnknownRoot</a>, or
in redefinitions of <a class="reference internal" href="#polprintintervalsprintexactzero">PolPrintIntervalsPrintExactZero</a> (used in the
@@ -2357,7 +2405,7 @@
<tt class="docutils literal"><span class="pre">\PolIfEndPointIsZero{A}{B}</span></tt>.</p>
</div>
<div class="section" id="polprintintervalsthevar">
-<h3><a class="toc-backref" href="#id145"><tt class="docutils literal">\PolPrintIntervalsTheVar</tt></a></h3>
+<h3><a class="toc-backref" href="#id146"><tt class="docutils literal">\PolPrintIntervalsTheVar</tt></a></h3>
<blockquote>
Expands to the name (default <tt class="docutils literal">Z</tt>) used for representing the roots,
which was passed as optional argument <tt class="docutils literal">varname</tt> to
@@ -2364,19 +2412,19 @@
<a class="reference internal" href="#polprintintervals-varname-sturmname">\PolPrintIntervals[varname]{sturmname}</a>.</blockquote>
</div>
<div class="section" id="polprintintervalstheindex">
-<h3><a class="toc-backref" href="#id146"><tt class="docutils literal">\PolPrintIntervalsTheIndex</tt></a></h3>
+<h3><a class="toc-backref" href="#id147"><tt class="docutils literal">\PolPrintIntervalsTheIndex</tt></a></h3>
<blockquote>
Expands to the index of the considered interval (indexing starting
at 1 for the leftmost interval).</blockquote>
</div>
<div class="section" id="polprintintervalsthesturmname">
-<h3><a class="toc-backref" href="#id147"><tt class="docutils literal">\PolPrintIntervalsTheSturmName</tt></a></h3>
+<h3><a class="toc-backref" href="#id148"><tt class="docutils literal">\PolPrintIntervalsTheSturmName</tt></a></h3>
<blockquote>
Expands to the argument which was passed as <tt class="docutils literal">sturmname</tt> to
<a class="reference internal" href="#polprintintervals-varname-sturmname">\PolPrintIntervals[varname]{sturmname}</a>.</blockquote>
</div>
<div class="section" id="polprintintervalstheleftendpoint">
-<h3><a class="toc-backref" href="#id148"><tt class="docutils literal">\PolPrintIntervalsTheLeftEndPoint</tt></a></h3>
+<h3><a class="toc-backref" href="#id149"><tt class="docutils literal">\PolPrintIntervalsTheLeftEndPoint</tt></a></h3>
<blockquote>
The left end point of the interval, as would be produced by
<a class="reference internal" href="#polsturmisolatedzeroleft">\PolSturmIsolatedZeroLeft</a> if it was
@@ -2385,7 +2433,7 @@
<a class="reference internal" href="#polprintintervalstheindex">\PolPrintIntervalsTheIndex</a>.</blockquote>
</div>
<div class="section" id="polprintintervalstherightendpoint">
-<h3><a class="toc-backref" href="#id149"><tt class="docutils literal">\PolPrintIntervalsTheRightEndPoint</tt></a></h3>
+<h3><a class="toc-backref" href="#id150"><tt class="docutils literal">\PolPrintIntervalsTheRightEndPoint</tt></a></h3>
<blockquote>
The right end point of the interval, as would be produced by
<a class="reference internal" href="#polsturmisolatedzeroright">\PolSturmIsolatedZeroRight</a> for
@@ -2392,7 +2440,7 @@
this Sturm chain name and index.</blockquote>
</div>
<div class="section" id="polprintintervalsthemultiplicity">
-<h3><a class="toc-backref" href="#id150"><tt class="docutils literal">\PolPrintIntervalsTheMultiplicity</tt></a></h3>
+<h3><a class="toc-backref" href="#id151"><tt class="docutils literal">\PolPrintIntervalsTheMultiplicity</tt></a></h3>
<blockquote>
The multiplicity of the unique root within the interval of index
<a class="reference internal" href="#polprintintervalstheindex">\PolPrintIntervalsTheIndex</a>. Makes sense only if the starred (or
@@ -2400,7 +2448,7 @@
</div>
</div>
<div class="section" id="poldectostring-decimal-number">
-<span id="poldectostring"></span><h2><a class="toc-backref" href="#id151"><tt class="docutils literal">\PolDecToString{decimal number}</tt></a></h2>
+<span id="poldectostring"></span><h2><a class="toc-backref" href="#id152"><tt class="docutils literal">\PolDecToString{decimal number}</tt></a></h2>
<blockquote>
<p>This is a utility macro to print decimal numbers. It has been
backported to <a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a> (release <tt class="docutils literal">1.3</tt> of <tt class="docutils literal">2018/03/01</tt>) under
@@ -2417,9 +2465,9 @@
</div>
</div>
<div class="section" id="booleans-with-default-setting-as-indicated">
-<h1><a class="toc-backref" href="#id152">Booleans (with default setting as indicated)</a></h1>
+<h1><a class="toc-backref" href="#id153">Booleans (with default setting as indicated)</a></h1>
<div class="section" id="xintverbosefalse">
-<h2><a class="toc-backref" href="#id153"><tt class="docutils literal">\xintverbosefalse</tt></a></h2>
+<h2><a class="toc-backref" href="#id154"><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
@@ -2432,13 +2480,13 @@
</blockquote>
</div>
<div class="section" id="poltypesetallfalse">
-<h2><a class="toc-backref" href="#id154"><tt class="docutils literal">\poltypesetallfalse</tt></a></h2>
+<h2><a class="toc-backref" href="#id155"><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="#id155"><tt class="docutils literal">\poltoexprallfalse</tt></a></h2>
+<h2><a class="toc-backref" href="#id156"><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>
@@ -2445,7 +2493,7 @@
</div>
</div>
<div class="section" id="polexprsetup">
-<h1><a class="toc-backref" href="#id156"><tt class="docutils literal">\polexprsetup</tt></a></h1>
+<h1><a class="toc-backref" href="#id157"><tt class="docutils literal">\polexprsetup</tt></a></h1>
<blockquote>
<p>Serves to customize the package. Currently only two keys are
recognized:</p>
@@ -2466,7 +2514,7 @@
</blockquote>
</div>
<div class="section" id="technicalities">
-<h1><a class="toc-backref" href="#id157">Technicalities</a></h1>
+<h1><a class="toc-backref" href="#id158">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
@@ -2523,7 +2571,7 @@
</ul>
</div>
<div class="section" id="change-log">
-<h1><a class="toc-backref" href="#id158">CHANGE LOG</a></h1>
+<h1><a class="toc-backref" href="#id159">CHANGE LOG</a></h1>
<ul>
<li><p class="first">v0.1 (2018/01/11): initial release. Features:</p>
<ul class="simple">
@@ -2768,7 +2816,11 @@
</li>
</ul>
</li>
-<li><p class="first">v0.7 (2018/12/08)</p>
+<li><p class="first">v0.7 (2018/12/08) (not released)</p>
+</li>
+<li><p class="first">v0.7.1 (2018/12/09) (bugfix)</p>
+</li>
+<li><p class="first">v0.7.2 (2018/12/09)</p>
<ul class="simple">
<li>breaking changes:<ul>
<li>although <a class="reference internal" href="#polprintintervals-varname-sturmname">\PolPrintIntervals[varname]{sturmname}</a> default output
@@ -2821,7 +2873,7 @@
</ul>
</div>
<div class="section" id="acknowledgments">
-<h1><a class="toc-backref" href="#id159">Acknowledgments</a></h1>
+<h1><a class="toc-backref" href="#id160">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
Modified: trunk/Master/texmf-dist/doc/latex/polexpr/polexpr.txt
===================================================================
--- trunk/Master/texmf-dist/doc/latex/polexpr/polexpr.txt 2018-12-10 22:58:31 UTC (rev 49377)
+++ trunk/Master/texmf-dist/doc/latex/polexpr/polexpr.txt 2018-12-10 22:58:44 UTC (rev 49378)
@@ -4,8 +4,8 @@
Package polexpr documentation
===============================
-0.7 (2018/12/08)
-================
+0.7.2 (2018/12/09)
+==================
.. contents::
@@ -334,7 +334,7 @@
\PolSturmIsolateZeros**{Q}
\PolPrintIntervals{Q}
- $Q_norr(x) = \PolTypeset{Q_norr}$
+ $Q_{norr}(x) = \PolTypeset{Q_norr}$
Here, all real roots are rational::
@@ -367,7 +367,7 @@
\PolSturmIsolateZeros{P} % finds 4 real roots
This polynomial has \PolSturmNbOfIsolatedZeros{P} distinct real roots:
\PolPrintIntervals{P}%
- % reports -2 < Z_1 < -1, 0 < Z_2 < 0.1, 0.1 < Z_3 < 0.2, 1 < Z_4 < 2
+ % reports -2 < Z_1 < -1, 0.09 < Z_2 < 0.10, 0.1 < Z_3 < 0.2, 1 < Z_4 < 2
Let us refine the second and third intervals to separate the corresponding
roots:
\PolRefineInterval*{P}{2}% will refine to 0.0999990 < Z_2 < 0.0999991
@@ -470,7 +470,57 @@
Z_9 = 8.9172502485...
Z_10 = 20.8469081014...
+The second Wilkinson polynomial
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+::
+
+ \documentclass{article}
+ \usepackage{polexpr}
+ \begin{document}
+ \poldef f(x) := mul(x - 2^-i, i = 1..20);
+
+ %\PolTypeset{f}
+
+ \PolToSturm{f}{f}
+ \PolSturmIsolateZeros**{f}
+ \PolPrintIntervals{f}
+ \end{document}
+
+This takes more time than the polynomial with 1, 2, .., 20 as roots but
+less than the latter modified by the ``2**-23`` change in one
+coefficient.
+
+Here is the output (with release 0.7.2)::
+
+ Z_1 = 0.00000095367431640625
+ Z_2 = 0.0000019073486328125
+ Z_3 = 0.000003814697265625
+ Z_4 = 0.00000762939453125
+ Z_5 = 0.0000152587890625
+ Z_6 = 0.000030517578125
+ Z_7 = 0.00006103515625
+ Z_8 = 0.0001220703125
+ Z_9 = 1/4096
+ Z_10 = 1/2048
+ Z_11 = 1/1024
+ Z_12 = 1/512
+ Z_13 = 1/256
+ Z_14 = 1/128
+ Z_15 = 0.015625
+ Z_16 = 0.03125
+ Z_17 = 0.0625
+ Z_18 = 0.125
+ Z_19 = 0.25
+ Z_20 = 0.5
+
+There is some incoherence in output format which has its source in the
+fact that some roots are found in branches which can only find decimal
+roots, whereas some are found in branches which could find general
+fractions and they use ``\xintIrr`` before storage of the found root.
+This may evolve in future.
+
+
The degree 41 polynomial with -2, -1.9, -1.8, ..., 0, 0.1, ..., 1.9, 2 as roots
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
@@ -961,7 +1011,7 @@
The polynomials ``sturmname_k`` main utility is for the execution of
`\\PolSturmIsolateZeros{sturmname}`_. Be careful not to use these
names ``sturmname_0``, ``sturmname_1``, etc... for defining other
- polynomials after having done ``\PolToSturm{polname}{sturmname} and
+ polynomials after having done ``\PolToSturm{polname}{sturmname}`` and
before executing ``\PolSturmIsolateZeros{sturmname}`` else the
latter will behave erroneously.
@@ -2448,8 +2498,12 @@
- `\\PolSturmNbWithMultOfRootsOf{sturmname}\\LessThanOrEqualTo{value}`_
- `\\PolSturmNbWithMultOfRootsOf{sturmname}\\LessThanOrEqualToExpr{expression}`_
-- v0.7 (2018/12/08)
+- v0.7 (2018/12/08) (not released)
+- v0.7.1 (2018/12/09) (bugfix)
+
+- v0.7.2 (2018/12/09)
+
* breaking changes:
- although `\\PolPrintIntervals[varname]{sturmname}`_ default output
Modified: trunk/Master/texmf-dist/tex/latex/polexpr/polexpr.sty
===================================================================
--- trunk/Master/texmf-dist/tex/latex/polexpr/polexpr.sty 2018-12-10 22:58:31 UTC (rev 49377)
+++ trunk/Master/texmf-dist/tex/latex/polexpr/polexpr.sty 2018-12-10 22:58:44 UTC (rev 49378)
@@ -1,7 +1,7 @@
% author: Jean-François Burnol
% License: LPPL 1.3c (author-maintained)
\ProvidesPackage{polexpr}%
- [2018/12/08 v0.7 Polynomial expressions with rational coefficients (JFB)]%
+ [2018/12/09 v0.7.2 Polynomial expressions with rational coefficients (JFB)]%
\RequirePackage{xintexpr}[2018/06/17]% xint 1.3c for \ifxintglobaldefs boolean
\edef\POL at restorecatcodes
{\catcode`\noexpand\_ \the\catcode`\_ %
@@ -914,12 +914,13 @@
{\let\POL at findrat@E\POL at findrat@Dexp} % aussi ok pour 1[0]
{\edef\POL at findrat@E{\the\numexpr\xintLen{\POL at findrat@Dint}%
+\POL at findrat@Dexp}}%
+% ATTENTION QUE LA CONVENTION DE SIGNE POUR \POL at findrat@E EST OPPOSÉE À CELLE
+% POUR LE CODE PLUS ANCIEN FAISANT "REFINE"
\POL at initarray\POL at IfMultIsKnown\xint_secondoftwo
\let\POL at findrat@nbofirrroots\POL at isolz@NbOfRoots
% find all rational roots, and their multiplicities,
% factor them out in passing from original (Sturm root) polynomial
\ifnum\POL at findrat@E<7
- \PolEnsureIntervalLength{\POL at sturmname}{1}{-\POL at findrat@E}%
\def\POL at findrat@index{1}%
\POL at findrat@loop at secondpass@direct
\else
@@ -928,7 +929,6 @@
\POL at findrat@loop at firstpass
% and now we do the final pass finding them all
\def\POL at findrat@index{1}%
- \PolEnsureIntervalLength{\POL at sturmname}{1}{-\POL at findrat@E}%
\POL at findrat@loop at secondpass
\fi
% declare the new polynomials
@@ -947,7 +947,7 @@
{}%
\ifnum\xintiloopindex<\POL at isolz@NbOfRoots\space
\repeat }%
- \begingroup\globaldefs1
+ \begingroup\globaldefs\@ne
% attention de ne surtout pas faire un \expandafter ici, car en cas d'un
% seul item, \xintAssignArray l'unbraces...
\xintAssignArray\POL at temp\to#1%
@@ -956,7 +956,7 @@
\def\POL at findrat@loop at firstpass{%
\PolSturmIfZeroExactlyKnown{\POL at sturmname}{\POL at findrat@index}%
\POL at findrat@loop at decimal% get its multiplicity
- \POL at findrat@loop at a
+ \POL at findrat@loop at aa % refine interval and check
\edef\POL at findrat@index{\the\numexpr\POL at findrat@index+\@ne}%
\ifnum\POL at findrat@index>\POL at isolz@NbOfRoots
\else
@@ -963,30 +963,14 @@
\expandafter\POL at findrat@loop at firstpass
\fi
}%
-\def\POL at findrat@loop at secondpass{%
+\def\POL at findrat@loop at aa{%
+ % we do a first pass to identify roots with denominators < 1000
+ \PolEnsureIntervalLength{\POL at sturmname}{\POL at findrat@index}{-6}%
+ % attention that perhaps now the root is known!
\PolSturmIfZeroExactlyKnown{\POL at sturmname}{\POL at findrat@index}%
- {}% nothing more to be done, already stored
- \POL at findrat@loop at b
- \edef\POL at findrat@index{\the\numexpr\POL at findrat@index+\@ne}%
- \ifnum\POL at findrat@index>\POL at isolz@NbOfRoots
- \else
- \PolEnsureIntervalLength
- {\POL at sturmname}{\POL at findrat@index}{-\POL at findrat@E}% dynamic
- \expandafter\POL at findrat@loop at secondpass
- \fi
+ \POL at findrat@loop at decimal
+ \POL at findrat@loop at a
}%
-\def\POL at findrat@loop at secondpass@direct{%
- \PolSturmIfZeroExactlyKnown{\POL at sturmname}{\POL at findrat@index}%
- \POL at findrat@loop at decimal
- \POL at findrat@loop at b
- \edef\POL at findrat@index{\the\numexpr\POL at findrat@index+\@ne}%
- \ifnum\POL at findrat@index>\POL at isolz@NbOfRoots
- \else
- \PolEnsureIntervalLength
- {\POL at sturmname}{\POL at findrat@index}{-\POL at findrat@E}% dynamic
- \expandafter\POL at findrat@loop at secondpass@direct
- \fi
-}%
\def\POL at findrat@loop at decimal{% we have an already found decimal root
% we do not go via @storeit, as it is already stored
% j'ai beaucoup hésité néanmoins, car je pourrais faire \xintIrr ici,
@@ -1002,6 +986,7 @@
\edef\POLuserpol at _findrat@oneterm{1.\noexpand\empty
{\xintiiOpp\POL at findrat@xN/1[0]}{\POL at findrat@xD/1[0]}}%
\POL at divide{\POL at sturmname\POL at sqfnorr}{_findrat at oneterm}% the one without mult.
+ %\expandafter\POL at split\POL at R;\POL at degR\POL at polR
\POL at findrat@loop at updatequotients
\POL at findrat@loop at getmultiplicity
}%
@@ -1018,10 +1003,7 @@
\def\POL at findrat@getE #1/1[#2]{#2}% /1 as it should be there.
% so an error will arise if not but cf \POL at refine@getE where I did not put it
\def\POL at findrat@loop at a{%
- % we do a first pass to identify roots with denominators < 1000
- \PolEnsureIntervalLength{\POL at sturmname}{\POL at findrat@index}{-6}%
% attention that the width may have been already smaller than 10^{-6}
- % also attention that one of the bound may be zero
\POL at get@IsoLeft at rawin
\POL at get@IsoRight at rawin
\edef\POL at findrat@localW
@@ -1046,29 +1028,21 @@
\fi
}%
\def\POL at findrat@gcdloop{%
- % we must be careful with sign (attention one of the bounds may be zero)
- \let\POL at findrat@IfNeg\xint_secondoftwo
+ % we must be careful with sign
+ % but we are certain no extremity is a root
+ \let\POL at findrat@ifnegative\xint_secondoftwo
\xintiiifSgn\POL at IsoLeft@Int
\POL at findrat@gcdloop at n
- \POL at findrat@gcdloop at zero
+ \POL at error@thisisimpossible
\POL at findrat@gcdloop at p
}%
\def\POL at findrat@gcdloop at n{%
- \let\POL at findrat@IfNeg\xint_firstoftwo
+ \let\POL at findrat@ifnegative\xint_firstoftwo
\let\POL at temp\POL at IsoRight@Int
\edef\POL at IsoRight@Int{\xintiiOpp{\POL at IsoLeft@Int}}%
\edef\POL at IsoLeft@Int{\xintiiOpp{\POL at temp}}%
- \xintiiifSgn\POL at IsoLeft@Int
- \POL at error % impossible branch
- \POL at findrat@gcdloop at zero
- \POL at findrat@gcdloop at p
+ \POL at findrat@gcdloop at p
}%
-\def\POL at findrat@gcdloop at zero{%
- % the continued fraction would be the one of 1/2, so only 1/2 to test...
- \edef\POL at findrat@x
- {1/2\romannumeral\xintreplicate{\POL at findrat@localW}{0}[0]}%
- \POL at findrat@gcdloop at testit
-}%
\def\POL at findrat@gcdloop at p{%
\edef\POL at findrat@gcdloop at Ap{\xintDec{\xintDouble\POL at IsoRight@Int}}%
\edef\POL at findrat@gcdloop at A
@@ -1140,7 +1114,7 @@
}%
\def\POL at findrat@gcdloop at testit{%
% zero should never occur here
- \POL at findrat@IfNeg{\edef\POL at findrat@x{-\POL at findrat@x}}{}%
+ \POL at findrat@ifnegative{\edef\POL at findrat@x{-\POL at findrat@x}}{}%
\POL at xintfrac@getNDE\POL at findrat@x\POL at findrat@xN\POL at findrat@xD\POL at _
\edef\POLuserpol at _findrat@oneterm{1.\noexpand\empty
{\xintiiOpp{\POL at findrat@xN}/1[0]}{\POL at findrat@xD/1[0]}}%
@@ -1151,10 +1125,36 @@
\POL at findrat@loop at updatequotients
\POL at findrat@loop at getmultiplicity % will continue updating the mult. one
\def\POL at findrat@gcdloop at done{1}%
- \else
\fi
}%
% This is second phase
+\def\POL at findrat@loop at secondpass{%
+ \PolSturmIfZeroExactlyKnown{\POL at sturmname}{\POL at findrat@index}%
+ {}% nothing more to be done, already stored
+ \POL at findrat@loop at bb % refine interval and check
+ \edef\POL at findrat@index{\the\numexpr\POL at findrat@index+\@ne}%
+ \ifnum\POL at findrat@index>\POL at isolz@NbOfRoots
+ \else
+ \expandafter\POL at findrat@loop at secondpass
+ \fi
+}%
+\def\POL at findrat@loop at secondpass@direct{%
+ \PolSturmIfZeroExactlyKnown{\POL at sturmname}{\POL at findrat@index}%
+ \POL at findrat@loop at decimal
+ \POL at findrat@loop at bb
+ \edef\POL at findrat@index{\the\numexpr\POL at findrat@index+\@ne}%
+ \ifnum\POL at findrat@index>\POL at isolz@NbOfRoots
+ \else
+ \expandafter\POL at findrat@loop at secondpass@direct
+ \fi
+}%
+\def\POL at findrat@loop at bb{%
+ \PolEnsureIntervalLength{\POL at sturmname}{\POL at findrat@index}{-\POL at findrat@E}%
+ % ATTENTION THAT PERHAPS NOW THE ROOT IS KNOWN!
+ \PolSturmIfZeroExactlyKnown{\POL at sturmname}{\POL at findrat@index}%
+ \POL at findrat@loop at decimal
+ \POL at findrat@loop at b
+}%
\def\POL at findrat@loop at b{%
\edef\POL at findrat@Lscaled{\xintMul{\POL at findrat@D}%
{\POL at xintexprGetVar{\POL at sturmname L_\POL at findrat@index}}}%
@@ -1206,11 +1206,13 @@
\let\csname POL_ZeroIsKnown\POL at sturmname\POL at findrat@index\endcsname
\xint_stop_atfirstoftwo
\begingroup\xintglobaldefstrue
- \xintdefvar
- \POL at sturmname L_\POL at findrat@index,%
- \POL at sturmname R_\POL at findrat@index,%
- \POL at sturmname Z_\POL at findrat@index _isknown
- := qfrac(\POL at findrat@x),qfrac(\POL at findrat@x),1;%
+ % skip some overhead of \xintdefvar...
+ \XINT_expr_defvar_one{\POL at sturmname L_\POL at findrat@index}%
+ {\csname .=\POL at findrat@x\endcsname}%
+ \XINT_expr_defvar_one{\POL at sturmname R_\POL at findrat@index}%
+ {\csname .=\POL at findrat@x\endcsname}%
+ \XINT_expr_defvar_one{\POL at sturmname Z_\POL at findrat@index _isknown}%
+ {\csname .=1\endcsname}%
\endgroup
}%
\def\POL at findrat@loop at updatequotients{%
@@ -1320,8 +1322,9 @@
% added at 0.7
\let\x\POL at isolz@NbOfRoots
\xintloop
- \xintdefvar \POL at sturmname M_\x
- := \csname POL_ZeroMult\POL at sturmname\x\endcsname ;%
+ % skip some overhead of \xintdefvar...
+ \XINT_expr_defvar_one{\POL at sturmname M_\x}%
+ {\csname .=\csname POL_ZeroMult\POL at sturmname\x\endcsname\endcsname}%
\edef\x{\the\numexpr\x-\@ne}%
\ifnum\x>\z@
\repeat
@@ -1694,11 +1697,11 @@
\repeat
}%
\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
+\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
@@ -1896,18 +1899,16 @@
\xint_stop_atfirstoftwo
\fi
\begingroup\xintglobaldefstrue
- \xintdefvar
- \POL at sturmname L_\POL at isolz@IntervalIndex,%
- \POL at sturmname R_\POL at isolz@IntervalIndex,%
+ % skip some overhead of \xintdefvar...
+ \XINT_expr_defvar_one{\POL at sturmname L_\POL at isolz@IntervalIndex}%
+ {\csname .=\POL at IsoLeft@rawout\endcsname}%
+ \XINT_expr_defvar_one{\POL at sturmname R_\POL at isolz@IntervalIndex}%
+ {\csname .=\POL at IsoRight@rawout\endcsname}%
% added at 0.7
- \POL at sturmname Z_\POL at isolz@IntervalIndex _isknown
- := qfrac(\POL at IsoLeft@rawout),%
- qfrac(\POL at IsoRight@rawout),%
- \ifnum\POL at IsoRightSign=\z@ 1\else 0\fi;%
+ \XINT_expr_defvar_one{\POL at sturmname Z_\POL at isolz@IntervalIndex _isknown}%
+ {\csname .=\ifnum\POL at IsoRightSign=\z@ 1\else 0\fi\endcsname}%
\endgroup
}%
-
-
%% \PolRefineInterval
\def\POL at xintexprGetVar#1{\expandafter\expandafter\expandafter
\XINT_expr_unlock\csname XINT_expr_var_#1\endcsname}%
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