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author | Norbert Preining <norbert@preining.info> | 2019-09-02 13:46:59 +0900 |
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committer | Norbert Preining <norbert@preining.info> | 2019-09-02 13:46:59 +0900 |
commit | e0c6872cf40896c7be36b11dcc744620f10adf1d (patch) | |
tree | 60335e10d2f4354b0674ec22d7b53f0f8abee672 /macros/latex/contrib/polexpr/polexpr.html |
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Diffstat (limited to 'macros/latex/contrib/polexpr/polexpr.html')
-rw-r--r-- | macros/latex/contrib/polexpr/polexpr.html | 2908 |
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diff --git a/macros/latex/contrib/polexpr/polexpr.html b/macros/latex/contrib/polexpr/polexpr.html new file mode 100644 index 0000000000..6fc48c2dd6 --- /dev/null +++ b/macros/latex/contrib/polexpr/polexpr.html @@ -0,0 +1,2908 @@ +<?xml version="1.0" encoding="utf-8" ?> +<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd"> +<html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en" lang="en"> +<head> +<meta http-equiv="Content-Type" content="text/html; charset=utf-8" /> +<meta name="generator" content="Docutils 0.14: http://docutils.sourceforge.net/" /> +<title>Package polexpr documentation</title> +<style type="text/css"> + +/* +:Author: David Goodger (goodger@python.org) +:Id: $Id: html4css1.css 7952 2016-07-26 18:15:59Z milde $ +:Copyright: This stylesheet has been placed in the public domain. + +Default cascading style sheet for the HTML output of Docutils. + +See 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white-space: pre } + +span.problematic { + color: red } + +span.section-subtitle { + /* font-size relative to parent (h1..h6 element) */ + font-size: 80% } + +table.citation { + border-left: solid 1px gray; + margin-left: 1px } + +table.docinfo { + margin: 2em 4em } + +table.docutils { + margin-top: 0.5em ; + margin-bottom: 0.5em } + +table.footnote { + border-left: solid 1px black; + margin-left: 1px } + +table.docutils td, table.docutils th, +table.docinfo td, table.docinfo th { + padding-left: 0.5em ; + padding-right: 0.5em ; + vertical-align: top } + +table.docutils th.field-name, table.docinfo th.docinfo-name { + font-weight: bold ; + text-align: left ; + white-space: nowrap ; + padding-left: 0 } + +/* "booktabs" style (no vertical lines) */ +table.docutils.booktabs { + border: 0px; + border-top: 2px solid; + border-bottom: 2px solid; + border-collapse: collapse; +} +table.docutils.booktabs * { + border: 0px; +} +table.docutils.booktabs th { + border-bottom: thin solid; + text-align: left; +} + +h1 tt.docutils, h2 tt.docutils, h3 tt.docutils, +h4 tt.docutils, h5 tt.docutils, h6 tt.docutils { + font-size: 100% } + +ul.auto-toc { + list-style-type: none } + +</style> +</head> +<body> +<div class="document" id="package-polexpr-documentation"> +<h1 class="title">Package polexpr documentation</h1> +<h2 class="subtitle" id="id1">0.7.4 (2019/02/12)</h2> + +<!-- comment: -*- fill-column: 72; mode: rst; -*- --> +<div class="contents topic" id="contents"> +<p class="topic-title first">Contents</p> +<ul class="simple"> +<li><a class="reference internal" href="#basic-syntax" id="id38">Basic syntax</a></li> +<li><a class="reference internal" href="#examples-of-localization-of-roots" id="id39">Examples of localization of roots</a><ul> +<li><a class="reference internal" href="#a-typical-example" id="id40">A typical example</a></li> +<li><a class="reference internal" href="#a-degree-four-polynomial-with-nearby-roots" id="id41">A degree four polynomial with nearby roots</a></li> +<li><a class="reference internal" href="#the-degree-nine-polynomial-with-0-99-0-999-0-9999-as-triple-roots" id="id42">The degree nine polynomial with 0.99, 0.999, 0.9999 as triple roots</a></li> +<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-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="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="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="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="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="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="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="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="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="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="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="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"> +<h1><a class="toc-backref" href="#id38">Basic syntax</a></h1> +<p>The syntax is:</p> +<pre class="literal-block"> +\poldef polname(x):= expression in variable x; +</pre> +<p>where:</p> +<ul class="simple"> +<li>in place of <tt class="docutils literal">x</tt> an arbitrary <em>dummy variable</em> is authorized, +i.e. per default any of <tt class="docutils literal"><span class="pre">[a-z|A-Z]</span></tt> (more letters can be declared +under Unicode engines.)</li> +<li><tt class="docutils literal">polname</tt> consists of letters, digits, and the <tt class="docutils literal">_</tt> and +<tt class="docutils literal">'</tt> characters. It must start with a letter.</li> +</ul> +<div class="admonition attention"> +<p class="first admonition-title">Attention!</p> +<p class="last">The <tt class="docutils literal">'</tt> is authorized since <tt class="docutils literal">0.5.1</tt>. As a result some constructs +recognized by the <tt class="docutils literal">\xintexpr</tt> parser, such as <tt class="docutils literal">var1 'and' var2</tt> +will get misinterpreted and cause errors. However these constructs +are unlikely to be frequently needed in polynomial expressions, and +the <tt class="docutils literal">\xintexpr</tt> syntax offers alternatives, so it was deemed a +small evil. Of course the <tt class="docutils literal">\xintexpr</tt> parser is modified only +temporarily during execution of <tt class="docutils literal">\poldef</tt>.</p> +</div> +<p>One can also issue:</p> +<pre class="literal-block"> +\PolDef{polname}{expression in variable x} +</pre> +<p>which admits an optional first argument to modify the variable letter +from its default <tt class="docutils literal">x</tt>.</p> +<dl class="docutils"> +<dt><tt class="docutils literal">\poldef <span class="pre">f(x):=</span> <span class="pre">1-x+x^2;</span></tt></dt> +<dd>defines polynomial <tt class="docutils literal">f</tt>. Polynomial names must start with a +letter and may contain letters, digits, underscores and the right +tick character. The +variable must be a single letter. The colon character is optional. +The semi-colon at end of expression is mandatory.</dd> +<dt><tt class="docutils literal"><span class="pre">\PolDef{f}{1-x+x^2}</span></tt></dt> +<dd>does the same as <tt class="docutils literal">\poldef <span class="pre">f(x):=</span> <span class="pre">1-x+x^2;</span></tt> To use another letter +than <tt class="docutils literal">x</tt> in the expression, one must pass it as an extra optional +argument to <tt class="docutils literal">\PolDef</tt>. Useful if the semi-colon has been assigned +some non-standard catcode by some package.</dd> +<dt><tt class="docutils literal"><span class="pre">\PolLet{g}={f}</span></tt></dt> +<dd>saves a copy of <tt class="docutils literal">f</tt> under name <tt class="docutils literal">g</tt>. Also usable without <tt class="docutils literal">=</tt>.</dd> +<dt><tt class="docutils literal">\poldef <span class="pre">f(z):=</span> <span class="pre">f(z)^2;</span></tt></dt> +<dd>redefines <tt class="docutils literal">f</tt> in terms of itself.</dd> +<dt><tt class="docutils literal">\poldef <span class="pre">f(T):=</span> <span class="pre">f(f(T));</span></tt></dt> +<dd>again redefines <tt class="docutils literal">f</tt> in terms of its (new) self.</dd> +<dt><tt class="docutils literal">\poldef <span class="pre">k(z):=</span> <span class="pre">f(z)-g(g(z)^2)^2;</span></tt></dt> +<dd>should now define the zero polynomial... Let's check: +<tt class="docutils literal">\[ k(z) = <span class="pre">\PolTypeset[z]{k}</span> \]</tt></dd> +<dt><tt class="docutils literal"><span class="pre">\PolDiff{f}{f'}</span></tt></dt> +<dd>sets <tt class="docutils literal">f'</tt> to the derivative of <tt class="docutils literal">f</tt>. The name doesn't have to be +<tt class="docutils literal">f'</tt> (in fact the <tt class="docutils literal">'</tt> is licit only since <tt class="docutils literal">0.5.1</tt>).</dd> +</dl> +<div class="admonition important"> +<p class="first admonition-title">Important</p> +<p class="last">This is not done automatically. If some new definition needs to use +the derivative of some available polynomial, that derivative +polynomial must have been defined via <tt class="docutils literal">\PolDiff</tt>: something like +<tt class="docutils literal"><span class="pre">T'(x)^2</span></tt> will not work without a prior <tt class="docutils literal"><span class="pre">\PolDiff{T}{T'}</span></tt>.</p> +</div> +<dl class="docutils"> +<dt><tt class="docutils literal"><span class="pre">\PolDiff{f'}{f''}</span></tt></dt> +<dd>obtains second derivative.</dd> +<dt><tt class="docutils literal"><span class="pre">\PolDiff[3]{f}{f'''}</span></tt></dt> +<dd>computes the third derivative.</dd> +</dl> +<pre class="literal-block"> +$f(z) = \PolTypeset[z]{f} $\newline +$f'(z) = \PolTypeset[z]{f'} $\newline +$f''(z) = \PolTypeset[z]{f''} $\newline +$f'''(z)= \PolTypeset[z]{f'''} $\par +</pre> +<div class="admonition important"> +<p class="first admonition-title">Important</p> +<p>The package does not currently know rational functions: <tt class="docutils literal">/</tt> in +a parsed polynomial expression does the Euclidean quotient:</p> +<pre class="literal-block"> +(1-x^2)/(1-x) +</pre> +<p>does give <tt class="docutils literal">1+x</tt> but</p> +<pre class="literal-block"> +(1/(1-x))*(1-x^2) +</pre> +<p>evaluates to zero. This will work as expected:</p> +<pre class="last literal-block"> +\poldef k(x):= (x-1)(x-2)(x-3)(x-4)/(x^2-5x+4); +</pre> +</div> +<div class="admonition attention" id="warningtacit"> +<p class="first admonition-title">Attention!</p> +<p><tt class="docutils literal">1/2 x^2</tt> skips the space and is treated like <tt class="docutils literal"><span class="pre">1/(2*x^2)</span></tt> because +of the tacit multiplication rules of xintexpr. But this means it +gives zero! Thus one must use <tt class="docutils literal">(1/2)x^2</tt> or <tt class="docutils literal">1/2*x^2</tt> or +<tt class="docutils literal"><span class="pre">(1/2)*x^2</span></tt> for disambiguation: <tt class="docutils literal">x - 1/2*x^2 + <span class="pre">1/3*x^3...</span></tt>. It is +even simpler to move the denominator to the right: <tt class="docutils literal">x - x^2/2 + +x^3/3 - ...</tt>.</p> +<p class="last">It is worth noting that <tt class="docutils literal"><span class="pre">1/2(x-1)(x-2)</span></tt> suffers the same issue: +<a class="reference external" href="http://www.ctan.org/pkg/xint">xint</a> tacit multiplication always "ties more", hence this gets +interpreted as <tt class="docutils literal"><span class="pre">1/(2*(x-1)*(x-2))</span></tt> which gives zero by polynomial +division. Thus, use one of <tt class="docutils literal"><span class="pre">(1/2)(x-1)(x-2)</span></tt>, <tt class="docutils literal"><span class="pre">1/2*(x-1)(x-2)</span></tt> or +<tt class="docutils literal"><span class="pre">(x-1)(x-2)/2</span></tt>.</p> +</div> +<p>After:</p> +<pre class="literal-block"> +\poldef f_1(x):= 25(x-1)(x^2-2)(x-3)(x-4)(x-5);% +\poldef f_2(x):= 37(x-1)(x^2-2)(x-6)(x-7)(x-8);% +</pre> +<p>the macro call <tt class="docutils literal"><span class="pre">\PolGCD{f_1}{f_2}{k}</span></tt> sets <tt class="docutils literal">k</tt> to the (unitary) GCD of +<tt class="docutils literal">f_1</tt> and <tt class="docutils literal">f_2</tt> (hence to the expansion of <tt class="docutils literal"><span class="pre">(x-1)(x^2-2)</span></tt>.)</p> +<dl class="docutils"> +<dt><tt class="docutils literal">\PolToExpr{k}</tt></dt> +<dd>will (expandably) give in this case <tt class="docutils literal"><span class="pre">x^3-x^2-2*x+2</span></tt>. This is +useful for console or file output (the syntax is Maple- and +PSTricks-compatible; the letter used in output can be +(non-expandably) changed via a redefinition of <a class="reference internal" href="#poltoexprvar">\PolToExprVar</a>.)</dd> +<dt><tt class="docutils literal"><span class="pre">\PolToExpr*{k}</span></tt></dt> +<dd>gives ascending powers: <tt class="docutils literal"><span class="pre">2-2*x-x^2+x^3</span></tt>.</dd> +</dl> +</div> +<div class="section" id="examples-of-localization-of-roots"> +<h1><a class="toc-backref" href="#id39">Examples of localization of roots</a></h1> +<ul> +<li><p class="first">To make printed decimal numbers more enjoyable than via +<tt class="docutils literal">\xintSignedFrac</tt>:</p> +<pre class="literal-block"> +\renewcommand\PolTypesetOne[1]{\PolDecToString{\xintREZ{#1}}}% +</pre> +<p><tt class="docutils literal">\PolDecToString</tt> will use decimal notation to incorporate the power +of ten part; and the <tt class="docutils literal">\xintREZ</tt> will have the effect to suppress +trailing zeros if present in raw numerator (if those digits end up +after decimal mark.) Notice that the above are expandable macros and +that one can also do:</p> +<pre class="literal-block"> +\renewcommand\PolToExprCmd[1]{\PolDecToString{\xintREZ{#1}}}% +</pre> +<p>to modify output of <a class="reference internal" href="#poltoexpr-polname">\PolToExpr{polname}</a>.</p> +</li> +<li><p class="first">For extra info in log file use <tt class="docutils literal">\xintverbosetrue</tt>.</p> +</li> +<li><p class="first">Only for some of these examples is the output included here.</p> +</li> +</ul> +<div class="section" id="a-typical-example"> +<h2><a class="toc-backref" href="#id40">A typical example</a></h2> +<p>In this example the polynomial is square-free.</p> +<pre class="literal-block"> +\poldef f(x) := x^7 - x^6 - 2x + 1; + +\PolToSturm{f}{f} +\PolSturmIsolateZeros{f} +The \PolTypeset{f} polynomial has \PolSturmNbOfIsolatedZeros{f} distinct real +roots which are located in the following intervals: +\PolPrintIntervals{f} +Here is the second root with ten more decimal digits: +\PolRefineInterval[10]{f}{2} +\[\PolSturmIsolatedZeroLeft{f}{2}<Z_2<\PolSturmIsolatedZeroRight{f}{2}\] +And here is the first root with twenty digits after decimal mark: +\PolEnsureIntervalLength{f}{1}{-20} +\[\PolSturmIsolatedZeroLeft{f}{1}<Z_1<\PolSturmIsolatedZeroRight{f}{1}\] +The first element of the Sturm chain has degree $\PolDegree{f_0}$. As +this is the original degreee $\PolDegree{f}$ we know that $f$ is square free. +Its derivative is up to a constant \PolTypeset{f_1} (in this example +it is identical with it). +\PolToSturm{f_1}{f_1}\PolSturmIsolateZeros{f_1}% +The derivative has \PolSturmNbOfIsolatedZeros{f_1} distinct real +roots: +\PolPrintIntervals[W]{f_1} +\PolEnsureIntervalLengths{f_1}{-10}% +Here they are with ten digits after decimal mark: +\PolPrintIntervals[W]{f_1} +\PolDiff{f_1}{f''} +\PolToSturm{f''}{f''} +\PolSturmIsolateZeros{f''} +The second derivative is \PolTypeset{f''}. +It has \PolSturmNbOfIsolatedZeros{f''} distinct real +roots: +\PolPrintIntervals[X]{f''} +Here is the positive one with 20 digits after decimal mark: +\PolEnsureIntervalLength{f''}{2}{-20}% +\[X_2 = \PolSturmIsolatedZeroLeft{f''}{2}\dots\] +The more mathematically advanced among our dear readers will be able +to give the exact value for $X_2$! +</pre> +</div> +<div class="section" id="a-degree-four-polynomial-with-nearby-roots"> +<h2><a class="toc-backref" href="#id41">A degree four polynomial with nearby roots</a></h2> +<p>Notice that this example is a bit outdated as <tt class="docutils literal">0.7</tt> release has +added <tt class="docutils literal"><span class="pre">\PolSturmIsolateZeros**{sturmname}</span></tt> which would find exactly +the roots. The steps here retain their interest when one is interested +in finding isolating intervals for example to prepare some demonstration +of dichotomy method.</p> +<pre class="literal-block"> +\PolDef{Q}{(x-1.050001)(x-1.105001)(x-1.110501)(x-1.111051)} +\PolTypeset{Q} +\PolToSturm{Q}{Q} % it is allowed to use same prefix for Sturm chain +\PolSturmIsolateZeros{Q} +\PolPrintIntervals{Q} +% reports 1.0 < Z_1 < 1.1, 1.10 < Z_2 < 1.11, 1.110 < Z_3 < 1.111, and 1.111 < Z_4 < 1.112 +% but the above bounds do not allow minimizing separation between roots +% so we refine: +\PolRefineInterval*{Q}{1} +\PolRefineInterval*{Q}{2} +\PolRefineInterval*{Q}{3} +\PolRefineInterval*{Q}{4} +\PolPrintIntervals{Q} +% reports 1.05 < Z_1 < 1.06, 1.105 < Z_2 < 1.106, 1.1105 < Z_3 < 1.1106, +% and 1.11105 < Z_4 < 1.11106. +\PolEnsureIntervalLengths{Q}{-6} +\PolPrintIntervals{Q} +% of course finds here all roots exactly +</pre> +</div> +<div class="section" id="the-degree-nine-polynomial-with-0-99-0-999-0-9999-as-triple-roots"> +<h2><a class="toc-backref" href="#id42">The degree nine polynomial with 0.99, 0.999, 0.9999 as triple roots</a></h2> +<pre class="literal-block"> +% define a user command (xinttools is loaded automatically by polexpr) +\newcommand\showmultiplicities[1]{% #1 = "sturmname" +\xintFor* ##1 in {\xintSeq{1}{\PolSturmNbOfIsolatedZeros{#1}}}\do{% + The multiplicity is \PolSturmIsolatedZeroMultiplicity{#1}{##1} + \PolSturmIfZeroExactlyKnown{#1}{##1}% + {at the root $x=\PolSturmIsolatedZeroLeft{#1}{##1}$} + {for the root such that + $\PolSturmIsolatedZeroLeft{#1}{##1}<x<\PolSturmIsolatedZeroRight{#1}{##1}$} + \par +}}% +\PolDef{f}{(x-0.99)^3(x-0.999)^3(x-0.9999)^3} +\renewcommand\PolTypesetOne[1]{\PolDecToString{\xintREZ{#1}}} +\PolTypeset{f}\par +\PolToSturm{f}{f}% it is allowed to use "polname" as "sturmname" too +\PolSturmIsolateZerosAndGetMultiplicities{f}% use the "sturmname" here +% or \PolSturmIsolateZeros*{f} which is exactly the same, but shorter.. + +\showmultiplicities{f} +</pre> +<p>In this example, the output will look like this (but using math mode):</p> +<pre class="literal-block"> +x^9 - 8.9667x^8 + 35.73400293x^7 - 83.070418400109x^6 + 124.143648875193123x^5 +- 123.683070924326075877x^4 + 82.149260397553075617891x^3 +- 35.07602992699900159127007x^2 + 8.7364078733314648368671733x +- 0.967100824643585986488103299 + +The multiplicity is 3 at the root x = 0.99 +The multiplicity is 3 at the root x = 0.999 +The multiplicity is 3 at the root x = 0.9999 +</pre> +<p>On first pass, these rational roots were found (due to their relative +magnitudes, using <tt class="docutils literal">\PolSturmIsolateZeros**</tt> was not needed here). But +multiplicity computation works also with (decimal) roots not yet +identified or with non-decimal or irrational roots.</p> +<p>It is fun to modify only a tiny bit the polynomial and see if polexpr +survives:</p> +<pre class="literal-block"> +\PolDef{g}{f(x)+1e-27} +\PolTypeset{g}\par +\PolToSturm{g}{g} +\PolSturmIsolateZeros*{g} + +\showmultiplicities{g} +</pre> +<p>This produces:</p> +<pre class="literal-block"> +x^9 - 8.9667x^8 + 35.73400293x^7 - 83.070418400109x^6 + 124.143648875193123x^5 +- 123.683070924326075877x^4 + 82.149260397553075617891x^3 +- 35.07602992699900159127007x^2 + 8.7364078733314648368671733x +- 0.967100824643585986488103298 + +The multiplicity is 1 for the root such that 0.98 < x < 0.99 +The multiplicity is 1 for the root such that 0.9991 < x < 0.9992 +The multiplicity is 1 for the root such that 0.9997 < x < 0.9998 +</pre> +<p>Which means that the multiplicity-3 roots each became a real and a pair of +complex ones. Let's see them better:</p> +<pre class="literal-block"> +\PolEnsureIntervalLengths{g}{-10} + +\showmultiplicities{g} +</pre> +<p>which produces:</p> +<pre class="literal-block"> +The multiplicity is 1 for the root such that 0.9899888032 < x < 0.9899888033 +The multiplicity is 1 for the root such that 0.9991447980 < x < 0.9991447981 +The multiplicity is 1 for the root such that 0.9997663986 < x < 0.9997663987 +</pre> +</div> +<div class="section" id="a-degree-five-polynomial-with-three-rational-roots"> +<h2><a class="toc-backref" href="#id43">A degree five polynomial with three rational roots</a></h2> +<pre class="literal-block"> +\poldef Q(x) := 1581755751184441 x^5 + -14907697165025339 x^4 + +48415668972339336 x^3 + -63952057791306264 x^2 + +46833913221154895 x + -49044360626280925; + +\PolToSturm{Q}{Q} +%\begin{flushleft} + \renewcommand\PolTypesetCmdPrefix[1]{\allowbreak\xintiiifSgn{#1}{}{+}{+}}% + $Q_0(x) = \PolTypeset{Q_0}$ +%\end{flushleft} +\PolSturmIsolateZeros**{Q} +\PolPrintIntervals{Q} + +$Q_{norr}(x) = \PolTypeset{Q_norr}$ +</pre> +<p>Here, all real roots are rational:</p> +<pre class="literal-block"> +Z_1 = 833719/265381 +Z_2 = 165707065/52746197 +Z_3 = 355/113 + +Q_norr(x) = x^2 + 1 +</pre> +<p>And let's get their decimal expansion too:</p> +<pre class="literal-block"> +% print decimal expansion of the found roots +\renewcommand\PolPrintIntervalsPrintExactZero + {\xintTrunc{20}{\PolPrintIntervalsTheLeftEndPoint}\dots} +\PolPrintIntervals{Q} + +Z_1 = 3.14159265358107777120... +Z_2 = 3.14159265358979340254... +Z_3 = 3.14159292035398230088... +</pre> +</div> +<div class="section" id="a-mignotte-type-polynomial"> +<h2><a class="toc-backref" href="#id44">A Mignotte type polynomial</a></h2> +<pre class="literal-block"> +\PolDef{P}{x^10 - (10x-1)^2}% +\PolTypeset{P} % prints it in expanded form +\PolToSturm{P}{P} % we can use same prefix for Sturm chain +\PolSturmIsolateZeros{P} % finds 4 real roots +This polynomial has \PolSturmNbOfIsolatedZeros{P} distinct real roots: +\PolPrintIntervals{P}% +% 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 +\PolRefineInterval*{P}{3}% will refine to 0.100001 < Z_3 < 0.100002 +\PolPrintIntervals{P}% +Let us now get to know all roots with 10 digits after decimal mark: +\PolEnsureIntervalLengths{P}{-10}% +\PolPrintIntervals{P}% now all roots are known 10 decimal digits after mark +Finally, we display 20 digits of the second root: +\PolEnsureIntervalLength{P}{2}{-20}% makes Z_2 known with 20 digits after mark +\[\PolSturmIsolatedZeroLeft{P}{2}<Z_2<\PolSturmIsolatedZeroRight{P}{2}\] +</pre> +<p>The last line produces:</p> +<pre class="literal-block"> +0.09999900004999650028 < Z_2 < 0.09999900004999650029 +</pre> +</div> +<div class="section" id="the-wilkinson-polynomial"> +<h2><a class="toc-backref" href="#id45">The Wilkinson polynomial</a></h2> +<p>See <a class="reference external" href="https://en.wikipedia.org/wiki/Wilkinson%27s_polynomial">Wilkinson polynomial</a>.</p> +<pre class="literal-block"> +\documentclass{article} +\usepackage{polexpr} +\begin{document} +%\xintverbosetrue % for the curious... + +\poldef f(x) := mul((x - i), i = 1..20); + +\renewcommand\PolTypesetCmdPrefix[1]{\allowbreak\xintiiifSgn{#1}{}{+}{+}}% +\renewcommand\PolTypesetOne[1]{\xintDecToString{#1}}% + +\noindent\PolTypeset{f} + +\PolToSturm{f}{f} +\PolSturmIsolateZeros{f} +\PolPrintIntervals{f} + +\clearpage + +\poldef g(x) := f(x) - 2**{-23} x**19; + +% be patient! +\PolToSturm{g}{g} +\noindent\PolTypeset{g_0}% integer coefficient primitive polynomial + +\PolSturmIsolateZeros{g} +\PolEnsureIntervalLengths{g}{-10} + +\renewcommand\PolPrintIntervalsPrintMultiplicity{} +\PolPrintIntervals*{g} + +\end{document} +</pre> +<p>The first polynomial:</p> +<pre class="literal-block"> +f(x) = x**20 +- 210 x**19 ++ 20615 x**18 +- 1256850 x**17 ++ 53327946 x**16 +- 1672280820 x**15 ++ 40171771630 x**14 +- 756111184500 x**13 ++ 11310276995381 x**12 +- 135585182899530 x**11 ++ 1307535010540395 x**10 +- 10142299865511450 x**9 ++ 63030812099294896 x**8 +- 311333643161390640 x**7 ++ 1206647803780373360 x**6 +- 3599979517947607200 x**5 ++ 8037811822645051776 x**4 +- 12870931245150988800 x**3 ++ 13803759753640704000 x**2 +- 8752948036761600000 x ++ 2432902008176640000 +</pre> +<p>is handled fast enough (a few seconds), but the modified one <tt class="docutils literal">f(x) - +<span class="pre">2**-23</span> <span class="pre">x**19</span></tt> takes about 20x longer (the Sturm chain polynomials +have integer coefficients with up to 321 digits, whereas (surprisingly +perhaps) those of the Sturm chain polynomials derived from <tt class="docutils literal">f</tt> never +have more than 21 digits ...).</p> +<p>Once the Sturm chain is computed and the zeros isolated, obtaining their +decimal digits is relatively faster. Here is for the ten real roots of +<tt class="docutils literal">f(x) - <span class="pre">2**-23</span> <span class="pre">x**19</span></tt> as computed by the code above:</p> +<pre class="literal-block"> +Z_1 = 0.9999999999... +Z_2 = 2.0000000000... +Z_3 = 2.9999999999... +Z_4 = 4.0000000002... +Z_5 = 4.9999999275... +Z_6 = 6.0000069439... +Z_7 = 6.9996972339... +Z_8 = 8.0072676034... +Z_9 = 8.9172502485... +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="#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> +<p>In the defining expression we could have used <tt class="docutils literal">i/10</tt> but this gives +less efficient internal form for the coefficients (the <tt class="docutils literal">10</tt>'s end up +in denominators). Using <tt class="docutils literal">\PolToExpr{P}</tt> after having done</p> +<pre class="literal-block"> +\renewcommand\PolToExprCmd[1]{\PolDecToString{\xintREZ{#1}}} +</pre> +<p>we get this expanded form:</p> +<pre class="literal-block"> +x^41 +-28.7*x^39 ++375.7117*x^37 +-2975.11006*x^35 ++15935.28150578*x^33 +-61167.527674162*x^31 ++173944.259366417394*x^29 +-373686.963560544648*x^27 ++613012.0665016658846445*x^25 +-771182.31133138163125495*x^23 ++743263.86672885754888959569*x^21 +-545609.076599482896371978698*x^19 ++301748.325708943677229642930528*x^17 +-123655.8987669450434698869844544*x^15 ++36666.1782054884005855608205864192*x^13 +-7607.85821367459445649518380016128*x^11 ++1053.15135918687298508885950223794176*x^9 +-90.6380005918141132650786081964032*x^7 ++4.33701563847327366842552218288128*x^5 +-0.0944770968420804735498178265088*x^3 ++0.00059190121813899276854174416896*x +</pre> +<p>which shows coefficients with up to 36 significant digits...</p> +<p>Stress test: not a hard challenge to <tt class="docutils literal">xint + polexpr</tt>, but be a bit patient!</p> +<pre class="literal-block"> +\PolDef{P}{mul((x-i*1e-1), i=-20..20)}% +\PolToSturm{P}{S} % dutifully computes S_0, ..., S_{41} +% the [1] optional argument limits the search to interval (-10,10) +\PolSturmIsolateZeros[1]{S} % finds *exactly* (but a bit slowly) all 41 roots! +\PolPrintIntervals{S} % nice, isn't it? +</pre> +<div class="admonition note"> +<p class="first admonition-title">Note</p> +<p>Release <tt class="docutils literal">0.5</tt> has <em>experimental</em> addition of optional argument +<tt class="docutils literal">E</tt> to <tt class="docutils literal">\PolSturmIsolateZeros</tt>. It instructs to search roots only +in interval <tt class="docutils literal"><span class="pre">(-10^E,</span> 10^E)</tt>. Important: the extremities are +<em>assumed to not be roots</em>. In this example, the <tt class="docutils literal">[1]</tt> in +<tt class="docutils literal"><span class="pre">\PolSturmIsolateZeros[1]{S}</span></tt> gives some speed gain; without it, it +turns out in this case that <tt class="docutils literal">polexpr</tt> would have started with +<tt class="docutils literal"><span class="pre">(-10^6,</span> 10^6)</tt> interval.</p> +<p class="last">Please note that this will probably get replaced in future by the +specification of a general interval. Do not rely on meaning of this +optional argument keeping the same.</p> +</div> +</div> +<div class="section" id="roots-of-chebyshev-polynomials"> +<h2><a class="toc-backref" href="#id48">Roots of Chebyshev polynomials</a></h2> +<pre class="literal-block"> +\newcount\mycount +\poldef T_0(x) := 1; +\poldef T_1(x) := x; +\mycount 2 +\xintloop + \poldef T_\the\mycount(x) := + 2x*T_\the\numexpr\mycount-1(x) + - T_\the\numexpr\mycount-2(x); +\ifnum\mycount<15 +\advance\mycount 1 +\repeat + +\[T_{15} = \PolTypeset[X]{T_15}\] +\PolToSturm{T_15}{T_15} +\PolSturmIsolateZeros{T_15} +\PolEnsureIntervalLengths{T_15}{-10} +\PolPrintIntervals{T_15} +</pre> +</div> +</div> +<div class="section" id="non-expandable-macros"> +<h1><a class="toc-backref" href="#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="#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, +under the user-chosen <tt class="docutils literal">polname</tt>. Of course the <em>expression</em> can +use other previously defined polynomials. Names must start with a +letter and are constituted of letters, digits, underscores and +(since <tt class="docutils literal">0.5.1</tt>) the right tick <tt class="docutils literal">'</tt>. +The whole <a class="reference external" href="http://www.ctan.org/pkg/xint">xintexpr</a> syntax is authorized:</p> +<pre class="literal-block"> +\poldef sin(z) := add((-1)^i z^(2i+1)/(2i+1)!, i = 0..10); +</pre> +<p>With fractional coefficients, beware the <a class="reference internal" href="#warningtacit">tacit multiplication issue</a>.</p> +<p>As a side effect the function <tt class="docutils literal">polname()</tt> is recognized as a +genuine <tt class="docutils literal"><span class="pre">\xintexpr...\relax</span></tt> function for (exact) numerical +evaluation (or within an <tt class="docutils literal">\xintdefvar</tt> assignment.) It computes +values not according to the original expression but via the Horner +scheme corresponding to the polynomial coefficients.</p> +<div class="admonition attention"> +<p class="first admonition-title">Attention!</p> +<p>Release <tt class="docutils literal">0.3</tt> also did the necessary set-up to let the +polynomial be known to the <tt class="docutils literal">\xintfloatexpr</tt> (or +<tt class="docutils literal">\xintdeffloatvar</tt>) parser.</p> +<p>Since <tt class="docutils literal">0.4</tt> this isn't done automatically. Even more, a +previously existing floating point variant of the same name will +be let undefined again, to avoid hard to debug mismatches between +exact and floating point polynomials. This also applies when the +polynomial is produced not via <tt class="docutils literal">\poldef</tt> or <tt class="docutils literal">\PolDef</tt> but as +a product of the other package macros.</p> +<p class="last">See <a class="reference internal" href="#polgenfloatvariant-polname">\PolGenFloatVariant{polname}</a>.</p> +</div> +<p>The original expression is lost after parsing, and in particular +the package provides no way to typeset it. This has to be done +manually, if needed.</p> +</blockquote> +</div> +<div class="section" id="poldef-letter-polname-expression-in-letter"> +<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 +semi-colon in presence of some packages.) In absence of the +<tt class="docutils literal">[letter]</tt> optional argument, the variable is assumed to be <tt class="docutils literal">x</tt>.</blockquote> +</div> +<div class="section" id="polgenfloatvariant-polname"> +<span id="polgenfloatvariant"></span><h2><a class="toc-backref" href="#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 +already pre-rounded to the float precision.</p> +<p>See also <a class="reference internal" href="#poltofloatexpr-polname">\PolToFloatExpr{polname}</a>.</p> +<div class="admonition attention"> +<p class="first admonition-title">Attention!</p> +<p>Release <tt class="docutils literal">0.3</tt> did this automatically on <tt class="docutils literal">\PolDef</tt> and +<tt class="docutils literal">\poldef</tt> but this was removed at <tt class="docutils literal">0.4</tt> for optimization.</p> +<p class="last">Any operation, for example generating the derivative polynomial, +or dividing two polynomials or using the <tt class="docutils literal">\PolLet</tt>, <strong>must</strong> be +followed by explicit usage of <tt class="docutils literal">\PolGenFloatVariant{polname}</tt> if +the new polynomial is to be used in <tt class="docutils literal">\xintfloatexpr</tt> or alike +context.</p> +</div> +</blockquote> +</div> +<div class="section" id="pollet-polname-2-polname-1"> +<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 +<tt class="docutils literal"><span class="pre">\PolDef{polname_2}{polname_1(x)}</span></tt> but with less overhead. The +<tt class="docutils literal">=</tt> is optional.</blockquote> +</div> +<div class="section" id="polgloballet-polname-2-polname-1"> +<span id="polgloballet"></span><h2><a class="toc-backref" href="#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="#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> +<ul class="simple"> +<li>Attention, coefficients here are indexed starting at 1.</li> +<li>With #1=-1, -2, ..., <tt class="docutils literal"><span class="pre">\macro{#1}</span></tt> returns leading coefficients.</li> +<li>With #1=0, returns the number of coefficients, i.e. <tt class="docutils literal">1 + deg f</tt> +for non-zero polynomials.</li> +<li>Out-of-range #1's return <tt class="docutils literal">0/1[0]</tt>.</li> +</ul> +<p>See also <a class="reference internal" href="#polnthcoeff-polname-number">\PolNthCoeff{polname}{number}</a>. The main difference is that +with <tt class="docutils literal">\PolAssign</tt>, <tt class="docutils literal">\macro</tt> is made a prefix to <tt class="docutils literal">1 + deg f</tt> +already defined (hidden to user) macros holding individually the +coefficients but <a class="reference internal" href="#polnthcoeff-polname-number">\PolNthCoeff{polname}{number}</a> does each time the job +to expandably recover the <tt class="docutils literal">Nth</tt> coefficient, and due to +expandability can not store it in a macro for future usage (of course, +it can be an argument in an <tt class="docutils literal">\edef</tt>.) The other difference +is the shift by one in indexing, mentioned above (negative +indices act the same in both.)</p> +</blockquote> +</div> +<div class="section" id="polget-polname-fromarray-macro"> +<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 +<tt class="docutils literal">\macro{number}</tt> gets expanded in an <tt class="docutils literal">\edef</tt> and then normalized +via <a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a>'s macro <tt class="docutils literal">\xintRaw</tt>.</p> +<p>The leading zeros are removed from the polynomial.</p> +<p>(contrived) Example:</p> +<pre class="literal-block"> +\xintAssignArray{1}{-2}{5}{-3}\to\foo +\PolGet{f}\fromarray\foo +</pre> +<p>This will define <tt class="docutils literal">f</tt> as would have <tt class="docutils literal">\poldef <span class="pre">f(x):=1-2x+5x^2-3x^3;</span></tt>.</p> +<div class="admonition note"> +<p class="first admonition-title">Note</p> +<p class="last">Prior to <tt class="docutils literal">0.5</tt>, coefficients were not normalized via +<tt class="docutils literal">\xintRaw</tt> for internal storage.</p> +</div> +</blockquote> +</div> +<div class="section" id="polfromcsv-polname-csv"> +<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 +item</em> gives the constant term, the <em>last item</em> gives the leading +coefficient, except if zero, then it is dropped (iteratively). List +items are each expanded in an <tt class="docutils literal">\edef</tt> and then put into normalized +form via <a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a>'s macro <tt class="docutils literal">\xintRaw</tt>.</p> +<p>As leading zero coefficients are removed:</p> +<pre class="literal-block"> +\PolFromCSV{f}{0, 0, 0, 0, 0, 0, 0, 0, 0, 0} +</pre> +<p>defines the zero polynomial, which holds only one coefficient.</p> +<p>See also expandable macro <a class="reference internal" href="#poltocsv-polname">\PolToCSV</a>.</p> +<div class="admonition note"> +<p class="first admonition-title">Note</p> +<p class="last">Prior to <tt class="docutils literal">0.5</tt>, coefficients were not normalized via +<tt class="docutils literal">\xintRaw</tt> for internal storage.</p> +</div> +</blockquote> +</div> +<div class="section" id="poltypeset-polname"> +<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> +<pre class="literal-block"> +\PolTypeset[z]{polname} +</pre> +<p>By default zero coefficients are skipped (issue <tt class="docutils literal">\poltypesetalltrue</tt> +to get all of them in output).</p> +<p>These commands (whose meanings will be found in the package code) +can be re-defined for customization. Their default definitions are +expandable, but this is not a requirement.</p> +</blockquote> +<div class="section" id="poltypesetcmd-raw-coeff"> +<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 +<a class="reference internal" href="#polifcoeffisplusorminusone-a-b">\PolIfCoeffIsPlusOrMinusOne{A}{B}</a>.</p> +<p>The actual printing of the coefficients, when not equal to plus or +minus one is handled by <a class="reference internal" href="#poltypesetone-raw-coeff">\PolTypesetOne{raw_coeff}</a>.</p> +</blockquote> +</div> +<div class="section" id="poltypesetone-raw-coeff"> +<span id="poltypesetone"></span><h3><a class="toc-backref" href="#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> +<p>One can do things such as for example: <a class="footnote-reference" href="#id5" id="id4">[1]</a></p> +<pre class="literal-block"> +\renewcommand\PolTypesetOne[1]{\num{\xintPFloat[5]{#1}}} +\renewcommand\PolTypesetOne[1]{\num{\xintRound{4}{#1}}} +</pre> +<p>where e.g. we used the <tt class="docutils literal">\num</tt> macro of <tt class="docutils literal">siunitx</tt> as it +understands floating point notation.</p> +<table class="docutils footnote" frame="void" id="id5" rules="none"> +<colgroup><col class="label" /><col /></colgroup> +<tbody valign="top"> +<tr><td class="label"><a class="fn-backref" href="#id4">[1]</a></td><td>the difference in the syntaxes of <tt class="docutils literal">\xintPFloat</tt> and +<tt class="docutils literal">\xintRound</tt> is explained from the fact that +<tt class="docutils literal">\xintPFloat</tt> by default uses the prevailing precision +hence the extra argument like here <tt class="docutils literal">5</tt> is an optional one.</td></tr> +</tbody> +</table> +<p>One can also give a try to using <a class="reference internal" href="#poldectostring-decimal-number">\PolDecToString{decimal number}</a> +which uses decimal notation (at least for the numerator part).</p> +</blockquote> +</div> +<div class="section" id="id6"> +<span id="poltypesetmonomialcmd"></span><h3><a class="toc-backref" href="#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 +for the constant term, <tt class="docutils literal">\PolVar</tt> for the first degree and +<tt class="docutils literal"><span class="pre">\PolVar^{\PolIndex}</span></tt> for higher degrees monomials. Beware that +<tt class="docutils literal">\PolIndex</tt> expands to digit tokens and needs termination in +<tt class="docutils literal">\ifnum</tt> tests.</blockquote> +</div> +<div class="section" id="poltypesetcmdprefix-raw-coeff"> +<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 +<tt class="docutils literal">\xintSignedFrac</tt> used by <a class="reference internal" href="#poltypesetcmd-raw-coeff">\PolTypesetCmd{raw_coeff}</a> will put +the <tt class="docutils literal">-</tt> sign in front of the fraction (if it is a fraction) and +this will thus serve as separator in the typeset formula. Not used +for the first term.</blockquote> +</div> +</div> +<div class="section" id="id8"> +<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="#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> +by <tt class="docutils literal">f'</tt>.</p> +<p>Coefficients of the result <tt class="docutils literal">polname_2</tt> are irreducible fractions +(see <a class="reference internal" href="#technicalities">Technicalities</a> for the whole story.)</p> +</blockquote> +</div> +<div class="section" id="poldiff-n-polname-1-polname-2"> +<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 +<tt class="docutils literal"><span class="pre">\PolLet{polname_2}={polname_1}</span></tt>. With negative <tt class="docutils literal">N</tt>, switches to +using <tt class="docutils literal">\PolAntiDiff</tt>.</blockquote> +</div> +<div class="section" id="polantidiff-polname-1-polname-2"> +<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> +<p>Coefficients of the result <tt class="docutils literal">polname_2</tt> are irreducible fractions +(see <a class="reference internal" href="#technicalities">Technicalities</a> for the whole story.)</p> +</blockquote> +</div> +<div class="section" id="polantidiff-n-polname-1-polname-2"> +<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="#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 +<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="#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="#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="#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> +and <tt class="docutils literal">polname_2</tt> vanish, then <tt class="docutils literal">polname_GCD</tt> is the zero +polynomial.</blockquote> +<!-- ``\PolIGCD{polname_1}{polname_2}{polname_iGCD}`` +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + + **NOT YET** + + This **assumes** that the two polynomials have integer coefficients. + It then computes the greatest common divisor in the integer + polynomial ring, normalized to have a positive leading coefficient + (if the inputs are not both zero). + +``\PolIContent{polname}`` +~~~~~~~~~~~~~~~~~~~~~~~~~ + + **NOT YET** + + This computes a positive rational number such that dividing the + polynomial with it returns an integer coefficients polynomial with + no common factor among the coefficients. --> +</div> +<div class="section" id="poltosturm-polname-sturmname"> +<span id="poltosturm"></span><h2><a class="toc-backref" href="#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 +of the) remainder in euclidean division, iteratively.</p> +<p>The last non-zero remainder <tt class="docutils literal">P_N_</tt> (where <tt class="docutils literal">N</tt> is obtainable as +<a class="reference internal" href="#polsturmchainlength-sturmname">\PolSturmChainLength{sturmname}</a>) is up to a factor +the GCD of <tt class="docutils literal">P</tt> and <tt class="docutils literal">P'</tt> hence it is a constant if and only if +<tt class="docutils literal">P</tt> is square-free.</p> +<div class="admonition note"> +<p class="first admonition-title">Note</p> +<ul class="last simple"> +<li>Since <tt class="docutils literal">0.5</tt> all these polynomials are divided by their rational +content, so they have integer coefficients with no common factor, +and the last one if a constant is either <tt class="docutils literal">1</tt> or <tt class="docutils literal"><span class="pre">-1</span></tt>.</li> +<li>After this normalization to primitive polynomials, they are +stored internally as <tt class="docutils literal">sturmname_k_</tt>, <tt class="docutils literal">k=0,1, ...</tt>.</li> +<li>These polynomials are used internally only. To keep them as +genuine declared polynomials also after the macro call, use the +starred variant <a class="reference internal" href="#id9">PolToSturm*</a>.</li> +</ul> +</div> +<div class="admonition note"> +<p class="first admonition-title">Note</p> +<p class="last">It is perfectly allowed to use the polynomial name as Sturm chain name: +<tt class="docutils literal"><span class="pre">\PolToSturm{f}(f}</span></tt>.</p> +</div> +<p>The macro then declares <tt class="docutils literal">sturmname_0</tt>, <tt class="docutils literal">sturmname_1</tt>, ..., which are +the (non-declared) <tt class="docutils literal">sturmname_k_</tt> divided by the last one. Division is +not done if this last one is the constant <tt class="docutils literal">1</tt> or <tt class="docutils literal"><span class="pre">-1</span></tt>, i.e. if the +original polynomial was square-free. These polynomials are primitive +polynomials too, i.e. with integer coefficients having no common factor.</p> +<p>Thus <tt class="docutils literal">sturmname_0</tt> has exactly the same real and complex roots as +polynomial <tt class="docutils literal">polname</tt>, but with each root now of multiplicity one: +i.e. it is the "square-free part" of original polynomial <tt class="docutils literal">polname</tt>.</p> +<p>Notice that <tt class="docutils literal">sturmname_1</tt> isn't necessarily the derivative of +<tt class="docutils literal">sturmname_0</tt> due to the various normalizations.</p> +<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></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> +</blockquote> +</div> +<div class="section" id="id10"> +<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 +polynomials <tt class="docutils literal">sturmname_k_</tt>, <tt class="docutils literal">k=0,1, <span class="pre">...,</span> N</tt>, with +<tt class="docutils literal">N</tt> being <a class="reference internal" href="#polsturmchainlength-sturmname">\PolSturmChainLength{sturmname}</a>.</p> +<div class="admonition note"> +<p class="first admonition-title">Note</p> +<p class="last">This behaviour was modified at <tt class="docutils literal">0.6</tt>, anyhow the macro was +broken at <tt class="docutils literal">0.5</tt>.</p> +</div> +<div class="admonition hint"> +<p class="first admonition-title">Hint</p> +<p class="last">The square-free part of <tt class="docutils literal">polname</tt> is <tt class="docutils literal">sturmname_0</tt>, and their +quotient is the polynomial with name +<tt class="docutils literal">sturname_\PolSturmChainLength{sturmname}_</tt>. It thus easy to +set-up a loop iteratively computing the latter until the last one +is a constant, thus obtaining the decomposition of an <tt class="docutils literal">f</tt> as +a product <tt class="docutils literal">c f_1 f_2 f_3 ...</tt> of a constant and square-free (primitive) +polynomials, where each <tt class="docutils literal">f_i</tt> divides its predecessor.</p> +</div> +</blockquote> +</div> +<div class="section" id="polsettosturmchainsignchangesat-macro-sturmname-fraction"> +<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> +(which must be in format as acceptable by the <a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a> macros.)</p> +<div class="admonition note"> +<p class="first admonition-title">Note</p> +<p>The author was lazy and did not provide rather an expandable +variant, where one would do <tt class="docutils literal"><span class="pre">\edef\macro{\PolNbOf...}</span></tt>.</p> +<p>This will presumably get added in a future release.</p> +<p class="last">After some hesitation it was decided the macro would by default +act globally. To make the scope of its macro definition local, +use <tt class="docutils literal">[\empty]</tt> as extra optional argument.</p> +</div> +</blockquote> +</div> +<div class="section" id="polsettonbofzeroswithin-macro-sturmname-value-a-value-b"> +<span id="polsettonbofzeroswithin"></span><h2><a class="toc-backref" href="#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, +value_b]</tt> (the macro first re-orders the value for <tt class="docutils literal">value_a <= +value_b</tt> to hold).</p> +<div class="admonition note"> +<p class="first admonition-title">Note</p> +<p>The author was lazy and did not provide rather an expandable +variant, where one would do <tt class="docutils literal"><span class="pre">\edef\macro{\PolNbOf...}</span></tt>.</p> +<p>This will presumably get added in future.</p> +<p class="last">After some hesitation it was decided the macro would by default +act globally. To make the scope of its macro definition local, +use <tt class="docutils literal">[\empty]</tt> as extra optional argument.</p> +</div> +<p>See also the expandable +<a class="reference internal" href="#polsturmnbofrootsof-sturmname-lessthanorequalto-value">\PolSturmNbOfRootsOf{sturmname}\LessThanOrEqualTo{value}</a>, from +which it is immediate (with <tt class="docutils literal">\numexpr</tt>) to create an expandable +variant of this macro. However the difference is that this macro +requires only <a class="reference internal" href="#poltosturm">\PolToSturm</a> to have been executed, +whereas the expandable variant requires prior execution of +<a class="reference internal" href="#polsturmisolatezeros">\PolSturmIsolateZeros</a>.</p> +<p>See also the expandable +<a class="reference internal" href="#polsturmnbwithmultofrootsof-sturmname-lessthanorequalto-value">\PolSturmNbWithMultOfRootsOf{sturmname}\LessThanOrEqualTo{value}</a> +which requires prior execution of +<a class="reference internal" href="#id11">\PolSturmIsolateZeros*</a>.</p> +</blockquote> +</div> +<div class="section" id="polsturmisolatezeros-sturmname"> +<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> +<div class="admonition important"> +<p class="first admonition-title">Important</p> +<p>The Sturm chain must have been produced by an earlier +<a class="reference internal" href="#poltosturm-polname-sturmname">\PolToSturm{polname}{sturmname}</a>.</p> +<p>Why does this macro ask for argument the name of Sturm chain, +rather than the name of a polynomial? well this is mainly for +legacy reason, and because it is accompanied by other macros for +which it is simpler to assume the argument will be the name of an +already computed Sturm chain.</p> +<p>Notice that <tt class="docutils literal"><span class="pre">\PolToSturm{f}{f}</span></tt> is perfectly legal (the +<tt class="docutils literal">sturmname</tt> can be same as the <tt class="docutils literal">polname</tt>): it defines +polynomials <tt class="docutils literal">f_0</tt>, <tt class="docutils literal">f_1</tt>, ... having <tt class="docutils literal">f</tt> has name prefix.</p> +<p class="last">Such a prior call +to <tt class="docutils literal">\PolToSturm</tt> must have been made at any rate for +<tt class="docutils literal">\PolSturmIsolateZeros</tt> to be usable.</p> +</div> +<p>After its execution they are two types of such intervals (stored in +memory and accessible via macros or <a class="reference external" href="http://www.ctan.org/pkg/xint">xintexpr</a> variables, see below):</p> +<ul class="simple"> +<li>singleton <tt class="docutils literal">{a}</tt>: then <tt class="docutils literal">a</tt> is a root, (necessarily a decimal +number, but not all such decimal numbers are exactly identified yet).</li> +<li>open intervals <tt class="docutils literal">(a,b)</tt>: then there is exactly one root <tt class="docutils literal">z</tt> +such that <tt class="docutils literal">a < z < b</tt>, and the end points are guaranteed to not +be roots.</li> +</ul> +<p>The interval boundaries are decimal numbers, originating +in iterated decimal subdivision from initial intervals +<tt class="docutils literal"><span class="pre">(-10^E,</span> 0)</tt> and <tt class="docutils literal">(0, 10^E)</tt> with <tt class="docutils literal">E</tt> chosen initially large +enough so that all roots are enclosed; if zero is a root it is always +identified as such. The non-singleton intervals are of the +type <tt class="docutils literal">(a/10^f, <span class="pre">(a+1)/10^f)</span></tt> with <tt class="docutils literal">a</tt> an integer, which is +neither <tt class="docutils literal">0</tt> nor <tt class="docutils literal"><span class="pre">-1</span></tt>. Hence either <tt class="docutils literal">a</tt> and <tt class="docutils literal">a+1</tt> are both positive +or they are both negative.</p> +<p>One does not <em>a priori</em> know what will be the lengths of these +intervals (except that they are always powers of ten), they +vary depending on how many digits two successive roots have in +common in their respective decimal expansions.</p> +<div class="admonition important"> +<p class="first admonition-title">Important</p> +<p>If some two consecutive intervals share an end-point, no +information is yet gained about the separation between the two +roots which could at this stage be arbitrarily small.</p> +<p class="last">See <a class="reference internal" href="#polrefineinterval-sturmname-index">\PolRefineInterval*{sturmname}{index}</a> which addresses +this issue.</p> +</div> +<!-- This procedure is covariant +with the independent variable ``x`` becoming ``-x``. +Hmm, pas sûr et trop fatigué --> +<p>The interval boundaries (and exactly found roots) are made available +for future computations in <tt class="docutils literal">\xintexpr</tt>-essions or polynomial +definitions as variables <tt class="docutils literal"><sturmname>L_1</tt>, +<tt class="docutils literal"><sturmname>L_2</tt>, etc..., for the left end-points and +<tt class="docutils literal"><sturmname>R_1</tt>, <tt class="docutils literal"><sturmname>R_2</tt>, ..., for the right +end-points.</p> +<p>Thus for example, if <tt class="docutils literal">sturmname</tt> is <tt class="docutils literal">f</tt>, one can use the +<a class="reference external" href="http://www.ctan.org/pkg/xint">xintexpr</a> variables <tt class="docutils literal">fL_1</tt>, <tt class="docutils literal">fL_2</tt>, ... to refer in expressions +to the left end-points (or to the exact root, if left and right end +points coincide). Additionally, <a class="reference external" href="http://www.ctan.org/pkg/xint">xintexpr</a> variable <tt class="docutils literal">fZ_1_isknown</tt> +will have value <tt class="docutils literal">1</tt> if the root in the first interval is known, +and <tt class="docutils literal">0</tt> otherwise. And similarly for the other intervals.</p> +<p>Also, macros <a class="reference internal" href="#polsturmisolatedzeroleft-sturmname-index">\PolSturmIsolatedZeroLeft{sturmname}{index}</a> and +<a class="reference internal" href="#polsturmisolatedzeroright-sturmname-index">\PolSturmIsolatedZeroRight{sturmname}{index}</a> are provided which +expand to these same values, written in decimal notation (i.e. +pre-processed by <a class="reference internal" href="#poldectostring">\PolDecToString</a>.) And there +is also <a class="reference internal" href="#polsturmifzeroexactlyknown-sturmname-index-a-b">\PolSturmIfZeroExactlyKnown{sturmname}{index}{A}{B}</a>.</p> +<div class="admonition important"> +<p class="first admonition-title">Important</p> +<p class="last">Trailing zeroes in the stored decimal numbers accessible via the +macros are significant: they are also present in the decimal +expansion of the exact root.</p> +</div> +<p>These variables and macros are automatically updated when one next +uses macros such as <a class="reference internal" href="#polrefineinterval-sturmname-index">\PolRefineInterval*{sturmname}{index}</a>.</p> +<p>The start of decimal expansion of a positive <tt class="docutils literal">k</tt>-th root is given +by <a class="reference internal" href="#polsturmisolatedzeroleft">\PolSturmIsolatedZeroLeft{sturmname}{k}</a>, and for a negative root it is given +by <a class="reference internal" href="#polsturmisolatedzeroright">PolSturmIsolatedZeroRight{sturmname}{k}</a>. These two decimal +numbers are either both zero or both of the same sign.</p> +<p>The number of distinct roots is obtainable expandably as +<a class="reference internal" href="#polsturmnbofisolatedzeros-sturmname">\PolSturmNbOfIsolatedZeros{sturmname}</a>.</p> +<p>Furthermore +<a class="reference internal" href="#polsturmnbofrootsof-sturmname-lessthanorequalto-value">\PolSturmNbOfRootsOf{sturmname}\LessThanOrEqualTo{value}</a> and +<a class="reference internal" href="#polsturmnbofrootsof-sturmname-lessthanorequaltoexpr-expression">\PolSturmNbOfRootsOf{sturmname}\LessThanOrEqualToExpr{expression}</a>. +will expandably compute respectively the number of real roots at +most equal to <tt class="docutils literal">value</tt> or <tt class="docutils literal">expression</tt>, and the same but with +multiplicities.</p> +<div class="admonition note"> +<p class="first admonition-title">Note</p> +<p class="last">In the current implementation the <a class="reference external" href="http://www.ctan.org/pkg/xint">xintexpr</a> variables +and <a class="reference external" href="http://www.ctan.org/pkg/xint">xinttools</a> arrays are globally defined. On the +other hand the Sturm sequence polynomials obey the current scope.</p> +</div> +<div class="admonition note"> +<p class="first admonition-title">Note</p> +<p>As all computations are done <em>exactly</em> there can be no errors... +apart those due to bad coding by author. The results are exact +bounds for the mathematically exact real roots.</p> +<p class="last">Future releases will perhaps also provide macros based on Newton +or Regula Falsi methods. Exact computations with such methods +lead however quickly to very big fractions, and this forces usage +of some rounding scheme for the abscissas if computation times +are to remain reasonable. This raises issues of its own, which +are studied in numerical mathematics.</p> +</div> +</blockquote> +</div> +<div class="section" id="id12"> +<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 +multiplicities (of the real roots): +after executing this macro, +<a class="reference internal" href="#polsturmisolatedzeromultiplicity-sturmname-index">\PolSturmIsolatedZeroMultiplicity{sturmname}{index}</a> will expand +to the multiplicity of the root located in the <tt class="docutils literal">index</tt>-th +interval (intervals are enumerated from left to right, with index +starting at <tt class="docutils literal">1</tt>).</p> +<p>Furthermore, if for example the <tt class="docutils literal">sturmname</tt> is <tt class="docutils literal">f</tt>, <a class="reference external" href="http://www.ctan.org/pkg/xint">xintexpr</a> +variables <tt class="docutils literal">fM_1</tt>, <tt class="docutils literal">fM_2</tt>... hold the multiplicities thus +computed.</p> +<div class="admonition note"> +<p class="first admonition-title">Note</p> +<p class="last">It is <strong>not</strong> necessary to have executed the <a class="reference internal" href="#id9">PolToSturm*</a> starred +variant, as the non-starred variant keeps internally the memory of the +original GCD (and even of the full non-normalized original Sturm +chain), even though it does not make the declarations as <em>user-level</em> +genuine polynomials.</p> +</div> +<p>See <a class="reference internal" href="#the-degree-nine-polynomial-with-0-99-0-999-0-9999-as-triple-roots">The degree nine polynomial with 0.99, 0.999, 0.9999 as triple +roots</a> for an example.</p> +</blockquote> +</div> +<div class="section" id="id14"> +<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> +roots.</p> +<div class="admonition note"> +<p class="first admonition-title">Note</p> +<p class="last">After execution of this macro, a root is "known" if and only if +it is rational.</p> +</div> +<p>Furthermore, primitive polynomial <tt class="docutils literal">sturmname_sqf_norr</tt> is created +to match the (square-free) <tt class="docutils literal">sturmname_0</tt> from which all rational +roots have been removed (see <a class="reference internal" href="#polexprsetup">\polexprsetup</a> for customizing this +name). The number of distinct rational roots is thus the difference +between the degrees of these two polynomials (see also +<a class="reference internal" href="#polsturmnbofrationalroots-sturmname">\PolSturmNbOfRationalRoots{sturmname}</a>).</p> +<p>And <tt class="docutils literal">sturmname_norr</tt> is <tt class="docutils literal">sturmname_0_</tt> from which all rational +roots have been removed (see <a class="reference internal" href="#polexprsetup">\polexprsetup</a>), i.e. it contains +the irrational roots of the original polynomial, with the same +multiplicities.</p> +<p>See <a class="reference internal" href="#a-degree-five-polynomial-with-three-rational-roots">A degree five polynomial with three rational +roots</a> for an example.</p> +</blockquote> +</div> +<div class="section" id="polsturmisolatezerosandgetmultiplicities-sturmname"> +<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="#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="#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 +<tt class="docutils literal">sturmname_norr</tt> with no rational roots) except that it does <em>not</em> +compute the multiplicities of the <em>non-rational</em> roots.</p> +<div class="admonition note"> +<p class="first admonition-title">Note</p> +<p class="last">There is no macro to find the rational roots but not compute +their multiplicities at the same time.</p> +</div> +<div class="admonition attention"> +<p class="first admonition-title">Attention!</p> +<p>This macro does <em>not</em> define <a class="reference external" href="http://www.ctan.org/pkg/xint">xintexpr</a> variables +<tt class="docutils literal">sturmnameM_1</tt>, <tt class="docutils literal">sturmnameM_2</tt>, ... holding the +multiplicities and it leaves the multiplicity array (whose accessor +is <a class="reference internal" href="#polsturmisolatedzeromultiplicity-sturmname-index">\PolSturmIsolatedZeroMultiplicity{sturmname}{index}</a>) into +a broken state, as all non-rational roots will supposedly have +multiplicity one. This means that the output of +<a class="reference internal" href="#id18">\PolPrintIntervals*</a> for example will be +erroneous for the intervals with irrational roots.</p> +<p class="last">I decided to document it because finding multiplicities of the +non rational roots is somewhat costly, and one may be interested +only into finding the rational roots (of course random +polynomials with integer coefficients will not have <em>any</em> +rational root anyhow).</p> +</div> +</blockquote> +</div> +<div class="section" id="polrefineinterval-sturmname-index"> +<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 +interval to have both its end-points distinct from the end-points of +the original interval. This means that the <tt class="docutils literal">k</tt>th root is then +strictly separated from the other roots.</blockquote> +</div> +<div class="section" id="polrefineinterval-n-sturmname-index"> +<span id="polrefineinterval-n"></span><h2><a class="toc-backref" href="#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 +<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="#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 +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="#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 +each of them to have length at most <tt class="docutils literal">10^E</tt> (length will be shorter +than <tt class="docutils literal">10^E</tt> in output only if it did not change or became zero.)</p> +<p>This means that decimal expansions of all roots will be known with +<tt class="docutils literal"><span class="pre">-E</span></tt> digits (for <tt class="docutils literal">E<0</tt>) after decimal mark.</p> +</blockquote> +</div> +<div class="section" id="polprintintervals-varname-sturmname"> +<span id="polprintintervals"></span><h2><a class="toc-backref" href="#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 +specify a replacement for the default <tt class="docutils literal">Z</tt>). This will be done (by +default) in a +math mode <tt class="docutils literal">array</tt>, one interval per row, and pattern <tt class="docutils literal">rcccl</tt>, +where the second and fourth column hold the <tt class="docutils literal"><</tt> sign, except when +the interval reduces to a singleton, which means the root is known +exactly.</p> +<div class="admonition attention"> +<p class="first admonition-title">Attention!</p> +<p class="last">This macro was refactored at 0.7, its default output remained +identical but the ways to customize it got completely +modified.</p> +</div> +<p>See next macros which govern its output.</p> +</blockquote> +<div class="section" id="polprintintervalsnorealroots"> +<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> +<pre class="literal-block"> +\newcommand\PolPrintIntervalsNoRealRoots{} +</pre> +</blockquote> +</div> +<div class="section" id="polprintintervalsbeginenv"> +<h3><a class="toc-backref" href="#id88"><tt class="docutils literal">\PolPrintIntervalsBeginEnv</tt></a></h3> +<blockquote> +<p>Default definition:</p> +<pre class="literal-block"> +\newcommand\PolPrintIntervalsBeginEnv{\[\begin{array}{rcccl}} +</pre> +</blockquote> +</div> +<div class="section" id="polprintintervalsendenv"> +<h3><a class="toc-backref" href="#id89"><tt class="docutils literal">\PolPrintIntervalsEndEnv</tt></a></h3> +<blockquote> +<p>Default definition:</p> +<pre class="literal-block"> +\newcommand\PolPrintIntervalsEndEnv{\end{array}\]} +</pre> +</blockquote> +</div> +<div class="section" id="polprintintervalsknownroot"> +<h3><a class="toc-backref" href="#id90"><tt class="docutils literal">\PolPrintIntervalsKnownRoot</tt></a></h3> +<blockquote> +<p>Default definition:</p> +<pre class="literal-block"> +\newcommand\PolPrintIntervalsKnownRoot{% + &&\PolPrintIntervalsTheVar_{\PolPrintIntervalsTheIndex}% + &=&\PolPrintIntervalsPrintExactZero +} +</pre> +</blockquote> +</div> +<div class="section" id="polprintintervalsunknownroot"> +<h3><a class="toc-backref" href="#id91"><tt class="docutils literal">\PolPrintIntervalsUnknownRoot</tt></a></h3> +<blockquote> +<p>Default definition:</p> +<pre class="literal-block"> +\newcommand\PolPrintIntervalsUnknownRoot{% + \PolPrintIntervalsPrintLeftEndPoint&<&% + \PolPrintIntervalsTheVar_{\PolPrintIntervalsTheIndex}&<&% + \PolPrintIntervalsPrintRightEndPoint +} +</pre> +</blockquote> +</div> +<div class="section" id="id15"> +<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"> +\newcommand\PolPrintIntervalsPrintExactZero{\PolPrintIntervalsTheLeftEndPoint} +</pre> +</blockquote> +</div> +<div class="section" id="id16"> +<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"> +\newcommand\PolPrintIntervalsPrintLeftEndPoint{\PolPrintIntervalsTheLeftEndPoint} +</pre> +</blockquote> +</div> +<div class="section" id="id17"> +<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"> +\newcommand\PolPrintIntervalsPrintRightEndPoint{\PolPrintIntervalsTheRightEndPoint} +</pre> +</blockquote> +</div> +</div> +<div class="section" id="id19"> +<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 +example of customization.</p> +<p>As replacement for <a class="reference internal" href="#polprintintervalsknownroot">\PolPrintIntervalsKnownRoot</a>, +<a class="reference internal" href="#polprintintervalsprintexactzero">\PolPrintIntervalsPrintExactZero</a>, +<a class="reference internal" href="#polprintintervalsunknownroot">\PolPrintIntervalsUnknownRoot</a> it uses its own +<tt class="docutils literal"><span class="pre">\POL@@PrintIntervals...</span></tt> macros. We only reproduce here one +definition:</p> +<pre class="literal-block"> +\newcommand\POL@@PrintIntervalsPrintExactZero{% + \displaystyle + \xintSignedFrac{\PolPrintIntervalsTheLeftEndPoint}% +}% +</pre> +<p>Multiplicities are printed using this auxiliary macro:</p> +</blockquote> +<div class="section" id="polprintintervalsprintmultiplicity"> +<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"> +\newcommand\PolPrintIntervalsPrintMultiplicity{(\mbox{mult. }\PolPrintIntervalsTheMultiplicity)} +</pre> +</blockquote> +</div> +</div> +<div class="section" id="polmapcoeffs-macro-polname"> +<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 +<tt class="docutils literal">\macro</tt>. The degree is adjusted as necessary if some leading +coefficients vanish after the operation. In replacement text of +<tt class="docutils literal">\macro</tt>, <tt class="docutils literal">\index</tt> expands to the coefficient index (which is +defined to be zero for the constant term).</p> +<p>Notice that <tt class="docutils literal">\macro</tt> will have to handle inputs of the shape +<tt class="docutils literal">A/B[N]</tt> (<a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a> internal notation). This means that it probably +will have to be expressed in terms of macros from <a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a> package.</p> +<p>Example:</p> +<pre class="literal-block"> +\def\foo#1{\xintMul{#1}{\the\numexpr\index*\index\relax}} +</pre> +<p>(or with <tt class="docutils literal"><span class="pre">\xintSqr{\index}</span></tt>) to replace <tt class="docutils literal">n</tt>-th coefficient +<tt class="docutils literal">f_n</tt> by <tt class="docutils literal">f_n*n^2</tt>.</p> +</blockquote> +</div> +<div class="section" id="polreducecoeffs-polname"> +<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 +polynomial function is used for computations.) This is a +one-argument macro, working 'in-place'.</blockquote> +</div> +<div class="section" id="id21"> +<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 +coefficient is internally <tt class="docutils literal">A/B[N]</tt>, then <tt class="docutils literal">A/B</tt> is reduced to +smallest terms, but the <tt class="docutils literal">10^N</tt> part is kept as is. Note: if the +polynomial is freshly defined directly via <a class="reference internal" href="#polfromcsv">\PolFromCSV</a> its coefficients might still be internally in some +format like <tt class="docutils literal">1.5e7</tt>; the macro will anyhow always first do the +needed conversion to strict format <tt class="docutils literal">A/B[N]</tt>.</p> +<p>Evaluations with polynomials treated by this can be much faster than +with those handled by the non-starred variant +<a class="reference internal" href="#polreducecoeffs-polname">\PolReduceCoeffs{polname}</a>: as the numerators and denominators +remain smaller, this proves very beneficial in favorable cases +(especially when the coefficients are decimal numbers) to the +expansion speed of the <a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a> macros used internally by +<a class="reference internal" href="#polevalat">\PolEval</a>.</p> +</blockquote> +</div> +<div class="section" id="polmakemonic-polname"> +<span id="polmakemonic"></span><h2><a class="toc-backref" href="#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 +done automatically, due to the case the original polynomial had integer +coefficients and we want to keep the leading one as common +denominator.</blockquote> +</div> +<div class="section" id="polmakeprimitive-polname"> +<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 +coefficient is not modified.</blockquote> +</div> +</div> +<div class="section" id="expandable-macros"> +<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="#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="#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="#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="#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 +an irreducible fraction.</blockquote> +</div> +<div class="section" id="polfloateval-polname-atexpr-numerical-expression"> +<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 +<a class="reference internal" href="#polgenfloatvariant-polname">\PolGenFloatVariant{polname}</a>), with already rounded +coefficients. <a class="footnote-reference" href="#id24" id="id22">[2]</a> To use the <em>exact coefficients</em> with <em>exactly +executed</em> additions and multiplications, just insert it in the float +expression as in this example: <a class="footnote-reference" href="#id25" id="id23">[3]</a></p> +<pre class="literal-block"> +\xintthefloatexpr 3.27*\xintexpr f(2.53)\relax^2\relax +</pre> +<p>The <tt class="docutils literal">f(2.53)</tt> is exactly computed then rounded at the time of +getting raised to the power <tt class="docutils literal">2</tt>. Moving the <tt class="docutils literal">^2</tt> inside, that +operation would also be treated exactly.</p> +<table class="docutils footnote" frame="void" id="id24" rules="none"> +<colgroup><col class="label" /><col /></colgroup> +<tbody valign="top"> +<tr><td class="label"><a class="fn-backref" href="#id22">[2]</a></td><td>Anyway each floating point operation starts by rounding its +operands to the floating point precision.</td></tr> +</tbody> +</table> +<table class="docutils footnote" frame="void" id="id25" rules="none"> +<colgroup><col class="label" /><col /></colgroup> +<tbody valign="top"> +<tr><td class="label"><a class="fn-backref" href="#id23">[3]</a></td><td>The <tt class="docutils literal">\xintexpr</tt> here could be <tt class="docutils literal">\xinttheexpr</tt> but that +would be less efficient. Cf. <a class="reference external" href="http://www.ctan.org/pkg/xint">xintexpr</a> documentation about +nested expressions.</td></tr> +</tbody> +</table> +</blockquote> +</div> +<div class="section" id="polfloateval-polname-at-fraction"> +<span id="polfloatevalat"></span><h2><a class="toc-backref" href="#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 +a floating point number.</blockquote> +</div> +<div class="section" id="polifcoeffisplusorminusone-a-b"> +<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 +used if needed in the execution of <a class="reference internal" href="#poltypesetmonomialcmd">\PolTypesetMonomialCmd</a>, +e.g. to insert a <tt class="docutils literal">\cdot</tt> in front of <tt class="docutils literal"><span class="pre">\PolVar^{\PolIndex}</span></tt> if +the coefficient is not plus or minus one.</p> +<p>The macro will execute <tt class="docutils literal">A</tt> if the coefficient has been found to be +plus or minus one, and <tt class="docutils literal">B</tt> if not.</p> +</blockquote> +</div> +<div class="section" id="polleadingcoeff-polname"> +<span id="polleadingcoeff"></span><h2><a class="toc-backref" href="#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="#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 +leading coefficients.</blockquote> +</div> +<div class="section" id="poldegree-polname"> +<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="#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 +with integer coefficients having no common prime divisor.</p> +<p>See <a class="reference internal" href="#polmakeprimitive">\PolMakePrimitive</a>.</p> +</blockquote> +</div> +<div class="section" id="poltoexpr-polname"> +<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"> +<colgroup><col class="label" /><col /></colgroup> +<tbody valign="top"> +<tr><td class="label"><a class="fn-backref" href="#id26">[4]</a></td><td>in a <tt class="docutils literal">\write</tt>, <tt class="docutils literal">\edef</tt>, or <tt class="docutils literal"><span class="pre">\csname...\endcsname</span></tt>, but +not under <tt class="docutils literal"><span class="pre">\romannumeral-`0</span></tt>.</td></tr> +</tbody> +</table> +<p>By default zero coefficients are skipped (issue <tt class="docutils literal">\poltoexpralltrue</tt> to +get all of them in output).</p> +<p>By default, no <tt class="docutils literal">+</tt> sign before negative coefficients, for +compliance with Maple input format (but see +<a class="reference internal" href="#poltoexprtermprefix-raw-coeff">\PolToExprTermPrefix{raw_coeff}</a>.) Also, like the default +behaviour of <a class="reference internal" href="#poltypeset-polname">\PolTypeset{polname}</a>, does not print (for the non +constant terms) coefficients equal to plus or minus one. The degree +one monomial is output as <tt class="docutils literal">x</tt>, not <tt class="docutils literal">x^1</tt>. Complete customization is +possible, see next macros.</p> +<p>Of course <tt class="docutils literal">\PolToExpr{f}</tt> can be inserted in a <tt class="docutils literal">\poldef</tt>, as the +latter expands token by token, hence will force complete expansion +of <tt class="docutils literal">\PolToExpr{f}</tt>, but a simple <tt class="docutils literal">f(x)</tt> is more efficient for +the identical result.</p> +</blockquote> +<div class="section" id="poltoexproneterm-raw-coeff-number"> +<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 +coefficients of absolute value <tt class="docutils literal">1</tt> not be printed explicitely +(except of course for the constant term). Also by default, the +monomial of degree one is <tt class="docutils literal">x</tt> not <tt class="docutils literal">x^1</tt>, and <tt class="docutils literal">x^0</tt> is skipped.</p> +<p>For compatibility with Maple input requirements, by default a <tt class="docutils literal">*</tt> +always precedes the <tt class="docutils literal">x^number</tt>, except if the coefficient is a one +or a minus one. See <a class="reference internal" href="#poltoexprtimes">\PolToExprTimes</a>.</p> +</blockquote> +</div> +<div class="section" id="poltoexpronetermstylea-raw-coeff-number"> +<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="#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"> +2*x^11/3+3*x^8/7-x^5-x^4/4-x^3-x^2/2-2*x+1 +</pre> +<p>issue <tt class="docutils literal">\let\PolToExprOneTerm\PolToExprOneTermStyleB</tt> before usage of +<tt class="docutils literal">\PolToExpr</tt>. Note that then <tt class="docutils literal">\PolToExprCmd</tt> isn't used at all. +To revert to package default, issue +<tt class="docutils literal">\let\PolToExprOneTerm\PolToExprOneTermStyleA</tt>.</p> +<p>To suppress the <tt class="docutils literal">*</tt>'s, cf. <a class="reference internal" href="#poltoexprtimes">\PolToExprTimes</a>.</p> +</blockquote> +</div> +<div class="section" id="poltoexprcmd-raw-coeff"> +<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 +equal to plus or minus one), and it defaults to +<tt class="docutils literal"><span class="pre">\xintPRaw{\xintRawWithZeros{#1}}</span></tt>. One will have to redefine it +to <tt class="docutils literal"><span class="pre">\xintIrr{#1}</span></tt> or to <tt class="docutils literal"><span class="pre">\xintPRaw{\xintIrr{#1}}</span></tt> to obtain in the +output forcefully reduced coefficients.</blockquote> +</div> +<div class="section" id="poltoexprtermprefix-raw-coeff"> +<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 +they don't carry one by themselves.</blockquote> +</div> +<div class="section" id="id28"> +<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 +is <tt class="docutils literal">x</tt>.</blockquote> +</div> +<div class="section" id="id29"> +<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 +<tt class="docutils literal">*</tt>. Redefine the macro to expand to nothing to get rid of it (but +this will give output incompatible with some professional computer +algebra software).</blockquote> +</div> +</div> +<div class="section" id="id31"> +<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="#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 +point format.</p> +<div class="admonition note"> +<p class="first admonition-title">Note</p> +<p>It is not necessary to have issued +<a class="reference internal" href="#polgenfloatvariant-polname">\PolGenFloatVariant{polname}</a>. The rounded coefficients are +not easily recoverable from the <tt class="docutils literal">\xintfloatexpr</tt> polynomial +function hence <tt class="docutils literal">\PolToFloatExprCmd</tt> operates from the <em>exact</em> +coefficients anew.</p> +<p class="last">Attention that both macros obey the prevailing float precision. +If it is changed between those macro calls, then a mismatch +exists between the coefficients as used in <tt class="docutils literal">\xintfloatexpr</tt> and +those output by <tt class="docutils literal">\PolToFloatExpr{polname}</tt>.</p> +</div> +</blockquote> +<div class="section" id="poltofloatexproneterm-raw-coeff-number"> +<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="#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> +<div class="admonition caution"> +<p class="first admonition-title">Caution!</p> +<p>Currently (<a class="reference external" href="http://www.ctan.org/pkg/xint">xint</a> <tt class="docutils literal">1.3c</tt>) <tt class="docutils literal">\xintFloat{0}</tt> outputs <tt class="docutils literal">0.e0</tt> +which is perfectly acceptable input for Python, but not for +Maple. Thus, one should better leave the <a class="reference internal" href="#poltoexprallfalse">\poltoexprallfalse</a> +toggle to its default <tt class="docutils literal">\iffalse</tt> state, if one intends to use +the output in a Maple worksheet.</p> +<p>But even then the zero polynomial will cause a problem. Workaround:</p> +<pre class="literal-block"> +\renewcommand\PolToFloatExprCmd[1]{\xintiiifZero{#1}{0.0}{\xintFloat{#1}}} +</pre> +<p class="last">Usage of <tt class="docutils literal">\xintiiifZero</tt> and not <tt class="docutils literal">\xintifZero</tt> is only for +optimization (I can't help it) because <tt class="docutils literal">#1</tt> is known to be +in <tt class="docutils literal">xintfrac</tt> raw format.</p> +</div> +</blockquote> +</div> +</div> +<div class="section" id="id35"> +<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="#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 +(the zero polynomial does give <tt class="docutils literal">{0/1[0]}</tt> and not an +empty output.)</blockquote> +</div> +<div class="section" id="poltocsv-polname"> +<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 +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="#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> +<p>See <a class="reference internal" href="#poltosturm-polname-sturmname">\PolToSturm{polname}{sturmname}</a>.</p> +</blockquote> +</div> +<div class="section" id="polsturmifzeroexactlyknown-sturmname-index-a-b"> +<span id="polsturmifzeroexactlyknown"></span><h2><a class="toc-backref" href="#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> +<div class="admonition note"> +<p class="first admonition-title">Note</p> +<p class="last"><tt class="docutils literal">index</tt> is allowed to be something like <tt class="docutils literal">1+2*3</tt> as it is fed +to <tt class="docutils literal"><span class="pre">\the\numexpr...\relax</span></tt>.</p> +</div> +</blockquote> +</div> +<div class="section" id="polsturmisolatedzeroleft-sturmname-index"> +<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> +<div class="admonition note"> +<p class="first admonition-title">Note</p> +<p class="last">Of course, this is kept updated by macros such as +<a class="reference internal" href="#polrefineinterval-n">\PolRefineInterval{sturmname}{index}</a>.</p> +</div> +<p>The value is pre-formatted using <a class="reference internal" href="#poldectostring">\PolDecTostring</a>.</p> +</blockquote> +</div> +<div class="section" id="polsturmisolatedzeroright-sturmname-index"> +<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 +possibly refined afterwards.</p> +<p>The value is pre-formatted using <a class="reference internal" href="#poldectostring">\PolDecTostring</a>.</p> +</blockquote> +</div> +<div class="section" id="polsturmisolatedzeromultiplicity-sturmname-index"> +<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> +<div class="admonition attention"> +<p class="first admonition-title">Attention!</p> +<p class="last">A prior execution of <a class="reference internal" href="#id12">\PolSturmIsolateZeros*{sturmname}</a> is mandatory.</p> +</div> +<p>See <a class="reference internal" href="#the-degree-nine-polynomial-with-0-99-0-999-0-9999-as-triple-roots">The degree nine polynomial with 0.99, 0.999, 0.9999 as triple +roots</a> for an example of use.</p> +</blockquote> +</div> +<div class="section" id="polsturmnbofisolatedzeros-sturmname"> +<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 +polynomial originally used to create the Sturm chain via +<a class="reference internal" href="#poltosturm-polname-sturmname">\PolToSturm{polname}{sturmname}</a>.</blockquote> +<div class="admonition warning"> +<p class="first admonition-title">Warning</p> +<p class="last">The next few macros counting roots, with or without multiplicities, +less than or equal to some value, are under evaluation and may be +removed from the package if their utility is judged to be not high +enough. They can be re-coded at user level on the basis of the other +documented package macros anyway.</p> +</div> +<div class="section" id="polsturmnbofrootsof-sturmname-lessthanorequalto-value"> +<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 +number of fraction recognizable by the <a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a> macros).</p> +<div class="admonition attention"> +<p class="first admonition-title">Attention!</p> +<p><a class="reference internal" href="#polsturmisolatezeros-sturmname">\PolSturmIsolateZeros{sturmname}</a> must have been executed +beforehand.</p> +<p class="last">And the argument is a <tt class="docutils literal">sturmname</tt>, not a <tt class="docutils literal">polname</tt> (this is +why the macro contains Sturm in its name), simply to be reminded +of the above constraint.</p> +</div> +</blockquote> +</div> +<div class="section" id="polsturmnbofrootsof-sturmname-lessthanorequaltoexpr-expression"> +<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 +given <tt class="docutils literal">expression</tt>.</p> +<div class="admonition attention"> +<p class="first admonition-title">Attention!</p> +<p class="last"><a class="reference internal" href="#polsturmisolatezeros-sturmname">\PolSturmIsolateZeros{sturmname}</a> must have been executed +beforehand.</p> +</div> +</blockquote> +</div> +<div class="section" id="polsturmnbwithmultofrootsof-sturmname-lessthanorequalto-value"> +<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 +or equal to the given <tt class="docutils literal">value</tt>.</p> +<div class="admonition attention"> +<p class="first admonition-title">Attention!</p> +<p class="last"><a class="reference internal" href="#id12">\PolSturmIsolateZeros*{sturmname}</a> (or the double starred +variant) must have been executed beforehand.</p> +</div> +</blockquote> +</div> +<div class="section" id="polsturmnbwithmultofrootsof-sturmname-lessthanorequaltoexpr-expression"> +<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> +<div class="admonition attention"> +<p class="first admonition-title">Attention!</p> +<p class="last"><a class="reference internal" href="#id12">\PolSturmIsolateZeros*{sturmname}</a> (or the double starred +variant) must have been executed beforehand.</p> +</div> +</blockquote> +</div> +</div> +<div class="section" id="polsturmnbofrationalroots-sturmname"> +<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"> +<p class="first admonition-title">Attention!</p> +<p class="last"><a class="reference internal" href="#id14">\PolSturmIsolateZeros**{sturmname}</a> must have been executed +beforehand.</p> +</div> +</blockquote> +</div> +<div class="section" id="polsturmnbofrationalrootswithmultiplicities-sturmname"> +<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"> +<p class="first admonition-title">Attention!</p> +<p class="last"><a class="reference internal" href="#id14">\PolSturmIsolateZeros**{sturmname}</a> must have been executed +beforehand.</p> +</div> +</blockquote> +</div> +<div class="section" id="polsturmrationalroot-sturmname-k"> +<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> +<div class="admonition attention"> +<p class="first admonition-title">Attention!</p> +<p class="last"><a class="reference internal" href="#id14">\PolSturmIsolateZeros**{sturmname}</a> must have been executed +beforehand.</p> +</div> +</blockquote> +</div> +<div class="section" id="polsturmrationalrootindex-sturmname-k"> +<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 +<a class="reference internal" href="#polsturmrationalroot-sturmname-k">\PolSturmRationalRoot{sturmname}{k}</a> is equivalent to this +nested call:</p> +<pre class="literal-block"> +\PolSturmIsolatedZeroLeft{sturmname}{\PolSturmRationalRootIndex{sturmname}{k}} +</pre> +<div class="admonition attention"> +<p class="first admonition-title">Attention!</p> +<p class="last"><a class="reference internal" href="#id14">\PolSturmIsolateZeros**{sturmname}</a> must have been executed +beforehand.</p> +</div> +</blockquote> +</div> +<div class="section" id="polsturmrationalrootmultiplicity-sturmname-k"> +<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"> +<p class="first admonition-title">Attention!</p> +<p class="last"><a class="reference internal" href="#id14">\PolSturmIsolateZeros**{sturmname}</a> must have been executed +beforehand.</p> +</div> +</blockquote> +</div> +<div class="section" id="polintervalwidth-sturmname-index"> +<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="#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 +default for the former) and of <a class="reference internal" href="#polprintintervalsprintleftendpoint">\PolPrintIntervalsPrintLeftEndPoint</a>, +<a class="reference internal" href="#polprintintervalsprintrightendpoint">\PolPrintIntervalsPrintRightEndPoint</a> (used in the default for the +latter).</p> +<div class="admonition attention"> +<p class="first admonition-title">Attention!</p> +<p class="last">Some macros formerly mentioned here got removed at 0.7: +<tt class="docutils literal">\PolPrintIntervalsTheEndPoint</tt>, +<tt class="docutils literal"><span class="pre">\PolIfEndPointIsPositive{A}{B}</span></tt>, +<tt class="docutils literal"><span class="pre">\PolIfEndPointIsNegative{A}{B}</span></tt>, +<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="#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 +<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="#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="#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="#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 +used with arguments the Sturm chain name and interval index returned +by <a class="reference internal" href="#polprintintervalsthesturmname">\PolPrintIntervalsTheSturmName</a> and +<a class="reference internal" href="#polprintintervalstheindex">\PolPrintIntervalsTheIndex</a>.</blockquote> +</div> +<div class="section" id="polprintintervalstherightendpoint"> +<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 +this Sturm chain name and index.</blockquote> +</div> +<div class="section" id="polprintintervalsthemultiplicity"> +<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 +double-starred) variant of <a class="reference internal" href="#polsturmisolatezeros">\PolSturmIsolateZeros</a> was used earlier.</blockquote> +</div> +</div> +<div class="section" id="poldectostring-decimal-number"> +<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 +the name <tt class="docutils literal">\xintDecToString</tt>, and the <tt class="docutils literal">polexpr</tt> macro is simply +now an alias to it.</p> +<p>For example +<tt class="docutils literal"><span class="pre">\PolDecToString{123.456e-8}</span></tt> will expand to <tt class="docutils literal">0.00000123456</tt> +and <tt class="docutils literal"><span class="pre">\PolDecToString{123.450e-8}</span></tt> to <tt class="docutils literal">0.00000123450</tt> which +illustrates that trailing zeros are not trimmed. To trim trailing +zeroes, one can use <tt class="docutils literal"><span class="pre">\PolDecToString{\xintREZ{#1}}</span></tt>.</p> +<p>The precise behaviour of this macro may evolve in future releases of +<a class="reference external" href="http://www.ctan.org/pkg/xint">xint</a>.</p> +</blockquote> +</div> +</div> +<div class="section" id="booleans-with-default-setting-as-indicated"> +<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="#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 +polynomials are defined.</p> +<div class="admonition caution"> +<p class="first admonition-title">Caution!</p> +<p class="last">The macro meanings as written to the log are to be considered +unstable and undocumented internal structures.</p> +</div> +</blockquote> +</div> +<div class="section" id="poltypesetallfalse"> +<h2><a class="toc-backref" href="#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="#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> +</div> +</div> +<div class="section" id="polexprsetup"> +<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> +<ul class="simple"> +<li><tt class="docutils literal">norr</tt>: the postfix that <a class="reference internal" href="#id14">\PolSturmIsolateZeros**{sturmname}</a> +should append to <tt class="docutils literal">sturmname</tt> to declare the primitive polynomial +obtained from original one after removal of all rational roots. +The default value is <tt class="docutils literal">_norr</tt> (standing for “no rational roots”).</li> +<li><tt class="docutils literal">sqfnorr</tt>: the postfix that <a class="reference internal" href="#id14">\PolSturmIsolateZeros**{sturmname}</a> +should append to <tt class="docutils literal">sturmname</tt> to declare the primitive polynomial +obtained from original one after removal of all rational roots and +suppression of all multiplicities. +The default value is <tt class="docutils literal">_sqf_norr</tt> (standing for “square-free with +no rational roots”).</li> +</ul> +<p>The package executes <tt class="docutils literal">\polexprsetup{norr=_norr, +sqfnorr=_sqf_norr}</tt> as default.</p> +</blockquote> +</div> +<div class="section" id="technicalities"> +<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 +whole thing was already part of a macro argument, in such cases one +can use <a class="reference internal" href="#id2">\PolDef{f}{P(x)}</a> +rather. The colon in <tt class="docutils literal">:=</tt> may be active with no consequences.</p> +</li> +<li><p class="first">As a consequence of <a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a> addition and subtraction always using +least common multiples for the denominators <a class="footnote-reference" href="#id37" id="id36">[5]</a>, user-chosen common +denominators survive additions and multiplications. For example, this:</p> +<pre class="literal-block"> +\poldef P(x):= 1/2 + 2/2*x + 3/2*x^3 + 4/2*x^4; +\poldef Q(x):= 1/3 + (2/3)x + (3/3)x^3 + (4/3)x^4; +\poldef PQ(x):= P(x)Q(x); +</pre> +<p>gives internally the polynomial:</p> +<pre class="literal-block"> +1/6+4/6*x^1+4/6*x^2+6/6*x^3+20/6*x^4+16/6*x^5+9/6*x^6+24/6*x^7+16/6*x^8 +</pre> +<p>where all coefficients have the same denominator 6. Notice though that +<tt class="docutils literal">\PolToExpr{PQ}</tt> outputs the <tt class="docutils literal">6/6*x^3</tt> as <tt class="docutils literal">x^3</tt> because (by +default) it recognizes and filters out coefficients equal to one or +minus one (since release <tt class="docutils literal">0.3</tt>). One can use for example +<tt class="docutils literal">\PolToCSV{PQ}</tt> to see the internally stored coefficients.</p> +<table class="docutils footnote" frame="void" id="id37" rules="none"> +<colgroup><col class="label" /><col /></colgroup> +<tbody valign="top"> +<tr><td class="label"><a class="fn-backref" href="#id36">[5]</a></td><td><p class="first last">prior to <tt class="docutils literal">0.4.1</tt>, <tt class="docutils literal">polexpr</tt> used to temporarily patch +during the parsing of polynomials the <a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a> macros. This +patch was backported to <a class="reference external" href="http://www.ctan.org/pkg/xint">xint</a> at release <tt class="docutils literal">1.3</tt>.</p> +</td></tr> +</tbody> +</table> +</li> +<li><p class="first"><a class="reference internal" href="#poldiff-polname-1-polname-2">\PolDiff{polname_1}{polname_2}</a> always applies <tt class="docutils literal">\xintIrr</tt> to the +resulting coefficients, except that the <em>power of ten</em> part <tt class="docutils literal">[N]</tt> +(for example an input in scientific notation such as <tt class="docutils literal">1.23e5</tt> gives +<tt class="docutils literal">123/1[3]</tt> internally in xintfrac) is not taken into account in the +reduction of the fraction. This is tentative and may change.</p> +<p>Same remark for <a class="reference internal" href="#polantidiff-polname-1-polname-2">\PolAntiDiff{polname_1}{polname_2}</a>.</p> +</li> +<li><p class="first">Currently, the package stores all coefficients from index <tt class="docutils literal">0</tt> to +index equal to the polynomial degree inside a single macro, as a list. +This data structure is obviously very inefficient for polynomials of +high degree and few coefficients (as an example with <tt class="docutils literal">\poldef +<span class="pre">f(x):=x^1000</span> + x^500;</tt> the subsequent definition <tt class="docutils literal">\poldef <span class="pre">g(x):=</span> +<span class="pre">f(x)^2;</span></tt> will do of the order of 1,000,000 multiplications and +additions involvings only zeroes... which does take time). This +may change in the future.</p> +</li> +<li><p class="first">As is to be expected internal structures of the package are barely +documented and unstable. Don't use them.</p> +</li> +</ul> +</div> +<div class="section" id="change-log"> +<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"> +<li>The <a class="reference internal" href="#poldef">\poldef</a> parser itself,</li> +<li>Differentiation and anti-differentiation,</li> +<li>Euclidean division and GCDs,</li> +<li>Various utilities such as <a class="reference internal" href="#polfromcsv">\PolFromCSV</a>, +<a class="reference internal" href="#polmapcoeffs">\PolMapCoeffs</a>, +<a class="reference internal" href="#poltocsv">\PolToCSV</a>, <a class="reference internal" href="#poltoexpr">\PolToExpr</a>, ...</li> +</ul> +<p>Only one-variable polynomials so far.</p> +</li> +<li><p class="first">v0.2 (2018/01/14)</p> +<ul class="simple"> +<li>Fix: <tt class="docutils literal">"README thinks \numexpr recognizes ^ operator"</tt>.</li> +<li>Convert README to reStructuredText markup.</li> +<li>Move main documentation from README to separate <tt class="docutils literal">polexpr.txt</tt> file.</li> +<li>Provide <tt class="docutils literal">polexpr.html</tt> as obtained via <a class="reference external" href="http://docutils.sourceforge.net/docs/index.html">DocUtils</a> <tt class="docutils literal">rst2html.py</tt>.</li> +<li>Convert README to (CTAN compatible) Markdown markup.</li> +</ul> +<p>Due to lack of available time the test suite might not be extensive +enough. Bug reports are very welcome!</p> +</li> +<li><p class="first">v0.3 (2018/01/17)</p> +<ul> +<li><p class="first">bug fixes:</p> +<ul> +<li><p class="first">the <tt class="docutils literal">0.1</tt> <a class="reference internal" href="#polevalat">\PolEval</a> accepted expressions for its second +argument, but this was removed by mistake at <tt class="docutils literal">0.2</tt>. Restored.</p> +<p><strong>Attention</strong>: at <tt class="docutils literal">0.4</tt> this has been reverted again, and +<a class="reference internal" href="#polevalatexpr">\PolEval{P}\AtExpr{foo}</a> syntax is needed for +using expressions in the second argument.</p> +</li> +</ul> +</li> +<li><p class="first">incompatible or breaking changes:</p> +<ul class="simple"> +<li><a class="reference internal" href="#poltoexpr">\PolToExpr</a> now by default uses <em>descending</em> +powers (it also treats differently coefficients equal to 1 or -1.) +Use <a class="reference internal" href="#id30">\PolToExpr*</a> for <em>ascending</em> powers.</li> +<li><a class="reference internal" href="#polevalat">\PolEval</a> reduced the output to smallest terms, +but as this is costly with big fractions and not needed if e.g. +wrapped in an <tt class="docutils literal">\xintRound</tt> or <tt class="docutils literal">\xintFloat</tt>, this step has been +removed; the former meaning is available as <a class="reference internal" href="#polevalreducedat">\PolEvalReduced</a>.</li> +</ul> +</li> +<li><p class="first">new (or newly documented) macros:</p> +<ul class="simple"> +<li><a class="reference internal" href="#poltypesetcmd">\PolTypesetCmd</a></li> +<li><a class="reference internal" href="#poltypesetcmdprefix">\PolTypesetCmdPrefix</a></li> +<li><a class="reference internal" href="#poltypesetmonomialcmd">\PolTypesetMonomialCmd</a></li> +<li><a class="reference internal" href="#polevalreducedat">\PolEvalReducedAt</a></li> +<li><a class="reference internal" href="#poltofloatexpr">\PolToFloatExpr</a></li> +<li><a class="reference internal" href="#poltoexproneterm">\PolToExprOneTerm</a></li> +<li><a class="reference internal" href="#poltofloatexproneterm">\PolToFloatExprOneTerm</a></li> +<li><a class="reference internal" href="#poltoexprcmd">\PolToExprCmd</a></li> +<li><a class="reference internal" href="#id33">\PolToFloatExprCmd</a></li> +<li><a class="reference internal" href="#poltoexprtermprefix">\PolToExprTermPrefix</a></li> +<li><a class="reference internal" href="#poltoexprvar">\PolToExprVar</a></li> +<li><a class="reference internal" href="#poltoexprtimes">\PolToExprTimes</a></li> +</ul> +</li> +<li><p class="first">improvements:</p> +<ul> +<li><p class="first">documentation has a table of contents, internal hyperlinks, +standardized signature notations and added explanations.</p> +</li> +<li><p class="first">one can do <tt class="docutils literal"><span class="pre">\PolLet{g}={f}</span></tt> or <tt class="docutils literal"><span class="pre">\PolLet{g}{f}</span></tt>.</p> +</li> +<li><p class="first"><tt class="docutils literal">\PolToExpr{f}</tt> is highly customizable.</p> +</li> +<li><p class="first"><a class="reference internal" href="#poldef">\poldef</a> and other defining macros prepare the polynomial +functions for usage within <tt class="docutils literal">\xintthefloatexpr</tt> (or +<tt class="docutils literal">\xintdeffloatvar</tt>). Coefficients are pre-rounded to the +floating point precision. Indispensible for numerical algorithms, +as exact fractions, even reduced, quickly become very big. See the +documentation about how to use the exact polynomials also in +floating point context.</p> +<p><strong>Attention</strong>: this has been reverted at <tt class="docutils literal">0.4</tt>. The macro +<a class="reference internal" href="#polgenfloatvariant">\PolGenFloatVariant</a> must be used for +generation floating point polynomial functions.</p> +</li> +</ul> +</li> +</ul> +</li> +<li><p class="first">v0.3.1 (2018/01/18)</p> +<p>Fixes two typos in example code included in the documentation.</p> +</li> +<li><p class="first">v0.4 (2018/02/16)</p> +<ul> +<li><p class="first">bug fixes:</p> +<ul class="simple"> +<li>when Euclidean division gave a zero remainder, the internal +representation of this zero polynomial could be faulty; this +could cause mysterious bugs in conjunction with other package +macros such as <a class="reference internal" href="#polmapcoeffs">\PolMapCoeffs</a>.</li> +<li><a class="reference internal" href="#polgcd">\PolGCD</a> was buggy in case of first polynomial being +of lesser degree than the second one.</li> +</ul> +</li> +<li><p class="first">breaking changes:</p> +<ul> +<li><p class="first">formerly <a class="reference internal" href="#polevalat">\PolEval{P}\At{foo}</a> allowed <tt class="docutils literal">foo</tt> to +be an expression, which was transparently handled via +<tt class="docutils literal">\xinttheexpr</tt>. Now, <tt class="docutils literal">foo</tt> must be a fraction (or a macro +expanding to such) in the format acceptable by <tt class="docutils literal">xintfrac.sty</tt> +macros. Use <a class="reference internal" href="#polevalatexpr">\PolEval{P}\AtExpr{foo}</a> for more +general arguments using expression syntax. E.g., if <tt class="docutils literal">foo</tt> is the +name of a variable known to <tt class="docutils literal">\xintexpr</tt>.</p> +<p>The same holds for <a class="reference internal" href="#polevalreducedat">\PolEvalReduced</a> +and <a class="reference internal" href="#polfloatevalat">\PolFloatEval</a>.</p> +</li> +<li><p class="first">the <tt class="docutils literal">3.0</tt> automatic generation of floating point variants has +been reverted. Not only do <em>not</em> the package macros automatically +generate floating point variants of newly created polynomials, +they actually make pre-existing such variant undefined.</p> +<p>See <a class="reference internal" href="#polgenfloatvariant">\PolGenFloatVariant</a>.</p> +</li> +</ul> +</li> +<li><p class="first">new non-expandable macros:</p> +<ul class="simple"> +<li><a class="reference internal" href="#polgenfloatvariant">\PolGenFloatVariant</a></li> +<li><a class="reference internal" href="#polgloballet">\PolGlobalLet</a></li> +<li><a class="reference internal" href="#poltypesetone">\PolTypesetOne</a></li> +<li><a class="reference internal" href="#polquo">\PolQuo</a></li> +<li><a class="reference internal" href="#polrem">\PolRem</a></li> +<li><a class="reference internal" href="#poltosturm">\PolToSturm</a></li> +<li><a class="reference internal" href="#id9">\PolToSturm*</a></li> +<li><a class="reference internal" href="#polsettosturmchainsignchangesat">\PolSetToSturmChainSignChangesAt</a></li> +<li><a class="reference internal" href="#polsettonbofzeroswithin">\PolSetToNbOfZerosWithin</a></li> +<li><a class="reference internal" href="#polsturmisolatezeros">\PolSturmIsolateZeros</a></li> +<li><a class="reference internal" href="#polrefineinterval">\PolRefineInterval*</a></li> +<li><a class="reference internal" href="#polrefineinterval-n">\PolRefineInterval[N]</a></li> +<li><a class="reference internal" href="#polensureintervallength">\PolEnsureIntervalLength</a></li> +<li><a class="reference internal" href="#polensureintervallengths">\PolEnsureIntervalLengths</a></li> +<li><a class="reference internal" href="#polprintintervals">\PolPrintIntervals</a></li> +<li><a class="reference internal" href="#polprintintervalsprintexactzero">\PolPrintIntervalsPrintExactZero</a></li> +<li><a class="reference internal" href="#polprintintervalsprintleftendpoint">\PolPrintIntervalsPrintLeftEndPoint</a></li> +<li><a class="reference internal" href="#polprintintervalsprintrightendpoint">\PolPrintIntervalsPrintRightEndPoint</a></li> +<li><a class="reference internal" href="#id20">\PolReduceCoeffs*</a></li> +<li><a class="reference internal" href="#polmakemonic">\PolMakeMonic</a></li> +</ul> +</li> +<li><p class="first">new expandable macros:</p> +<ul class="simple"> +<li><a class="reference internal" href="#poltoexpronetermstylea">\PolToExprOneTermStyleA</a></li> +<li><a class="reference internal" href="#polifcoeffisplusorminusone">\PolIfCoeffIsPlusOrMinusOne</a></li> +<li><a class="reference internal" href="#polleadingcoeff">\PolLeadingCoeff</a></li> +<li><a class="reference internal" href="#polsturmchainlength">\PolSturmChainLength</a></li> +<li><a class="reference internal" href="#polsturmnbofisolatedzeros">\PolSturmNbOfIsolatedZeros</a></li> +<li><a class="reference internal" href="#polsturmifzeroexactlyknown">\PolSturmIfZeroExactlyKnown</a></li> +<li><a class="reference internal" href="#polsturmisolatedzeroleft">\PolSturmIsolatedZeroLeft</a></li> +<li><a class="reference internal" href="#polsturmisolatedzeroright">\PolSturmIsolatedZeroRight</a></li> +<li><tt class="docutils literal">\PolPrintIntervalsTheEndPoint</tt> (removed at 0.7)</li> +<li><a class="reference internal" href="#polprintintervalstheindex">\PolPrintIntervalsTheIndex</a></li> +<li><tt class="docutils literal">\PolIfEndPointIsPositive</tt> (removed at 0.7)</li> +<li><tt class="docutils literal">\PolIfEndPointIsNegative</tt> (removed at 0.7)</li> +<li><tt class="docutils literal">\PolIfEndPointIsZero</tt> (removed at 0.7)</li> +<li><a class="reference internal" href="#polintervalwidth">\PolIntervalWidth</a></li> +<li><a class="reference internal" href="#poldectostring">\PolDecToString</a></li> +</ul> +</li> +<li><p class="first">improvements:</p> +<p>The main new feature is implementation of the <a class="reference external" href="https://en.wikipedia.org/wiki/Sturm%27s_theorem">Sturm algorithm</a> +for localization of the real roots of polynomials.</p> +</li> +</ul> +</li> +<li><p class="first">v0.4.1 (2018/03/01)</p> +<p>Synced with xint 1.3.</p> +</li> +<li><p class="first">v0.4.2 (2018/03/03)</p> +<p>Documentation fix.</p> +</li> +<li><p class="first">v0.5 (2018/04/08)</p> +<ul class="simple"> +<li>bug fixes:<ul> +<li><a class="reference internal" href="#polget-polname-fromarray-macro">\PolGet{polname}\fromarray\macro</a> crashed when <tt class="docutils literal">\macro</tt> was +an <a class="reference external" href="http://www.ctan.org/pkg/xint">xinttools</a> array macro with no items. It now produces the zero +polynomial.</li> +</ul> +</li> +<li>breaking changes:<ul> +<li><a class="reference internal" href="#poltosturm">\PolToSturm</a> creates primitive integer coefficients polynomials. +This speeds up localization of roots via +<a class="reference internal" href="#polsturmisolatezeros">\PolSturmIsolateZeros</a>. In case of user protests the author +will make available again the code producing the bona fide Sturm +polynomials as used formerly.</li> +<li>polynomials created from <a class="reference internal" href="#polfromcsv">\PolFromCSV</a> or <a class="reference internal" href="#polget">\PolGet</a> +get their coefficients normalized via <a class="reference external" href="http://www.ctan.org/pkg/xint">xintfrac</a>'s <tt class="docutils literal">\xintRaw</tt>.</li> +</ul> +</li> +<li>experimental change:<ul> +<li>optional argument to <a class="reference internal" href="#polsturmisolatezeros">\PolSturmIsolateZeros</a> (see <a class="reference internal" href="#the-degree-41-polynomial-with-2-1-9-1-8-0-0-1-1-9-2-as-roots">The +degree 41 polynomial with -2, -1.9, -1.8, ..., 0, 0.1, ..., 1.9, 2 +as roots</a> for usage). It will presumably be replaced in future by +an interval specification.</li> +</ul> +</li> +<li>new non-expandable macro:<ul> +<li><a class="reference internal" href="#polmakeprimitive">\PolMakePrimitive</a></li> +</ul> +</li> +<li>new expandable macro:<ul> +<li><a class="reference internal" href="#policontent">\PolIContent</a></li> +</ul> +</li> +</ul> +</li> +<li><p class="first">v0.5.1 (2018/04/22)</p> +<ul class="simple"> +<li>new feature:<ul> +<li>the character <tt class="docutils literal">'</tt> can be used in polynomial names.</li> +</ul> +</li> +</ul> +</li> +<li><p class="first">v0.6 (2018/11/20)</p> +<ul class="simple"> +<li>bugfix:<ul> +<li>the starred variant <a class="reference internal" href="#id10">\PolToSturm*{polname}{sturmname}</a> was +broken. On the occasion of the fix, its meaning has been modified, +see its documentation.</li> +<li>using <a class="reference internal" href="#poltosturm">\PolToSturm</a> with a constant polynomial +caused a division by zero error.</li> +</ul> +</li> +<li>new macro:<ul> +<li><a class="reference internal" href="#id11">\PolSturmIsolateZeros*</a> +acts like the <a class="reference internal" href="#polsturmisolatezeros">non-starred variant</a> then computes all the multiplicities.</li> +</ul> +</li> +<li>new expandable macros:<ul> +<li><a class="reference internal" href="#polsturmisolatedzeromultiplicity-sturmname-index">\PolSturmIsolatedZeroMultiplicity{sturmname}{index}</a></li> +<li><a class="reference internal" href="#polsturmnbofrootsof-sturmname-lessthanorequalto-value">\PolSturmNbOfRootsOf{sturmname}\LessThanOrEqualTo{value}</a></li> +<li><a class="reference internal" href="#polsturmnbofrootsof-sturmname-lessthanorequaltoexpr-expression">\PolSturmNbOfRootsOf{sturmname}\LessThanOrEqualToExpr{expression}</a></li> +<li><a class="reference internal" href="#polsturmnbwithmultofrootsof-sturmname-lessthanorequalto-value">\PolSturmNbWithMultOfRootsOf{sturmname}\LessThanOrEqualTo{value}</a></li> +<li><a class="reference internal" href="#polsturmnbwithmultofrootsof-sturmname-lessthanorequaltoexpr-expression">\PolSturmNbWithMultOfRootsOf{sturmname}\LessThanOrEqualToExpr{expression}</a></li> +</ul> +</li> +</ul> +</li> +<li><p class="first">v0.7 (2018/12/08), v0.7.1 (bugfix), v0.7.2 (2nd bugfix) (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 +remains the same, some auxiliary macros for user-customization +have been removed: <tt class="docutils literal">\PolPrintIntervalsTheEndPoint</tt>, +<tt class="docutils literal"><span class="pre">\PolIfEndPointIsPositive{A}{B}</span></tt>, +<tt class="docutils literal"><span class="pre">\PolIfEndPointIsNegative{A}{B}</span></tt>, and +<tt class="docutils literal"><span class="pre">\PolIfEndPointIsZero{A}{B}</span></tt>.</li> +</ul> +</li> +<li>bugfix:<ul> +<li>it could happen that, contrarily to documentation, an interval +computed by <a class="reference internal" href="#polsturmisolatezeros-sturmname">\PolSturmIsolateZeros{sturmname}</a> had zero as an +endpoint,</li> +<li><a class="reference internal" href="#polensureintervallength-sturmname-index-e">\PolEnsureIntervalLength{sturmname}{index}{E}</a> could under +certain circumstances erroneously replace a non-zero root by +zero,</li> +<li><a class="reference internal" href="#polensureintervallengths-sturmname-e">\PolEnsureIntervalLengths{sturmname}{E}</a> crashed when used with +a polynomial with no real roots, hence for which no isolation intervals +existed (thanks to Thomas Söll for report).</li> +</ul> +</li> +<li>new macros:<ul> +<li><a class="reference internal" href="#id14">\PolSturmIsolateZeros**{sturmname}</a></li> +<li><a class="reference internal" href="#polsturmisolatezerosgetmultiplicitiesandrationalroots-sturmname">\PolSturmIsolateZerosGetMultiplicitiesAndRationalRoots{sturmname}</a></li> +<li><a class="reference internal" href="#polsturmisolatezerosandfindrationalroots-sturmname">\PolSturmIsolateZerosAndFindRationalRoots{sturmname}</a></li> +<li><a class="reference internal" href="#polexprsetup">\polexprsetup</a></li> +<li><a class="reference internal" href="#id18">\PolPrintIntervals*</a></li> +<li><a class="reference internal" href="#polprintintervalsnorealroots">\PolPrintIntervalsNoRealRoots</a></li> +<li><a class="reference internal" href="#polprintintervalsbeginenv">\PolPrintIntervalsBeginEnv</a></li> +<li><a class="reference internal" href="#polprintintervalsendenv">\PolPrintIntervalsEndEnv</a></li> +<li><a class="reference internal" href="#polprintintervalsknownroot">\PolPrintIntervalsKnownRoot</a></li> +<li><a class="reference internal" href="#polprintintervalsunknownroot">\PolPrintIntervalsUnknownRoot</a></li> +<li><a class="reference internal" href="#polprintintervalsprintmultiplicity">\PolPrintIntervalsPrintMultiplicity</a></li> +</ul> +</li> +<li>new expandable macros:<ul> +<li><a class="reference internal" href="#polsturmnbofrationalroots-sturmname">\PolSturmNbOfRationalRoots{sturmname}</a></li> +<li><a class="reference internal" href="#polsturmnbofrationalrootswithmultiplicities-sturmname">\PolSturmNbOfRationalRootsWithMultiplicities{sturmname}</a></li> +<li><a class="reference internal" href="#polsturmrationalroot-sturmname-k">\PolSturmRationalRoot{sturmname}{k}</a></li> +<li><a class="reference internal" href="#polsturmrationalrootindex-sturmname-k">\PolSturmRationalRootIndex{sturmname}{k}</a></li> +<li><a class="reference internal" href="#polsturmrationalrootmultiplicity-sturmname-k">\PolSturmRationalRootMultiplicity{sturmname}{k}</a></li> +<li><a class="reference internal" href="#polprintintervalsthevar">\PolPrintIntervalsTheVar</a></li> +<li><a class="reference internal" href="#polprintintervalsthesturmname">\PolPrintIntervalsTheSturmName</a></li> +<li><a class="reference internal" href="#polprintintervalsthemultiplicity">\PolPrintIntervalsTheMultiplicity</a></li> +</ul> +</li> +</ul> +</li> +<li><p class="first">v0.7.3 (2019/02/04)</p> +<ul class="simple"> +<li>bugfix:<ul> +<li>Debugging information not destined to user showed in log if root +finding was done under <tt class="docutils literal">\xintverbosetrue</tt> regime.</li> +<li><a class="reference internal" href="#polprintintervalsthevar">\PolPrintIntervalsTheVar</a> remained defined after +<a class="reference internal" href="#polprintintervals">\PolPrintIntervals</a> but was left undefined after +<a class="reference internal" href="#id18">\PolPrintIntervals*</a> (reported by Jürgen Gilg). Now remains +defined in both cases, and <a class="reference internal" href="#polprintintervalsthesturmname">\PolPrintIntervalsTheSturmName</a> +also.</li> +<li>Polynomial names ending in digits caused errors (reported by Thomas +Söll).</li> +</ul> +</li> +</ul> +</li> +<li><p class="first">v0.7.4 (2019/02/12)</p> +<ul class="simple"> +<li>bugfix:<ul> +<li>20000000000 is too big for <tt class="docutils literal">\numexpr</tt>, shouldn't I know that? +Thanks to Jürgen Gilg for report.</li> +</ul> +</li> +</ul> +</li> +</ul> +</div> +<div class="section" id="acknowledgments"> +<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 +concrete problems.</p> +<p>Renewed thanks to them on occasion of the <tt class="docutils literal">0.6</tt> and <tt class="docutils literal">0.7</tt> releases for their +continued interest.</p> +<p>See README.md for the License.</p> +</div> +</div> +</body> +</html> |