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-rw-r--r--Master/texmf-dist/tex/latex/polexpr/polexpr.sty237
1 files changed, 159 insertions, 78 deletions
diff --git a/Master/texmf-dist/tex/latex/polexpr/polexpr.sty b/Master/texmf-dist/tex/latex/polexpr/polexpr.sty
index 7e132905392..02d3ecdec76 100644
--- a/Master/texmf-dist/tex/latex/polexpr/polexpr.sty
+++ b/Master/texmf-dist/tex/latex/polexpr/polexpr.sty
@@ -1,7 +1,7 @@
% author: Jean-François Burnol
% License: LPPL 1.3c (author-maintained)
\ProvidesPackage{polexpr}%
- [2018/03/03 v0.4.2 Polynomial expressions with rational coefficients (JFB)]%
+ [2018/04/08 v0.5 Polynomial expressions with rational coefficients (JFB)]%
\RequirePackage{xintexpr}[2018/03/01]% xint 1.3
\edef\POL@restorecatcodes
{\catcode`\noexpand\_ \the\catcode`\_ \catcode0 \the\catcode0\relax}%
@@ -24,7 +24,7 @@
\else\expandafter\xint_secondoftwo
\fi}%
\def\POL@split#1.#2;#3#4% separates degree and list of coefficients
-% The \empty token is to avoid brace removal issues for degree 0 polynomials
+% The \expandafter chain removes the \empty token
{\def#3{#1}\expandafter\def\expandafter#4\expandafter{#2}}%
%
\def\POL@resultfromarray #1{% ATTENTION, **MUST** be executed with
@@ -64,8 +64,8 @@
\let\xintPow\POL@pow
\let\xintOpp\POL@opp
\def\xintSub ##1##2{\xintAdd{##1}{\xintOpp{##2}}}%
- % \xintAdd to get \POL@result defined even if numerical only expression
- % I could also test \ifPOL@pol, but well, this is very small overhead
+ % \xintAdd{0} to get \POL@result defined even if numerical only expression
+ % I could also test \ifPOL@pol, but this is anyhow small overhead
\xintAdd{0}%
{\csname XINT_expr_userfunc___pol\endcsname
{\global\POL@poltrue\def\POL@result{1.\empty{0/1[0]}{1/1[0]}}}}%
@@ -209,32 +209,45 @@
}%
-\newcommand\PolGet{}
+\newcommand\PolGet{}%
\def\PolGet#1#2\fromarray#3{%
\begingroup % closed in \POL@getfromarray
\POL@getfromarray{#1}{#3}%
\POL@newpol{#1}%
}%
\def\POL@getfromarray#1#2{%
- \count@=#2{0} %<- intentional space,
- % must be > 0, else could create infinite loop
+ \count@=#2{0} %<- intentional space
+ \ifnum\count@=\z@
+ \def\POL@result{-1.\empty{0/1[0]}}% 0.5 fix for empty array
+ \else
+ \xintloop
+ \edef\POL@tmp{#2{\count@}}%
+ \edef\POL@tmp{\xintRaw{\POL@tmp}}%
+% sadly xinttools (current 1.3a) arrays have no setters for individual items...
+ \expandafter\let\csname POL@tmparray\the\count@\endcsname\POL@tmp
+ \if0\xintiiSgn{\POL@tmp}%
+ \advance\count@\m@ne
+ \repeat
+% dans le cas particulier d'un array avec que des éléments nuls, \count@ est
+% ici devenu 0 et la boucle s'est arrêtée car #2{0} était au moins 1. De plus
+% \POL@tmparray1 est bien 0/1[0] donc ok pour polynôme nul dans \POL@result
+ \count\tw@\count@
\xintloop
+% on mouline tous les coeffs via \xintRaw
+ \ifnum\count@>\@ne
+ \advance\count@\m@ne
\edef\POL@tmp{#2{\count@}}%
- \xintiiifZero{\POL@tmp}%
- {\iftrue}%
- {\iffalse}%
- \advance\count@\m@ne
+ \edef\POL@tmp{\xintRaw{\POL@tmp}}%
+ \expandafter\let\csname POL@tmparray\the\count@\endcsname\POL@tmp
\repeat
-% should I use \xintRaw ? but if #2 expands only in an \edef, problem
-% (but it is not very probable the #2 macro does not already give completely
-% expanded contents), I would need to proceed in two steps. Or the \xintRaw
-% could get injected at level of \POL@newpol
- \def\POL@tmp##1.{{#2{##1}}}%
+ \count@\count\tw@
+ \def\POL@tmp##1.{{\csname POL@tmparray##1\endcsname}}%
\edef\POL@result{\the\numexpr\count@-\@ne.\noexpand\empty
\xintiloop[1+1]%
\expandafter\POL@tmp\xintiloopindex.%
\ifnum\xintiloopindex<\count@
\repeat}%
+ \fi
\expandafter
\endgroup
\expandafter
@@ -325,14 +338,14 @@
}%
\def\POL@mapcoeffs#1#2{%
\begingroup
- \def\POL@map@macro{#1}%
+ \def\POL@mapcoeffs@macro{#1}%
\expandafter\expandafter\expandafter\POL@split
- \csname POLuserpol@#2\endcsname;\POL@var@deg\POL@var@coeffs
+ \csname POLuserpol@#2\endcsname;\POL@mapcoeffs@deg\POL@mapcoeffs@coeffs
% attention à ne pas faire un \expandafter ici, car brace removal si 1 item
- \xintAssignArray\POL@var@coeffs\to\POL@arrayA
+ \xintAssignArray\POL@mapcoeffs@coeffs\to\POL@arrayA
\def\index{0}%
\count@\z@
- \expandafter\POL@map@loop\expandafter.\POL@var@coeffs\relax
+ \expandafter\POL@map@loop\expandafter.\POL@mapcoeffs@coeffs\relax
\xintloop
% this abuses that \POL@arrayA0 is never 0.
\xintiiifZero{\csname POL@arrayA\the\count@\endcsname}%
@@ -349,7 +362,7 @@
}%
\def\POL@map@loop.#1{\if\relax#1\expandafter\xint_gob_til_dot\fi
\advance\count@\@ne
- \edef\POL@map@coeff{\POL@map@macro{#1}}%
+ \edef\POL@map@coeff{\POL@mapcoeffs@macro{#1}}%
\expandafter
\let\csname POL@arrayA\the\count@\endcsname\POL@map@coeff
\edef\index{\the\numexpr\index+\@ne}%
@@ -620,28 +633,61 @@
\fi
}%
+%% IContent and \PolMakePrimitive (0.5)
+\def\POL@aux@mgcd@loop#1#2{%
+ \if\relax#2\expandafter\POL@aux@mgcd@exit\fi
+ \expandafter
+ \POL@aux@mgcd@loop\romannumeral0\POL@aux@gcd#1.#2.%
+}%
+\def\POL@aux@mgcd@exit
+ \expandafter
+ \POL@aux@mgcd@loop\romannumeral0\POL@aux@gcd#1.\relax.{\xintiiabs{#1}}%
+\def\POL@aux@gcd#1.#2.{%
+ \if0\xintiiSgn{#1}\expandafter\POL@aux@gcd@exit\fi
+ \expandafter\POL@aux@gcd\romannumeral0\xintmod {#2}{#1}.#1.}%
+\def\POL@aux@gcd@exit
+ \expandafter\POL@aux@gcd\romannumeral0\xintmod #1#2.#3.{{#1}}%
+
+\def\POL@icontent #1{\romannumeral0\expandafter
+ \POL@aux@mgcd@loop\romannumeral`^^@#1\relax}%
+
+\newcommand\PolIContent[1]{\romannumeral0\expandafter
+ \POL@aux@mgcd@loop\romannumeral`^^@\PolToList{#1}\relax}%
+
+
+\def\POL@makeprim@macro#1%
+ {\xintREZ{\xintNum{\xintDiv{#1}{\POL@makeprim@icontent}}}}%
+\newcommand\PolMakePrimitive[1]{%
+ \edef\POL@makeprim@icontent{\PolIContent{#1}}%
+ \PolMapCoeffs\POL@makeprim@macro{#1}%
+}%
+
%% Sturm Algorithm (polexpr 0.4)
+%% 0.5 uses primitive polynomials for faster evaluations afterwards
\newcommand\PolToSturm{%
\@ifstar{\POL@sturm@normalizefalse}{\POL@sturm@normalizetrue}%
\POL@ToSturm
}%
-\newcommand\POL@ToSturm[2]{%
+\def\POL@aux@toint#1{\xintREZ{\xintNum{#1}}}%
+\def\POL@ToSturm#1#2{%
\edef\POL@sturmname{#2}%
\POL@let{\POL@sturmname _0}{#1}%
- \POL@mapcoeffs{\xintPIrr}{\POL@sturmname _0}%
+ \PolMakePrimitive{\POL@sturmname _0}%
\POL@Diff@@one{\POL@sturmname _0}{\POL@sturmname _1}%
+ % re-utiliser \POL@varcoeffs directement?
+ \PolMakePrimitive{\POL@sturmname _1}%
\POL@count\@ne
\xintloop
\POL@divide{\POL@sturmname _\the\numexpr\POL@count-\@ne}%
{\POL@sturmname _\the\POL@count}%
\expandafter\POL@split\POL@R;\POL@degR\POL@polR
\unless\ifnum\POL@degR=\m@ne
- % suis-je sûr format raw certain?
- \edef\POL@polR{\xintApply{\POL@dooppandirr}{\POL@polR}}%
\advance\POL@count\@ne
- \expandafter\edef\csname POLuserpol@\POL@sturmname _\the\POL@count\endcsname
- {\POL@degR.\noexpand\empty\POL@polR}%
+ \expandafter\let
+ \csname POLuserpol@\POL@sturmname _\the\POL@count\endcsname\POL@R
+ \edef\POL@makeprim@icontent{-\POL@icontent\POL@polR}%
+ \POL@mapcoeffs\POL@makeprim@macro{\POL@sturmname _\the\POL@count}%
\repeat
\edef\POL@sturm@N{\the\POL@count}%
\ifPOL@sturm@normalize
@@ -650,12 +696,10 @@
\advance\POL@count\m@ne
\POL@divide{\POL@sturmname _\the\POL@count}%
{\POL@sturmname _\POL@sturm@N}%
- \expandafter\POL@split\POL@Q;\POL@degQ\POL@polQ
- \edef\POL@polQ % suis-je sûr format raw certain?
- {\xintApply{\expandafter\XINT_pirr_start\xint_firstofone}{\POL@polQ}}%
- \edef\POL@Q{\POL@degQ.\noexpand\empty\POL@polQ}%
- \expandafter\edef\csname POLuserpol@\POL@sturmname _\the\POL@count\endcsname
- {\POL@degQ.\noexpand\empty\POL@polQ}%
+ \expandafter\let
+ \csname POLuserpol@\POL@sturmname _\the\POL@count\endcsname\POL@Q
+ % quotient actually belongs to Z[X]
+ \POL@mapcoeffs{\POL@aux@toint}{\POL@sturmname _\the\POL@count}%
\ifnum\POL@count>\z@
\repeat
\@namedef{POLuserpol@\POL@sturmname _\POL@sturm@N}{0.\empty{1/1[0]}}%
@@ -667,9 +711,10 @@
\unless\ifnum\POL@sturm@N=\POL@count
\advance\POL@count\@ne
\repeat
- \expandafter\let\csname PolSturmChainLength_\POL@sturmname \endcsname\POL@sturm@N
+ \expandafter\let
+ \csname PolSturmChainLength_\POL@sturmname \endcsname\POL@sturm@N
}%
-\def\POL@dooppandirr#1{\xintiiOpp{\XINT_pirr_start#1}}%
+
\newcommand\PolSturmChainLength[1]
{\romannumeral`^^@\csname PolSturmChainLength_#1\endcsname}%
\newcommand\PolSetToSturmChainSignChangesAt[4][\global]{%
@@ -681,8 +726,8 @@
}%
\def\POL@sturmchain@getSV@at#1{% ATTENTION USES \POL@count
\def\POL@sturmchain@SV{0}%
- \edef\POL@isolz@lastsign{\xintiiSgn{\PolEval{\POL@sturmname _0}\At{#1}}}%
- \let\POL@IsoRightSign\POL@isolz@lastsign % needed only for SturmIsolate etc...
+ \edef\POL@sturmchain@sign{\xintiiSgn{\PolEval{\POL@sturmname _0}\At{#1}}}%
+ \let\POL@isolz@lastsign\POL@sturmchain@sign
\POL@count \z@
\ifnum\POL@isolz@lastsign=\z@
\edef\POL@isolz@lastsign
@@ -717,17 +762,63 @@
}%
-\newcommand\PolSturmIsolateZeros[1]{%
- % #1 name of Sturm chain (already pre-computed from a given polynomial)
- \edef\POL@sturmname{#1}%
- \edef\POL@sturmlength{\PolSturmChainLength{#1}}%
+\newcommand\PolSturmIsolateZeros[2][\empty]{%
+ % #1 optional E such that roots are searched in -10^E < x < 10^E
+ % both -10^E and +10^E must not be roots!
+ % #2 name of Sturm chain (already pre-computed from a given polynomial)
+ \edef\POL@sturmname{#2}%
+ \edef\POL@sturmlength{\PolSturmChainLength{#2}}%
+ \ifx\empty#1\relax
+ \POL@isolz@getsignchanges@plusinf
+ \POL@isolz@getsignchanges@minusinf
+ \else
+ \edef\POL@isolz@E{\the\numexpr\xint_zapspaces #1 \xint_gobble_i\relax}%
+ \POL@sturmchain@getSV@at{1[\POL@isolz@E]}%
+ \let\POL@isolz@plusinf@SV \POL@sturmchain@SV
+ \let\POL@isolz@plusinf@sign\POL@sturmchain@sign
+ \POL@sturmchain@getSV@at{-1[\POL@isolz@E]}%
+ \let\POL@isolz@minusinf@SV \POL@sturmchain@SV
+ \let\POL@isolz@minusinf@sign\POL@sturmchain@sign
+ \ifnum\POL@isolz@plusinf@sign=\z@
+ \PackageError{polexpr}%
+{The polynomial #2 vanishes at set upper bound 10^\POL@isolz@E}%
+{Compile again with a bigger exponent in source. (X to abort).}%
+ \fi
+ \ifnum\POL@isolz@minusinf@sign=\z@
+ \PackageError{polexpr}%
+{The polynomial #2 vanishes at set lower bound -10^\POL@isolz@E}%
+{Compile again with a bigger exponent in source. (X to abort).}%
+ \fi
+ \fi
+ \edef\POL@isolz@NbOfRoots
+ {\the\numexpr\POL@isolz@minusinf@SV-\POL@isolz@plusinf@SV}%
+ \ifnum\POL@isolz@NbOfRoots=\z@
+ \begingroup\globaldefs\@ne
+ \expandafter\xintAssignArray\expandafter\to\csname POL_ZeroInt#2L\endcsname
+ \expandafter\xintAssignArray\expandafter\to\csname POL_ZeroInt#2R\endcsname
+ \endgroup
+ \else
+ \begingroup\globaldefs\@ne
+ \expandafter\POL@isolz@initarray\csname POL_ZeroInt#2L\endcsname
+ \expandafter\POL@isolz@initarray\csname POL_ZeroInt#2R\endcsname
+ \endgroup
+ \ifx\empty#1\relax\expandafter\POL@isolz@getaprioribound\fi
+ \expandafter\POL@isolz@main
+ \fi
+}%
+\def\POL@isolz@initarray#1{%
+ \expandafter\xintAssignArray
+ \romannumeral\xintreplicate{\POL@isolz@NbOfRoots}{{0}}\to#1%
+}%
+\def\POL@isolz@getsignchanges@plusinf{%
% Count number of sign changes at plus infinity in Sturm sequence
\def\POL@isolz@plusinf@SV{0}%
- \edef\POL@isolz@lastsign{\xintiiSgn{\PolLeadingCoeff{#1_0}}}%
+ \edef\POL@isolz@lastsign{\xintiiSgn{\PolLeadingCoeff{\POL@sturmname _0}}}%
\let\POL@isolz@plusinf@sign\POL@isolz@lastsign
\POL@count\@ne
\xintloop
- \edef\POL@isolz@newsign{\xintiiSgn{\PolLeadingCoeff{#1_\the\POL@count}}}%
+ \edef\POL@isolz@newsign
+ {\xintiiSgn{\PolLeadingCoeff{\POL@sturmname _\the\POL@count}}}%
\unless\ifnum\POL@isolz@newsign=\POL@isolz@lastsign
\edef\POL@isolz@plusinf@SV{\the\numexpr\POL@isolz@plusinf@SV+\@ne}%
\fi
@@ -735,17 +826,20 @@
\ifnum\POL@sturmlength>\POL@count
\advance\POL@count\@ne
\repeat
+}%
+\def\POL@isolz@getsignchanges@minusinf{%
% Count number of sign changes at minus infinity in Sturm sequence
\def\POL@isolz@minusinf@SV{0}%
- \edef\POL@isolz@lastsign{\xintiiSgn{\PolLeadingCoeff{#1_0}}}%
- \ifodd\PolDegree{#1_0}
+ \edef\POL@isolz@lastsign{\xintiiSgn{\PolLeadingCoeff{\POL@sturmname _0}}}%
+ \ifodd\PolDegree{\POL@sturmname _0}
\edef\POL@isolz@lastsign{\xintiiOpp{\POL@isolz@lastsign}}%
\fi
\let\POL@isolz@minusinf@sign\POL@isolz@lastsign
\POL@count\@ne
\xintloop
- \edef\POL@isolz@newsign{\xintiiSgn{\PolLeadingCoeff{#1_\the\POL@count}}}%
- \ifodd\PolDegree{#1_\the\POL@count}
+ \edef\POL@isolz@newsign
+ {\xintiiSgn{\PolLeadingCoeff{\POL@sturmname _\the\POL@count}}}%
+ \ifodd\PolDegree{\POL@sturmname _\the\POL@count}
\edef\POL@isolz@newsign{\xintiiOpp{\POL@isolz@newsign}}%
\fi
\unless\ifnum\POL@isolz@newsign=\POL@isolz@lastsign
@@ -755,24 +849,6 @@
\ifnum\POL@sturmlength>\POL@count
\advance\POL@count\@ne
\repeat
- \edef\POL@isolz@NbOfRoots
- {\the\numexpr\POL@isolz@minusinf@SV-\POL@isolz@plusinf@SV}%
- \ifnum\POL@isolz@NbOfRoots=\z@
- \begingroup\globaldefs\@ne
- \expandafter\xintAssignArray\expandafter\to\csname POL_ZeroInt#1L\endcsname
- \expandafter\xintAssignArray\expandafter\to\csname POL_ZeroInt#1R\endcsname
- \endgroup
- \else
- \begingroup\globaldefs\@ne
- \expandafter\POL@isolz@initarray\csname POL_ZeroInt#1L\endcsname
- \expandafter\POL@isolz@initarray\csname POL_ZeroInt#1R\endcsname
- \endgroup
- \expandafter\POL@isolz@getaprioribound
- \fi
-}%
-\def\POL@isolz@initarray#1{%
- \expandafter\xintAssignArray
- \romannumeral\xintreplicate{\POL@isolz@NbOfRoots}{{0}}\to#1%
}%
% utility macro for a priori bound on root decimal exponent, via Float Rounding
\def\POL@isolz@updateE #1e#2;%
@@ -800,7 +876,6 @@
% \ifxintverbose\xintMessage{polexpr}{Info}%
% {Roots a priori bounded in absolute value by 10 to the \POL@isolz@E.}%
% \fi
- \POL@isolz@main
}%
\def\POL@IsoRight@raw{\POL@IsoRight@Int/1[\POL@isolz@E]}%
\def\POL@IsoLeft@raw {\POL@IsoLeft@Int/1[\POL@isolz@E]}%
@@ -817,8 +892,9 @@
\global\POL@isolz@nextwillneedrefinefalse
\def\POL@IsoRight@Int{0}%
\POL@sturmchain@getSV@at\POL@IsoRight@raw
- \let\POL@IsoRightSV\POL@sturmchain@SV
- \let\POL@IsoAtZeroSV\POL@IsoRightSV
+ \let\POL@IsoRightSV \POL@sturmchain@SV
+ \let\POL@IsoRightSign\POL@sturmchain@sign
+ \let\POL@IsoAtZeroSV \POL@IsoRightSV
\let\POL@IsoAtZeroSign\POL@IsoRightSign
\ifnum\POL@IsoAtZeroSign=\z@
\xdef\POL@isolz@IntervalIndex
@@ -830,7 +906,7 @@
{\xintiiOpp{\xintiiSgn{\PolEval{\POL@sturmname _1}\At{0/1[0]}}}}%
\fi
\def\POL@IsoLeft@Int{-1}% -10^E isn't a root!
- \let\POL@IsoLeftSV\POL@isolz@minusinf@SV
+ \let\POL@IsoLeftSV \POL@isolz@minusinf@SV
\let\POL@IsoLeftSign\POL@isolz@minusinf@sign
\edef\POL@isolz@NbOfNegRoots{\the\numexpr\POL@IsoLeftSV-\POL@IsoRightSV}%
\gdef\POL@isolz@IntervalIndex{0}%
@@ -842,7 +918,8 @@
\xintloop
\edef\POL@isolz@E{\the\numexpr\POL@isolz@E-\@ne}%
\POL@sturmchain@getSV@at\POL@IsoRight@raw
- \let\POL@IsoRightSV\POL@sturmchain@SV
+ \let\POL@IsoRightSV \POL@sturmchain@SV
+ \let\POL@IsoRightSign\POL@sturmchain@sign
% would an \ifx test be quicker? (to be checked)
\ifnum\POL@IsoRightSV=\POL@IsoLeftSV\space
% no roots in-between, sign and SV kept
@@ -854,7 +931,8 @@
\edef\POL@IsoRight@Int{\the\numexpr\POL@IsoLeft@Int+\@ne}%
% we could arguably do a more efficient dichotomy here
\POL@sturmchain@getSV@at\POL@IsoRight@raw
- \let\POL@IsoRightSV\POL@sturmchain@SV
+ \let\POL@IsoRightSV \POL@sturmchain@SV
+ \let\POL@IsoRightSign\POL@sturmchain@sign
\POL@isolz@check
\ifnum\POL@isolz@IntervalIndex=\POL@isolz@NbOfNegRoots\space
\expandafter\xintbreakloop
@@ -883,7 +961,7 @@
\fi
\endgroup
\def\POL@IsoLeft@Int{0}%
- \let\POL@IsoLeftSV\POL@IsoAtZeroSV
+ \let\POL@IsoLeftSV \POL@IsoAtZeroSV
\let\POL@IsoLeftSign\POL@IsoAtZeroSign
\ifnum\POL@IsoLeftSign=\z@
\xdef\POL@isolz@IntervalIndex{\the\numexpr\POL@isolz@IntervalIndex+\@ne}%
@@ -891,8 +969,8 @@
\else
\global\POL@isolz@nextwillneedrefinefalse
\fi
- \let\POL@@IsoRightSV=\POL@isolz@plusinf@SV
- \let\POL@@IsoRightSign=\POL@isolz@plusinf@sign % 10^E not a root!
+ \let\POL@@IsoRightSV \POL@isolz@plusinf@SV
+ \let\POL@@IsoRightSign\POL@isolz@plusinf@sign % 10^E not a root!
\edef\POL@isolz@NbOfPosRoots
{\the\numexpr\POL@IsoLeftSV-\POL@@IsoRightSV}% attention @@
\ifnum\POL@isolz@NbOfPosRoots>\z@
@@ -900,7 +978,8 @@
\xintloop
\edef\POL@isolz@E{\the\numexpr\POL@isolz@E-\@ne}%
\POL@sturmchain@getSV@at\POL@IsoRight@raw
- \let\POL@IsoRightSV\POL@sturmchain@SV
+ \let\POL@IsoRightSV \POL@sturmchain@SV
+ \let\POL@IsoRightSign\POL@sturmchain@sign
\ifnum\POL@IsoRightSV=\POL@@IsoRightSV\space
\let\POL@@IsoRightSign\POL@IsoRightSign % root here possible!
\repeat
@@ -914,7 +993,8 @@
% we could arguably do a more efficient dichotomy here
\edef\POL@IsoRight@Int{\the\numexpr\POL@IsoLeft@Int+\@ne}%
\POL@sturmchain@getSV@at\POL@IsoRight@raw
- \let\POL@IsoRightSV\POL@sturmchain@SV
+ \let\POL@IsoRightSV \POL@sturmchain@SV
+ \let\POL@IsoRightSign\POL@sturmchain@sign
\POL@isolz@check
\let\POL@IsoLeft@Int\POL@IsoRight@Int
\let\POL@IsoLeftSign\POL@IsoRightSign
@@ -925,7 +1005,7 @@
\ifnum\POL@IsoLeft@Int < \xint_c_ix
\repeat
\ifnum\POL@isolz@IntervalIndex<\POL@isolz@NbOfRoots\space
- % get now the last, right most, root (or roots)
+ % get now the last, rightmost, root (or roots)
\def\POL@IsoRight@Int{10}%
\let\POL@IsoRightSign\POL@@IsoRightSign
\let\POL@IsoRightSV\POL@@IsoRightSV
@@ -976,7 +1056,8 @@
\xintiloop[1+1]
\edef\POL@IsoRight@Int{\xintInc{\POL@IsoLeft@Int}}%
\POL@sturmchain@getSV@at\POL@IsoRight@raw
- \let\POL@IsoRightSV\POL@sturmchain@SV
+ \let\POL@IsoRightSV \POL@sturmchain@SV
+ \let\POL@IsoRightSign\POL@sturmchain@sign
\POL@isolz@check
\let\POL@IsoLeft@Int\POL@IsoRight@Int
\let\POL@IsoLeftSV\POL@IsoRightSV