diff options
Diffstat (limited to 'macros/latex/contrib/polexpr/polexpr.sty')
-rw-r--r-- | macros/latex/contrib/polexpr/polexpr.sty | 3164 |
1 files changed, 0 insertions, 3164 deletions
diff --git a/macros/latex/contrib/polexpr/polexpr.sty b/macros/latex/contrib/polexpr/polexpr.sty deleted file mode 100644 index 30fef28914..0000000000 --- a/macros/latex/contrib/polexpr/polexpr.sty +++ /dev/null @@ -1,3164 +0,0 @@ -% author: Jean-François Burnol -% License: LPPL 1.3c (author-maintained) -\ProvidesPackage{polexpr}% - [2020/01/31 v0.7.5 Polynomial expressions with rational coefficients (JFB)]% -\RequirePackage{xintexpr}[2020/01/31]% xint 1.4 -\edef\POL@restorecatcodes % TODO: think better about what is reasonable here - {\catcode`\noexpand\_ \the\catcode`\_ % - \catcode`\noexpand\! \the\catcode`\! % - \catcode`\noexpand\* \the\catcode`\* % - \catcode`\noexpand\~ \the\catcode`\~ % - \catcode`\noexpand\: \the\catcode`\: % - \catcode0 \the\catcode0\relax}% -\catcode`\_ 11 \catcode0 12 \catcode`\* 12 -\long\def\xint_stop_atfirstoftwo #1#2{ #1}% not yet in xint 1.3c -\long\def\xint_stop_atsecondoftwo #1#2{ #2}% - -%% 0.7.5 VERY SERIOUS TROUBLES TO GET polexpr TO WORK WITH xintexpr 1.4 - -%% I hesitated about incorporating it directly into xint 1.4 -%% Don't do this at home, only xint gurus are allowed. -\let\POL@originalXINT_expr_redefinemacros\XINT_expr_redefinemacros -\def\XINT_expr_redefinemacros -{% - \POL@originalXINT_expr_redefinemacros - \POL@activateNEhook -}% -%% Using \def's and not \let's to get better readable trace -%% in case I need to debug but this never happens -\def\POL@activateNEhook@xint % done in a group -{% - \def\POL@NEhook@polfunc{\POL@NE@polfunc}% -}% -\def\POL@activateNEhook@pol -{% - \def\POL@NEhook@polfunc{\POL@NP@polfunc}% -}% -\def\POL@activateNEhook{\POL@activateNEhook@xint}% -% -% -\catcode`~ 12 -\catcode`! 11 -\catcode`: 11 -% We drop consideration of \XINT_global matters -% because we have other more urgent and arduous problems -\def\POL@defpolfunc #1#2% -{% - \expandafter\POL@defpolfunc_a - \csname XINT_#2_func_#1\expandafter\endcsname - \csname XINT_#2_polfunc_#1\endcsname {#1}{#2}% -}% -\def\POL@defpolfunc_a #1#2#3#4% -{% - \protected % xintexpr 1.4 does things such as \expandafter\xintAdd\expanded - \expandafter\def\expandafter#2\expandafter##\expandafter1\expandafter - {% - #2{##1}% - }% - \def#1##1##2##3% - {% - % put it directly at the correct level of bracing - % don't worry for now about minimizing how many times ##3 is grabbed - \expandafter##1\expandafter##2\expandafter{\expandafter - {\romannumeral`^^@\POL@NEhook@polfunc{XINT_#4_polfunc_#3}#2{##3}}}% - }% -}% -% -\def\POL@polfunc@go #1#2#3{#2#3}% brace stripping intentional -\def\POL@NEhook@polfunc{\POL@polfunc@go}% default for pure numerics -% -% Hook for expansion in \poldef -\def\POL@NP@polfunc #1{% -\def\POL@NP@polfunc ##1##2##3% -{% - \if0\expandafter\XINT:NE:hastilde\detokenize{##3}~!\relax - \expandafter\XINT:NE:hashash\detokenize{##3}#1!\relax 0% - \expandafter\POL@polfunc@go - \else - \expandafter\POL:NP:polfunc:p - \fi {##1}{##2}{##3}% -}}\expandafter\POL@NP@polfunc\string#% -\def\POL:NP:polfunc:p #1#2#3% -{% - ~romannumeral~POL:usepolfunc:pol{#1}{#3}% -}% -\def\POL:usepolfunc:pol #1%#2% -{% -% Here we are in the core of \poldef and we really -% need to get rid of some \expanded tokens so -% we accept being exposed to \expanded but arrange to -% remain invariant. Then we will try to speed up -% polynomial composition (at this time the \#1 -% is a nested Horner type macro) by «pre-expanding» -% the argument, but this means using the \POL@get -% methods inside an \hbox -% -% \POL@applypolfunc will be defined \protected -% - \expandafter\xint_c_\expandafter\POL@applypolfunc -% This will be \protected - \csname#1\endcsname -% #2% brace stripping is deliberate -}% -% -% Hook for expansion in \xintexpr -\def\POL@NE@polfunc #1{% -\def\POL@NE@polfunc ##1##2##3% -{% - \if0\expandafter\XINT:NE:hastilde\detokenize{##3}~!\relax - \expandafter\XINT:NE:hashash\detokenize{##3}#1!\relax 0% - \expandafter\POL@polfunc@go - \else - \expandafter\POL:NE:polfunc:p - \fi {##1}{##2}{##3}% -}}\expandafter\POL@NE@polfunc\string#% -\def\POL:NE:polfunc:p #1#2#3% -{% - ~romannumeral~POL:usepolfunc:xint{#1}{#3}% -}% -\def\POL:usepolfunc:xint #1% -{% -% This is done to overcome \protected and is useful -% in case the polynomial function ends up nested -% in some non-polynomial user declared function -% as the latter (and other things) tries to pre-expand -% its arguments (as they may be used multiple time) -% using \expanded, but \#1 is protected. -% And this works recursively. We are inside braces. -% However we have a very big problem with constant -% polynomial functions. We have to handle them -% in a special way. - -`0\csname#1\expandafter\endcsname\expanded -}% -\catcode`~ 13 -\catcode`: 12 - - -%% Start defining some \protected ones here -\protected\def\POL@empty{}% -\newif\ifPOL@pol -\protected\def\POL@polglobaltrue {\global\let\ifPOL@pol\iftrue}% -\protected\def\POL@polglobalfalse{\global\let\ifPOL@pol\iffalse}% - - -%% Patch xintexpr to authorize ' in names (0.5.1) -%% Adapted 0.7.5 to follow-up on xintexpr 1.4 internal changes -%% (much simpler than previous stuff...) -%% This allows ' as a character in a polynomial name (not initial one) -\def\POL@XINT_expr_scanfunc_b #1% -{% - \ifcat \relax#1\xint_dothis{\iffalse{\fi}(_#1}\fi - \if (#1\xint_dothis{\iffalse{\fi}(`}\fi - \if 1\ifcat a#10\fi - \ifnum\xint_c_ix<1\string#1 0\fi - \if @#10\fi - \if _#10\fi - \if '#10\fi - 1% - \xint_dothis{\iffalse{\fi}(_#1}\fi - \xint_orthat {#1\XINT_expr_scanfunc_a}% -}% - - -%% Activate polexp's modified xintexpr (only during definitions -%% of polynomials) -\def\POL@hackxintexpr {% - \let\POL@originalXINT_expr_scanfunc_b\XINT_expr_scanfunc_b - \let\XINT_expr_scanfunc_b\POL@XINT_expr_scanfunc_b - \def\POL@activateNEhook{\POL@activateNEhook@pol}% -}% -\def\POL@restorexintexpr {% - \let\XINT_expr_scanfunc_b\POL@originalXINT_expr_scanfunc_b - \def\POL@activateNEhook{\POL@activateNEhook@xint}% -}% - - -%% AUXILIARIES -\catcode`! 3 -%% added at 0.7 -\newcommand\polexprsetup[1]{\POL@setup_parsekeys #1,=!,\xint_bye}% -\def\POL@setup_parsekeys #1=#2#3,{% - \ifx!#2\expandafter\xint_bye\fi - \csname POL@setup_setkey_\xint_zapspaces #1 \xint_gobble_i\endcsname - \xint_firstoftwo - {\PackageWarning{polexpr}{The \detokenize{#1} key is unknown! ignoring}}% - {\xintZapLastSpaces{#2#3}}% - \POL@setup_parsekeys -}% -\catcode`! 11 -\def\POL@setup_setkey_norr #1#2{\edef\POL@norr}% -\def\POL@setup_setkey_sqfnorr #1#2{\edef\POL@sqfnorr}% -\polexprsetup{norr=_norr, sqfnorr=_sqf_norr} - -\newcount\POL@count -\newif\ifxintveryverbose -\newif\ifpoltypesetall -\newif\ifPOL@tosturm@makefirstprimitive -\POL@tosturm@makefirstprimitivetrue -\newif\ifPOL@isolz@nextwillneedrefine -\newif\ifpoltoexprall -%% the main exchange structure (stored in macros \POLuserpol@<name>) -%% is: degree.\POL@empty{coeff0}{coeff1}....{coeffN} -%% (degree=N except zero polynomial recognized from degree set to -1 -%% but it has always the {0/1[0]} coeff0.) -\def\POL@ifZero#1{\expandafter\POL@ifZero@aux#1;}% -\def\POL@ifZero@aux #1#2;{\if-#1\expandafter\xint_firstoftwo - \else\expandafter\xint_secondoftwo - \fi}% -\def\POL@split#1.#2;#3#4% separates degree and list of coefficients -% 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 -% \count@ set to 1 + degree (\count@ = 0 for zero polynomial) -% Attention to the \protected here at 0.7.5 -% They are many all over the place - \protected\edef\POL@result{\ifnum\count@>\z@ - \the\numexpr\count@-\@ne.\POL@empty - \xintiloop [1+1]% - \expandafter\POL@braceit\csname POL@array#1\xintiloopindex\endcsname - \ifnum\xintiloopindex<\count@ - \repeat -% Attention to this \protected\POL@empty -% They are many all over the place - \else-1.\POL@empty{0/1[0]}\fi}% -}% -\def\POL@braceit#1{{#1}}% needed as \xintiloopindex can not "see" through braces - - -\newcommand\PolDef[3][x]{\poldef #2(#1):=#3;}% -\def\poldef{\edef\POL@restoresemicolon{\catcode59=\the\catcode59\relax}% - \catcode59 12 \POL@defpol}% -\def\POL@defpol #1(#2)#3=#4;{% - \POL@restoresemicolon - \edef\POL@tmp{\ifxintverbose1\else0\fi}% - \unless\ifxintveryverbose\xintverbosefalse\fi - \POL@hackxintexpr - \xintdeffunc __pol(#2):=0+(#4);% force conversion to raw if a constant - \POL@restorexintexpr - \if1\POL@tmp\xintverbosetrue\fi - \edef\POL@polname{\xint_zapspaces #1 \xint_gobble_i}% - \begingroup - \setbox0\hbox{% - \let\xintScalarAdd\xintAdd - \let\xintScalarSub\xintSub - \let\xintScalarMul\xintMul - \let\xintScalarDiv\xintDiv - \let\xintScalarPow\xintPow - \let\xintScalarOpp\xintOpp - \let\xintAdd\POL@add - \let\xintMul\POL@mul - \let\xintDiv\POL@div - \let\xintPow\POL@pow - \let\xintOpp\POL@opp - \def\xintSub ##1##2{\xintAdd{##1}{\xintOpp{##2}}}% - % \xintAdd{0} to get \POL@result defined even if numerical only expression - % I could also test \ifPOL@pol, but this is anyhow small overhead -% Attention that xintexpr 1.4 has braces all over the place - \expandafter\xintAdd\expandafter{\expandafter0\expandafter}% - \romannumeral0\csname XINT_expr_userfunc___pol\endcsname - {\POL@polglobaltrue\protected\def\POL@result{1.\POL@empty{0/1[0]}{1/1[0]}}}% - \expandafter}\expandafter - \endgroup\expandafter - \def\csname POLuserpol@\POL@polname\expandafter\endcsname - \expandafter{\POL@result}% - \expandafter\POL@newpol\expandafter{\POL@polname}% -}% -% - - -\def\POL@newpol#1{% -%% We must handle specially constant polynomials because they must -%% be made to work expandably in \poldef of other polynomials due -%% to complicated matters having to do with the \POL@ifpol conditional - \ifnum\PolDegree{#1}<\@ne - % non-zero constant - % I am defining this one only for the Info message, no time now - \expandafter\edef\csname XINT_expr_polfunc_#1\endcsname - ##1{\PolNthCoeff{#1}{0}}% - % No hooks here! - \expandafter\edef\csname XINT_expr_func_#1\endcsname ##1##2##3% - {##1##2{{\PolNthCoeff{#1}{0}}}}% - \else - % polynomial of degree at least 1. This means that mechanism - % to get \POL@result will get activated and we must be very careful - % to never \edef when the Horner macro will be converted to - % a polynomial - \POL@newpolhorner{#1}% - \POL@defpolfunc{#1}{expr}% - \fi - \expandafter\let\csname XINT_flexpr_func_#1\endcsname\@undefined - \ifxintverbose\POL@info{#1}\fi -}% -\def\POL@newfloatpol#1{% -%% We must handle specially constant polynomials because they must -%% be made to work expandably in \poldef of other polynomials due -%% to complicated matters having to do with the \POL@ifpol conditional - \ifnum\PolDegree{#1}<\@ne - % non-zero constant - % I am defining this one only for the Info message, no time now - \expandafter\edef\csname XINT_flexpr_polfunc_#1\endcsname - ##1{\PolNthCoeff{#1}{0}}% - % No hooks here! - \expandafter\edef\csname XINT_flexpr_func_#1\endcsname ##1##2##3% - {##1##2{{\PolNthCoeff{#1}{0}}}}% - \else - % polynomial of degree at least 1. This means that mechanism - % to get \POL@result will get activated and we must be very careful - % to never \edef when the Horner macro will be converted to - % a polynomial - \POL@newfloatpolhorner{#1}% - \POL@defpolfunc{#1}{flexpr}% - \fi - \ifxintverbose\POL@floatinfo{#1}\fi -}% -\def\POL@info #1{% - \xintMessage {polexpr}{Info}% - {Function #1 for the \string\xintexpr\space parser is - associated to \string\XINT_expr_polfunc_#1\space - whose meaning uses Horner scheme: - \expandafter\meaning - \csname XINT_expr_polfunc_#1\endcsname}% -}% -\def\POL@floatinfo #1{% - \xintMessage {polexpr}{Info}% - {Function #1 for the \string\xintfloatexpr\space parser is - associated to \string\XINT_flexpr_polfunc_#1\space - whose meaning uses Horner scheme: - \expandafter\meaning - \csname XINT_flexpr_polfunc_#1\endcsname}% -}% -% -\def\POL@newpolhorner#1{% - %% redefine function to expand by Horner scheme. Is this useful? - %% perhaps bad idea for numerical evaluation of thing such as (1+x)^10? -% note: I added {0/1[0]} item to zero polynomial also to facilitate this - \expandafter\expandafter\expandafter\POL@split - \csname POLuserpol@#1\endcsname;\POL@var@deg\POL@var@coeffs - \edef\POL@var@coeffs{\xintRevWithBraces{\POL@var@coeffs}}% - \begingroup - \expandafter\POL@newpol@horner\POL@var@coeffs\relax - \expandafter - \endgroup - \expandafter\def\csname XINT_expr_polfunc_#1\expandafter\endcsname - \expandafter##\expandafter1\expandafter{\POL@tmp{##1}}% -}% -\def\POL@newfloatpolhorner#1{% - %% redefine function to expand by Horner scheme. Is this useful? - %% perhaps bad idea for numerical evaluation of thing such as (1+x)^10? -% note: I added {0/1[0]} item to zero polynomial also to facilitate this - \expandafter\expandafter\expandafter\POL@split - \csname POLuserpol@#1\endcsname;\POL@var@deg\POL@var@coeffs - \edef\POL@var@coeffs{\xintRevWithBraces{\POL@var@coeffs}}% - \begingroup - \expandafter\POL@newpol@floathorner\POL@var@coeffs\relax - \expandafter - \endgroup - \expandafter\def\csname XINT_flexpr_polfunc_#1\expandafter\endcsname - \expandafter##\expandafter1\expandafter{\POL@tmp{##1}}% -}% -\def\POL@newpol@horner#1{\let\xintAdd\relax\let\xintMul\relax - \def\POL@tmp##1{#1}\POL@newpol@horner@loop.}% -\def\POL@newpol@horner@loop.#1{% - \if\relax#1\expandafter\xint_gob_til_dot\fi - \edef\POL@tmp##1{\xintiiifZero{#1} - {\@firstofone}{\xintAdd{#1}}% - {\xintMul{##1}{\POL@tmp{##1}}}}% - \POL@newpol@horner@loop.% -}% -\def\POL@newpol@floathorner#1{\let\XINTinFloatAdd\relax\let\XINTinFloatMul\relax - \def\xintAdd{\XINTinFloatAdd}\def\xintMul{\XINTinFloatMul}% - \edef\POL@tmp##1{\XINTinFloatdigits{#1}}% - \POL@newpol@floathorner@loop.}% -\def\POL@newpol@floathorner@loop.#1{% - \if\relax#1\expandafter\xint_gob_til_dot\fi - \edef\POL@tmp##1{\xintiiifZero{#1} - {\@firstofone}{\xintAdd{\XINTinFloatdigits{#1}}}% - {\xintMul{##1}{\POL@tmp{##1}}}}% - \POL@newpol@floathorner@loop.% -}% - - -\newcommand\PolGenFloatVariant[1]{\POL@newfloatpol{#1}}% - - -\newcommand\PolLet[2]{\if=\noexpand#2\expandafter\xint_firstoftwo - \else\expandafter\xint_secondoftwo\fi - \POL@@let\POL@let{#1}{#2}}% -\def\POL@@let#1#2#3{\POL@let{#1}{#3}}% -\def\POL@let#1#2{% - \expandafter\let\csname POLuserpol@#1\expandafter\endcsname - \csname POLuserpol@#2\endcsname - \expandafter\let\csname XINT_expr_polfunc_#1\expandafter\endcsname - \csname XINT_expr_polfunc_#2\endcsname - \POL@defpolfunc{#1}{expr}% - \ifxintverbose\POL@info{#1}\fi -}% -\newcommand\PolGlobalLet[2]{\begingroup - \globaldefs\@ne - \if=\noexpand#2\expandafter\xint_firstoftwo - \else\expandafter\xint_secondoftwo\fi -% do I need to check something here relative to \xintNewExpr? - \POL@@globallet\POL@globallet {#1}{#2}}% -\def\POL@@globallet#1#2#3{\POL@globallet{#1}{#3}}% -\def\POL@globallet#1#2{\POL@let{#1}{#2}\endgroup}% - -\newcommand\PolAssign[1]{\def\POL@polname{#1}\POL@assign}% zap spaces in #1? -\def\POL@assign#1\toarray#2{% - \expandafter\expandafter\expandafter\POL@split - \csname POLuserpol@\POL@polname\endcsname;\POL@var@deg\POL@var@coeffs - \xintAssignArray\POL@var@coeffs\to#2% - % modify \#200 macro to return 0/1[0] for out of range indices - \@namedef{\xint_arrayname00}##1##2##3{% - \@namedef{\xint_arrayname00}####1{% - \ifnum####1>##1 \xint_dothis{ 0/1[0]}\fi - \ifnum####1>\m@ne \xint_dothis - {\expandafter\expandafter\expandafter##3% - \csname##2####1\endcsname}\fi - \unless\ifnum-####1>##1 \xint_dothis - {\expandafter\expandafter\expandafter##3% - \csname##2\the\numexpr##1+####1+\@ne\endcsname}\fi - \xint_orthat{ 0/1[0]}}% space stops a \romannumeral0 - }% - \csname\xint_arrayname00\expandafter\expandafter\expandafter\endcsname - \expandafter\expandafter\expandafter - {\csname\xint_arrayname0\expandafter\endcsname\expandafter}\expandafter - {\xint_arrayname}{ }% -}% - - -\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 - \ifnum\count@=\z@ - \protected\def\POL@result{-1.\POL@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@}}% - \edef\POL@tmp{\xintRaw{\POL@tmp}}% - \expandafter\let\csname POL@tmparray\the\count@\endcsname\POL@tmp - \repeat - \count@\count\tw@ - \def\POL@tmp##1.{{\csname POL@tmparray##1\endcsname}}% - \protected\edef\POL@result{\the\numexpr\count@-\@ne.\POL@empty - \xintiloop[1+1]% - \expandafter\POL@tmp\xintiloopindex.% - \ifnum\xintiloopindex<\count@ - \repeat}% - \fi - \expandafter - \endgroup - \expandafter - \def\csname POLuserpol@#1\expandafter\endcsname - \expandafter{\POL@result}% -}% - - -\newcommand\PolFromCSV[2]{% - \begingroup % closed in \POL@getfromarray - \xintAssignArray\xintCSVtoList{#2}\to\POL@arrayA - \POL@getfromarray{#1}\POL@arrayA - \POL@newpol{#1}% -% semble un peu indirect et sous-optimal -% mais je veux élaguer les coefficients nuls. Peut-être à revoir. -}% - - -\newcommand\PolTypesetCmdPrefix[1]{\xintiiifSgn{#1}{}{+}{+}}% -\newcommand\PolTypesetCmd[1]{\xintifOne{\xintiiAbs{#1}}% - {\ifnum\PolIndex=\z@\xintiiSgn{#1}\else - \xintiiifSgn{#1}{-}{}{}\fi - \let\PolIfCoeffIsPlusOrMinusOne\@firstoftwo}% - {\PolTypesetOne{#1}% - \let\PolIfCoeffIsPlusOrMinusOne\@secondoftwo}% - }% -\newcommand\PolTypesetOne{\xintSignedFrac}% -\newcommand\PolTypesetMonomialCmd{% - \ifcase\PolIndex\space - % - \or\PolVar - \else\PolVar^{\PolIndex}% - \fi -}% -\newcommand\PolTypeset{\@ifstar - {\def\POL@ts@ascending{1}\POL@Typeset}% - {\def\POL@ts@ascending{0}\POL@Typeset}% -}% -\newcommand\POL@Typeset[2][x]{% LaTeX \newcommand forces optional argument first - \ensuremath{% - \expandafter\expandafter\expandafter\POL@split - \csname POLuserpol@#2\endcsname;\POL@var@deg\POL@var@coeffs - \if\POL@ts@ascending1% - \def\PolIndex{0}% - \let\POL@ts@reverse\@firstofone - \let\POL@@ne@or@m@ne\@ne - \else - \let\PolIndex\POL@var@deg - \ifnum\PolIndex<\z@\def\PolIndex{0}\fi - \let\POL@ts@reverse\xintRevWithBraces - \let\POL@@ne@or@m@ne\m@ne - \fi - \def\PolVar{#1}% - \ifnum\POL@var@deg<\z@ - \PolTypesetCmd{0/1[0]}\PolTypesetMonomialCmd - \else - \ifnum\POL@var@deg=\z@ - \expandafter\PolTypesetCmd\POL@var@coeffs\PolTypesetMonomialCmd - \else - \def\POL@ts@prefix##1{\let\POL@ts@prefix\PolTypesetCmdPrefix}% - \expandafter\POL@ts@loop - \romannumeral-`0\POL@ts@reverse{\POL@var@coeffs}\relax - \fi - \fi - }% -}% -\def\POL@ts@loop{\ifpoltypesetall\expandafter\xint_firstoftwo - \else\expandafter\xint_secondoftwo\fi - {\POL@ts@nocheck}{\POL@ts@check}.% -}% -\def\POL@ts@check.#1{% - \if\relax#1\expandafter\xint_gob_til_dot\fi - \xintiiifZero{#1}% - {}% - {\POL@ts@prefix{#1}\PolTypesetCmd{#1}\PolTypesetMonomialCmd}% - \edef\PolIndex{\the\numexpr\PolIndex+\POL@@ne@or@m@ne}\POL@ts@check.% -}% -\def\POL@ts@nocheck.#1{% - \if\relax#1\expandafter\xint_gob_til_dot\fi - \POL@ts@prefix{#1}\PolTypesetCmd{#1}\PolTypesetMonomialCmd - \edef\PolIndex{\the\numexpr\PolIndex+\POL@@ne@or@m@ne}\POL@ts@nocheck.% -}% - - -\newcommand\PolMapCoeffs[2]{% #1 = macro, #2 = name - \POL@mapcoeffs{#1}{#2}% - \POL@newpol{#2}% -}% -\def\POL@mapcoeffs#1#2{% - \begingroup - \def\POL@mapcoeffs@macro{#1}% - \expandafter\expandafter\expandafter\POL@split - \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@mapcoeffs@coeffs\to\POL@arrayA - \def\index{0}% - \count@\z@ - \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}% - {\iftrue}% - {\iffalse}% - \advance\count@\m@ne - \repeat -% donc en sortie \count@ est 0 ssi pol nul. - \POL@resultfromarray A% - \expandafter - \endgroup - \expandafter - \def\csname POLuserpol@#2\expandafter\endcsname\expandafter{\POL@result}% -}% -\def\POL@map@loop.#1{\if\relax#1\expandafter\xint_gob_til_dot\fi - \advance\count@\@ne - \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}% - \POL@map@loop.}% -\def\POL@xintIrr#1{\xintIrr{#1}[0]}% -\newcommand\PolReduceCoeffs{\@ifstar\POL@sreducecoeffs\POL@reducecoeffs}% -\def\POL@reducecoeffs#1{\PolMapCoeffs{\POL@xintIrr}{#1}}% -\def\POL@sreducecoeffs#1{\PolMapCoeffs{\xintPIrr}{#1}}% - - -%% EUCLIDEAN DIVISION -\newcommand\PolDivide[4]{% #3=quotient, #4=remainder of #1 by #2 - \POL@divide{#1}{#2}% - \expandafter\let\csname POLuserpol@#3\endcsname\POL@Q - \POL@newpol{#3}% - \expandafter\let\csname POLuserpol@#4\endcsname\POL@R - \POL@newpol{#4}% -}% -\newcommand\PolQuo[3]{% #3=quotient of #1 by #2 - \POL@divide{#1}{#2}% - \expandafter\let\csname POLuserpol@#3\endcsname\POL@Q - \POL@newpol{#3}% -}% -\newcommand\PolRem[3]{% #3=remainder of #1 by #2 - \POL@divide{#1}{#2}% - \expandafter\let\csname POLuserpol@#3\endcsname\POL@R - \POL@newpol{#3}% -}% -\newcommand\POL@divide[2]{% - \begingroup - \let\xintScalarSub\xintSub - \let\xintScalarAdd\xintAdd - \let\xintScalarMul\xintMul - \let\xintScalarDiv\xintDiv - \expandafter\let\expandafter\POL@A\csname POLuserpol@#1\endcsname - \expandafter\let\expandafter\POL@B\csname POLuserpol@#2\endcsname - \POL@div@c - \let\POL@Q\POL@result - \ifnum\POL@degQ<\z@ - \let\POL@R\POL@A - \else - \count@\numexpr\POL@degR+\@ne\relax - \POL@resultfromarray R% - \let\POL@R\POL@result - \fi - \expandafter - \endgroup - \expandafter - \def\csname POL@Q\expandafter\expandafter\expandafter\endcsname - \expandafter\expandafter\expandafter{\expandafter\POL@Q\expandafter}% - \expandafter - \def\csname POL@R\expandafter\endcsname\expandafter{\POL@R}% -}% - - -%% GCD -\newcommand\PolGCD[3]{% sets #3 to the (unitary) G.C.D. of #1 and #2 - \POL@GCD{#1}{#2}{#3}% - \POL@newpol{#3}% -}% -\def\POL@GCD #1#2#3{% - \begingroup - \let\xintScalarSub\xintSub - \let\xintScalarAdd\xintAdd - \let\xintScalarMul\xintMul - \let\xintScalarDiv\xintDiv - \expandafter\let\expandafter\POL@A\csname POLuserpol@#1\endcsname - \expandafter\let\expandafter\POL@B\csname POLuserpol@#2\endcsname - \expandafter\POL@split\POL@A;\POL@degA\POL@polA - \expandafter\POL@split\POL@B;\POL@degB\POL@polB - \ifnum\POL@degA<\z@ - \expandafter\xint_firstoftwo\else\expandafter\xint_secondoftwo - \fi - {\ifnum\POL@degB<\z@ - \expandafter\xint_firstoftwo\else\expandafter\xint_secondoftwo - \fi - {\protected\def\POL@result{-1.\POL@empty{0/1[0]}}}% - {\xintAssignArray\POL@polB\to\POL@arrayB - \POL@normalize{B}% - \POL@gcd@exit BA}}% - {\ifnum\POL@degB<\z@ - \expandafter\xint_firstoftwo\else\expandafter\xint_secondoftwo - \fi - {\xintAssignArray\POL@polA\to\POL@arrayA - \POL@normalize{A}% - \POL@gcd@exit AB}% - {\ifnum\POL@degA<\POL@degB\space - \let\POL@tmp\POL@B\let\POL@B\POL@A\let\POL@A\POL@tmp - \let\POL@tmp\POL@degB\let\POL@degB\POL@degA\let\POL@degA\POL@tmp - \let\POL@tmp\POL@polB\let\POL@polB\POL@polA\let\POL@polA\POL@tmp - \fi - \xintAssignArray\POL@polA\to\POL@arrayA - \xintAssignArray\POL@polB\to\POL@arrayB - \POL@gcd AB% - }}% - \expandafter - \endgroup - \expandafter\def\csname POLuserpol@#3\expandafter\endcsname - \expandafter{\POL@result}% -}% -\def\POL@normalize#1{% - \expandafter\def\expandafter\POL@tmp\expandafter - {\csname POL@array#1\csname POL@array#10\endcsname\endcsname}% - \edef\POL@normalize@leading{\POL@tmp}% - \expandafter\def\POL@tmp{1/1[0]}% - \count@\csname POL@deg#1\endcsname\space - \xintloop - \ifnum\count@>\z@ - \expandafter\edef\csname POL@array#1\the\count@\endcsname - {\xintIrr{\xintScalarDiv - {\csname POL@array#1\the\count@\endcsname}% - {\POL@normalize@leading}}[0]}% - \advance\count@\m@ne - \repeat -}% -\def\POL@gcd#1#2{% - \POL@normalize{#2}% - \edef\POL@degQ{\the\numexpr\csname POL@deg#1\endcsname - -\csname POL@deg#2\endcsname}% - \count@\numexpr\csname POL@deg#1\endcsname+\@ne\relax - \count\tw@\numexpr\POL@degQ+\@ne\relax - \xintloop - \POL@gcd@getremainder@loopbody#1#2% - \ifnum\count\tw@>\z@ - \repeat - \expandafter\def\csname POL@array#10\endcsname{1}% - \xintloop - \xintiiifZero{\csname POL@array#1\the\count@\endcsname}% - {\iftrue}% - {\iffalse}% - \advance\count@\m@ne - \repeat - \expandafter\edef\csname POL@deg#1\endcsname{\the\numexpr\count@-\@ne}% - \ifnum\count@<\@ne - \expandafter\POL@gcd@exit - \else - \expandafter\edef\csname POL@array#10\endcsname{\the\count@}% - \expandafter\POL@gcd - \fi{#2}{#1}% -}% -\def\POL@gcd@getremainder@loopbody#1#2{% - \edef\POL@gcd@ratio{\csname POL@array#1\the\count@\endcsname}% - \advance\count@\m@ne - \advance\count\tw@\m@ne - \count4 \count@ - \count6 \csname POL@deg#2\endcsname\space - \xintloop - \ifnum\count6>\z@ - \expandafter\edef\csname POL@array#1\the\count4\endcsname - {\xintScalarSub - {\csname POL@array#1\the\count4\endcsname}% - {\xintScalarMul - {\POL@gcd@ratio}% - {\csname POL@array#2\the\count6\endcsname}}}% - \advance\count4 \m@ne - \advance\count6 \m@ne - \repeat -}% -\def\POL@gcd@exit#1#2{% - \count@\numexpr\csname POL@deg#1\endcsname+\@ne\relax - \POL@resultfromarray #1% -}% - - -%% TODO: BEZOUT - - -%% DIFFERENTIATION -\def\POL@diff@loop@one #1/#2[#3]#4% - {\xintIrr{\xintiiMul{#4}{#1}/#2[0]}[#3]}% -\def\POL@diff#1{\POL@diff@loop1.}% -\def\POL@diff@loop#1.#2{% - \if\relax#2\expandafter\xint_gob_til_dot\fi - {\expandafter\POL@diff@loop@one\romannumeral0\xintraw{#2}{#1}}% - \expandafter\POL@diff@loop\the\numexpr#1+\@ne.% -}% -\newcommand\PolDiff[1][1]{% - % optional parameter is how many times to derivate - % first mandatory arg is name of polynomial function to derivate, - % same name as in \NewPolExpr - % second mandatory arg name of derivative - \edef\POL@iterindex{\the\numexpr#1\relax}% - \ifnum\POL@iterindex<\z@ - \expandafter\@firstoftwo - \else - \expandafter\@secondoftwo - \fi - {\PolAntiDiff[-\POL@iterindex]}{\POL@Diff}% -}% -\def\POL@Diff{% - \ifcase\POL@iterindex\space - \expandafter\POL@Diff@no - \or\expandafter\POL@Diff@one - \else\xint_afterfi{\POL@Iterate\POL@Diff@one}% - \fi -}% -\def\POL@Diff@no #1#2{\POL@let{#2}{#1}}% -\def\POL@Diff@one #1#2{\POL@Diff@@one {#1}{#2}\POL@newpol{#2}}% -\def\POL@Diff@@one#1#2{% - \expandafter\expandafter\expandafter\POL@split - \csname POLuserpol@#1\endcsname;\POL@var@deg\POL@var@coeffs - \ifnum\POL@var@deg<\@ne - \@namedef{POLuserpol@#2}{-1.\POL@empty{0/1[0]}}% - \else - \edef\POL@var@coeffs{\expandafter\POL@diff\POL@var@coeffs\relax}% - \expandafter\edef\csname POLuserpol@#2\endcsname - {\the\numexpr\POL@var@deg-\@ne.\POL@empty\POL@var@coeffs}% - \fi -}% -% lazy way but allows to share with AntiDiff -\def\POL@Iterate#1#2#3{% - \begingroup - \xintverbosefalse - #1{#2}{#3}% - \xintloop - \ifnum\POL@iterindex>\tw@ - #1{#3}{#3}% - \edef\POL@iterindex{\the\numexpr\POL@iterindex-\@ne}% - \repeat - \expandafter - \endgroup\expandafter - \def\csname POLuserpol@#3\expandafter\endcsname - \expandafter{\romannumeral`^^@\csname POLuserpol@#3\endcsname}% - #1{#3}{#3}% -}% - - -%% ANTI-DIFFERENTIATION -\def\POL@antidiff@loop@one #1/#2[#3]#4% - {\xintIrr{#1/\xintiiMul{#4}{#2}[0]}[#3]}% -\def\POL@antidiff{\POL@antidiff@loop1.}% -\def\POL@antidiff@loop#1.#2{% - \if\relax#2\expandafter\xint_gob_til_dot\fi - {\expandafter\POL@antidiff@loop@one\romannumeral0\xintraw{#2}{#1}}% - \expandafter\POL@antidiff@loop\the\numexpr#1+\@ne.% -}% -\newcommand\PolAntiDiff[1][1]{% - % optional parameter is how many times to derivate - % first mandatory arg is name of polynomial function to derivate, - % same name as in \NewPolExpr - % second mandatory arg name of derivative - \edef\POL@iterindex{\the\numexpr#1\relax}% - \ifnum\POL@iterindex<\z@ - \expandafter\@firstoftwo - \else - \expandafter\@secondoftwo - \fi - {\PolDiff[-\POL@iterindex]}{\POL@AntiDiff}% -}% -\def\POL@AntiDiff{% - \ifcase\POL@iterindex\space - \expandafter\POL@AntiDiff@no - \or\expandafter\POL@AntiDiff@one - \else\xint_afterfi{\POL@Iterate\POL@AntiDiff@one}% - \fi -}% -\let\POL@AntiDiff@no\POL@Diff@no -\def\POL@AntiDiff@one #1#2{\POL@AntiDiff@@one{#1}{#2}\POL@newpol{#2}}% -\def\POL@AntiDiff@@one#1#2{% - \expandafter\expandafter\expandafter\POL@split - \csname POLuserpol@#1\endcsname;\POL@var@deg\POL@var@coeffs - \ifnum\POL@var@deg<\z@ - \@namedef{POLuserpol@#2}{-1.\POL@empty{0/1[0]}}% - \else - \edef\POL@var@coeffs{\expandafter\POL@antidiff\POL@var@coeffs\relax}% - \expandafter\edef\csname POLuserpol@#2\endcsname - {\the\numexpr\POL@var@deg+\@ne.\POL@empty{0/1[0]}\POL@var@coeffs}% - \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]{% - % This does not need a full user declared polynomial on input, only - % a \POLuserpol@name macro, but on output it is fully declared - \edef\POL@makeprim@icontent{\PolIContent{#1}}% - \PolMapCoeffs\POL@makeprim@macro{#1}% -}% -\def\POL@makeprimitive#1{% - % Avoids declaring the polynomial, internal usage in \PolToSturm - \edef\POL@makeprim@icontent{\PolIContent{#1}}% - \POL@mapcoeffs\POL@makeprim@macro{#1}% -}% - - -%% Sturm Algorithm (polexpr 0.4) -%% 0.5 uses primitive polynomials for faster evaluations afterwards -%% 0.6 corrects misuse of \@ifstar! (mumble). \PolToSturm* was broken. -%% 0.6's \PolToSturm* defines both normalized and unnormalized, the -%% unnormalized using two underscores, so both are available -%% Sole difference is that \PolToSturm* also declares them as -%% user polynomials, whereas the non-starred only keeps the macros -%% holding the coefficients in memory -%% 0.6 fixes the case of a constant polynomial P which caused division -%% by zero error from P'. -\newcommand\PolToSturm{\@ifstar{\PolToSturm@@}{\PolToSturm@}}% -\def\POL@aux@toint#1{\xintREZ{\xintNum{#1}}}% for polynomials with int. coeffs! -%% Attention that some macros rely upon this one setting \POL@sturmname -%% and \POL@sturm@N as it does -\def\PolToSturm@#1#2{% - \edef\POL@sturmname{#2}% - % 0.6 uses 2 underscores (one before index, one after) to keep in memory - % the unnormalized chain - % This supposes #1 to be a genuine polynomial, not only a name with - % a \POLuserpol@#1 macro - \POL@let{\POL@sturmname _0_}{#1}% - \ifnum\PolDegree{#1}=\z@ - \def\POL@sturm@N{0}% - \POL@count\z@ - % if I applied the same as for positive degree, I should make it -1 - % if constant is negative. I also don't worry if polynomial is zero. - \@namedef{POLuserpol@\POL@sturmname _0}{0.\POL@empty{1/1[0]}}% - \else - \ifPOL@tosturm@makefirstprimitive\POL@makeprimitive{\POL@sturmname _0_}\fi - \POL@tosturm@dosturm - \fi - \expandafter - \let\csname PolSturmChainLength_\POL@sturmname\endcsname\POL@sturm@N - % declare the normalized ones as full-fledged polynomials - % \POL@count\z@ - \xintloop - \POL@newpol{\POL@sturmname _\the\POL@count}% - \unless\ifnum\POL@sturm@N=\POL@count - \advance\POL@count\@ne - \repeat -}% -\def\PolToSturm@@#1#2{\PolToSturm@{#1}{#2}\POL@tosturm@declareunnormalized}% -\def\POL@tosturm@declareunnormalized{% - % optionally declare also the unnormalized ones - \POL@count\z@ - \xintloop - \POL@newpol{\POL@sturmname _\the\POL@count _}% - \unless\ifnum\POL@sturm@N=\POL@count - \advance\POL@count\@ne - \repeat -}% -\def\POL@tosturm@dosturm{% - \POL@Diff@@one{\POL@sturmname _0_}{\POL@sturmname _1_}% - % re-utiliser \POL@varcoeffs directement? - \POL@makeprimitive{\POL@sturmname _1_}% does not do \POL@newpol - \POL@count\@ne - \xintloop - \POL@divide{\POL@sturmname _\the\numexpr\POL@count-\@ne\relax _}% - {\POL@sturmname _\the\POL@count _}% - \expandafter\POL@split\POL@R;\POL@degR\POL@polR - \unless\ifnum\POL@degR=\m@ne - \advance\POL@count\@ne - \expandafter\let - \csname POLuserpol@\POL@sturmname _\the\POL@count _\endcsname\POL@R - \edef\POL@makeprim@icontent{-\POL@icontent\POL@polR}% - % this avoids the \POL@newpol from \PolMapCoeffs - \POL@mapcoeffs\POL@makeprim@macro{\POL@sturmname _\the\POL@count _}% - \repeat - \edef\POL@sturm@N{\the\POL@count}% - % normalize (now always done even by starred variant) - \ifnum\PolDegree{\POL@sturmname _\POL@sturm@N _}>\z@ - % \POL@count\POL@sturm@N\relax - \xintloop - \advance\POL@count\m@ne - \POL@divide{\POL@sturmname _\the\POL@count _}% - {\POL@sturmname _\POL@sturm@N _}% - \expandafter - \let\csname POLuserpol@\POL@sturmname _\the\POL@count\endcsname\POL@Q - % quotient actually belongs to Z[X] and is primitive - \POL@mapcoeffs\POL@aux@toint{\POL@sturmname _\the\POL@count}% - \ifnum\POL@count>\z@ - \repeat - \@namedef{POLuserpol@\POL@sturmname _\POL@sturm@N}{0.\POL@empty{1/1[0]}}% - \else % they are already normalized - \advance\POL@count\@ne % attention to include last one also - \xintloop - \advance\POL@count\m@ne - \expandafter\let - \csname POLuserpol@\POL@sturmname _\the\POL@count\expandafter\endcsname - \csname POLuserpol@\POL@sturmname _\the\POL@count _\endcsname - \ifnum\POL@count>\z@ - \repeat - \fi - % Back to \PolToSturm@, \POL@count holds 0 -}% -\newcommand\PolSturmChainLength[1] - {\romannumeral`^^@\csname PolSturmChainLength_#1\endcsname}% - -\newcommand\PolSetToSturmChainSignChangesAt[4][\global]{% - \edef\POL@sturmchain@X{\xintREZ{#4}}% - \edef\POL@sturmname{#3}% - \edef\POL@sturmlength{\PolSturmChainLength{\POL@sturmname}}% - \POL@sturmchain@getSV@at\POL@sturmchain@X - #1\let#2\POL@sturmchain@SV -}% -\def\POL@sturmchain@getSV@at#1{% ATTENTION USES \POL@count - \def\POL@sturmchain@SV{0}% - \edef\POL@sturmchain@sign{\xintiiSgn{\POL@eval{\POL@sturmname _0}{#1}}}% - \let\POL@isolz@lastsign\POL@sturmchain@sign - \POL@count \z@ - \ifnum\POL@isolz@lastsign=\z@ - \edef\POL@isolz@lastsign - {\xintiiSgn{\POL@eval{\POL@sturmname _1}{#1}}}% - \POL@count \@ne - \fi - \xintloop - \unless\ifnum\POL@sturmlength=\POL@count - \advance\POL@count \@ne - \edef\POL@isolz@newsign - {\xintiiSgn{\POL@eval{\POL@sturmname _\the\POL@count}{#1}}}% - \ifnum\POL@isolz@newsign=\numexpr-\POL@isolz@lastsign\relax - \edef\POL@sturmchain@SV{\the\numexpr\POL@sturmchain@SV+\@ne}% - \let\POL@isolz@lastsign=\POL@isolz@newsign - \fi - \repeat -}% -\newcommand\PolSetToNbOfZerosWithin[5][\global]{% - \edef\POL@tmpA{\xintREZ{#4}}% - \edef\POL@tmpB{\xintREZ{#5}}% - \edef\POL@sturmname{#3}% - \edef\POL@sturmlength{\PolSturmChainLength{\POL@sturmname}}% - \POL@sturmchain@getSV@at\POL@tmpA - \let\POL@SVA\POL@sturmchain@SV - \POL@sturmchain@getSV@at\POL@tmpB - \let\POL@SVB\POL@sturmchain@SV - \ifnum\POL@SVA<\POL@SVB\space - #1\edef#2{\the\numexpr\POL@SVB-\POL@SVA}% - \else - #1\edef#2{\the\numexpr\POL@SVA-\POL@SVB}% - \fi -}% - - -% 0.6 added starred variant to count multiplicities -% 0.7 added double starred variant to locate all rational roots -\newcommand\PolSturmIsolateZeros{\@ifstar - {\PolSturmIsolateZerosAndGetMultiplicities}% - {\PolSturmIsolateZeros@}% -}% -\newcommand\PolSturmIsolateZerosAndGetMultiplicities{\@ifstar - {\PolSturmIsolateZerosGetMultiplicitiesAndRationalRoots}% - {\PolSturmIsolateZerosAndGetMultiplicities@}% -}% -% on aurait besoin de ça dans xint, mais il aurait un \xintRaw{#1} alors -\def\POL@xintfrac@getNDE #1% - {\expandafter\POL@xintfrac@getNDE@i\romannumeral`^^@#1}% -\def\POL@xintfrac@getNDE@i #1/#2[#3]#4#5#6{\def#4{#1}\def#5{#2}\def#6{#3}}% -\newcommand\PolSturmIsolateZerosGetMultiplicitiesAndRationalRoots[2][\empty]{% - \PolSturmIsolateZerosAndFindRationalRoots[#1]{#2}% - \ifnum\POL@isolz@NbOfRoots>\z@ - % get multiplicities of irrational (real) roots, if any - \ifnum\POL@findrat@nbofirrroots>\z@ - \POL@findrat@getirrmult - \fi - \POL@isolzmult@defvar@M - \fi -}% -% added at 0.7 -\newcommand\PolSturmIsolateZerosAndFindRationalRoots[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) - \edef\POL@sturmname{#2}% - \edef\POL@sturm@N{\@nameuse{PolSturmChainLength_\POL@sturmname}}% - % isolate the roots (detects case of constant polynomial) - \PolSturmIsolateZeros@{\POL@sturmname}% - \ifnum\POL@isolz@NbOfRoots=\z@ - % no real roots, define empty arrays nevertheless - \begingroup\globaldefs\@ne - \expandafter\xintAssignArray\expandafter\to\csname POL_ZM\POL@sturmname*\endcsname - \expandafter\xintAssignArray\expandafter\to\csname POL_RI\POL@sturmname*\endcsname - \endgroup - \else - % all we currently know is that multiplicities are at least one - \begingroup\globaldefs\@ne - \expandafter\POL@initarray\csname POL_ZM\POL@sturmname*\endcsname{1}% - \endgroup - % on ne va pas utiliser de Horner, mais des divisions par X - x, et ces - % choses vont évoluer, ainsi que le coefficient dominant entier - % (pour \POL@divide entre autres if faut des noms de user pol) - \expandafter\let - \csname POLuserpol@\POL@sturmname\POL@sqfnorr\expandafter\endcsname - \csname POLuserpol@\POL@sturmname _0\endcsname - \expandafter\let - \csname POLuserpol@\POL@sturmname\POL@norr\expandafter\endcsname - \csname POLuserpol@\POL@sturmname _0_\endcsname - % attention formé avec\xintREZ d'où le \xintAbs pas \xintiiAbs - % D and its exponent E will get updated along the way - \edef\POL@findrat@D{\xintAbs{\PolLeadingCoeff{\POL@sturmname _0}}}% - \POL@xintfrac@getNDE\POL@findrat@D\POL@findrat@Dint\POL@_\POL@findrat@Dexp - \xintiiifOne{\POL@findrat@Dint} - {\let\POL@findrat@E\POL@findrat@Dexp} % aussi ok pour 1[0] - {\edef\POL@findrat@E{\the\numexpr\xintLen{\POL@findrat@Dint}% - +\POL@findrat@Dexp}}% -% ATTENTION QUE LA CONVENTION DE SIGNE POUR \POL@findrat@E EST OPPOSÉE À CELLE -% POUR LE CODE PLUS ANCIEN FAISANT "REFINE" - \POL@initarray\POL@IfMultIsKnown\xint_secondoftwo - \let\POL@findrat@nbofirrroots\POL@isolz@NbOfRoots - % find all rational roots, and their multiplicities, - % factor them out in passing from original (Sturm root) polynomial - \ifnum\POL@findrat@E<7 - \def\POL@findrat@index{1}% - \POL@findrat@loop@secondpass@direct - \else - % we do a first pass scanning for "small" roots p/q (i.e. q < 1000) - \def\POL@findrat@index{1}% - \POL@findrat@loop@firstpass - % and now we do the final pass finding them all - \def\POL@findrat@index{1}% - \POL@findrat@loop@secondpass - \fi - % declare the new polynomials - \POL@newpol{\POL@sturmname\POL@sqfnorr}% without multiplicities - \POL@newpol{\POL@sturmname\POL@norr}% with multiplicities - % declare the array holding the interval indices for the rational roots - \expandafter\POL@findrat@doRRarray\csname POL_RI\POL@sturmname*\endcsname - \fi -}% -\def\POL@findrat@doRRarray#1{% - % il faudrait un \xintAssignArray* qui fasse même expansion que \xintFor* - \edef\POL@temp{% - \xintiloop[1+1] - \romannumeral0\csname POL_ZK\POL@sturmname*\xintiloopindex\endcsname - \xintbracediloopindex % I should have named it \xintiloopbracedindex... - {}% - \ifnum\xintiloopindex<\POL@isolz@NbOfRoots\space - \repeat }% - \begingroup\globaldefs\@ne - % attention de ne surtout pas faire un \expandafter ici, car en cas d'un - % seul item, \xintAssignArray l'unbraces... - \xintAssignArray\POL@temp\to#1% - \endgroup -}% -\def\POL@findrat@loop@firstpass{% - \PolSturmIfZeroExactlyKnown{\POL@sturmname}{\POL@findrat@index}% - \POL@findrat@loop@decimal% get its multiplicity - \POL@findrat@loop@aa % refine interval and check - \edef\POL@findrat@index{\the\numexpr\POL@findrat@index+\@ne}% - \ifnum\POL@findrat@index>\POL@isolz@NbOfRoots - \else - \expandafter\POL@findrat@loop@firstpass - \fi -}% -\def\POL@findrat@loop@aa{% - % we do a first pass to identify roots with denominators < 1000 - \PolEnsureIntervalLength{\POL@sturmname}{\POL@findrat@index}{-6}% - % attention that perhaps now the root is known! - \PolSturmIfZeroExactlyKnown{\POL@sturmname}{\POL@findrat@index}% - \POL@findrat@loop@decimal - \POL@findrat@loop@a -}% -\def\POL@findrat@loop@decimal{% we have an already found decimal root - % we do not go via @storeit, as it is already stored - % j'ai beaucoup hésité néanmoins, car je pourrais faire \xintIrr ici, - % mais attention aussi à l'interaction avec le \PolDecToString. Les racines - % trouvées directement (qui peuvent être des nombres décimaux) sont elles - % stockées comme fraction irréductibles (modulo action additionnelle de - % \PolDecToString). - \POL@xintfrac@getNDE - {\xintIrr{\POL@xintexprGetVar{\POL@sturmname L_\POL@findrat@index}}[0]}% - \POL@findrat@xN\POL@findrat@xD\POl@_ - % we can't move this to updatequotients because other branch will - % need to do the division first anyhow - \edef\POLuserpol@_findrat@oneterm{1.\POL@empty - {\xintiiOpp\POL@findrat@xN/1[0]}{\POL@findrat@xD/1[0]}}% - \POL@divide{\POL@sturmname\POL@sqfnorr}{_findrat@oneterm}% the one without mult. - %\expandafter\POL@split\POL@R;\POL@degR\POL@polR - \POL@findrat@loop@updatequotients - \POL@findrat@loop@getmultiplicity -}% -% lacking from xint 1.3c, but \xintSgn has overhead, so we define ii version -\def\xintiiifNeg{\romannumeral0\xintiiifneg }% -\def\xintiiifneg #1% -{% - \ifcase \xintiiSgn{#1} - \expandafter\xint_stop_atsecondoftwo - \or\expandafter\xint_stop_atsecondoftwo - \else\expandafter\xint_stop_atfirstoftwo - \fi -}% -\def\POL@findrat@getE #1/1[#2]{#2}% /1 as it should be there. -% so an error will arise if not but cf \POL@refine@getE where I did not put it -\def\POL@findrat@loop@a{% - % attention that the width may have been already smaller than 10^{-6} - \POL@get@IsoLeft@rawin - \POL@get@IsoRight@rawin - \edef\POL@findrat@localW - {\the\numexpr-\expandafter\POL@findrat@getE - % do I really need the \xintREZ? - \romannumeral0\xintrez - {\xintSub{\POL@IsoRight@rawin}{\POL@IsoLeft@rawin}}% - }% at least 6, maybe larger - \expandafter\POL@get@Int@aux - \POL@IsoLeft@rawin\POL@IsoLeft@Int{-\POL@findrat@localW}% - \expandafter\POL@get@Int@aux - \POL@IsoRight@rawin\POL@IsoRight@Int{-\POL@findrat@localW}% - % in case of odd, some waste here - \edef\POL@findrat@halflocalW{\the\numexpr(\POL@findrat@localW+1)/2-1}% - % Legendre Theorem will be used now but we separate a branch where - % everything can be done with \numexpr - \ifnum\POL@findrat@localW>9 - % not implemented yet by lazyness! - % this root will be handled in second pass only - \else - \POL@findrat@gcdloop - \fi -}% -\def\POL@findrat@gcdloop{% - % we must be careful with sign - % but we are certain no extremity is a root - \let\POL@findrat@ifnegative\xint_secondoftwo - \xintiiifSgn\POL@IsoLeft@Int - \POL@findrat@gcdloop@n - \POL@error@thisisimpossible - \POL@findrat@gcdloop@p -}% -\def\POL@findrat@gcdloop@n{% - \let\POL@findrat@ifnegative\xint_firstoftwo - \let\POL@temp\POL@IsoRight@Int - \edef\POL@IsoRight@Int{\xintiiOpp{\POL@IsoLeft@Int}}% - \edef\POL@IsoLeft@Int{\xintiiOpp{\POL@temp}}% - \POL@findrat@gcdloop@p -}% -\def\POL@findrat@gcdloop@p{% - \edef\POL@findrat@gcdloop@Ap{\xintDec{\xintDouble\POL@IsoRight@Int}}% - \edef\POL@findrat@gcdloop@A - % at most 2e9: this is acceptable to \numexpr - {2\romannumeral\xintreplicate\POL@findrat@localW{0}}% - \xintAssign - \xintiiDivision\POL@findrat@gcdloop@Ap\POL@findrat@gcdloop@A - \to\POL@findrat@gcdloop@B\POL@findrat@gcdloop@An - % on fait de la tambouille pour n'utiliser que \numexpr par la suite - % le reste @An est < 2.10^9 au pire donc ok pour \numexpr - % we will drop integral part in our updating P - \let\POL@findrat@gcdloop@Binitial\POL@findrat@gcdloop@B - \def\POL@findrat@gcdloop@B{0}% do as if B1 = 0 - \def\POL@findrat@gcdloop@Pp{1}% P0 - \def\POL@findrat@gcdloop@P{0}% P1 - \def\POL@findrat@gcdloop@Qp{0}% Q0 - \def\POL@findrat@gcdloop@Q{1}% Q1 - % A2=An can not be zero, as Ap (=A0) is odd and A (=A1=200...000) is even - % first Binitial + P1/Q1 ( = Binitial) can not be root - \let\POL@findrat@gcdloop@Ap\POL@findrat@gcdloop@A % A1 - \let\POL@findrat@gcdloop@A\POL@findrat@gcdloop@An % A2 - \def\next{\POL@findrat@gcdloop@update}% - \def\POL@findrat@gcdloop@done{0}% - \POL@findrat@gcdloop@body -}% -\def\POL@findrat@gcdloop@body{% - % annoying that \numexpr has no divmod... use counts? but groups annoying - \edef\POL@findrat@gcdloop@B - {\the\numexpr(\POL@findrat@gcdloop@Ap+\POL@findrat@gcdloop@A/2)/% - \POL@findrat@gcdloop@A - \@ne}% - \edef\POL@findrat@gcdloop@An - {\the\numexpr\POL@findrat@gcdloop@Ap-% - \POL@findrat@gcdloop@B*\POL@findrat@gcdloop@A}% - \edef\POL@findrat@gcdloop@Pn - {\the\numexpr\POL@findrat@gcdloop@Pp+% - \POL@findrat@gcdloop@B*\POL@findrat@gcdloop@P}% - \edef\POL@findrat@gcdloop@Qn - {\the\numexpr\POL@findrat@gcdloop@Qp+% - \POL@findrat@gcdloop@B*\POL@findrat@gcdloop@Q}% - \ifnum\expandafter\xintLength\expandafter{\POL@findrat@gcdloop@Qn}% - >\POL@findrat@halflocalW\space - \let\next\empty % no solution was found - \else - % with these conditions on denom, only candidates are by Legendre - % theorem among the convergents as computed here - \ifnum\POL@findrat@gcdloop@Qn>\POL@findrat@gcdloop@An\space - % means that P/Q is in interval and is thus a candidate - % it is automatically irreducible - \edef\POL@findrat@x{\xintiiAdd - {\xintiiMul{\POL@findrat@gcdloop@Qn}{\POL@findrat@gcdloop@Binitial}}% - {\POL@findrat@gcdloop@Pn}/\POL@findrat@gcdloop@Qn[0]}% - \POL@findrat@gcdloop@testit - \if1\POL@findrat@gcdloop@done - \let\next\empty % a solution was found - \fi - \fi - \fi - \next -}% -\def\POL@findrat@gcdloop@update{% - \ifnum\POL@findrat@gcdloop@An>\z@ - \let\POL@findrat@gcdloop@Ap\POL@findrat@gcdloop@A - \let\POL@findrat@gcdloop@A\POL@findrat@gcdloop@An - \let\POL@findrat@gcdloop@Pp\POL@findrat@gcdloop@P - \let\POL@findrat@gcdloop@P\POL@findrat@gcdloop@Pn - \let\POL@findrat@gcdloop@Qp\POL@findrat@gcdloop@Q - \let\POL@findrat@gcdloop@Q\POL@findrat@gcdloop@Qn - \expandafter\POL@findrat@gcdloop@body - \fi -}% -\def\POL@findrat@gcdloop@testit{% - % zero should never occur here - \POL@findrat@ifnegative{\edef\POL@findrat@x{-\POL@findrat@x}}{}% - \POL@xintfrac@getNDE\POL@findrat@x\POL@findrat@xN\POL@findrat@xD\POL@_ - \edef\POLuserpol@_findrat@oneterm{1.\POL@empty - {\xintiiOpp{\POL@findrat@xN}/1[0]}{\POL@findrat@xD/1[0]}}% - \POL@divide{\POL@sturmname\POL@sqfnorr}{_findrat@oneterm}% the one without mult. - \expandafter\POL@split\POL@R;\POL@degR\POL@polR - \ifnum\POL@degR=\m@ne % found a root - \POL@findrat@loop@storeit - \POL@findrat@loop@updatequotients - \POL@findrat@loop@getmultiplicity % will continue updating the mult. one - \def\POL@findrat@gcdloop@done{1}% - \fi -}% -% This is second phase -\def\POL@findrat@loop@secondpass{% - \PolSturmIfZeroExactlyKnown{\POL@sturmname}{\POL@findrat@index}% - {}% nothing more to be done, already stored - \POL@findrat@loop@bb % refine interval and check - \edef\POL@findrat@index{\the\numexpr\POL@findrat@index+\@ne}% - \ifnum\POL@findrat@index>\POL@isolz@NbOfRoots - \else - \expandafter\POL@findrat@loop@secondpass - \fi -}% -\def\POL@findrat@loop@secondpass@direct{% - \PolSturmIfZeroExactlyKnown{\POL@sturmname}{\POL@findrat@index}% - \POL@findrat@loop@decimal - \POL@findrat@loop@bb - \edef\POL@findrat@index{\the\numexpr\POL@findrat@index+\@ne}% - \ifnum\POL@findrat@index>\POL@isolz@NbOfRoots - \else - \expandafter\POL@findrat@loop@secondpass@direct - \fi -}% -\def\POL@findrat@loop@bb{% - \PolEnsureIntervalLength{\POL@sturmname}{\POL@findrat@index}{-\POL@findrat@E}% - % ATTENTION THAT PERHAPS NOW THE ROOT IS KNOWN! - \PolSturmIfZeroExactlyKnown{\POL@sturmname}{\POL@findrat@index}% - \POL@findrat@loop@decimal - \POL@findrat@loop@b -}% -\def\POL@findrat@loop@b{% - \edef\POL@findrat@Lscaled{\xintMul{\POL@findrat@D}% - {\POL@xintexprGetVar{\POL@sturmname L_\POL@findrat@index}}}% - \edef\POL@findrat@Rscaled{\xintMul{\POL@findrat@D}% - {\POL@xintexprGetVar{\POL@sturmname R_\POL@findrat@index}}}% - \xintiiifNeg{\POL@findrat@Lscaled}% using ii version is an abuse - {% negative interval (right bound possibly zero!) - % truncate towards zero (i.e. to the right) the left bound - \edef\POL@findrat@Num{\xintNum{\POL@findrat@Lscaled}/1[0]}% - % interval boundaries are not root hence in case that was exact - % this will not be found as a root; check if in interval - \xintifLt\POL@findrat@Num\POL@findrat@Rscaled - \POL@findrat@loop@c - {}% iterate - }% - {% positive interval (left bound possibly zero!) - % truncate towards zero (i.e. to the left) the right bound - \edef\POL@findrat@Num{\xintNum{\POL@findrat@Rscaled}/1[0]}% - % check if in interval - \xintifGt\POL@findrat@Num\POL@findrat@Lscaled - \POL@findrat@loop@c - {}% iterate - }% -}% -\def\POL@findrat@loop@c{% - % safer to do the edef as \POL@findrat@x used later in storeit - \edef\POL@findrat@x{\xintIrr{\xintDiv\POL@findrat@Num\POL@findrat@D}[0]}% - \POL@xintfrac@getNDE\POL@findrat@x\POL@findrat@xN\POL@findrat@xD\POL@_ - \edef\POLuserpol@_findrat@oneterm{1.\POL@empty - {\xintiiOpp{\POL@findrat@xN}/1[0]}{\POL@findrat@xD/1[0]}}% - \POL@divide{\POL@sturmname\POL@sqfnorr}{_findrat@oneterm}% the one without mult. - \expandafter\POL@split\POL@R;\POL@degR\POL@polR - \ifnum\POL@degR=\m@ne % found a root - \POL@findrat@loop@storeit - \POL@findrat@loop@updatequotients - \POL@findrat@loop@getmultiplicity % will continue updating the mult. one - \fi - % iterate -}% -\def\POL@findrat@loop@storeit{% - % update storage, I can not use storeleftandright here (due to rawout etc...) - \expandafter - \xdef\csname POL_ZL\POL@sturmname*\POL@findrat@index\endcsname - {\PolDecToString{\POL@findrat@x}}% - \global\expandafter - \let\csname POL_ZR\POL@sturmname*\POL@findrat@index\expandafter\endcsname - \csname POL_ZL\POL@sturmname*\POL@findrat@index\endcsname - \global\expandafter - \let\csname POL_ZK\POL@sturmname*\POL@findrat@index\endcsname - \xint_stop_atfirstoftwo - \begingroup\xintglobaldefstrue - % skip some overhead of \xintdefvar... - \XINT_expr_defvar_one{\POL@sturmname L_\POL@findrat@index}% - {\POL@findrat@x}% - \XINT_expr_defvar_one{\POL@sturmname R_\POL@findrat@index}% - {\POL@findrat@x}% - \XINT_expr_defvar_one{\POL@sturmname Z_\POL@findrat@index _isknown}% - {1}% - \endgroup -}% -\def\POL@findrat@loop@updatequotients{% - % attention last division must have been one testing vanishing of\POL@sqfnorr - \expandafter\let\csname POLuserpol@\POL@sturmname\POL@sqfnorr\endcsname\POL@Q - % quotient belongs to Z[X] and is primitive - \POL@mapcoeffs\POL@aux@toint{\POL@sturmname\POL@sqfnorr}% - % update the one with multiplicities - \POL@divide{\POL@sturmname\POL@norr}{_findrat@oneterm}% - \expandafter\let\csname POLuserpol@\POL@sturmname\POL@norr\endcsname\POL@Q - \POL@mapcoeffs\POL@aux@toint{\POL@sturmname\POL@norr} - % updating of \POL@findrat@D at end of execution of getmultiplicity -}% -\def\POL@findrat@loop@getmultiplicity{% - % the one without multiplicity must not be divided again! - % check if we have remaining multiplicity - \POL@divide{\POL@sturmname\POL@norr}{_findrat@oneterm}% - \expandafter\POL@split\POL@R;\POL@degR\POL@polR - \ifnum\POL@degR=\m@ne % yes - \expandafter\let\csname POLuserpol@\POL@sturmname\POL@norr\endcsname\POL@Q - \POL@mapcoeffs\POL@aux@toint{\POL@sturmname\POL@norr}% - \expandafter - \xdef - \csname POL_ZM\POL@sturmname*\POL@findrat@index\endcsname - {\the\numexpr - \csname POL_ZM\POL@sturmname*\POL@findrat@index\endcsname+\@ne}% - \expandafter\POL@findrat@loop@getmultiplicity - \else - % done with multiplicity for this rational root, update stuff - \edef\POL@findrat@nbofirrroots - {\the\numexpr\POL@findrat@nbofirrroots-\@ne}% - \@namedef{POL@IfMultIsKnown\POL@findrat@index}{\xint_firstoftwo}% - \edef\POL@findrat@D{\xintAbs{\PolLeadingCoeff{\POL@sturmname\POL@sqfnorr}}}% - \POL@xintfrac@getNDE\POL@findrat@D\POL@findrat@Dint\POL@_\POL@findrat@Dexp - \xintiiifOne{\POL@findrat@Dint} - {\let\POL@findrat@E\POL@findrat@Dexp} % aussi ok pour 1[0] - {\edef\POL@findrat@E{\the\numexpr\xintLen{\POL@findrat@Dint}% - +\POL@findrat@Dexp}}% - \fi -}% -\def\POL@findrat@getirrmult{% - % first get the GCD of remaining pol with its derivative - \POL@divide{\POL@sturmname\POL@norr}{\POL@sturmname\POL@sqfnorr}% - \expandafter\let - % attention au _ (cf. grosse astuce pour \POL@isolzmult@loop) - \csname POLuserpol@@_1\POL@sturmname _\endcsname\POL@Q - \ifnum\PolDegree{@_1\POL@sturmname _}>\z@ - % il reste des multiplicités (mais peut-être pour des racines complexes) - % (ou pour des racines en-dehors de l'intervalle optionnel) - % attention recyclage ici de \POL@isolzmult@loop qui dépend de - % la grosse astuce avec \@gobble - \POL@makeprimitive{@_1\POL@sturmname _}% - \let\POL@originalsturmname\POL@sturmname - % trick to get isolzmult@loop to define @@lastGCD to @_1sturmname_ - % because it will do \POL@sturmname _\POL@sturm@N _ - \edef\POL@sturmname{@_1\POL@sturmname}% - \let\POL@sturm@N\@gobble% ! - \let\POL@isolz@NbOfRoots@with_unknown_mult\POL@findrat@nbofirrroots - \POL@tosturm@makefirstprimitivefalse - \POL@isolzmult@loop - \POL@tosturm@makefirstprimitivetrue - \let\POL@sturmname\POL@originalsturmname - \fi -}% - - -\newcommand\PolSturmIsolateZerosAndGetMultiplicities@[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) - \edef\POL@sturmname{#2}% - \edef\POL@sturm@N{\@nameuse{PolSturmChainLength_\POL@sturmname}}% - % isolate the roots (detects case of constant polynomial) - \PolSturmIsolateZeros@{\POL@sturmname}% - \ifnum\POL@isolz@NbOfRoots=\z@ - % no roots, define empty array nevertheless - \begingroup\globaldefs\@ne - \expandafter\xintAssignArray\expandafter\to\csname POL_ZM\POL@sturmname*\endcsname - \endgroup - \else - % all we currently know is that multiplicities are at least one - \begingroup\globaldefs\@ne - \expandafter\POL@initarray\csname POL_ZM\POL@sturmname*\endcsname{1}% - \endgroup - % check if GCD had positive degree (hence some roots, maybe complex, have - % multiplicity) - \ifnum\PolDegree{\POL@sturmname _\POL@sturm@N _}>\z@ - % scratch array of flags to signal known multiplicities - \POL@initarray\POL@IfMultIsKnown\xint_secondoftwo - % this count has utility for the case there are other roots - % either complex or outside interval (in case of optional argument) - \let\POL@isolz@NbOfRoots@with_unknown_mult\POL@isolz@NbOfRoots - % store Sturm chain name, it is needed and altered in isolzmult@loop - \let\POL@originalsturmname\POL@sturmname - \POL@tosturm@makefirstprimitivefalse - \POL@isolzmult@loop - \POL@tosturm@makefirstprimitivetrue - \let\POL@sturmname\POL@originalsturmname - \fi - \POL@isolzmult@defvar@M - \fi -}% -\def\POL@isolzmult@defvar@M{% - % Attention that is used not only in ...GetMultiplicities@ but also - % in FindRationalRoots - \begingroup\xintglobaldefstrue - % added at 0.7 - \let\x\POL@isolz@NbOfRoots - \xintloop - % skip some overhead of \xintdefvar... - \XINT_expr_defvar_one{\POL@sturmname M_\x}% - {\csname POL_ZM\POL@sturmname*\x\endcsname}% - \edef\x{\the\numexpr\x-\@ne}% - \ifnum\x>\z@ - \repeat - \endgroup -}% -\def\POL@isolzmult@loop{% - % we are here only if last iteration gave a new GCD still of degree > 0 - % \POL@sturm@N is the one from last iteration - % Attention to not use \POL@sturmname directly in first arg. of \PolToSturm - % Attention that we need for the case of known roots also to have the last - % GCD (with its multiplicities) known as a genuine polynomial - % - because of usage of \POL@eval in @isknown branch - % - because \PolToSturm@ does a \POL@let which would be anomalous - % if the extended structure is not existing - \edef\POL@isolzmult@lastGCD{\POL@sturmname _\POL@sturm@N _}% - \edef\POL@isolzmult@newsturmname{@_1\POL@sturmname}% - \POL@newpol{\POL@isolzmult@lastGCD}% - \PolToSturm@{\POL@isolzmult@lastGCD}{\POL@isolzmult@newsturmname}% - % now both \POL@sturmname and \POL@sturm@N have changed - \edef\POL@isolzmult@newGCDdegree{\PolDegree{\POL@sturmname _\POL@sturm@N _}}% - \let\POL@isolzmult@index\POL@isolz@NbOfRoots - \xintloop - % ATTENTION that this executes macros which also modifies \POL@sturmname! - % (but not \POL@sturm@N) - \POL@isolzmult@doone - \edef\POL@isolzmult@index{\the\numexpr\POL@isolzmult@index-\@ne}% - \if1\ifnum\POL@isolz@NbOfRoots@with_unknown_mult=\z@ 0\fi - \ifnum\POL@isolzmult@index=\z@ 0\fi 1% - \repeat - \let\POL@sturmname\POL@isolzmult@newsturmname - \if1\ifnum\POL@isolz@NbOfRoots@with_unknown_mult=\z@ 0\fi - % (if new GCD is constant, time to abort) - \ifnum\POL@isolzmult@newGCDdegree=\z@ 0\fi 1% - \expandafter\POL@isolzmult@loop - \fi -}% -\def\POL@isolzmult@doone{% - \csname POL@IfMultIsKnown\POL@isolzmult@index\endcsname - {}% nothing to do - {\POL@SturmIfZeroExactlyKnown{\POL@originalsturmname}% - {\POL@isolzmult@index}% - \POL@isolzmult@loop@isknown - \POL@isolzmult@loop@isnotknown - \POL@isolzmult@loop@sharedbody - }% -}% -\def\POL@isolzmult@loop@isknown{% - \xintifZero - % attention that \POL@eval requires a declared polynomial - {\POL@eval{\POL@isolzmult@lastGCD}% - {\POL@xintexprGetVar{\POL@originalsturmname L_\POL@isolzmult@index}}}% - {\let\POL@isolzmult@haszero\@ne}% - {\let\POL@isolzmult@haszero\z@}% -}% -\def\POL@isolzmult@loop@isnotknown{% - \edef\POL@isolzmult@loop@A - {\POL@xintexprGetVar{\POL@originalsturmname L_\POL@isolzmult@index}} - \edef\POL@isolzmult@loop@B - {\POL@xintexprGetVar{\POL@originalsturmname - R_\POL@isolzmult@index}} - % attention that \PolSetToNbOfZerosWithin sets \POL@sturmname to 2nd argument - \PolSetToNbOfZerosWithin - \POL@isolzmult@haszero % nb of zeros A < x <= B, here 0 or 1 - \POL@isolzmult@newsturmname - \POL@isolzmult@loop@A - \POL@isolzmult@loop@B -}% -\def\POL@isolzmult@loop@sharedbody{% - \ifnum\POL@isolzmult@haszero>\z@ - \expandafter - \xdef - \csname POL_ZM\POL@originalsturmname*\POL@isolzmult@index\endcsname - {\the\numexpr - \csname POL_ZM\POL@originalsturmname - *\POL@isolzmult@index\endcsname+\@ne}% - \else - % multiplicity now known, no need to check this index in future - \@namedef{POL@IfMultIsKnown\POL@isolzmult@index}{\xint_firstoftwo}% - \edef\POL@isolz@NbOfRoots@with_unknown_mult - {\the\numexpr\POL@isolz@NbOfRoots@with_unknown_mult-\@ne}% - \fi -}% - - -\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) - % For reasons I have forgotten (no time now) this code **must** be used - % with a *normalized* Sturm chain. - \edef\POL@sturmname{#2}% - \edef\POL@sturmlength{\PolSturmChainLength{#2}}% - % attention to constant polynomial, we must redefine the arrays then - \ifnum\POL@sturmlength>\z@ - \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}% - \else - % constant polynomial - \def\POL@isolz@NbOfRoots{0}% - \fi - \ifnum\POL@isolz@NbOfRoots=\z@ - \begingroup\globaldefs\@ne - \expandafter\xintAssignArray\expandafter\to\csname POL_ZL#2*\endcsname - \expandafter\xintAssignArray\expandafter\to\csname POL_ZR#2*\endcsname - \expandafter\xintAssignArray\expandafter\to\csname POL_ZK#2*\endcsname - \endgroup - \else - \begingroup\globaldefs\@ne - \expandafter\POL@initarray\csname POL_ZL#2*\endcsname{0}% - \expandafter\POL@initarray\csname POL_ZR#2*\endcsname{0}% - \expandafter\POL@initarray\csname POL_ZK#2*\endcsname - \xint_stop_atsecondoftwo - \endgroup - \ifx\empty#1\relax\expandafter\POL@isolz@getaprioribound\fi - \expandafter\POL@isolz@main - \fi -}% -\def\POL@initarray#1#2{% -% ATTENTION, if only one item, \xintAssignArray UNBRACES IT -% so we use an \empty trick to avoid that. Maybe considered a bug of xinttools? - \expandafter\xintAssignArray\expandafter\empty - \romannumeral\xintreplicate{\POL@isolz@NbOfRoots}{{#2}}\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{\POL@sturmname _0}}}% - \let\POL@isolz@plusinf@sign\POL@isolz@lastsign - \POL@count\@ne - \xintloop - \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 - \let\POL@isolz@lastsign=\POL@isolz@newsign - \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{\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{\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 - \edef\POL@isolz@minusinf@SV{\the\numexpr\POL@isolz@minusinf@SV+\@ne}% - \fi - \let\POL@isolz@lastsign=\POL@isolz@newsign - \ifnum\POL@sturmlength>\POL@count - \advance\POL@count\@ne - \repeat -}% -% utility macro for a priori bound on root decimal exponent, via Float Rounding -\def\POL@isolz@updateE #1e#2;% - {\unless\ifnum#2<\POL@isolz@E\space\edef\POL@isolz@E{\the\numexpr#2+\@ne}\fi}% -\def\POL@isolz@getaprioribound{% - \PolAssign{\POL@sturmname _0}\toarray\POL@arrayA - \edef\POL@isolz@leading{\POL@arrayA{\POL@arrayA{0}}}% - \POL@count\z@ - \xintloop - \advance\POL@count\@ne - \ifnum\POL@arrayA{0}>\POL@count - \expandafter\edef\csname POL@arrayA\the\POL@count\endcsname - {\xintDiv{\POL@arrayA\POL@count}\POL@isolz@leading}% - \repeat - \def\POL@isolz@E{1}% WE SEEK SMALLEST E SUCH HAT -10^E < roots < +10^E - \advance\POL@count\m@ne - \xintloop - \ifnum\POL@count>\z@ - \expandafter\POL@isolz@updateE - % use floating point to get decimal exponent - \romannumeral0\xintfloat[4]% should I use with [2] rather? (should work) - {\xintAdd{1/1[0]}{\xintAbs{\POL@arrayA\POL@count}}};% - \advance\POL@count\m@ne - \repeat - % \ifxintverbose\xintMessage{polexpr}{Info}% - % {Roots a priori bounded in absolute value by 10 to the \POL@isolz@E.}% - % \fi -}% -\def\POL@IsoRight@raw{\POL@IsoRight@Int/1[\POL@isolz@E]}% -\def\POL@IsoLeft@raw {\POL@IsoLeft@Int/1[\POL@isolz@E]}% -\def\POL@IsoRight@rawout{% - \ifnum\POL@IsoRightSign=\z@\expandafter\xintREZ\fi\POL@IsoRight@raw -}% -\def\POL@IsoLeft@rawout{% - \ifnum\POL@IsoRightSign=\z@ - \expandafter\xint_firstoftwo\else\expandafter\xint_secondoftwo - \fi{\xintREZ\POL@IsoRight@raw}% - {\POL@IsoLeft@Int/1[\POL@isolz@E]}% -}% -\def\POL@isolz@main {% -% NOTE 2018/02/16. THIS WILL PRESUMABLY BE RE-ORGANIZED IN FUTURE TO DO -% FIRST POSITIVE ROOTS THEN NEGATIVE ROOTS VIA CHANGE OF VARIABLE TO OPPOSITE. - \global\POL@isolz@nextwillneedrefinefalse - \def\POL@IsoRight@Int{0}% - \POL@sturmchain@getSV@at\POL@IsoRight@raw - \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 - {\the\numexpr\POL@isolz@minusinf@SV-\POL@IsoRightSV}% - \POL@refine@storeleftandright % store zero root, \POL@IsoRightSign is zero - \edef\POL@IsoRightSV{\the\numexpr\POL@IsoRightSV+\@ne}% -% subtlety here if original polynomial had multiplicities, but ok. I checked! - \edef\POL@IsoRightSign % evaluated twice, but that's not so bad - {\xintiiOpp{\xintiiSgn{\POL@eval{\POL@sturmname _1}{0/1[0]}}}}% - \fi - \def\POL@IsoLeft@Int{-1}% -10^E isn't a root! - \let\POL@IsoLeftSV \POL@isolz@minusinf@SV - \let\POL@IsoLeftSign\POL@isolz@minusinf@sign - % \POL@IsoRight@SV was modified if zero is a root - \edef\POL@isolz@NbOfNegRoots{\the\numexpr\POL@IsoLeftSV-\POL@IsoRightSV}% - \gdef\POL@isolz@IntervalIndex{0}% - \let\POL@isolz@@E\POL@isolz@E - \ifnum\POL@isolz@NbOfNegRoots>\z@ -% refactored at 0.7 to fix cases leading to an intervals with zero as end-point - \POL@isolz@findroots@neg - \fi - \let\POL@isolz@E\POL@isolz@@E - \def\POL@IsoLeft@Int{0}% - \let\POL@IsoLeftSV \POL@IsoAtZeroSV % véritable SV en zéro - \let\POL@IsoLeftSign\POL@IsoAtZeroSign% véritable signe en zéro - \ifnum\POL@IsoLeftSign=\z@ - \xdef\POL@isolz@IntervalIndex{\the\numexpr\POL@isolz@IntervalIndex+\@ne}% - \fi - \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@ - % always do that to avoid zero as end-point whether it is a root or not - \global\POL@isolz@nextwillneedrefinetrue - \POL@isolz@findroots@pos - \fi -}% -\def\POL@isolz@findroots@neg{% - \def\POL@IsoRight@Int{-1}% - \POL@isolz@findnextzeroboundeddecade@neg - \def\POL@IsoLeft@Int{-10}% - \let\POL@@IsoRightSign\POL@IsoRightSign % a zero there is possible - \let\POL@@IsoRightSV \POL@IsoRightSV - % this will do possibly recursive \POL@isolz@check's - \POL@isolz@explorenexteightsubdecades@neg - \ifnum\POL@isolz@IntervalIndex<\POL@isolz@NbOfNegRoots\space - % above did not explore -2, -1 for this optimization (SV known at Right) - \def\POL@IsoRight@Int{-1}% - \let\POL@IsoRightSign\POL@@IsoRightSign - \let\POL@IsoRightSV \POL@@IsoRightSV - \POL@isolz@check - \ifnum\POL@isolz@IntervalIndex<\POL@isolz@NbOfNegRoots\space - \def\POL@IsoLeft@Int{-1}% - \let\POL@IsoLeftSign\POL@@IsoRightSign - \let\POL@IsoLeftSV \POL@@IsoRightSV - % I don't like being inside TeX conditionals - \expandafter\expandafter\expandafter\POL@isolz@findroots@neg - \fi - \fi -}% -\def\POL@isolz@findnextzeroboundeddecade@neg{% - \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@IsoRightSign\POL@sturmchain@sign - % would an \ifx test be quicker? (to be checked) - \ifnum\POL@IsoRightSV=\POL@IsoLeftSV\space - % no roots in-between, iterate - \repeat -}% -\def\POL@isolz@explorenexteightsubdecades@neg{% - \xintloop - \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@IsoRightSign\POL@sturmchain@sign - \POL@isolz@check % may recurse if multiple roots are to be found - \ifnum\POL@isolz@IntervalIndex=\POL@isolz@NbOfNegRoots\space - \expandafter\xintbreakloop - \fi - \let\POL@IsoLeft@Int\POL@IsoRight@Int - \let\POL@IsoLeftSign\POL@IsoRightSign - \let\POL@IsoLeftSV\POL@IsoRightSV - \ifnum\POL@IsoRight@Int < -\tw@ - \repeat -}% -\def\POL@isolz@findroots@pos{% - % remark (2018/12/08), this needs some refactoring, I hardly understand - % the logic and it hides most into the recursion done by \POL@isolz@check - % It would probably make more sense to proceed like done for the negative - % but here finding the largest roots first. - \def\POL@IsoRight@Int{1}% - \POL@isolz@findnextzeroboundeddecade@pos - \unless\ifnum\POL@IsoRightSV=\POL@IsoLeftSV\space - % this actually explores the whole of some interval (0, 10^{e-1}] - % in a context where some roots are known to be in (10^{e-1}, 10^{e}] - % and none are larger - \POL@isolz@check % will recurse inside groups if needed with modified E - \fi - % we know get the roots in the last 9 decades from 10^{e-1} to 10^{e} - % we should arguably do a more efficient dichotomy here - \def\POL@IsoLeft@Int{1}% - \let\POL@IsoLeftSV\POL@IsoRightSV - \let\POL@IsoLeftSign\POL@IsoRightSign - \xintloop - \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@IsoRightSign\POL@sturmchain@sign - \POL@isolz@check % recurses in needed - \let\POL@IsoLeft@Int\POL@IsoRight@Int - \let\POL@IsoLeftSign\POL@IsoRightSign - \let\POL@IsoLeftSV\POL@IsoRightSV - \ifnum\POL@isolz@IntervalIndex=\POL@isolz@NbOfRoots\space - \expandafter\xintbreakloop - \fi - \ifnum\POL@IsoLeft@Int < \xint_c_ix - \repeat - \ifnum\POL@isolz@IntervalIndex<\POL@isolz@NbOfRoots\space - % get now the last, rightmost, root (or roots) - \def\POL@IsoRight@Int{10}% - \let\POL@IsoRightSign\POL@@IsoRightSign - \let\POL@IsoRightSV\POL@@IsoRightSV - \POL@isolz@check - \fi -}% -\def\POL@isolz@findnextzeroboundeddecade@pos{% - \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@IsoRightSign\POL@sturmchain@sign - \ifnum\POL@IsoRightSV=\POL@@IsoRightSV\space - \let\POL@@IsoRightSign\POL@IsoRightSign % root here possible! - \repeat -}% -\def\POL@isolz@check{% \POL@IsoRightSign must be ready for use here -% \ifxintverbose -% \xintMessage{polexpr}{Info}% -% {\the\numexpr\POL@IsoLeftSV-\POL@IsoRightSV\relax\space roots -% in (\POL@IsoLeft@raw,\POL@IsoRight@raw] (E = \POL@isolz@E)}% -% \fi - \ifcase\numexpr\POL@IsoLeftSV-\POL@IsoRightSV\relax - % no root in ]left, right] - \global\POL@isolz@nextwillneedrefinefalse - \or - % exactly one root in ]left, right] - \xdef\POL@isolz@IntervalIndex{\the\numexpr\POL@isolz@IntervalIndex+\@ne}% - \ifnum\POL@IsoRightSign=\z@ - % if right boundary is a root, ignore previous flag - \global\POL@isolz@nextwillneedrefinefalse - \fi - % if left boundary is known to have been a root we refine interval - \ifPOL@isolz@nextwillneedrefine - \expandafter\expandafter\expandafter\POL@isolz@refine - \else - % \POL@IsoRightSign is zero iff root now exactly known - \POL@refine@storeleftandright - \ifnum\POL@IsoRightSign=\z@ - \global\POL@isolz@nextwillneedrefinetrue - \fi - \fi - \else - % more than one root, we need to recurse - \expandafter\POL@isolz@recursedeeper - \fi -}% -\def\POL@isolz@recursedeeper{% -% NOTE 2018/02/16. I SHOULD DO A REAL BINARY DICHOTOMY HERE WHICH ON AVERAGE -% SHOULD BRING SOME GAIN (LIKE WHAT IS ALREADY DONE FOR THE "refine" MACROS. -% THUS IN FUTURE THIS MIGHT BE REFACTORED. -\begingroup - \edef\POL@isolz@E{\the\numexpr\POL@isolz@E-\@ne}% - \edef\POL@@IsoRight@Int{\xintDSL{\POL@IsoRight@Int}}% - \let\POL@@IsoRightSign \POL@IsoRightSign - \let\POL@@IsoRightSV \POL@IsoRightSV - \edef\POL@IsoLeft@Int {\xintDSL{\POL@IsoLeft@Int}}% - \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@IsoRightSign\POL@sturmchain@sign - \POL@isolz@check - \let\POL@IsoLeft@Int\POL@IsoRight@Int - \let\POL@IsoLeftSV\POL@IsoRightSV - \let\POL@IsoLeftSign\POL@IsoRightSign% not used, actually - \ifnum\POL@IsoLeftSV=\POL@@IsoRightSV\space - \expandafter\xintbreakiloop - \fi - \ifnum\xintiloopindex < \xint_c_ix - \repeat - \let\POL@IsoRight@Int\POL@@IsoRight@Int - \let\POL@IsoRightSign\POL@@IsoRightSign - \let\POL@IsoRightSV \POL@@IsoRightSV - % if we exited the loop via breakiloop this is superfluous - % but it only costs one \ifnum - \POL@isolz@check -\endgroup -}% -\def\POL@isolz@refine{% - % starting point is first root = left < unique second root < right - % even if we hit exactly via refinement second root, we set flag false as - % processing will continue with original right end-point, which isn't a root - \global\POL@isolz@nextwillneedrefinefalse -\begingroup - \let\POL@@IsoRightSign\POL@IsoRightSign % already evaluated - \xintloop - \edef\POL@isolz@E{\the\numexpr\POL@isolz@E-\@ne}% - \edef\POL@IsoLeft@Int {\xintDSL{\POL@IsoLeft@Int}}% - \edef\POL@IsoRight@Int{\xintInc{\POL@IsoLeft@Int}}% - \edef\POL@IsoRightSign - {\xintiiSgn{\POL@eval{\POL@sturmname _0}{\POL@IsoRight@raw}}}% - \ifnum\POL@IsoRightSign=\POL@@IsoRightSign\space - \repeat - % now second root has been separated from the one at left end point -% we update the storage of the root at left for it to have the same number -% of digits in mantissa. No, I decided not to do that to avoid complications. - % \begingroup - % \let\POL@IsoRight@Int\POL@IsoLeft@Int - % \def\POL@IsoRightSign{0}% - % \edef\POL@isolz@IntervalIndex{\the\numexpr\POL@isolz@IntervalIndex-\@ne}% - % \POL@refine@storeleftandright - % \endgroup - \edef\POL@@IsoRight@Int{\xintDSL{\xintInc{\xintDSR{\POL@IsoLeft@Int}}}}% - \let\POL@IsoLeft@Int\POL@IsoRight@Int - \let\POL@IsoLeftSign\POL@IsoRightSign - \ifnum\POL@IsoRightSign=\z@ % check if new Left is actually a root - \else - \edef\POL@IsoRight@Int{\xintDec{\POL@@IsoRight@Int}}% - \edef\POL@IsoRightSign - {\xintiiSgn{\POL@eval{\POL@sturmname _0}{\POL@IsoRight@raw}}}% - \ifnum\POL@IsoRightSign=\POL@@IsoRightSign\space - \POL@refine@doonce % we need to locate in interval (1, 9) in local scale - \else - \let\POL@IsoLeft@Int\POL@IsoRight@Int - \ifnum\POL@IsoRightSign=\z@ - \def\POL@IsoLeftSign{0}% - \else - \let\POL@IsoRight@Int\POL@@IsoRight@Int - % the IsoRightSign is now wrong but here we don't care - \fi\fi - \fi - % on exit, exact root found iff \POL@IsoRightSign is zero - \POL@refine@storeleftandright -\endgroup -}% -\def\POL@refine@doonce{% if exact root is found, always in IsoRight on exit -% NOTE: FUTURE REFACTORING WILL GET RID OF \xintiiAdd WHICH ARE A BIT COSTLY -% BUT BASICALLY NEEDED TO HANDLE BOTH NEGATIVE AND POSITIVE HERE. -% I WILL RE-ORGANIZE THE WHOLE THING IN FUTURE TO GET ROOTS STARTING FROM -% THE ORIGIN AND SIMPLY RE-LABEL THE NEGATIVE ONE AT THE END. 2018/02/16. - \let\POL@@IsoRight@Int\POL@IsoRight@Int % 9 - \let\POL@@IsoRightSign\POL@IsoRightSign - \edef\POL@IsoRight@Int{\xintiiAdd{4}{\POL@IsoLeft@Int}}% 5 - \edef\POL@IsoRightSign - {\xintiiSgn{\POL@eval{\POL@sturmname _0}{\POL@IsoRight@raw}}}% - \ifnum\POL@IsoRightSign=\POL@IsoLeftSign\space - \let\POL@IsoLeft@Int\POL@IsoRight@Int % 5 - \edef\POL@IsoRight@Int{\xintiiAdd{2}{\POL@IsoLeft@Int}}% - \edef\POL@IsoRightSign - {\xintiiSgn{\POL@eval{\POL@sturmname _0}{\POL@IsoRight@raw}}}% - \ifnum\POL@IsoRightSign=\POL@IsoLeftSign\space - \let\POL@IsoLeft@Int\POL@IsoRight@Int % 7 - \edef\POL@IsoRight@Int{\xintInc{\POL@IsoLeft@Int}}% - \edef\POL@IsoRightSign - {\xintiiSgn{\POL@eval{\POL@sturmname _0}{\POL@IsoRight@raw}}}% - \ifnum\POL@IsoRightSign=\POL@IsoLeftSign\space - \let\POL@IsoLeft@Int\POL@IsoRight@Int % 8 - \let\POL@IsoRight@Int\POL@@IsoRight@Int % 9 - \let\POL@IsoRightSign\POL@@IsoRightSign % opposite of one at left - \fi % else 7, 8 with possible root at 8 - \else - \ifnum\POL@IsoRightSign=\z@ - \let\POL@IsoLeft@Int\POL@IsoRight@Int % root at 7 - \def\POL@IsoLeftSign{0}% - \else - \let\POL@@IsoRight@Int\POL@IsoRight@Int % 7 - \edef\POL@IsoRight@Int{\xintInc{\POL@IsoLeft@Int}}% 6 - \edef\POL@IsoRightSign - {\xintiiSgn{\POL@eval{\POL@sturmname _0}{\POL@IsoRight@raw}}}% - \ifnum\POL@IsoRightSign=\POL@IsoLeftSign\space - \let\POL@IsoLeft@Int\POL@IsoRight@Int % 6 - \let\POL@IsoRight@Int\POL@@IsoRight@Int % 7 - \let\POL@IsoRightSign\POL@@IsoRightSign - \fi % else 5, 6 with possible root at 6 - \fi\fi - \else - \ifnum\POL@IsoRightSign=\z@ - \let\POL@IsoLeft@Int\POL@IsoRight@Int % root at 5 - \def\POL@IsoLeftSign{0}% - \else - \let\POL@@IsoRight@Int\POL@IsoRight@Int % 5 - \edef\POL@IsoRight@Int{\xintiiAdd{2}{\POL@IsoLeft@Int}}% - \edef\POL@IsoRightSign - {\xintiiSgn{\POL@eval{\POL@sturmname _0}{\POL@IsoRight@raw}}}% - \ifnum\POL@IsoRightSign=\POL@IsoLeftSign\space - \let\POL@IsoLeft@Int\POL@IsoRight@Int % 3 - \edef\POL@IsoRight@Int{\xintInc{\POL@IsoLeft@Int}}% 4 - \edef\POL@IsoRightSign - {\xintiiSgn{\POL@eval{\POL@sturmname _0}{\POL@IsoRight@raw}}}% - \ifnum\POL@IsoRightSign=\POL@IsoLeftSign\space - \let\POL@IsoLeft@Int\POL@IsoRight@Int % 4 - \let\POL@IsoRight@Int\POL@@IsoRight@Int % 5 - \let\POL@IsoRightSign\POL@@IsoRightSign - \fi % else 3, 4 with possible root at 4 - \else - \ifnum\POL@IsoRightSign=\z@ - \let\POL@IsoLeft@Int\POL@IsoRight@Int % root at 3 - \def\POL@IsoLeftSign{0}% - \else - \let\POL@@IsoRight@Int\POL@IsoRight@Int % 3 - \edef\POL@IsoRight@Int{\xintInc{\POL@IsoLeft@Int}}% 2 - \edef\POL@IsoRightSign - {\xintiiSgn{\POL@eval{\POL@sturmname _0}{\POL@IsoRight@raw}}}% - \ifnum\POL@IsoRightSign=\POL@IsoLeftSign\space - \let\POL@IsoLeft@Int\POL@IsoRight@Int % 2 - \let\POL@IsoRight@Int\POL@@IsoRight@Int % 3 - \let\POL@IsoRightSign\POL@@IsoRightSign - \fi % else 1, 2 with possible root at 2 - \fi\fi - \fi\fi -}% -\def\POL@refine@storeleftandright{% - \expandafter - \xdef\csname POL_ZL\POL@sturmname*\POL@isolz@IntervalIndex\endcsname - {\PolDecToString{\POL@IsoLeft@rawout}}% - \expandafter - \xdef\csname POL_ZR\POL@sturmname*\POL@isolz@IntervalIndex\endcsname - {\PolDecToString{\POL@IsoRight@rawout}}% - % added at 0.6 - \ifnum\POL@IsoRightSign=\z@ - \global - \expandafter - \let\csname POL_ZK\POL@sturmname*\POL@isolz@IntervalIndex\endcsname - \xint_stop_atfirstoftwo - \fi - \begingroup\xintglobaldefstrue - % skip some overhead of \xintdefvar... - \XINT_expr_defvar_one{\POL@sturmname L_\POL@isolz@IntervalIndex}% - {\POL@IsoLeft@rawout}% - \XINT_expr_defvar_one{\POL@sturmname R_\POL@isolz@IntervalIndex}% - {\POL@IsoRight@rawout}% - % added at 0.7 - \XINT_expr_defvar_one{\POL@sturmname Z_\POL@isolz@IntervalIndex _isknown}% - {\ifnum\POL@IsoRightSign=\z@ 1\else 0\fi}% - \endgroup -}% -%% \PolRefineInterval -\def\POL@xintexprGetVar#1{\csname XINT_expr_varvalue_#1\endcsname}% -% attention, also used by \POL@findrat@loop@a -\def\POL@get@IsoLeft@rawin{% - \edef\POL@IsoLeft@rawin - {\POL@xintexprGetVar{\POL@sturmname L_\POL@isolz@IntervalIndex}}% -}% -% attention, also used by \POL@findrat@loop@a -\def\POL@get@IsoRight@rawin{% - \edef\POL@IsoRight@rawin - {\POL@xintexprGetVar{\POL@sturmname R_\POL@isolz@IntervalIndex}}% -}% -% attention, also used by \POL@findrat@loop@a -\def\POL@get@Int@aux #1/1[#2]#3#4{\edef#3{\xintDSH{#4-#2}{#1}}}% -\def\POL@get@IsoLeft@Int{% - \expandafter\POL@get@Int@aux\POL@IsoLeft@rawin\POL@IsoLeft@Int\POL@isolz@E -}% -\newcommand\PolRefineInterval{\@ifstar\POL@srefine@start\POL@refine@start}% -\newcommand\POL@refine@start[3][1]{% - \edef\POL@isolz@IntervalIndex{\the\numexpr#3}% - \edef\POL@sturmname{#2}% - \expandafter\POL@refine@sharedbody\expandafter - {\expandafter\POL@refine@loop\expandafter{\the\numexpr#1}}% -}% -\def\POL@srefine@start#1#2{% - \edef\POL@isolz@IntervalIndex{\the\numexpr#2}% - \edef\POL@sturmname{#1}% - \POL@refine@sharedbody - {\let\POL@refine@left@next\POL@refine@main % we want to recurse if needed - \let\POL@refine@right@next\POL@refine@main % we want to recurse if needed - \POL@refine@main}% -}% -\def\POL@refine@sharedbody#1{% - \POL@get@IsoLeft@rawin - \edef\POL@IsoLeftSign - {\xintiiSgn{\POL@eval{\POL@sturmname _0}{\POL@IsoLeft@rawin}}}% - \ifnum\POL@IsoLeftSign=\z@ - % do nothing if that interval was already a singleton - \else - % else both end-points are not roots and there is a single one in-between - \POL@get@IsoRight@rawin - \edef\POL@IsoRightSign{\the\numexpr-\POL@IsoLeftSign}% - \edef\POL@isolz@E{\expandafter\POL@refine@getE - % je pense que le xintrez ici est superflu - \romannumeral0\xintrez{\xintSub{\POL@IsoRight@rawin}{\POL@IsoLeft@rawin}}}% - \POL@get@IsoLeft@Int - \edef\POL@IsoRight@Int{\xintInc{\POL@IsoLeft@Int}}% - #1% - \POL@refine@storeleftandright % \POL@IsoRightSign not zero - \fi -}% -\def\POL@refine@loop#1{% - \let\POL@refine@left@next \@empty % no recursion at end sub-intervals - \let\POL@refine@right@next\@empty - \xintiloop[1+1] - \POL@refine@main - \ifnum\POL@IsoRightSign=\z@ - \expandafter\xintbreakiloop - \fi - \ifnum\xintiloopindex<#1 - \repeat -}% -\def\POL@refine@main{% - \edef\POL@isolz@E{\the\numexpr\POL@isolz@E-\@ne}% - \edef\POL@IsoLeft@Int{\xintDSL{\POL@IsoLeft@Int}}% - \edef\POL@IsoRight@Int{\xintDSL{\POL@IsoRight@Int}}% - \let\POL@@IsoRight@Int\POL@IsoRight@Int - \let\POL@@IsoRightSign\POL@IsoRightSign - \edef\POL@IsoRight@Int{\xintInc{\POL@IsoLeft@Int}}% - \edef\POL@IsoRightSign - {\xintiiSgn{\POL@eval{\POL@sturmname _0}{\POL@IsoRight@raw}}}% - \ifnum\POL@IsoRightSign=\z@ - \let\POL@IsoLeft@Int\POL@IsoRight@Int % root at 1 - \def\POL@IsoLeftSign{0}% - \let\POL@next\@empty - \else - \ifnum\POL@IsoRightSign=\POL@@IsoRightSign\space - \let\POL@next\POL@refine@left@next % may be \@empty or \POL@refine@main for recursion - \let\POL@refine@right@next\@empty - \else - \let\POL@IsoLeft@Int\POL@IsoRight@Int - \edef\POL@IsoRight@Int{\xintDec{\POL@@IsoRight@Int}}% - \edef\POL@IsoRightSign - {\xintiiSgn{\POL@eval{\POL@sturmname _0}{\POL@IsoRight@raw}}}% - \ifnum\POL@IsoRightSign=\z@ - \let\POL@IsoLeft@Int\POL@IsoRight@Int % root at 9 - \def\POL@IsoLeftSign{0}% - \let\POL@next\@empty - \else - \ifnum\POL@IsoRightSign=\POL@@IsoRightSign\space - \let\POL@next\POL@refine@doonce - \else - \let\POL@IsoLeft@Int\POL@IsoRight@Int - \let\POL@IsoRight@Int\POL@@IsoRight@Int - \let\POL@IsoRightSign\POL@@IsoRightSign - \let\POL@next\POL@refine@right@next - \let\POL@refine@left@next\@empty - \fi - \fi - \fi\fi - \POL@next -}% -% lacking pre-defined xintfrac macro here (such as an \xintRawExponent) -\def\POL@refine@getE#1[#2]{#2}% \xintREZ already applied, for safety - - -\newcommand\PolIntervalWidth[2]{% -% le \xintRez est à cause des E positifs, car trailing zéros explicites -% si je travaillais à partir des variables xintexpr directement ne devrait -% pas être nécessaire, mais trop fragile par rapport à chgt internes possibles - \romannumeral0\xintrez{\xintSub{\@nameuse{POL_ZR#1*}{#2}}% - {\@nameuse{POL_ZL#1*}{#2}}} -}% - - -\newcommand\PolEnsureIntervalLengths[2]{% #1 = Sturm chain name, - % localize roots in intervals of length at most 10^{#2} - \edef\POL@sturmname{#1}% - \edef\POL@ensure@targetE{\the\numexpr#2}% - \edef\POL@nbofroots{\csname POL_ZL\POL@sturmname*0\endcsname}% - \ifnum\POL@nbofroots>\z@ - \expandafter\POL@ensureintervallengths - \fi -}% -\def\POL@ensureintervallengths{% - \POL@count\z@ - % \POL@count used by \POL@sturmchain@getSV@at but latter not used - \xintloop - \advance\POL@count\@ne - \edef\POL@isolz@IntervalIndex{\the\POL@count}% - \POL@ensure@one - \ifnum\POL@nbofroots>\POL@count - \repeat -}% -\newcommand\PolEnsureIntervalLength[3]{% #1 = Sturm chain name, - % #2 = index of interval - % localize roots in intervals of length at most 10^{#3} - \edef\POL@sturmname{#1}% - \edef\POL@ensure@targetE{\the\numexpr#3}% - \edef\POL@isolz@IntervalIndex{\the\numexpr#2}% -% peut-être autoriser -1, -2, ... ? - \ifnum\POL@isolz@IntervalIndex>\z@ -% 0.7, add this safeguard but attention means this structure must be in place - \ifnum\csname POL_ZL\POL@sturmname*0\endcsname>\z@ -% je ne fais pas les \expandafter mais je préfèrerai ne pas être à l'intérieur - \POL@ensure@one - \fi - \fi -}% -\def\POL@ensure@one{% - \POL@get@IsoLeft@rawin - \POL@get@IsoRight@rawin - \edef\POL@ensure@delta{\xintREZ{\xintSub{\POL@IsoRight@rawin}{\POL@IsoLeft@rawin}}}% - \xintiiifZero{\POL@ensure@delta} - {} - {\edef\POL@isolz@E{\expandafter\POL@refine@getE\POL@ensure@delta}% - \POL@get@IsoLeft@Int - \edef\POL@IsoRight@Int{\xintInc{\POL@IsoLeft@Int}}% - \ifnum\POL@isolz@E>\POL@ensure@targetE\space - \edef\POL@IsoLeftSign - {\xintiiSgn{\POL@eval{\POL@sturmname _0}{\POL@IsoLeft@raw}}}% - % at start left and right are not roots, and values of opposite signs - % \edef\POL@IsoRightSign{\the\numexpr-\POL@IsoLeftSign}% - \xintloop - \POL@ensure@Eloopbody % decreases E by one at each iteration - % if separation level is still too coarse we recurse at deeper level - \ifnum\POL@isolz@E>\POL@ensure@targetE\space - \repeat - % will check if right is at a zero, it needs \POL@IsoRightSign set up - \POL@refine@storeleftandright - \fi - }% -}% -\def\POL@ensure@Eloopbody {% - \edef\POL@isolz@E{\the\numexpr\POL@isolz@E-\@ne}% - \edef\POL@IsoLeft@Int{\xintDSL{\POL@IsoLeft@Int}}% - % this will loop at most ten times - \xintloop - \edef\POL@IsoRight@Int{\xintInc{\POL@IsoLeft@Int}}% - \edef\POL@IsoRightSign - {\xintiiSgn{\POL@eval{\POL@sturmname _0}{\POL@IsoRight@raw}}}% - % if we have found a zero at right boundary the \ifnum test will fail - % and we exit the loop - % else we exit the loop if sign at right boundary is opposite of - % sign at left boundary (the latter is +1 or -1, never 0) - % this is a bit wasteful if we go ten times to the right, because - % we know that there the sign will be opposite, evaluation was superfluous - \ifnum\POL@IsoLeftSign=\POL@IsoRightSign\space - \let\POL@IsoLeft@Int\POL@IsoRight@Int - \repeat - % check for case when we exited the inner loop because we actually - % found a zero, then we force exit from the main (E decreasing) loop - \ifnum\POL@IsoRightSign=\z@ - \expandafter\xintbreakloop - \fi -}% - - -\catcode`_ 8 -\newcommand\PolPrintIntervals - {\@ifstar{\PolPrintIntervals@@}{\PolPrintIntervals@}}% -\newcommand\PolPrintIntervals@@{% - \begingroup - \def\POL@AfterPrintIntervals{\endgroup}% - \def\arraystretch{2}% - \let\PolPrintIntervalsPrintExactZero\POL@@PrintIntervalsPrintExactZero - \let\PolPrintIntervalsUnknownRoot\POL@@PrintIntervalsUnknownRoot - \let\PolPrintIntervalsKnownRoot\POL@@PrintIntervalsKnownRoot - \def\PolPrintIntervalsBeginEnv{\[\begin{array}{cl}}%\] - \def\PolPrintIntervalsEndEnv{\end{array}\]}% - \PolPrintIntervals@ -}% -\newcommand\PolPrintIntervals@[2][Z]{\POL@PrintIntervals{#1}{#2}}% -\newcommand\POL@PrintIntervals[2]{% - \def\PolPrintIntervalsTheVar{#1}% - \def\PolPrintIntervalsTheSturmName{#2}% - \ifnum\@nameuse{POL_ZL#2*}{0}=\z@ - \PolPrintIntervalsNoRealRoots - \else - \gdef\PolPrintIntervalsTheIndex{1}% - \POL@PrintIntervals@DoDefs - \begingroup\edef\POL@tmp{\endgroup - \unexpanded\expandafter{\PolPrintIntervalsBeginEnv}% - \unexpanded\expandafter{\POL@PrintIntervals@Loop}% - \unexpanded\expandafter{\PolPrintIntervalsEndEnv}% - }\POL@tmp - \fi - \POL@AfterPrintIntervals - \def\PolPrintIntervalsTheVar{#1}% - \def\PolPrintIntervalsTheSturmName{#2}% -}% -\let\POL@AfterPrintIntervals\@empty -\newcommand\PolPrintIntervalsNoRealRoots{}% -\newcommand\PolPrintIntervalsBeginEnv{\[\begin{array}{rcccl}}% -\newcommand\PolPrintIntervalsEndEnv{\end{array}\]}% -\newcommand\PolPrintIntervalsKnownRoot{% - &&\PolPrintIntervalsTheVar_{\PolPrintIntervalsTheIndex}% - &=&\PolPrintIntervalsPrintExactZero -}% -\newcommand\PolPrintIntervalsUnknownRoot{% - \PolPrintIntervalsPrintLeftEndPoint&<&% - \PolPrintIntervalsTheVar_{\PolPrintIntervalsTheIndex}&<&% - \PolPrintIntervalsPrintRightEndPoint -}% -\newcommand\PolPrintIntervalsPrintExactZero {\PolPrintIntervalsTheLeftEndPoint}% -\newcommand\PolPrintIntervalsPrintLeftEndPoint {\PolPrintIntervalsTheLeftEndPoint}% -\newcommand\PolPrintIntervalsPrintRightEndPoint{\PolPrintIntervalsTheRightEndPoint}% -\newcommand\PolPrintIntervalsPrintMultiplicity{(\mbox{mult. }\PolPrintIntervalsTheMultiplicity)}% -% -\newcommand\POL@@PrintIntervalsKnownRoot{% - \PolPrintIntervalsPrintMultiplicity&% - \PolPrintIntervalsTheVar_{\PolPrintIntervalsTheIndex}=% - \PolPrintIntervalsPrintExactZero -}% -\newcommand\POL@@PrintIntervalsPrintExactZero{% - \displaystyle - \xintSignedFrac{\PolPrintIntervalsTheLeftEndPoint}% -}% -\newcommand\POL@@PrintIntervalsUnknownRoot{% - \PolPrintIntervalsPrintMultiplicity&% - \xintifSgn{\PolPrintIntervalsTheLeftEndPoint}% - {\xintifSgn{\PolPrintIntervalsTheRightEndPoint} - {\PolPrintIntervalsTheVar_{\PolPrintIntervalsTheIndex}=% - \PolPrintIntervalsPrintRightEndPoint\dots}% - {0>\PolPrintIntervalsTheVar_{\PolPrintIntervalsTheIndex}>% - \PolPrintIntervalsPrintLeftEndPoint}% - {\PolErrorThisShouldNotHappenPleaseReportToAuthorA}}% - {\xintifSgn{\PolPrintIntervalsTheRightEndPoint} - {\PolErrorThisShouldNotHappenPleaseReportToAuthorB}% - {\PolErrorThisShouldNotHappenPleaseReportToAuthorC}% - {0<\PolPrintIntervalsTheVar_{\PolPrintIntervalsTheIndex}<% - \PolPrintIntervalsPrintRightEndPoint}}% - {\xintifSgn{\PolPrintIntervalsTheRightEndPoint} - {\PolErrorThisShouldNotHappenPleaseReportToAuthorD}% - {\PolErrorThisShouldNotHappenPleaseReportToAuthorE}% - {\PolPrintIntervalsTheVar_{\PolPrintIntervalsTheIndex}=% - \PolPrintIntervalsPrintLeftEndPoint\dots}}% -}% -% -\catcode`_ 11 -\def\POL@PrintIntervals@Loop{% - \POL@SturmIfZeroExactlyKnown\PolPrintIntervalsTheSturmName - \PolPrintIntervalsTheIndex - \PolPrintIntervalsKnownRoot - \PolPrintIntervalsUnknownRoot - \xdef\PolPrintIntervalsTheIndex{\the\numexpr\PolPrintIntervalsTheIndex+\@ne}% - \unless\ifnum\PolPrintIntervalsTheIndex> - \@nameuse{POL_ZL\PolPrintIntervalsTheSturmName*0} - \POL@PrintIntervals@DoDefs - \xint_afterfi{\\\POL@PrintIntervals@Loop}% - \fi -}% -\def\POL@PrintIntervals@DoDefs{% - \xdef\PolPrintIntervalsTheLeftEndPoint{% - \csname POL_ZL\PolPrintIntervalsTheSturmName*\PolPrintIntervalsTheIndex - \endcsname - }% - \xdef\PolPrintIntervalsTheRightEndPoint{% - \csname POL_ZR\PolPrintIntervalsTheSturmName*\PolPrintIntervalsTheIndex - \endcsname - }% - \xdef\PolPrintIntervalsTheMultiplicity{% - \ifcsname POL_ZM\PolPrintIntervalsTheSturmName*\PolPrintIntervalsTheIndex - \endcsname - \csname POL_ZM\PolPrintIntervalsTheSturmName*\PolPrintIntervalsTheIndex - \endcsname - \else - ?% or use 0 ? - \fi - }% -}% - - -\newcommand\PolSturmIfZeroExactlyKnown[2]{% #1 = sturmname, #2=index - \romannumeral0\csname POL_ZK#1*\endcsname{#2}% -}% -\newcommand\POL@SturmIfZeroExactlyKnown[2]{% #1 = sturmname, #2=index - \romannumeral0\csname POL_ZK#1*\the\numexpr#2\endcsname -}% -\newcommand\PolSturmIsolatedZeroMultiplicity[2]{% - \romannumeral`^^@\csname POL_ZM#1*\endcsname{#2}% -}% -\newcommand\PolSturmIsolatedZeroLeft[2]{% - \romannumeral`^^@\csname POL_ZL#1*\endcsname{#2}% -}% -\newcommand\PolSturmIsolatedZeroRight[2]{% - \romannumeral`^^@\csname POL_ZR#1*\endcsname{#2}% -}% -\newcommand\PolSturmNbOfIsolatedZeros[1]{% - \romannumeral`^^@\csname POL_ZL#1*0\endcsname -}% -\newcommand\PolSturmRationalRoot[2]{% - \romannumeral`^^@\csname POL_ZL#1*% - \csname POL_RI#1*\endcsname{#2}\endcsname -}% -\newcommand\PolSturmRationalRootIndex[2]{% - \romannumeral`^^@\csname POL_RI#1*\endcsname{#2}% -}% -\newcommand\PolSturmRationalRootMultiplicity[2]{% - \romannumeral`^^@\csname POL_ZM#1% - *\csname POL_RI#1*\endcsname{#2}\endcsname -}% -\newcommand\PolSturmNbOfRationalRoots[1]{% - \romannumeral`^^@\csname POL_RI#1*0\endcsname -}% -\newcommand\PolSturmNbOfRationalRootsWithMultiplicities[1]{% -% means the \POL@norr must not have been changed in-between... - \the\numexpr\PolDegree{#1}-\PolDegree{#1\POL@norr}\relax -}% - - -\let\PolDecToString\xintDecToString - - -\newcommand\PolMakeMonic[1]{% - \edef\POL@leadingcoeff{\PolLeadingCoeff{#1}}% - \edef\POL@leadingcoeff@inverse{\xintDiv{1/1[0]}{\POL@leadingcoeff}}% - \PolMapCoeffs{\xintMul{\POL@leadingcoeff@inverse}}{#1}% -}% - - -%% CORE ALGEBRA MACROS -%% We do this non-expandably, but in a nestable way... this is the whole -%% point because \xintdeffunc as used by \poldef creates a big nested macro. -%% The idea is to execute it with another meaning given to \xintAdd etc.., -%% so that it operates on "polynomials". This is a mixture of expandable -%% and non-expandable techniques. -%% -%% And it was complicated to let it work with xintexpr 1.4 -%% -\def\POL@get#1#2#3{% - \relax %!! part de la tambouille pour fonctionner en xint 1.4 - \POL@polglobalfalse - \begingroup - \protected\def\POL@result{#3}% - #3% - \expandafter - \endgroup - \expandafter\def\expandafter#1\expandafter{\POL@result}% - \unless\ifPOL@pol - % avoid expanding more than twice #3 - % #3 must be purely numerical or at least compatible with \edef - % this is why at 0.7.5 I had to handle especially constant - % polynomial functions to remove any protection from them - % (because the protection triggers the COMPOSITION when - % the polynomial is found as argument of another one and - % this is not expandable) - \edef#1{#3}% - \xintiiifZero{#1}% - {\def#1{-1.\POL@empty{0/1[0]}}}% - {\edef#1{0.\POL@empty{#1}}}% - \fi - #2% -}% - -%% COMPOSITION -%% This did not exist before 0.7.5 and is part of its adaptation to xint 1.4 -%% We thus took up this opportunity to speed up substantially composition. -%% Very serious difficulties with constant polynomials. Had to handle them -%% especially. -%% OK, that was really tough, but advantage now is that composition -%% at 0.7.5 should be more efficient than before. However when polynomials -%% become big via composition, coefficients also are big and the time -%% taken by arithmetic dominates. No time to test really, though, relieved -%% I can release xint 1.4 at last. My basic polexpr test suite passes, -%% but it goes back already to old releases. -\protected\def\POL@applypolfunc#1#2% -{% -% This #2 may be also invoing \POL@applypolfunc... - \POL@get\POL@A\POL@applypolfunc@b#2#1% -}% -\def\POL@applypolfunc@b #1% -{% -% and now the have our Horner scheme nested macro -% which hopefully will do its job with \POL@add, \POL@mul etc... - \POL@polglobalfalse - \expandafter#1\expanded - {{\POL@polglobaltrue\protected\def\noexpand\POL@result{\POL@A}}}% - \unless\ifPOL@pol - \odef\POL@result{#1{0}}% - \xintiiifZero{\POL@result}% - {\protected\def\POL@result{-1.\POL@empty{0/1[0]}}}% - {\protected\edef\POL@result{0.\POL@empty{\POL@result}}}% - \fi -}% - -%% ADDITION -\def\POL@add {\POL@get\POL@A\POL@add@b}% -\def\POL@add@b{\POL@get\POL@B\POL@add@c}% -\def\POL@add@c{% - \POL@polglobaltrue - \POL@ifZero\POL@A - {\let\POL@result\POL@B}% - {\POL@ifZero\POL@B - {\let\POL@result\POL@A}% - {\POL@@add}}% -}% -\def\POL@@add{% - \expandafter\POL@split\POL@A;\POL@degA\POL@polA - \expandafter\POL@split\POL@B;\POL@degB\POL@polB - \ifnum\POL@degA>\POL@degB\relax - \xintAssignArray\POL@polA\to\POL@arrayA - \xintAssignArray\POL@polB\to\POL@arrayB - \else - \xintAssignArray\POL@polB\to\POL@arrayA - \xintAssignArray\POL@polA\to\POL@arrayB - \let\POL@tmp\POL@degB\let\POL@degB\POL@degA\let\POL@degA\POL@tmp - \fi - \count@\z@ - \xintloop - \advance\count@\@ne - \expandafter\edef\csname POL@arrayA\the\count@\endcsname - {\xintScalarAdd{\@nameuse{POL@arrayA\the\count@}}% - {\@nameuse{POL@arrayB\the\count@}}}% - \unless\ifnum\POL@degB<\count@ - \repeat - \count@\@nameuse{POL@arrayA0} % 1+\POL@degA - % trim zero leading coefficients (we could check for equal degrees, - % but would not bring much as anyhow loop exists immediately if not) - \xintloop - % this abuses that \POL@arrayA0 is never zero - \xintiiifZero{\@nameuse{POL@arrayA\the\count@}}% - {\iftrue}% - {\iffalse}% - \advance\count@\m@ne - \repeat - \POL@resultfromarray A% attention that \POL@arrayA0 not updated -}% - -%% MULTIPLICATION -\def\POL@mul {\POL@get\POL@A\POL@mul@b}% -\def\POL@mul@b{\POL@get\POL@B\POL@mul@c}% -\def\POL@mul@c{% - \POL@polglobaltrue - \POL@ifZero\POL@A - {\protected\def\POL@result{-1.\POL@empty{0/1[0]}}}% - {\POL@ifZero\POL@B - {\protected\def\POL@result{-1.\POL@empty{0/1[0]}}}% - {\POL@@mul}}% -}% -\def\POL@@mul{% - \expandafter\POL@split\POL@A;\POL@degA\POL@polA - \expandafter\POL@split\POL@B;\POL@degB\POL@polB - \ifnum\POL@degA>\POL@degB\relax - \xintAssignArray\POL@polA\to\POL@arrayA - \xintAssignArray\POL@polB\to\POL@arrayB - \else - \xintAssignArray\POL@polB\to\POL@arrayA - \xintAssignArray\POL@polA\to\POL@arrayB - \let\POL@tmp\POL@degB - \let\POL@degB\POL@degA - \let\POL@degA\POL@tmp - \fi - \count@\z@ - \xintloop - \POL@@mul@phaseIloopbody - \unless\ifnum\POL@degB<\count@ - \repeat - \xintloop - \unless\ifnum\POL@degA<\count@ % car attention au cas de mêmes degrés - \POL@@mul@phaseIIloopbody - \repeat - \edef\POL@degC{\the\numexpr\POL@degA+\POL@degB}% - \xintloop - \unless\ifnum\POL@degC<\count@ - \POL@@mul@phaseIIIloopbody - \repeat - %\count@\the\numexpr\POL@degC+\@ne\relax % never zero polynomial here - \POL@resultfromarray C% -}% -\def\POL@@mul@phaseIloopbody{% - \advance\count@\@ne - \def\POL@tmp{0[0]}% - \count\tw@\z@ - \xintloop - \advance\count\tw@\@ne - \edef\POL@tmp{% - \xintScalarAdd - {\POL@tmp}% - {\xintScalarMul - {\@nameuse{POL@arrayA\the\count\tw@}}% - {\@nameuse{POL@arrayB\the\numexpr\count@+\@ne-\count\tw@}}% - }% - }% - \ifnum\count\tw@<\count@ - \repeat - \expandafter\let\csname POL@arrayC\the\count@\endcsname\POL@tmp -}% -\def\POL@@mul@phaseIIloopbody{% - \advance\count@\@ne - \def\POL@tmp{0[0]}% - \count\tw@\count@ - \advance\count\tw@-\@nameuse{POL@arrayB0} % - \xintloop - \ifnum\count\tw@<\count@ - \advance\count\tw@\@ne - \edef\POL@tmp{% - \xintScalarAdd - {\POL@tmp}% - {\xintScalarMul - {\@nameuse{POL@arrayA\the\count\tw@}}% - {\@nameuse{POL@arrayB\the\numexpr\count@+\@ne-\count\tw@}}% - }% - }% - \repeat - \expandafter\let\csname POL@arrayC\the\count@\endcsname\POL@tmp -}% -\def\POL@@mul@phaseIIIloopbody{% - \advance\count@\@ne - \def\POL@tmp{0[0]}% - \count\tw@\count@ - \advance\count\tw@-\@nameuse{POL@arrayB0} % - \xintloop - \advance\count\tw@\@ne - \edef\POL@tmp{% - \xintScalarAdd{\POL@tmp}% - {\xintScalarMul - {\@nameuse{POL@arrayA\the\count\tw@}}% - {\@nameuse{POL@arrayB\the\numexpr\count@+\@ne-\count\tw@}}% - }% - }% - \ifnum\@nameuse{POL@arrayA0}>\count\tw@ - \repeat - \expandafter\let\csname POL@arrayC\the\count@\endcsname\POL@tmp -}% - -%% POWERS (SCALAR EXPONENT...) -\def\POL@pow #1#2{% - \POL@polglobalfalse - \begingroup - \protected\def\POL@result{#1}% - #1% - \expandafter - \endgroup - \expandafter\def\expandafter\POL@A\expandafter{\POL@result}% - \unless\ifPOL@pol - \edef\POL@A{\xintScalarPow{#1}{#2}}% no error check - \xintiiifZero{\POL@A}% - {\protected\def\POL@result{-1.\POL@empty{0/1[0]}}}% - {\protected\edef\POL@result{0.\POL@empty{\POL@A}}}% - \else - \edef\POL@B{\numexpr\xintNum{#2}\relax}% no check on exponent >= 0 - \ifcase\POL@B - \protected\def\POL@result{0.\POL@empty{1/1[0]}}% - \or - \let\POL@result\POL@A - \else - \POL@@pow@check - \fi - \fi - \POL@polglobaltrue -}% -\def\POL@@pow@check {% -% no problem here with leftover tokens! -% should I have used that I-don't-care technique more elsewhere? - \ifnum\@ne>\POL@A - % polynomial is a constant, must get rid of dot and \empty (\POL@empty) - \edef\POL@A{\expandafter\xintScalarPow\romannumeral`^^@% - \expandafter\xint_gob_til_dot\POL@A{\POL@B}}% - \xintiiifZero{\POL@A}% - {\protected\def\POL@result{-1.\POL@empty{0/1[0]}}}% - {\protected\edef\POL@result{0.\POL@empty{\POL@A}}}% - \else - \ifnum\@ne=\POL@A - % perhaps a constant times X, check constant term - \xintiiifZero - {\expandafter\xint_firstoftwo\romannumeral`^^@% - \expandafter\xint_gob_til_dot\POL@A} - {\protected\edef\POL@result - {\the\POL@B.% here at least 2. - \POL@empty - \romannumeral\xintreplicate{\POL@B}{{0/1[0]}}% - {\xintScalarPow - {\expandafter\xint_secondoftwo\romannumeral`^^@% - \expandafter\xint_gob_til_dot\POL@A}% - {\POL@B}}}}% - {\POL@@pow}% not constant times X, use general recursion - \else - \POL@@pow% general recursion - \fi\fi -}% -\def\POL@@pow@recurse#1#2{% - \begingroup - #1% - \expandafter - \endgroup - \expandafter\def\expandafter\POL@A\expandafter{\POL@result}% - \edef\POL@B{\numexpr\xintNum{#2}\relax}% - \ifcase\POL@B - \POL@thisshouldneverhappen - \or - \let\POL@result\POL@A - \else - \expandafter\POL@@pow - \fi -}% -\def\POL@@pow {% - \let\POL@pow@exp\POL@B - \let\POL@B\POL@A - \POL@@mul - \let\POL@sqA\POL@result - \ifodd\POL@pow@exp\space - \expandafter\POL@@pow@odd - \the\numexpr(\POL@pow@exp+\@ne)/\tw@-\@ne\expandafter.% - \else - \expandafter\POL@@pow@even - \the\numexpr(\POL@pow@exp+\@ne)/\tw@-\@ne\expandafter.% - \fi -}% -\def\POL@@pow@even#1.{% - \expandafter\POL@@pow@recurse\expandafter - {\expandafter\def\expandafter\POL@result\expandafter{\POL@sqA}}% - {#1}% -}% -\def\POL@@pow@odd#1.{% - \expandafter\POL@@pow@odd@i\expandafter{\POL@A}{#1}% -}% -\def\POL@@pow@odd@i #1#2{% - \expandafter\POL@@pow@recurse\expandafter - {\expandafter\def\expandafter\POL@result\expandafter{\POL@sqA}}% - {#2}% - \expandafter\POL@mul\expandafter - {\expandafter\def\expandafter\POL@result\expandafter - {\POL@result}\POL@polglobaltrue}% - {\protected\def\POL@result{#1}\POL@polglobaltrue}% -}% - -%% DIVISION -%% no check on divisor being non-zero -\def\POL@div {\POL@get\POL@A\POL@div@b}% -\def\POL@div@b{\POL@get\POL@B\POL@div@c}% -\def\POL@div@c{% - \POL@polglobaltrue - \expandafter\POL@split\POL@A;\POL@degA\POL@polA - \expandafter\POL@split\POL@B;\POL@degB\POL@polB - \ifnum\POL@degA<\POL@degB\space - \@namedef{POL@arrayQ1}{0/1[0]}% - \def\POL@degQ{-1}% - \else - \xintAssignArray\POL@polA\to\POL@arrayR - \xintAssignArray\POL@polB\to\POL@arrayB - \POL@@div - \fi - \count@\numexpr\POL@degQ+\@ne\relax - \POL@resultfromarray Q% -}% -\def\POL@@div{% - \xintAssignArray\POL@polA\to\POL@arrayR - \xintAssignArray\POL@polB\to\POL@arrayB - \edef\POL@B@leading{\csname POL@arrayB\the\numexpr\POL@degB+\@ne\endcsname}% - \edef\POL@degQ{\the\numexpr\POL@degA-\POL@degB}% - \count@\numexpr\POL@degA+\@ne\relax - \count\tw@\numexpr\POL@degQ+\@ne\relax - \xintloop - \POL@@div@loopbody - \ifnum\count\tw@>\z@ - \repeat - %%\expandafter\def\csname POL@arrayR0\endcsname{1}% - \xintloop - \xintiiifZero{\csname POL@arrayR\the\count@\endcsname}% - {\iftrue}% - {\iffalse}% - \advance\count@\m@ne - \repeat - \edef\POL@degR{\the\numexpr\count@-\@ne}% -}% -\def\POL@@div@loopbody{% - \edef\POL@@div@ratio{% - \xintScalarDiv{\csname POL@arrayR\the\count@\endcsname}% - {\POL@B@leading}}% - \expandafter\let\csname POL@arrayQ\the\count\tw@\endcsname - \POL@@div@ratio - \advance\count@\m@ne - \advance\count\tw@\m@ne - \count4 \count@ - \count6 \POL@degB\space - \xintloop - \ifnum\count6>\z@ - \expandafter\edef\csname POL@arrayR\the\count4\endcsname - {\xintScalarSub - {\csname POL@arrayR\the\count4\endcsname}% - {\xintScalarMul - {\POL@@div@ratio}% - {\csname POL@arrayB\the\count6\endcsname}}}% - \advance\count4 \m@ne - \advance\count6 \m@ne - \repeat -}% - -%% MINUS SIGN AS UNARY OPERATOR -\def\POL@opp #1{% - \POL@polglobalfalse - \begingroup - \protected\def\POL@result{#1}% - #1% - \expandafter - \endgroup - \expandafter\def\expandafter\POL@A\expandafter{\POL@result}% - \unless\ifPOL@pol - \edef\POL@A{\xintScalarOpp{#1}}% - \xintiiifZero{\POL@A}% - {\protected\def\POL@result{-1.\POL@empty{0/1[0]}}}% - {\protected\edef\POL@result{0.\POL@empty{\POL@A}}}% - \else - \edef\POL@B{0.\POL@empty{-1/1[0]}}% - \POL@@mul - \fi - \POL@polglobaltrue -}% - - -%% EXPANDABLE MACROS -\def\POL@eval@fork#1\At#2#3\krof{#2}% -\newcommand\PolEval[3]{\romannumeral`^^@\POL@eval@fork - #2\PolEvalAt - \At\PolEvalAtExpr\krof {#1}{#3}% -}% -\newcommand\PolEvalAt[2] - {\xintpraw{\csname XINT_expr_polfunc_#1\endcsname{#2}}}% -\newcommand\POL@eval[2] - {\csname XINT_expr_polfunc_#1\endcsname{#2}}% -\newcommand\PolEvalAtExpr[2]{\xinttheexpr #1(#2)\relax}% -% -\newcommand\PolEvalReduced[3]{\romannumeral`^^@\POL@eval@fork - #2\PolEvalReducedAt - \At\PolEvalReducedAtExpr\krof {#1}{#3}% -}% -\newcommand\PolEvalReducedAt[2]{% - \xintpraw % in order not to print denominator if the latter equals 1 - {\xintIrr{\csname XINT_expr_polfunc_#1\endcsname{#2}}[0]}% -}% -\newcommand\PolEvalReducedAtExpr[2]{% - \xintpraw - {\expandafter\xintIrr\romannumeral`^^@\xintthebareeval#1(#2)\relax[0]}% -}% -% -\newcommand\PolFloatEval[3]{\romannumeral`^^@\POL@eval@fork - #2\PolFloatEvalAt - \At\PolFloatEvalAtExpr\krof {#1}{#3}% -}% -\newcommand\PolFloatEvalAt[2] - {\xintpfloat{\csname XINT_flexpr_polfunc_#1\endcsname{#2}}}% -\newcommand\PolFloatEvalAtExpr[2]{\xintthefloatexpr #1(#2)\relax}% - - -\newcommand\PolSturmIntervalIndex[3]{\the\numexpr\POL@eval@fork - #2\PolSturmIntervalIndexAt - \At\PolSturmIntervalIndexAtExpr\krof {#1}{#3}% -}% -\newcommand\PolSturmIntervalIndexAtExpr[2] - {\PolSturmIntervalIndexAt{#1}{\xinttheexpr#2\relax}}% -\newcommand\PolSturmIntervalIndexAt[2] - {\expandafter\POL@sturm@index@at\romannumeral`^^@#2!{#1}\xint_bye\relax}% -\def\POL@sturm@index@at#1!#2% -{% - \expandafter\POL@sturm@index@at@iloop - \romannumeral`^^@\PolSturmNbOfIsolatedZeros{#2}!{#2}{#1}% -}% -% implementation is sub-optimal as it should use some kind of binary tree -% search rather than comparing to the intervals from right to left as here -\def\POL@sturm@index@at@iloop #1!% -{% - \ifnum #1=\z@ 0\expandafter\xint_bye\fi - \POL@sturm@index@at@iloop@a #1!% -}% -\def\POL@sturm@index@at@iloop@a #1!#2#3% -{% #1 = index, #2 = sturmname, #3 value - \PolSturmIfZeroExactlyKnown{#2}{#1} - {\xintifCmp{#3}{\POL@xintexprGetVar{#2L_#1}}% - {}% - {#1\xint_bye}% - {0\xint_bye}% - }% - {\xintifGt{#3}{\POL@xintexprGetVar{#2L_#1}}% - {\xintifLt{#3}{\POL@xintexprGetVar{#2R_#1}}% - {#1\xint_bye}% - {0\xint_bye}% - }% - {}% - }% - % catcode of ! is 11 in polexpr.sty - \expandafter\POL@sturm@index@at@iloop\the\numexpr#1-\@ne !{#2}{#3}% -}% - - -\def\POL@leq@fork#1\LessThanOrEqualTo#2#3\krof{#2}% -\newcommand\PolSturmNbOfRootsOf[3]{\romannumeral`^^@\POL@leq@fork - #2\PolNbOfRootsLessThanOrEqualTo - \LessThanOrEqualTo\PolNbOfRootsLessThanOrEqualToExpr\krof {#1}{#3}% -}% -\newcommand\PolNbOfRootsLessThanOrEqualToExpr[2] - {\PolNbOfRootsLessThanOrEqualTo{#1}{\xinttheexpr#2\relax}}% -\newcommand\PolNbOfRootsLessThanOrEqualTo[1]{% - \ifnum\PolSturmNbOfIsolatedZeros{#1}=\z@ - \expandafter\xint_firstofthree\expandafter0% - \else - \expandafter\PolNbOfRootsLessThanOrEqualTo@% - \fi {#1}% -}% -\def\PolNbOfRootsLessThanOrEqualTo@ #1#2% -{% - \expandafter\POL@nbofrootsleq@prep\romannumeral`^^@#2!{#1}% -}% -\def\POL@nbofrootsleq@prep#1!#2% -{% - \expandafter\POL@nbofrootsleq@iloop\expandafter 1\expandafter !% - \romannumeral0\xintsgn{\POL@eval{#2_0}{#1}}!% - #1!{#2}% -}% -\def\POL@nbofrootsleq@iloop#1!#2!#3!#4% -{% #1 = index, #2 = sign of evaluation at value, #3 = value, #4 = sturmname - \xintifCmp{#3}{\POL@xintexprGetVar{#4L_#1}}% - {\POL@nbofrootsleq@return #1-\@ne !}% - {\POL@nbofrootsleq@return - \PolSturmIfZeroExactlyKnown{#4}{#1}{#1}{#1-\@ne}!% - }% - % in third branch we are sure that if root is exactly known - % the test \xintifLt will be negative - {\xintifLt{#3}{\POL@xintexprGetVar{#4R_#1}}% - {\POL@nbofrootsleq@return - #1\ifnum#2=\xintSgn{\POL@eval{#4_0}{\POL@xintexprGetVar{#4L_#1}}} - -\@ne\fi !% - }% - {\ifnum#1=\PolSturmNbOfIsolatedZeros{#4} - \expandafter\POL@nbofrootsleq@rightmost - \fi \expandafter\POL@nbofrootsleq@iloop \the\numexpr\@ne+% - }% - }% - #1!#2!#3!{#4}% -}% -\def\POL@nbofrootsleq@return #1!#2!#3!#4!#5{\the\numexpr #1\relax}% -\def\POL@nbofrootsleq@rightmost\expandafter\POL@nbofrootsleq@iloop - \the\numexpr\@ne+#1!#2!#3!#4{#1}% - - -\newcommand\PolSturmNbWithMultOfRootsOf[3] -{\the\numexpr0\POL@leq@fork - #2\PolNbWithMultOfRootsLessThanOrEqualTo - \LessThanOrEqualTo\PolNbWithMultOfRootsLessThanOrEqualToExpr\krof {#1}{#3}% -}% -\newcommand\PolNbWithMultOfRootsLessThanOrEqualToExpr[2] - {\PolNbWithMultOfRootsLessThanOrEqualTo{#1}{\xinttheexpr#2\relax}}% -\newcommand\PolNbWithMultOfRootsLessThanOrEqualTo[1]{% - \ifnum\PolSturmNbOfIsolatedZeros{#1}=\z@ - \expandafter\POL@nbwmofroots@noroots - \else - \expandafter\PolNbWithMultOfRootsLessThanOrEqualTo@% - \fi {#1}% -}% -\def\POL@nbwmofroots@noroots#1#2{\relax}% -\def\PolNbWithMultOfRootsLessThanOrEqualTo@ #1#2% -{% - \expandafter\POL@nbwmofrootsleq@prep\romannumeral`^^@#2!{#1}% -}% -\def\POL@nbwmofrootsleq@prep#1!#2% -{% - \expandafter\POL@nbwmofrootsleq@iloop\expandafter 1\expandafter !% - \romannumeral0\xintsgn{\POL@eval{#2_0}{#1}}!% - #1!{#2}% -}% -\def\POL@nbwmofrootsleq@iloop#1!#2!#3!#4% -{% #1 = index, #2 = sign of evaluation at value, #3 = value, #4 = sturmname - \xintifCmp{#3}{\POL@xintexprGetVar{#4L_#1}}% - {\POL@nbwmofrootsleq@return !}% - {\POL@nbwmofrootsleq@return - \PolSturmIfZeroExactlyKnown{#4}{#1}% - {+\PolSturmIsolatedZeroMultiplicity{#4}{#1}}{}!% - }% - % in third branch we are sure that if root is exactly known - % the test \xintifLt will be negative - {\xintifLt{#3}{\POL@xintexprGetVar{#4R_#1}}% - {\POL@nbwmofrootsleq@return - \unless - \ifnum#2=\xintSgn{\POL@eval{#4_0}{\POL@xintexprGetVar{#4L_#1}}} - +\PolSturmIsolatedZeroMultiplicity{#4}{#1}\fi !% - }% - {+\PolSturmIsolatedZeroMultiplicity{#4}{#1}% - \ifnum#1=\PolSturmNbOfIsolatedZeros{#4} - \expandafter\POL@nbwmofrootsleq@return\expandafter !% - \fi - \expandafter\POL@nbwmofrootsleq@iloop \the\numexpr\@ne+% - }% - }% - #1!#2!#3!{#4}% -}% -\def\POL@nbwmofrootsleq@return #1!#2!#3!#4!#5{#1\relax}% - - -\newcommand\PolLeadingCoeff[1]{% - \romannumeral`^^@\expandafter\expandafter\expandafter\xintlastitem - \expandafter\expandafter\expandafter - {\csname POLuserpol@#1\endcsname}% -}% -% -\newcommand\PolNthCoeff[2]{\romannumeral`^^@% - \expandafter\POL@nthcoeff - \romannumeral0\xintnthelt{\ifnum\numexpr#2<\z@#2\else(#2)+1\fi}% - {\expandafter\expandafter\expandafter - \xint_gob_til_dot\csname POLuserpol@#1\endcsname}@% -}% -\def\POL@nthcoeff#1@{\if @#1@\expandafter\xint_firstoftwo - \else\expandafter\xint_secondoftwo\fi - {0/1[0]}{#1}}% -% -% returns -1 for zero polynomial for context of numerical expression -% should it return -\infty? -\newcommand\PolDegree[1]{\romannumeral`^^@\expandafter\expandafter\expandafter - \POL@degree\csname POLuserpol@#1\endcsname;}% -\def\POL@degree #1.#2;{#1}% -% -\newcommand\PolToList[1]{\romannumeral`^^@\expandafter\expandafter\expandafter - \xint_gob_til_dot\csname POLuserpol@#1\endcsname}% -% -\newcommand\PolToCSV[1]{\romannumeral0\xintlistwithsep{, }{\PolToList{#1}}}% - - -\newcommand\PolToExprCmd[1]{\xintPRaw{\xintRawWithZeros{#1}}}% -\newcommand\PolToFloatExprCmd[1]{\xintFloat{#1}}% -\let\PolToExprTermPrefix\PolTypesetCmdPrefix -\newcommand\PolToExprOneTermStyleA[2]{% - \ifnum#2=\z@ - \PolToExprCmd{#1}% - \else - \xintifOne{\xintiiAbs{#1}} - {\xintiiifSgn{#1}{-}{}{}}% + from \PolToExprTermPrefix - {\PolToExprCmd{#1}\PolToExprTimes}% - \fi - \ifcase\xintiiAbs{#2} %<-- space here mandatory - \or\PolToExprVar - \else\PolToExprVar^\xintiiAbs{#2}% - \fi -}% -\let\PolToExprOneTerm\PolToExprOneTermStyleA -\newcommand\PolToExprOneTermStyleB[2]{% - \ifnum#2=\z@ - \xintNumerator{#1}% - \else - \xintifOne{\xintiiAbs{\xintNumerator{#1}}} - {\xintiiifSgn{#1}{-}{}{}}% + from \PolToExprTermPrefix - {\xintNumerator{#1}\PolToExprTimes}% - \fi - \ifcase\xintiiAbs{#2} %<-- space here mandatory - \or\PolToExprVar - \else\PolToExprVar^\xintiiAbs{#2}% - \fi - \xintiiifOne{\xintDenominator{#1}}{}{/\xintDenominator{#1}}% -}% -\newcommand\PolToFloatExprOneTerm[2]{% - \ifnum#2=\z@ - \PolToFloatExprCmd{#1}% - \else - \PolToFloatExprCmd{#1}\PolToExprTimes - \fi - \ifcase\xintiiAbs{#2} %<-- space here mandatory - \or\PolToExprVar - \else\PolToExprVar^\xintiiAbs{#2}% - \fi -}% -\newcommand\PolToExprTimes{*}% -\newcommand\PolToExprVar{x}% -\newcommand\PolToExpr[1]{% - \if*\noexpand#1\expandafter\xint_firstoftwo\else - \expandafter\xint_secondoftwo\fi - \PolToExprAscending\PolToExprDescending{#1}}% -\newcommand\PolToFloatExpr[1]{% - \if*\noexpand#1\expandafter\xint_firstoftwo\else - \expandafter\xint_secondoftwo\fi - \PolToFloatExprAscending\PolToFloatExprDescending{#1}}% -\newcommand\PolToExprAscending[2]{% - \expandafter\POL@toexpr\csname POLuserpol@#2\endcsname - \PolToExprOneTerm\POL@toexprA}% -\newcommand\PolToFloatExprAscending[2]{% - \expandafter\POL@toexpr\csname POLuserpol@#2\endcsname - \PolToFloatExprOneTerm\POL@toexprA}% -\newcommand\PolToExprDescending[1]{% - \expandafter\POL@toexpr\csname POLuserpol@#1\endcsname - \PolToExprOneTerm\POL@toexprD}% -\newcommand\PolToFloatExprDescending[1]{% - \expandafter\POL@toexpr\csname POLuserpol@#1\endcsname - \PolToFloatExprOneTerm\POL@toexprD}% -% -\def\POL@toexpr#1#2#3{\expandafter\POL@toexpr@ - \expandafter#3\expandafter#2#1\relax}% -\def\POL@toexpr@#1#2#3.{% - \ifnum#3<\z@ - #2{0/1[0]}{0}\expandafter\xint_gobble_v - \else - \expandafter#1% - \fi {#3}#2}% -% -\def\POL@toexprA #1#2\POL@empty#3{% - \ifpoltoexprall\expandafter\POL@toexprall@b - \else\expandafter\POL@toexpr@b - \fi {#3}#2{0}1.% -}% -\def\POL@toexprD #1#2#3\relax{% #3 has \empty (\POL@empty) to prevent brace removal - \expandafter\POL@toexprD@a\expandafter#2% - \the\numexpr #1\expandafter.\romannumeral0\xintrevwithbraces{#3}\relax -}% -\def\POL@toexprD@a #1#2.#3{% - \ifpoltoexprall\expandafter\POL@toexprall@b - \else\expandafter\POL@toexpr@b - \fi{#3}#1{-#2}\the\numexpr\@ne+-#2.% -}% -\def\POL@toexpr@b #1#2#3{% - \xintiiifZero{#1}% - {\expandafter\POL@toexpr@loop\expandafter\POL@toexpr@b}% - {#2{#1}{#3}% - \expandafter\POL@toexpr@loop\expandafter\POL@toexpr@c}% - \expandafter#2% -}% -\def\POL@toexpr@c #1#2#3{% - \xintiiifZero{#1}% - {}% - {\PolToExprTermPrefix{#1}#2{#1}{#3}}% - \expandafter\POL@toexpr@loop\expandafter\POL@toexpr@c - \expandafter#2% -}% -\def\POL@toexprall@b #1#2#3{% - #2{#1}{#3}% - \expandafter\POL@toexpr@loop\expandafter\POL@toexprall@c - \expandafter#2% -}% -\def\POL@toexprall@c #1#2#3{% - \PolToExprTermPrefix{#1}#2{#1}{#3}% - \expandafter\POL@toexpr@loop\expandafter\POL@toexprall@c - \expandafter#2% -}% -\def\POL@toexpr@loop#1#2#3.#4{% - \if\relax#4\expandafter\xint_gob_til_dot\fi - #1{#4}#2{#3}\the\numexpr\@ne+#3.% -}% - - -\POL@restorecatcodes -\endinput |