% author: Jean-François Burnol
% License: LPPL 1.3c (author-maintained)
% Usage: \input polexpr.sty (Plain or other macro formats)
% or \usepackage{polexpr} (LaTeX macro format)
% Release 0.8.7a (2022/05/19) of polexpr.sty. This file inputs
% polexprcore.tex
% polexprexpr.tex
% polexprsturm.tex
\begingroup\catcode61\catcode48\catcode32=10\relax%
\catcode13=5 % ^^M
\endlinechar=13 %
\catcode123=1 % {
\catcode125=2 % }
\catcode64=11 % @
\catcode35=6 % #
\catcode44=12 % ,
\catcode45=12 % -
\catcode46=12 % .
\catcode58=12 % :
\def\z {\endgroup}%
\expandafter\let\expandafter\x\csname ver@polexpr.sty\endcsname
\expandafter\let\expandafter\w\csname ver@xintexpr.sty\endcsname
\expandafter
\ifx\csname PackageInfo\endcsname\relax
\def\y#1#2{\immediate\write-1{Package #1 Info: #2.}}%
\else
\def\y#1#2{\PackageInfo{#1}{#2}}%
\fi
\expandafter
% I don't think engine exists providing \expanded but not \numexpr
\ifx\csname expanded\endcsname\relax
\y{polexpr}{\expanded not available, aborting input}%
\aftergroup\endinput
\else
\ifx\x\relax % plain-TeX, first loading of polexpr.sty
\ifx\w\relax % but xintexpr.sty not yet loaded.
\expandafter\def\expandafter\z\expandafter
{\z\input xintexpr.sty\relax}%
\fi
\else
\def\empty {}%
\ifx\x\empty % LaTeX, first loading,
% variable is initialized, but \ProvidesPackage not yet seen
\ifx\w\relax % xintexpr.sty not yet loaded.
\expandafter\def\expandafter\z\expandafter
{\z\RequirePackage{xintexpr}[2021/03/29]}%
\fi
\else
\aftergroup\endinput % polexpr already loaded.
\fi
\fi
\fi
\z%
\XINTsetupcatcodes% (does \endlinechar13 in particular)
\XINT_providespackage
\ProvidesPackage{polexpr}%
[2022/05/19 v0.8.7a Polynomial expressions with rational coefficients (JFB)]%
\begingroup
\def\x#1/#2/#3 #4\xint:{#1#2#3}%
\ifnum\expandafter\x\expanded{\csname ver@xintexpr.sty\endcsname}\xint:
<20210527 % xint 1.4h
\errmessage{Package polexpr error: xintexpr too old, aborting input}%
\else\expandafter\xint_gobble_i
\fi
\endinput\endgroup
\let\PolDecToString\xintDecToString
\long\def\POL@ifstar#1#2%
{%
\begingroup\def\@tempa{#1}\def\@tempb{#2}%
\futurelet\@let@token\POL@@ifstar
}%
\def\POL@@ifstar
{%
\xint_firstofone{\ifx} \@let@token\def\next{\POL@@again\POL@@ifstar}\else
\ifx*\@let@token\def\next##1{\expandafter\endgroup\@tempa}\else
\def\next{\expandafter\endgroup\@tempb}\fi\fi\next
}%
\xint_firstofone{\def\POL@@again#1} {\futurelet\@let@token#1}%
\long\def\POL@chkopt#1[#2]%
{%
\begingroup\def\@tempa{#1}\def\@tempb{#1[#2]}%
\futurelet\@let@token\POL@@ifopt
}%
\def\POL@@ifopt
{%
\xint_firstofone{\ifx} \@let@token\def\next{\POL@@again\POL@@ifopt}\else
\ifx[\@let@token\def\next{\expandafter\endgroup\@tempa}\else %]
\def\next{\expandafter\endgroup\@tempb}\fi\fi\next
}%
% \polexprsetup added at 0.7
\catcode`! 3 %
\def\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
}%
\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}%
\catcode`! 11 % special catcode for ! as used in xintexpr.sty
%
\newif\ifxintveryverbose
\newif\ifpolnewpolverbose
\newif\ifpoltypesetall
\newif\ifpoltoexprall
%%
%% Main data format for non-expandable manipulations
%%
%% The main exchange structure is:
%% N.\empty{coeff0}{coeff1}....{coeffN}
%% It is stored in macros \POLuserpol@<name of polynomial>
%% The \empty is basically there to avoid brace-stripping
%% in some grabbing contexts (maybe I should revisit this)
%%
%% The zero polynomial is stored as -1.\empty{0/1[0]}
%% Degree zero polynomials are 0.\empty{numeric value}
%%
%% Depending on input path the numeric values coeff0, coeff1, ...., coeffN
%% may have been or not already converted into A/B[n] format.
%% As a rule, computations are not followed with reducing the fractions
%% to smallest terms; the innocent may be unaware that computing
%% with fractions quickly give gigantic numbers. There is \PolReduceCoeffs
%% to do that.
%%
%% This base structure is maintained at 0.8 for legacy reasons but perhaps I
%% need to revisit this. A characteristic of the package so far is that it
%% thus stores and manipulate polynomials basically as the complete sequence
%% of coefficients, (using the xintfrac "zero" for missing coefficients) which
%% means that it will handle poorly polynomials of high degrees such as X^500.
%%
%% Test if zero
\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}%
%% Split into degree and coefficients
% The \expandafter chain removes the \empty token
\def\POL@split#1.#2;#3#4%
{\def#3{#1}\expandafter\def\expandafter#4\expandafter{#2}}%
%% Define from values stored in a "macros-array"
\def\POL@resultfromarray #1{%
\edef\POL@result{\ifnum\count@>\z@
\the\numexpr\count@-\@ne.\noexpand\empty
\xintiloop [1+1]%
\expandafter\POL@braceit\csname POL@array#1\xintiloopindex\endcsname
\ifnum\xintiloopindex<\count@
\repeat
\else-1.\noexpand\empty{0/1[0]}\fi}%
}%
\def\POL@braceit#1{{#1}}% needed as \xintiloopindex can not "see" through braces
%%
%% Conversion between legacy data storage and the one used for the
%% the novel polexpr 0.8 notion of \xintexpr polynomial variables
%%
%% The 0.8 expandable implementation of core algebra is also manipulating
%% the complete list of coefficients. The internal data structure is
%% (this is the numeric leaf in xintexpr ople terminology) currently:
%% PN.{coeff0}{coeff1}....{coeffN}
%% where the P letter identifies the polynomial type.
%% Here the degree N is *always* at least 1: if some evaluation ends
%% up in a constant polynomial it will always be output as a genuine
%% scalar numeric variable, as a rule in in A/B[n] format
%%
%% This is not definitive and I need to think about it more (in particular
%% in the distant perspective of supporting multi-variable polynomials).
%% However modifying this will be costly labor at this stage.
%%
\input polexprcore.tex\relax % load expandable algebra
\def\POL@vartolegacy #1% \romannumeral\POL@vartolegacy ... \xint:
{%
\if 0#1\xint_dothis\POL@vartolegacy@zero\fi
\if P#1\xint_dothis\POL@vartolegacy@pol\fi
\xint_orthat\POL@vartolegacy@scalar #1%
}%
\def\POL@vartolegacy@zero #1\xint:{\xint_c_ -1.\empty{0/1[0]}}%
\def\POL@vartolegacy@scalar #1\xint:{\xint_c_ 0.\empty{#1}}%
\def\POL@vartolegacy@pol P#1.#2\xint:{\xint_c_ #1.\empty#2}%
%
\def\POL@tovar#1{\romannumeral\expandafter\expandafter\expandafter
\POL@legacytovar\csname POLuserpol@#1\endcsname}%
\def\POL@legacytovar #1.% \romannumeral\POL@legacytovar N.\empty{c0}...
{%
\ifnum #1<\xint_c_i\xint_dothis\POL@legacytovar@scalar\fi
\xint_orthat\POL@legacytovar@pol #1.%
}%
\def\POL@legacytovar@scalar #1.\empty#2{\xint_c_ #2}%
\def\POL@legacytovar@pol #1.\empty{\xint_c_ P#1.}%
%%
%% Extend \xintexpr (\xintdefvar, \xintdeffunc) to recognize the new
%% polynomial type
%%
%% **** It does NOT apply to \xintfloatexpr context
%%
\input polexprexpr.tex\relax
%%
%% \poldef
%%
%% Ever since 1.0, catcode sanitisation was minimal and only handled
%% the semicolon. At last 0.8.7 uses \xintexprSafeCatcodes to enhance
%% compatibility with hostile contexts such as babel+french. This
%% adds overhead but at least is coherent with \xintdefvar/\xintdeffunc
\def\PolDef{\xintexprSafeCatcodes\POL@chkopt\POL@oPolDef[x]}%
\def\POL@oPolDef[#1]#2#3{\POL@defpol #2(#1):={#3};}%
\def\poldef{\xintexprSafeCatcodes\POL@defpol}%
\def\POL@defpol #1(#2)#3=#4;{%
\xintexprRestoreCatcodes
\edef\POL@polname{\xint_zapspaces #1 \xint_gobble_i}%
\begingroup
\unless\ifxintveryverbose\xintverbosefalse\fi
%% RADICAL CHANGE AT 0.8:
%% we define a **variable** not a **function**
%% ever since polexpr initial version, a function was defined and
%% the associated macros was then deconstructed in further analysis
%% via non-expandable approach. At 0.8 the polynomial algebra has
%% been implemented expandably allowing direct plug-in into \xintexpr
\xintdefvar_a __pol = subs(#4,#2=qraw({{P1.{0/1[0]}{1/1[0]}}}));%
\expandafter
\endgroup
\expandafter\def\expandafter\POL@result\expandafter
{\romannumeral0\expandafter\xint_stop_atfirstofone
\romannumeral0\csname XINT_expr_varvalue___pol\endcsname}%
\XINT_global\expandafter\def\csname POLuserpol@\POL@polname\expandafter\endcsname
\expandafter{\romannumeral\expandafter\POL@vartolegacy\POL@result\xint:}%
\expandafter\POL@newpol\expandafter{\POL@polname}%
}%
\def\POL@newpol#1{%
% 0.7.5 had some complicated special handling of constant
% polynomials, but these are complications of the past
% First a variable usable in \poldef but not in \xintexpr for arithmetic
% only for special dedicated functions such as coeff(), deg()
% (when they will be implemented). In \poldef, composition of polynomials
% in P(Q) syntax will be more efficient than P(Q(x)).
% This will use \XINT_global and obey \xintverbose... setting
\XINT_expr_defvar_one{#1}{{\POL@tovar{#1}}}%
% Second a function usable not only in \poldef but also in \xintexpr
% Will use \XINT_global
\POL@newpolhorner{#1}%
\POL@defpolfunc{#1}{expr}%
\XINT_global\expandafter\let\csname XINT_flexpr_func_#1\endcsname\@undefined
\ifpolnewpolverbose\POL@info{#1}\fi
}%
\def\POL@newfloatpol#1{%
\POL@newfloatpolhorner{#1}%
\POL@defpolfunc{#1}{flexpr}%
\ifpolnewpolverbose\POL@floatinfo{#1}%
\else
\ifxintverbose\POL@floatinfo{#1}\fi
\fi
}%
\def\POL@info #1{%
\xintMessage {polexpr}{Info}%
{Function #1 for the \string\xintexpr\space parser is
\ifxintglobaldefs(globally) \fi
associated to \string\XINT_expr_polfunc_#1\space
with meaning:
\expandafter\meaning
\csname XINT_expr_polfunc_#1\endcsname}%
}%
\def\POL@floatinfo #1{%
\xintMessage {polexpr}{Info}%
{Function #1 for the \string\xintfloatexpr\space parser is
\ifxintglobaldefs(globally) \fi
associated to \string\XINT_flexpr_polfunc_#1\space
with meaning:
\expandafter\meaning
\csname XINT_flexpr_polfunc_#1\endcsname}%
}%
%
\def\POL@newpolhorner#1{%
\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\XINT_global
\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\xintPolAdd\relax\let\xintPolMul\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}
{\xint_firstofone}{\xintPolAdd{#1}}%
{\xintPolMul{##1}{\POL@tmp{##1}}}}%
\POL@newpol@horner@loop.%
}%
\def\POL@newpol@floathorner#1{\let\XINTinFloatAdd\relax\let\XINTinFloatMul\relax
\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}
{\xint_firstofone}{\XINTinFloatAdd{\XINTinFloatdigits{#1}}}%
{\XINTinFloatMul{##1}{\POL@tmp{##1}}}}%
\POL@newpol@floathorner@loop.%
}%
%%
%% Non-expandable polynomial manipulations
%%
\def\PolGenFloatVariant#1{\POL@newfloatpol{#1}}%
%
\def\PolLet#1#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{%
\XINT_global
\expandafter\let\csname POLuserpol@#1\expandafter\endcsname
\csname POLuserpol@#2\endcsname
\XINT_expr_defvar_one{#1}{{\POL@tovar{#1}}}%
\XINT_global
\expandafter\let\csname XINT_expr_polfunc_#1\expandafter\endcsname
\csname XINT_expr_polfunc_#2\endcsname
\POL@defpolfunc{#1}{expr}%
\ifpolnewpolverbose\POL@info{#1}\fi
}%
\def\PolGlobalLet#1#2{\begingroup\xintglobaldefstrue\PolLet{#1}{#2}\endgroup}%
%
\def\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}{ }%
}%
\def\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@
\def\POL@result{-1.\empty{0/1[0]}}% 0.5 fix for empty array
\else
\xintloop
\edef\POL@tmp{#2{\count@}}%
\edef\POL@tmp{\xintRaw{\POL@tmp}}%
% sadly xinttools (current 1.3a) arrays have no setters for individual items...
\expandafter\let\csname POL@tmparray\the\count@\endcsname\POL@tmp
\if0\xintiiSgn{\POL@tmp}%
\advance\count@\m@ne
\repeat
\count\tw@\count@
\xintloop
\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}}%
\edef\POL@result{\the\numexpr\count@-\@ne.\noexpand\empty
\xintiloop[1+1]%
\expandafter\POL@tmp\xintiloopindex.%
\ifnum\xintiloopindex<\count@
\repeat}%
\fi
\expandafter
\endgroup
\expandafter
\XINT_global
\expandafter
\def\csname POLuserpol@#1\expandafter\endcsname
\expandafter{\POL@result}%
}%
%
\def\PolFromCSV#1#2{%
\begingroup % closed in \POL@getfromarray
\xintAssignArray\xintCSVtoList{#2}\to\POL@arrayA
\POL@getfromarray{#1}\POL@arrayA
\POL@newpol{#1}%
}%
%
\def\PolMapCoeffs#1#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 Ã 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
\XINT_global
\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]}%
\def\PolReduceCoeffs{\POL@ifstar\POL@sreducecoeffs\POL@reducecoeffs}%
\def\POL@reducecoeffs#1{\PolMapCoeffs{\POL@xintIrr}{#1}}%
\def\POL@sreducecoeffs#1{\PolMapCoeffs{\xintPIrr}{#1}}%
%
\def\PolMakeMonic#1{%
\edef\POL@leadingcoeff{\PolLeadingCoeff{#1}}%
\edef\POL@leadingcoeff@inverse{\xintDiv{1/1[0]}{\POL@leadingcoeff}}%
\PolMapCoeffs{\xintMul{\POL@leadingcoeff@inverse}}{#1}%
}%
%
%% \PolMakePrimitive (0.5)
% This uses expandable \PolIContent
% Note: the integer coefficients stored in A/1[n] form with
% A not having trailing zeroes, due to usage of \xintREZ here.
\def\POL@makeprim@macro#1%
{\xintREZ{\xintNum{\xintDiv{#1}{\POL@makeprim@icontent}}}}%
\def\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}%
}%
%
%% Euclidean division
% since 0.8 based on the expandable routine from polexprcore.tex
%
\def\PolDivide#1#2#3#4{% #3=quotient, #4=remainder of #1 by #2
\POL@divide{#1}{#2}%
\XINT_global\expandafter\let\csname POLuserpol@#3\endcsname\POL@Q
\POL@newpol{#3}%
\XINT_global\expandafter\let\csname POLuserpol@#4\endcsname\POL@R
\POL@newpol{#4}%
}%
\def\PolQuo#1#2#3{% #3=quotient of #1 by #2
\POL@divide{#1}{#2}%
\XINT_global\expandafter\let\csname POLuserpol@#3\endcsname\POL@Q
\POL@newpol{#3}%
}%
\def\PolRem#1#2#3{% #3=remainder of #1 by #2
\POL@divide{#1}{#2}%
\XINT_global\expandafter\let\csname POLuserpol@#3\endcsname\POL@R
\POL@newpol{#3}%
}%
\def\POL@divide#1#2{%
% much simpler at 0.8 thanks to our expandable macros
\xintAssign\xintPolQuoRem{\POL@tovar{#1}}{\POL@tovar{#2}}\to\POL@Q\POL@R
\odef\POL@Q{\romannumeral\expandafter\POL@vartolegacy\POL@Q\xint:}%
\odef\POL@R{\romannumeral\expandafter\POL@vartolegacy\POL@R\xint:}%
}%
%% Euclidean special pseudo-remainder
\def\POL@getprem#1#2{%
\let\POL@Q\undefined % trap errors in Sturm code update to use \POL@prem
% this was simpler before I converted \xintPolPRem into returning a tuple...
\odef\POL@R{\romannumeral\expandafter\POL@vartolegacy
\romannumeral0\expandafter\xint_stop_atsecondoftwo
\romannumeral`&&@\xintPolPRem{\POL@tovar{#1}}{\POL@tovar{#2}}%
\xint:}%
}%
%
%%%%%%%%%%%%
%%
%% Things are currenly implemented twice : here the legacy macros
%% such as GCD or Diff, and in polexprcore.tex the expandable
%% support macros for the \xinteval interface.
%%
%% Soon, I will probably remove all legacy code (like I did already
%% for division) and make the user macros simple wrappers to the
%% expandable code.
%%
%% But for 0.8 release, I preferred not to yet, as I did not have
%% really the time to compare speed. Usage of the "special
%% pseudo euclidean remainder" (expandable) code in Sturm chain
%% construction proved very beneficial as it divided by 3 the
%% \PolToSturm execution time on the Wilkinson perturbed type 1
%% example in the documentation.
%%
%%%%%%%%%%%%
%
%% GCD
%
% It seems I didn't even use here the (now deleted) macros implementing
% division, and I redid here what was needed: this code, which I leave
% standing as I have other priorities, does not use the \POL@divide !
%
\def\PolGCD#1#2#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
\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
{\def\POL@result{-1.\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
\XINT_global
\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{\xintDiv
{\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
{\xintSub
{\csname POL@array#1\the\count4\endcsname}%
{\xintMul
{\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%
}%
%
%% 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.%
}%
\def\PolDiff{\POL@chkopt\POL@oPolDiff[1]}%
\def\POL@oPolDiff[#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\xint_firstoftwo
\else
\expandafter\xint_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
\XINT_global\@namedef{POLuserpol@#2}{-1.\empty{0/1[0]}}%
\else
\edef\POL@var@coeffs{\expandafter\POL@diff\POL@var@coeffs\relax}%
\XINT_global\expandafter\edef\csname POLuserpol@#2\endcsname
{\the\numexpr\POL@var@deg-\@ne.\noexpand\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
\XINT_global
\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.%
}%
\def\PolAntiDiff{\POL@chkopt\POL@oPolAntiDiff[1]}%
\def\POL@oPolAntiDiff[#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\xint_firstoftwo
\else
\expandafter\xint_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@
\XINT_global\@namedef{POLuserpol@#2}{-1.\empty{0/1[0]}}%
\else
\edef\POL@var@coeffs{\expandafter\POL@antidiff\POL@var@coeffs\relax}%
\XINT_global\expandafter\edef\csname POLuserpol@#2\endcsname
{\the\numexpr\POL@var@deg+\@ne.\noexpand\empty{0/1[0]}\POL@var@coeffs}%
\fi
}%
%
%%
%% Localization of roots
%%
% this is big. It provides also output macros, of both expandable and
% non-expandable type
\input polexprsturm.tex\relax
%
%
%% Non-expandable output macros
%
\def\PolTypesetCmdPrefix#1{\xintiiifSgn{#1}{}{+}{+}}%
\def\PolTypesetCmd#1{\xintifOne{\xintiiAbs{#1}}%
{\ifnum\PolIndex=\z@\xintiiSgn{#1}\else
\xintiiifSgn{#1}{-}{}{}\fi
\let\PolIfCoeffIsPlusOrMinusOne\xint_firstoftwo}%
{\PolTypesetOne{#1}%
\let\PolIfCoeffIsPlusOrMinusOne\xint_secondoftwo}%
}%
\ifdefined\frac
\def\PolTypesetOne{\xintTeXsignedFrac}%
\else
\def\PolTypesetOne{\xintTeXsignedOver}%
\fi
\catcode`^ 7 %
\def\PolTypesetMonomialCmd{%
\ifcase\PolIndex\space
%
\or\PolVar
\else\PolVar^{\PolIndex}%
\fi
}%
\catcode`^ 11 % normal xint catcode
\def\PolTypeset{\POL@ifstar
{\def\POL@ts@ascending{1}\POL@Typeset}%
{\def\POL@ts@ascending{0}\POL@Typeset}%
}%
%%
%% \PolTypeset
%%
%% extended at 0.8 to handle arbitrary expressions on input
%%
\def\POL@Typeset{\POL@chkopt\POL@oPOL@Typeset[x]}%
\def\POL@oPOL@Typeset[#1]#2{%
\ifmmode\let\POL@endtypeset\empty\else$\def\POL@endtypeset{$}\fi
\ifcsname POLuserpol@#2\endcsname
\expandafter\expandafter\expandafter\POL@split
\csname POLuserpol@#2\endcsname;\POL@var@deg\POL@var@coeffs
\else
\xintAssign\expandafter\xint_firstofone\romannumeral0\xintbareeval
subs((deg(x),coeffs(x)),x=subs(#2,\PolToExprInVar=pol([0,1])))\relax
\to\POL@var@deg\POL@var@coeffs
\fi
\if\POL@ts@ascending1%
\def\PolIndex{0}%
\let\POL@ts@reverse\xint_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
\POL@endtypeset
}%
\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.%
}%
%
%%
%% Expandable output macros (legacy)
%%
\def\POL@eval@fork#1\At#2#3\krof{#2}%
\def\PolEval#1#2#3{\romannumeral`&&@\POL@eval@fork
#2\PolEvalAt
\At\PolEvalAtExpr\krof {#1}{#3}%
}%
\def\PolEvalAt#1#2{%
\xintpraw{\csname XINT_expr_polfunc_#1\endcsname{#2}}%
}%
\def\POL@eval#1#2{%
\csname XINT_expr_polfunc_#1\endcsname{#2}%
}%
\def\PolEvalAtExpr#1#2{\xinttheexpr #1(#2)\relax}%
%
\def\PolEvalReduced#1#2#3{\romannumeral`&&@\POL@eval@fork
#2\PolEvalReducedAt
\At\PolEvalReducedAtExpr\krof {#1}{#3}%
}%
\def\PolEvalReducedAt#1#2{%
\xintpraw % in order not to print denominator if the latter equals 1
{\xintIrr{\csname XINT_expr_polfunc_#1\endcsname{#2}}[0]}%
}%
\def\PolEvalReducedAtExpr#1#2{%
\xintpraw
{\expandafter\xintIrr\romannumeral`&&@\xintthebareeval#1(#2)\relax[0]}%
}%
%
\def\PolFloatEval#1#2#3{\romannumeral`&&@\POL@eval@fork
#2\PolFloatEvalAt
\At\PolFloatEvalAtExpr\krof {#1}{#3}%
}%
\def\PolFloatEvalAt#1#2{%
\xintpfloat{\csname XINT_flexpr_polfunc_#1\endcsname{#2}}%
}%
\def\PolFloatEvalAtExpr#1#2{\xintthefloatexpr #1(#2)\relax}%
\def\PolLeadingCoeff#1{%
\romannumeral`&&@\expandafter\expandafter\expandafter\xintlastitem
\expandafter\expandafter\expandafter
{\csname POLuserpol@#1\endcsname}%
}%
%
\def\PolNthCoeff#1#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?
\def\PolDegree#1{\romannumeral`&&@\expandafter\expandafter\expandafter
\POL@degree\csname POLuserpol@#1\endcsname;}%
\def\POL@degree #1.#2;{#1}%
%
\def\PolToList#1{\romannumeral`&&@\expandafter\expandafter\expandafter
\xint_gob_til_dot\csname POLuserpol@#1\endcsname}%
%
\def\PolToCSV#1{\romannumeral0\xintlistwithsep{, }{\PolToList{#1}}}%
%
% \PolIContent (0.5)
% Why did I call this IContent and not Content? Ah, I see, Maple terminology!
% But I realize now I misread in 2018 the Maple doc, its icontent() is the gcd
% of all coeffs of a multivariate polynomial. Whereas content(,) second argument
% specifies which variable to consider expression as being univariate in it.
% Refactored at 0.8 as xint 1.4 has a backported fractional gcd
% (itself refactored at 1.4d)
\def\POL@icontent#1{\romannumeral0\expandafter\XINT_fgcd_out
\romannumeral0\expandafter\XINT_fgcdof\romannumeral`&&@#1^}%
% Since xintexpr 1.4d, \xintGCDof always outputs an irreducible fraction A/B.
% (with B=1 if A/B integer).
\def\PolIContent#1{\xintGCDof{\PolToList{#1}}}%
%
\def\PolToExprCmd#1{\xintPRaw{\xintRawWithZeros{#1}}}%
\def\PolToFloatExprCmd#1{\xintPFloat{#1}}% CHANGED AT 0.8.2! was \xintFloat
% \def\PolTypesetCmdPrefix#1{\xintiiifSgn{#1}{}{+}{+}}%
\let\PolToExprTermPrefix\PolTypesetCmdPrefix
\def\PolToExprOneTermStyleA#1#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\PolToExprCaret\xintiiAbs{#2}%
\fi
}%
\let\PolToExprOneTerm\PolToExprOneTermStyleA
\def\PolToExprOneTermStyleB#1#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\PolToExprCaret\xintiiAbs{#2}%
\fi
\xintiiifOne{\xintDenominator{#1}}{}{/\xintDenominator{#1}}%
}%
\def\PolToFloatExprOneTerm#1#2{%
\ifnum#2=\z@
\PolToFloatExprCmd{#1}%
\else
\PolToFloatExprCmd{#1}\PolToExprTimes
\fi
\ifcase\xintiiAbs{#2} %<-- space here mandatory
\or\PolToExprVar
\else\PolToExprVar\PolToExprCaret\xintiiAbs{#2}%
\fi
}%
\def\PolToExprTimes{*}%
\def\PolToExprVar{x}%
\def\PolToExprInVar{x}%
\edef\PolToExprCaret{\string ^}%
%%
%% \PolToExpr
%%
%% extended at 0.8 to handle arbitrary expressions on input
%%
\def\PolToExpr#1{%
\if*\noexpand#1\expandafter\xint_firstoftwo\else
\expandafter\xint_secondoftwo\fi
\PolToExprAscending\PolToExprDescending{#1}}%
\def\PolToFloatExpr#1{%
\if*\noexpand#1\expandafter\xint_firstoftwo\else
\expandafter\xint_secondoftwo\fi
\PolToFloatExprAscending\PolToFloatExprDescending{#1}}%
\def\PolToExpr@getit#1%
{%
\ifcsname XINT_expr_varvalue_#1\endcsname
\csname XINT_expr_varvalue_#1\expandafter\endcsname
\else
\expandafter\xint_firstofone\romannumeral0%
\xintbareeval subs(#1,\PolToExprInVar=pol([0,1]))\expandafter\relax
\fi
}%
\def\PolToExprAscending#1#2{%
\expandafter\POL@toexpr\romannumeral0\PolToExpr@getit{#2}%
\PolToExprOneTerm\POL@toexprA
}%
\def\PolToFloatExprAscending#1#2{%
\expandafter\POL@toexpr\romannumeral0\PolToExpr@getit{#2}%
\PolToFloatExprOneTerm\POL@toexprA
}%
\def\PolToExprDescending#1{%
\expandafter\POL@toexpr\romannumeral0\PolToExpr@getit{#1}%
\PolToExprOneTerm\POL@toexprD
}%
\def\PolToFloatExprDescending#1{%
\expandafter\POL@toexpr\romannumeral0\PolToExpr@getit{#1}%
\PolToFloatExprOneTerm\POL@toexprD
}%
\def\POL@toexpr#1#2#3{\POL@toexpr@fork#3#2#1\relax}%
\def\POL@toexpr@fork #1#2#3{%
\POL_Pfork
#3\POL@toexpr@pol
P\POL@toexpr@cst
\krof #1#2#3%
}%
\def\POL@toexpr@cst#1#2#3\relax{#2{#3}{0}}%
\def\POL@toexpr@pol#1#2P#3.{#1{#3}#2\empty}%
% now back to legacy pre 0.8 code
\def\POL@toexprA #1#2\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 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.%
}%
\XINTrestorecatcodesendinput%