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% ======================================================
% compatibility with PGF 3.0.1a
% ======================================================
%
%%% This file is a copy of some part of PGF/Tikz.
%%% It has been copied here to provide :
%%% - compatibility with older PGF versions
%%% - availability of PGF contributions by Christian Feuersaenger
%%% which are necessary or helpful for pgfplots.
%%%
%%% It contains a couple of patches such that selected changes which
%%% are also part of PGF/TikZ (and can be found in the development
%%% version of PGF/TikZ) are available within pgfplots.
%%%
%%% this file contains solvers which have been written by Christian Feuersaenger (primarily, but not only, for pgfplots)
% Solves a linear equation system of size 2x2 using gauss elimination.
%
% It employs TeX register arithmetics to do so.
% #1: should contain 4 sets of braces with matrix entries,
% {<a11>}{<a12>}
% {<a21>}{<a22>}
% where each entry should be a number without unit.
% It is acceptable if '#1' is a macro which expands to the expected
% format.
% #2: should contain 2 sets of braces with the right-hand-side,
% {<r1>}{<r2>}
% where each entry should be a number without unit.
% It is acceptable if '#2' is a macro which expands to the expected
% format.
%
% It will assign \pgfmathresult to contain two sets of braces with the
% result.
%
% Example:
% \pgfutilsolvetwotwoleq{
% {0.24}{1}
% {-0.97}{0}
% }{
% {-7}
% {18}
% }
% -> yields \pgfmathresult={−18.55618}{−2.54642}
%
% The algorithm employs column pivotisation.
%
% If the matrix is singular, the routine will return {}.
\def\pgfutilsolvetwotwoleq#1#2{%
\begingroup
\dimendef\aa=0
\dimendef\ab=1
\dimendef\ba=2
\dimendef\bb=3
\dimendef\ra=4
\dimendef\rb=5
\dimendef\tmpa=6
\dimendef\tmpb=7
\edef\pgf@temp{#1}%
\expandafter\pgfutilsolvetwotwoleq@A\pgf@temp
\edef\pgf@temp{#2}%
\expandafter\pgfutilsolvetwotwoleq@r\pgf@temp
%
\pgfutilsolvetwotwoleq@ifislarger\aa\ba{%
% identity "permutation":
\def\Pa{a}%
\def\Pb{b}%
}{%
% permutation matrix: switch rows!
\def\Pa{b}%
\def\Pb{a}%
}%
% \pivot := 1/aa
\tmpa=\csname m\Pa a\endcsname pt %
\ifdim\tmpa<0pt \tmpa=-\tmpa\fi
\ifdim\tmpa<0.0001pt
% singular matrix!
\let\pgfmathresult=\pgfutil@empty
\else
\pgfmathreciprocal@
{\csname m\Pa a\endcsname}%
\let\pivot=\pgfmathresult
%
% \factor := 1/aa * ba
\csname \Pb a\endcsname=\pivot\csname \Pb a\endcsname
\edef\factor{\expandafter\pgf@sys@tonumber\csname \Pb a\endcsname}%
%
% bb -= ba/aa * ab
\tmpa=-\factor\csname \Pa b\endcsname
\advance\csname \Pb b\endcsname by\tmpa
%
% rb -= ba/aa * ra
\tmpa=-\factor\csname r\Pa\endcsname
\advance\csname r\Pb\endcsname by\tmpa
%
\tmpa=\csname \Pb b\endcsname%
\ifdim\tmpa<0pt \tmpa=-\tmpa\fi
\ifdim\tmpa<0.0001pt
% singular matrix!
\let\pgfmathresult=\pgfutil@empty
\else
% xb := rb / bb (the modified rb and modified bb!)
\pgfmathdivide@
{\expandafter\pgf@sys@tonumber\csname r\Pb\endcsname}
{\expandafter\pgf@sys@tonumber\csname \Pb b\endcsname}%
\expandafter\let\csname pgfmathresult\Pb\endcsname=\pgfmathresult
%
% ra := ra - xb * ab
\tmpa=\csname pgfmathresult\Pb\endcsname\csname \Pa b\endcsname
\advance\csname r\Pa\endcsname by-\tmpa
%
% xa := 1/aa * ra (the modified ra!)
\tmpa=\pivot\csname r\Pa\endcsname
\expandafter\edef\csname pgfmathresult\Pa\endcsname{\pgf@sys@tonumber\tmpa}%
%
\edef\pgfmathresult{%
{\csname pgfmathresult\Pa\endcsname}%
{\csname pgfmathresult\Pb\endcsname}%
}%
\fi
\fi
\pgfmath@smuggleone\pgfmathresult
\endgroup
}%
\def\pgfutilsolvetwotwoleq@ifislarger#1#2#3#4{%
\tmpa=#1
\ifdim\tmpa<0pt
\multiply\tmpa by-1
\fi
\tmpb=#2
\ifdim\tmpb<0pt
\multiply\tmpb by-1
\fi
\ifdim\tmpa>\tmpb
#3%
\else
#4%
\fi
}%
\def\pgfutilsolvetwotwoleqfloat@ifislarger#1#2#3#4{%
\pgfmathfloatabs@{#1}\let\tmpa=\pgfmathresult
\pgfmathfloatabs@{#2}\let\tmpb=\pgfmathresult
\pgfmathfloatlessthan@{\tmpb}{\tmpa}%
\ifpgfmathfloatcomparison
#3%
\else
#4%
\fi
}%
\def\pgfutilsolvetwotwoleq@A#1#2#3#4{%
\def\maa{#1}\def\mab{#2}%
\def\mba{#3}\def\mbb{#3}%
\aa=#1pt \ab=#2pt
\ba=#3pt \bb=#4pt
}
\def\pgfutilsolvetwotwoleq@r#1#2{%
\ra=#1pt \rb=#2pt
}%
\def\pgfutilsolvetwotwoleqfloat@A#1#2#3#4{%
\pgfmathfloatparsenumber{#1}\let\maa=\pgfmathresult
\pgfmathfloatparsenumber{#2}\let\mab=\pgfmathresult
\pgfmathfloatparsenumber{#3}\let\mba=\pgfmathresult
\pgfmathfloatparsenumber{#4}\let\mbb=\pgfmathresult
}
\def\pgfutilsolvetwotwoleqfloat@r#1#2{%
\pgfmathfloatparsenumber{#1}\let\ra=\pgfmathresult
\pgfmathfloatparsenumber{#2}\let\rb=\pgfmathresult
}%
% Same as \pgfutilsolvetwotwoleq, but using floating point
% arithmetics. The return value is still in fixed point.
\def\pgfutilsolvetwotwoleqfloat#1#2{%
\begingroup
\pgfmathfloatcreate{1}{1.0}{-4}% FIXME : use a smaller threshold for FPU?
\let\thresh=\pgfmathresult
%
\edef\pgf@temp{#1}%
\expandafter\pgfutilsolvetwotwoleqfloat@A\pgf@temp
\edef\pgf@temp{#2}%
\expandafter\pgfutilsolvetwotwoleqfloat@r\pgf@temp
%
\pgfutilsolvetwotwoleqfloat@ifislarger\maa\mba{%
% identity "permutation":
\def\Pa{a}%
\def\Pb{b}%
}{%
% permutation matrix: switch rows!
\def\Pa{b}%
\def\Pb{a}%
}%
% \pivot := 1/aa
\expandafter\pgfmathfloatabs@\expandafter{\csname m\Pa a\endcsname}%
\let\tmpa=\pgfmathresult
\pgfmathfloatlessthan@{\tmpa}{\thresh}%
\ifpgfmathfloatcomparison
% singular matrix!
\let\pgfmathresult=\pgfutil@empty
\else
\expandafter\pgfmathfloatreciprocal@\expandafter{\csname m\Pa a\endcsname}%
\let\pivot=\pgfmathresult
%
% \factor := 1/aa * ba
\expandafter\pgfmathfloatmultiply@\expandafter{\csname m\Pb a\endcsname}{\pivot}%
\let\factor=\pgfmathresult
\expandafter\let\csname m\Pb a\endcsname=\factor
%
% bb -= ba/aa * ab
\expandafter\pgfmathfloatmultiply@\expandafter{\csname m\Pa b\endcsname}{\factor}%
\let\tmpa=\pgfmathresult
\expandafter\pgfmathfloatsubtract@\expandafter{\csname m\Pb b\endcsname}{\tmpa}%
\expandafter\let\csname m\Pb b\endcsname=\pgfmathresult
%
% rb -= ba/aa * ra
\expandafter\pgfmathfloatmultiply@\expandafter{\csname r\Pa\endcsname}{\factor}%
\let\tmpa=\pgfmathresult
\expandafter\pgfmathfloatsubtract@\expandafter{\csname r\Pb\endcsname}{\tmpa}%
\expandafter\let\csname r\Pb\endcsname=\pgfmathresult
%
\expandafter\pgfmathfloatabs@\expandafter{\csname m\Pb b\endcsname}%
\let\tmpa=\pgfmathresult
\pgfmathfloatlessthan@{\tmpa}{\thresh}%
\ifpgfmathfloatcomparison
% singular matrix!
\let\pgfmathresult=\pgfutil@empty
\else
% xb := rb / bb (the modified rb and modified bb!)
\edef\pgf@marshal{
\noexpand\pgfmathfloatdivide@
{\csname r\Pb\endcsname}
{\csname m\Pb b\endcsname}%
}%
\pgf@marshal
\expandafter\let\csname pgfmathresult\Pb\endcsname=\pgfmathresult
\let\tmpa=\pgfmathresult
%
% ra := ra - xb * ab
\expandafter\pgfmathfloatmultiply@\expandafter{\csname m\Pa b\endcsname}{\tmpa}%
\let\tmpa=\pgfmathresult
\expandafter\pgfmathfloatsubtract@\expandafter{\csname r\Pa\endcsname}{\tmpa}%
\expandafter\let\csname r\Pa\endcsname=\pgfmathresult
%
% xa := 1/aa * ra (the modified ra!)
\expandafter\pgfmathfloatmultiply@\expandafter{\csname r\Pa\endcsname}{\pivot}%
\expandafter\let\csname pgfmathresult\Pa\endcsname=\pgfmathresult
%
\edef\pgfmathresult{%
{\csname pgfmathresult\Pa\endcsname}%
{\csname pgfmathresult\Pb\endcsname}%
}%
\expandafter\pgfutilsolvetwotwoleqfloat@to@pgf@range\pgfmathresult
\fi
\fi
\pgfmath@smuggleone\pgfmathresult
\endgroup
}%
\def\pgfutilsolvetwotwoleqfloat@to@pgf@range#1#2{%
\pgfmathfloatcreate{1}{1.6}{4}\let\pgfutilsolvetwotwoleqfloat@to@pgf@range@max=\pgfmathresult
\pgfmathfloatabs@{#1}%
\expandafter\pgfmathfloatlessthan@\expandafter{\pgfmathresult}{\pgfutilsolvetwotwoleqfloat@to@pgf@range@max}%
\ifpgfmathfloatcomparison
\pgfmathfloatabs@{#2}%
\expandafter\pgfmathfloatlessthan@\expandafter{\pgfmathresult}{\pgfutilsolvetwotwoleqfloat@to@pgf@range@max}%
\ifpgfmathfloatcomparison
% ok.
\pgfmathfloattofixed{#1}\let\tmpa=\pgfmathresult
\pgfmathfloattofixed{#2}%
\edef\pgfmathresult{{\tmpa}{\pgfmathresult}}%
\else
% singular (because PGF cannot represent its result
\let\pgfmathresult=\pgfutil@empty
\fi
\else
% singular (because PGF cannot represent its result
\let\pgfmathresult=\pgfutil@empty
\fi
}%
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