% ======================================================
% 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
}%