path: root/Master/texmf-dist/tex/latex/gauss/gauss.sty

% TODO: 
%  *  ALLOW FINE TUNING OF ARROW ETC ARRANGEMENT BY PARAMETERS
%     (Suggestion by Antonis Tsolomitis)
%  *  BUG: ROWOPMINSIZE AND COLOPMINSIZE DONT WORK IN \swap01 etc.
%     (MAXIMUM COMPUTATION MISSIG)
%  *  BUG: ARROWHT HAS TWO CONTRADICTING MEANINGS
%
% \title{\texttt{gauss.sty} -- A Package for Typesetting Matrix Operations}
% \author{Manuel Kauers}
% \maketitle
% 
% \MakeShortVerb{\|}
%
% \newenvironment{example}
% {\par\goodbreak\medskip
%  \begin{minipage}[c]{.45\textwidth}
%   \def\switch{\end{minipage}\begin{minipage}[c]{.45\textwidth}\hfil$}
%   \obeylines
% }{$\hfil\end{minipage}\medskip\goodbreak\par}
% 
% \begin{abstract}
%  This package provides \LaTeX-macros for typesetting operations on a matrix.
%  By an ``operation on a matrix'' we understand a \textit{row operation}
%  or a \textit{column operation}. 
%
%  The user interface of the package is very straightforward and easy to understand
%  while the results look quite pretty.
% \end{abstract}
%
% \tableofcontents
%
% \section{Usage}
%
% If you find yourself in search of a package that enables you to easily typeset 
% constructions like
% \[
%  \begin{gmatrix}[v]
%   1 & 0 & 5 & 7 & 2 \\
%   3 & 1 & 1 & 5 & 1 \\
%   1 & 0 & -7 & 1 & 4 \\
%   4 & 3 & 6 & 5 & 4\\
%   1 & 7 & 9 & 4 & 3 \\
%   0 & 0 & 8 & 0 & -1
%  \rowops
%   \add[-3]01
%   \add[-1]02
%   \swap34
%   \mult5{\cdot   0}
%   \add[x^2-1]53
%  \colops
%   \swap01
%   \mult3{\cdot   1}
%   \add[0]24
%  \end{gmatrix} = \begin{gmatrix}[v]
%   0 & 1 & 5 & 7 & 2 \\
%   1 & 0 & -14 & -16 & -5 \\
%   0 & 0 & -12 & -6 & 2 \\
%   7 & 1 & 9 & 4 & 3 \\
%   3 & 4 & 6 & 5 & 4\\
%   0 & 0 & 0 & 0 & 0
%  \end{gmatrix},
% \]
% then this package is what you need. 
% It defines a new matrix environment which is extended by comprehensive macros for
% typesetting so-called ``operations'' on the matrix.
% An operation is either a row operation or a column operation, and may involve one or 
% two lines. 
% Examples of such operations arise in the context of Gaussian elimination for solving
% systems of linear equations in linear algebra: swaping rows, adding the multiple of one
% row to another, and multiply a row by a constant factor.
%
% \subsection{How to typeset matrix operations}
%
% \begin{environment}{gmatrix}
% The package defines a new matrix environment |gmatrix| which
% behaves just like \LaTeX's and \AmS\LaTeX's |matrix|. It takes an optional 
% parameter \meta{delimtype} to select the matrix delimiters. So, |gmatrix[p]|
% corresponds to |pmatrix|, |gmatrix[v]| to |vmatrix|, and so on.
%
% The |gmatrix| environment can be used exaclty like its brothers and sisters
% defined by \LaTeX\ and \AmS\LaTeX, for instance:
% \begin{example}
%  |\begin{gmatrix}[p]|
%  | a & b \\|
%  | c & d|
%  |\end{gmatrix}|
%  \switch
%  \begin{gmatrix}[p] a&b\\ c&d\end{gmatrix}
% \end{example}
% The content of the |gmatrix| environment consists of three parts: matrix, row operations,
% and column operations. The latter two are optional parts, and the ordering of them is
% arbitrary (i.e.\ row operations may be stated before column operations and vice versa). 
% The matrix part is required, and it must be the first one.
% \end{environment}
% 
% \begin{macro}{\rowops}
% \begin{macro}{\colops}
% To skip to the next section, there are two comands |\rowops| which swiches to the row 
% operation section, and |\colops| which switches to the column operation section.
% \end{macro}
% \end{macro}
%
% \begin{macro}{\mult}
% \begin{macro}{\add}
% \begin{macro}{\swap}
% Within the operation sections, you have to state the sequence of operations which are to
% be typeset. There are the three commands |\mult|, |\add|, and |\swap| to do this. These
% commands are specified as follows:
%
% \begin{enumerate}
% \item |\mult{i}{\cdot   b}| typesets the operation ``multiply the $i$th row (or column)
%   by~$b$'',
% \item |\swap[a][b]{i}{j}| typesets the operation ``swap the $i$th and the $j$th row 
%   (or column)''.
%   $a$~and~$b$ are labels to typeset at the end of the arrows, similar to the $\cdot   b$ of
%   the |\mult| command. The command does nothing if $i=j$.
% \item |\add[a][b]{i}{j}| typesets the operation ``add the $a$-fold of row (or column)~$i$ to
%   row (or column)~$j$. $b$~is a label for the $j$th line. The command does nothing if $i=j$.
% \end{enumerate}
% 
% In the standard implementation, optional arguments of |\swap| and the second optional
% argument of |\add| are ignored. See Section~\ref{ssec:atp} for how to enable them. 
%
% Rows are counted top-down, and columns are counted from left to right. The uppermost row
% and the leftmost column have the index~0. There is also the posibility to use |*| as index
% which causes the typesetting of several operations where |*| runs over all indices. For
% example, |\mult{*}{5}| in the |\rowops| section of a $n\times   n$ matrix is equivalent to
% state |\mult{0}{5}|,\dots,|\mult{|$n-1$|}{5}|.
% \end{macro}
% \end{macro}
% \end{macro}
%
% \subsection{Examples}
%
% \begin{itemize}
% \item A matrix with row operations
% \begin{example}
%  |\begin{gmatrix}[p]|
%  | 1 & 2 & 3 \\|
%  | 4 & 5 & 6 \\|
%  | 7 & 8 & 9|
%  |\rowops|
%  | \swap{0}{1}|
%  | \mult{0}{\cdot   7}|
%  | \add[5]{1}{2}|
%  |\end{gmatrix}|
% \switch
%  \begin{gmatrix}[p] 1&2&3\\4&5&6\\7&8&9 \rowops
%  \swap01\mult0{\cdot   7}\add[5]12
%  \end{gmatrix}
% \end{example}
% \item The same operations in an other ordering
% \begin{example}
%  |\begin{gmatrix}[p]|
%  | 1 & 2 & 3 \\|
%  | 4 & 5 & 6 \\|
%  | 7 & 8 & 9|
%  |\rowops|
%  | \add[5]{1}{2}|
%  | \swap{0}{1}|
%  | \mult{0}{\cdot   7}|
%  |\end{gmatrix}|
% \switch
%  \begin{gmatrix}[p] 1&2&3\\4&5&6\\7&8&9 \rowops
%  \add[5]12\swap01\mult0{\cdot   7}
%  \end{gmatrix}
% \end{example}
% \item A matrix with column operations
% \begin{example}
%  |\begin{gmatrix}[p]|
%  | 1 & 2 & 3 \\|
%  | 4 & 5 & 6 \\|
%  | 7 & 8 & 9|
%  |\colops|
%  | \swap{0}{1}|
%  | \mult{0}{\cdot   7}|
%  | \add[5]{1}{2}|
%  |\end{gmatrix}|
% \switch
%  \begin{gmatrix}[p] 1&2&3\\4&5&6\\7&8&9 \colops
%  \swap01\mult0{\cdot   7}\add[5]12
%  \end{gmatrix}
% \end{example}
% \item A matrix with both row and column operations
% \begin{example}
%  |\begin{gmatrix}[v]|
%  | 1 & 2 & 3 \\|
%  | 4 & 5 & 6 \\|
%  | 7 & 8 & 9|
%  |\rowops|
%  | \swap{1}{2}|
%  | \mult{2}{\cdot   3}|
%  | \add[-5]{1}{0}|
%  | \add[-3]{1}{2}|
%  |\colops|
%  | \swap{0}{1}|
%  | \mult{0}{\cdot   7}|
%  | \add[5]{1}{2}|
%  |\end{gmatrix}|
% \switch
%  \begin{gmatrix}[v] 1&2&3\\4&5&6\\7&8&9 \rowops
%  \swap12\mult2{\cdot   3}\add[-5]10\add[-3]12 \colops
%  \swap01\mult0{\cdot   7}\add[5]12
%  \end{gmatrix}
% \end{example}
% \item Multiple operations using the |*| index
% \begin{example}
%  |\begin{gmatrix}[p]|
%  | 1&2&3&4\\|
%  | 5&6&7&8\\|
%  | 9&10&11&12\\|
%  | 13&14&15&16|
%  |\rowops|
%  | \add[x]{0}{*}|
%  |\end{gmatrix}|
% \switch
%  \begin{gmatrix}[p]
%   1&2&3&4\\
%   5&6&7&8\\
%   9&10&11&12\\
%   13&14&15&16
%  \rowops \add[x]0*
%  \end{gmatrix}
% \end{example}
%  Note that the first row is not added to itself, because |\add[x]{0}{0}| has no effect.
%  You can also use two stars:
% \begin{example}
%  |\begin{gmatrix}[p]|
%  | 1&2&3\\|
%  | 4&5&6\\|
%  | 7&8&9|
%  |\rowops|
%  | \add{*}{*}|
%  |\end{gmatrix}|
% \switch
%  \kern-1.5em\begin{gmatrix}[p]
%   1&2&3\\ 4&5&6\\ 7&8&9
%  \rowops \add**
%  \end{gmatrix}\kern-2em
% \end{example}
% \item The package clearly also handels a matrix with larger entries correctly:
% \[
%  \begin{gmatrix}[p]
%   a & b & c & d & e \\
%   0 & 0 & \displaystyle\int\limits_a^b f(x)\,dx & 0 & 0 \\
%   a & b & c & d & e 
%  \rowops
%   \mult{1}{:\displaystyle\int\limits_a^b f(x)\,dx}%
%   \add[-c]10 \add[-1]02 
%  \end{gmatrix}
% \]
% Even nested |gmatrix|es are possible:
% \[
% \def\littleA#1#2#3#4{\begin{gmatrix}[p]#1&#2\\#3&#4\rowops \add[-#3/#1]01\end{gmatrix}}
% \def\littleB#1#2#3#4#5{\begin{gmatrix}[p]#1&#2\\#3&#4\rowops \mult0{\cdot   #5}\end{gmatrix}}
% \def\littleC#1#2#3#4{\begin{gmatrix}[p]#1&#2\\#3&#4\rowops \swap01\end{gmatrix}}
% \kern-1.5em
% \begin{gmatrix}[v]
%  \littleA 2233 & \littleC 1234 & \littleA abcd \\
%  \rule[-20pt]{0pt}{45pt}\littleB 0110\pi & \littleC vwxy & \littleC 1xx{x^2} \\
%  \littleB 12345 & \littleA \cdot   \cdot   \cdot   \cdot   & \littleB 54321
% \rowops
%  \add[\pi^2/6]01
%  \mult1{\cdot   42}
%  \swap02
% \end{gmatrix}
% \kern-1.5em
% \]
% \end{itemize}
%
% \subsection{Adapting the package}\label{ssec:atp}
%
% \subsubsection{Distances and dimensions}
%
% The appearance of the operation lines and arrows depends strongly on the values of the
% dimension parameters described in this section.
%
% \def\test{\begin{gmatrix}[p]a&b&c\\d&e&f\\g&h&i\rowops
%  \add[x]01\add[y]12\mult1{\cdot   y}\swap02\end{gmatrix}}%
% \def\ttest#1=#2.{\[#1=#2\relax\test\]}%
%
% \begin{macro}{\rowarrowsep}
% \begin{macro}{\colarrowsep}
%  |\rowarrowsep| denotes the distance from the matrix to the operations. 
%  For example, |\rowarrowsep=10pt| yields
%  \ttest\rowarrowsep=5pt.
%  and |\rowarrowsep=50pt| yiels
%  \ttest\rowarrowsep=50pt.
%  The corresponding dimension for column operations is |\colarrowsep|.
% \end{macro}
% \end{macro}
%  \begin{macro}{\opskip}
%   |\opskip| is the distance between two consecutive operations.
%  For example, |\opskip=6pt| yields
%  \ttest\opskip=6pt.
%  and |\opskip=30pt| yields
%  \ttest\opskip=25pt.
%  The |\opskip| length is responsible for both row and column operations.
% \end{macro}
% \begin{macro}{\labelskip}
%  |\labelskip| is the distance between an operation arrow and its labels.
%  For example, |\labelskip=3pt| yields
%  \ttest\labelskip=3pt.
%  and |\labelskip=15pt| yields
%  \ttest\labelskip=15pt.
%  The |\labelskip| length is responsible for both row and column operations.
% \end{macro}
% \begin{macro}{\rowopminsize}
% \begin{macro}{\colopminsize}
%  The length |\rowopminsize| is the minimum amount of a horizontal operation
%  segment to go to the right. 
%  For example, |\rowopminsize=3pt| yields
%  \ttest\rowopminsize=3pt.
%  If the horizontal segment ends with an arrow tip and |\rowopminsize| is less than 
%  the width of |\leftarrow|, then the width of |\leftarrow| is taken. In the above 
%  example, this is the case in the |\add[x]{0}{1}| operation. An example for an 
%  exact use of a small value of |\rowopminsize| is the upper horizontal line of 
%  |\add[y]{1}{2}|.
%  For comparation, |\rowopminsize=30pt| yields
%  \ttest\rowopminsize=30pt.
%  The corresponding value for column operations is |\colopminsize|.
% \end{macro}
% \end{macro}
%
% \subsubsection{Labels}
%
% The typesetting of a label can be changed by redefining the macros which are responsible
% for label typesetting. Each label parameter of |\mult|, |\add|, and |\swap| is passed to
% special ``fontifier'' macros which take one argument and fontify it according to the
% semantical requirements. Here is a list of those fontifier macros and their default 
% definitions:
%
% \begin{macro}{\rowmultlabel}
%  |\rowmultlabel| is the label of a |\mult| operation in the |\rowops| section.
%  Its default definition is \verb?{|\,#1}?.
% \end{macro}
% \begin{macro}{\colmultlabel}
%  |\colmultlabel| is the respective macro for the |\colops| section. It is defined
%  to 
% \begin{example}
%  |\underline{\hbox to 1.2em{$\hss\mathstrut{}#1\hss$}}|\kern-20em
% \switch
% \end{example}
% \noindent by default.
% \end{macro}
% \begin{macro}{\rowswapfromlabel}
%  |\rowswapfromlabel| is the label of a |\swap| operation in the |\rowops| section
%  which is to place at the first of the two rows. It is defaultly defined to |{}|, i.e.\
%  the label parameter is ignored.
% \end{macro}
% \begin{macro}{\colswapfromlabel}
%  |\colswapfromlabel| is the respective macro for the |\colops| section which is
%  also empty by default.
% \end{macro}
% \begin{macro}{\rowswaptolabel}
%  |\rowswaptolabel| is like |\rowswapfromlabel|, but for the other row. It is empy
%  by default.
% \end{macro}
% \begin{macro}{\colswaptolabel}
%  |\colswaptolabel| is |\rowswaptolabel|'s brother for the |\colops| section.
% \end{macro}
% \begin{macro}{\rowaddfromlabel}
%  |\rowaddfromlabel| is the macro for the label of the from-line of an |\add| command.
%  It is defined to |{\scriptstyle#1}| by default.
% \end{macro}
% \begin{macro}{\coladdfromlabel}
%  |\coladdfromlabel| is respective macro for the column operations.
% \end{macro}
% \begin{macro}{\rowaddtolabel}
%  |\rowaddtolabel| fontifies the label of the to-line of an |\add| command. This macro
%  is defined to |{\scriptscriptstyle +}| by default, i.e.\ it ignores the parameter.
% \end{macro}
% \begin{macro}{\coladdtolabel}
%  |\coladdtolabel| is the respective command for the column operation. It behaves
%  likewise.
% \end{macro}
%
% For the following example, all of the above labels were defined to |{#1}|, i.e.\ to identity.
%
% \begin{example}
%  |\begin{gmatrix}[p]|
%  | a & b & c \\|
%  | d & e & f \\|
%  | g & h & i|
%  |\colops|
%  | \mult0{m}|
%  | \add[af][at]01|
%  | \swap[sf][st]02|
%  |\rowops|
%  | \mult0{m}|
%  | \add[af][at]01|
%  | \swap[sf][st]02|
%  |\end{gmatrix}|
% \switch
% \def\rowmultlabel#1{#1}
% \def\colmultlabel#1{#1}
% \def\rowswapfromlabel#1{#1}
% \def\colswapfromlabel#1{#1}
% \def\rowswaptolabel#1{#1}
% \def\colswaptolabel#1{#1}
% \def\rowaddfromlabel#1{#1}
% \def\coladdfromlabel#1{#1}
% \def\rowaddtolabel#1{#1}
% \def\coladdtolabel#1{#1}
%  \begin{gmatrix}[p]
%   a & b & c \\
%   d & e & f \\
%   g & h & i 
%  \colops
%   \mult0{m}
%   \add[af][at]01
%   \swap[sf][st]02
%  \rowops
%   \mult0{m}
%   \add[af][at]01
%   \swap[sf][st]02
%  \end{gmatrix}
% \end{example}
% 
% \subsubsection{Matrix delimiters}
%
% \begin{macro}{\newmatrix}
%  It is possible to define new delimiter specifiers to |gmatrix|, say |gmatrix[X]|,
%  by defining a matrix environment |Xmatrix|.
%  A definition of |Xmatrix| which fulfills the requirements needed for compatibility
%  with |gmatrix| is provided automatically by the call of
%
%  \begin{example}
%  |\newmatrix{|\meta{left-delim}|}{|\meta{right-delim}|}{X}|,\kern-20em
%  \switch
%  \end{example}
%
%  which defines the environment |Xmatrix|. The arguments \meta{left-delim} and 
%  \meta{right-delim} need to be compatible to the |\left|-|\right| mechanism of \TeX.
%  As soon as |Xmatrix| exists, it is also possible to use |X| as optional argument
%  to |gmatrix|.
%
%  By convention, the suffix is one single character. If you try to enter |g@| or
%  the empty string as suffix, nothing is done, otherwise, the definition works
%  as well.
% \end{macro}
%
% \subsection{Features}
%
% \begin{itemize}
%  \item You need not care about the width or height of some macro cells, 
%   operations are always aligned well, i.e. centered to the column or row.
%  \item Operation elements will not intersect each other, unless you give
%   some very huge labels.
%  \item There is no restriction to the order of operation commands, so you can
%   choose an arbitrary order to achive the best typographic result.
%  \item If no operations are given, the result is exactly the result of
%   the \AmS-\TeX\ |matrix| environment. 
%  \item Unlike \AmS's |matrix| environment, there is no limit to the matrix' size
%   in our reimplementation |gmatrix|.
%  \item Nested |gmatrix|'s are possible.
% \end{itemize}
%
% \subsection{Trap doors and hints}
%
% \begin{itemize}
%  \item The last row \emph{must not} end with an |\\|, but each other line 
%   should end with |\\|.
%  \item The last row is used internally to measure the column's widths.
%   Therefore, if you want to point to a column~$i$, then the last row must have
%   at least $i+1$ entries.
%  \item In row operations, the package considers the width of labels
%   (that is, the width of factors in |\mult| and |\add|). But you have to
%   take care that your labels are not higher than the corresponding line,
%   otherwise they may intersect with other arrows or labels.
%  \item analogously for column operations.
%  \item The package should also run without the |amsmath| package, but if you
%   use that package (which is assumed to be the usual situation), you have to
%   load |gauss| after |amsmath|.
% \end{itemize}
%
% \subsection{Bug parade}
%
% A list of submitted bugs and suggested work-arounds or fixes.
% If you face any bug that is not in the list below, feel free to contact me
% at |manuel@kauers.de|.
%
% \begin{itemize}
% \item Hans Frederik Nordhaug faced problems with versions of \AmS-\LaTeX\ 
%  that don't define |*matrix| environments as expected (e.g.\ version 2.13).
%  The current version of |gauss| therefore redefines all those environments 
%  using our |\newmatrix| tool, and requires |amsmath| to be loaded prior to 
%  the |gauss| package.
% \item Morten H{\o}gholm suggested the introduction of fontifying macros and
%  the use of changeable lengths as discussed in Section~\ref{ssec:atp}. 
%  He also suggested some very fine typographic tunings. 
% \item Herbert Voss found that a |\unitlength=1pt| was missing to make the
%  behaviour of the package independent of redefinitions of |\unitlength|
%  outside |gmatrix|. (Fixed.)
% \item Michael Hagedorn noticed that signs in entries a treated like
%  binary relations, i.e., wrong spacing is used. (Fixed.)
% \end{itemize}
%
% \StopEventually
% 
% \section{Implementation}
%
%    \begin{macrocode}
\ProvidesPackage{gauss}[2003/01/14]
\RequirePackage{amsmath}
\makeatletter
%    \end{macrocode}
%
% To avoid naming conflicts with other packages, our private control 
% sequences all start with |\g@|. 
% Permanently public are only the |gmatrix| environment, the fontifying macros (like
% |\rowaddfromlabel|), and the dimensions (like |\opskip|).
%
% The |amsmath| package is not necessarily needed, but if used, it has to be
% loaded prior to the |gauss| package. This is forced by the |\RequirePackage|
% command.
% 
% \subsection{Allocation of registers and definition of common macros}
%
% Boxes,\dots
%    \begin{macrocode}
\newbox\g@trash
\newbox\g@matrixbox
\newbox\g@eastbox
\newbox\g@northbox
\newbox\g@label
\newbox\g@b@tmp
\newbox\g@b@tmpa
\newbox\g@b@tmpb
%    \end{macrocode}
% \dots counters,\dots
%    \begin{macrocode}
\newcount\g@maxrow 
\newcount\g@maxcol
\newcount\g@maxrow@old
\newcount\g@maxcol@old
\newcount\g@c@tmp
\newcount\g@c@tmpa
%    \end{macrocode}
% \dots and dimensions \dots
%    \begin{macrocode}
\newdimen\g@axisHeight
\newdimen\g@linethickness
\newdimen\g@tab
\newdimen\g@arrowht
\newdimen\g@arrowwd
\newdimen\g@d@tmp
\newdimen\g@d@tmpa
\newdimen\g@d@tmpb
\newdimen\g@d@tmpc
\newdimen\g@d@tmpd
\newdimen\g@d@tmpe
%    \end{macrocode}
% are allocated.
%
% \begin{macro}{\g@for}
% For frequent use, we define a special loop mechanism, which allowes to
% iterate over a given interval from a lower bound to a higher one, or
% reversely. The code to execute is given as the third argument of |\g@for|,
% using |#1| for the iteration variable that is substituted by |\the\loopCount|
% for each value in the given bounds.
%
% Each of the bounds is also visited. Example: The following code prints out
% the numbers from 1 to 37, inclusively:
%
% \begin{example}
% |\g@for1\to37\do{#1 }|
% \switch
% \end{example}
%
% We first need some more control sequences: |\g@loopContent| is defined to the loop's
% body when the loop is entered.
% |\g@loopCount| is the variable to increment or decrement with each 
% iteration. |\g@loopEnd| marks the value at which to stop the loop,
% and |\g@loopStep| contains the direction, i.e. $|\g@loopStep|=-1$ iff
% $|\g@loopEnd| < \meta{start value}$.
%    \begin{macrocode}
\def\g@loopContent#1{}
\newcount\g@loopCount\g@loopCount=0
\newcount\g@loopEnd\g@loopEnd=1
\newcount\g@loopStep\g@loopStep=1
%    \end{macrocode}
% The |\g@loop| command executes the loop initialized by |\g@for|.
% Each iteration is executed in its own group to avoid side effects and
% expecially to provide nested loops.
%    \begin{macrocode}
\def\g@loop{%
 % base case?
 \ifnum\g@loopCount=\g@loopEnd\else
  % no: execute loop body
  {\expandafter\g@loopContent\expandafter{\the\g@loopCount}}%
  % increment or decrement the loop variable
  \advance\g@loopCount\g@loopStep
  % call \g@loop recursivly.
  \g@loop
 \fi
}
%    \end{macrocode}
% Finally, here is the definition of |\g@for|. Each value in the interval
% from |#1| to |#2|, including |#1| and |#2| is visited exactly one time.
%    \begin{macrocode}
\def\g@for#1\to#2\do#3{%
 \def\g@loopContent##1{#3}%
 \g@loopCount=#1
 \g@loopEnd=#2
 \ifnum\g@loopEnd>\g@loopCount%
  \g@loopStep=1
  \else\g@loopStep=-1
 \fi
 \advance\g@loopEnd\g@loopStep % inclusive upper bound
 \g@loop
}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\g@checkBounds}
% The next tool is used by the generic operation commands to check whether or not
% a given index is valid. If $|#2|\leq|#3|\leq|#4|$ does not hold, a package
% error is thrown that tells the user what happened.
%
% Parameter |#1| contains `r' or `c' to denote ``rows'' or ``columns'', 
% respectively. This piece of information is only used within the construction of
% the error message.
%
% \begin{macro}{\ifg@indexCorrect}
% The result of |\g@checkBounds| is returned via |\ifg@indexCorrect|.
%    \begin{macrocode}
\newif\ifg@indexCorrect
\def\g@checkBounds#1#2#3#4{%
 \g@indexCorrectfalse
 \ifnum#2>#3%
  \PackageError{gauss}{\g@shorterror{#1} #3<#2}{\g@longerror{#1}}
 \else
  \ifnum#3>#4%
   \PackageError{gauss}{\g@shorterror{#1} #3>#4}{\g@longerror{#1}}
  \else
   \g@indexCorrecttrue
  \fi
 \fi
}
%    \end{macrocode}
% We skip the definitions of |\g@shorterror| and |\g@longerror| which serve to 
% produce error messages. 
\def\g@shorterror#1{\ifx r#1 Row \else Column \fi index out of bounds: }
\def\g@longerror#1{%
 An index of an operation points to a \ifx r#1 row \else column \fi %
 that does not exist.\MessageBreak
 Note that the index of the %
 \ifx r#1 bottom row \else leftmost column \fi is 0 while the index of the %
 \ifx r#1 top row \else rightmost column \fi is <number of %
 \ifx r#1 rows\else columns\fi - 1> .%
}
% \end{macro}\end{macro}
%
% \begin{macro}{\g@downarrow}
% For drawing horizontal arrows of arbitrary length, we use the construction
%
% \begin{example} 
% |\hbox to|\meta{width}|{$\leftarrowfill$}|\kern-20em
% \switch\end{example}
%
% \noindent which uses Plain-\TeX's |\leftarrowfill|. Unfortunately, there is no
% vertical correspondence to that mechanism and thus, we have construct something
% like this by ourselves. We will do so by reimplementing a mechanism that is used
% by |\left| and |\right| to construct delimiters of arbitrary height.
%
%    \begin{macrocode}
\DeclareMathSymbol{\g@downarrowSymb}{\mathord}{largesymbols}{`\y}
\DeclareMathSymbol{\g@vertlineSymb}{\mathord}{largesymbols}{`\?}
\def\g@vertline{\hbox{$\g@vertlineSymb$}\kern-\lineskip}%
%    \end{macrocode}
%
% After allocating the basic symbols, we define |\g@downarrow| by a recursion
% which fills up a vbox with the necessary number of |\g@vertline|'s and a
% final |\g@downarrowSymb|. 
%
% The resulting vbox has exactly the height given in |#1| (as \TeX-length), and
% no depth. If |#1| is less than a minimum height, then it is set to that minimum
% height.
%
%    \begin{macrocode}
\def\g@downarrow#1{\vbox{%
 \vfill
 \baselineskip\z@\relax
 \g@d@tmpc=#1\relax
 \ifdim \g@d@tmpc<\g@arrowht
  \g@d@tmpc\g@arrowht\relax
 \fi
 \g@vlineRec
 \kern\g@d@tmpc 
 \setbox\g@trash=\hbox{$\g@downarrowSymb$}%
 \hbox{\raise\dp\g@trash\box\g@trash}%
}}
\def\g@vlineRec{%
 \advance\g@d@tmpc-\g@arrowht
 \ifdim \g@d@tmpc<\z@ \else
  \g@vertline
  \g@vlineRec
 \fi
}
%    \end{macrocode}
%
% \end{macro}
%
% \subsection{Converting floasts and lengths to each other}
%
% \begin{macro}{\g@defdim}%
% \begin{macro}{\g@defdouble}%
% \begin{macro}{\g@dim}%
% \begin{macro}{\g@double}%
% The typesetting of matrix operations is done by use of the |picture| 
% environment of \LaTeX. The macros of that environment require plain
% numbers, possibly containing a decimal point. Though it is not clearly
% correct, we will call that data format \emph{float} or \emph{double}.
% 
% |picture|'s macros do not work if you give them dimensions as input.
% And since the results of measuring a matrix are necessarily dimensions,
% we need a mechanism to convert dimensions to floats and vice versa.
%
% This mechanism is the topic of the current section.
%
% In fact, we almost provide our own data structure whose values can be shown
% as \TeX\ dimensions or as floats. You can ``construct a new instance'' of
% that structure either by a dimension (using |\g@defdim|) or by a double
% (using |\g@defdouble|). In both cases, a macro is defined to be the 
% corresponding double value. 
%
% Given an instance of our data structure, i.e.\ given a double, you can get
% its double representation using |\g@double| (this just typesets the double
% representation), and you can store its value into a \TeX\ dimension using
% |\g@dim|.
%
% Macros for manipulation on floats are defined in the following section.
% 
% \medskip
% We first need a macro that cuts away the ``pt''. This is rather tricky because
% the ``pt'' that arises in the result of some |\the|\meta{counter} has not the
% catcodes as expected. We can redefine them temporarily but we have to note that
% constructions like |\g@defdim{|\meta{identifier}|}{12pt}| (i.e.\ giving the length
% directly) are no longer possible, since the ``pt'' of a directly given length
% has the ``normal'' catcodes.
%    \begin{macrocode}
\edef\redo#1{\catcode`p=#1\catcode`t=#1\relax}
\redo{12}
\def\g@del#1pt{#1}
\redo{11}
%    \end{macrocode}
% Defining a float by a dimension. The first argument expects an idetifier
% (identifiers are arbitrary strings), and the second argument expects a
% \TeX\ dimension \emph{register}, i.e. some control sequence |\cs| that 
% evaluates to ``\dots pt'' if you say |\the\cs|. 
%
% It is not possible to specify a double by directly give a length. Use
% |\g@defdouble| below in that case.
%    \begin{macrocode}
\def\g@defdim#1#2{%
 \edef\g@defdim@arg{\the #2}%
 \edef\g@defdim@arg{\expandafter\g@del\g@defdim@arg}%
 \g@defdouble{#1}{\g@defdim@arg}%
}
%    \end{macrocode}
% And here is |\g@defdouble|. |#1| should be an identifier and |#2| should
% be the value to store in float |#1|. To avoid naming conflics with other
% macros, |#2| is stored into a macro based on |g@@| and the content of |#1|.
%    \begin{macrocode}
\def\g@defdouble#1#2{%
 \expandafter\expandafter\expandafter\global
 \expandafter\edef\csname g@@#1 \endcsname{#2}%
}
%    \end{macrocode}
% We now come to the macros for ``reading'' a float. These are |\g@dim| (to
% read the dimensional representation) and |\g@double| (for the double 
% representation).
%
% An error will occur if you try to read the value of a float that was not
% previously defined. (``Missing number, treated as zero.'')
%
% First |\g@dim|: Let |#1| be the identifier and |#2| the \TeX\ dimension
% registern to store the value of |#1| in.
%    \begin{macrocode}
\def\g@dim#1#2{%
 \edef\g@dim@arg{\g@double{#1}}%
 #2=\g@dim@arg\p@\relax
}
%    \end{macrocode}
% And |\g@double| is even simpler:
%    \begin{macrocode}
\def\g@double#1{%
 \csname g@@#1 \endcsname
}
%    \end{macrocode}
% \end{macro}\end{macro}\end{macro}\end{macro}
%
% \subsection{Macros for calculus on floats}
%
% We need some macros that provide simple arithmetic calculation on 
% floats. Those are defined now.
%
% \begin{macro}{\g@advance}
% Given a float $f_1$, the following macro performs $f_1 := f_1 + f_2$
% where $f_2$ may be either a \TeX\ dimension or a float:
% If |\csname|$f_2$|\encsname| does not evaluate to some control sequence, 
% it is assumed to denote a \TeX\ dimension (e.g. |5pt|, or |\the\something|)
%    \begin{macrocode}
\def\g@advance#1#2{%
 \g@dim{#1}{\g@d@tmpa}% f_1 := #1
 \expandafter\ifx\csname g@@#2 \endcsname\relax
  \g@d@tmpb=#2% f_2 := #2 (TeX dimension)
 \else
  \g@dim{#2}{\g@d@tmpb}% f_2 := #2 (float)
 \fi
 \advance\g@d@tmpa\g@d@tmpb\relax% f_1 += f_2
 \g@defdim{#1}{\g@d@tmpa}% #1 := f_1
}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\g@min}\begin{macro}{\g@minD}
% Given two floats $f_1, f_2$ and a \TeX\ dimension $d_3$, 
% the following macro performs $d_3 := \min\{f_1, f_2\}$.
%    \begin{macrocode}
\def\g@min#1#2#3{%
 \g@dim{#1}{\g@d@tmpa}% f_1 := #1
 \g@dim{#2}{\g@d@tmpb}% f_2 := #2
 \ifdim \g@d@tmpa<\g@d@tmpb
   #3=\g@d@tmpa
 \else
   #3=\g@d@tmpb
 \fi
 \relax
}
%    \end{macrocode}
% There is a so called $D$-version of the latter macro. By use of |\g@min|, 
% this macro also calculates $\min\{f_1,f_2\}$, but its result is translated
% into a double representation which is then stored in control sequence |#3|.
%    \begin{macrocode}
\def\g@minD#1#2#3{%
 \g@min{#1}{#2}{\g@d@tmpc}%
 \edef\g@minD@arg{\the\g@d@tmpc}%
 \edef\g@minD@arg{\expandafter\g@del\g@minD@arg}%
 \edef#3{\g@minD@arg}%
}
%    \end{macrocode}
% \end{macro}\end{macro}
% \begin{macro}{\g@max}\begin{macro}{\g@maxD}
% And here is are the two opposite macros of the preceeding two.
%    \begin{macrocode}
\def\g@max#1#2#3{%
 \g@dim{#1}{\g@d@tmpa}% 
 \g@dim{#2}{\g@d@tmpb}% 
 \ifdim \g@d@tmpa<\g@d@tmpb
   #3=\g@d@tmpb
 \else
   #3=\g@d@tmpa
 \fi
 \relax
}
\def\g@maxD#1#2#3{%
 \g@max{#1}{#2}{\g@d@tmpc}%
 \edef\g@maxD@arg{\the\g@d@tmpc}%
 \edef\g@maxD@arg{\expandafter\g@del\g@maxD@arg}%
 \edef#3{\g@maxD@arg}%
}
%    \end{macrocode}
% \end{macro}\end{macro}
% \begin{macro}{\g@dist}\begin{macro}{\g@distD}
% Given two floats $f_1, f_2$ and a \TeX\ dimension $d_3$, the following
% macro performs $d_3 := f_1 - f_2$. 
%    \begin{macrocode}
\def\g@dist#1#2#3{%
 \g@dim{#1}{\g@d@tmpa}% f_1 := #1
 \g@dim{#2}{\g@d@tmpb}% f_2 := #2
 \ifdim \g@d@tmpa<\g@d@tmpb
   #3=\g@d@tmpb
   \advance#3 by-\g@d@tmpa
 \else
   #3=\g@d@tmpa
   \advance#3 by-\g@d@tmpb
 \fi
 \relax
}
%    \end{macrocode}
% Again, we have a $D$-version, where the result is given in double 
% representation.
%    \begin{macrocode}
\def\g@distD#1#2#3{%
 \g@dist{#1}{#2}{\g@d@tmpc}%
 \edef\g@distD@arg{\the\g@d@tmpc}%
 \edef\g@distD@arg{\expandafter\g@del\g@distD@arg}%
 \edef#3{\g@distD@arg}%
}
%    \end{macrocode}
% \end{macro}\end{macro}
%
% \begin{macro}{\g@updateArea}\begin{macro}{\g@update}
% While the macros that we have seen in this section so far are mainly used
% for elementary drawing purposes, we now define a slightly more sophisticated 
% macro.
% It is needed to update the leftmost $x$-values of the so-far matrix operation
% set (in terms of row operations). It is invoked after adding a new operation
% to the set.
%
% To update a float $f_1$ with respect to $f_2$ is defined to execute
% $f_1 := \max\{f_1, f_2\}$. This is what the following macro does.
%    \begin{macrocode}
\def\g@update#1#2{%
 \g@dim{#2}{\g@d@tmpe}
 \g@dim{#1}{\g@d@tmpb}
 \ifdim \g@d@tmpe>\g@d@tmpb
  \g@defdim{#1}{\g@d@tmpe}%
 \fi
}
%    \end{macrocode}
%
% The matrix dimensions are stored in floats named
% $\meta{name} + \meta{index}$ where \meta{name} spcifies the dimension 
% (e.g. ``cy'' for the current $y$ values of a \emph{c}olumn) and \meta{index}
% is the index of the row/column to which the float's value belongs.
%
% Now, the following macro iterates over $i\in\{|#3|,\dots,|#4|\}$ and updates
% all the floats with name $|#2| + i$ with respect to float |#1|.
%    \begin{macrocode}
\def\g@updateArea#1#2#3#4{\g@for#3\to#4\do{\g@update{#2##1}{#1}}}
%    \end{macrocode}
% \end{macro}\end{macro}
%
%
%
% \subsection{Macros for measurements}
%
% The macros defined in this section are used to measure the dimensions
% of a given matrix and store the measured values into floats.
%
% For each row~$i$ of the matrix, the $y$-position of the center of
% row~$i$ with respect to the bottom of the matrix is stored in a float
% named $|ry| + i$. Another float $|rx| + i$ is initialized to~$0$. This latter
% value will always contain the leftmost position at which a new row operation can
% start without intersecting previous operations that crossed row~$i$.
%
% For each row~$j$ of the matrix we similarly define the values
% $|cx| + i$ and $|cy| + i$. Note that $|cx| + i$ corresponds to $|ry| + i$
% and $|cy| + i$ corresponds to $|rx| + i$, since column operations grow
% vertically whereas row operations grow horizontally.
%
% \begin{macro}{\g@measureRows}
% We first consider row measuring. The following macro assumes that the current
% box is a |\vbox| that only contains a copy of the matrix, 
% i.e. one |\hbox| per row including all the intermediate glues and kerns and 
% whatever. You can initialize what we assume to have by saying
%
% \medskip
% {\obeylines |\vbox{\halign{|\dots|}}| (typeset the matrix)
% |\box0=\lastbox| (save the matrix)
% |\vbox{\unhcopy0\g@measureRows}| (measure the row's heights)
% |\box0| (restore the matrix)}
% \medskip
%
% \noindent Caution: The following macros will not work if the matrix was not 
% constructed with an |\halign| because special knowledge about the structure 
% of |\halign|'s result is used.
%
% It is assumed that |\g@d@tmp| initially contains the $y$-position of the
% matrix's bottom. It is further assumed that |\g@maxrow| contains the total
% number of rows. These two counters will be modified during the recursion.
%    \begin{macrocode}
\def\g@measureRows{%
 \setbox\g@trash\lastbox
 \ifnum\g@maxrow<0% base case: this box is not part of the matrix
 \else 
  \ifdim\ht\g@trash=0pt%
   \advance\g@d@tmp\lastskip\unskip
   \advance\g@d@tmp\lastkern\unkern
   \unpenalty
  \else
   \advance\g@d@tmp\dp\g@trash
   \advance\g@d@tmp\g@axisHeight
   \g@defdim{ry\the\g@maxrow}{\g@d@tmp}%
   \g@defdim{rx\the\g@maxrow}{\z@}%
   \advance\g@d@tmp-\g@axisHeight
   \advance\g@d@tmp\ht\g@trash
   \advance\g@maxrow-1%
  \fi
  \g@measureRows
 \fi
}
%    \end{macrocode}
% \end{macro}
% \begin{macro}{\g@measureCols}
% In fact, the row measurement is the easier case. The measurement of column
% widths is more complicated by two reasons: 1.\ The number of columns is
% unknown, and 2.\ we will meet the cells in reverse order.
%
% This is why column measurement is implemented in two main steps. First the
% width of each cell and the distance between two preceeding cells is 
% measured and stored into temporary floats $|ct| + \meta{index}$ (distance) and
% $|cy| + \meta{index}$ (width), where \meta{index} is counted from back to front.
% By the way, we count the number of columns.
%
% In the base case of the recursion we start a second recursion that will
% calculate the final results out of the intermediate results and that will
% arange the indexing properly.
%
% What input do we expect? It is assumed that the current box is an |\hbox|
% whose first item is an |\hbox| of width 100cm (to detect the base case),
% followed by a copy of the last row of the matrix to measure. See the 
% definition of |g@matrix| to see how such a situation can be constructed.
% 
% We further assume that |g@d@tmp| is initialized to 0.0pt.
%
%    \begin{macrocode}
\def\g@measureCols{%
 \setbox\g@trash\lastbox
 \ifdim \wd\g@trash=100cm%
  % base case. Invert the ordering and sum the dimensions.
  \g@defdouble{ct\the\g@maxcol}{0}%
  \g@defdouble{cy\the\g@maxcol}{0}%
  \global\g@maxcol\g@maxcol
  \g@c@tmp\g@maxcol
  \advance\g@c@tmp-1%
  \g@measureColsSucc
  \global\advance\g@maxcol-1%
 \else
  \ifdim \ht\g@trash=0pt%
   \advance\g@d@tmp\lastskip\unskip
   \advance\g@d@tmp\lastkern\unkern
   \unpenalty
  \else
   % use ct temporaryly to store the skip between
   % colnr + 1 and colnr.
   \g@defdim{ct\the\g@maxcol}{\g@d@tmp}%
   \g@d@tmp\z@
   % use cy temporaryly to store the cell's width.
   \g@defdim{cy\the\g@maxcol}{\wd\g@trash}%
   \advance\g@maxcol1%
  \fi
  \g@measureCols
 \fi
}
%    \end{macrocode}
% Now, the macro for the second step of the measurement algorithm is defined.
% This is easier, since we now already know the total number of columns that
% have been measured. Roughly speaking, we sum their width's from left to right
% to obtain the $x$-values of the horizontal center of each column. Furthermore,
% the $y$-values are now initialized to~$0$, and the order is inverted.
%
% Knowledge about the implementation of |g@matrix| is used!
%
%    \begin{macrocode}
\def\g@measureColsSucc{%
 \ifnum \g@c@tmp<0\else
  \g@c@tmpa=\g@maxcol
  \advance\g@c@tmpa-\g@c@tmp
  \advance\g@c@tmpa-1
  \g@dim{cy\the\g@c@tmp}{\g@d@tmpa}% width of this cell
  \g@dim{ct\the\g@c@tmp}{\g@d@tmpb}% glue right to this cell
  \advance\g@d@tmp.5\g@d@tmpa%
  \g@defdouble{cy\the\g@c@tmp}{0}%
  \g@defdim{cx\the\g@c@tmpa}{\g@d@tmp}%
  \advance\g@d@tmp.5\g@d@tmpa
  \advance\g@d@tmp\g@d@tmpb
  \ifnum \g@c@tmpa=0%
   \advance\g@d@tmp.5\g@tab
  \fi
  \advance\g@c@tmp-1
  \g@measureColsSucc
 \fi
}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\g@measureAxis}
% This is an easier macro. It measures and defines some common lengths,
% e.g.\ the distance between bottom line and math axis, and the dimensions
% of math arrows which are used for drawing arrows in operations.
%    \begin{macrocode}
\def\g@measureAxis{%
 % 1. Where is the math axis relative to the ground line?
 \setbox\g@trash=\hbox{$\vcenter{\hbox to 5pt{}}$}%
 \global\g@axisHeight=\ht\g@trash
 % 2. What is the minimum width of a horizontal arrow?
 \setbox\g@trash=\hbox{$\leftarrow$}%
 \global\g@arrowwd\wd\g@trash
 % 3. What is the minimum height of a vertical arrow?
 \setbox\g@trash=\vbox{\g@vertline}%
 \global\g@arrowht=\ht\g@trash
 \global\advance\g@arrowht\dp\g@trash
 \global\advance\g@arrowht\lineskip
 % 4. What should be the thickness of ordinary lines?
 \global\g@linethickness=\fboxrule\relax
}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\g@measureArea}
% The last marco of this subsection provides the measurement of 
% a set of floats. (Therefore, it is rather a calculus macro.)
%
% Assuming that |#4| is a float identifier and for all $i\in I:=\{|#2|,\dots,|#3|\}$
% $|#1|+i$ is a float identifier, the macro does
% \[
%  |#4| := \max_{i\in I}\{|#1| + i\}
% \]
%    \begin{macrocode}
\def\g@measureArea#1#2#3#4{%
 \g@defdim{#4}{\z@}%
 \g@for#2\to#3\do{%
  \g@max{#1##1}{#4}{\g@d@tmpe}%
  \g@defdim{#4}{\g@d@tmpe}%
 }%
}
%    \end{macrocode}
% \end{macro}
%
%
%
% \subsection{Macros for drawing purposes}
%
% This Section defines low level macros for drawing purposes within a
% |picture| environment by use of floats.
% 
% \begin{macro}{\g@vline}
% Let $f_1, f_2$ and~$f_3$ be floats. Then, 
% 
% \begin{example}
%  |\g@vline{|$f_1$|}{|$f_2$|}{|$f_3$|}|
% \switch\end{example}
% 
% \noindent draws a vertical line from $(f_1,f_2)$ to $(f_2, f_3)$.
%    \begin{macrocode}
\def\g@vline#1#2#3{%
 \g@minD{#2}{#3}{\min}
 \g@distD{#2}{#3}{\dist}
 \put(\g@double{#1},\min){\line(0,1){\dist}}
}
%    \end{macrocode}
% \end{macro}
% \begin{macro}{\g@vvline}
% Let $f_1, f_2$ and~$f_3$ be floats. Then,
% 
% \begin{example}
% |\g@vvline{|$f_1$|}{|$f_2$|}{|$f_3$|}|
% \switch\end{example}
% 
% \noindent draws a vertical line of length~$|f_3|$, starting at $(f_1,f_2)$, i.e.\
% a line from $(f_1,f_2)$ to $(f_1, f_2+f_3)$.
%    \begin{macrocode}
\def\g@vvline#1#2#3{%
 \put(\g@double{#1},\g@double{#2}){\line(0,1){\g@double{#3}}}
}
%    \end{macrocode}
% \end{macro}
% \begin{macro}{\g@varrow}
% Let $f_1, f_2$ and~$f_3$ be floats. Then,
%
% \begin{example}
% |\g@varrow{|$f_1$|}{|$f_2$|}{|$f_3$|}|
% \switch\end{example}
%
% \noindent draws an arrow from $(f_1, \max\{f_2,f_3\})$ to $(f_1, \min\{f_2,f_3\})$.
%    \begin{macrocode}
\def\g@varrow#1#2#3{%
 \g@dim{#2}{\g@d@tmpa}%
 \g@dim{#3}{\g@d@tmpb}%
 \advance\g@d@tmpb-\g@d@tmpa
 \g@cbox{#1}{#2}{\g@downarrow{\g@d@tmpb}}%
}
%    \end{macrocode}
% \end{macro}
% \begin{macro}{\g@hline}
% Let $f_1, f_2$ and~$f_3$ be floats. Then, 
% 
% \begin{example}
% |\g@hline{|$f_1$|}{|$f_2$|}{|$f_3$|}|
% \switch\end{example}
% 
% \noindent draws a horizontal line from $(f_1,f_2)$ to $(f_3,f_2)$.
%    \begin{macrocode}
\def\g@hline#1#2#3{%
 \g@minD{#1}{#3}{\min}%
 \g@distD{#1}{#3}{\dist}%
 \put(\min,\g@double{#2}){\line(1,0){\dist}}%
}
%    \end{macrocode}
% \end{macro}
% \begin{macro}{\g@hhline}
% Let $f_1, f_2$ and~$f_3$ be floats. Then,
% 
% \begin{example}
% |\g@hhline{|$f_1$|}{|$f_2$|}{|$f_3$|}|
% \switch\end{example}
% 
% \noindent draws a horizontal line of length~$|f_3|$, starting at $(f_1,f_2)$, 
% i.e.\ a line from $(f_1,f_2)$ to $(f_1+f_3, f_2)$.
%    \begin{macrocode}
\def\g@hhline#1#2#3{%
 \put(\g@double{#1},\g@double{#2}){\line(1,0){\g@double{#3}}}%
}
%    \end{macrocode}
% \end{macro}
% \begin{macro}{\g@harrow}
% Let $f_1, f_2$ and~$f_3$ be floats. Then,
%
% \begin{example}
% |\g@harrow{|$f_1$|}{|$f_2$|}{|$f_3$|}|
% \switch\end{example}
%
% \noindent draws an arrow from $(\max\{f_1,f_3\},f_2)$ to $(\min\{f_1,f_3\},f_2)$.
%    \begin{macrocode}
\def\g@harrow#1#2#3{%
 \g@dim{#1}{\g@d@tmpa}%
 \g@dim{#3}{\g@d@tmpb}%
 \advance\g@d@tmpb-\g@d@tmpa
 \advance\g@d@tmpb2\p@
 \g@rbox{#1}{#2}{\hbox to\g@d@tmpb{\leftarrowfill}}%
}
%    \end{macrocode}
% \end{macro}
%
% The remaining two macros allow to put arbitrary math material to a
% specified position. Those are used for typesetting so called labels within
% matrix operations, for example, the factor at an |\add| arrow.
%
% \begin{macro}{\g@rbox}
% The first one is intended to use for row operations.
% Assuming that |#1|, |#2| are float identifiers and |#3| is math material,
% we put |#3| into an |\hbox| and put that box to point $(|#1|,|#2|)$.
% 
% The box will be vertically aligned to |#2| (i.e., the math axis of |#3| will
% be at height |#2|), and horizontally start at |#1|.
%
% The macro puts the math material of |#3| into |\g@label| and just copies its content when
% using, so you can reuse |\g@label| (e.g.\ for measuring purposes).
%    \begin{macrocode}
\def\g@rbox#1#2#3{%
 \setbox\g@label=\hbox{$\relax#3\relax$}%
 \ht\g@label\z@\dp\g@label\z@
 \setbox\g@label=\hbox{$\mathstrut\box\g@label$}%  
 \put(\g@double{#1},\g@double{#2})%
 {\makebox(0,0)[l]{\kern-\p@\copy\g@label}}%
}
%    \end{macrocode}
% \end{macro}
% \begin{macro}{\g@cbox}
% The last macro of this section does the corresponding job for columns.
%
% Here, |#3| will be centered horizontally to |#1|, whereas |#2| denotes the
% height of the label's bottom.
%
% Again, you can reuse the constructed box, it remains in register |\g@label|.
%    \begin{macrocode}
\def\g@cbox#1#2#3{%
 \setbox\g@label=\hbox{$\relax#3\relax$}%
 \setbox\g@label=\hbox{\raise\dp\g@label\box\g@label}%
 \put(\g@double{#1},\g@double{#2})%
  {\makebox(0,0)[b]{\copy\g@label}}%
}
%    \end{macrocode}
% \end{macro}
%
%
%
% \subsection{Generic operation commands}
%
% Before |\halign| begins, the matrix construction macro defines, what to do
% if the matrix is finished. This is defined in |\g@endregion| (see the next
% section for further information).
%
% The |\rowops| and |\colops| commands are temporarily set to |\g@east| or 
% |\g@north|, respectively. Thus, when entering an operation part, the first
% thing to do is to invoke |\g@endregion| to do the things that have to be
% done when the matrix input finishes. After that, |\g@endregion| has to be
% redefined to avoid doing the same process two times. Fortunately, |\g@north|
% and |\g@east| can easily reuse |\g@endregion| and store there those things
% that have to be done at the end of a region.
%
% Hence, each switching to another part of the matrix input consists of three
% parts:
% \begin{enumerate}
%  \item Invoke |\g@endregion| to finish the current input part.
%  \item Redefine |\g@endregion| to do the stuff that has to be done at the end
%   of the region that is now starting. The result of the region is stored into
%   a special box register which is used in the |gmatrix| environment.
%  \item Initialize the new region.
% \end{enumerate}
% You can imagine that it is easy to define further regions (e.g.\ for operations
% to the right or below the matrix).
%
% The two predefined regions |\rowops| and |\colops| are very similar, so we will
% show just one of them in this documentation.
%
% \begin{macro}{\g@north}
% The |\g@north| macro is the generic version of |\colops|, its corresponding
% part is |\g@east|.
%  
%    \begin{macrocode}
\def\g@north{%
%    \end{macrocode}
% 1.\ Finish the current region
%    \begin{macrocode}
 \g@endregion
%    \end{macrocode}
% 2.\ Redefine |\g@endregion| and prevent |\colops| from being called again.
%    \begin{macrocode}
 \gdef\colops{\PackageError{gauss}
   {Two sets of column operations are specified in %
    just one matrix. This is not allowed.}}%
 \gdef\g@endregion{%
   \end{picture}\egroup
   \g@measureArea{cy}{0}{\the\g@maxcol}{sum}%
   \g@dim{sum}{\ht\g@northbox}%
   \global\setbox\g@northbox=\hbox{%
    \raise\colarrowsep\box\g@northbox}%
 }%
%    \end{macrocode}
% 3.\ Initialization of the |\colops| region: Define the operation macros to be 
% the corresponding private versions of this region (see below), set $|sum|:=0$ 
% and start the |picture| environment where the operations are painted in.
%    \begin{macrocode}
 \def\swap{\g@north@arrow11\colswapfromlabel\colswaptolabel}%
 \def\add{\g@north@arrow01\coladdfromlabel\coladdtolabel}%
 \let\mult\g@north@mult
 \g@defdim{sum}{\z@}%
 \global\setbox\g@northbox=\hbox\bgroup
   \begin{picture}(\g@double{w},0)(0,0)
     \linethickness{\g@linethickness}%
}
%    \end{macrocode}
% \end{macro}
% \begin{macro}{\g@north@mult}
% The multiplication macro is the simplest one because it affects only one single
% column.
%    \begin{macrocode}
\def\g@north@mult#1#2{%
 \ifx *#1
%    \end{macrocode}
% Reduce |*| to a set of numbers.
%    \begin{macrocode}
  \g@for0\to\g@maxcol\do{\g@north@mult{##1}{#2}}%
 \else
%    \end{macrocode}
% Now |#1| is a number. Is it an index?
%    \begin{macrocode}
  \g@checkBounds{c}{0}{#1}{\the\g@maxcol}%
  \ifg@indexCorrect
%    \end{macrocode}
% Yes, it is. Typeset the operation.
%    \begin{macrocode}
   \g@cbox{cx#1}{cy#1}{\colmultlabel{#2}}%
   \g@dim{cy#1}{\g@d@tmpc}%
   \advance\g@d@tmpc\ht\g@label
   \g@defdim{cy#1}{\g@d@tmpc}%
   \g@advance{cy#1}{\the\opskip}%
   \g@update{sum}{cx#1}%
  \fi
 \fi
}
%    \end{macrocode}
% \end{macro}
% \begin{macro}{\g@north@arrow}
% 
% The |\g@north@arrow| macro is a generalisation of |\swap| and |\add|. 
% It takes the following eight parameters:
%
% \begin{itemize}
% \item |#1|: 0 to make the `from' line non-arrowed, 1 to get an arrow tip
% \item |#2|: 0 to make the `to' line non-arrowed, 1 to get an arrow tip
% \item |#3|: macro for `from' label rendering
% \item |#4|: macro for `to' label rendering
% \item |#5|: [opt] label of the `from' row
% \item |#6|: [opt] label of the `to' row
% \item |#7|: index of the `from' row
% \item |#8|: index of the `to' row
% \end{itemize}
%
% If only one of the two optional arguments is given, then it is taken as |#5|
% and |#6| is taken empty. If both are missing, both are taken empty.
%
% In |\g@north| above, |\add| is defined to 
% \begin{example}
%  |\g@north@arrow01\coladdfromlabel\coladdtolabel|\kern-20em
% \switch\end{example}
% and |\swap| is defined as
% \begin{example}
%  |\g@north@arrow11\colswapfromlabel\colswaptolabel|\kern-20em
% \switch\end{example}
%
%    \begin{macrocode}
\def\g@north@arrow#1#2#3#4{%
 \@ifnextchar[%
 {\g@north@arrow@a{#1}{#2}{#3}{#4}}%
 {\g@north@arrow@b{#1}{#2}{#3}{#4}{}[]}%
}
\def\g@north@arrow@a#1#2#3#4[#5]{%
 \@ifnextchar[%
 {\g@north@arrow@b{#1}{#2}{#3}{#4}{#5}}%
 {\g@north@arrow@b{#1}{#2}{#3}{#4}{#5}[]}%
}
\def\g@north@arrow@b#1#2#3#4#5[#6]#7#8{%
 \ifx *#7
%    \end{macrocode}
% Reduce star indices to loops of number indices.
% |**| needs a special handling. 
%    \begin{macrocode}
  \ifx *#8
   \g@for0\to\g@maxcol\do{%
    \g@north@arrow@b{#1}{#2}{#3}{#4}{#5}[#6]{##1}{*}}%
  \else
%    \end{macrocode}
% Two loops rather than one because going from |#8| down
% to 0 looks better than going from 0 to |#8|
%    \begin{macrocode}
   \g@for#8\to0\do{%
    \g@north@arrow@b{#1}{#2}{#3}{#4}{#5}[#6]{##1}{#8}}%
   \g@for#8\to\g@maxcol\do{%
    \g@north@arrow@b{#1}{#2}{#3}{#4}{#5}[#6]{##1}{#8}}%
  \fi
 \else
  \ifx *#8
%    \end{macrocode}
% Reduce star indices to loops of number indices.
%    \begin{macrocode}
   \g@for#7\to0\do{%
    \g@north@arrow@b{#1}{#2}{#3}{#4}{#5}[#6]{#7}{##1}}%
   \g@for#7\to\g@maxcol\do{%
    \g@north@arrow@b{#1}{#2}{#3}{#4}{#5}[#6]{#7}{##1}}%
  \else
%    \end{macrocode}
% Now, |#7| and |#8| are numbers.
%    \begin{macrocode}
   \ifnum #7=#8\else
    \g@checkBounds{c}{0}{#7}{\the\g@maxcol}%
    \ifg@indexCorrect
     \g@checkBounds{c}{0}{#8}{\the\g@maxcol}%
     \ifg@indexCorrect
%    \end{macrocode}
% Now, |#7| and |#8| are different from each other, and
% both of them are legal indices.
% Store the current hights of the operations tower
% above column |#7| and |#8| into |tmp1| and |tmp2|,
% respectively.
%    \begin{macrocode}
      \g@defdouble{tmp1}{\g@double{cy#7}}%
      \g@defdouble{tmp2}{\g@double{cy#8}}%
%    \end{macrocode}
% Find out the height of the horizontal connection
% line.
% First increment |#7| and |#8| by the minimum amounts.
%    \begin{macrocode}
      \ifx0#1
       \g@advance{cy#7}{\the\colopminsize}%
      \else
       \g@advance{cy#7}{\the\g@arrowht}%
      \fi
      \ifx0#2
       \g@advance{cy#8}{\the\colopminsize}%
      \else
       \g@advance{cy#8}{\the\g@arrowht}%
      \fi      
%    \end{macrocode}
% Incorporate the columns between |#7| and |#8| into
% the height. Then |sum| denotes the level of the 
% horizontal line.
%    \begin{macrocode}
      \g@measureArea{cy}{#7}{#8}{sum}%
%    \end{macrocode}
% Draw arrows and/or vertical lines from |#7|'s and
% |#8|'s height up to |sum|.
%    \begin{macrocode}
      \ifx0#1
       \g@vline{cx#7}{tmp1}{sum}%
      \else
       \g@varrow{cx#7}{tmp1}{sum}%
      \fi
      \ifx0#2
       \g@vline{cx#8}{tmp2}{sum}%
      \else
       \g@varrow{cx#8}{tmp2}{sum}%
      \fi
%    \end{macrocode}
% Draw the horizontal line.
%    \begin{macrocode}
      \g@hline{cx#7}{sum}{cx#8}%
%    \end{macrocode}
% Insert space between the horizontal line and 
% the labels if at least one of the labels |#5| and |#6| is not empty.
% Typeset the labels into boxes and measure them. 
%    \begin{macrocode}
      \setbox\g@b@tmpa=\hbox{$#3{#5}$}%
      \setbox\g@b@tmpb=\hbox{$#4{#6}$}%
      \ifdim\ht\g@b@tmpa>\z@
       \g@advance{sum}{\the\labelskip}%
      \else
       \ifdim\ht\g@b@tmpb>\z@
        \g@advance{sum}{\the\labelskip}%
       \fi
      \fi
%    \end{macrocode}
% Draw the `from' label if there is one
%    \begin{macrocode}
      \g@d@tmpc\z@
      \ifdim\ht\g@b@tmpa>\z@
       \g@cbox{cx#7}{sum}{\kern-\p@\vcenter{\box\g@b@tmpa}}%
       \g@d@tmpc=\ht\g@label
      \fi
%    \end{macrocode}
% Draw the `to' label if there is one
%    \begin{macrocode}
      \ifdim\ht\g@b@tmpb>\z@
       \g@cbox{cx#8}{sum}{\kern-\p@\vcenter{\box\g@b@tmpb}}%
       \ifdim \ht\g@label>\g@d@tmpc
        \g@d@tmpc=\ht\g@label
       \fi
      \fi
%    \end{macrocode}
% Advance the sum by the maximum height of the two 
% labels and the desired space between two consecutive
% operations
%    \begin{macrocode}
      \g@advance{sum}{\the\g@d@tmpc}%
      \g@advance{sum}{\the\opskip}%
%    \end{macrocode}
% Update all column tower heights between |#7| and |#8| to
% |sum|.
%    \begin{macrocode}
      \g@updateArea{sum}{cy}{#7}{#8}%
%    \end{macrocode}
% That's it.
%    \begin{macrocode}
     \fi
    \fi
   \fi
  \fi
 \fi
}
%    \end{macrocode}
% \end{macro}
% 
% \begin{macro}{\g@east}
% \begin{macro}{\g@east@mult}
%
% The corresponding eastern macros are very similar to the 
% above defined northern versions. Maybe there is a way
% to define generic operation commands once for all regions,
% but this would at least lead to less comprehesive definitions.
% 
% We skip the definitions of |\g@east|, |\g@east@mult| and |\g@east@arrow|
% in this documentation.
%
\def\g@east{%
 \g@endregion 
 \def\swap{\g@east@arrow11\rowswapfromlabel\rowswaptolabel}
 \def\add{\g@east@arrow01\rowaddfromlabel\rowaddtolabel}
 \let\mult\g@east@mult
 \g@defdim{sum}{\z@}%
 \gdef\rowops{\PackageError{gauss}{Two sets of row operations were specified in %
  just one matrix. This is not allowed.}}
 \gdef\g@endregion{%
   \end{picture}\egroup
   \g@measureArea{rx}{0}{\the\g@maxrow}{sum}%
   \g@dim{sum}{\wd\g@eastbox}%
 }%
 \global\setbox\g@eastbox=\hbox\bgroup
  \begin{picture}(0,\g@double{h})(0,0)
   \linethickness{\g@linethickness}%
}
\def\g@east@mult#1#2{%
 \ifx *#1 
  \g@for0\to\g@maxrow\do{\g@east@mult{##1}{#2}}%
 \else
  \g@checkBounds{r}{0}{#1}{\the\g@maxrow}%
  \ifg@indexCorrect
   \g@rbox{rx#1}{ry#1}{\rowmultlabel{#2}}
   \g@dim{rx#1}{\g@d@tmpc}\advance\g@d@tmpc\wd\g@label
   \g@defdim{rx#1}{\g@d@tmpc}%
   \g@advance{rx#1}{\the\labelskip}%
   \g@update{sum}{rx#1}%
  \fi
 \fi
}
%
\def\g@east@arrow#1#2#3#4{%
 \@ifnextchar[%
 {\g@east@arrow@a{#1}{#2}{#3}{#4}}%
 {\g@east@arrow@b{#1}{#2}{#3}{#4}{}[]}%
}
\def\g@east@arrow@a#1#2#3#4[#5]{%
 \@ifnextchar[%
 {\g@east@arrow@b{#1}{#2}{#3}{#4}{#5}}%
 {\g@east@arrow@b{#1}{#2}{#3}{#4}{#5}[]}%
}
\def\g@east@arrow@b#1#2#3#4#5[#6]#7#8{%
 \ifx *#7
  \ifx *#8
   \g@for0\to\g@maxrow\do{\g@east@arrow@b{#1}{#2}{#3}{#4}{#5}[#6]{##1}{*}}%
  \else
   \g@for#8\to0\do{\g@east@arrow@b{#1}{#2}{#3}{#4}{#5}[#6]{##1}{#8}}%
   \g@for#8\to\g@maxrow\do{\g@east@arrow@b{#1}{#2}{#3}{#4}{#5}[#6]{##1}{#8}}%
  \fi
 \else
  \ifx *#8
   \g@for#7\to0\do{\g@east@arrow@b{#1}{#2}{#3}{#4}{#5}[#6]{#7}{##1}}%
   \g@for#7\to\g@maxrow\do{\g@east@arrow@b{#1}{#2}{#3}{#4}{#5}[#6]{#7}{##1}}%
  \else
   \ifnum #7=#8\else
    \g@checkBounds{r}{0}{#7}{\the\g@maxrow}%
    \ifg@indexCorrect
     \g@checkBounds{r}{0}{#8}{\the\g@maxrow}%
     \ifg@indexCorrect
      \g@defdouble{tmp1}{\g@double{rx#7}}%
      \g@defdouble{tmp2}{\g@double{rx#8}}%
      \ifx0#1
       \g@advance{rx#7}{\the\rowopminsize}%
      \else
       \g@advance{rx#7}{\the\g@arrowwd}%
      \fi
      \ifx0#2
       \g@advance{rx#8}{\the\rowopminsize}%
      \else
       \g@advance{rx#8}{\the\g@arrowwd}%
      \fi
      \g@measureArea{rx}{#7}{#8}{sum}%
      \ifx0#1
       \g@hline{tmp1}{ry#7}{sum}%
      \else
       \g@harrow{tmp1}{ry#7}{sum}%
      \fi
      \ifx0#2
       \g@hline{tmp2}{ry#8}{sum}%
      \else
       \g@harrow{tmp2}{ry#8}{sum}%
      \fi
      \g@vline{sum}{ry#7}{ry#8}%
      \setbox\g@b@tmpa=\hbox{$#3{#5}$}%
      \setbox\g@b@tmpb=\hbox{$#4{#6}$}%
      \ifdim\wd\g@b@tmpa>\z@
       \g@advance{sum}{\the\labelskip}%
      \else
       \ifdim\wd\g@b@tmpb>\z@
        \g@advance{sum}{\the\labelskip}%
       \fi 
      \fi 
      \g@d@tmpc\z@
      \ifdim\wd\g@b@tmpa>\z@
       \g@rbox{sum}{ry#7}{\kern-\p@\vcenter{\box\g@b@tmpa}}%
       \g@d@tmpc=\wd\g@label
      \fi
      \ifdim\wd\g@b@tmpb>\z@
       \g@rbox{sum}{ry#8}{\kern-\p@\vcenter{\box\g@b@tmpb}}%
       \ifdim \wd\g@label>\g@d@tmpc
        \g@d@tmpc=\wd\g@label
       \fi
      \fi
      \g@advance{sum}{\the\g@d@tmpc}%
      \g@advance{sum}{\the\opskip}%
      \g@updateArea{sum}{rx}{#7}{#8}%
     \fi
    \fi
   \fi
  \fi
 \fi
}
% \end{macro}\end{macro}
%
%
%
% \subsection{The \texttt{gmatrix} environment}
%
% |gmatrix| calls |#1matrix| where |matrix| is redefined to |g@matrix|.
% |g@matrix| typesets the matrix using |\halign| and stores the 
% operations into box registers |\g@northbox| and |\g@eastbox|, respectively.
% The matrix itself is stored into |\g@matrixbox|.
% 
% The ``real'' typesetting is done at the end of |gmatrix|.
%
% \begin{environment}{gmatrix}
% \dots and here is |gmatrix|:
%    \begin{macrocode}
\newenvironment{gmatrix}[1][]
{\unitlength=1pt\def\g@environment{#1matrix}%
 \begin{g@matrix}%
}{%
 \end{g@matrix}%
%    \end{macrocode}
% Delete temporarily the definition of |matrix|.
%    \begin{macrocode}
 \let\matrix\@empty
 \let\endmatrix\@empty
%    \end{macrocode}
% Find out sizes of the matrix. Set |\g@d@tmp| to the height of the matrix.
%    \begin{macrocode}
 \g@d@tmpa\ht\g@matrixbox \advance\g@d@tmpa\p@
 \g@d@tmpb\dp\g@matrixbox \advance\g@d@tmpb\p@
 \g@d@tmp\ht\g@northbox \ht\g@northbox\z@
 \dp\g@northbox\z@
 \ifdim \g@d@tmp>\z@
  \advance\g@d@tmp-\opskip
 \fi
 \advance\g@d@tmp.5\ht\g@matrixbox
 \advance\g@d@tmp.5\dp\g@matrixbox
%    \end{macrocode}
% Start the matrix environment to get the left delimiter.
%    \begin{macrocode}
 \begin{\g@environment}%
%    \end{macrocode}
% Typeset the column operations to the north of the matrix,
% and the matrix itself.
%    \begin{macrocode}
  \vcenter{\hbox{%
   \rlap{\raise\ht\g@matrixbox\box\g@northbox}% north
   % 1 additional pt above and below the matrix
   \rule\z@\g@d@tmpa\lower\g@d@tmpb\null
   \box\g@matrixbox% the matrix itself
  }}%
%    \end{macrocode}
% Close the matrix environment to get now the right delimiter.
%    \begin{macrocode}
 \end{\g@environment}%
%    \end{macrocode}
% Finally typeset the eastern operations.
% Insert vertical space of |\g@d@tmp| (the height
% of the matrix) and horizontal space of |\rowarrowsep| before.
%    \begin{macrocode}
 \rule\rowarrowsep\z@
 \rule\z@\g@d@tmp
 \g@dim{d}{\g@d@tmpa}%
 \vcenter{\hbox{\lower\g@d@tmpa\box\g@eastbox}}%
}
%    \end{macrocode}
% \end{environment}
% Here is the definition of |\g@endmatrix|. This is the initial |\g@endregion|
% which is defined within |\begin{gmatrix}| to finish the matrix input.
%    \begin{macrocode}
\def\g@endmatrix{%
   \mathstrut\crcr
  \egroup % end of \halign
 \egroup % end of \vbox, this contains the matrix
%    \end{macrocode}
% Save the matrix into matrixbox.
%    \begin{macrocode}
 \global\setbox\g@matrixbox\lastbox
%    \end{macrocode}
% Measure the matrix' dimensions.
%    \begin{macrocode}
 \g@measureAxis
 \setbox\g@trash=\vbox{%
  \unvcopy\g@matrixbox
%    \end{macrocode}
% Copy the last row of the matrix into |\g@eastbox| and reinsert it to the vbox.
%    \begin{macrocode}
  \global\setbox\g@eastbox=\lastbox
  \copy\g@eastbox
  \g@d@tmp\z@ {\g@measureRows}% measure rows
 }%
 \setbox\g@trash=\hbox{%
%    \end{macrocode}
% Insert a box of width 100cm to recognize the beginning of the hbox within the 
% measurement recursion.
%    \begin{macrocode}
  \hbox to 100cm{.\hfill.}%
  \unhbox\g@eastbox
  \g@d@tmp\z@ {\g@measureCols}% measure columns
 }%
%    \end{macrocode}
% Determine global dimensions of the matrix (total height, etc.).
%    \begin{macrocode}
 \g@d@tmpa=\ht\g@matrixbox\advance\g@d@tmpa\dp\g@matrixbox
 \g@defdim{h}{\g@d@tmpa}%
 \g@defdim{w}{\wd\g@matrixbox}%
 \g@defdim{d}{\dp\g@matrixbox}%
}%
%    \end{macrocode}
% \begin{environment}{g@matrix}
% Finally, we have the following definition of |g@matrix|:
%    \begin{macrocode}
\edef\g@prae{\hfil\noexpand\mathstrut$\relax}
\edef\g@post{\relax$\hfil} 
\newenvironment{g@matrix}
{\setbox\g@trash=\hbox\bgroup
  \global\g@maxrow@old\g@maxrow
  \global\g@maxcol@old\g@maxcol
  \global\g@maxrow0%
  \global\g@maxcol0%
  \let\rowops\g@east
  \let\colops\g@north
  \vbox\bgroup
   % count rows while typesetting
   \def\\{\mathstrut\cr\global\advance\g@maxrow1\relax}%
   \global\let\g@endregion\g@endmatrix
   \global\g@tab=2\arraycolsep
   \ialign\bgroup\g@prae##\g@post&&\kern\g@tab\g@prae##\g@post\cr
}{%
  \g@endregion
 \egroup % end of \hbox
 % enable nested gmatrixes (for DniQ :-)
 \global\g@maxrow\g@maxrow@old
 \global\g@maxcol\g@maxcol@old
 \global\let\g@endregion\g@endmatrix
 \global\let\rowops\g@east
 \global\let\colops\g@north
}
%    \end{macrocode}
% \end{environment}
% 
%
%
% \subsection{Public tools}
%
% \begin{macro}{\newmatrix}
% The |\newmatrix| command allows to define new matrix environments with
% special delimiters as described in Section~1.
%
%    \begin{macrocode}
\def\newmatrix#1#2#3{%
 \ifx g#3 \else
  \ifx {#3}{g@} \else
   \expandafter\ifx\csname#3matrix\endcsname\relax
    \newenvironment{#3matrix}%
     {\left#1\begin{matrix}}{\end{matrix}\right#2}%
   \else
    \renewenvironment{#3matrix}%
     {\left#1\begin{matrix}}{\end{matrix}\right#2}%
   \fi
  \fi
 \fi
}
%    \end{macrocode}
%
% For compatibility reasons, we redefine predefined matrix environments such
% as |pmatrix|. This is necessary to avoid problems that arise when dealing with
% earlier \AmS\TeX\ versions.
%
%    \begin{macrocode}
\newmatrix()p
\newmatrix[]b
\newmatrix\lbrace\rbrace B
\newmatrix\lvert\rvert v
\newmatrix\lVert\rVert V
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\rowmultlabel}\begin{macro}{\colmultlabel}
% \begin{macro}{\rowaddfromlabel}\begin{macro}{\coladdfromlabel}
% \begin{macro}{\rowaddtolabel}\begin{macro}{\coladdtolabel}
% \begin{macro}{\rowswapfromlabel}\begin{macro}{\colswapfromlabel}
% \begin{macro}{\rowswaptolabel}\begin{macro}{\colswaptolabel}
%  Labels of operations are typeset using the so-called fontifying macros
%  described in Section~\ref{ssec:atp}.
%  All of them take exaclty one argument, and they are called within math
%  mode. The user may redefine them to adjust the appearence of operations
%  according to his needs. The following is the standard definition:

%    \begin{macrocode}
\def\rowmultlabel#1{|\,#1}
\def\rowswapfromlabel#1{}
\def\rowswaptolabel#1{}
\def\rowaddfromlabel#1{\scriptstyle #1}
\def\rowaddtolabel#1{\scriptscriptstyle +}
\def\colmultlabel#1{%
 \underline{\hbox to 1.2em{$\hss\mathstrut{}#1\hss$}}%
}
\def\colswapfromlabel#1{}
\def\colswaptolabel#1{}
\def\coladdfromlabel#1{\scriptstyle #1}
\def\coladdtolabel#1{\scriptscriptstyle +}
%    \end{macrocode}
% \end{macro}\end{macro}
% \end{macro}\end{macro}
% \end{macro}\end{macro}
% \end{macro}\end{macro}
% \end{macro}\end{macro}
%
% \begin{macro}{\colarrowsep}
% \begin{macro}{\rowarrowsep}
% \begin{macro}{\labelskip}
% \begin{macro}{\opskip}
% \begin{macro}{\colopminsize}
% \begin{macro}{\rowopminsize}
% Finally, we define the public lengths of Section~\ref{ssec:atp}:
%    \begin{macrocode}
\newdimen\colarrowsep\colarrowsep=.5em
\newdimen\rowarrowsep\rowarrowsep=.5em
\newdimen\opskip\opskip=5pt
\newdimen\labelskip\labelskip=4pt
\newdimen\colopminsize\colopminsize=3pt
\newdimen\rowopminsize\rowopminsize=3pt
%    \end{macrocode}
% \end{macro}\end{macro}
% \end{macro}\end{macro}
% \end{macro}\end{macro}
%
% And that's all.
%    \begin{macrocode}
\makeatother
%    \end{macrocode}
% \CheckSum{1188}
% \Finale
\endinput