\documentstyle[11pt,reduce,fancyheadings]{article}
\title{A Linear Algebra package for \REDUCE{}}
\author{Matt Rebbeck \\
Konrad-Zuse-Zentrum f\"ur Informationstechnik Berlin}
\date{July 1994}
\def\foottitle{The Linear Algebra Package}
\pagestyle{fancy}
\lhead[]{{\footnotesize\leftmark}{}}
\rhead[]{\thepage}
\setlength{\headrulewidth}{0.6pt}
\setlength{\footrulewidth}{0.6pt}
\cfoot{}
\rfoot{\small\foottitle}
%decided against using these.
%\def\ltri{$\triangleleft$}
%\def\rtri{$\triangleright$}
%\newcommand{\tribound}[1]{\rtri#1\ltri}
\def\exprlist {expr$_{1}$,expr$_{2}$, \ldots ,expr$_{{\tt n}}$}
\def\lineqlist {lin\_eqn$_{1}$,lin\_eqn$_{2}$, \ldots ,lin\_eqn$_{n}$}
\def\matlist {mat$_{1}$,mat$_{2}$, \ldots ,mat$_{n}$}
\def\veclist {vec$_{1}$,vec$_{2}$, \ldots ,vec$_{n}$}
\def\lazyfootnote{\footnote{If you're feeling lazy then the \{\}'s can
be omitted.}}
\renewcommand{\thefootnote}{\fnsymbol{footnote}}
\begin{document}
\maketitle
\index{Linear Algebra package}
\section{Introduction}
This package provides a selection of functions that are useful
in the world of linear algebra. These functions are described
alphabetically in section 3 of this document and are labelled 3.1 to
3.51. They can be classified into four sections(n.b: the numbers after
the dots signify the function label in section 3).
\subsection{Basic matrix handling}
\begin{center}
\begin{tabular}{l l l l l l}
add\_columns & \ldots & 3.1 &
add\_rows & \ldots & 3.2 \\
add\_to\_columns & \ldots & 3.3 &
add\_to\_rows & \ldots & 3.4 \\
augment\_columns & \ldots & 3.5 &
char\_poly & \ldots & 3.9 \\
column\_dim & \ldots & 3.12 &
copy\_into & \ldots & 3.14 \\
diagonal & \ldots & 3.15 &
extend & \ldots & 3.16 \\
find\_companion & \ldots & 3.17 &
get\_columns & \ldots & 3.18 \\
get\_rows & \ldots & 3.19 &
hermitian\_tp & \ldots & 3.21 \\
matrix\_augment & \ldots & 3.28 &
matrix\_stack & \ldots & 3.30 \\
minor & \ldots & 3.31 &
mult\_columns & \ldots & 3.32 \\
mult\_rows & \ldots & 3.33 &
pivot & \ldots & 3.34 \\
remove\_columns & \ldots & 3.37 &
remove\_rows & \ldots & 3.38 \\
row\_dim & \ldots & 3.39 &
rows\_pivot & \ldots & 3.40 \\
stack\_rows & \ldots & 3.43 &
sub\_matrix & \ldots & 3.44 \\
swap\_columns & \ldots & 3.46 &
swap\_entries & \ldots & 3.47 \\
swap\_rows & \ldots & 3.48 &
\end{tabular}
\end{center}
\subsection{Constructors}
Functions that create matrices.
\begin{center}
\begin{tabular}{l l l l l l}
band\_matrix & \ldots & 3. 6 &
block\_matrix & \ldots & 3. 7 \\
char\_matrix & \ldots & 3. 8 &
coeff\_matrix & \ldots & 3. 11 \\
companion & \ldots & 3. 13 &
hessian & \ldots & 3. 22 \\
hilbert & \ldots & 3. 23 &
jacobian & \ldots & 3. 24 \\
jordan\_block & \ldots & 3. 25 &
make\_identity & \ldots & 3. 27 \\
random\_matrix & \ldots & 3. 36 &
toeplitz & \ldots & 3. 50 \\
Vandermonde & \ldots & 3. 51 &
Kronecker\_Product & \ldots & 3. 52
\end{tabular}
\end{center}
\subsection{High level algorithms}
\begin{center}
\begin{tabular}{l l l l l l}
char\_poly & \ldots & 3.9 &
cholesky & \ldots & 3.10 \\
gram\_schmidt & \ldots & 3.20 &
lu\_decom & \ldots & 3.26 \\
pseudo\_inverse & \ldots & 3.35 &
simplex & \ldots & 3.41 \\
svd & \ldots & 3.45 &
\end{tabular}
\end{center}
\vspace*{5mm}
There is a separate {\small NORMFORM}[1] package for computing
the following matrix normal forms in \REDUCE.
\begin{center}
smithex, smithex\_int, frobenius, ratjordan, jordansymbolic, jordan.
\end{center}
\subsection{Predicates}
\begin{center}
\begin{tabular}{l l l l l l}
matrixp & \ldots & 3.29 &
squarep & \ldots & 3.42 \\
symmetricp & \ldots & 3.49 &
\end{tabular}
\end{center}
\subsection*{Note on examples:}
In the examples the matrix ${\cal A}$ will be
\begin{flushleft}
\begin{math}
{\cal A} = \left( \begin{array}{ccc} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9
\end{array} \right)
\end{math}
\end{flushleft}
\subsection*{Notation}
Throughout ${\cal I}$ is used to indicate the identity matrix and
${\cal A}^T$ to indicate the transpose of the matrix ${\cal A}$.
\section{Getting started}
If you have not used matrices within {\REDUCE} before then the
following may be helpful.
\subsection*{Creating matrices}
Initialisation of matrices takes the following syntax:
{\tt mat1 := mat((a,b,c),(d,e,f),(g,h,i));}
will produce
\begin{flushleft}
\begin{math}
mat1 := \left( \begin{array}{ccc} a & b & c \\ d & e & f \\ g & h & i
\end{array} \right)
\end{math}
\end{flushleft}
\subsection*{Getting at the entries}
The (i,$\,$j)'th entry can be accessed by:
{\tt mat1(i,j);}
\subsection*{Loading the linear\_algebra package}
The package is loaded by:
{\tt load\_package linalg;}
\section{What's available}
\subsection{add\_columns, add\_rows}
%{\bf How to use it:}
\hspace*{0.175in} {\tt add\_columns(${\cal A}$,c1,c2,expr);}
\hspace*{0.1in}
\begin{tabular}{l l l}
${\cal A}$ & :- & a matrix. \\
c1,c2 & :- & positive integers. \\
expr & :- & a scalar expression.
\end{tabular}
{\bf Synopsis:} %{\bf What it does:}
\begin{addtolength}{\leftskip}{0.22in}
\parbox[t]{0.95\linewidth}{{\tt add\_columns} replaces column c2 of
${\cal A}$ by expr $*$ column(${\cal A}$,c1) $+$ column(${\cal A}$,c2).}
{\tt add\_rows} performs the equivalent task on the rows of ${\cal A}$.
\end{addtolength}
{\bf Examples:}
\begin{flushleft}
\begin{math}
\hspace*{0.16in}
\begin{array}{ccc}
{\tt add\_columns}({\cal A},1,2,x) & = &
\left( \begin{array}{ccc} 1 & x+2 & 3 \\ 4 & 4*x+5 & 6 \\ 7 & 7*x+8 & 9
\end{array} \right)
\end{array}
\end{math}
\end{flushleft}
\vspace*{0.1in}
\begin{flushleft}
\hspace*{0.1in}
\begin{math}
\begin{array}{ccc}
{\tt add\_rows}({\cal A},2,3,5) & = &
\left( \begin{array}{ccc} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 27 & 33 & 39
\end{array} \right)
\end{array}
\end{math}
\end{flushleft}
{\bf Related functions:}
\hspace*{0.175in} {\tt add\_to\_columns}, {\tt add\_to\_rows},
{\tt mult\_columns}, {\tt mult\_rows}.
\subsection{add\_rows}
\hspace*{0.175in} see: {\tt add\_columns}.
\subsection{add\_to\_columns, add\_to\_rows}
%{\bf How to use it:}
\hspace*{0.175in} {\tt add\_to\_columns(${\cal A}$,column\_list,expr);}
\hspace*{0.1in}
\begin{tabular}{l l l}
${\cal A}$ &:-& a matrix. \\
column\_list &:-& a positive integer or a list of positive integers. \\
expr &:-& a scalar expression.
\end{tabular}
{\bf Synopsis:} %{\bf What it does:}
\begin{addtolength}{\leftskip}{0.22in}
{\tt add\_to\_columns} adds expr to each column specified in
column\_list of ${\cal A}$.
{\tt add\_to\_rows} performs the equivalent task on the rows of
${\cal A}$.
\end{addtolength}
{\bf Examples:}
\begin{flushleft}
\hspace*{0.175in}
\begin{math}
\begin{array}{ccc}
{\tt add\_to\_columns}({\cal A},\{1,2\},10) & = &
\left( \begin{array}{ccc} 11 & 12 & 3 \\ 14 & 15 & 6 \\ 17 & 18 & 9
\end{array} \right)
\end{array}
\end{math}
\end{flushleft}
\vspace*{0.1in}
\begin{flushleft}
\hspace*{0.175in}
\begin{math}
\begin{array}{ccc}
{\tt add\_to\_rows}({\cal A},2,-x) & = &
\left( \begin{array}{ccc} 1 & 2 & 3 \\ -x+4 & -x+5 & -x+6 \\ 7 & 8 & 9
\end{array} \right)
\end{array}
\end{math}
\end{flushleft}
{\bf Related functions:}
\hspace*{0.175in}
{\tt add\_columns}, {\tt add\_rows}, {\tt mult\_rows},
{\tt mult\_columns}.
\subsection{add\_to\_rows}
\hspace*{0.175in} see: {\tt add\_to\_columns}.
\subsection{augment\_columns, stack\_rows}
%{\bf How to use it:}
\hspace*{0.175in} {\tt augment\_columns(${\cal A}$,column\_list);}
\hspace*{0.1in}
\begin{tabular}{l l l}
${\cal A}$ &:-& a matrix. \\
column\_list &:-& either a positive integer or a list of positive
integers.
\end{tabular}
{\bf Synopsis:} %{\bf What it does:}
\begin{addtolength}{\leftskip}{0.22in}
{\tt augment\_columns} gets hold of the columns of ${\cal A}$ specified
in column\_list and sticks them together.
{\tt stack\_rows} performs the same task on rows of
${\cal A}$.
\end{addtolength}
{\bf Examples:}
\begin{flushleft}
\hspace*{0.1in}
\begin{math}
\begin{array}{ccc}
{\tt augment\_columns}({\cal A},\{1,2\}) & = &
\left( \begin{array}{cc} 1 & 2 \\ 4 & 5 \\ 7 & 8
\end{array} \right)
\end{array}
\end{math}
\end{flushleft}
\vspace*{0.1in}
\begin{flushleft}
\hspace*{0.1in}
\begin{math}
\begin{array}{ccc}
{\tt stack\_rows}({\cal A},\{1,3\}) & = &
\left( \begin{array}{ccc} 1 & 2 & 3 \\ 7 & 8 & 9
\end{array} \right)
\end{array}
\end{math}
\end{flushleft}
{\bf Related functions:}
\hspace*{0.175in} {\tt get\_columns}, {\tt get\_rows},
{\tt sub\_matrix}.
\subsection{band\_matrix}
%{\bf How to use it:}
\hspace*{0.175in} {\tt band\_matrix(expr\_list,square\_size);}
\hspace*{0.1in}
\begin{tabular}{l l l}
expr\_list \hspace*{0.088in} &:-& \parbox[t]{.72\linewidth}
{either a single scalar expression or a list of an odd number of scalar
expressions.}
\end{tabular}
\vspace*{0.04in}
\hspace*{0.1in}
\begin{tabular}{l l l}
square\_size &:-& a positive integer.
\end{tabular}
{\bf Synopsis:} %{\bf What it does:}
\begin{addtolength}{\leftskip}{0.22in}
{\tt band\_matrix} creates a square matrix of dimension
square\_size. The diagonal consists of the middle expr
of the expr\_list. The exprs to the left of this fill
the required number of sub\_diagonals and the exprs
to the right the super\_diagonals.
\end{addtolength}
{\bf Examples:}
\begin{flushleft}
\hspace*{0.1in}
\begin{math}
\begin{array}{ccc}
{\tt band\_matrix}(\{x,y,z\},6) & = &
\left( \begin{array}{cccccc} y & z & 0 & 0 & 0 & 0 \\ x & y & z & 0 & 0
& 0 \\ 0 & x & y & z & 0 & 0 \\ 0 & 0 & x & y & z & 0 \\ 0 & 0 & 0 & x &
y & z \\ 0 & 0 & 0 & 0 & x & y
\end{array} \right)
\end{array}
\end{math}
\end{flushleft}
{\bf Related functions:}
\hspace*{0.175in} {\tt diagonal}.
\subsection{block\_matrix}
%{\bf How to use it:}
\hspace*{0.175in} {\tt block\_matrix(r,c,matrix\_list);}
\hspace*{0.1in}
\begin{tabular}{l l l}
r,c &:-& positive integers. \\
matrix\_list &:-& a list of matrices.
\end{tabular}
{\bf Synopsis:} %{\bf What it does:}
\begin{addtolength}{\leftskip}{0.22in}
{\tt block\_matrix} creates a matrix that consists of r by c matrices
filled from the matrix\_list row wise.
\end{addtolength}
{\bf Examples:}
\begin{flushleft}
\hspace*{0.1in}
\begin{math}
\begin{array}{ccc}
{\cal B} = \left( \begin{array}{cc} 1 & 0 \\ 0 & 1
\end{array} \right), &
{\cal C} = \left( \begin{array}{c} 5 \\ 5
\end{array} \right), &
{\cal D} = \left( \begin{array}{cc} 22 & 33 \\ 44 & 55
\end{array} \right)
\end{array}
\end{math}
\end{flushleft}
\vspace*{0.175in}
\begin{flushleft}
\hspace*{0.1in}
\begin{math}
\begin{array}{ccc}
{\tt block\_matrix}(2,3,\{{\cal B,C,D,D,C,B}\}) & = &
\left( \begin{array}{ccccc} 1 & 0 & 5 & 22 & 33 \\ 0 & 1 & 5 & 44 & 55
\\
22 & 33 & 5 & 1 & 0 \\ 44 & 55 & 5 & 0 & 1
\end{array} \right)
\end{array}
\end{math}
\end{flushleft}
\subsection{char\_matrix}
%{\bf How to use it:}
\hspace*{0.175in} {\tt char\_matrix(${\cal A},\lambda$);}
\hspace*{0.1in}
\begin{tabular}{l l l}
${\cal A}$ &:-& a square matrix. \\
$\lambda$ &:-& a symbol or algebraic expression.
\end{tabular}
{\bf Synopsis:} %{\bf What it does:}
\begin{addtolength}{\leftskip}{0.22in}
{\tt char\_matrix} creates the characteristic matrix ${\cal C}$ of
${\cal A}$.
This is ${\cal C} = \lambda * {\cal I} - {\cal A}$.
\end{addtolength}
{\bf Examples:}
\begin{flushleft}
\hspace*{0.1in}
\begin{math}
\begin{array}{ccc}
{\tt char\_matrix}({\cal A},x) & = &
\left( \begin{array}{ccc} x-1 & -2 & -3 \\ -4 & x-5 & -6 \\ -7 & -8 &
x-9
\end{array} \right)
\end{array}
\end{math}
\end{flushleft}
{\bf Related functions:}
\hspace*{0.175in} {\tt char\_poly}.
\subsection{char\_poly}
%{\bf How to use it:}
\hspace*{0.175in} {\tt char\_poly(${\cal A},\lambda$);}
\hspace*{0.1in}
\begin{tabular}{l l l}
${\cal A}$ &:-& a square matrix. \\
$\lambda$ &:-& a symbol or algebraic expression.
\end{tabular}
{\bf Synopsis:} %{\bf What it does:}
\begin{addtolength}{\leftskip}{0.22in}
{\tt char\_poly} finds the characteristic polynomial of
${\cal A}$.
This is the determinant of $\lambda * {\cal I} - {\cal A}$.
\end{addtolength}
{\bf Examples:}
\hspace*{0.175in}
{\tt char\_poly({\cal A},$x$) $= x^3-15*x^2-18*x$}
{\bf Related functions:}
\hspace*{0.175in} {\tt char\_matrix}.
\subsection{cholesky}
%{\bf How to use it:}
\hspace*{0.175in} {\tt cholesky(${\cal A}$);}
\hspace*{0.1in}
\begin{tabular}{l l l}
${\cal A}$ &:-& a positive definite matrix containing numeric entries.
\end{tabular}
{\bf Synopsis:} %{\bf What it does:}
\begin{addtolength}{\leftskip}{0.22in}
{\tt cholesky} computes the cholesky decomposition of ${\cal A}$.
It returns \{${\cal L,U}$\} where ${\cal L}$
is a lower matrix, ${\cal U}$ is an upper matrix, \\ ${\cal A} =
{\cal LU}$, and ${\cal U} = {\cal L}^T$.
\end{addtolength}
{\bf Examples:}
\begin{flushleft}
\hspace*{0.175in}
\begin{math}
{\cal F} = \left( \begin{array}{ccc} 1 & 1 & 0 \\ 1 & 3 & 1 \\ 0 & 1 &
1
\end{array} \right)
\end{math}
\end{flushleft}
\begin{flushleft}
\hspace*{0.1in}
\begin{math}
\begin{array}{ccc}
${\tt cholesky}$({\cal F}) & = &
\left\{ \left( \begin{array}{ccc} 1 & 0 & 0 \\ 1 & \sqrt{2} & 0 \\
0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{array} \right), \left(
\begin{array}{ccc} 1 & 1 & 0 \\ 0 & \sqrt{2} & \frac{1}{\sqrt{2}} \\ 0
& 0 & \frac{1}{\sqrt{2}} \end{array} \right)
\right\} \end{array}
\end{math}
\end{flushleft}
{\bf Related functions:}
\hspace*{0.175in} {\tt lu\_decom}.
\subsection{coeff\_matrix}
%{\bf How to use it:}
\hspace*{0.175in} {\tt coeff\_matrix(\{\lineqlist{}\});}
\lazyfootnote{}
\hspace*{0.1in}
\begin{tabular}{l l l}
\lineqlist &:-& \parbox[t]{.435\linewidth}{linear equations. Can be
of the form {\it equation $=$ number} or just {\it equation}.}
\end{tabular}
{\bf Synopsis:} %{\bf What it does:}
\begin{addtolength}{\leftskip}{0.22in}
{\tt coeff\_matrix} creates the coefficient matrix
${\cal C}$ of the linear equations.
It returns \{${\cal C,X,B}$\} such that ${\cal CX} = {\cal B}$.
\end{addtolength}
{\bf Examples:}
\begin{math}
\hspace*{0.175in}
{\tt coeff\_matrix}(\{x+y+4*z=10,y+x-z=20,x+y+4\}) =
\end{math}
\vspace*{0.1in}
\begin{flushleft}
\hspace*{0.175in}
\begin{math}
\left\{ \left( \begin{array}{ccc} 4 & 1 & 1 \\ -1 & 1 & 1 \\ 0 & 1 & 1
\end{array} \right), \left( \begin{array}{c} z \\ y \\ x \end{array}
\right), \left( \begin{array}{c} 10 \\ 20 \\ -4
\end{array} \right) \right\}
\end{math}
\end{flushleft}
\subsection{column\_dim, row\_dim}
%{\bf How to use it:}
\hspace*{0.175in} {\tt column\_dim(${\cal A}$);}
\hspace*{0.1in}
\begin{tabular}{l l l}
${\cal A}$ &:-& a matrix.
\end{tabular}
{\bf Synopsis:} %{\bf What it does:}
\hspace*{0.175in} {\tt column\_dim} finds the column dimension of
${\cal A}$.
\hspace*{0.175in} {\tt row\_dim} finds the row dimension of ${\cal A}$.
{\bf Examples:}
\hspace*{0.175in}
{\tt column\_dim}(${\cal A}$) = 3
\subsection{companion}
%{\bf How to use it:}
\hspace*{0.175in} {\tt companion(poly,x);}
\hspace*{0.1in}
\begin{tabular}{l l l}
poly &:-& a monic univariate polynomial in x. \\
x &:-& the variable.
\end{tabular}
{\bf Synopsis:} %{\bf What it does:}
\begin{addtolength}{\leftskip}{0.22in}
{\tt companion} creates the companion matrix ${\cal C}$
of poly.
This is the square matrix of dimension n, where n is the degree of poly
w.r.t. x.
The entries of ${\cal C}$ are:
${\cal C}$(i,n) = -coeffn(poly,x,i-1) for i = 1
\ldots n, ${\cal C}$(i,i-1) = 1 for i = 2 \ldots n and
the rest are 0.
\end{addtolength}
{\bf Examples:}
\begin{flushleft}
\hspace*{0.1in}
\begin{math}
\begin{array}{ccc}
{\tt companion}(x^4+17*x^3-9*x^2+11,x) & = &
\left( \begin{array}{cccc} 0 & 0 & 0 & -11 \\ 1 & 0 & 0 & 0 \\
0 & 1 & 0 & 9 \\ 0 & 0 & 1 & -17
\end{array} \right)
\end{array}
\end{math}
\end{flushleft}
{\bf Related functions:}
\hspace*{0.175in} {\tt find\_companion}.
\subsection{copy\_into}
%{\bf How to use it:}
\hspace*{0.175in} {\tt copy\_into(${\cal A,B}$,r,c);}
\hspace*{0.1in}
\begin{tabular}{l l l}
${\cal A,B}$ &:-& matrices. \\
r,c &:-& positive integers.
\end{tabular}
{\bf Synopsis:} %{\bf What it does:}
\hspace*{0.175in} {\tt copy\_into} copies matrix ${\cal A}$ into
${\cal B}$ with ${\cal A}$(1,1) at ${\cal B}$(r,c).
{\bf Examples:}
\begin{flushleft}
\hspace*{0.175in}
\begin{math}
{\cal G} = \left( \begin{array}{cccc} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0
\end{array} \right)
\end{math}
\end{flushleft}
\begin{flushleft}
\hspace*{0.1in}
\begin{math}
\begin{array}{ccc}
{\tt copy\_into}({\cal A,G},1,2) & = &
\left( \begin{array}{cccc} 0 & 1 & 2 & 3 \\ 0 & 4 & 5 & 6 \\ 0 & 7 & 8
& 9 \\ 0 & 0 & 0 & 0
\end{array} \right)
\end{array}
\end{math}
\end{flushleft}
{\bf Related functions:}
\begin{addtolength}{\leftskip}{0.22in}
{\tt augment\_columns}, {\tt extend}, {\tt matrix\_augment},
{\tt matrix\_stack}, {\tt stack\_rows}, {\tt sub\_matrix}.
\end{addtolength}
\subsection{diagonal}
%{\bf How to use it:}
\hspace*{0.175in} {\tt diagonal(\{\matlist{}\});}\lazyfootnote{}
\hspace*{0.1in}
\begin{tabular}{l l l}
\matlist &:-& \parbox[t]{.58\linewidth}{each can be either a scalar
expr or a square matrix. }
\end{tabular}
{\bf Synopsis:} %{\bf What it does:}
\hspace*{0.175in} {\tt diagonal} creates a matrix that contains the
input on the diagonal.
{\bf Examples:}
\begin{flushleft}
\hspace*{0.175in}
\begin{math}
{\cal H} = \left( \begin{array}{cc} 66 & 77 \\ 88 & 99
\end{array} \right)
\end{math}
\end{flushleft}
\begin{flushleft}
\hspace*{0.1in}
\begin{math}
\begin{array}{ccc}
{\tt diagonal}(\{{\cal A},x,{\cal H}\}) & = &
\left( \begin{array}{cccccc} 1 & 2 & 3 & 0 & 0 & 0 \\ 4 & 5 & 6 & 0 & 0
& 0 \\ 7 & 8 & 9 & 0 & 0 & 0 \\ 0 & 0 & 0 & x & 0 & 0 \\ 0 & 0 & 0 & 0
& 66 & 77 \\ 0 & 0 & 0 & 0 & 88 & 99
\end{array} \right)
\end{array}
\end{math}
\end{flushleft}
{\bf Related functions:}
\hspace*{0.175in} {\tt jordan\_block}.
\subsection{extend}
%{\bf How to use it:}
\hspace*{0.175in} {\tt extend(${\cal A}$,r,c,expr);}
\hspace*{0.1in}
\begin{tabular}{l l l}
${\cal A}$ &:-& a matrix. \\
r,c &:-& positive integers. \\
expr &:-& algebraic expression or symbol.
\end{tabular}
{\bf Synopsis:} %{\bf What it does:}
\begin{addtolength}{\leftskip}{0.22in}
{\tt extend} returns a copy of ${\cal A}$ that has been
extended by r rows and c columns. The new entries are
made equal to expr.
\end{addtolength}
{\bf Examples:}
\begin{flushleft}
\hspace*{0.1in}
\begin{math}
\begin{array}{ccc}
{\tt extend}({\cal A},1,2,x) & = &
\left( \begin{array}{ccccc} 1 & 2 & 3 & x & x \\ 4 & 5 & 6 & x & x
\\ 7 & 8 & 9 & x & x \\ x & x & x & x & x
\end{array} \right)
\end{array}
\end{math}
\end{flushleft}
{\bf Related functions:}
\begin{addtolength}{\leftskip}{0.22in}
\parbox[t]{0.95\linewidth}{{\tt copy\_into}, {\tt matrix\_augment},
{\tt matrix\_stack}, {\tt remove\_columns}, {\tt remove\_rows}.}
\end{addtolength}
\subsection{find\_companion}
\hspace*{0.175in} {\tt find\_companion(${\cal A}$,x);}
\hspace*{0.1in}
\begin{tabular}{l l l}
${\cal A}$ &:-& a matrix. \\
x &:-& the variable.
\end{tabular}
{\bf Synopsis:}
\begin{addtolength}{\leftskip}{0.22in}
Given a companion matrix, {\tt find\_companion} finds the polynomial
from which it was made.
\end{addtolength}
{\bf Examples:}
\begin{flushleft}
\hspace*{0.175in}
\begin{math}
{\cal C} = \left( \begin{array}{cccc} 0 & 0 & 0 & -11 \\ 1 & 0 & 0 & 0
\\ 0 & 1 & 0 & 9 \\ 0 & 0 & 1 & -17
\end{array} \right)
\end{math}
\end{flushleft}
\vspace*{3mm}
\begin{flushleft}
\hspace*{0.175in}
\begin{math}
{\tt find\_companion}({\cal C},x) = x^4+17*x^3-9*x^2+11
\end{math}
\end{flushleft}
\vspace*{3mm}
{\bf Related functions:}
\hspace*{0.175in} {\tt companion}.
\subsection{get\_columns, get\_rows}
%{\bf How to use it:}
\hspace*{0.175in} {\tt get\_columns(${\cal A}$,column\_list);}
\hspace*{0.1in}
\begin{tabular}{l l l}
${\cal A}$ &:-& a matrix. \\
c &:-& either a positive integer or a list of positive
integers.
\end{tabular}
{\bf Synopsis:} %{\bf What it does:}
\begin{addtolength}{\leftskip}{0.22in}
{\tt get\_columns} removes the columns of ${\cal A}$ specified in
column\_list and returns them as a list of column
matrices.
\end{addtolength}
\hspace*{0.175in} {\tt get\_rows} performs the same task on the rows of
${\cal A}$.
{\bf Examples:}
\begin{flushleft}
\hspace*{0.1in}
\begin{math}
\begin{array}{ccc}
{\tt get\_columns}({\cal A},\{1,3\}) & = &
\left\{
\left( \begin{array}{c} 1 \\ 4 \\ 7 \end{array} \right),
\left( \begin{array}{c} 3 \\ 6 \\ 9 \end{array} \right)
\right\}
\end{array}
\end{math}
\end{flushleft}
\vspace*{0.1in}
\begin{flushleft}
\hspace*{0.1in}
\begin{math}
\begin{array}{ccc}
{\tt get\_rows}({\cal A},2) & = &
\left\{
\left( \begin{array}{ccc} 4 & 5 & 6 \end{array} \right)
\right\}
\end{array}
\end{math}
\end{flushleft}
{\bf Related functions:}
\hspace*{0.175in} {\tt augment\_columns}, {\tt stack\_rows},
{\tt sub\_matrix}.
\subsection{get\_rows}
\hspace*{0.175in} see: {\tt get\_columns}.
\subsection{gram\_schmidt}
%{\bf How to use it:}
\hspace*{0.175in} {\tt gram\_schmidt(\{\veclist{}\});} \lazyfootnote{}
\hspace*{0.1in}
\begin{tabular}{l l l}
\veclist &:-& \parbox[t]{.62\linewidth}{linearly independent vectors.
Each vector must be written as a list,
eg:\{1,0,0\}. }
\end{tabular}
{\bf Synopsis:} %{\bf What it does:}
\begin{addtolength}{\leftskip}{0.22in}
{\tt gram\_schmidt} performs the gram\_schmidt
orthonormalisation on the input vectors.
It returns a list of orthogonal normalised vectors.
\end{addtolength}
{\bf Examples:}
\hspace*{0.175in}
{\tt gram\_schmidt(\{\{1,0,0\},\{1,1,0\},\{1,1,1\}\})} =
\{\{1,0,0\},\{0,1,0\},\{0,0,1\}\}
\hspace*{0.175in}
{\tt gram\_schmidt(\{\{1,2\},\{3,4\}\})} $=
\{\{ \frac{1}{{\sqrt{5}}} , \frac{2}{\sqrt{5}} \},
\{ \frac{2*\sqrt{5}}{5} , \frac{-\sqrt{5}}{5} \}\}$
\subsection{hermitian\_tp}
%{\bf How to use it:}
\hspace*{0.175in} {\tt hermitian\_tp(${\cal A}$);}
\hspace*{0.1in}
\begin{tabular}{l l l}
${\cal A}$ &:-& a matrix.
\end{tabular}
{\bf Synopsis:} %{\bf What it does:}
\begin{addtolength}{\leftskip}{0.22in}
{\tt hermitian\_tp} computes the hermitian transpose of
${\cal A}$.
This is a matrix in which the (i,$\,$j)'th entry is the conjugate of
the (j,$\,$i)'th entry of ${\cal A}$.
\end{addtolength}
{\bf Examples:}
\begin{flushleft}
\hspace*{0.175in}
\begin{math}
{\cal J} = \left( \begin{array}{ccc} i+1 & i+2 & i+3 \\ 4 & 5 & 2 \\ 1 &
i & 0
\end{array} \right)
\end{math}
\end{flushleft}
\vspace*{0.1in}
\begin{flushleft}
\hspace*{0.1in}
\begin{math}
\begin{array}{ccc}
{\tt hermitian\_tp}({\cal J}) & = &
\left( \begin{array}{ccc} -i+1 & 4 & 1 \\ -i+2 & 5 & -i \\-i+3 & 2 & 0
\end{array} \right)
\end{array}
\end{math}
\end{flushleft}
{\bf Related functions:}
\hspace*{0.175in} {\tt tp}\footnote{standard reduce call for the
transpose of a matrix - see {\REDUCE} User's Manual[2].}.
\subsection{hessian}
%{\bf How to use it:}
\hspace*{0.175in} {\tt hessian(expr,variable\_list);}
\hspace*{0.1in}
\begin{tabular}{l l l}
expr &:-& a scalar expression. \\
variable\_list &:-& either a single variable or a list of variables.
\end{tabular}
{\bf Synopsis:} %{\bf What it does:}
\begin{addtolength}{\leftskip}{0.22in}
{\tt hessian} computes the hessian matrix of expr w.r.t.
the varibles in variable\_list.
This is an n by n matrix
where n is the number of variables and the (i,$\,$j)'th
entry is df(expr,variable\_list(i),variable\_list(j)).
\end{addtolength}
{\bf Examples:}
\begin{flushleft}
\hspace*{0.1in}
\begin{math}
\begin{array}{ccc}
{\tt hessian}(x*y*z+x^2,\{w,x,y,z\}) & = &
\left( \begin{array}{cccc} 0 & 0 & 0 & 0 \\ 0 & 2 & z & y \\ 0 & z & 0
& x \\ 0 & y & x & 0
\end{array} \right)
\end{array}
\end{math}
\end{flushleft}
{\bf Related functions:}
\hspace*{0.175in} {\tt df}\footnote{standard reduce call for
differentiation - see {\REDUCE} User's Manual[2]}.
\subsection{hilbert}
%{\bf How to use it:}
\hspace*{0.175in} {\tt hilbert(square\_size,expr);}
\hspace*{0.1in}
\begin{tabular}{l l l}
square\_size &:-& a positive integer. \\
expr &:-& an algebraic expression.
\end{tabular}
{\bf Synopsis:} %{\bf What it does:}
\begin{addtolength}{\leftskip}{0.22in}
{\tt hilbert} computes the square hilbert matrix of
dimension square\_size.
This is the symmetric matrix in
which the (i,$\,$j)'th entry is 1/(i+j-expr).
\end{addtolength}
{\bf Examples:}
\begin{flushleft}
\hspace*{0.1in}
\begin{math}
\begin{array}{ccc}
{\tt hilbert}(3,y+x) & = &
\left( \begin{array}{ccc} \frac{-1}{x+y-2} & \frac{-1}{x+y-3}
& \frac{-1}{x+y-4} \\ \frac{-1}{x+y-3} & \frac{-1}{x+y-4} &
\frac{-1}{x+y-5} \\ \frac{-1}{x+y-4} & \frac{-1}{x+y-5} &
\frac{-1}{x+y-6}
\end{array} \right)
\end{array}
\end{math}
\end{flushleft}
\subsection{jacobian}
%{\bf How to use it:}
\hspace*{0.175in} {\tt jacobian(expr\_list,variable\_list);}
\hspace*{0.1in}
\begin{tabular}{l l l}
expr\_list \hspace*{0.175in} &:-& \parbox[t]{.72\linewidth}{either a
single algebraic expression or a list of algebraic expressions.}
\end{tabular}
\vspace*{0.04in}
\hspace*{0.1in}
\begin{tabular}{l l l}
variable\_list &:-& either a single variable or a list of variables.
\end{tabular}
{\bf Synopsis:} %{\bf What it does:}
\begin{addtolength}{\leftskip}{0.22in}
{\tt jacobian} computes the jacobian matrix of expr\_list w.r.t.
variable\_list.
This is a matrix whose (i,$\,$j)'th entry is df(expr\_list(i),
variable\_list(j)).
The matrix is n by m where n is the
number of variables and m the number of expressions.
\end{addtolength}
{\bf Examples:}
\hspace*{0.175in}
{\tt jacobian(\{$x^4,x*y^2,x*y*z^3$\},\{$w,x,y,z$\})} =
\vspace*{0.1in}
\begin{flushleft}
\hspace*{0.175in}
\begin{math}
\left( \begin{array}{cccc} 0 & 4*x^3 & 0 & 0 \\ 0 & y^2 & 2*x*y & 0 \\
0 & y*z^3 & x*z^3 & 3*x*y*z^2
\end{array} \right)
\end{math}
\end{flushleft}
{\bf Related functions:}
\hspace*{0.175in} {\tt hessian}, {\tt df}\footnote{standard reduce call
for differentiation - see {\REDUCE} User's Manual[2].}.
\subsection{jordan\_block}
%{\bf How to use it:}
\hspace*{0.175in} {\tt jordan\_block(expr,square\_size);}
\hspace*{0.1in}
\begin{tabular}{l l l}
expr &:-& an algebraic expression or symbol. \\
square\_size &:-& a positive integer.
\end{tabular}
{\bf Synopsis:} %{\bf What it does:}
\begin{addtolength}{\leftskip}{0.22in}
{\tt jordan\_block} computes the square jordan block matrix ${\cal J}$
of dimension square\_size.
The entries of ${\cal J}$ are:
${\cal J}$(i,i) = expr for i=1
\ldots n, ${\cal J}$(i,i+1) = 1 for i=1
\ldots n-1, and all other entries are 0.
\end{addtolength}
{\bf Examples:}
\begin{flushleft}
\hspace*{0.1in}
\begin{math}
\begin{array}{ccc}
{\tt jordan\_block(x,5)} & = &
\left( \begin{array}{ccccc} x & 1 & 0 & 0 & 0 \\ 0 & x & 1 & 0 & 0 \\ 0
& 0 & x & 1 & 0 \\ 0 & 0 & 0 & x & 1 \\ 0 & 0 & 0 & 0 & x
\end{array} \right)
\end{array}
\end{math}
\end{flushleft}
{\bf Related functions:}
\hspace*{0.175in} {\tt diagonal}, {\tt companion}.
\subsection{lu\_decom}
%{\bf How to use it:}
\hspace*{0.175in} {\tt lu\_decom(${\cal A}$);}
\hspace*{0.1in}
\begin{tabular}{l l l}
${\cal A}$ &:-& \parbox[t]{.848\linewidth}{a matrix containing either
numeric entries or imaginary entries with numeric coefficients.}
\end{tabular}
{\bf Synopsis:} %{\bf What it does:}
\begin{addtolength}{\leftskip}{0.22in}
{\tt lu\_decom} performs LU decomposition on ${\cal A}$,
ie: it returns \{${\cal L,U}$\} where ${\cal L}$
is a lower diagonal matrix, ${\cal U}$ an upper diagonal
matrix and ${\cal A} = {\cal LU}$.
\end{addtolength}
{\bf caution:}
\begin{addtolength}{\leftskip}{0.22in}
The algorithm used can swap the rows of ${\cal A}$
during the calculation. This means that ${\cal LU}$ does
not equal ${\cal A}$ but a row equivalent of it. Due to
this, {\tt lu\_decom} returns \{${\cal L,U}$,vec\}. The
call {\tt convert(${\cal A}$,vec)} will return the
matrix that has been decomposed, ie: ${\cal LU} = $
{\tt convert(${\cal A}$,vec)}.
\end{addtolength}
{\bf Examples:}
\begin{flushleft}
\hspace*{0.175in}
\begin{math}
{\cal K} = \left( \begin{array}{ccc} 1 & 3 & 5 \\ -4 & 3 & 7 \\ 8 & 6 &
4
\end{array} \right)
\end{math}
\end{flushleft}
\begin{flushleft}
\hspace*{0.1in}
\begin{math}
\begin{array}{cccc}
${\tt lu} := {\tt lu\_decom}$({\cal K}) & = &
\left\{
\left( \begin{array}{ccc} 8 & 0 & 0 \\ -4 & 6 & 0 \\ 1 & 2.25 &
1.125 1 \end{array} \right),
\left( \begin{array}{ccc} 1 & 0.75 & 0.5 \\ 0 & 1 & 1.5 \\ 0 &
0 & 1 \end{array} \right),
[\; 3 \; 2 \; 3 \; ]
\right\}
\end{array}
\end{math}
\end{flushleft}
\vspace*{0.1in}
\begin{flushleft}
\hspace*{0.1in}
\begin{math}
\begin{array}{ccc}
${\tt first lu * second lu}$ & = &
\left( \begin{array}{ccc} 8 & 6 & 4 \\ -4 & 3 & 7 \\ 1 & 3 & 5
\end{array} \right)
\end{array}
\end{math}
\end{flushleft}
\begin{flushleft}
\hspace*{0.1in}
\begin{math}
\begin{array}{ccc}
${\tt convert(${\cal K}$,third lu}$) \hspace*{0.055in} & = &
\left( \begin{array}{ccc} 8 & 6 & 4 \\ -4 & 3 & 7 \\ 1 & 3 & 5
\end{array} \right)
\end{array}
\end{math}
\end{flushleft}
\vspace*{0.5in}
\begin{flushleft}
\hspace*{0.175in}
\begin{math}
{\cal P} = \left( \begin{array}{ccc} i+1 & i+2 & i+3 \\ 4 & 5 & 2 \\ 1
& i & 0
\end{array} \right)
\end{math}
\end{flushleft}
\begin{eqnarray}
\hspace*{0.22in}
{\tt lu} := {\tt lu\_decom}({\cal P}) & = &
\left\{
\left( \begin{array}{ccc} 1 & 0 & 0 \\ 4 & -4*i+5 & 0 \\ i+1 &
3 & 0.41463*i+2.26829 \end{array} \right), \right. \nonumber \\ & &
\left. \: \; \, \left( \begin{array}{ccc} 1 & i & 0 \\ 0 & 1 &
0.19512*i+0.24390 \\ 0 & 0 & 1 \end{array} \right), \hspace*{0.05in}
[\; 3 \; 2 \; 3 \;] \hspace*{0.05in}
\right\} \nonumber
\end{eqnarray}
\vspace*{0.1in}
\begin{flushleft}
\hspace*{0.1in}
\begin{math}
\begin{array}{ccc}
${\tt first lu * second lu}$ & = &
\left( \begin{array}{ccc} 1 & i & 0 \\ 4 & 5 & 2 \\ i+1 & i+2 &
i+3
\end{array} \right)
\end{array}
\end{math}
\end{flushleft}
\begin{flushleft}
\hspace*{0.1in}
\begin{math}
\begin{array}{ccc}
${\tt convert({\cal P},third lu}$) \hspace*{0.1in} & = & \left(
\begin{array}{c c c} 1 & i & 0 \\ 4 & 5 & 2 \\ i+1 & i+2 & i+3
\end{array} \right)
\end{array}
\end{math}
\end{flushleft}
{\bf Related functions:}
\hspace*{0.175in} {\tt cholesky}.
\subsection{make\_identity}
%{\bf How to use it:}
\hspace*{0.175in} {\tt make\_identity(square\_size);}
\hspace*{0.1in}
\begin{tabular}{l l l}
square\_size &:-& a positive integer.
\end{tabular}
{\bf Synopsis:} %{\bf What it does:}
\hspace*{0.175in} {\tt make\_identity} creates the identity matrix of
dimension square\_size.
{\bf Examples:}
\begin{flushleft}
\hspace*{0.1in}
\begin{math}
\begin{array}{ccc}
{\tt make\_identity}(4) & = &
\left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0
& 0 & 1 & 0 \\ 0 & 0 & 0 & 1
\end{array} \right)
\end{array}
\end{math}
\end{flushleft}
{\bf Related functions:}
\hspace*{0.175in} {\tt diagonal}.
\subsection{matrix\_augment, matrix\_stack}
%{\bf How to use it:}
\hspace*{0.175in} {\tt matrix\_augment(\{\matlist\});}\lazyfootnote{}
\hspace*{0.1in}
\begin{tabular}{l l l}
\matlist &:-& matrices.
\end{tabular}
{\bf Synopsis:} %{\bf What it does:}
\hspace*{0.175in} {\tt matrix\_augment} sticks the matrices in
matrix\_list together horizontally.
\hspace*{0.175in}
{\tt matrix\_stack} sticks the matrices in matrix\_list
together vertically.
{\bf Examples:}
\begin{flushleft}
\hspace*{0.1in}
\begin{math}
\begin{array}{ccc}
{\tt matrix\_augment}(\{{\cal A,A}\}) & = &
\left( \begin{array}{cccccc} 1 & 2 & 3 & 1 & 2 & 3 \\ 4 & 4 & 6
& 4 & 5 & 6 \\ 7 & 8 & 9 & 7 & 8 & 9
\end{array} \right)
\end{array}
\end{math}
\end{flushleft}
\vspace*{0.1in}
\begin{flushleft}
\hspace*{0.1in}
\begin{math}
\begin{array}{ccc}
{\tt matrix\_stack}(\{{\cal A,A}\}) & = &
\left( \begin{array}{ccc} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9
\\ 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9
\end{array} \right)
\end{array}
\end{math}
\end{flushleft}
{\bf Related functions:}
\hspace*{0.175in} {\tt augment\_columns}, {\tt stack\_rows},
{\tt sub\_matrix}.
\subsection{matrixp}
%{\bf How to use it:}
\hspace*{0.175in} {\tt matrixp(test\_input);}
\hspace*{0.1in}
\begin{tabular}{l l l}
test\_input &:-& anything you like.
\end{tabular}
{\bf Synopsis:} %{\bf What it does:}
\begin{addtolength}{\leftskip}{0.22in}
{\tt matrixp} is a boolean function that returns t if
the input is a matrix and nil otherwise.
\end{addtolength}
{\bf Examples:}
\hspace*{0.175in} {\tt matrixp}(${\cal A}$) = t
\hspace*{0.175in} {\tt matrixp}(doodlesackbanana) = nil
{\bf Related functions:}
\hspace*{0.175in} {\tt squarep}, {\tt symmetricp}.
\subsection{matrix\_stack}
\hspace*{0.175in} see: {\tt matrix\_augment}.
\subsection{minor}
%{\bf How to use it:}
\hspace*{0.175in} {\tt minor(${\cal A}$,r,c);}
\hspace*{0.1in}
\begin{tabular}{l l l}
${\cal A}$ &:-& a matrix. \\
r,c &:-& positive integers.
\end{tabular}
{\bf Synopsis:} %{\bf What it does:}
\begin{addtolength}{\leftskip}{0.22in}
{\tt minor} computes the (r,c)'th minor of ${\cal A}$.
This is created by removing the r'th row and the c'th
column from ${\cal A}$.
\end{addtolength}
{\bf Examples:}
\begin{flushleft}
\hspace*{0.1in}
\begin{math}
\begin{array}{ccc}
{\tt minor}({\cal A},1,3) & = &
\left( \begin{array}{cc} 4 & 5 \\ 7 & 8
\end{array} \right)
\end{array}
\end{math}
\end{flushleft}
{\bf Related functions:}
\hspace*{0.175in} {\tt remove\_columns}, {\tt remove\_rows}.
\subsection{mult\_columns, mult\_rows}
%{\bf How to use it:}
\hspace*{0.175in} {\tt mult\_columns(${\cal A}$,column\_list,expr);}
\hspace*{0.1in}
\begin{tabular}{l l l}
${\cal A}$ &:-& a matrix. \\
column\_list &:-& a positive integer or a list of positive integers. \\
expr &:-& an algebraic expression.
\end{tabular}
{\bf Synopsis:} %{\bf What it does:}
\begin{addtolength}{\leftskip}{0.22in}
{\tt mult\_columns} returns a copy of ${\cal A}$ in which
the columns specified in column\_list have been
multiplied by expr.
{\tt mult\_rows} performs the same task on the rows of ${\cal A}$.
\end{addtolength}
{\bf Examples:}
\begin{flushleft}
\hspace*{0.1in}
\begin{math}
\begin{array}{ccc}
{\tt mult\_columns}({\cal A},\{1,3\},x) & = &
\left( \begin{array}{ccc} x & 2 & 3*x \\ 4*x & 5 & 6*x \\ 7*x &
8 & 9*x
\end{array} \right)
\end{array}
\end{math}
\end{flushleft}
\vspace*{0.1in}
\begin{flushleft}
\hspace*{0.1in}
\begin{math}
\begin{array}{ccc}
{\tt mult\_rows}({\cal A},2,10) & = &
\left( \begin{array}{ccc} 1 & 2 & 3 \\ 40 & 50 & 60 \\ 7 & 8 &
9 \end{array} \right)
\end{array}
\end{math}
\end{flushleft}
{\bf Related functions:}
\hspace*{0.175in} {\tt add\_to\_columns}, {\tt add\_to\_rows}.
\subsection{\tt mult\_rows}
\hspace*{0.175in} see: {\tt mult\_columns}.
\subsection{pivot}
%{\bf How to use it:}
\hspace*{0.175in} {\tt pivot(${\cal A}$,r,c);}
\hspace*{0.1in}
\begin{tabular}{l l l}
${\cal A}$ &:-& a matrix. \\
r,c &:-& positive integers such that ${\cal A}$(r,c) neq 0.
\end{tabular}
{\bf Synopsis:} %{\bf What it does:}
\begin{addtolength}{\leftskip}{0.22in}
{\tt pivot} pivots ${\cal A}$ about its (r,c)'th entry.
To do this, multiples of the r'th row are added to every
other row in the matrix.
This means that the c'th column
will be 0 except for the (r,c)'th entry.
\end{addtolength}
{\bf Examples:}
\begin{flushleft}
\hspace*{0.1in}
\begin{math}
\begin{array}{ccc}
{\tt pivot}({\cal A},2,3) & = &
\left( \begin{array}{ccc} -1 & -0.5 & 0 \\ 4 & 5 & 6 \\ 1 & 0.5
& 0
\end{array} \right)
\end{array}
\end{math}
\end{flushleft}
{\bf Related functions:}
\hspace*{0.175in} {\tt rows\_pivot}.
\subsection{pseudo\_inverse}
%{\bf How to use it:}
\hspace*{0.175in} {\tt pseudo\_inverse(${\cal A}$);}
\hspace*{0.1in}
\begin{tabular}{l l l}
${\cal A}$ &:-& a matrix.
\end{tabular}
{\bf Synopsis:} %{\bf What it does:}
\begin{addtolength}{\leftskip}{0.22in}
{\tt pseudo\_inverse}, also known as the Moore-Penrose inverse, computes
the pseudo inverse of ${\cal A}$.
Given the singular value decomposition of ${\cal A}$, i.e: ${\cal A} =
{\cal U}
\sum {\cal V}^T$, then the pseudo inverse ${\cal A}^{-1}$ is defined
by ${\cal A}^{-1} = {\cal V}^T \sum^{-1} {\cal U}$.
Thus ${\cal A}$ $ * $ {\tt pseudo\_inverse}$({\cal A}) = {\cal I}$.
\end{addtolength}
{\bf Examples:}
% \begin{flushleft}
% \hspace*{0.175in}
% \begin{math}
% {\cal R} = \left( \begin{array}{cccc} 1 & 2 & 3 & 4 \\ 9 & 8 & 7 & 6
% \end{array} \right)
% \end{math}
% \end{flushleft}
\begin{flushleft}
\hspace*{0.1in}
\begin{math}
\begin{array}{ccc}
{\tt pseudo\_inverse}({\cal A}) & = &
\left( \begin{array}{cc} -0.2 & 0.1 \\ -0.05 & 0.05 \\ 0.1 & 0
\\ 0.25 & -0.05
\end{array} \right)
\end{array}
\end{math}
\end{flushleft}
{\bf Related functions:}
\hspace*{0.175in} {\tt svd}.
\subsection{random\_matrix}
%{\bf How to use it:}
\hspace*{0.175in} {\tt random\_matrix(r,c,limit);}
\hspace*{0.1in}
\begin{tabular}{l l l}
r,c,$\,$limit &:-& positive integers. \\
\end{tabular}
{\bf Synopsis:} %{\bf What it does:}
\begin{addtolength}{\leftskip}{0.22in}
{\tt random\_matrix} creates an r by c matrix with random
entries in the range $-$limit $<$ entry $<$ limit.
\end{addtolength}
{\bf switches:}
\hspace*{0.1in}
\begin{tabular}{l l l}
{\tt imaginary} \hspace*{0.175in} &:-& \parbox[t]{0.685\linewidth}{if
on then matrix entries are x+i$*$y where $-$limit $<$ x,y $<$ limit.}
\end{tabular}
\vspace*{0.04in}
\hspace*{0.1in}
\begin{tabular}{l l l}
{\tt not\_negative} &:-& \parbox[t]{0.685\linewidth}{if on then 0 $<$
entry $<$ limit. In the imaginary case we have 0 $<$ x,y $<$ limit.}
\end{tabular}
\vspace*{0.04in}
\hspace*{0.1in}
\begin{tabular}{l l l}
{\tt only\_integer} &:-& \parbox[t]{0.685\linewidth}{if on then each
entry is an integer. In the imaginary case x and y are integers.}
\end{tabular}
\vspace*{0.04in}
\hspace*{0.1in}
\begin{tabular}{l l l}
{\tt symmetric} &:-& if on then the matrix is symmetric. \\
{\tt upper\_matrix} &:-& \parbox[t]{0.685\linewidth}{if on then the
matrix is upper triangular.} \\
{\tt lower\_matrix} &:-& if on then the matrix is lower triangular.
\end{tabular}
{\bf Examples:}
\begin{flushleft}
\hspace*{0.1in}
\begin{math}
\begin{array}{ccc}
{\tt random\_matrix}(3,3,10) & = &
\left( \begin{array}{ccc} -4.729721 & 6.987047 & 7.521383 \\
- 5.224177 & 5.797709 & - 4.321952 \\
- 9.418455 & - 9.94318 & - 0.730980
\end{array} \right)
\end{array}
\end{math}
\end{flushleft}
\vspace*{0.2in}
\hspace*{0.165in}
{\tt on only\_integer, not\_negative, upper\_matrix, imaginary;}
\begin{flushleft}
\hspace*{0.12in}
\begin{math}
\begin{array}{ccc}
{\tt random\_matrix}(4,4,10) & = &
\left( \begin{array}{cccc} 2*i+5 & 3*i+7 & 7*i+3 & 6 \\ 0 & 2*i+5 &
5*i+1 & 2*i+1 \\ 0 & 0 & 8 & i \\ 0 & 0 & 0& 5*i+9
\end{array} \right)
\end{array}
\end{math}
\end{flushleft}
\subsection{remove\_columns, remove\_rows}
%{\bf How to use it:}
\hspace*{0.175in} {\tt remove\_columns(${\cal A}$,column\_list);}
\hspace*{0.1in}
\begin{tabular}{l l l}
${\cal A}$ &:-& a matrix. \\
column\_list &:-& either a positive integer or a list of
positive integers.
\end{tabular}
{\bf Synopsis:} %{\bf What it does:}
\hspace*{0.175in} {\tt remove\_columns} removes the columns specified in
column\_list from ${\cal A}$.
\hspace*{0.175in} {\tt remove\_rows} performs the same task on the rows
of ${\cal A}$.
{\bf Examples:}
\begin{flushleft}
\hspace*{0.1in}
\begin{math}
\begin{array}{ccc}
{\tt remove\_columns}({\cal A},2) & = &
\left( \begin{array}{cc} 1 & 3 \\ 4 & 6 \\ 7 & 9
\end{array} \right)
\end{array}
\end{math}
\end{flushleft}
\vspace*{0.1in}
\begin{flushleft}
\hspace*{0.1in}
\begin{math}
\begin{array}{ccc}
{\tt remove\_rows}({\cal A},\{1,3\}) & = &
\left( \begin{array}{ccc} 4 & 5 & 6
\end{array} \right)
\end{array}
\end{math}
\end{flushleft}
{\bf Related functions:}
\hspace*{0.175in} {\tt minor}.
\subsection{remove\_rows}
\hspace*{0.175in} see: {\tt remove\_columns}.
\subsection{row\_dim}
\hspace{0.175in} see: {\tt column\_dim}.
\subsection{rows\_pivot}
%{\bf How to use it:}
\hspace*{0.175in} {\tt rows\_pivot(${\cal A}$,r,c,\{row\_list\});}
\hspace*{0.1in}
\begin{tabular}{l l l}
${\cal A}$ &:-& a matrix. \\
r,c &:-& positive integers such that ${\cal A}$(r,c) neq 0.\\
row\_list &:-& positive integer or a list of positive integers.
\end{tabular}
{\bf Synopsis:} %{\bf What it does:}
\begin{addtolength}{\leftskip}{0.22in}
{\tt rows\_pivot} performs the same task as {\tt pivot} but applies
the pivot only to the rows specified in row\_list.
\end{addtolength}
{\bf Examples:}
\begin{flushleft}
\hspace*{0.175in}
\begin{math}
{\cal N} = \left( \begin{array}{ccc} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 &
9 \\1 & 2 & 3 \\ 4 & 5 & 6
\end{array} \right)
\end{math}
\end{flushleft}
\begin{flushleft}
\hspace*{0.1in}
\begin{math}
\begin{array}{ccc}
{\tt rows\_pivot}({\cal N},2,3,\{4,5\}) & = & \left( \begin{array}
{c c c}1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \\ -0.75 & 0 & 0.75 \\
-0.375 & 0 & 0.375
\end{array} \right)
\end{array}
\end{math}
\end{flushleft}
{\bf Related functions:}
\hspace*{0.175in} {\tt pivot}.
\subsection{simplex}
%{\bf How to use it:}
\hspace*{0.175in} {\tt simplex(max/min,objective function,\{linear
inequalities\});}
\hspace*{0.1in}
\begin{tabular}{l l l}
max/min & :- & \parbox[t]{.63\linewidth}{either max or min
(signifying maximise and minimise).} \\
objective function & :- & the function you are maximising or
minimising. \\
linear inequalities & :- & \parbox[t]{.63\linewidth}{the constraint
inequalities. Each one must be of the form
{\it sum of variables ($<=,=,>=$) number}.}
\end{tabular}
{\bf Synopsis:} %{\bf What it does:}
\begin{addtolength}{\leftskip}{0.22in}
{\tt simplex} applies the revised simplex algorithm to find the
optimal(either maximum or minimum) value of the objective function
under the linear inequality constraints.
It returns \{optimal value,\{ values of variables at this optimal\}\}.
The algorithm implies that all the variables are non-negative.
\end{addtolength}
{\bf Examples:}
\begin{addtolength}{\leftskip}{0.22in}
%\begin{math}
{\tt simplex($max,x+y,\{x>=10,y>=20,x+y<=25\}$);}
%\end{math}
{\tt ***** Error in simplex: Problem has no feasible solution.}
\vspace*{0.2in}
\parbox[t]{0.96\linewidth}{\tt simplex($max,10x+5y+5.5z,\{5x+3z<=200,
x+0.1y+0.5z<=12$,\\
\hspace*{0.55in} $0.1x+0.2y+0.3z<=9, 30x+10y+50z<=1500\}$);}
\vspace*{0.1in}
{\tt $\{525.0,\{x=40.0,y=25.0,z=0\}$\}}
\end{addtolength}
\subsection{squarep}
%{\bf How to use it:}
\hspace*{0.175in} {\tt squarep(${\cal A}$);}
\hspace*{0.1in}
\begin{tabular}{l l l}
${\cal A}$ &:-& a matrix.
\end{tabular}
{\bf Synopsis:} %{\bf What it does:}
\begin{addtolength}{\leftskip}{0.22in}
{\tt squarep} is a boolean function that returns t if
the matrix is square and nil otherwise.
\end{addtolength}
{\bf Examples:}
\begin{flushleft}
\hspace*{0.175in}
\begin{math}
{\cal L} = \left( \begin{array}{ccc} 1 & 3 & 5
\end{array} \right)
\end{math}
\end{flushleft}
\vspace*{0.1in}
\hspace*{0.175in} {\tt squarep}(${\cal A}$) = t
\hspace*{0.175in} {\tt squarep}(${\cal L}$) = nil
{\bf Related functions:}
\hspace*{0.175in} {\tt matrixp}, {\tt symmetricp}.
\subsection{stack\_rows}
\hspace*{0.175in} see: {\tt augment\_columns}.
\subsection{sub\_matrix}
%{\bf How to use it:}
\hspace*{0.175in} {\tt sub\_matrix(${\cal A}$,row\_list,column\_list);}
\hspace*{0.1in}
\begin{tabular}{l l l}
${\cal A}$ &:-& a matrix. \\
row\_list, column\_list &:-& \parbox[t]{.605\linewidth}{either a
positive integer or a list of positive integers.}
\end{tabular}
{\bf Synopsis:} %{\bf What it does:}
\begin{addtolength}{\leftskip}{0.22in}
{\tt sub\_matrix} produces the matrix consisting of the
intersection of the rows specified in row\_list and the
columns specified in column\_list.
\end{addtolength}
{\bf Examples:}
\begin{flushleft}
\hspace*{0.1in}
\begin{math}
\begin{array}{ccc}
{\tt sub\_matrix}({\cal A},\{1,3\},\{2,3\}) & = &
\left( \begin{array}{cc} 2 & 3 \\ 8 & 9
\end{array} \right)
\end{array}
\end{math}
\end{flushleft}
{\bf Related functions:}
\hspace*{0.175in} {\tt augment\_columns}, {\tt stack\_rows}.
\subsection{svd (singular value decomposition)}
%{\bf How to use it:}
\hspace*{0.175in} {\tt svd(${\cal A}$);}
\hspace*{0.1in}
\begin{tabular}{l l l}
${\cal A}$ &:-& a matrix containing only numeric entries.
\end{tabular}
{\bf Synopsis:} %{\bf What it does:}
\begin{addtolength}{\leftskip}{0.22in}
{\tt svd} computes the singular value decomposition of ${\cal A}$.
It returns \{${\cal U},\sum,{\cal V}$\} where ${\cal A} = {\cal U}
\sum {\cal V}^T$ and $\sum = diag(\sigma_{1}, \ldots ,\sigma_{n}). \;
\sigma_{i}$ for $i= (1 \ldots n)$ are the singular values of ${\cal A}$.
n is the column dimension of ${\cal A}$.
The singular values of ${\cal A}$ are the non-negative square roots of
the eigenvalues of ${\cal A}^T {\cal A}$.
${\cal U}$ and ${\cal V}$ are such that ${\cal UU}^T = {\cal VV}^T =
{\cal V}^T {\cal V} = {\cal I}_n$.
\end{addtolength}
{\bf Examples:}
\begin{flushleft}
\hspace*{0.175in}
\begin{math}
{\cal Q} = \left( \begin{array}{cc} 1 & 3 \\ -4 & 3
\end{array} \right)
\end{math}
\end{flushleft}
\begin{eqnarray}
\hspace*{0.1in}
{\tt svd({\cal Q})} & = &
\left\{
\left( \begin{array}{cc} 0.289784 & 0.957092 \\ -0.957092 &
0.289784 \end{array} \right), \left( \begin{array}{cc} 5.149162 & 0 \\
0 & 2.913094 \end{array} \right), \right. \nonumber \\ & & \left. \: \;
\, \left( \begin{array}{cc} -0.687215 & 0.726453 \\ -0.726453 &
-0.687215 \end{array} \right)
\right\} \nonumber
\end{eqnarray}
\subsection{swap\_columns, swap\_rows}
%{\bf How to use it:}
\hspace*{0.175in} {\tt swap\_columns(${\cal A}$,c1,c2);}
\hspace*{0.1in}
\begin{tabular}{l l l}
${\cal A}$ &:-& a matrix. \\
c1,c1 &:-& positive integers.
\end{tabular}
{\bf Synopsis:} %{\bf What it does:}
\hspace*{0.175in}
{\tt swap\_columns} swaps column c1 of ${\cal A}$ with column c2.
\hspace*{0.175in} {\tt swap\_rows} performs the same task on 2 rows of
${\cal A}$.
{\bf Examples:}
\begin{flushleft}
\hspace*{0.1in}
\begin{math}
\begin{array}{ccc}
{\tt swap\_columns}({\cal A},2,3) & = &
\left( \begin{array}{ccc} 1 & 3 & 2 \\ 4 & 6 & 5 \\ 7 & 9 & 8
\end{array} \right)
\end{array}
\end{math}
\end{flushleft}
{\bf Related functions:}
\hspace*{0.175in} {\tt swap\_entries}.
\subsection{swap\_entries}
%{\bf How to use it:}
\hspace*{0.175in} {\tt swap\_entries(${\cal A}$,\{r1,c1\},\{r2,c2\});}
\hspace*{0.1in}
\begin{tabular}{l l l}
${\cal A}$ &:-& a matrix. \\
r1,c1,r2,c2 &:-& positive integers.
\end{tabular}
{\bf Synopsis:} %{\bf What it does:}
\hspace*{0.175in} {\tt swap\_entries} swaps ${\cal A}$(r1,c1) with
${\cal A}$(r2,c2).
{\bf Examples:}
\begin{flushleft}
\hspace*{0.1in}
\begin{math}
\begin{array}{ccc}
{\tt swap\_entries}({\cal A},\{1,1\},\{3,3\}) & = &
\left( \begin{array}{ccc} 9 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 1
\end{array} \right)
\end{array}
\end{math}
\end{flushleft}
{\bf Related functions:}
\hspace*{0.175in} {\tt swap\_columns}, {\tt swap\_rows}.
\subsection{swap\_rows}
\hspace*{0.175in} see: {\tt swap\_columns}.
\subsection{symmetricp}
%{\bf How to use it:}
\hspace*{0.175in} {\tt symmetricp(${\cal A}$);}
\hspace*{0.1in}
\begin{tabular}{l l l}
${\cal A}$ &:-& a matrix.
\end{tabular}
{\bf Synopsis:} %{\bf What it does:}
\begin{addtolength}{\leftskip}{0.22in}
{\tt symmetricp} is a boolean function that returns t if the
matrix is symmetric and nil otherwise.
\end{addtolength}
{\bf Examples:}
\begin{flushleft}
\hspace*{0.175in}
\begin{math}
{\cal M} = \left( \begin{array}{cc} 1 & 2 \\ 2 & 1
\end{array} \right)
\end{math}
\end{flushleft}
\vspace*{0.1in}
\hspace*{0.175in} {\tt symmetricp}(${\cal A}$) = nil
\hspace*{0.175in} {\tt symmetricp}(${\cal M}$) = t
{\bf Related functions:}
\hspace*{0.175in} {\tt matrixp}, {\tt squarep}.
\subsection{toeplitz}
%{\bf How to use it:}
\hspace*{0.175in} {\tt toeplitz(\{\exprlist{}\});} \lazyfootnote{}
\hspace*{0.1in}
\begin{tabular}{l l l}
\exprlist{} &:-& algebraic expressions.
\end{tabular}
{\bf Synopsis:} %{\bf What it does:}
\begin{addtolength}{\leftskip}{0.22in}
{\tt toeplitz} creates the toeplitz matrix from the
expression list.
This is a square symmetric matrix in
which the first expression is placed on the diagonal
and the i'th expression is placed on the (i-1)'th sub
and super diagonals.
It has dimension n where n is the
number of expressions.
\end{addtolength}
{\bf Examples:}
\begin{flushleft}
\hspace*{0.1in}
\begin{math}
\begin{array}{ccc}
{\tt toeplitz}(\{w,x,y,z\}) & = &
\left( \begin{array}{cccc} w & x & y & z \\ x & w & x & y \\
y & x & w & x \\ z & y & x & w
\end{array} \right)
\end{array}
\end{math}
\end{flushleft}
\subsection{Vandermonde}
%{\bf How to use it:}
\hspace*{0.175in} {\tt vandermonde}(\{\exprlist{}\}); \addtocounter
{footnote}{-1}\footnotemark
%\lazyfootnote{}
\hspace*{0.1in}
\begin{tabular}{l l l}
\exprlist{} &:-& algebraic expressions.
\end{tabular}
{\bf Synopsis:} %{\bf What it does:}
\begin{addtolength}{\leftskip}{0.22in}
{\tt Vandermonde} creates the Vandermonde matrix from
the expression list.
This is the square matrix in which
the (i,$\,$j)'th entry is expr\_list(i) $^{(j-1)}$.
It has dimension n where n is the number of expressions.
\end{addtolength}
{\bf Examples:}
\begin{flushleft}
\hspace*{0.1in}
\begin{math}
\begin{array}{ccc}
{\tt vandermonde}(\{x,2*y,3*z\}) & = &
\left( \begin{array}{ccc} 1 & x & x^2 \\ 1 & 2*y & 4*y^2 \\ 1
& 3*z & 9*z^2
\end{array} \right)
\end{array}
\end{math}
\end{flushleft}
\subsection{kronecker\_product}
\hspace*{0.175in} {\tt kronecker\_product}($Mat_1,Mat_2$)
\hspace*{0.1in}
\begin{tabular}{l l l}
$Mat_1,Mat_2$ &:-& Matrices
\end{tabular}
{\bf Synopsis:}
\begin{addtolength}{\leftskip}{0.22in}
{\tt kronecker\_product} creates a matrix containing the Kronecker product
(also called {\tt direct product} or {\tt tensor product}) of its arguments.
\end{addtolength}
{\bf Examples:}
\begin{verbatim}
a1 := mat((1,2),(3,4),(5,6))$
a2 := mat((1,1,1),(2,z,2),(3,3,3))$
kronecker_product(a1,a2);
\end{verbatim}
\begin{flushleft}
\hspace*{0.1in}
\begin{math}
\begin{array}{ccc}
\left( \begin{array}{cccccc} 1 & 1 & 1 & 2 & 2 & 2 \\
2 & z & 2 & 4 &2*z &4 \\
3 & 3 & 3 & 6 & 6 &6 \\
3 & 3 & 3 & 4 & 4 &4 \\
6 & 3*z& 6 & 8 &4*z &8 \\
9 & 9 & 9 & 12 &12 &12\\
5 & 5 & 5 & 6 & 6 &6 \\
10 &5*z& 10& 12 &6*z &12 \\
15 &15 & 15& 18 &18 &18 \end{array} \right)
\end{array}
\end{math}
\end{flushleft}
\section{Fast Linear Algebra}
By turning the {\tt fast\_la} switch on, the speed of the following
functions will be increased:
\begin{tabular}{l l l l}
add\_columns & add\_rows & augment\_columns & column\_dim \\
copy\_into & make\_identity & matrix\_augment & matrix\_stack\\
minor & mult\_column & mult\_row & pivot \\
remove\_columns & remove\_rows & rows\_pivot & squarep \\
stack\_rows & sub\_matrix & swap\_columns & swap\_entries\\
swap\_rows & symmetricp
\end{tabular}
The increase in speed will be insignificant unless you are making a
significant number(i.e: thousands) of calls. When using this switch,
error checking is minimised. This means that illegal input may give
strange error messages. Beware.
\newpage
\section{Acknowledgments}
Many of the ideas for this package came from the Maple[3] Linalg package
[4].
The algorithms for {\tt cholesky}, {\tt lu\_decom}, and {\tt svd} are
taken from the book Linear Algebra - J.H. Wilkinson \& C. Reinsch[5].
The {\tt gram\_schmidt} code comes from Karin Gatermann's Symmetry
package[6] for {\REDUCE}.
\begin{thebibliography}{}
\bibitem{matt} Matt Rebbeck: NORMFORM: A {\REDUCE} package for the
computation of various matrix normal forms. ZIB, Berlin. (1993)
\bibitem{Reduce} Anthony C. Hearn: {\REDUCE} User's Manual 3.6.
RAND (1995)
\bibitem{Maple} Bruce W. Char\ldots [et al.]: Maple (Computer
Program). Springer-Verlag (1991)
\bibitem{linalg} Linalg - a linear algebra package for Maple[3].
\bibitem{WiRe} J. H. Wilkinson \& C. Reinsch: Linear Algebra
(volume II). Springer-Verlag (1971)
\bibitem{gat} Karin Gatermann: Symmetry: A {\REDUCE} package for the
computation of linear representations of groups. ZIB, Berlin. (1992)
\end{thebibliography}
\end{document}