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%%\VignetteIndexEntry{Introduction to the Matrix Package}
\title{Introduction to the Matrix package}
\author{Douglas Bates\\R Core Development Group\\\email{}}
  Linear algebra is at the core of many areas of statistical computing
  and from its inception the \Slang{} has supported numerical linear
  algebra via a matrix data type and several functions and operators,
  such as \code{\%*\%}, \code{qr}, \code{chol}, and \code{solve}.
  However, these data types and functions do not provide direct access
  to all of the facilities for efficient manipulation of dense
  matrices, as provided by the Lapack subroutines, and they do not
  provide for manipulation of sparse matrices.

  The \code{Matrix} package provides a set of S4 classes for dense and
  sparse matrices that extend the basic matrix data type.  Methods for
  a wide variety of functions and operators applied to objects from
  these classes provide efficient access to BLAS (Basic Linear Algebra
  Subroutines), Lapack (dense matrix), TAUCS (sparse matrix) and
  UMFPACK (sparse matrix) routines.  One notable characteristic of the
  package is that whenever a matrix is factored, the factorization is
  stored as part of the original matrix so that further operations on
  the matrix can reuse this factorization.
<<preliminaries, echo=FALSE>>=


Linear algebra is at the core of many statistical computing techniques
and, from its inception, the \Slang{} has supported numerical linear
algebra via a matrix data type and several functions and operators,
such as \code{\%*\%}, \code{qr}, \code{chol}, and \code{solve}.
Initially the numerical linear algebra functions in \RR{} called
underlying Fortran routines from the Linpack~\citep{Linpack} and
Eispack~\cite{Eispack} libraries but over the years most of these
functions have been switched to use routines from the
Lapack~\cite{Lapack} library.  Furthermore, \RR{} can be configured to
use accelerated BLAS (Basic Linear Algebra Subroutines), such as those
from the Atlas~\cite{Atlas} project or Goto's BLAS~\cite{GotosBLAS}.

Lapack provides routines for operating on several special forms of
matrices, such as triangular matrices and symmetric matrices.
Furthermore,matrix decompositions like the QR decompositions produce
multiple output components that should be regarded as parts of a
single object.  There is some support in R for operations on special
forms of matrices (e.g. the \code{backsolve}, \code{forwardsolve} and
\code{chol2inv} functions) and for special structures (e.g. a QR
structure is implicitly defined as a list by the \code{qr},
\code{qr.qy}, \code{qr.qty}, and related functions) but it is not as
fully developed as it could be.

Also there is no direct support for sparse matrices in R although
\citet{koen:ng:2003} have developed a contributed package for sparse
matrices based on SparseKit.

The \code{Matrix} package provides S4 classes and methods for dense
and sparse matrices.  The methods for dense matrices use Lapack and
BLAS.  The sparse matrix methods use TAUCS~\citep{Taucs},
UMFPACK~\citep{Umfpack}, and Metis~\citep{Metis}.

\section{Classes for dense matrices}

The \code{Matrix} package will provide classes for real (stored as
double precision) and complex (stored as double precision complex)
dense matrices.  At present only the real classes have been
implemented.  These classes are
\item[geMatrix] Real matrices in general storage mode
\item[syMatrix] Symmetric real matrices in non-packed storage
\item[trMatrix] Triangular real matrices in non-packaged storage
\item[poMatrix] Positive semi-definite symmetric real matrices in
  non-package storage
Methods for these classes include coersion between these classes, when
appropriate, and coersion to the \code{matrix} class; methods for
matrix multiplication (\code{\%*\%}); cross products
(\code{crossprod}), matrix norm (\code{norm}); reciprocal condition
number (\code{rcond}); LU factorization (\code{lu}) or, for the
\code{poMatrix} class, the Cholesky decomposition (\code{chol}); and
solutions of linear systems of equations (\code{solve}).

Whenever a factorization or a decomposition is calculated it is
preserved as a (list) element in the \code{factors} slot of the
original object.  In this way a sequence of operations, such as
determining the condition number of a matrix then solving a linear
system based on the matrix, do not require multiple factorizations of
the same matrix nor do they require the user to store the intermediate

\section{Classes for sparse matrices}

\subsection{Representations of sparse matrices}

Conceptually, the simplest representation of a sparse matrix is as a
triplet of an integer vector \code{i} giving the row numbers, an
integer vector \code{j} giving the column numbers, and a numeric
vector \code{x} giving the non-zero values in the matrix.  An S4 class
definition might be
         representation(i = "integer", j = "integer", x = "numeric",
                        Dim = "integer"))
The triplet representation is row-oriented if elements in the same row
were adjacent and column-oriented if elements in the same column were
adjacent.  The compressed sparse row (csr) (or compressed sparse
column - csc) representation is similar to row-oriented triplet
(column-oriented triplet) except that \code{i} (\code{j}) just stores
the index of the first element in the row (column).  (There are a
couple of other details but that is the gist of it.)  These compressed
representations remove the redundant row (column) indices and provide
faster access to a given location in the matrix because you only need
to check one row (column).
The preferred representation of sparse matrices in the SparseM package
is csr.  Matlab uses csc.  We hope that Octave will also use this
representation. There are certain advantages to csc in systems like R
and Matlab where dense matrices are stored in column-major order.  For
example, Sivan Toledo's TAUCS~\cite{Taucs} library and Tim Davis's
UMFPACK~\cite{Umfpack} library are both based on csc and can both use
level-3 BLAS in certain sparse matrix computations.

The Matrix package provides the following classes for sparse matrices
\item[tripletMatrix] general, real, sparse matrices in (a possibly
  redundant) triplet form.  This can be a convenient form in which to
  construct sparse matrices.
\item[cscMatrix] general, real, sparse matrices in the (sorted) compressed
  sparse column format.
\item[sscMatrix] symmetric, real, sparse matrices in the (sorted)
  compressed sparse column format.  Only the upper or the lower triangle is
  stored.  Although there is provision for both forms, the lower
  triangle form works best with TAUCS.
\item[tscMatrix] triangular, real, sparse matrices in the (sorted)
  compressed sparse column format.

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