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M[...] <- "sparseVector" etc, see ChangeLog
\usepackage{fullpage}% save trees ;-)
\newcommand{\noFootnote}[1]{{\small (\textit{#1})}}
%%\VignetteIndexEntry{2nd Introduction to the Matrix Package}
%								          ^^^^^^^^^^^^^^^^
\title{2nd Introduction to the Matrix package}
\author{Martin Maechler and Douglas Bates\\ R Core Development Team
  \\\email{}, \email{}}
\date{September 2006 ({\tiny typeset on \tiny\today})}
% \emph{\Large Why should you want to work with this package and what
%   does it do for you?}

  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 \pkg{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), CHOLMOD including AMD and COLAMD and
  \code{Csparse} (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.

%% Note: These are explained in '?RweaveLatex' :
<<preliminaries, echo=FALSE>>=

The most automatic way to use the \pkg{Matrix} package is via the
\Rfun{Matrix} function which is very similar to the standard \RR\ function

M <- Matrix(10 + 1:28, 4, 7)
tM <- t(M)
@ %def
Such a matrix can be appended to (using \Rfun{cBind} or \Rfun{rBind}
with capital ``B'') or indexed,
(M2 <- cBind(-1, M))
M[2, 1]
M[4, ]
where the last two statements show customary matrix indexing, returning a
simple numeric vector each\footnote{because there's an additional default
  argument to indexing, \code{drop = TRUE}.  If you add
  \hbox{``\code{\ ,\ drop = FALSE}\ ''} you will get submatrices instead of
  simple vectors.}.
We assign 0 to some columns and rows to ``sparsify'' it, and some \code{NA}s
(typically ``missing values'' in data analysis) in order to demonstrate how
they are dealt with; note how we can \emph{``subassign''} as usual,
for classical \RR{} matrices (i.e., single entries or whole slices at once),
M2[, c(2,4:6)] <- 0
M2[2, ] <- 0
M2 <- rBind(0, M2, 0)
M2[1:2,2] <- M2[3,4:5] <- NA
and then coerce it to a sparse matrix,
sM <- as(M2, "sparseMatrix")
10 * sM
identical(sM * 2, sM + sM)
is(sM / 10  +  M2 %/% 2, "sparseMatrix")
@ %def
where the last three calls show that multiplication by a scalar keeps
sparcity, as does other arithmetic, but addition to a ``dense'' object does not,
as you might have expected after some thought about ``sensible'' behavior:
sM + 10
@ %def

Operations on our classed matrices include
(componentwise) arithmetic ($+$, $-$, $*$, $/$, etc) as partly seen above,
comparison ($>$, $\le$, etc), e.g.,
Mg2 <- (sM > 2)
returning a logical sparse matrix.  When interested in the internal
\textbf{str}ucture, \Rfun{str} comes handy, and we have been using it
ourselves more regulary than \Rfun{print}ing (or \Rfun{show}ing as it
happens) our matrices; alternatively, \Rfun{summary} gives output similar
to Matlab's printing of sparse matrices.
As you see from both of these, \code{Mg2} contains ``extra zero'' (here
\code{FALSE}) entries; such sparse matrices may be created for different reasons,
and you can use \Rfun{drop0} to remove (``drop'') these extra zeros.  This
should \emph{never} matter for functionality, and does not even show
differently for logical sparse matrices, but the internal structure is more
Mg2 <- drop0(Mg2)
str(Mg2@x) # length 13, was 16
For large sparse matrices, visualization (of the sparsity pattern) is important,
and we provide \Rfun{image} methods for that, e.g.,
print(image(CAex, main = "image(CAex)")) # print(.) needed for Sweave


Further, i.e., in addition to the above implicitly mentioned \code{"Ops"} operators
(\code{+}, \code{*},\dots, \code{<=},\code{>},\dots, \code{\&} which all
work with our matrices, notably in conjunction with scalars and traditional
matrices), the \code{"Math"}-operations (such as \Rfun{exp}, \Rfun{sin} or \Rfun{gamma})
and \code{"Math2"} (\Rfun{round} etc) and the \code{"Summary"} group of
functions, \Rfun{min}, \Rfun{range}, \Rfun{sum}, all work on our matrices
as they should.  Note that all these are implemented via so called
\emph{group methods}, see e.g., \code{?Arith} in \RR.
The intention is that sparse matrices remain sparse whenever sensible,
given the matrix \emph{classes} and operators involved,
but not content specifically. E.g.,  <sparse> + <dense> gives <dense> even
for the rare cases where it would be advantageous to get a <sparse>

These classed matrices can be ``indexed'' (more technically ``subset'') as
traditional \Slang{} (and hence \RR) matrices, as partly seen above. This also includes the idiom
\code{M [ M \myOp{\mathit{op}} \myOp{\mathrm{num}}~]}
which returns simple vectors,
sM[sM > 2]
sml <- sM[sM <= 2]
@ %def
and \emph{``subassign''}ment similarly works in the same generality as for
traditional \Slang{} matrices.
%% We have seen that already above!

%% This was the 2005 - Introduction vignette's first section:
\subsection{\pkg{Matrix} package for numerical linear algebra}

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~\citep{Eispack} libraries but over the years most of these
functions have been switched to use routines from the
Lapack~\citep{Lapack} library which is the state-of-the-art implementation
of numerical dense linear algebra.
Furthermore, \RR{} can be configured to
use accelerated BLAS (Basic Linear Algebra Subroutines), such as those
from the Atlas~\citep{Atlas} project or other ones, see the \RR~manual
``Installation and Administration''.

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 \RR{} 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 \RR{} although
\citet{koen:ng:2003} have developed the \pkg{SparseM} package for sparse
matrices based on SparseKit.

The \pkg{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
CHOLMOD~\citep{Cholmod}, CSparse~\citep{Csparse}
and other parts (AMD, COLAMD) of Tim Davis' ``SuiteSparse'' collection of
sparse matrix libraries, many of which also use BLAS.

\TODO{\Rfun{triu}, \Rfun{tril}, \Rfun{diag}, ...
  and  \command{as(.,.)} , but of course only when they've seen a few
  different ones.}

\TODO{matrix operators include \code{\%*\%}, \Rfun{crossprod},
  \Rfun{tcrossprod}, \Rfun{solve}}

\TODO{\Rfun{expm} is the matrix exponential ... ...}

\TODO{\Rfun{symmpart} and \Rfun{skewpart} compute the symmetric part,
  \code{(x + t(x))/2} and the skew-symmetric part,
  \code{(x - t(x))/2} of a matrix \code{x}.}

\TODO{factorizations include \Rfun{Cholesky} (or \Rfun{chol}), \Rfun{lu}, \Rfun{qr} (not yet for dense)}

\TODO{Although generally the result of an operation on dense matrices is
  a dgeMatrix, certain operations return matrices of special types.}
\TODO{E.g. show the distinction between \code{t(mm) \%*\% mm}
  and \code{crossprod(mm)}.}

% \bigskip

% ... ... ...  The following is the old \file{Introduction.Rnw} ... FIXME ... ...


\section{Matrix Classes}
The \pkg{Matrix} package provides classes for real (stored as
double precision), logical and so-called ``pattern'' (binary) dense and
sparse matrices. There are provisions to also provide integer and complex
(stored as double precision complex) matrices.

Note that in \RR, \code{logical} means entries
\code{TRUE}, \code{FALSE}, or \code{NA}.
To store just the non-zero pattern for typical sparse matrix algorithms,
the pattern matrices are \emph{binary}, i.e., conceptually just \code{TRUE}
or \code{FALSE}.  In \pkg{Matrix}, the pattern matrices all have class
names starting with \code{"n"} (patter\textbf{n}).

\subsection{Classes for dense matrices}

For the sake of brevity, we restrict ourselves to the
\emph{real} (\textbf{d}ouble) classes, but they are paralleled by
\textbf{l}ogical and patter\textbf{n} matrices for all but the positive
definite ones.
\item[dgeMatrix] Real matrices in general storage mode
\item[dsyMatrix] Symmetric real matrices in non-packed storage
\item[dspMatrix] Symmetric real matrices in packed storage (one triangle only)
\item[dtrMatrix] Triangular real matrices in non-packed storage
\item[dtpMatrix] Triangular real matrices in packed storage (triangle only)
\item[dpoMatrix] Positive semi-definite symmetric real matrices in
  non-packed storage
\item[dppMatrix] \ \ ditto \ \ in packed storage
Methods for these classes include coercion between these classes, when
appropriate, and coercion 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}).

%-- mentioned above already:
% Further, group methods have been defined for the \code{Arith} (basic
% arithmetic, including with scalar numbers) and the \code{Math} (basic
% mathematical functions) group..

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

\subsection{Classes for sparse matrices}

Used for large matrices in which most of the elements are known to
be zero.

\TODO{E.g. model matrices created from factors with a large number of levels}

\TODO{ or from spline basis functions  (e.g. COBS, package \pkg{cobs}), etc.}

\TODO{Other uses include representations of graphs.
  indeed; good you mentioned it!
  particularly since we still have the interface to the \pkg{graph} package.
  I think I'd like to draw one graph in that article --- maybe the
  undirected graph corresponding to a crossprod() result of
  dimension ca. $50^2$}

 \TODO{Specialized algorithms can give substantial savings in amount of
   storage used and execution time of operations.}

 \TODO{Our implementation is based on the CHOLMOD and CSparse libraries by
   Tim Davis.}

\subsection{Representations of sparse matrices}

\subsubsection{Triplet representation (\class{TsparseMatrix})}
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.\footnote{For efficiency
reasons, we use ``zero-based'' indexing in the \pkg{Matrix} package, i.e.,
the row indices \code{i} are in \code{0:(nrow(.)-1)} and the column indices
\code{j} accordingly.}  In \pkg{Matrix}, the \class{TsparseMatrix} class is the
virtual class of all sparse matrices in triplet representation.
Its main use is for easy input or transfer to other classes.

As for the dense matrices, the class of the \code{x} slot may vary, and the
subclasses may be triangular, symmetric or unspecified (``general''), such that
the \class{TsparseMatrix} class has several\footnote{the $3 \times 3$
  actual subclasses of \class{TsparseMatrix} are the three structural
  kinds, namely \textbf{t}riangular, \textbf{s}ymmetric and \textbf{g}eneral,
  times three entry classes, \textbf{d}ouble, \textbf{l}ogical, and
`actual'' subclasses,
the most typical (numeric, general) is \class{dgTMatrix}:
getClass("TsparseMatrix") # (i,j, Dim, Dimnames) slots are common to all
Note that the \emph{order} of the entries in the \code{(i,j,x)} vectors
does not matter; consequently, such matrices are not unique in their
representation.  \footnote{
  Furthermore, there can be \emph{repeated} \code{(i,j)} entries with the
  customary convention that the corresponding \code{x} entries are
  \emph{added} to form the matrix element $m_{ij}$.
%% 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.

\subsubsection{Compressed representations: \class{CsparseMatrix} }
% (and \class{RsparseMatrix})} -- but don't advertize them

For most sparse operations we use the compressed column-oriented
representation (virtual class \class{CsparseMatrix}) (also known as
``csc'', ``compressed sparse column'').  Here, instead of storing all
column indices \code{j}, only the \emph{start} index of every column is stored.

Analogously, there is also a compressed sparse row (csr) representation,
which e.g. is used in in the \pkg{SparseM} package, and we provide the
\class{RsparseMatrix} for compatibility and completeness purposes, in
addition to basic coercion (\code({as(., \textit{<cl>})} between the classes.
%% (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).

There are certain advantages \footnote{routines can make use of
  high-level (``level-3'') BLAS in certain sparse matrix computations}
to csc in systems like \RR{}, Octave and Matlab where dense matrices are
stored in column-major order, therefore it is used in sparse matrix
libraries such as CHOLMOD or CSparse of which we make use.  For this
reason, the \class{CsparseMatrix} class and subclasses are the
principal classes for sparse matrices in the \pkg{Matrix} package.

The Matrix package provides the following classes for sparse matrices
\FIXME{many more --- maybe explain naming scheme?}
\item[dgTMatrix] general, numeric, sparse matrices in (a possibly
  redundant) triplet form.  This can be a convenient form in which to
  construct sparse matrices.
\item[dgCMatrix] general, numeric, sparse matrices in the (sorted) compressed
  sparse column format.
\item[dsCMatrix] 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[dtCMatrix] triangular, real, sparse matrices in the (sorted)
  compressed sparse column format.

\TODO{Can also read and write the Matrix Market and read the Harwell-Boeing

\TODO{Can convert from a dense matrix to a sparse matrix (or use the
  Matrix function) but going through an intermediate dense matrix may
  cause problems with the amount of memory required.}

\TODO{similar range of operations as for the dense matrix classes.}

\section{More detailed examples of ``Matrix'' operations}

Have seen \texttt{drop0()} above, %(p.3); only with logical
showe a nice double example (where you see ``.'' and ``0'').

Show the use of \code{dim<-} for \emph{resizing} a (sparse) matrix.

Maybe mention  \Rfun{nearPD}.

\TODO{Solve a sparse least squares problem and demonstrate memory / speed gain}

\TODO{mention \code{lme4} and \Rfun{lmer}, maybe use one example to show the
  matrix sizes.}

\section{Notes about S4 classes and methods implementation}

Maybe we could % even here (for R News, not only for JSS)
give   some glimpses of implementations at least on the \RR{} level ones?

 \TODO{The class hierarchy: a non-trivial tree where only the leaves
    are ``actual'' classes.}

 \TODO{The main advantage of the multi-level hierarchy is that
    methods can often be defined on a higher (virtual class) level
    which ensures consistency [and saves from ``cut \& paste'' and
    forgetting things]}

 \TODO{Using Group Methods}

\section{Session Info}

<<sessionInfo, results=tex>>=


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