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dense matrices.  At present only the real classes have been
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dense matrices.  At present only the real classes have been
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implemented.  These classes are
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implemented.  These classes are
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\begin{description}
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\begin{description}
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\item[dgeMatrix] Real matrices in general storage mode
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\item[dgeMatrix] Real matrices in general storage mode
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\item[dsyMatrix] Symmetric real matrices in non-packed storage
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\item[dsyMatrix] Symmetric real matrices in non-packed storage
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\item[dspMatrix] Symmetric real matrices in packed storage (one triangle only)
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\item[dtrMatrix] Triangular real matrices in non-packaged storage
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\item[dtrMatrix] Triangular real matrices in non-packed storage
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\item[dtpMatrix] Triangular real matrices in packed storage (triangle only)
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\item[dpoMatrix] Positive semi-definite symmetric real matrices in
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\item[dpoMatrix] Positive semi-definite symmetric real matrices in
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  non-package storage
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  non-packed storage
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\item[dppMatrix] \ \ dito \ \ in packed storage
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\end{description}
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\end{description}
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Methods for these classes include coercion between these classes, when
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Methods for these classes include coercion between these classes, when
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appropriate, and coercion to the \code{matrix} class; methods for
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appropriate, and coercion to the \code{matrix} class; methods for
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matrix multiplication (\code{\%*\%}); cross products
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matrix multiplication (\code{\%*\%}); cross products
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(\code{crossprod}), matrix norm (\code{norm}); reciprocal condition
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(\code{crossprod}), matrix norm (\code{norm}); reciprocal condition
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number (\code{rcond}); LU factorization (\code{lu}) or, for the
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number (\code{rcond}); LU factorization (\code{lu}) or, for the
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\code{poMatrix} class, the Cholesky decomposition (\code{chol}); and
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\code{poMatrix} class, the Cholesky decomposition (\code{chol}); and
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solutions of linear systems of equations (\code{solve}).
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solutions of linear systems of equations (\code{solve}).
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Further, group methods have been defined for the \code{Arith} (basic
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arithmetic, including with scalar numbers) and the \code{Math} (basic
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mathematical functions) group..
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Whenever a factorization or a decomposition is calculated it is
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Whenever a factorization or a decomposition is calculated it is
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preserved as a (list) element in the \code{factors} slot of the
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preserved as a (list) element in the \code{factors} slot of the
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original object.  In this way a sequence of operations, such as
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original object.  In this way a sequence of operations, such as
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determining the condition number of a matrix then solving a linear
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determining the condition number of a matrix then solving a linear
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system based on the matrix, do not require multiple factorizations of
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system based on the matrix, do not require multiple factorizations of
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\begin{Sinput}
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\begin{Sinput}
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setClass("dgTMatrix",
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setClass("dgTMatrix",
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         representation(i = "integer", j = "integer", x = "numeric",
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         representation(i = "integer", j = "integer", x = "numeric",
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                        Dim = "integer"))
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                        Dim = "integer"))
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\end{Sinput}
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\end{Sinput}
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\end{Schunk}
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\end{Schunk}
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The triplet representation is row-oriented if elements in the same row
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The triplet representation is row-oriented if elements in the same row
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were adjacent and column-oriented if elements in the same column were
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were adjacent and column-oriented if elements in the same column were
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adjacent.  The compressed sparse row (csr) (or compressed sparse
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adjacent.  The compressed sparse row (csr) (or compressed sparse
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column - csc) representation is similar to row-oriented triplet
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column - csc) representation is similar to row-oriented triplet
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(column-oriented triplet) except that \code{i} (\code{j}) just stores
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(column-oriented triplet) except that \code{i} (\code{j}) just stores
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the index of the first element in the row (column).  (There are a
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the index of the first element in the row (column).  (There are a
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couple of other details but that is the gist of it.)  These compressed
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couple of other details but that is the gist of it.)  These compressed
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representations remove the redundant row (column) indices and provide
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representations remove the redundant row (column) indices and provide
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faster access to a given location in the matrix because you only need
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faster access to a given location in the matrix because you only need
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to check one row (column).
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to check one row (column).
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The preferred representation of sparse matrices in the SparseM package
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The preferred representation of sparse matrices in the SparseM package
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is csr.  Matlab uses csc.  We hope that Octave will also use this
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is csr.  Matlab uses csc.  We hope that Octave will also use this
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representation. There are certain advantages to csc in systems like R
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representation. There are certain advantages to csc in systems like R
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and Matlab where dense matrices are stored in column-major order.  For
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and Matlab where dense matrices are stored in column-major order.  For
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example, Sivan Toledo's TAUCS~\cite{Taucs} library and Tim Davis's
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example, Sivan Toledo's TAUCS~\cite{Taucs} library and Tim Davis's