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\title{Design Issues in Matrix package Development}
\author{Martin Maechler and Douglas Bates\\R Core Development Team
  \\\email{maechler@stat.math.ethz.ch}, \email{bates@r-project.org}}
\date{Spring 2008 ({\tiny typeset on \tiny\today})}
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\begin{document}
\maketitle
\begin{abstract}
This is a (\textbf{currently very incomplete}) write-up of the many smaller and
larger design decisions we have made in organizing functionalities in the
Matrix package.

Classes: There's a rich hierarchy of matrix classes, which you can
visualize as a set of trees whose inner (and ``upper'') nodes are
\emph{virtual} classes and only the leaves are non-virtual ``actual'' classes.

Functions and Methods:

- setAs()

- others

\end{abstract}
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\section{The Matrix class structures}
\label{sec:classes}

Take Martin's DSC 2007 talk to depict class hierarchy.
\\ --- --- --- %% \hrule[1pt]{\textwidth}

\subsection{Diagonal Matrices}
\label{ssec:diagMat}
The class of diagonal matrices is worth mentioning for several reasons.
First, we have wanted such a class, because \emph{multiplication}
methods are particularly simple with diagonal matrices.
The typical constructor is \Rfun{Diagonal} whereas the accessor
(as for traditional matrices), \Rfun{diag} simply returns the
\emph{vector} of diagonal entries:
<<diag-class>>=
(D4 <- Diagonal(4, 10*(1:4)))
str(D4)
diag(D4)
@
We can \emph{modify} the diagonal in the traditional way
(via method definition for \Rfun{diag<-}):
<<diag-2>>=
diag(D4) <- diag(D4) + 1:4
D4
@

Note that \textbf{unit-diagonal} matrices (the identity matrices of linear algebra)
with slot \code{diag = "U"} can have an empty \code{x} slot, very
analogously to the unit-diagonal triangular matrices:
<<unit-diag>>=
str(I3 <- Diagonal(3)) ## empty 'x' slot

getClass("diagonalMatrix") ## extending "denseMatrix"
@
We have implemented diagonal matrices as \emph{dense} rather than sparse
matrices, for the following reasons:
\begin{enumerate}
\item The \code{diag()}onal (vector) is the basic part of such a matrix,
  and this is simply the \code{x} slot unless the \code{diag} slot is \code{"U"},
  the unit-diagonal case, which is the identity matrix.
\item given '1)', it does not make much sense to store \emph{where} the
  matrix is non-zero.  This contrasts with all sparse matrix
  implementations, and hence we did not want to call a diagonal matrix ``sparse''.
\end{enumerate}



\section{Matrix Transformations}
\label{sec:trafos}

\subsection{Coercions between Matrix classes}
\label{ssec:coerce}

You may need to transform Matrix objects into specific shape (triangular,
symmetric), content type (double, logical, \dots) or storage structure
(dense or sparse).
Every useR should use \code{as(x, <superclass>)} to this end, where
\code{<superclass>} is a \emph{virtual} Matrix super class, such as
\code{"triangularMatrix"} \code{"dMatrix"}, or \code{"sparseMatrix"}.

In other words, the user should \emph{not} coerce directly to a specific
desired class such as \code{"dtCMatrix"}, even though that may
occasionally work as well.

Here is a set of rules to which the Matrix developers and the users
should typically adhere:
\begin{description}

\item[Rule~1]:  \code{as(M, "matrix")} should work for \textbf{all} Matrix
  objects \code{M}.

\item[Rule~2]:  \code{Matrix(x)} should also work for matrix like
objects \code{x} and always return a ``classed'' Matrix.

Applied to a \code{"matrix"} object \code{m}, \code{M <- Matrix(m)} can be
considered a kind of inverse of \code{m <- as(M, "matrix")}.

\item[Rule~3]: All the following coercions to \emph{virtual} matrix
  classes should work:\\
  \begin{enumerate}
  \item \code{as(m, "dMatrix")}
  \item \code{as(m, "lMatrix")}
  \item \code{as(m, "nMatrix")}

  \item \code{as(m, "denseMatrix")}
  \item \code{as(m, "sparseMatrix")}

  \item \code{as(m, "generalMatrix")}
  \end{enumerate}
  whereas the next ones should work under some assumptions:

  \begin{enumerate}
  \item \code{as(m1, "triangularMatrix")} \\
       should work when \code{m1} is a triangular matrix, i.e. the upper or
       lower triangle of \code{m1} contains only zeros.

  \item \code{as(m2, "symmetricMatrix")}
       should work when \code{m2} is a symmetric matrix in the sense of
       \code{isSymmetric(m2)} returning \code{TRUE}.
       Note that this is typically equivalent to something like
       \code{isTRUE(all.equal(m2, t(m2)))}, i.e., the lower and upper
       triangle of the matrix have to be equal \emph{up to small
       numeric fuzz}.
  \end{enumerate}

\end{description}



\section{Session Info}

<<sessionInfo, results=tex>>=
toLatex(sessionInfo())
@

%not yet
%\bibliography{Matrix}

\end{document}