\HeaderA{svd}{Singular Value Decomposition of a Matrix}{svd}
\aliasA{La.svd}{svd}{La.svd}
\keyword{algebra}{svd}
\keyword{array}{svd}
\begin{Description}\relax
Compute the singular-value decomposition of a rectangular matrix.
\end{Description}
\begin{Usage}
\begin{verbatim}
svd(x, nu = min(n, p), nv = min(n, p), LINPACK = FALSE)

La.svd(x, nu = min(n, p), nv = min(n, p))
\end{verbatim}
\end{Usage}
\begin{Arguments}
\begin{ldescription}
\item[\code{x}] a real or complex matrix whose SVD decomposition is to be computed.
\item[\code{nu}] the number of left  singular vectors to be computed.
This must between \code{0} and \code{n = nrow(x)}.
\item[\code{nv}] the number of right singular vectors to be computed.
This must be between \code{0} and \code{p = ncol(x)}.
\item[\code{LINPACK}] logical. Should LINPACK be used (for compatibility with
\R{} \textless{} 1.7.0)?  In this case \code{nu} must be \code{0},
\code{nrow(x)} or \code{ncol(x)}.
\end{ldescription}
\end{Arguments}
\begin{Details}\relax
The singular value decomposition plays an important role in many
statistical techniques.  \code{svd} and \code{La.svd} provide two
slightly different interfaces.  The main functions used are
the LAPACK routines DGESDD and ZGESVD; \code{svd(LINPACK = TRUE)}
provides an interface to the LINPACK routine DSVDC, purely for
backwards compatibility.

Computing the singular vectors is the slow part for large matrices.
The computation will be more efficient if \code{nu <= min(n, p)} and
\code{nv <= min(n, p)}, and even more efficient if one or both are zero.

Unsuccessful results from the underlying LAPACK code will result in an
error giving a positive error code: these can only be interpreted by
detailed study of the FORTRAN code.
\end{Details}
\begin{Value}
The SVD decomposition of the matrix as computed by LAPACK/LINPACK,
\deqn{ \bold{X = U D V'},}{} where \eqn{\bold{U}}{} and \eqn{\bold{V}}{} are
orthogonal, \eqn{\bold{V'}}{} means \emph{V transposed}, and
\eqn{\bold{D}}{} is a diagonal matrix with the singular
values \eqn{D_{ii}}{}.  Equivalently, \eqn{\bold{D = U' X V}}{},
which is verified in the examples, below.

The returned value is a list with components
\begin{ldescription}
\item[\code{d}] a vector containing the singular values of \code{x}, of
length \code{min(n, p)}.
\item[\code{u}] a matrix whose columns contain the left singular vectors of
\code{x}, present if \code{nu > 0}.  Dimension \code{c(n, nu)}.
\item[\code{v}] a matrix whose columns contain the right singular vectors of
\code{x}, present if \code{nv > 0}.  Dimension \code{c(p, nv)}.
\end{ldescription}


For \code{La.svd} the return value replaces \code{v} by \code{vt}, the
(conjugated if complex) transpose of \code{v}.
\end{Value}
\begin{References}\relax
Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988)
\emph{The New S Language}.
Wadsworth \& Brooks/Cole.

Dongarra, J. J., Bunch, J. R., Moler, C. B. and Stewart, G. W. (1978)
\emph{LINPACK Users Guide.}  Philadelphia: SIAM Publications.

Anderson. E. and ten others (1999)
\emph{LAPACK Users' Guide}. Third Edition. SIAM.\\
Available on-line at
\url{http://www.netlib.org/lapack/lug/lapack_lug.html}.
\end{References}
\begin{SeeAlso}\relax
\code{\LinkA{eigen}{eigen}}, \code{\LinkA{qr}{qr}}.
\end{SeeAlso}
\begin{Examples}
\begin{ExampleCode}
hilbert <- function(n) { i <- 1:n; 1 / outer(i - 1, i, "+") }
X <- hilbert(9)[,1:6]
(s <- svd(X))
D <- diag(s$d)
s$u %*% D %*% t(s$v) #  X = U D V'
t(s$u) %*% X %*% s$v #  D = U' X V
\end{ExampleCode}
\end{Examples}

