\Header{svd}{Singular Value Decomposition of a Matrix}
\keyword{algebra}{svd}
\keyword{array}{svd}
\begin{Usage}
\begin{verbatim}
svd(x, nu=min(n,p), nv=min(n,p))
\end{verbatim}
\end{Usage}
\begin{Arguments}
\begin{ldescription}
\item[\texttt{x}] a matrix whose SVD decomposition is to be computed.
\item[\texttt{nu}] the number of left eigenvectors to be computed.
This must be one of \texttt{0}, \texttt{nrow(x)} and \texttt{ncol(x)}.
\item[\texttt{nv}] the number of right eigenvectors to be computed.
This must be one of \texttt{0}, and \texttt{ncol(x)}.
\end{ldescription}
\end{Arguments}
\begin{Description}
\texttt{svd} provides an interface to the LINPACK routine DSVDC.
The singular value decompostion plays an important role in many
statistical techniques.\end{Description}
\begin{Value}
The SVD decomposition of the matrix as computed by LINPACK,
\deqn{ \bold{X = U D V}}{}, where \eqn{\bold{U}}{} and \eqn{\bold{V}}{} are
orthogonal, and \eqn{\bold{D}}{} is a diagonal matrix with the singular
values \eqn{D_{ii}}{D[i,i]}.

The components in the returned value correspond directly
to the values returned by DSVDC.
\begin{ldescription}
\item[\texttt{d}] a vector containing the singular values of \texttt{x}.
\item[\texttt{u}] a matrix whose columns contain the left eigenvectors of \texttt{x}.
\item[\texttt{v}] a matrix whose columns contain the right eigenvectors of \texttt{x}.
\end{ldescription}
\end{Value}
\begin{References}
Dongarra, J. J., J. R. Bunch, C. B. Moler and G. W. Stewart (1978).
\emph{LINPACK Users Guide}, SIAM Publications, Philadelphia.\end{References}
\begin{SeeAlso}
\texttt{\Link{eigen}}, \texttt{\Link{qr}}.\end{SeeAlso}
\begin{Examples}
\begin{ExampleCode}
hilbert <- function(n) { i <- 1:n; 1 / outer(i - 1, i, "+") }
str(X <- hilbert(9)[,1:6])
str(s <- svd(X))
Eps <- 10 * .Machine$double.eps

all(abs(diag(s$d) - t(s$u)%*% X %*% s$v) < Eps)# TRUE:  D = U' X V

X <- cbind(1,1:7)
str(s <- svd(X))
all(abs(diag(s$d) - t(s$u)%*% X %*% s$v) < Eps)# TRUE:  D = U' X V
\end{ExampleCode}
\end{Examples}

