The R Project SVN R

\name{svd}
\alias{svd}
\alias{La.svd}
\title{Singular Value Decomposition of a Matrix}
\usage{
svd(x, nu = min(n, p), nv = min(n, p), LINPACK = FALSE)

La.svd(x, nu = min(n, p), nv = min(n, p),
       method = c("dgesdd", "dgesvd"))
}
\arguments{
  \item{x}{a matrix whose SVD decomposition is to be computed.}
  \item{nu}{the number of left  singular vectors to be computed.
    This must be one of \code{0}, \code{nrow(x)} and \code{ncol(x)},
    except for \code{method = "dgesdd"}.}
  \item{nv}{the number of right singular vectors to be computed.
    This must be one of \code{0} and \code{ncol(x)}.}
  \item{LINPACK}{logical. Should LINPACK be used (for compatibility with
    \R < 1.7.0)?}
  \item{method}{The LAPACK routine to use in the real case.}
}
\description{
  Compute the singular-value decomposition of a rectangular matrix.
}
\details{
  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.
  
  \code{La.svd} provides an interface to both the LAPACK routines
  DGESVD and DGESDD.  The latter is usually substantially faster
  if singular vectors are required: see
  \url{http://www.cs.berkeley.edu/~demmel/DOE2000/Report0100.html}.
  Most benefit is seen with an optimized BLAS system.

  Computing the singular vectors is the slow part for large matrices.

  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.
}
\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, \eqn{\bold{V'}} means \emph{V transposed}, and
  \eqn{\bold{D}} is a diagonal matrix with the singular
  values \eqn{D_{ii}}{D[i,i]}.  Equivalently, \eqn{\bold{D = U' X V}},
  which is verified in the examples, below.

  The returned value is a list with components
  \item{d}{a vector containing the singular values of \code{x}.}
  \item{u}{a matrix whose columns contain the left singular vectors of
    \code{x}, present if \code{nu > 0}}
  \item{v}{a matrix whose columns contain the right singular vectors of
    \code{x}, present if \code{nv > 0}.}

  For \code{La.svd} the return value replaces \code{v} by \code{vt}, the
  (conjugated if complex) transpose of \code{v}.
}
\references{
  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.\cr
  Available on-line at
  \url{http://www.netlib.org/lapack/lug/lapack_lug.html}.
}
\seealso{
  \code{\link{eigen}}, \code{\link{qr}}.

  \code{\link{capabilities}} to test for IEEE 754 arithmetic.
}
\examples{
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
}
\keyword{algebra}
\keyword{array}