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% File src/library/base/man/svd.Rd
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% File src/library/base/man/svd.Rd
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% Part of the R package, https://www.R-project.org
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% Part of the R package, https://www.R-project.org
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% Copyright 1995-2014 R Core Team
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% Copyright 1995-2017 R Core Team
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% Distributed under GPL 2 or later
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% Distributed under GPL 2 or later
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\name{svd}
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\name{svd}
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\alias{svd}
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\alias{svd}
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\alias{La.svd}
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\alias{La.svd}
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\value{
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\value{
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The SVD decomposition of the matrix as computed by LAPACK, \deqn{
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The SVD decomposition of the matrix as computed by LAPACK, \deqn{
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\bold{X = U D V'},} where \eqn{\bold{U}} and \eqn{\bold{V}} are
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\bold{X = U D V'},} where \eqn{\bold{U}} and \eqn{\bold{V}} are
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orthogonal, \eqn{\bold{V'}} means \emph{V transposed} (and conjugated
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orthogonal, \eqn{\bold{V'}} means \emph{V transposed} (and conjugated
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for complex input), and \eqn{\bold{D}} is a diagonal matrix with the
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for complex input), and \eqn{\bold{D}} is a diagonal matrix with the
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singular values \eqn{D_{ii}}{D[i,i]}. Equivalently, \eqn{\bold{D = U'
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(non-negative) singular values \eqn{D_{ii}}{D[i,i]} in decreasing
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order. Equivalently, \eqn{\bold{D = U' X V}}, which is verified in
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X V}}, which is verified in the examples.
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the examples.
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The returned value is a list with components
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The returned value is a list with components
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\item{d}{a vector containing the singular values of \code{x}, of
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\item{d}{a vector containing the singular values of \code{x}, of
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length \code{min(n, p)}.}
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length \code{min(n, p)}, sorted decreasingly.}
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\item{u}{a matrix whose columns contain the left singular vectors of
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\item{u}{a matrix whose columns contain the left singular vectors of
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\code{x}, present if \code{nu > 0}. Dimension \code{c(n, nu)}.}
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\code{x}, present if \code{nu > 0}. Dimension \code{c(n, nu)}.}
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\item{v}{a matrix whose columns contain the right singular vectors of
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\item{v}{a matrix whose columns contain the right singular vectors of
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\code{x}, present if \code{nv > 0}. Dimension \code{c(p, nv)}.}
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\code{x}, present if \code{nv > 0}. Dimension \code{c(p, nv)}.}
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Anderson. E. and ten others (1999)
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Anderson. E. and ten others (1999)
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\emph{LAPACK Users' Guide}. Third Edition. SIAM.\cr
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\emph{LAPACK Users' Guide}. Third Edition. SIAM.\cr
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Available on-line at
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Available on-line at
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\url{http://www.netlib.org/lapack/lug/lapack_lug.html}.
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\url{http://www.netlib.org/lapack/lug/lapack_lug.html}.
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The \href{https://en.wikipedia.org/wiki/Singular-value_decomposition}{%
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\sQuote{Singular-value decomposition}} Wikipedia article.
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Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988)
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Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988)
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\emph{The New S Language}.
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\emph{The New S Language}.
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Wadsworth & Brooks/Cole.
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Wadsworth & Brooks/Cole.
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}
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}
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The R Project SVN
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