svd {base}

R Documentation

Singular Value Decomposition of a Matrix

Description

Compute the singular-value decomposition of a rectangular 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))

Arguments

x	a real or complex matrix whose SVD decomposition is to be computed.
nu	the number of left singular vectors to be computed. This must between 0 and n = nrow(x).
nv	the number of right singular vectors to be computed. This must be between 0 and p = ncol(x).
LINPACK	logical. Should LINPACK be used (for compatibility with R < 1.7.0)? In this case nu must be 0, nrow(x) or ncol(x).

Details

The singular value decomposition plays an important role in many statistical techniques. svd and La.svd provide two slightly different interfaces. The main functions used are the LAPACK routines DGESDD and ZGESVD; 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 nu <= min(n, p) and 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.

Value

The SVD decomposition of the matrix as computed by LAPACK/LINPACK,

\bold{X = U D V'},

where \bold{U} and \bold{V} are orthogonal, \bold{V'} means V transposed, and \bold{D} is a diagonal matrix with the singular values D_{ii}. Equivalently, \bold{D = U' X V}, which is verified in the examples, below.

The returned value is a list with components

d	a vector containing the singular values of x, of length min(n, p).
u	a matrix whose columns contain the left singular vectors of x, present if nu > 0. Dimension c(n, nu).
v	a matrix whose columns contain the right singular vectors of x, present if nv > 0. Dimension c(p, nv).

For La.svd the return value replaces v by vt, the (conjugated if complex) transpose of v.

References

Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988) The New S Language. Wadsworth & Brooks/Cole.

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

Anderson. E. and ten others (1999) LAPACK Users' Guide. Third Edition. SIAM.
Available on-line at http://www.netlib.org/lapack/lug/lapack_lug.html.

See Also

eigen, qr.

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

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