svd {base}R Documentation

Singular Value Decomposition of a Matrix


Compute the singular-value decomposition of a rectangular matrix.


svd(x, nu = min(n, p), nv = min(n, p), LINPACK = FALSE)

La.svd(x, nu = min(n, p), nv = min(n, p))


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).


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.


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

X = U D V',

where U and V are orthogonal, V' means V transposed, and D is a diagonal matrix with the singular values D[i,i]. Equivalently, 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.


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

See Also

eigen, qr.


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

[Package base version 2.5.0 Index]