svd {base} | R Documentation |
Compute the singular-value decomposition of a rectangular matrix.
svd(x, nu = min(n, p), nv = min(n, p)) La.svd(x, nu = min(n, p), nv = min(n, p))
x |
a matrix whose SVD decomposition is to be computed. |
nu |
the number of left singular vectors to be computed.
This must be one of 0 , nrow(x) and ncol(x) . |
nv |
the number of right singular vectors to be computed.
This must be one of 0 and ncol(x) . |
The singular value decomposition plays an important role in many statistical techniques.
svd
provides an interface to the LINPACK routine DSVDC.
La.svd
provides an interface to the LAPACK routine DGESVD.
La.svd
is preferred to svd
for new projects, but it is
not an exact replacement as it returns the transpose of the right
singular vector matrix, and the signs of the singular vectors may differ
from those given by svd
.
Both functions handle complex matrices via LAPACK routine ZGESVD.
The SVD decomposition of the matrix as computed by 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 components in the returned value correspond directly
to the values returned by DSVDC.
d |
a vector containing the singular values of x . |
u |
a matrix whose columns contain the left singular vectors of x . |
v |
a matrix whose columns contain the right singular vectors of
x . |
For La.svd
the return value replaces v
by vt
, the
(conjugated if complex) transpose of v
.
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 (1995) LAPACK Users' Guide. Second Edition. SIAM.
hilbert <- function(n) { i <- 1:n; 1 / outer(i - 1, i, "+") } str(X <- hilbert(9)[,1:6]) str(s <- svd(X)) Eps <- 100 * .Machine$double.eps D <- diag(s$d) stopifnot(abs(X - s$u %*% D %*% t(s$v)) < Eps)# X = U D V' stopifnot(abs(D - t(s$u) %*% X %*% s$v) < Eps)# D = U' X V X <- cbind(1, 1:7) str(s <- svd(X)); D <- diag(s$d) stopifnot(abs(X - s$u %*% D %*% t(s$v)) < Eps)# X = U D V' stopifnot(abs(D - t(s$u) %*% X %*% s$v) < Eps)# D = U' X V