svd {base}  R Documentation 
Compute the singularvalue 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 numeric or complex matrix whose SVD decomposition is to be computed. Logical matrices are coerced to numeric. 
nu 
the number of left singular vectors to be computed.
This must between 
nv 
the number of right singular vectors to be computed.
This must be between 
LINPACK 
logical. Defunct and ignored (with a warning for true values). 
The singular value decomposition plays an important role in many
statistical techniques. svd
and La.svd
provide two
slightly different interfaces.
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 (most often 1
): these can
only be interpreted by detailed study of the FORTRAN code but mean
that the algorithm failed to converge.
The SVD decomposition of the matrix as computed by LAPACK,
\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[i,i]. 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 
u 
a matrix whose columns contain the left singular vectors of

v 
a matrix whose columns contain the right singular vectors of

For La.svd
the return value replaces v
by vt
, the
(conjugated if complex) transpose of v
.
The main functions used are the LAPACK routines DGESDD
and
ZGESVD
.
LAPACK is from http://www.netlib.org/lapack and its guide is listed in the references.
Anderson. E. and ten others (1999)
LAPACK Users' Guide. Third Edition. SIAM.
Available online at
http://www.netlib.org/lapack/lug/lapack_lug.html.
Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988) The New S Language. Wadsworth & Brooks/Cole.
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