svd                   package: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,

                            *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'.

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 <URL:
     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

