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))
     La.svd(x, nu = min(n, p), nv = min(n, p), method = c("dgesdd", "dgesvd"))

Arguments:

       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)', except for
          `method = "dgesdd"'.

      nv: the number of right singular vectors to be computed. This
          must be one of `0' and `ncol(x)'.

  method: The LAPACK routine to use in the real case.

Details:

     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 routines DGESVD and DGESDD. 
     The latter is usually substantially faster if singular vectors are
     required: see <URL:
     http://www.cs.berkeley.edu/~demmel/DOE2000/Report0100.html>. Most
     benefit is seen with an optimized BLAS system.

     `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'.  (They may also differ between
     methods and between platforms.)

     Both functions handle complex matrices via LAPACK routine ZGESVD.

     Computing the singular vectors is the slow part for large
     matrices.

     Using `method="dgesdd"' requires IEEE 754 arithmetic.  Should this
     not be supported on your platform, `method="dgesvd"' is used, with
     a warning.

Value:

     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'.

References:

     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'.

     `capabilities' to test for IEEE 754 arithmetic.

Examples:

     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

