svd {svd}

Generic Singular Value Decomposition of a Matrix

Description

Compute the singular-value decomposition of a real or complex rectangular matrix.

Usage

propack.svd(X, neig = min(m, n), opts = list())
trlan.svd(X, neig = min(m, n), opts = list(), lambda = NULL, U = NULL)
ztrlan.svd(X, neig = min(m, n), opts = list(), lambda = NULL, U = NULL)

Arguments

X

the matrix to be decomposed. This can be either normal matrix or 'external matrix' object (e.g. one, created via 'extmat' function).

neig

number of desired eigentriples

opts

different options for eigensolver. See 'Details' section for more information

lambda

set of already computed singular values (used for continuation of the decomposition).

U

matrix of already computed eigenvectors (used for continuation of the decomposition).

Details

These routines provides an interface to two state-of-art implementations of truncated SVD.

PROPACK does this via the implicitly restarted Lanczos bidiagonalization with partial reorthogonalization. nu-TRLAN does the thick-restart Lanczos eigendecomposition of cross-product matrix.

'opts' is a list of different options which can be passed to the routines. Note that by default more or less suitable values for these options are set by the routines automatically.

The options for PROPACK are:

kmax

integer, maximum number of iterations.

dim

integer, dimension of Krylov subspace.

p

integer, number of shifts per restart.

maxiter

integer. maximum number of restarts.

tol

numeric, tolerance level.

verbose

logical, if 'TRUE', provide verbose output.

The options for nu-TRLAN are:

kmax

integer, maximum number of iterations.

maxiter

integer. maximum number of matrix-vector products.

tol

numeric, tolerance level.

verbose

integer, verboseness level.

Value

The returned value is a list with components

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' (only for 'propack.svd')

References

Wu, K. and Simon, H. (2000). Thick-restart Lanczos method for large symmetric eigenvalue problems. SIAM J. Matrix Anal. Appl. 22, 2, 602-616.

Yamazaki, I., Bai, Z., Simon, H., Wang, L.-W., and Wu, K. (2008). Adaptive projection subspace dimension for the thick restart Lanczos method. Tech. rep., Lawrence Berkeley National Laboratory, University of California, One Cyclotron road, Berkeley, California 94720.

Larsen, R. M. (1998). Efficient algorithms for helioseismic inversion. Ph.D. thesis, University of Aarhus, Denmark.

Korobeynikov, A. (2010) Computation- and space-efficient implementation of SSA. Statistics and Its Interface, Vol. 3, No. 3, Pp. 257-268