gsvd {geigen}

Generalized Singular Value Decomposition

Description

Computes the generalized singular value decomposition of a pair of matrices.

Usage

gsvd(A,B)

Arguments

A

a matrix with m rows and n columns.

B

a matrix with p rows and n columns.

Details

The matrix A is a m-by-n matrix and the matrix B is a p-by-n matrix. This function decomposes both matrices; if either one is complex than the other matrix is coerced to be complex.

The Generalized Singular Value Decomposition of numeric matrices A and B is given as

A = U D_1 [0\, R] Q^T

and

B = V D_2 [0\, R] Q^T

where

U

an m\times m orthogonal matrix.

V

a p\times p orthogonal matrix.

Q

an n\times n orthogonal matrix.

R

an r-by-r upper triangular non singular matrix and the matrix [0\, R] is an r-by-n matrix. The quantity r is the rank of the matrix \left( \begin{array}{c} A \\B \end{array} \right) with r \le n.

D_1,D_2

are quasi diagonal matrices and nonnegative and satisfy D_1^T D_1 + D_2^T D_2 = I. D_1 is an m-by-r matrix and D_2 is a p-by-r matrix.

The Generalized Singular Value Decomposition of complex matrices A and B is given as

A = U D_1 [0\, R] Q^H

and

B = V D_2 [0\, R] Q^H

where

U

an m\times m unitary matrix.

V

a p\times p unitary matrix.

Q

an n\times n unitary matrix.

R

an r-by-r upper triangular non singular matrix and the matrix [0\, R] is an r-by-n matrix. The quantity r is the rank of the matrix \left( \begin{array}{c} A \\B \end{array} \right) with r \le n.

D_1,D_2

are quasi diagonal matrices and nonnegative and satisfy D_1^T D_1 + D_2^T D_2 = I. D_1 is an m-by-r matrix and D_2 is a p-by-r matrix.

For details on this decomposition and the structure of the matrices D_1 and D_2 see http://www.netlib.org/lapack/lug/node36.html.

Value

The return value is a list containing the following components

A

the upper triangular matrix or a part of R.

B

lower part of the triangular matrix R if k+l>m (see below).

m

number of rows of A.

k

r{-}l. The number of rows of the matrix R is k{+}l. The first k generalized singular values are infinite.

l

effective rank of the input matrix B. The number of finite generalized singular values after the first k infinite ones.

alpha

a numeric vector with length n containing the numerators of the generalized singular values in the first (k{+}l) entries.

beta

a numeric vector with length n containing the denominators of the generalized singular value in the first (k{+}l) entries.

U

the matrix U.

V

the matrix V.

Q

the matrix Q.

For a detailed description of these items see http://www.netlib.org/lapack/lug/node36.html. Auxiliary functions are provided for extraction and manipulation of the various items.

Source

gsvd uses the LAPACK routines DGGSVD3 and ZGGSVD3 from Lapack 3.8.0. LAPACK is from http://www.netlib.org/lapack. The decomposition is fully explained in http://www.netlib.org/lapack/lug/node36.html.

References

Anderson. E. and ten others (1999) LAPACK Users' Guide. Third Edition. SIAM.
Available on-line at http://www.netlib.org/lapack/lug/lapack_lug.html. See the section Generalized Eigenvalue and Singular Value Problems (http://www.netlib.org/lapack/lug/node33.html) and the section Generalized Singular Value Decomposition (GSVD) (http://www.netlib.org/lapack/lug/node36.html).

See Also

gsvd.aux

Examples

A <- matrix(c(1,2,3,3,2,1,4,5,6,7,8,8), nrow=2, byrow=TRUE)
B <- matrix(1:18,byrow=TRUE, ncol=6)
A
B

z <- gsvd(A,B)
z