DGGSVP(l) — LAPACK routine (version 2.0)

NAME

DGGSVP - compute orthogonal matrices U, V and Q such that N-K-L K L U’∗A∗Q = K ( 0 A12 A13 ) if M-K-L >= 0

SYNOPSIS

SUBROUTINE DGGSVP(
JOBU, JOBV, JOBQ, M, P, N, A, LDA, B, LDB, TOLA, TOLB, K, L, U, LDU, V, LDV, Q, LDQ, IWORK, TAU, WORK, INFO )

CHARACTER JOBQ, JOBU, JOBV

INTEGER INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N, P

DOUBLE PRECISION TOLA, TOLB

INTEGER IWORK( ∗ )

DOUBLE PRECISION A( LDA, ∗ ), B( LDB, ∗ ), Q( LDQ, ∗ ), TAU( ∗ ), U( LDU, ∗ ), V( LDV, ∗ ), WORK( ∗ )

PURPOSE

DGGSVP computes orthogonal matrices U, V and Q such that
              L ( 0     0   A23 )
          M-K-L ( 0     0    0 )

                 N-K-L K    L
        =     K ( 0    A12 A13 ) if M-K-L < 0;
            M-K ( 0     0   A23 )

               N-K-L K    L
V’∗B∗Q =   L ( 0     0   B13 )
          P-L ( 0     0    0 )

where the K-by-K matrix A12 and L-by-L matrix B13 are nonsingular upper triangular; A23 is L-by-L upper triangular if M-K-L >= 0, otherwise A23 is (M-K)-by-L upper trapezoidal. K+L = the effective numerical rank of the (M+P)-by-N matrix (A’,B’)’. Z’ denotes the transpose of Z.

This decomposition is the preprocessing step for computing the Generalized Singular Value Decomposition (GSVD), see subroutine DGGSVD.

ARGUMENTS

JOBU (input) CHARACTER∗1
= ’U’: Orthogonal matrix U is computed;
= ’N’: U is not computed.

JOBV (input) CHARACTER∗1
= ’V’: Orthogonal matrix V is computed;
= ’N’: V is not computed.

JOBQ (input) CHARACTER∗1
= ’Q’: Orthogonal matrix Q is computed;
= ’N’: Q is not computed.

M (input) INTEGER
The number of rows of the matrix A. M >= 0.

P (input) INTEGER
The number of rows of the matrix B. P >= 0.

N (input) INTEGER
The number of columns of the matrices A and B. N >= 0.

A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the M-by-N matrix A. On exit, A contains the triangular (or trapezoidal) matrix described in the Purpose section.

LDA (input) INTEGER
The leading dimension of the array A. LDA >= max(1,M).

B (input/output) DOUBLE PRECISION array, dimension (LDB,N)
On entry, the P-by-N matrix B. On exit, B contains the triangular matrix described in the Purpose section.

LDB (input) INTEGER
The leading dimension of the array B. LDB >= max(1,P).

TOLA (input) DOUBLE PRECISION
TOLB (input) DOUBLE PRECISION TOLA and TOLB are the thresholds to determine the effective numerical rank of matrix B and a subblock of A. Generally, they are set to TOLA = MAX(M,N)∗norm(A)∗MAZHEPS, TOLB = MAX(P,N)∗norm(B)∗MAZHEPS. The size of TOLA and TOLB may affect the size of backward errors of the decomposition.

K (output) INTEGER
L (output) INTEGER On exit, K and L specify the dimension of the subblocks described in Purpose. K + L = effective numerical rank of (A’,B’)’.

U (output) DOUBLE PRECISION array, dimension (LDU,M)
If JOBU = ’U’, U contains the orthogonal matrix U. If JOBU = ’N’, U is not referenced.

LDU (input) INTEGER
The leading dimension of the array U. LDU >= max(1,M) if JOBU = ’U’; LDU >= 1 otherwise.

V (output) DOUBLE PRECISION array, dimension (LDV,M)
If JOBV = ’V’, V contains the orthogonal matrix V. If JOBV = ’N’, V is not referenced.

LDV (input) INTEGER
The leading dimension of the array V. LDV >= max(1,P) if JOBV = ’V’; LDV >= 1 otherwise.

Q (output) DOUBLE PRECISION array, dimension (LDQ,N)
If JOBQ = ’Q’, Q contains the orthogonal matrix Q. If JOBQ = ’N’, Q is not referenced.

LDQ (input) INTEGER
The leading dimension of the array Q. LDQ >= max(1,N) if JOBQ = ’Q’; LDQ >= 1 otherwise.

IWORK (workspace) INTEGER array, dimension (N)

TAU (workspace) DOUBLE PRECISION array, dimension (N)

WORK (workspace) DOUBLE PRECISION array, dimension (max(3∗N,M,P))

INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value.

FURTHER DETAILS

The subroutine uses LAPACK subroutine DGEQPF for the QR factorization with column pivoting to detect the effective numerical rank of the a matrix. It may be replaced by a better rank determination strategy.