zggsvp(3)

NAME

ZGGSVP - compute unitary 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  ZGGSVP(  JOBU, JOBV, JOBQ, M, P, N, A, LDA, B,
LDB, TOLA, TOLB, K, L, U, LDU, V, LDV, Q, LDQ, IWORK, RWORK, 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 RWORK( * )
    COMPLEX*16     A( LDA, * ), B( LDB, * ), Q( LDQ, *  ),
TAU( * ), U( LDU, * ), V( LDV, * ), WORK( * )

PURPOSE

ZGGSVP computes unitary matrices U, V and Q such that N-K

L K L U'*A*Q = K ( 0 A12 A13 ) if M-K-L >= 0; 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 non

singular 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 conjugate transpose of Z.

This decomposition is the preprocessing step for computing

the Generalized Singular Value Decomposition (GSVD), see subrou

tine ZGGSVD.

ARGUMENTS

JOBU (input) CHARACTER*1

= 'U': Unitary matrix U is computed;
= 'N': U is not computed.

JOBV (input) CHARACTER*1

= 'V': Unitary matrix V is computed;
= 'N': V is not computed.

JOBQ (input) CHARACTER*1

= 'Q': Unitary 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) COMPLEX*16 array, dimension (LDA,N)

On entry, the M-by-N matrix A. On exit, A con

tains 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) COMPLEX*16 array, dimension (LDB,N)

On entry, the P-by-N matrix B. On exit, B con

tains 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 ma

trix 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 section. K +

L = effective numerical rank of (A',B')'.

U (output) COMPLEX*16 array, dimension (LDU,M)

If JOBU = 'U', U contains the unitary 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) COMPLEX*16 array, dimension (LDV,M)

If JOBV = 'V', V contains the unitary 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) COMPLEX*16 array, dimension (LDQ,N)

If JOBQ = 'Q', Q contains the unitary 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)

RWORK (workspace) DOUBLE PRECISION array, dimension

(2*N)

TAU (workspace) COMPLEX*16 array, dimension (N)

WORK (workspace) COMPLEX*16 array, dimension

(max(3*N,M,P))

INFO (output) INTEGER

= 0: successful exit
< 0: if INFO = -i, the i-th argument had an ille

gal value.

FURTHER DETAILS

The subroutine uses LAPACK subroutine ZGEQPF for the QR

factorization with column pivoting to detect the effective numer

ical rank of the a matrix. It may be replaced by a better rank

determination strategy.

LAPACK version 3.0 15 June 2000

ZGGSVP(3)