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NAME
ZGGSVP - compute unitary matrices U, V and Q such that
U'*A*Q = ( 0 A12 A13 ) K , V'*B*Q = ( 0 0 B13 ) L ( 0 0 A23
) L ( 0 0 0 ) P-L ( 0 0 0 ) M-K-L N-K-L K L N-K-L K L
where the K-by-K matrix A12 and L-by-L matrix B13 are non-
singular upper triangular
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 A23 is
upper trapezoidal. K+L = the effective 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
subroutine 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 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) COMPLEX*16 array, dimension (LDB,N)
On entry, the P-by-N matrix B. On exit, B contains
the triangular matrix described in the Purpose sec-
tion.
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 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 decom-
position.
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).
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).
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).
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 illegal
value.
FURTHER DETAILS
The subroutine uses LAPACK subroutine ZGEQPF for the QR fac-
torization with column pivoting to detect the effective
numerical rank of the a matrix. It may be replaced by a
better rank determination strategy.