ZGGSVP(1) LAPACK routine (version 3.2) ZGGSVP(1)NAME
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
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
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 conjugate transpose of
Z.
This decomposition is the preprocessing step for computing the General‐
ized 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 trian‐
gular (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 trian‐
gular 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 thresh‐
olds 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 section. K + L = effec‐
tive 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,P)
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 illegal value.
FURTHER DETAILS
The subroutine uses LAPACK subroutine ZGEQPF 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.
LAPACK routine (version 3.2) November 2008 ZGGSVP(1)
