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 trape
zoidal. 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
subroutine ZGGSVD.
ARGUMENTS
JOBU (input) CHARACTER*1
= 'U': Unitary matrix U is computed;
= 'N': U is not computed.
= '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 Pur
pose 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 numeri
cal rank of matrix B and a subblock of A. Gener
ally, 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 Pur
pose 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.
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
numerical rank of the a matrix. It may be replaced by a
better rank determination strategy.
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