CGGSVP(l) ) CGGSVP(l)NAME
CGGSVP - 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 CGGSVP( 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
REAL TOLA, TOLB
INTEGER IWORK( * )
REAL RWORK( * )
COMPLEX A( LDA, * ), B( LDB, * ), Q( LDQ, * ), TAU( * ), U(
LDU, * ), V( LDV, * ), WORK( * )
PURPOSE
CGGSVP 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 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 CGGSVD.
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 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 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) REAL
TOLB (input) REAL TOLA and TOLB are the thresholds to deter‐
mine the effective numerical rank of matrix B and a subblock of
A. Generally, they are set to TOLA = MAX(M,N)*norm(A)*MACHEPS,
TOLB = MAX(P,N)*norm(B)*MACHEPS. 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 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 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 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) REAL array, dimension (2*N)
TAU (workspace) COMPLEX array, dimension (N)
WORK (workspace) COMPLEX 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 CGEQPF 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 version 3.0 15 June 2000 CGGSVP(l)