CGGSVD(l) LAPACK driver routine (version 1.1) CGGSVD(l)
NAME
CGGSVD - compute the generalized singular value decomposition (GSVD) of the
M-by-N complex matrix A and P-by-N complex matrix B
SYNOPSIS
SUBROUTINE CGGSVD( JOBU, JOBV, JOBQ, M, N, P, K, L, A, LDA, B, LDB, ALPHA,
BETA, U, LDU, V, LDV, Q, LDQ, WORK, RWORK, IWORK, INFO )
CHARACTER JOBQ, JOBU, JOBV
INTEGER INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N, P
INTEGER IWORK( * )
REAL ALPHA( * ), BETA( * ), RWORK( * )
COMPLEX A( LDA, * ), B( LDB, * ), Q( LDQ, * ), U( LDU, * ), V(
LDV, * ), WORK( * )
PURPOSE
CGGSVD computes the generalized singular value decomposition (GSVD) of the
M-by-N complex matrix A and P-by-N complex matrix B:
U'*A*Q = D1*( 0 R ), V'*B*Q = D2*( 0 R ) (1)
where U, V and Q are unitary matrices, R is an upper triangular matrix, and
Z' means the conjugate transpose of Z. Let K+L = the numerical effective
rank of the matrix (A',B')', then D1 and D2 are M-by-(K+L) and P-by-(K+L)
"diagonal" matrices and of the following structures, respectively:
If M-K-L >= 0,
U'*A*Q = D1*( 0 R )
= K ( I 0 ) * ( 0 R11 R12 ) K
L ( 0 C ) ( 0 0 R22 ) L
M-K-L ( 0 0 ) N-K-L K L
K L
V'*B*Q = D2*( 0 R )
= L ( 0 S ) * ( 0 R11 R12 ) K
P-L ( 0 0 ) ( 0 0 R22 ) L
K L N-K-L K L
where
C = diag( ALPHA(K+1), ... , ALPHA(K+L) ),
S = diag( BETA(K+1), ... , BETA(K+L) ), C**2 + S**2 = I.
The nonsingular triangular matrix R = ( R11 R12 ) is stored
( 0 R22 )
in A(1:K+L,N-K-L+1:N) on exit.
If M-K-L < 0,
U'*A*Q = D1*( 0 R )
= K ( I 0 0 ) * ( 0 R11 R12 R13 ) K
M-K ( 0 C 0 ) ( 0 0 R22 R23 ) M-K
K M-K K+L-M ( 0 0 0 R33 ) K+L-M
N-K-L K M-K K+L-M
V'*B*Q = D2*( 0 R )
= M-K ( 0 S 0 ) * ( 0 R11 R12 R13 ) K
K+L-M ( 0 0 I ) ( 0 0 R22 R23 ) M-K
P-L ( 0 0 0 ) ( 0 0 0 R33 ) K+L-M
K M-K K+L-M N-K-L K M-K K+L-M where
C = diag( ALPHA(K+1), ... , ALPHA(M) ),
S = diag( BETA(K+1), ... , BETA(M) ), C**2 + S**2 = I.
R = ( R11 R12 R13 ) is a nonsingular upper triangular matrix,
( 0 R22 R23 )
( 0 0 R33 )
(R11 R12 R13 ) is stored in A(1:M, N-K-L+1:N), and R33 is stored
( 0 R22 R23 )
in B(M-K+1:L,N+M-K-L+1:N) on exit.
The routine computes C, S, R, and optionally the unitary
transformation matrices U, V and Q.
In particular, if B is an N-by-N nonsingular matrix, then the GSVD of A and
B implicitly gives the SVD of the matrix A*inv(B):
A*inv(B) = U*(D1*inv(D2))*V'.
If ( A',B')' has orthnormal columns, then the GSVD of A and B is also equal
to the CS decomposition of A and B. Furthermore, the GSVD can be used to
derive the solution of the eigenvalue problem:
A'*A x = lambda* B'*B x.
In some literature, the GSVD of A and B is presented in the form
U'*A*X = ( 0 D1 ), V'*B*X = ( 0 D2 ) (2) where
U and V are orthogonal and X is nonsingular, and D1 and D2 are ``diago-
nal''. It is easy to see that the GSVD form (1) can be converted to the
form (2) by taking the nonsingular matrix X as
X = Q*( I 0 )
( 0 inv(R) )
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.
N (input) INTEGER
The number of columns of the matrices A and B. N >= 0.
P (input) INTEGER
The number of rows of the matrix B. P >= 0.
K (output) INTEGER
L (output) INTEGER On exit, K and L specify the dimension of
the subblocks described in Purpose. K + L = effective numerical
rank of (A',B')'.
A (input/output) COMPLEX array, dimension (LDA,N)
On entry, the M-by-N matrix A. On exit, A contains the triangular
matrix R, or part of R. See Purpose for details.
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 part of the
triangular matrix R if M-K-L < 0. See Purpose for details.
LDB (input) INTEGER
The leading dimension of the array B. LDB >= max(1,P).
ALPHA (output) REAL array, dimension (N)
BETA (output) REAL array, dimension (N) On exit, ALPHA and BETA
contain the generalized singular value pairs of A and B; if M-K-L
>= 0, ALPHA(1:K) = ONE, ALPHA(K+1:K+L) = C,
BETA(1:K) = ZERO, BETA(K+1:K+L) = S; or if M-K-L < 0, ALPHA(1:K)=
ONE, ALPHA(K+1:M)= C, ALPHA(M+1:K+L)= ZERO,
BETA(1:K) = ZERO, BETA(K+1:M) = S, BETA(M+1:K+L) = ONE. and
ALPHA(K+L+1:N) = ZERO
BETA(K+L+1:N) = ZERO
U (output) COMPLEX array, dimension (LDU,M)
If JOBU = 'U', U contains the M-by-M 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 array, dimension (LDV,P)
If JOBU = 'V', V contains the P-by-P 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 array, dimension (LDQ,N)
If JOBU = 'Q', Q contains the N-by-N unitary matrix Q. If JOBQ =
'N', Q is not referenced.
LDQ (input) INTEGER
The leading dimension of the array Q. LDQ >= max(1,N).
WORK (workspace) COMPLEX array, dimension (MAX(3*N,M,P)+N)
RWORK (workspace) REAL array, dimension (2*N)
IWORK (workspace) INTEGER array, dimension (N)
INFO (output)INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
> 0: if INFO = 1, the Jacobi-type procedure failed to converge.
For further details, see subroutine CTGSJA.
PARAMETERS
TOLA REAL
TOLB REAL TOLA and TOLB are the thresholds to determine the
effective rank of (A',B')'. 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.
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