SYNOPSIS
- SUBROUTINE ZGGSVD(
- 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( * )
- DOUBLE PRECISION ALPHA( * ), BETA( * ), RWORK( * )
- COMPLEX*16 A( LDA, * ), B( LDB, * ), Q( LDQ, * ), U( LDU, * ), V( LDV, * ), WORK( * )
PURPOSE
ZGGSVD computes the generalized singular value decomposition (GSVD) of an 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 )
where U, V and Q are unitary matrices, and Z' means the conjugate transpose of Z. Let K+L = the effective numerical rank of the matrix (A',B')', then R is a (K+L)-by-(K+L) nonsingular upper triangular matrix, 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,
K L
D1 = K ( I 0 )
L ( 0 C )
M-K-L ( 0 0 )
K L
D2 = L ( 0 S )
P-L ( 0 0 )
N-K-L K L
( 0 R ) = K ( 0 R11 R12 )
L ( 0 0 R22 )
where
C = diag( ALPHA(K+1), ... , ALPHA(K+L) ),
S = diag( BETA(K+1), ... , BETA(K+L) ),
C**2 + S**2 = I.
R is stored in A(1:K+L,N-K-L+1:N) on exit.
If M-K-L < 0,
K M-K K+L-M
D1 = K ( I 0 0 )
M-K ( 0 C 0 )
K M-K K+L-M
D2 = M-K ( 0 S 0 )
K+L-M ( 0 0 I )
P-L ( 0 0 0 )
N-K-L K M-K K+L-M
( 0 R ) = K ( 0 R11 R12 R13 )
M-K ( 0 0 R22 R23 )
K+L-M ( 0 0 0 R33 )
where
C = diag( ALPHA(K+1), ... , ALPHA(M) ),
S = diag( BETA(K+1), ... , BETA(M) ),
C**2 + S**2 = I.
(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 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 )
where U and V are orthogonal and X is nonsingular, and D1 and D2 are ``diagonal''. The former GSVD form can be converted to the latter form 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*16 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*16 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) DOUBLE PRECISION array, dimension (N)
-
BETA (output) DOUBLE PRECISION array, dimension (N)
On exit, ALPHA and BETA contain the generalized singular
value pairs of A and B;
ALPHA(1:K) = 1,
BETA(1:K) = 0, and if M-K-L >= 0, ALPHA(K+1:K+L) = C,
BETA(K+1:K+L) = S, or if M-K-L < 0, ALPHA(K+1:M)= C, ALPHA(M+1:K+L)= 0
BETA(K+1:M) = S, BETA(M+1:K+L) = 1 and ALPHA(K+L+1:N) = 0
BETA(K+L+1:N) = 0 - U (output) COMPLEX*16 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) if JOBU = 'U'; LDU >= 1 otherwise.
- V (output) COMPLEX*16 array, dimension (LDV,P)
- If JOBV = '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) if JOBV = 'V'; LDV >= 1 otherwise.
- Q (output) COMPLEX*16 array, dimension (LDQ,N)
- If JOBQ = '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) if JOBQ = 'Q'; LDQ >= 1 otherwise.
- WORK (workspace) COMPLEX*16 array, dimension (max(3*N,M,P)+N)
- RWORK (workspace) DOUBLE PRECISION array, dimension (2*N)
- IWORK (workspace/output) INTEGER array, dimension (N)
- On exit, IWORK stores the sorting information. More precisely, the following loop will sort ALPHA for I = K+1, min(M,K+L) swap ALPHA(I) and ALPHA(IWORK(I)) endfor such that ALPHA(1) >= ALPHA(2) >= ... >= ALPHA(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 ZTGSJA.
PARAMETERS
- TOLA DOUBLE PRECISION
-
TOLB DOUBLE PRECISION
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)*MAZHEPS,
TOLB = MAX(P,N)*norm(B)*MAZHEPS.
The size of TOLA and TOLB may affect the size of backward
errors of the decomposition.
Further Details
===============
2-96 Based on modifications by
Ming Gu and Huan Ren, Computer Science Division, University of
California at Berkeley, USA