ZGGSVD(l) LAPACK driver routine (version 1.1) ZGGSVD(l)
NAME
ZGGSVD - compute the generalized singular value decomposition (GSVD) of the
M-by-N complex matrix A and P-by-N complex matrix B
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 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*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; 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*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).
V (output) COMPLEX*16 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*16 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*16 array, dimension (MAX(3*N,M,P)+N)
RWORK (workspace) DOUBLE PRECISION 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 ZTGSJA.
PARAMETERS
TOLA DOUBLE PRECISION
TOLB DOUBLE PRECISION TOLA and TOLB are the thresholds to deter-
mine 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.
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