CXML
LAPACK version 3.0
cggsvd(3)
PURPOSE
CGGSVD - compute the generalized singular value decomposition (GSVD) of an
M-by-N complex matrix A and P-by-N complex matrix B
SYNTAX
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( * )
DESCRIPTION
CGGSVD 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 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; 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 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 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 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 array, dimension (max(3*N,M,P)+N)
RWORK (workspace) REAL 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 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.
Further Details ===============
2-96 Based on modifications by Ming Gu and Huan Ren, Computer
Science Division, University of California at Berkeley, USA
CXML Home Page Index of CXML Routines