DGGSVD - compute the generalized singular value decomposi
tion (GSVD) of an M-by-N real matrix A and P-by-N real
matrix B
SYNOPSIS
SUBROUTINE DGGSVD( JOBU, JOBV, JOBQ, M, N, P, K, L, A,
LDA, B, LDB, ALPHA, BETA, U, LDU, V,
LDV, Q, LDQ, WORK, IWORK, INFO )
CHARACTER JOBQ, JOBU, JOBV
INTEGER INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M,
N, P
INTEGER IWORK( * )
DOUBLE PRECISION A( LDA, * ), ALPHA( * ), B(
LDB, * ), BETA( * ), Q( LDQ, * ), U(
LDU, * ), V( LDV, * ), WORK( * )
PURPOSE
DGGSVD computes the generalized singular value decomposi
tion (GSVD) of an M-by-N real matrix A and P-by-N real
matrix B:
U'*A*Q = D1*( 0 R ), V'*B*Q = D2*( 0 R )
where U, V and Q are orthogonal matrices, and Z' is the
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 struc
tures, 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.
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 orthogo
nal 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 orthonormal 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, D1 and
D2 are ``diagonal''. The former GSVD form can be con
verted to the latter form by taking the nonsingular matrix
X as
X = Q*( I 0 )
( 0 inv(R) ).
ARGUMENTS
= 'U': Orthogonal matrix U is computed;
= 'N': U is not computed.
JOBV (input) CHARACTER*1
= 'V': Orthogonal matrix V is computed;
= 'N': V is not computed.
JOBQ (input) CHARACTER*1
= 'Q': Orthogonal 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 the
Purpose section. K + L = effective numerical rank
of (A',B')'.
A (input/output) DOUBLE PRECISION array, dimension
(LDA,N)
On entry, the M-by-N matrix A. On exit, A con
tains 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) DOUBLE PRECISION array, dimension
(LDB,N)
On entry, the P-by-N matrix B. On exit, B con
tains the triangular matrix R if M-K-L < 0. See
Purpose for details.
LDB (input) INTEGER
The leading dimension of the array B. LDA >=
max(1,P).
ALPHA (output) DOUBLE PRECISION array, dimension (N)
BETA (output) DOUBLE PRECISION array, dimension
(N) On exit, ALPHA and BETA contain the general
ized 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)
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) DOUBLE PRECISION array, dimension (LDU,M)
If JOBU = 'U', U contains the M-by-M orthogonal
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) DOUBLE PRECISION array, dimension (LDV,P)
If JOBV = 'V', V contains the P-by-P orthogonal
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) DOUBLE PRECISION array, dimension (LDQ,N)
If JOBQ = 'Q', Q contains the N-by-N orthogonal
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) DOUBLE PRECISION array,
dimension (max(3*N,M,P)+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 ille
gal value.
> 0: if INFO = 1, the Jacobi-type procedure
failed to converge. For further details, see sub
routine DTGSJA.
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(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 Cal
ifornia at Berkeley, USA
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