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dggsvd.z
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dggsvd
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1996-03-14
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265 lines
DDDDGGGGGGGGSSSSVVVVDDDD((((3333FFFF)))) DDDDGGGGGGGGSSSSVVVVDDDD((((3333FFFF))))
NNNNAAAAMMMMEEEE
DGGSVD - compute the generalized singular value decomposition (GSVD) of
an M-by-N real matrix A and P-by-N real matrix B
SSSSYYYYNNNNOOOOPPPPSSSSIIIISSSS
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( * )
PPPPUUUURRRRPPPPOOOOSSSSEEEE
DGGSVD computes the generalized singular value decomposition (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 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.
PPPPaaaaggggeeee 1111
DDDDGGGGGGGGSSSSVVVVDDDD((((3333FFFF)))) DDDDGGGGGGGGSSSSVVVVDDDD((((3333FFFF))))
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 orthogonal
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 converted to the latter form
by taking the nonsingular matrix X as
X = Q*( I 0 )
( 0 inv(R) ).
AAAARRRRGGGGUUUUMMMMEEEENNNNTTTTSSSS
JOBU (input) CHARACTER*1
= 'U': Orthogonal matrix U is computed;
= 'N': U is not computed.
PPPPaaaaggggeeee 2222
DDDDGGGGGGGGSSSSVVVVDDDD((((3333FFFF)))) DDDDGGGGGGGGSSSSVVVVDDDD((((3333FFFF))))
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 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) DOUBLE PRECISION array, dimension (LDB,N)
On entry, the P-by-N matrix B. On exit, B contains 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 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) 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.
PPPPaaaaggggeeee 3333
DDDDGGGGGGGGSSSSVVVVDDDD((((3333FFFF)))) DDDDGGGGGGGGSSSSVVVVDDDD((((3333FFFF))))
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) 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 DTGSJA.
PPPPAAAARRRRAAAAMMMMEEEETTTTEEEERRRRSSSS
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.
PPPPaaaaggggeeee 4444