Contents
dggsvd - compute the generalized singular value decomposi-
tion (GSVD) of an M-by-N real matrix A and P-by-N real
matrix B
SUBROUTINE DGGSVD(JOBU, JOBV, JOBQ, M, N, P, K, L, A, LDA, B, LDB,
ALPHA, BETA, U, LDU, V, LDV, Q, LDQ, WORK, IWORK3, INFO)
CHARACTER * 1 JOBU, JOBV, JOBQ
INTEGER M, N, P, K, L, LDA, LDB, LDU, LDV, LDQ, INFO
INTEGER IWORK3(*)
DOUBLE PRECISION A(LDA,*), B(LDB,*), ALPHA(*), BETA(*),
U(LDU,*), V(LDV,*), Q(LDQ,*), WORK(*)
SUBROUTINE DGGSVD_64(JOBU, JOBV, JOBQ, M, N, P, K, L, A, LDA, B, LDB,
ALPHA, BETA, U, LDU, V, LDV, Q, LDQ, WORK, IWORK3, INFO)
CHARACTER * 1 JOBU, JOBV, JOBQ
INTEGER*8 M, N, P, K, L, LDA, LDB, LDU, LDV, LDQ, INFO
INTEGER*8 IWORK3(*)
DOUBLE PRECISION A(LDA,*), B(LDB,*), ALPHA(*), BETA(*),
U(LDU,*), V(LDV,*), Q(LDQ,*), WORK(*)
F95 INTERFACE
SUBROUTINE GGSVD(JOBU, JOBV, JOBQ, [M], [N], [P], K, L, A, [LDA], B,
[LDB], ALPHA, BETA, U, [LDU], V, [LDV], Q, [LDQ], [WORK], IWORK3,
[INFO])
CHARACTER(LEN=1) :: JOBU, JOBV, JOBQ
INTEGER :: M, N, P, K, L, LDA, LDB, LDU, LDV, LDQ, INFO
INTEGER, DIMENSION(:) :: IWORK3
REAL(8), DIMENSION(:) :: ALPHA, BETA, WORK
REAL(8), DIMENSION(:,:) :: A, B, U, V, Q
SUBROUTINE GGSVD_64(JOBU, JOBV, JOBQ, [M], [N], [P], K, L, A, [LDA],
B, [LDB], ALPHA, BETA, U, [LDU], V, [LDV], Q, [LDQ], [WORK],
IWORK3, [INFO])
CHARACTER(LEN=1) :: JOBU, JOBV, JOBQ
INTEGER(8) :: M, N, P, K, L, LDA, LDB, LDU, LDV, LDQ, INFO
INTEGER(8), DIMENSION(:) :: IWORK3
REAL(8), DIMENSION(:) :: ALPHA, BETA, WORK
REAL(8), DIMENSION(:,:) :: A, B, U, V, Q
C INTERFACE
#include <sunperf.h>
void dggsvd(char jobu, char jobv, char jobq, int m, int n,
int p, int *k, int *l, double *a, int lda, double
*b, int ldb, double *alpha, double *beta, double
*u, int ldu, double *v, int ldv, double *q, int
ldq, int *iwork3, int *info);
void dggsvd_64(char jobu, char jobv, char jobq, long m, long
n, long p, long *k, long *l, double *a, long lda,
double *b, long ldb, double *alpha, double *beta,
double *u, long ldu, double *v, long ldv, double
*q, long ldq, long *iwork3, long *info);
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.
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. Further-
more, 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) ).
JOBU (input)
= 'U': Orthogonal matrix U is computed;
= 'N': U is not computed.
JOBV (input)
= 'V': Orthogonal matrix V is computed;
= 'N': V is not computed.
JOBQ (input)
= 'Q': Orthogonal matrix Q is computed;
= 'N': Q is not computed.
M (input) The number of rows of the matrix A. M >= 0.
N (input) The number of columns of the matrices A and B. N
>= 0.
P (input) The number of rows of the matrix B. P >= 0.
K (output)
On exit, K and L specify the dimension of the sub-
blocks described in the Purpose section. K + L =
effective numerical rank of (A',B')'.
L (output)
See the description of K.
A (input/output)
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)
The leading dimension of the array A. LDA >=
max(1,M).
B (input/output)
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)
The leading dimension of the array B. LDA >=
max(1,P).
ALPHA (output)
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
BETA (output)
See the description of ALPHA.
U (output)
If JOBU = 'U', U contains the M-by-M orthogonal
matrix U. If JOBU = 'N', U is not referenced.
LDU (input)
The leading dimension of the array U. LDU >=
max(1,M) if JOBU = 'U'; LDU >= 1 otherwise.
V (output)
If JOBV = 'V', V contains the P-by-P orthogonal
matrix V. If JOBV = 'N', V is not referenced.
LDV (input)
The leading dimension of the array V. LDV >=
max(1,P) if JOBV = 'V'; LDV >= 1 otherwise.
Q (output)
If JOBQ = 'Q', Q contains the N-by-N orthogonal
matrix Q. If JOBQ = 'N', Q is not referenced.
LDQ (input)
The leading dimension of the array Q. LDQ >=
max(1,N) if JOBQ = 'Q'; LDQ >= 1 otherwise.
WORK (workspace)
dimension (max(3*N,M,P)+N)
IWORK3 (output)
dimension(N) On exit, IWORK3 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(IWORK3(I)) endfor such that
ALPHA(1) >= ALPHA(2) >= ... >= ALPHA(N).
INFO (output)
= 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 STGSJA.