SUBROUTINE SGGSVD( 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(*) REAL A(LDA,*), B(LDB,*), ALPHA(*), BETA(*), U(LDU,*), V(LDV,*), Q(LDQ,*), WORK(*) SUBROUTINE SGGSVD_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(*) REAL A(LDA,*), B(LDB,*), ALPHA(*), BETA(*), U(LDU,*), V(LDV,*), Q(LDQ,*), WORK(*)
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, DIMENSION(:) :: ALPHA, BETA, WORK REAL, 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, DIMENSION(:) :: ALPHA, BETA, WORK REAL, DIMENSION(:,:) :: A, B, U, V, Q
void sggsvd(char jobu, char jobv, char jobq, int m, int n, int p, int *k, int *l, float *a, int lda, float *b, int ldb, float *alpha, float *beta, float *u, int ldu, float *v, int ldv, float *q, int ldq, int *iwork3, int *info);
void sggsvd_64(char jobu, char jobv, char jobq, long m, long n, long p, long *k, long *l, float *a, long lda, float *b, long ldb, float *alpha, float *beta, float *u, long ldu, float *v, long ldv, float *q, long ldq, long *iwork3, long *info);
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. 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) ).
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