SUBROUTINE DTGSJA( JOBU, JOBV, JOBQ, M, P, N, K, L, A, LDA, B, LDB, * TOLA, TOLB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ, WORK, NCYCLE, * INFO) CHARACTER * 1 JOBU, JOBV, JOBQ INTEGER M, P, N, K, L, LDA, LDB, LDU, LDV, LDQ, NCYCLE, INFO DOUBLE PRECISION TOLA, TOLB DOUBLE PRECISION A(LDA,*), B(LDB,*), ALPHA(*), BETA(*), U(LDU,*), V(LDV,*), Q(LDQ,*), WORK(*) SUBROUTINE DTGSJA_64( JOBU, JOBV, JOBQ, M, P, N, K, L, A, LDA, B, * LDB, TOLA, TOLB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ, WORK, * NCYCLE, INFO) CHARACTER * 1 JOBU, JOBV, JOBQ INTEGER*8 M, P, N, K, L, LDA, LDB, LDU, LDV, LDQ, NCYCLE, INFO DOUBLE PRECISION TOLA, TOLB DOUBLE PRECISION A(LDA,*), B(LDB,*), ALPHA(*), BETA(*), U(LDU,*), V(LDV,*), Q(LDQ,*), WORK(*)
SUBROUTINE TGSJA( JOBU, JOBV, JOBQ, M, P, N, K, L, A, [LDA], B, [LDB], * TOLA, TOLB, ALPHA, BETA, U, [LDU], V, [LDV], Q, [LDQ], [WORK], * NCYCLE, [INFO]) CHARACTER(LEN=1) :: JOBU, JOBV, JOBQ INTEGER :: M, P, N, K, L, LDA, LDB, LDU, LDV, LDQ, NCYCLE, INFO REAL(8) :: TOLA, TOLB REAL(8), DIMENSION(:) :: ALPHA, BETA, WORK REAL(8), DIMENSION(:,:) :: A, B, U, V, Q SUBROUTINE TGSJA_64( JOBU, JOBV, JOBQ, M, P, N, K, L, A, [LDA], B, * [LDB], TOLA, TOLB, ALPHA, BETA, U, [LDU], V, [LDV], Q, [LDQ], * [WORK], NCYCLE, [INFO]) CHARACTER(LEN=1) :: JOBU, JOBV, JOBQ INTEGER(8) :: M, P, N, K, L, LDA, LDB, LDU, LDV, LDQ, NCYCLE, INFO REAL(8) :: TOLA, TOLB REAL(8), DIMENSION(:) :: ALPHA, BETA, WORK REAL(8), DIMENSION(:,:) :: A, B, U, V, Q
void dtgsja(char jobu, char jobv, char jobq, int m, int p, int n, int k, int l, double *a, int lda, double *b, int ldb, double tola, double tolb, double *alpha, double *beta, double *u, int ldu, double *v, int ldv, double *q, int ldq, int *ncycle, int *info);
void dtgsja_64(char jobu, char jobv, char jobq, long m, long p, long n, long k, long l, double *a, long lda, double *b, long ldb, double tola, double tolb, double *alpha, double *beta, double *u, long ldu, double *v, long ldv, double *q, long ldq, long *ncycle, long *info);
On entry, it is assumed that matrices A and B have the following forms, which may be obtained by the preprocessing subroutine SGGSVP from a general M-by-N matrix A and P-by-N matrix B:
N-K-L K L A = K ( 0 A12 A13 ) if M-K-L >= 0; L ( 0 0 A23 ) M-K-L ( 0 0 0 ) N-K-L K L A = K ( 0 A12 A13 ) if M-K-L < 0; M-K ( 0 0 A23 ) N-K-L K L B = L ( 0 0 B13 ) P-L ( 0 0 0 )
where the K-by-K matrix A12 and L-by-L matrix B13 are nonsingular upper triangular; A23 is L-by-L upper triangular if M-K-L >= 0, otherwise A23 is (M-K)-by-L upper trapezoidal.
On exit,
U'*A*Q = D1*( 0 R ), V'*B*Q = D2*( 0 R ),
where U, V and Q are orthogonal matrices, Z' denotes the transpose of Z, R is a nonsingular upper triangular matrix, and D1 and D2 are ``diagonal'' matrices, which are of the following structures:
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 ) K L ( 0 0 R22 ) L
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 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.
R = ( 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 computation of the orthogonal transformation matrices U, V or Q is optional. These matrices may either be formed explicitly, or they may be postmultiplied into input matrices U1, V1, or Q1.
STGSJA essentially uses a variant of Kogbetliantz algorithm to reduce min(L,M-K)-by-L triangular (or trapezoidal) matrix A23 and L-by-L matrix B13 to the form:
U1'*A13*Q1 = C1*R1; V1'*B13*Q1 = S1*R1,
where U1, V1 and Q1 are orthogonal matrix, and Z' is the transpose of Z. C1 and S1 are diagonal matrices satisfying
C1**2 + S1**2 = I,
and R1 is an L-by-L nonsingular upper triangular matrix.
A23 = A(K+1:MIN(K+L,M),N-L+1:N) and B13 = B(1:L,N-L+1:N) of A and B, whose GSVD is going to be computed by STGSJA. See Further details.
BETA(1:K) = 0, and if M-K-L >= 0, ALPHA(K+1:K+L) = diag(C),
BETA(K+1:K+L) = diag(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. Furthermore, if K+L < N, ALPHA(K+L+1:N) = 0 and
BETA(K+L+1:N) = 0.