cggsvd - compute the generalized singular value decomposition (GSVD) of an M-by-N complex matrix A and P-by-N complex matrix B
SUBROUTINE CGGSVD(JOBU, JOBV, JOBQ, M, N, P, K, L, A, LDA, B, LDB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ, WORK, WORK2, IWORK3, INFO) CHARACTER*1 JOBU, JOBV, JOBQ COMPLEX A(LDA,*), B(LDB,*), U(LDU,*), V(LDV,*), Q(LDQ,*), WORK(*) INTEGER M, N, P, K, L, LDA, LDB, LDU, LDV, LDQ, INFO INTEGER IWORK3(*) REAL ALPHA(*), BETA(*), WORK2(*) SUBROUTINE CGGSVD_64(JOBU, JOBV, JOBQ, M, N, P, K, L, A, LDA, B, LDB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ, WORK, WORK2, IWORK3, INFO) CHARACTER*1 JOBU, JOBV, JOBQ COMPLEX A(LDA,*), B(LDB,*), U(LDU,*), V(LDV,*), Q(LDQ,*), WORK(*) INTEGER*8 M, N, P, K, L, LDA, LDB, LDU, LDV, LDQ, INFO INTEGER*8 IWORK3(*) REAL ALPHA(*), BETA(*), WORK2(*) 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, WORK2, IWORK3, INFO) CHARACTER(LEN=1) :: JOBU, JOBV, JOBQ COMPLEX, DIMENSION(:) :: WORK COMPLEX, DIMENSION(:,:) :: A, B, U, V, Q INTEGER :: M, N, P, K, L, LDA, LDB, LDU, LDV, LDQ, INFO INTEGER, DIMENSION(:) :: IWORK3 REAL, DIMENSION(:) :: ALPHA, BETA, WORK2 SUBROUTINE GGSVD_64(JOBU, JOBV, JOBQ, M, N, P, K, L, A, LDA, B, LDB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ, WORK, WORK2, IWORK3, INFO) CHARACTER(LEN=1) :: JOBU, JOBV, JOBQ COMPLEX, DIMENSION(:) :: WORK COMPLEX, DIMENSION(:,:) :: A, B, U, V, Q INTEGER(8) :: M, N, P, K, L, LDA, LDB, LDU, LDV, LDQ, INFO INTEGER(8), DIMENSION(:) :: IWORK3 REAL, DIMENSION(:) :: ALPHA, BETA, WORK2 C INTERFACE #include <sunperf.h> void cggsvd(char jobu, char jobv, char jobq, int m, int n, int p, int *k, int *l, complex *a, int lda, complex *b, int ldb, float *alpha, float *beta, complex *u, int ldu, complex *v, int ldv, complex *q, int ldq, int *iwork3, int *info); void cggsvd_64(char jobu, char jobv, char jobq, long m, long n, long p, long *k, long *l, complex *a, long lda, complex *b, long ldb, float *alpha, float *beta, complex *u, long ldu, complex *v, long ldv, complex *q, long ldq, long *iwork3, long *info);
Oracle Solaris Studio Performance Library cggsvd(3P) NAME cggsvd - compute the generalized singular value decomposition (GSVD) of an M-by-N complex matrix A and P-by-N complex matrix B SYNOPSIS SUBROUTINE CGGSVD(JOBU, JOBV, JOBQ, M, N, P, K, L, A, LDA, B, LDB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ, WORK, WORK2, IWORK3, INFO) CHARACTER*1 JOBU, JOBV, JOBQ COMPLEX A(LDA,*), B(LDB,*), U(LDU,*), V(LDV,*), Q(LDQ,*), WORK(*) INTEGER M, N, P, K, L, LDA, LDB, LDU, LDV, LDQ, INFO INTEGER IWORK3(*) REAL ALPHA(*), BETA(*), WORK2(*) SUBROUTINE CGGSVD_64(JOBU, JOBV, JOBQ, M, N, P, K, L, A, LDA, B, LDB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ, WORK, WORK2, IWORK3, INFO) CHARACTER*1 JOBU, JOBV, JOBQ COMPLEX A(LDA,*), B(LDB,*), U(LDU,*), V(LDV,*), Q(LDQ,*), WORK(*) INTEGER*8 M, N, P, K, L, LDA, LDB, LDU, LDV, LDQ, INFO INTEGER*8 IWORK3(*) REAL ALPHA(*), BETA(*), WORK2(*) 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, WORK2, IWORK3, INFO) CHARACTER(LEN=1) :: JOBU, JOBV, JOBQ COMPLEX, DIMENSION(:) :: WORK COMPLEX, DIMENSION(:,:) :: A, B, U, V, Q INTEGER :: M, N, P, K, L, LDA, LDB, LDU, LDV, LDQ, INFO INTEGER, DIMENSION(:) :: IWORK3 REAL, DIMENSION(:) :: ALPHA, BETA, WORK2 SUBROUTINE GGSVD_64(JOBU, JOBV, JOBQ, M, N, P, K, L, A, LDA, B, LDB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ, WORK, WORK2, IWORK3, INFO) CHARACTER(LEN=1) :: JOBU, JOBV, JOBQ COMPLEX, DIMENSION(:) :: WORK COMPLEX, DIMENSION(:,:) :: A, B, U, V, Q INTEGER(8) :: M, N, P, K, L, LDA, LDB, LDU, LDV, LDQ, INFO INTEGER(8), DIMENSION(:) :: IWORK3 REAL, DIMENSION(:) :: ALPHA, BETA, WORK2 C INTERFACE #include <sunperf.h> void cggsvd(char jobu, char jobv, char jobq, int m, int n, int p, int *k, int *l, complex *a, int lda, complex *b, int ldb, float *alpha, float *beta, complex *u, int ldu, complex *v, int ldv, complex *q, int ldq, int *iwork3, int *info); void cggsvd_64(char jobu, char jobv, char jobq, long m, long n, long p, long *k, long *l, complex *a, long lda, complex *b, long ldb, float *alpha, float *beta, complex *u, long ldu, complex *v, long ldv, complex *q, long ldq, long *iwork3, long *info); PURPOSE cggsvd computes the generalized singular value decomposition (GSVD) of an M-by-N complex matrix A and P-by-N complex matrix B: U'*A*Q = D1*( 0 R ), V'*B*Q = D2*( 0 R ) where U, V and Q are unitary matrices, and Z' means the conjugate 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 unitary 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 orthnormal 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, and 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) ) ARGUMENTS JOBU (input) = 'U': Unitary matrix U is computed; = 'N': U is not computed. JOBV (input) = 'V': Unitary matrix V is computed; = 'N': V is not computed. JOBQ (input) = 'Q': Unitary 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 subblocks described in Purpose. K + L = effective numerical rank of (A',B')'. L (output) On exit, K and L specify the dimension of the subblocks described in Purpose. K + L = effective numerical rank of (A',B')'. A (input/output) On entry, the M-by-N matrix A. On exit, A contains the tri- angular 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 contains part of the triangular matrix R if M-K-L < 0. See Purpose for details. LDB (input) The leading dimension of the array B. LDB >= 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 description of ALPHA. U (output) If JOBU = 'U', U contains the M-by-M unitary 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 unitary 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 unitary 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) WORK2 (workspace) dimension(2*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 illegal value. > 0: if INFO = 1, the Jacobi-type procedure failed to con- verge. For further details, see subroutine CTGSJA. 7 Nov 2015 cggsvd(3P)