cggsvd(3S)
CGGSVD - compute the generalized singular value decomposition (GSVD) of an M-by-N complex matrix A and P-by-N complex matrix B
As shipped in IRIX 6.5.15. Added in IRIX 6.5.15.
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, RWORK, IWORK, INFO ) CHARACTER JOBQ, JOBU, JOBV INTEGER INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N, P INTEGER IWORK( * ) REAL ALPHA( * ), BETA( * ), RWORK( * ) COMPLEX A( LDA, * ), B( LDB, * ), Q( LDQ, * ), U( LDU, * ), V( LDV, * ), WORK( * ) IMPLEMENTATION These routines are part of the SCSL Scientific Library and can be loaded using either the -lscs or the -lscs_mp option. The -lscs_mp option directs the linker to use the multi-processor version of the library. When linking to SCSL with -lscs or -lscs_mp, the default integer size is 4 bytes (32 bits). Another version of SCSL is available in which integers are 8 bytes (64 bits). This version allows the user access to larger memory sizes and helps when porting legacy Cray codes. It can be loaded by using the -lscs_i8 option or the -lscs_i8_mp option. A program may use only one of the two versions; 4-byte integer and 8-byte integer library calls cannot be mixed. 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) CHARACTER*1 = 'U': Unitary matrix U is computed; = 'N': U is not computed. JOBV (input) CHARACTER*1 = 'V': Unitary matrix V is computed; = 'N': V is not computed. JOBQ (input) CHARACTER*1 = 'Q': Unitary 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 Purpose. K + L = effective numerical rank of (A',B')'. A (input/output) COMPLEX 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) COMPLEX array, dimension (LDB,N) 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) INTEGER The leading dimension of the array B. LDB >= max(1,P). ALPHA (output) REAL array, dimension (N) BETA (output) REAL 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) COMPLEX array, dimension (LDU,M) If JOBU = 'U', U contains the M-by-M unitary matrix U. If JOBU = 'N', U is not referenced. LDU (input) INTEGER The leading dimension of the array U. LDU >= max(1,M) if JOBU = 'U'; LDU >= 1 otherwise. V (output) COMPLEX array, dimension (LDV,P) If JOBV = 'V', V contains the P-by-P unitary 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) COMPLEX array, dimension (LDQ,N) If JOBQ = 'Q', Q contains the N-by-N unitary 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) COMPLEX array, dimension (max(3*N,M,P)+N) RWORK (workspace) REAL array, dimension (2*N) IWORK (workspace/output) INTEGER array, dimension (N) On exit, IWORK 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(IWORK(I)) endfor such that ALPHA(1) >= ALPHA(2) >= ... >= ALPHA(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 CTGSJA. PARAMETERS TOLA REAL TOLB REAL 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)*MACHEPS, TOLB = MAX(P,N)*norm(B)*MACHEPS. The size of TOLA and TOLB may affect the size of backward errors of the decomposition. Further Details =============== 2-96 Based on modifications by Ming Gu and Huan Ren, Computer Science Division, University of California at Berkeley, USA SEE ALSO INTRO_LAPACK(3S), INTRO_SCSL(3S) This man page is available only online.