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NAME ZGGSVD - compute the generalized singular value decomposi- tion (GSVD) of the M-by-N complex matrix A and P-by-N com- plex matrix B SYNOPSIS SUBROUTINE ZGGSVD( 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( * ) DOUBLE PRECISION ALPHA( * ), BETA( * ), RWORK( * ) COMPLEX*16 A( LDA, * ), B( LDB, * ), Q( LDQ, * ), U( LDU, * ), V( LDV, * ), WORK( * ) PURPOSE ZGGSVD computes the generalized singular value decomposition (GSVD) of the 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 ) (1) where U, V and Q are unitary matrices, R is an upper tri- angular matrix, and Z' means the conjugate transpose of Z. Let K+L = the numerical effective rank of the matrix (A',B')', then D1 and D2 are M-by-(K+L) and P-by-(K+L) "diagonal" matrices and of the following structures, respec- tively: If M-K-L >= 0, U'*A*Q = D1*( 0 R ) = K ( I 0 ) * ( 0 R11 R12 ) K L ( 0 C ) ( 0 0 R22 ) L M-K-L ( 0 0 ) N-K-L K L K L V'*B*Q = D2*( 0 R ) = L ( 0 S ) * ( 0 R11 R12 ) K P-L ( 0 0 ) ( 0 0 R22 ) L K L N-K-L K L where C = diag( ALPHA(K+1), ... , ALPHA(K+L) ), S = diag( BETA(K+1), ... , BETA(K+L) ), C**2 + S**2 = I. The nonsingular triangular matrix R = ( R11 R12 ) is stored ( 0 R22 ) in A(1:K+L,N-K-L+1:N) on exit. If M-K-L < 0, U'*A*Q = D1*( 0 R ) = K ( I 0 0 ) * ( 0 R11 R12 R13 ) K M-K ( 0 C 0 ) ( 0 0 R22 R23 ) M-K K M-K K+L-M ( 0 0 0 R33 ) K+L-M N-K-L K M-K K+L-M V'*B*Q = D2*( 0 R ) = M-K ( 0 S 0 ) * ( 0 R11 R12 R13 ) K K+L-M ( 0 0 I ) ( 0 0 R22 R23 ) M-K P-L ( 0 0 0 ) ( 0 0 0 R33 ) K+L-M K M-K K+L-M N-K-L K M-K K+L-M 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 a nonsingular upper triangular matrix, ( 0 R22 R23 ) ( 0 0 R33 ) (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 the matrix 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 ) (2) where U and V are orthogonal and X is nonsingular, and D1 and D2 are ``diagonal''. It is easy to see that the GSVD form (1) can be converted to the form (2) by taking the non- singular 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*16 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*16 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) DOUBLE PRECISION array, dimension (N) BETA (output) DOUBLE PRECISION array, dimension (N) On exit, ALPHA and BETA contain the generalized singular value pairs of A and B; if M-K-L >= 0, ALPHA(1:K) = ONE, ALPHA(K+1:K+L) = C, BETA(1:K) = ZERO, BETA(K+1:K+L) = S; or if M-K-L < 0, ALPHA(1:K)= ONE, ALPHA(K+1:M)= C, ALPHA(M+1:K+L)= ZERO, BETA(1:K) = ZERO, BETA(K+1:M) = S, BETA(M+1:K+L) = ONE. and ALPHA(K+L+1:N) = ZERO BETA(K+L+1:N) = ZERO U (output) COMPLEX*16 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). V (output) COMPLEX*16 array, dimension (LDV,P) If JOBU = '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). Q (output) COMPLEX*16 array, dimension (LDQ,N) If JOBU = '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). WORK (workspace) COMPLEX*16 array, dimension (MAX(3*N,M,P)+N) RWORK (workspace) DOUBLE PRECISION array, dimension (2*N) IWORK (workspace) INTEGER array, dimension (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 ZTGSJA. PARAMETERS TOLA DOUBLE PRECISION TOLB DOUBLE PRECISION 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)*MAZHEPS, TOLB = MAX(P,N)*norm(B)*MAZHEPS. The size of TOLA and TOLB may affect the size of backward errors of the decom- position.