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NAME ZGGSVP - compute unitary matrices U, V and Q such that U'*A*Q = ( 0 A12 A13 ) K , V'*B*Q = ( 0 0 B13 ) L ( 0 0 A23 ) L ( 0 0 0 ) P-L ( 0 0 0 ) M-K-L N-K-L K L N-K-L K L where the K-by-K matrix A12 and L-by-L matrix B13 are non- singular upper triangular SYNOPSIS SUBROUTINE ZGGSVP( JOBU, JOBV, JOBQ, M, P, N, A, LDA, B, LDB, TOLA, TOLB, K, L, U, LDU, V, LDV, Q, LDQ, IWORK, RWORK, TAU, WORK, INFO ) CHARACTER JOBQ, JOBU, JOBV INTEGER INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N, P DOUBLE PRECISION TOLA, TOLB INTEGER IWORK( * ) DOUBLE PRECISION RWORK( * ) COMPLEX*16 A( LDA, * ), B( LDB, * ), Q( LDQ, * ), TAU( * ), U( LDU, * ), V( LDV, * ), WORK( * ) PURPOSE ZGGSVP computes unitary matrices U, V and Q such that A23 is upper trapezoidal. K+L = the effective rank of the (M+P)- by-N matrix (A',B')'. Z' denotes the conjugate transpose of Z. This decomposition is the preprocessing step for computing the Generalized Singular Value Decomposition (GSVD), see subroutine ZGGSVD. 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. P (input) INTEGER The number of rows of the matrix B. P >= 0. N (input) INTEGER The number of columns of the matrices A and B. N >= 0. A (input/output) COMPLEX*16 array, dimension (LDA,N) On entry, the M-by-N matrix A. On exit, A contains the triangular (or trapezoidal) matrix described in the Purpose section. 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 the triangular matrix described in the Purpose sec- tion. LDB (input) INTEGER The leading dimension of the array B. LDB >= max(1,P). TOLA (input) DOUBLE PRECISION TOLB (input) DOUBLE PRECISION TOLA and TOLB are the thresholds to determine the effective rank of matrix B and a subblock of A. 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. K (output) INTEGER L (output) INTEGER On exit, K and L specify the dimension of the subblocks described in Purpose section. K + L = effective numerical rank of (A',B')'. U (output) COMPLEX*16 array, dimension (LDU,M) If JOBU = 'U', U contains the 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,M) If JOBV = 'V', V contains the 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 JOBQ = 'Q', Q contains the unitary matrix Q. If JOBQ = 'N', Q is not referenced. LDQ (input) INTEGER The leading dimension of the array Q. LDQ >= max(1,N). IWORK (workspace) INTEGER array, dimension (N) RWORK (workspace) DOUBLE PRECISION array, dimension (2*N) TAU (workspace) COMPLEX*16 array, dimension (N) WORK (workspace) COMPLEX*16 array, dimension (MAX(3*N,M,P)) INFO (output) INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value. FURTHER DETAILS The subroutine uses LAPACK subroutine ZGEQPF for the QR fac- torization with column pivoting to detect the effective numerical rank of the a matrix. It may be replaced by a better rank determination strategy.