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NAME SGGSVP - compute orthogonal 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 SGGSVP( JOBU, JOBV, JOBQ, M, P, N, A, LDA, B, LDB, TOLA, TOLB, K, L, U, LDU, V, LDV, Q, LDQ, IWORK, TAU, WORK, INFO ) CHARACTER JOBQ, JOBU, JOBV INTEGER INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N, P REAL TOLA, TOLB INTEGER IWORK( * ) REAL A( LDA, * ), B( LDB, * ), Q( LDQ, * ), TAU( * ), U( LDU, * ), V( LDV, * ), WORK( * ) PURPOSE SGGSVP computes orthogonal matrices U, V and Q such that A23 is upper trapezoidal. K+L = the effective rank of (M+P)- by-N matrix (A',B')'. Z' denotes the transpose of Z. This decomposition is the preprocessing step for computing the Generalized Singular Value Decomposition (GSVD), see subroutine SGGSVD. ARGUMENTS JOBU (input) CHARACTER*1 = 'U': Orthogonal matrix U is computed; = 'N': U is not computed. JOBV (input) CHARACTER*1 = 'V': Orthogonal matrix V is computed; = 'N': V is not computed. JOBQ (input) CHARACTER*1 = 'Q': Orthogonal 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) REAL 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) REAL 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) REAL TOLB (input) REAL TOLA and TOLB are the thres- holds to determine the effective rank of matrix B and a subblock of A. 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 decom- position. 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')'. U (output) REAL array, dimension (LDU,M) If JOBU = 'U', U contains the orthogonal 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) REAL array, dimension (LDV,M) If JOBV = 'V', V contains the orthogonal 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) REAL array, dimension (LDQ,N) If JOBQ = 'Q', Q contains the orthogonal 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) TAU (workspace) REAL array, dimension (N) WORK (workspace) REAL 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 SGEQPF 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.