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NAME SGGSVD - compute the generalized singular value decomposi- tion (GSVD) of the M-by-N matrix A and P-by-N matrix B SYNOPSIS SUBROUTINE SGGSVD( JOBU, JOBV, JOBQ, M, N, P, K, L, A, LDA, B, LDB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ, WORK, IWORK, INFO ) CHARACTER JOBQ, JOBU, JOBV INTEGER INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N, P INTEGER IWORK( * ) REAL A( LDA, * ), ALPHA( * ), B( LDB, * ), BETA( * ), Q( LDQ, * ), U( LDU, * ), V( LDV, * ), WORK( * ) PURPOSE SGGSVD computes the generalized singular value decomposition (GSVD) of the M-by-N matrix A and P-by-N matrix B: U'*A*Q = D1*( 0 R ), V'*B*Q = D2*( 0 R ) (1) where U, V and Q are orthogonal matrices, and Z' is the transpose of Z. Let K+L = the numerical effective rank of the matrix (A',B')', then R is a K+L-by-K+L nonsingular upper tridiagonal matrix, D1 and D2 are "diagonal" matrices, and of the following structures, respectively: 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 orthogonal 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. Further- more, 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, 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': 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. 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 the Pur- pose section. K + L = effective numerical rank of (A',B')'. A (input/output) REAL 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) REAL array, dimension (LDB,N) On entry, the P-by-N matrix B. On exit, B contains the triangular matrix R if necessary. See Purpose for details. LDB (input) INTEGER The leading dimension of the array B. LDA >= MAX(1,P). ALPHA (output) REAL arrays, dimension (N) BETA (output) REAL 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) REAL array, dimension (LDU,M) If JOBU = 'U', U contains the M-by-M 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,P) If JOBV = 'V', V contains the P-by-P orthogonal matrix V. If JOBV = 'N', V is not referenced. LDV (input) INTEGER The leading dimension of the array V. LDA >= MAX(1,P). Q (output) REAL array, dimension (LDQ,N) If JOBQ = 'Q', Q contains the N-by-N orthogonal 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) REAL array, dimension (MAX(3*N,M,P)+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 STGSJA. 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.