SUBROUTINE DGGSVD( JOBU, JOBV, JOBQ, M, N, P, K, L, A, LDA, B, \$ LDB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ, WORK, \$ IWORK, INFO ) * * -- LAPACK driver routine (version 3.2) -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * November 2006 * * .. Scalar Arguments .. CHARACTER JOBQ, JOBU, JOBV INTEGER INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N, P * .. * .. Array Arguments .. INTEGER IWORK( * ) DOUBLE PRECISION A( LDA, * ), ALPHA( * ), B( LDB, * ), \$ BETA( * ), Q( LDQ, * ), U( LDU, * ), \$ V( LDV, * ), WORK( * ) * .. * * Purpose * ======= * * DGGSVD computes the generalized singular value decomposition (GSVD) * of an M-by-N real matrix A and P-by-N real matrix B: * * U'*A*Q = D1*( 0 R ), V'*B*Q = D2*( 0 R ) * * where U, V and Q are orthogonal matrices, and Z' is the 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 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 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 ) * where U and V are orthogonal and X is nonsingular, 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': 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 Purpose section. * K + L = effective numerical rank of (A',B')'. * * A (input/output) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension (LDB,N) * On entry, the P-by-N matrix B. * On exit, B contains 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; * 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) DOUBLE PRECISION 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) if * JOBU = 'U'; LDU >= 1 otherwise. * * V (output) DOUBLE PRECISION 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. LDV >= max(1,P) if * JOBV = 'V'; LDV >= 1 otherwise. * * Q (output) DOUBLE PRECISION 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) if * JOBQ = 'Q'; LDQ >= 1 otherwise. * * WORK (workspace) DOUBLE PRECISION array, * dimension (max(3*N,M,P)+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 DTGSJA. * * Internal 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 decomposition. * * Further Details * =============== * * 2-96 Based on modifications by * Ming Gu and Huan Ren, Computer Science Division, University of * California at Berkeley, USA * * ===================================================================== * * .. Local Scalars .. LOGICAL WANTQ, WANTU, WANTV INTEGER I, IBND, ISUB, J, NCYCLE DOUBLE PRECISION ANORM, BNORM, SMAX, TEMP, TOLA, TOLB, ULP, UNFL * .. * .. External Functions .. LOGICAL LSAME DOUBLE PRECISION DLAMCH, DLANGE EXTERNAL LSAME, DLAMCH, DLANGE * .. * .. External Subroutines .. EXTERNAL DCOPY, DGGSVP, DTGSJA, XERBLA * .. * .. Intrinsic Functions .. INTRINSIC MAX, MIN * .. * .. Executable Statements .. * * Test the input parameters * WANTU = LSAME( JOBU, 'U' ) WANTV = LSAME( JOBV, 'V' ) WANTQ = LSAME( JOBQ, 'Q' ) * INFO = 0 IF( .NOT.( WANTU .OR. LSAME( JOBU, 'N' ) ) ) THEN INFO = -1 ELSE IF( .NOT.( WANTV .OR. LSAME( JOBV, 'N' ) ) ) THEN INFO = -2 ELSE IF( .NOT.( WANTQ .OR. LSAME( JOBQ, 'N' ) ) ) THEN INFO = -3 ELSE IF( M.LT.0 ) THEN INFO = -4 ELSE IF( N.LT.0 ) THEN INFO = -5 ELSE IF( P.LT.0 ) THEN INFO = -6 ELSE IF( LDA.LT.MAX( 1, M ) ) THEN INFO = -10 ELSE IF( LDB.LT.MAX( 1, P ) ) THEN INFO = -12 ELSE IF( LDU.LT.1 .OR. ( WANTU .AND. LDU.LT.M ) ) THEN INFO = -16 ELSE IF( LDV.LT.1 .OR. ( WANTV .AND. LDV.LT.P ) ) THEN INFO = -18 ELSE IF( LDQ.LT.1 .OR. ( WANTQ .AND. LDQ.LT.N ) ) THEN INFO = -20 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DGGSVD', -INFO ) RETURN END IF * * Compute the Frobenius norm of matrices A and B * ANORM = DLANGE( '1', M, N, A, LDA, WORK ) BNORM = DLANGE( '1', P, N, B, LDB, WORK ) * * Get machine precision and set up threshold for determining * the effective numerical rank of the matrices A and B. * ULP = DLAMCH( 'Precision' ) UNFL = DLAMCH( 'Safe Minimum' ) TOLA = MAX( M, N )*MAX( ANORM, UNFL )*ULP TOLB = MAX( P, N )*MAX( BNORM, UNFL )*ULP * * Preprocessing * CALL DGGSVP( JOBU, JOBV, JOBQ, M, P, N, A, LDA, B, LDB, TOLA, \$ TOLB, K, L, U, LDU, V, LDV, Q, LDQ, IWORK, WORK, \$ WORK( N+1 ), INFO ) * * Compute the GSVD of two upper "triangular" matrices * CALL DTGSJA( JOBU, JOBV, JOBQ, M, P, N, K, L, A, LDA, B, LDB, \$ TOLA, TOLB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ, \$ WORK, NCYCLE, INFO ) * * Sort the singular values and store the pivot indices in IWORK * Copy ALPHA to WORK, then sort ALPHA in WORK * CALL DCOPY( N, ALPHA, 1, WORK, 1 ) IBND = MIN( L, M-K ) DO 20 I = 1, IBND * * Scan for largest ALPHA(K+I) * ISUB = I SMAX = WORK( K+I ) DO 10 J = I + 1, IBND TEMP = WORK( K+J ) IF( TEMP.GT.SMAX ) THEN ISUB = J SMAX = TEMP END IF 10 CONTINUE IF( ISUB.NE.I ) THEN WORK( K+ISUB ) = WORK( K+I ) WORK( K+I ) = SMAX IWORK( K+I ) = K + ISUB ELSE IWORK( K+I ) = K + I END IF 20 CONTINUE * RETURN * * End of DGGSVD * END