Provided by: liblapack-doc_3.3.1-1_all bug

NAME

       LAPACK-3  - computes the generalized singular value decomposition (GSVD) of an M-by-N real
       matrix A and P-by-N real matrix B

SYNOPSIS

       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 )

           CHARACTER      JOBQ, JOBU, JOBV

           INTEGER        INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N, P

           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**T*A*Q = D1*( 0 R ),    V**T*B*Q = D2*( 0 R )
        where U, V and Q are orthogonal matrices.
        Let K+L = the effective numerical rank of the matrix (A**T,B**T)**T,
        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**T.
        If ( A**T,B**T)**T  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**T*A x = lambda* B**T*B x.
        In some literature, the GSVD of A and B is presented in the form
                         U**T*A*X = ( 0 D1 ),   V**T*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**T,B**T)**T.

        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.

PARAMETERS

        TOLA    DOUBLE PRECISION
                TOLB    DOUBLE PRECISION
                TOLA and TOLB are the thresholds to determine the effective
                rank of (A',B')**T. 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

 LAPACK driver routine (version 3.3.1)      April 2011                            DGGSVD(3lapack)