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


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


                          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( * )


       SGGSVD 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 )
          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  )
          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) ).


        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) 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 M-K-L < 0.
                See Purpose for details.

        LDB     (input) INTEGER
                The leading dimension of the array B. LDB >= max(1,P).

        ALPHA   (output) REAL array, dimension (N)
                BETA    (output) REAL 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
                ALPHA(K+L+1:N) = 0
                BETA(K+L+1:N)  = 0

        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) if
                JOBU = 'U'; LDU >= 1 otherwise.

        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. LDV >= max(1,P) if
                JOBV = 'V'; LDV >= 1 otherwise.

        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) if
                JOBQ = 'Q'; LDQ >= 1 otherwise.

        WORK    (workspace) REAL 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))
                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 STGSJA.


        TOLA    REAL
                TOLB    REAL
                TOLA and TOLB are the thresholds to determine the effective
                rank of (A**T,B**T)**T. 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.
                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                            SGGSVD(3lapack)