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NAME

       sggsvd.f -

SYNOPSIS

   Functions/Subroutines
       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)
            SGGSVD computes the singular value decomposition (SVD) for OTHER matrices

Function/Subroutine Documentation

   subroutine sggsvd (characterJOBU, characterJOBV, characterJOBQ, integerM, integerN, integerP,
       integerK, integerL, real, dimension( lda, * )A, integerLDA, real, dimension( ldb, * )B,
       integerLDB, real, dimension( * )ALPHA, real, dimension( * )BETA, real, dimension( ldu, *
       )U, integerLDU, real, dimension( ldv, * )V, integerLDV, real, dimension( ldq, * )Q,
       integerLDQ, real, dimension( * )WORK, integer, dimension( * )IWORK, integerINFO)
        SGGSVD computes the singular value decomposition (SVD) for OTHER matrices

       Purpose:

            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 )

            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) ).

       Parameters:
           JOBU

                     JOBU is CHARACTER*1
                     = 'U':  Orthogonal matrix U is computed;
                     = 'N':  U is not computed.

           JOBV

                     JOBV is CHARACTER*1
                     = 'V':  Orthogonal matrix V is computed;
                     = 'N':  V is not computed.

           JOBQ

                     JOBQ is CHARACTER*1
                     = 'Q':  Orthogonal matrix Q is computed;
                     = 'N':  Q is not computed.

           M

                     M is INTEGER
                     The number of rows of the matrix A.  M >= 0.

           N

                     N is INTEGER
                     The number of columns of the matrices A and B.  N >= 0.

           P

                     P is INTEGER
                     The number of rows of the matrix B.  P >= 0.

           K

                     K is INTEGER

           L

                     L is INTEGER

                     On exit, K and L specify the dimension of the subblocks
                     described in Purpose.
                     K + L = effective numerical rank of (A**T,B**T)**T.

           A

                     A is 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

                     LDA is INTEGER
                     The leading dimension of the array A. LDA >= max(1,M).

           B

                     B is 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

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

           ALPHA

                     ALPHA is REAL array, dimension (N)

           BETA

                     BETA is 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
                     and
                       ALPHA(K+L+1:N) = 0
                       BETA(K+L+1:N)  = 0

           U

                     U is 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

                     LDU is INTEGER
                     The leading dimension of the array U. LDU >= max(1,M) if
                     JOBU = 'U'; LDU >= 1 otherwise.

           V

                     V is 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

                     LDV is INTEGER
                     The leading dimension of the array V. LDV >= max(1,P) if
                     JOBV = 'V'; LDV >= 1 otherwise.

           Q

                     Q is 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

                     LDQ is INTEGER
                     The leading dimension of the array Q. LDQ >= max(1,N) if
                     JOBQ = 'Q'; LDQ >= 1 otherwise.

           WORK

                     WORK is REAL array,
                                 dimension (max(3*N,M,P)+N)

           IWORK

                     IWORK is 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

                     INFO is 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.

       Internal Parameters:

             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.

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           November 2011

       Contributors:
           Ming Gu and Huan Ren, Computer Science Division, University of California at Berkeley,
           USA

       Definition at line 331 of file sggsvd.f.

Author

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