sggsvd.f(3)

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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