dtgsja.f(3)

NAME
dtgsja.f -

SYNOPSIS

  Functions/Subroutines

subroutine 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)
DTGSJA

Function/Subroutine Documentation

  subroutine dtgsja (characterJOBU, characterJOBV, characterJOBQ, integerM, integerP, integerN, integerK, integerL, double precision, dimension( lda, * )A, integerLDA, double precision, dimension( ldb, * )B, integerLDB, double precisionTOLA, double precisionTOLB, double precision, dimension( * )ALPHA, double precision, dimension( * )BETA, double precision, dimension( ldu, * )U, integerLDU, double precision, dimension( ldv, * )V, integerLDV, double precision, dimension( ldq, * )Q, integerLDQ, double precision, dimension( * )WORK, integerNCYCLE, integerINFO)

DTGSJA

Purpose:

     DTGSJA computes the generalized singular value decomposition (GSVD)
     of two real upper triangular (or trapezoidal) matrices A and B.

     On entry, it is assumed that matrices A and B have the following
     forms, which may be obtained by the preprocessing subroutine DGGSVP
     from a general M-by-N matrix A and P-by-N matrix B:

                  N-K-L  K    L
        A =    K ( 0    A12  A13 ) if M-K-L >= 0;
               L ( 0     0   A23 )
           M-K-L ( 0     0    0  )

                N-K-L  K    L
        A =  K ( 0    A12  A13 ) if M-K-L < 0;
           M-K ( 0     0   A23 )

                N-K-L  K    L
        B =  L ( 0     0   B13 )
           P-L ( 0     0    0  )

     where the K-by-K matrix A12 and L-by-L matrix B13 are nonsingular
     upper triangular; A23 is L-by-L upper triangular if M-K-L >= 0,
     otherwise A23 is (M-K)-by-L upper trapezoidal.

     On exit,

            U**T *A*Q = D1*( 0 R ),    V**T *B*Q = D2*( 0 R ),

     where U, V and Q are orthogonal matrices.
     R is a nonsingular upper triangular matrix, and D1 and D2 are
     ``diagonal'' matrices, which are of the following structures:

     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 ) K
                 L (  0    0   R22 ) L

     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.

     R = ( 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 computation of the orthogonal transformation matrices U, V or Q
     is optional.  These matrices may either be formed explicitly, or they
     may be postmultiplied into input matrices U1, V1, or Q1.

Parameters:
    JOBU

              JOBU is CHARACTER*1
              = 'U':  U must contain an orthogonal matrix U1 on entry, and
                      the product U1*U is returned;
              = 'I':  U is initialized to the unit matrix, and the
                      orthogonal matrix U is returned;
              = 'N':  U is not computed.

    JOBV

              JOBV is CHARACTER*1
              = 'V':  V must contain an orthogonal matrix V1 on entry, and
                      the product V1*V is returned;
              = 'I':  V is initialized to the unit matrix, and the
                      orthogonal matrix V is returned;
              = 'N':  V is not computed.

    JOBQ

              JOBQ is CHARACTER*1
              = 'Q':  Q must contain an orthogonal matrix Q1 on entry, and
                      the product Q1*Q is returned;
              = 'I':  Q is initialized to the unit matrix, and the
                      orthogonal matrix Q is returned;
              = 'N':  Q is not computed.

    M

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

    P

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

    N

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

    K

              K is INTEGER

    L

              L is INTEGER

              K and L specify the subblocks in the input matrices A and B:
              A23 = A(K+1:MIN(K+L,M),N-L+1:N) and B13 = B(1:L,N-L+1:N)
              of A and B, whose GSVD is going to be computed by DTGSJA.
              See Further Details.

    A

              A is DOUBLE PRECISION array, dimension (LDA,N)
              On entry, the M-by-N matrix A.
              On exit, A(N-K+1:N,1:MIN(K+L,M) ) 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 DOUBLE PRECISION array, dimension (LDB,N)
              On entry, the P-by-N matrix B.
              On exit, if necessary, B(M-K+1:L,N+M-K-L+1:N) contains
              a part of R.  See Purpose for details.

    LDB

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

    TOLA

              TOLA is DOUBLE PRECISION

    TOLB

              TOLB is DOUBLE PRECISION

              TOLA and TOLB are the convergence criteria for the Jacobi-
              Kogbetliantz iteration procedure. Generally, they are the
              same as used in the preprocessing step, say
                  TOLA = max(M,N)*norm(A)*MAZHEPS,
                  TOLB = max(P,N)*norm(B)*MAZHEPS.

    ALPHA

              ALPHA is DOUBLE PRECISION array, dimension (N)

    BETA

              BETA is 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) = diag(C),
                BETA(K+1:K+L)  = diag(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.
              Furthermore, if K+L < N,
                ALPHA(K+L+1:N) = 0 and
                BETA(K+L+1:N)  = 0.

    U

              U is DOUBLE PRECISION array, dimension (LDU,M)
              On entry, if JOBU = 'U', U must contain a matrix U1 (usually
              the orthogonal matrix returned by DGGSVP).
              On exit,
              if JOBU = 'I', U contains the orthogonal matrix U;
              if JOBU = 'U', U contains the product U1*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 DOUBLE PRECISION array, dimension (LDV,P)
              On entry, if JOBV = 'V', V must contain a matrix V1 (usually
              the orthogonal matrix returned by DGGSVP).
              On exit,
              if JOBV = 'I', V contains the orthogonal matrix V;
              if JOBV = 'V', V contains the product V1*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 DOUBLE PRECISION array, dimension (LDQ,N)
              On entry, if JOBQ = 'Q', Q must contain a matrix Q1 (usually
              the orthogonal matrix returned by DGGSVP).
              On exit,
              if JOBQ = 'I', Q contains the orthogonal matrix Q;
              if JOBQ = 'Q', Q contains the product Q1*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 DOUBLE PRECISION array, dimension (2*N)

    NCYCLE

              NCYCLE is INTEGER
              The number of cycles required for convergence.

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value.
              = 1:  the procedure does not converge after MAXIT cycles.

  Internal Parameters
  ===================

  MAXIT   INTEGER
          MAXIT specifies the total loops that the iterative procedure
          may take. If after MAXIT cycles, the routine fails to
          converge, we return INFO = 1..fi

Author:
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Date:
    November 2011

Further Details:

      DTGSJA essentially uses a variant of Kogbetliantz algorithm to reduce
      min(L,M-K)-by-L triangular (or trapezoidal) matrix A23 and L-by-L
      matrix B13 to the form:

               U1**T *A13*Q1 = C1*R1; V1**T *B13*Q1 = S1*R1,

      where U1, V1 and Q1 are orthogonal matrix, and Z**T is the transpose
      of Z.  C1 and S1 are diagonal matrices satisfying

                    C1**2 + S1**2 = I,

      and R1 is an L-by-L nonsingular upper triangular matrix.

Definition at line 377 of file dtgsja.f.

Author

Generated automatically by Doxygen for LAPACK from the source code.