SRC/dtgsja.f(3)

NAME
SRC/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 (character jobu, character jobv, character jobq, integer m, integer p, integer n, integer k, integer l, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( ldb, * ) b, integer ldb, double precision tola, double precision tolb, double precision, dimension( * ) alpha, double precision, dimension( * ) beta, double precision, dimension( ldu, * ) u, integer ldu, double precision, dimension( ldv, * ) v, integer ldv, double precision, dimension( ldq, * ) q, integer ldq, double precision, dimension( * ) work, integer ncycle, integer info)

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.

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 375 of file dtgsja.f.

Author

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