SRC/dbbcsd.f(3)

NAME
SRC/dbbcsd.f

SYNOPSIS

  Functions/Subroutines
subroutine dbbcsd (jobu1, jobu2, jobv1t, jobv2t, trans, m, p, q, theta, phi, u1, ldu1, u2, ldu2, v1t, ldv1t, v2t, ldv2t, b11d, b11e, b12d, b12e, b21d, b21e, b22d, b22e, work, lwork, info)
DBBCSD

Function/Subroutine Documentation

  subroutine dbbcsd (character jobu1, character jobu2, character jobv1t, character jobv2t, character trans, integer m, integer p, integer q, double precision, dimension( * ) theta, double precision, dimension( * ) phi, double precision, dimension( ldu1, * ) u1, integer ldu1, double precision, dimension( ldu2, * ) u2, integer ldu2, double precision, dimension( ldv1t, * ) v1t, integer ldv1t, double precision, dimension( ldv2t, * ) v2t, integer ldv2t, double precision, dimension( * ) b11d, double precision, dimension( * ) b11e, double precision, dimension( * ) b12d, double precision, dimension( * ) b12e, double precision, dimension( * ) b21d, double precision, dimension( * ) b21e, double precision, dimension( * ) b22d, double precision, dimension( * ) b22e, double precision, dimension( * ) work, integer lwork, integer info)

DBBCSD

Purpose:

     DBBCSD computes the CS decomposition of an orthogonal matrix in
     bidiagonal-block form,
         [ B11 | B12 0  0 ]
         [  0  |  0 -I  0 ]
     X = [----------------]
         [ B21 | B22 0  0 ]
         [  0  |  0  0  I ]
                                   [  C | -S  0  0 ]
                       [ U1 |    ] [  0 |  0 -I  0 ] [ V1 |    ]**T
                     = [---------] [---------------] [---------]   .
                       [    | U2 ] [  S |  C  0  0 ] [    | V2 ]
                                   [  0 |  0  0  I ]
     X is M-by-M, its top-left block is P-by-Q, and Q must be no larger
     than P, M-P, or M-Q. (If Q is not the smallest index, then X must be
     transposed and/or permuted. This can be done in constant time using
     the TRANS and SIGNS options. See DORCSD for details.)
     The bidiagonal matrices B11, B12, B21, and B22 are represented
     implicitly by angles THETA(1:Q) and PHI(1:Q-1).
     The orthogonal matrices U1, U2, V1T, and V2T are input/output.
     The input matrices are pre- or post-multiplied by the appropriate
     singular vector matrices.

Parameters
    JOBU1

              JOBU1 is CHARACTER
              = 'Y':      U1 is updated;
              otherwise:  U1 is not updated.

    JOBU2

              JOBU2 is CHARACTER
              = 'Y':      U2 is updated;
              otherwise:  U2 is not updated.

    JOBV1T

              JOBV1T is CHARACTER
              = 'Y':      V1T is updated;
              otherwise:  V1T is not updated.

    JOBV2T

              JOBV2T is CHARACTER
              = 'Y':      V2T is updated;
              otherwise:  V2T is not updated.

    TRANS

              TRANS is CHARACTER
              = 'T':      X, U1, U2, V1T, and V2T are stored in row-major
                          order;
              otherwise:  X, U1, U2, V1T, and V2T are stored in column-
                          major order.

    M

              M is INTEGER
              The number of rows and columns in X, the orthogonal matrix in
              bidiagonal-block form.

    P

              P is INTEGER
              The number of rows in the top-left block of X. 0 <= P <= M.

    Q

              Q is INTEGER
              The number of columns in the top-left block of X.
              0 <= Q <= MIN(P,M-P,M-Q).

    THETA

              THETA is DOUBLE PRECISION array, dimension (Q)
              On entry, the angles THETA(1),...,THETA(Q) that, along with
              PHI(1), ...,PHI(Q-1), define the matrix in bidiagonal-block
              form. On exit, the angles whose cosines and sines define the
              diagonal blocks in the CS decomposition.

    PHI

              PHI is DOUBLE PRECISION array, dimension (Q-1)
              The angles PHI(1),...,PHI(Q-1) that, along with THETA(1),...,
              THETA(Q), define the matrix in bidiagonal-block form.

    U1

              U1 is DOUBLE PRECISION array, dimension (LDU1,P)
              On entry, a P-by-P matrix. On exit, U1 is postmultiplied
              by the left singular vector matrix common to [ B11 ; 0 ] and
              [ B12 0 0 ; 0 -I 0 0 ].

    LDU1

              LDU1 is INTEGER
              The leading dimension of the array U1, LDU1 >= MAX(1,P).

    U2

              U2 is DOUBLE PRECISION array, dimension (LDU2,M-P)
              On entry, an (M-P)-by-(M-P) matrix. On exit, U2 is
              postmultiplied by the left singular vector matrix common to
              [ B21 ; 0 ] and [ B22 0 0 ; 0 0 I ].

    LDU2

              LDU2 is INTEGER
              The leading dimension of the array U2, LDU2 >= MAX(1,M-P).

    V1T

              V1T is DOUBLE PRECISION array, dimension (LDV1T,Q)
              On entry, a Q-by-Q matrix. On exit, V1T is premultiplied
              by the transpose of the right singular vector
              matrix common to [ B11 ; 0 ] and [ B21 ; 0 ].

    LDV1T

              LDV1T is INTEGER
              The leading dimension of the array V1T, LDV1T >= MAX(1,Q).

    V2T

              V2T is DOUBLE PRECISION array, dimension (LDV2T,M-Q)
              On entry, an (M-Q)-by-(M-Q) matrix. On exit, V2T is
              premultiplied by the transpose of the right
              singular vector matrix common to [ B12 0 0 ; 0 -I 0 ] and
              [ B22 0 0 ; 0 0 I ].

    LDV2T

              LDV2T is INTEGER
              The leading dimension of the array V2T, LDV2T >= MAX(1,M-Q).

    B11D

              B11D is DOUBLE PRECISION array, dimension (Q)
              When DBBCSD converges, B11D contains the cosines of THETA(1),
              ..., THETA(Q). If DBBCSD fails to converge, then B11D
              contains the diagonal of the partially reduced top-left
              block.

    B11E

              B11E is DOUBLE PRECISION array, dimension (Q-1)
              When DBBCSD converges, B11E contains zeros. If DBBCSD fails
              to converge, then B11E contains the superdiagonal of the
              partially reduced top-left block.

    B12D

              B12D is DOUBLE PRECISION array, dimension (Q)
              When DBBCSD converges, B12D contains the negative sines of
              THETA(1), ..., THETA(Q). If DBBCSD fails to converge, then
              B12D contains the diagonal of the partially reduced top-right
              block.

    B12E

              B12E is DOUBLE PRECISION array, dimension (Q-1)
              When DBBCSD converges, B12E contains zeros. If DBBCSD fails
              to converge, then B12E contains the subdiagonal of the
              partially reduced top-right block.

    B21D

              B21D is DOUBLE PRECISION  array, dimension (Q)
              When DBBCSD converges, B21D contains the negative sines of
              THETA(1), ..., THETA(Q). If DBBCSD fails to converge, then
              B21D contains the diagonal of the partially reduced bottom-left
              block.

    B21E

              B21E is DOUBLE PRECISION  array, dimension (Q-1)
              When DBBCSD converges, B21E contains zeros. If DBBCSD fails
              to converge, then B21E contains the subdiagonal of the
              partially reduced bottom-left block.

    B22D

              B22D is DOUBLE PRECISION  array, dimension (Q)
              When DBBCSD converges, B22D contains the negative sines of
              THETA(1), ..., THETA(Q). If DBBCSD fails to converge, then
              B22D contains the diagonal of the partially reduced bottom-right
              block.

    B22E

              B22E is DOUBLE PRECISION  array, dimension (Q-1)
              When DBBCSD converges, B22E contains zeros. If DBBCSD fails
              to converge, then B22E contains the subdiagonal of the
              partially reduced bottom-right block.

    WORK

              WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
              On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

    LWORK

              LWORK is INTEGER
              The dimension of the array WORK. LWORK >= MAX(1,8*Q).
              If LWORK = -1, then a workspace query is assumed; the
              routine only calculates the optimal size of the WORK array,
              returns this value as the first entry of the work array, and
              no error message related to LWORK is issued by XERBLA.

    INFO

              INFO is INTEGER
              = 0:  successful exit.
              < 0:  if INFO = -i, the i-th argument had an illegal value.
              > 0:  if DBBCSD did not converge, INFO specifies the number
                    of nonzero entries in PHI, and B11D, B11E, etc.,
                    contain the partially reduced matrix.

Internal Parameters:

      TOLMUL  DOUBLE PRECISION, default = MAX(10,MIN(100,EPS**(-1/8)))
              TOLMUL controls the convergence criterion of the QR loop.
              Angles THETA(i), PHI(i) are rounded to 0 or PI/2 when they
              are within TOLMUL*EPS of either bound.

References:
    [1] Brian D. Sutton. Computing the complete CS decomposition. Numer. Algorithms, 50(1):33-65, 2009.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 328 of file dbbcsd.f.

Author

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