sbbcsd.f(3)

NAME
sbbcsd.f -

SYNOPSIS

  Functions/Subroutines

subroutine sbbcsd (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)
SBBCSD

Function/Subroutine Documentation

  subroutine sbbcsd (characterJOBU1, characterJOBU2, characterJOBV1T, characterJOBV2T, characterTRANS, integerM, integerP, integerQ, real, dimension( * )THETA, real, dimension( * )PHI, real, dimension( ldu1, * )U1, integerLDU1, real, dimension( ldu2, * )U2, integerLDU2, real, dimension( ldv1t, * )V1T, integerLDV1T, real, dimension( ldv2t, * )V2T, integerLDV2T, real, dimension( * )B11D, real, dimension( * )B11E, real, dimension( * )B12D, real, dimension( * )B12E, real, dimension( * )B21D, real, dimension( * )B21E, real, dimension( * )B22D, real, dimension( * )B22E, real, dimension( * )WORK, integerLWORK, integerINFO)

SBBCSD

Purpose:

     SBBCSD 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 SORCSD 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 REAL 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 REAL 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 REAL array, dimension (LDU1,P)
              On entry, an LDU1-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.

    U2

              U2 is REAL array, dimension (LDU2,M-P)
              On entry, an LDU2-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.

    V1T

              V1T is REAL array, dimension (LDV1T,Q)
              On entry, a LDV1T-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.

    V2T

              V2T is REAL array, dimenison (LDV2T,M-Q)
              On entry, a LDV2T-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.

    B11D

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

    B11E

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

    B12D

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

    B12E

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

    B21D

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

    B21E

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

    B22D

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

    B22E

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

    WORK

              WORK is REAL 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 SBBCSD did not converge, INFO specifies the number
                    of nonzero entries in PHI, and B11D, B11E, etc.,
                    contain the partially reduced matrix.

Internal Parameters:

      TOLMUL  REAL, 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.

Date:
    November 2013

Definition at line 330 of file sbbcsd.f.

Author

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