SRC/dorbdb.f(3)

NAME
SRC/dorbdb.f

SYNOPSIS

  Functions/Subroutines
subroutine dorbdb (trans, signs, m, p, q, x11, ldx11, x12, ldx12, x21, ldx21, x22, ldx22, theta, phi, taup1, taup2, tauq1, tauq2, work, lwork, info)
DORBDB

Function/Subroutine Documentation

  subroutine dorbdb (character trans, character signs, integer m, integer p, integer q, double precision, dimension( ldx11, * ) x11, integer ldx11, double precision, dimension( ldx12, * ) x12, integer ldx12, double precision, dimension( ldx21, * ) x21, integer ldx21, double precision, dimension( ldx22, * ) x22, integer ldx22, double precision, dimension( * ) theta, double precision, dimension( * ) phi, double precision, dimension( * ) taup1, double precision, dimension( * ) taup2, double precision, dimension( * ) tauq1, double precision, dimension( * ) tauq2, double precision, dimension( * ) work, integer lwork, integer info)

DORBDB

Purpose:

    !>
    !> DORBDB simultaneously bidiagonalizes the blocks of an M-by-M
    !> partitioned orthogonal matrix X:
    !>
    !>                                 [ B11 | B12 0  0 ]
    !>     [ X11 | X12 ]   [ P1 |    ] [  0  |  0 -I  0 ] [ Q1 |    ]**T
    !> X = [-----------] = [---------] [----------------] [---------]   .
    !>     [ X21 | X22 ]   [    | P2 ] [ B21 | B22 0  0 ] [    | Q2 ]
    !>                                 [  0  |  0  0  I ]
    !>
    !> X11 is P-by-Q. Q must be no larger than P, M-P, or M-Q. (If this is
    !> not the case, 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 orthogonal matrices P1, P2, Q1, and Q2 are P-by-P, (M-P)-by-
    !> (M-P), Q-by-Q, and (M-Q)-by-(M-Q), respectively. They are
    !> represented implicitly by Householder vectors.
    !>
    !> B11, B12, B21, and B22 are Q-by-Q bidiagonal matrices represented
    !> implicitly by angles THETA, PHI.
    !>

Parameters
    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.
    !>

    SIGNS

    !>          SIGNS is CHARACTER
    !>          = 'O':      The lower-left block is made nonpositive (the
    !>                       convention);
    !>          otherwise:  The upper-right block is made nonpositive (the
    !>                       convention).
    !>

    M

    !>          M is INTEGER
    !>          The number of rows and columns in X.
    !>

    P

    !>          P is INTEGER
    !>          The number of rows in X11 and X12. 0 <= P <= M.
    !>

    Q

    !>          Q is INTEGER
    !>          The number of columns in X11 and X21. 0 <= Q <=
    !>          MIN(P,M-P,M-Q).
    !>

    X11

    !>          X11 is DOUBLE PRECISION array, dimension (LDX11,Q)
    !>          On entry, the top-left block of the orthogonal matrix to be
    !>          reduced. On exit, the form depends on TRANS:
    !>          If TRANS = 'N', then
    !>             the columns of tril(X11) specify reflectors for P1,
    !>             the rows of triu(X11,1) specify reflectors for Q1;
    !>          else TRANS = 'T', and
    !>             the rows of triu(X11) specify reflectors for P1,
    !>             the columns of tril(X11,-1) specify reflectors for Q1.
    !>

    LDX11

    !>          LDX11 is INTEGER
    !>          The leading dimension of X11. If TRANS = 'N', then LDX11 >=
    !>          P; else LDX11 >= Q.
    !>

    X12

    !>          X12 is DOUBLE PRECISION array, dimension (LDX12,M-Q)
    !>          On entry, the top-right block of the orthogonal matrix to
    !>          be reduced. On exit, the form depends on TRANS:
    !>          If TRANS = 'N', then
    !>             the rows of triu(X12) specify the first P reflectors for
    !>             Q2;
    !>          else TRANS = 'T', and
    !>             the columns of tril(X12) specify the first P reflectors
    !>             for Q2.
    !>

    LDX12

    !>          LDX12 is INTEGER
    !>          The leading dimension of X12. If TRANS = 'N', then LDX12 >=
    !>          P; else LDX11 >= M-Q.
    !>

    X21

    !>          X21 is DOUBLE PRECISION array, dimension (LDX21,Q)
    !>          On entry, the bottom-left block of the orthogonal matrix to
    !>          be reduced. On exit, the form depends on TRANS:
    !>          If TRANS = 'N', then
    !>             the columns of tril(X21) specify reflectors for P2;
    !>          else TRANS = 'T', and
    !>             the rows of triu(X21) specify reflectors for P2.
    !>

    LDX21

    !>          LDX21 is INTEGER
    !>          The leading dimension of X21. If TRANS = 'N', then LDX21 >=
    !>          M-P; else LDX21 >= Q.
    !>

    X22

    !>          X22 is DOUBLE PRECISION array, dimension (LDX22,M-Q)
    !>          On entry, the bottom-right block of the orthogonal matrix to
    !>          be reduced. On exit, the form depends on TRANS:
    !>          If TRANS = 'N', then
    !>             the rows of triu(X22(Q+1:M-P,P+1:M-Q)) specify the last
    !>             M-P-Q reflectors for Q2,
    !>          else TRANS = 'T', and
    !>             the columns of tril(X22(P+1:M-Q,Q+1:M-P)) specify the last
    !>             M-P-Q reflectors for P2.
    !>

    LDX22

    !>          LDX22 is INTEGER
    !>          The leading dimension of X22. If TRANS = 'N', then LDX22 >=
    !>          M-P; else LDX22 >= M-Q.
    !>

    THETA

    !>          THETA is DOUBLE PRECISION array, dimension (Q)
    !>          The entries of the bidiagonal blocks B11, B12, B21, B22 can
    !>          be computed from the angles THETA and PHI. See Further
    !>          Details.
    !>

    PHI

    !>          PHI is DOUBLE PRECISION array, dimension (Q-1)
    !>          The entries of the bidiagonal blocks B11, B12, B21, B22 can
    !>          be computed from the angles THETA and PHI. See Further
    !>          Details.
    !>

    TAUP1

    !>          TAUP1 is DOUBLE PRECISION array, dimension (P)
    !>          The scalar factors of the elementary reflectors that define
    !>          P1.
    !>

    TAUP2

    !>          TAUP2 is DOUBLE PRECISION array, dimension (M-P)
    !>          The scalar factors of the elementary reflectors that define
    !>          P2.
    !>

    TAUQ1

    !>          TAUQ1 is DOUBLE PRECISION array, dimension (Q)
    !>          The scalar factors of the elementary reflectors that define
    !>          Q1.
    !>

    TAUQ2

    !>          TAUQ2 is DOUBLE PRECISION array, dimension (M-Q)
    !>          The scalar factors of the elementary reflectors that define
    !>          Q2.
    !>

    WORK

    !>          WORK is DOUBLE PRECISION array, dimension (LWORK)
    !>

    LWORK

    !>          LWORK is INTEGER
    !>          The dimension of the array WORK. LWORK >= M-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.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

    !>
    !>  The bidiagonal blocks B11, B12, B21, and B22 are represented
    !>  implicitly by angles THETA(1), ..., THETA(Q) and PHI(1), ...,
    !>  PHI(Q-1). B11 and B21 are upper bidiagonal, while B21 and B22 are
    !>  lower bidiagonal. Every entry in each bidiagonal band is a product
    !>  of a sine or cosine of a THETA with a sine or cosine of a PHI. See
    !>  [1] or DORCSD for details.
    !>
    !>  P1, P2, Q1, and Q2 are represented as products of elementary
    !>  reflectors. See DORCSD for details on generating P1, P2, Q1, and Q2
    !>  using DORGQR and DORGLQ.
    !>

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

Definition at line 284 of file dorbdb.f.

Author

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