Provided by: liblapack-doc_3.3.1-1_all bug

NAME

       LAPACK-3 - computes the CS decomposition of a unitary matrix in bidiagonal-block form,

SYNOPSIS

       SUBROUTINE CBBCSD( 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, RWORK, LRWORK, INFO )

           IMPLICIT       NONE

           CHARACTER      JOBU1, JOBU2, JOBV1T, JOBV2T, TRANS

           INTEGER        INFO, LDU1, LDU2, LDV1T, LDV2T, LRWORK, M, P, Q

           REAL           B11D( * ), B11E( * ), B12D( * ), B12E( * ), B21D( * ), B21E( * ), B22D(
                          * ), B22E( * ), PHI( * ), THETA( * ), RWORK( * )

           COMPLEX        U1( LDU1, * ), U2( LDU2, * ), V1T( LDV1T, * ), V2T( LDV2T, * )

PURPOSE

       CBBCSD computes the CS decomposition of a unitary 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 |    ]**H
                        = [---------] [---------------] [---------]   .
                          [    | 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 CUNCSD for details.)
        The bidiagonal matrices B11, B12, B21, and B22 are represented
        implicitly by angles THETA(1:Q) and PHI(1:Q-1).
        The unitary matrices U1, U2, V1T, and V2T are input/output.
        The input matrices are pre- or post-multiplied by the appropriate
        singular vector matrices.

ARGUMENTS

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

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

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

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

        TRANS   (input) 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       (input) INTEGER
                The number of rows and columns in X, the unitary matrix in
                bidiagonal-block form.

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

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

        THETA   (input/output) 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     (input/workspace) 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      (input/output) COMPLEX 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    (input) INTEGER
                The leading dimension of the array U1.

        U2      (input/output) COMPLEX 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    (input) INTEGER
                The leading dimension of the array U2.

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

        LDV1T   (input) INTEGER
                The leading dimension of the array V1T.

        V2T     (input/output) COMPLEX array, dimenison (LDV2T,M-Q)
                On entry, a LDV2T-by-(M-Q) matrix. On exit, V2T is
                premultiplied by the conjugate transpose of the right
                singular vector matrix common to [ B12 0 0 ; 0 -I 0 ] and
                [ B22 0 0 ; 0 0 I ].

        LDV2T   (input) INTEGER
                The leading dimension of the array V2T.

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

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

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

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

        RWORK   (workspace) REAL array, dimension (MAX(1,LWORK))
                On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

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

        INFO    (output) INTEGER
                = 0:  successful exit.
                < 0:  if INFO = -i, the i-th argument had an illegal value.
                > 0:  if CBBCSD did not converge, INFO specifies the number
                of nonzero entries in PHI, and B11D, B11E, etc.,
                contain the partially reduced matrix.
                Reference
                =========
                [1] Brian D. Sutton. Computing the complete CS decomposition. Numer.
                Algorithms, 50(1):33-65, 2009.

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.

 LAPACK routine (version 3.3.0)             April 2011                            CBBCSD(3lapack)