CBBCSD(3)      LAPACK routine of NEC Numeric Library Collection      CBBCSD(3)



NAME
       CBBCSD

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)



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

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

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

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

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

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

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

           THETA     (input/output)
                     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       (input/output)
                     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        (input/output)
                     U1 is 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)
                     LDU1 is INTEGER
                     The leading dimension of the array U1.

           U2        (input/output)
                     U2 is 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)
                     LDU2 is INTEGER
                     The leading dimension of the array U2.

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

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

           B11D      (output)
                     B11D is 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)
                     B11E is 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)
                     B12D is 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)
                     B12E is 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.

           B21D      (output)
                     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      (output)
                     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      (output)
                     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      (output)
                     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.

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

           LRWORK    (input)
                     LRWORK is 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)
                     INFO is 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.



       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.



LAPACK routine                  31 October 2017                      CBBCSD(3)