bbcsd(3)

NAME
bbcsd - bbcsd: ??

SYNOPSIS

  Functions

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)
CBBCSD
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
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
subroutine zbbcsd (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)
ZBBCSD

Detailed Description

Function Documentation

  subroutine cbbcsd (character jobu1, character jobu2, character jobv1t, character jobv2t, character trans, integer m, integer p, integer q, real, dimension( * ) theta, real, dimension( * ) phi, complex, dimension( ldu1, * ) u1, integer ldu1, complex, dimension( ldu2, * ) u2, integer ldu2, complex, dimension( ldv1t, * ) v1t, integer ldv1t, complex, dimension( ldv2t, * ) v2t, integer ldv2t, 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( * ) rwork, integer lrwork, integer info)

CBBCSD

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.
    !>

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 unitary 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 COMPLEX 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 COMPLEX 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 COMPLEX array, dimension (LDV1T,Q)
    !>          On entry, a Q-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

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

    V2T

    !>          V2T is COMPLEX array, dimension (LDV2T,M-Q)
    !>          On entry, an (M-Q)-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

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

    B11D

    !>          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

    !>          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

    !>          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

    !>          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

    !>          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.
    !>

    RWORK

    !>          RWORK is REAL array, dimension (MAX(1,LRWORK))
    !>          On exit, if INFO = 0, RWORK(1) returns the optimal LRWORK.
    !>

    LRWORK

    !>          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

    !>          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.
    !>

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.

  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.

  subroutine sbbcsd (character jobu1, character jobu2, character jobv1t, character jobv2t, character trans, integer m, integer p, integer q, real, dimension( * ) theta, real, dimension( * ) phi, real, dimension( ldu1, * ) u1, integer ldu1, real, dimension( ldu2, * ) u2, integer ldu2, real, dimension( ldv1t, * ) v1t, integer ldv1t, real, dimension( ldv2t, * ) v2t, integer ldv2t, 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, integer lwork, integer info)

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, 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 REAL 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 REAL 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 REAL 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 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 SBBCSD converges, B21D contains the negative sines of
    !>          THETA(1), ..., THETA(Q). If SBBCSD fails to converge, then
    !>          B21D contains the diagonal of the partially reduced bottom-left
    !>          block.
    !>

    B21E

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

    B22D

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

    B22E

    !>          B22E is REAL array, dimension (Q-1)
    !>          When SBBCSD converges, B22E contains zeros. If SBBCSD 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.

  subroutine zbbcsd (character jobu1, character jobu2, character jobv1t, character jobv2t, character trans, integer m, integer p, integer q, double precision, dimension( * ) theta, double precision, dimension( * ) phi, complex*16, dimension( ldu1, * ) u1, integer ldu1, complex*16, dimension( ldu2, * ) u2, integer ldu2, complex*16, dimension( ldv1t, * ) v1t, integer ldv1t, complex*16, 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( * ) rwork, integer lrwork, integer info)

ZBBCSD

Purpose:

    !>
    !> ZBBCSD 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 ZUNCSD 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.
    !>

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 unitary 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 COMPLEX*16 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 COMPLEX*16 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 COMPLEX*16 array, dimension (LDV1T,Q)
    !>          On entry, a Q-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

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

    V2T

    !>          V2T is COMPLEX*16 array, dimension (LDV2T,M-Q)
    !>          On entry, an (M-Q)-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

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

    B11D

    !>          B11D is DOUBLE PRECISION array, dimension (Q)
    !>          When ZBBCSD converges, B11D contains the cosines of THETA(1),
    !>          ..., THETA(Q). If ZBBCSD 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 ZBBCSD converges, B11E contains zeros. If ZBBCSD fails
    !>          to converge, then B11E contains the superdiagonal of the
    !>          partially reduced top-left block.
    !>

    B12D

    !>          B12D is DOUBLE PRECISION array, dimension (Q)
    !>          When ZBBCSD converges, B12D contains the negative sines of
    !>          THETA(1), ..., THETA(Q). If ZBBCSD 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 ZBBCSD converges, B12E contains zeros. If ZBBCSD fails
    !>          to converge, then B12E contains the subdiagonal of the
    !>          partially reduced top-right block.
    !>

    B21D

    !>          B21D is DOUBLE PRECISION array, dimension (Q)
    !>          When ZBBCSD converges, B21D contains the negative sines of
    !>          THETA(1), ..., THETA(Q). If ZBBCSD 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 ZBBCSD converges, B21E contains zeros. If ZBBCSD fails
    !>          to converge, then B21E contains the subdiagonal of the
    !>          partially reduced bottom-left block.
    !>

    B22D

    !>          B22D is DOUBLE PRECISION array, dimension (Q)
    !>          When ZBBCSD converges, B22D contains the negative sines of
    !>          THETA(1), ..., THETA(Q). If ZBBCSD 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 ZBBCSD converges, B22E contains zeros. If ZBBCSD fails
    !>          to converge, then B22E contains the subdiagonal of the
    !>          partially reduced bottom-right block.
    !>

    RWORK

    !>          RWORK is DOUBLE PRECISION array, dimension (MAX(1,LRWORK))
    !>          On exit, if INFO = 0, RWORK(1) returns the optimal LRWORK.
    !>

    LRWORK

    !>          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

    !>          INFO is INTEGER
    !>          = 0:  successful exit.
    !>          < 0:  if INFO = -i, the i-th argument had an illegal value.
    !>          > 0:  if ZBBCSD 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.

Author

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