bbcsd - Man Page

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

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:

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 328 of file cbbcsd.f.

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:

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 328 of file dbbcsd.f.

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:

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 328 of file sbbcsd.f.

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:

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 328 of file zbbcsd.f.