table of contents
| bbcsd(3) | LAPACK | 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¶
Generated automatically by Doxygen for LAPACK from the source code.
| Wed Feb 7 2024 11:30:40 | Version 3.12.0 |