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.
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.
Arguments
| JOBU1 |
(input) CHARACTER
= 'Y': U1 is updated;
otherwise: U1 is not updated. |
| JOBU2 |
(input) CHARACTER
= 'Y': U2 is updated;
otherwise: U2 is not updated. |
| JOBV1T |
(input) CHARACTER
= 'Y': V1T is updated;
otherwise: V1T is not updated. |
| JOBV2T |
(input) CHARACTER
= 'Y': V2T is updated;
otherwise: V2T is not updated. |
| TRANS |
(input) CHARACTER
= 'T': X, U1, U2, V1T, and V2T are stored in row-major
order; otherwise: X, U1, U2, V1T, and V2T are stored in column- major order. |
| M |
(input) INTEGER
The number of rows and columns in X, the orthogonal matrix in
bidiagonal-block form. |
| P |
(input) INTEGER
The number of rows in the top-left block of X. 0 <= P <= M.
|
| Q |
(input) INTEGER
The number of columns in the top-left block of X.
0 <= Q <= MIN(P,M-P,M-Q). |
| THETA |
(input/output) REAL array, dimension (Q)
On entry, the angles THETA(1),...,THETA(Q) that, along with
PHI(1), ...,PHI(Q-1), define the matrix in bidiagonal-block form. On exit, the angles whose cosines and sines define the diagonal blocks in the CS decomposition. |
| PHI |
(input/workspace) REAL array, dimension (Q-1)
The angles PHI(1),...,PHI(Q-1) that, along with THETA(1),...,
THETA(Q), define the matrix in bidiagonal-block form. |
| U1 |
(input/output) REAL array, dimension (LDU1,P)
On entry, an LDU1-by-P matrix. On exit, U1 is postmultiplied
by the left singular vector matrix common to [ B11 ; 0 ] and [ B12 0 0 ; 0 -I 0 0 ]. |
| LDU1 |
(input) INTEGER
The leading dimension of the array U1.
|
| U2 |
(input/output) REAL array, dimension (LDU2,M-P)
On entry, an LDU2-by-(M-P) matrix. On exit, U2 is
postmultiplied by the left singular vector matrix common to [ B21 ; 0 ] and [ B22 0 0 ; 0 0 I ]. |
| LDU2 |
(input) INTEGER
The leading dimension of the array U2.
|
| V1T |
(input/output) REAL array, dimension (LDV1T,Q)
On entry, a LDV1T-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 |
(input) INTEGER
The leading dimension of the array V1T.
|
| V2T |
(input/output) REAL array, dimenison (LDV2T,M-Q)
On entry, a LDV2T-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 |
(input) INTEGER
The leading dimension of the array V2T.
|
| B11D |
(output) REAL array, dimension (Q)
When SBBCSD converges, B11D contains the cosines of THETA(1),
contains the diagonal of the partially reduced top-left block. |
| B11E |
(output) 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 |
(output) 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 |
(output) 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. |
| WORK |
(workspace) REAL array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
|
| LWORK |
(input) 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 |
(output) 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. |
Reference
[1] Brian D. Sutton. Computing the complete CS decomposition. Numer.
Algorithms, 50(1):33-65, 2009.
Algorithms, 50(1):33-65, 2009.
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.
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.