SBBCSD(3) LAPACK routine of NEC Numeric Library Collection SBBCSD(3)
NAME
SBBCSD
SYNOPSIS
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)
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.
ARGUMENTS
JOBU1 (input)
JOBU1 is CHARACTER
= 'Y': U1 is updated;
otherwise: U1 is not updated.
JOBU2 (input)
JOBU2 is CHARACTER
= 'Y': U2 is updated;
otherwise: U2 is not updated.
JOBV1T (input)
JOBV1T is CHARACTER
= 'Y': V1T is updated;
otherwise: V1T is not updated.
JOBV2T (input)
JOBV2T is CHARACTER
= 'Y': V2T is updated;
otherwise: V2T is not updated.
TRANS (input)
TRANS is CHARACTER
= 'T': X, U1, U2, V1T, and V2T are stored in row-major
order;
otherwise: X, U1, U2, V1T, and V2T are stored in column-
major order.
M (input)
M is INTEGER
The number of rows and columns in X, the orthogonal matrix in
bidiagonal-block form.
P (input)
P is INTEGER
The number of rows in the top-left block of X. 0 <= P <= M.
Q (input)
Q is INTEGER
The number of columns in the top-left block of X.
0 <= Q <= MIN(P,M-P,M-Q).
THETA (input/output)
THETA is REAL array, dimension (Q)
On entry, the angles THETA(1),...,THETA(Q) that, along with
PHI(1), ...,PHI(Q-1), define the matrix in bidiagonal-block
form. On exit, the angles whose cosines and sines define the
diagonal blocks in the CS decomposition.
PHI (input/output)
PHI is REAL array, dimension (Q-1)
The angles PHI(1),...,PHI(Q-1) that, along with THETA(1),...,
THETA(Q), define the matrix in bidiagonal-block form.
U1 (input/output)
U1 is 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)
LDU1 is INTEGER
The leading dimension of the array U1.
U2 (input/output)
U2 is 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)
LDU2 is INTEGER
The leading dimension of the array U2.
V1T (input/output)
V1T is 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)
LDV1T is INTEGER
The leading dimension of the array V1T.
V2T (input/output)
V2T is 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)
LDV2T is INTEGER
The leading dimension of the array V2T.
B11D (output)
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 (output)
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 (output)
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 (output)
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 (output)
B21D is REAL array, dimension (Q)
When CBBCSD converges, B21D contains the negative sines of
THETA(1), ..., THETA(Q). If CBBCSD fails to converge, then
B21D contains the diagonal of the partially reduced bottom-left
block.
B21E (output)
B21E is REAL array, dimension (Q-1)
When CBBCSD converges, B21E contains zeros. If CBBCSD fails
to converge, then B21E contains the subdiagonal of the
partially reduced bottom-left block.
B22D (output)
B22D is REAL array, dimension (Q)
When CBBCSD converges, B22D contains the negative sines of
THETA(1), ..., THETA(Q). If CBBCSD fails to converge, then
B22D contains the diagonal of the partially reduced bottom-right
block.
B22E (output)
B22E is REAL array, dimension (Q-1)
When CBBCSD converges, B22E contains zeros. If CBBCSD fails
to converge, then B22E contains the subdiagonal of the
partially reduced bottom-right block.
WORK (output)
WORK is REAL array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK (input)
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 (output)
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.
LAPACK routine 31 October 2017 SBBCSD(3)