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sbbcsd.f(3) LAPACK sbbcsd.f(3)

NAME

sbbcsd.f

SYNOPSIS

Functions/Subroutines


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

Function/Subroutine Documentation

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.

Date:

June 2016

Definition at line 334 of file sbbcsd.f.

Author

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Tue Nov 14 2017 Version 3.8.0