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zbbcsd.f(3) LAPACK zbbcsd.f(3)

NAME

zbbcsd.f -

SYNOPSIS

Functions/Subroutines


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

Function/Subroutine Documentation

subroutine zbbcsd (characterJOBU1, characterJOBU2, characterJOBV1T, characterJOBV2T, characterTRANS, integerM, integerP, integerQ, double precision, dimension( * )THETA, double precision, dimension( * )PHI, complex*16, dimension( ldu1, * )U1, integerLDU1, complex*16, dimension( ldu2, * )U2, integerLDU2, complex*16, dimension( ldv1t, * )V1T, integerLDV1T, complex*16, dimension( ldv2t, * )V2T, integerLDV2T, 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, integerLRWORK, integerINFO)

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, 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


LDU1 is INTEGER
The leading dimension of the array U1.

U2


U2 is COMPLEX*16 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


LDU2 is INTEGER
The leading dimension of the array U2.

V1T


V1T is COMPLEX*16 array, dimension (LDV1T,Q)
On entry, a LDV1T-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.

V2T


V2T is COMPLEX*16 array, dimenison (LDV2T,M-Q)
On entry, a LDV2T-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.

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 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

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.

Date:

November 2011

Definition at line 330 of file zbbcsd.f.

Author

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Tue Sep 25 2012 Version 3.4.2