DLASD2 - merge the two sets of singular values together
into a single sorted set
SYNOPSIS
SUBROUTINE DLASD2( NL, NR, SQRE, K, D, Z, ALPHA, BETA, U,
LDU, VT, LDVT, DSIGMA, U2, LDU2, VT2,
LDVT2, IDXP, IDX, IDXC, IDXQ, COLTYP,
INFO )
INTEGER INFO, K, LDU, LDU2, LDVT, LDVT2, NL,
NR, SQRE
DOUBLE PRECISION ALPHA, BETA
INTEGER COLTYP( * ), IDX( * ), IDXC( * ), IDXP(
* ), IDXQ( * )
DOUBLE PRECISION D( * ), DSIGMA( * ), U( LDU,
* ), U2( LDU2, * ), VT( LDVT, * ), VT2(
LDVT2, * ), Z( * )
PURPOSE
DLASD2 merges the two sets of singular values together
into a single sorted set. Then it tries to deflate the
size of the problem. There are two ways in which deflation
can occur: when two or more singular values are close
together or if there is a tiny entry in the Z vector. For
each such occurrence the order of the related secular
equation problem is reduced by one.
DLASD2 is called from DLASD1.
ARGUMENTS
NL (input) INTEGER
The row dimension of the upper block. NL >= 1.
NR (input) INTEGER
The row dimension of the lower block. NR >= 1.
SQRE (input) INTEGER
= 0: the lower block is an NR-by-NR square matrix.
= 1: the lower block is an NR-by-(NR+1) rectangular
matrix.
The bidiagonal matrix has N = NL + NR + 1 rows and
M = N + SQRE >= N columns.
K (output) INTEGER
Contains the dimension of the non-deflated matrix,
This is the order of the related secular equation.
1 <= K <=N.
On entry D contains the singular values of the two
submatrices to be combined. On exit D contains the
trailing (N-K) updated singular values (those which
were deflated) sorted into increasing order.
ALPHA (input) DOUBLE PRECISION
Contains the diagonal element associated with the
added row.
BETA (input) DOUBLE PRECISION
Contains the off-diagonal element associated with
the added row.
U (input/output) DOUBLE PRECISION array, dimen
sion(LDU,N)
On entry U contains the left singular vectors of
two submatrices in the two square blocks with cor
ners at (1,1), (NL, NL), and (NL+2, NL+2), (N,N).
On exit U contains the trailing (N-K) updated left
singular vectors (those which were deflated) in its
last N-K columns.
LDU (input) INTEGER
The leading dimension of the array U. LDU >= N.
Z (output) DOUBLE PRECISION array, dimension(N)
On exit Z contains the updating row vector in the
secular equation.
DSIGMA (output) DOUBLE PRECISION array, dimension
(N) Contains a copy of the diagonal elements (K-1
singular values and one zero) in the secular equa
tion.
U2 (output) DOUBLE PRECISION array, dimension(LDU2,N)
Contains a copy of the first K-1 left singular vec
tors which will be used by DLASD3 in a matrix mul
tiply (DGEMM) to solve for the new left singular
vectors. U2 is arranged into four blocks. The first
block contains a column with 1 at NL+1 and zero
everywhere else; the second block contains non-zero
entries only at and above NL; the third contains
non-zero entries only below NL+1; and the fourth is
dense.
LDU2 (input) INTEGER
The leading dimension of the array U2. LDU2 >= N.
VT (input/output) DOUBLE PRECISION array, dimen
sion(LDVT,M)
On entry VT' contains the right singular vectors of
two submatrices in the two square blocks with cor
ners at (1,1), (NL+1, NL+1), and (NL+2, NL+2),
updated right singular vectors (those which were
deflated) in its last N-K columns. In case SQRE
=1, the last row of VT spans the right null space.
LDVT (input) INTEGER
The leading dimension of the array VT. LDVT >= M.
VT2 (output) DOUBLE PRECISION array, dimension(LDVT2,N)
VT2' contains a copy of the first K right singular
vectors which will be used by DLASD3 in a matrix
multiply (DGEMM) to solve for the new right singu
lar vectors. VT2 is arranged into three blocks. The
first block contains a row that corresponds to the
special 0 diagonal element in SIGMA; the second
block contains non-zeros only at and before NL +1;
the third block contains non-zeros only at and
after NL +2.
LDVT2 (input) INTEGER
The leading dimension of the array VT2. LDVT2 >=
M.
IDXP (workspace) INTEGER array, dimension(N)
This will contain the permutation used to place
deflated values of D at the end of the array. On
output IDXP(2:K)
points to the nondeflated D-values and IDXP(K+1:N)
points to the deflated singular values.
IDX (workspace) INTEGER array, dimension(N)
This will contain the permutation used to sort the
contents of D into ascending order.
IDXC (output) INTEGER array, dimension(N)
This will contain the permutation used to arrange
the columns of the deflated U matrix into three
groups: the first group contains non-zero entries
only at and above NL, the second contains non-zero
entries only below NL+2, and the third is dense.
COLTYP (workspace/output) INTEGER array, dimen
sion(N) As workspace, this will contain a label
which will indicate which of the following types a
column in the U2 matrix or a row in the VT2 matrix
is:
1 : non-zero in the upper half only
2 : non-zero in the lower half only
3 : dense
4 : deflated
On exit, it is an array of dimension 4, with
COLTYP(I) being the dimension of the I-th type
columns.
This contains the permutation which separately
sorts the two sub-problems in D into ascending
order. Note that entries in the first hlaf of this
permutation must first be moved one position back
ward; and entries in the second half must first
have NL+1 added to their values.
INFO (output) INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an ille
gal value.
FURTHER DETAILS
Based on contributions by
Ming Gu and Huan Ren, Computer Science Division, Uni
versity of
California at Berkeley, USA
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