DLASD2(3S)DLASD2(3S)NAME
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( * )
IMPLEMENTATION
These routines are part of the SCSL Scientific Library and can be loaded
using either the -lscs or the -lscs_mp option. The -lscs_mp option
directs the linker to use the multi-processor version of the library.
When linking to SCSL with -lscs or -lscs_mp, the default integer size is
4 bytes (32 bits). Another version of SCSL is available in which integers
are 8 bytes (64 bits). This version allows the user access to larger
memory sizes and helps when porting legacy Cray codes. It can be loaded
by using the -lscs_i8 option or the -lscs_i8_mp option. A program may use
only one of the two versions; 4-byte integer and 8-byte integer library
calls cannot be mixed.
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.
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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.
D (input/output) DOUBLE PRECISION array, dimension(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, dimension(LDU,N)
On entry U contains the left singular vectors of two submatrices
in the two square blocks with corners 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 equation.
U2 (output) DOUBLE PRECISION array, dimension(LDU2,N)
Contains a copy of the first K-1 left singular vectors which will
be used by DLASD3 in a matrix multiply (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, dimension(LDVT,M)
On entry VT' contains the right singular vectors of two
submatrices in the two square blocks with corners at (1,1), (NL+1,
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DLASD2(3S)DLASD2(3S)
NL+1), and (NL+2, NL+2), (M,M). On exit VT' contains the trailing
(N-K) 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 singular 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, dimension(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.
IDXQ (input) INTEGER array, dimension(N)
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
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DLASD2(3S)DLASD2(3S)
backward; 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 illegal value.
FURTHER DETAILS
Based on contributions by
Ming Gu and Huan Ren, Computer Science Division, University of
California at Berkeley, USA
SEE ALSOINTRO_LAPACK(3S), INTRO_SCSL(3S)
This man page is available only online.
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