DLALS0 - applie back the multiplying factors of either the
left or the right singular vector matrix of a diagonal
matrix appended by a row to the right hand side matrix B
in solving the least squares problem using the divide-and-
conquer SVD approach
SYNOPSIS
SUBROUTINE DLALS0( ICOMPQ, NL, NR, SQRE, NRHS, B, LDB, BX,
LDBX, PERM, GIVPTR, GIVCOL, LDGCOL,
GIVNUM, LDGNUM, POLES, DIFL, DIFR, Z,
K, C, S, WORK, INFO )
INTEGER GIVPTR, ICOMPQ, INFO, K, LDB, LDBX,
LDGCOL, LDGNUM, NL, NR, NRHS, SQRE
DOUBLE PRECISION C, S
INTEGER GIVCOL( LDGCOL, * ), PERM( * )
DOUBLE PRECISION B( LDB, * ), BX( LDBX, * ),
DIFL( * ), DIFR( LDGNUM, * ), GIVNUM(
LDGNUM, * ), POLES( LDGNUM, * ), WORK(
* ), Z( * )
PURPOSE
DLALS0 applies back the multiplying factors of either the
left or the right singular vector matrix of a diagonal
matrix appended by a row to the right hand side matrix B
in solving the least squares problem using the divide-and-
conquer SVD approach. For the left singular vector
matrix, three types of orthogonal matrices are involved:
(1L) Givens rotations: the number of such rotations is
GIVPTR; the
pairs of columns/rows they were applied to are stored
in GIVCOL;
and the C- and S-values of these rotations are stored
in GIVNUM.
(2L) Permutation. The (NL+1)-st row of B is to be moved to
the first
row, and for J=2:N, PERM(J)-th row of B is to be
moved to the
J-th row.
(3L) The left singular vector matrix of the remaining
matrix.
For the right singular vector matrix, four types of
orthogonal matrices are involved:
(1R) The right singular vector matrix of the remaining
matrix.
the right
null space.
(3R) The inverse transformation of (2L).
(4R) The inverse transformation of (1L).
ARGUMENTS
ICOMPQ (input) INTEGER Specifies whether singular vectors
are to be computed in factored form:
= 0: Left singular vector matrix.
= 1: Right singular vector matrix.
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 row dimension N = NL + NR
+ 1, and column dimension M = N + SQRE.
NRHS (input) INTEGER
The number of columns of B and BX. NRHS must be at
least 1.
B (input/output) DOUBLE PRECISION array, dimension
( LDB, NRHS )
On input, B contains the right hand sides of the
least squares problem in rows 1 through M. On out
put, B contains the solution X in rows 1 through N.
LDB (input) INTEGER
The leading dimension of B. LDB must be at least
max(1,MAX( M, N ) ).
BX (workspace) DOUBLE PRECISION array, dimension (
LDBX, NRHS )
LDBX (input) INTEGER
The leading dimension of BX.
PERM (input) INTEGER array, dimension ( N )
The permutations (from deflation and sorting)
applied to the two blocks.
GIVPTR (input) INTEGER The number of Givens
GIVCOL (input) INTEGER array, dimension ( LDGCOL, 2
) Each pair of numbers indicates a pair of
rows/columns involved in a Givens rotation.
LDGCOL (input) INTEGER The leading dimension of
GIVCOL, must be at least N.
GIVNUM (input) DOUBLE PRECISION array, dimension (
LDGNUM, 2 ) Each number indicates the C or S value
used in the corresponding Givens rotation.
LDGNUM (input) INTEGER The leading dimension of
arrays DIFR, POLES and GIVNUM, must be at least K.
POLES (input) DOUBLE PRECISION array, dimension ( LDGNUM,
2 )
On entry, POLES(1:K, 1) contains the new singular
values obtained from solving the secular equation,
and POLES(1:K, 2) is an array containing the poles
in the secular equation.
DIFL (input) DOUBLE PRECISION array, dimension ( K ).
On entry, DIFL(I) is the distance between I-th
updated (undeflated) singular value and the I-th
(undeflated) old singular value.
DIFR (input) DOUBLE PRECISION array, dimension ( LDGNUM,
2 ).
On entry, DIFR(I, 1) contains the distances between
I-th updated (undeflated) singular value and the
I+1-th (undeflated) old singular value. And DIFR(I,
2) is the normalizing factor for the I-th right
singular vector.
Z (input) DOUBLE PRECISION array, dimension ( K )
Contain the components of the deflation-adjusted
updating row vector.
K (input) INTEGER
Contains the dimension of the non-deflated matrix,
This is the order of the related secular equation.
1 <= K <=N.
C (input) DOUBLE PRECISION
C contains garbage if SQRE =0 and the C-value of a
Givens rotation related to the right null space if
SQRE = 1.
S (input) DOUBLE PRECISION
S contains garbage if SQRE =0 and the S-value of a
Givens rotation related to the right null space if
SQRE = 1.
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 Ren-Cang Li, Computer Science Division,
University of
California at Berkeley, USA
Osni Marques, LBNL/NERSC, USA
Man(1) output converted with
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