# dlals0 (l) - Linux Man Pages

## NAME

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

## 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. (2R) If SQRE = 1, one extra Givens rotation to generate 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 output, 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
PERM (input) INTEGER array, dimension ( N )
The permutations (from deflation and sorting) applied to the two blocks. GIVPTR (input) INTEGER The number of Givens rotations which took place in this subproblem. 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.
WORK (workspace) DOUBLE PRECISION array, dimension ( K )
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 Ren-Cang Li, Computer Science Division, University of
California at Berkeley, USA

Osni Marques, LBNL/NERSC, USA