slals0(3)

NAME

SLALS0 - applie back the multiplying factors of either the

left or the right singular vector matrix of a diagonal matrix ap

pended 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 SLALS0( 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
    REAL           C, S
    INTEGER        GIVCOL( LDGCOL, * ), PERM( * )
    REAL           B( LDB, * ), BX( LDBX, * ), DIFL( *  ),
DIFR( LDGNUM, * ), GIVNUM( LDGNUM, * ), POLES( LDGNUM, * ), WORK(
* ), Z( * )

PURPOSE

SLALS0 applies back the multiplying factors of either the

left or the right singular vector matrix of a diagonal matrix ap

pended 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 ma

trix.

For the right singular vector matrix, four types of or

thogonal matrices are involved:

(1R) The right singular vector matrix of the remaining ma

trix.

(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) REAL 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) REAL array, dimension ( LDBX, NRHS )

LDBX (input) INTEGER

The leading dimension of BX.

PERM (input) INTEGER array, dimension ( N )

The permutations (from deflation and sorting) ap

plied to the two blocks.

GIVPTR (input) INTEGER The number of Givens rota

tions 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) REAL 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 ar

rays DIFR, POLES and GIVNUM, must be at least K.

POLES (input) REAL 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) REAL array, dimension ( K ).

On entry, DIFL(I) is the distance between I-th up

dated (undeflated) singular value and the I-th (undeflated) old

singular value.

DIFR (input) REAL array, dimension ( LDGNUM, 2 ).

On entry, DIFR(I, 1) contains the distances between

I-th updated (undeflated) singular value and the I+1-th (unde

flated) old singular value. And DIFR(I, 2) is the normalizing

factor for the I-th right singular vector.

Z (input) REAL 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) REAL

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) REAL

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) REAL array, dimension ( K )

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

LAPACK version 3.0 15 June 2000

SLALS0(3)