clalsa(3)

NAME

CLALSA - i an itermediate step in solving the least

squares problem by computing the SVD of the coefficient matrix in

compact form (The singular vectors are computed as products of

simple orthorgonal matrices.)

SYNOPSIS

SUBROUTINE  CLALSA(  ICOMPQ,  SMLSIZ, N, NRHS, B, LDB, BX,
LDBX, U, LDU, VT, K, DIFL, DIFR, Z, POLES, GIVPTR,  GIVCOL,  LDGCOL, PERM, GIVNUM, C, S, RWORK, IWORK, INFO )
    INTEGER         ICOMPQ,  INFO, LDB, LDBX, LDGCOL, LDU,
N, NRHS, SMLSIZ
    INTEGER        GIVCOL(  LDGCOL,  *  ),  GIVPTR(  *  ),
IWORK( * ), K( * ), PERM( LDGCOL, * )
    REAL           C( * ), DIFL( LDU, * ), DIFR( LDU, * ),
GIVNUM( LDU, * ), POLES( LDU, * ), RWORK( * ), S( * ), U( LDU,  *
), VT( LDU, * ), Z( LDU, * )
    COMPLEX        B( LDB, * ), BX( LDBX, * )

PURPOSE

CLALSA is an itermediate step in solving the least squares

problem by computing the SVD of the coefficient matrix in compact

form (The singular vectors are computed as products of simple or

thorgonal matrices.). If ICOMPQ = 0, CLALSA applies the inverse

of the left singular vector matrix of an upper bidiagonal matrix

to the right hand side; and if ICOMPQ = 1, CLALSA applies the

right singular vector matrix to the right hand side. The singular

vector matrices were generated in compact form by CLALSA.

ARGUMENTS

ICOMPQ (input) INTEGER Specifies whether the left or the

right singular vector matrix is involved. = 0: Left singular

vector matrix
= 1: Right singular vector matrix

SMLSIZ (input) INTEGER The maximum size of the subproblems

at the bottom of the computation tree.

N (input) INTEGER

The row and column dimensions of the upper bidiago

nal matrix.

NRHS (input) INTEGER

The number of columns of B and BX. NRHS must be at

least 1.

B (input) COMPLEX 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 in the calling subpro

gram. LDB must be at least max(1,MAX( M, N ) ).

BX (output) COMPLEX array, dimension ( LDBX, NRHS )

On exit, the result of applying the left or right

singular vector matrix to B.

LDBX (input) INTEGER

The leading dimension of BX.

U (input) REAL array, dimension ( LDU, SMLSIZ ).

On entry, U contains the left singular vector ma

trices of all subproblems at the bottom level.

LDU (input) INTEGER, LDU = > N.

The leading dimension of arrays U, VT, DIFL, DIFR,

POLES, GIVNUM, and Z.

VT (input) REAL array, dimension ( LDU, SMLSIZ+1 ).

On entry, VT' contains the right singular vector

matrices of all subproblems at the bottom level.

K (input) INTEGER array, dimension ( N ).

DIFL (input) REAL array, dimension ( LDU, NLVL ).

where NLVL = INT(log_2 (N/(SMLSIZ+1))) + 1.

DIFR (input) REAL array, dimension ( LDU, 2 * NLVL ).

On entry, DIFL(*, I) and DIFR(*, 2 * I -1) record

distances between singular values on the I-th level and singular

values on the (I -1)-th level, and DIFR(*, 2 * I) record the nor

malizing factors of the right singular vectors matrices of sub

problems on I-th level.

Z (input) REAL array, dimension ( LDU, NLVL ).

On entry, Z(1, I) contains the components of the

deflation- adjusted updating row vector for subproblems on the I

th level.

POLES (input) REAL array, dimension ( LDU, 2 * NLVL ).

On entry, POLES(*, 2 * I -1: 2 * I) contains the

new and old singular values involved in the secular equations on

the I-th level.

GIVPTR (input) INTEGER array, dimension ( N ). On

entry, GIVPTR( I ) records the number of Givens rotations per

formed on the I-th problem on the computation tree.

GIVCOL (input) INTEGER array, dimension ( LDGCOL, 2

* NLVL ). On entry, for each I, GIVCOL(*, 2 * I - 1: 2 * I)

records the locations of Givens rotations performed on the I-th

level on the computation tree.

LDGCOL (input) INTEGER, LDGCOL = > N. The leading

dimension of arrays GIVCOL and PERM.

PERM (input) INTEGER array, dimension ( LDGCOL, NLVL ).

On entry, PERM(*, I) records permutations done on

the I-th level of the computation tree.

GIVNUM (input) REAL array, dimension ( LDU, 2 *

NLVL ). On entry, GIVNUM(*, 2 *I -1 : 2 * I) records the C- and

S- values of Givens rotations performed on the I-th level on the

computation tree.

C (input) REAL array, dimension ( N ).

On entry, if the I-th subproblem is not square, C(

I ) contains the C-value of a Givens rotation related to the

right null space of the I-th subproblem.

S (input) REAL array, dimension ( N ).

On entry, if the I-th subproblem is not square, S(

I ) contains the S-value of a Givens rotation related to the

right null space of the I-th subproblem.

RWORK (workspace) REAL array, dimension at least

max ( N, (SMLSZ+1)*NRHS*3 ).

IWORK (workspace) INTEGER array.

The dimension must be at least 3 * N

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

CLALSA(3)