clar1v(3)

NAME

CLAR1V - compute the (scaled) r-th column of the inverse

of the sumbmatrix in rows B1 through BN of the tridiagonal matrix

L D L^T - sigma I

SYNOPSIS

SUBROUTINE CLAR1V( N, B1, BN, SIGMA, D, L, LD,  LLD,  GERSCH, Z, ZTZ, MINGMA, R, ISUPPZ, WORK )
    INTEGER        B1, BN, N, R
    REAL           MINGMA, SIGMA, ZTZ
    INTEGER        ISUPPZ( * )
    REAL            D(  * ), GERSCH( * ), L( * ), LD( * ),
LLD( * ), WORK( * )
    COMPLEX        Z( * )

PURPOSE

CLAR1V computes the (scaled) r-th column of the inverse of

the sumbmatrix in rows B1 through BN of the tridiagonal matrix L

D L^T - sigma I. The following steps accomplish this computation

: (a) Stationary qd transform, L D L^T - sigma I = L(+) D(+)

L(+)^T, (b) Progressive qd transform, L D L^T - sigma I = U(-)

D(-) U(-)^T, (c) Computation of the diagonal elements of the in

verse of

L D L^T - sigma I by combining the above transforms,

and choosing
r as the index where the diagonal of the inverse is

(one of the)
largest in magnitude.

(d) Computation of the (scaled) r-th column of the inverse

using the

twisted factorization obtained by combining the top

part of the
the stationary and the bottom part of the progressive

transform.

ARGUMENTS

N (input) INTEGER

The order of the matrix L D L^T.

B1 (input) INTEGER

First index of the submatrix of L D L^T.

BN (input) INTEGER

Last index of the submatrix of L D L^T.

SIGMA (input) REAL

The shift. Initially, when R = 0, SIGMA should be

a good approximation to an eigenvalue of L D L^T.

L (input) REAL array, dimension (N-1)

The (n-1) subdiagonal elements of the unit bidi

agonal matrix L, in elements 1 to N-1.

D (input) REAL array, dimension (N)

The n diagonal elements of the diagonal matrix D.

LD (input) REAL array, dimension (N-1)

The n-1 elements L(i)*D(i).

LLD (input) REAL array, dimension (N-1)

The n-1 elements L(i)*L(i)*D(i).

GERSCH (input) REAL array, dimension (2*N)

The n Gerschgorin intervals. These are used to

restrict the initial search for R, when R is input as 0.

Z (output) COMPLEX array, dimension (N)

The (scaled) r-th column of the inverse. Z(R) is

returned to be 1.

ZTZ (output) REAL

The square of the norm of Z.

MINGMA (output) REAL

The reciprocal of the largest (in magnitude) di

agonal element of the inverse of L D L^T - sigma I.

R (input/output) INTEGER

Initially, R should be input to be 0 and is then

output as the index where the diagonal element of the inverse is

largest in magnitude. In later iterations, this same value of R

should be input.

ISUPPZ (output) INTEGER array, dimension (2)

The support of the vector in Z, i.e., the vector

Z is nonzero only in elements ISUPPZ(1) through ISUPPZ( 2 ).

WORK (workspace) REAL array, dimension (4*N)

FURTHER DETAILS

Based on contributions by

Inderjit Dhillon, IBM Almaden, USA
Osni Marques, LBNL/NERSC, USA
Ken Stanley, Computer Science Division, University of

California at Berkeley, USA

LAPACK version 3.0 15 June 2000

CLAR1V(3)