
CLAR1V(l) ) CLAR1V(l)
NAME
CLAR1V  compute the (scaled) rth 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) rth 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 inverse 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) rth 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 ei
genvalue of L D L^T.
L (input) REAL array, dimension (N1)
The (n1) subdiagonal elements of the unit bidiagonal matrix L, in elements 1 to
N1.
D (input) REAL array, dimension (N)
The n diagonal elements of the diagonal matrix D.
LD (input) REAL array, dimension (N1)
The n1 elements L(i)*D(i).
LLD (input) REAL array, dimension (N1)
The n1 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) rth 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) diagonal 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(l) 
