CLAR1V(l)								 )								 CLAR1V(l)

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 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) 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 bidiagonal 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) 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  mag-
		nitude. 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)