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CLAR1V(l)) CLAR1V(l)NAMECLAR1V - 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 ISYNOPSISSUBROUTINE 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( * )PURPOSECLAR1V 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.ARGUMENTSN (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 (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 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 DETAILSBased on contributions by Inderjit Dhillon, IBM Almaden, USA Osni Marques, LBNL/NERSC, USA Ken Stanley, Computer Science Division, University of California at Berkeley, USALAPACK version 3.015 June 2000 CLAR1V(l)

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