NAME¶

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

SYNOPSIS¶

SUBROUTINE CLAR1V(

N, B1, BN, LAMBDA, D, L, LD, LLD, PIVMIN, GAPTOL, Z, WANTNC, NEGCNT, ZTZ, MINGMA, R, ISUPPZ, NRMINV, RESID, RQCORR, WORK )

LOGICAL WANTNC INTEGER B1, BN, N, NEGCNT, R REAL GAPTOL, LAMBDA, MINGMA, NRMINV, PIVMIN, RESID, RQCORR, ZTZ INTEGER ISUPPZ( * ) REAL D( * ), 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. When sigma is close to an eigenvalue, the computed vector is an accurate eigenvector. Usually, r corresponds to the index where the eigenvector is largest in magnitude. 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.

LAMBDA (input) REAL

The shift. In order to compute an accurate eigenvector, LAMBDA 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).

PIVMIN (input) REAL

The minimum pivot in the Sturm sequence.

GAPTOL (input) REAL

Tolerance that indicates when eigenvector entries are negligible w.r.t. their contribution to the residual.

Z (input/output) COMPLEX array, dimension (N)

On input, all entries of Z must be set to 0. On output, Z contains the (scaled) r-th column of the inverse. The scaling is such that Z(R) equals 1.

WANTNC (input) LOGICAL

Specifies whether NEGCNT has to be computed.

NEGCNT (output) INTEGER

If WANTNC is .TRUE. then NEGCNT = the number of pivots < pivmin in the matrix factorization L D L^T, and NEGCNT = -1 otherwise.

ZTZ (output) REAL

The square of the 2-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

The twist index for the twisted factorization used to compute Z. On input, 0 <= R <= N. If R is input as 0, R is set to the index where (L D L^T - sigma I)^{-1} is largest in magnitude. If 1 <= R <= N, R is unchanged. On output, R contains the twist index used to compute Z. Ideally, R designates the position of the maximum entry in the eigenvector.

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 ).

NRMINV (output) REAL

NRMINV = 1/SQRT( ZTZ )

RESID (output) REAL

The residual of the FP vector. RESID = ABS( MINGMA )/SQRT( ZTZ )

RQCORR (output) REAL

The Rayleigh Quotient correction to LAMBDA. RQCORR = MINGMA*TMP

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

FURTHER DETAILS¶

Based on contributions by
Beresford Parlett, University of California, Berkeley, USA
Jim Demmel, University of California, Berkeley, USA
Inderjit Dhillon, University of Texas, Austin, USA
Osni Marques, LBNL/NERSC, USA
Christof Voemel, University of California, Berkeley, USA