# clar1v (l) - Linux Man Pages

## 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

## 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

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

## 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 byBeresford 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