# zlar1v.f man page

zlar1v.f

## Synopsis

### Functions/Subroutines

subroutine **zlar1v** (**N**, B1, BN, LAMBDA, D, L, LD, LLD, PIVMIN, GAPTOL, Z, WANTNC, NEGCNT, ZTZ, MINGMA, R, ISUPPZ, NRMINV, RESID, RQCORR, WORK)**ZLAR1V** computes the (scaled) r-th column of the inverse of the submatrix in rows b1 through bn of the tridiagonal matrix LDLT - λI.

## Function/Subroutine Documentation

### subroutine zlar1v (integer N, integer B1, integer BN, double precision LAMBDA, double precision, dimension( * ) D, double precision, dimension( * ) L, double precision, dimension( * ) LD, double precision, dimension( * ) LLD, double precision PIVMIN, double precision GAPTOL, complex*16, dimension( * ) Z, logical WANTNC, integer NEGCNT, double precision ZTZ, double precision MINGMA, integer R, integer, dimension( * ) ISUPPZ, double precision NRMINV, double precision RESID, double precision RQCORR, double precision, dimension( * ) WORK)

**ZLAR1V** computes the (scaled) r-th column of the inverse of the submatrix in rows b1 through bn of the tridiagonal matrix LDLT - λI.

**Purpose:**

ZLAR1V 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.

**Parameters:**-
*N*N is INTEGER The order of the matrix L D L**T.

*B1*B1 is INTEGER First index of the submatrix of L D L**T.

*BN*BN is INTEGER Last index of the submatrix of L D L**T.

*LAMBDA*LAMBDA is DOUBLE PRECISION The shift. In order to compute an accurate eigenvector, LAMBDA should be a good approximation to an eigenvalue of L D L**T.

*L*L is DOUBLE PRECISION array, dimension (N-1) The (n-1) subdiagonal elements of the unit bidiagonal matrix L, in elements 1 to N-1.

*D*D is DOUBLE PRECISION array, dimension (N) The n diagonal elements of the diagonal matrix D.

*LD*LD is DOUBLE PRECISION array, dimension (N-1) The n-1 elements L(i)*D(i).

*LLD*LLD is DOUBLE PRECISION array, dimension (N-1) The n-1 elements L(i)*L(i)*D(i).

*PIVMIN*PIVMIN is DOUBLE PRECISION The minimum pivot in the Sturm sequence.

*GAPTOL*GAPTOL is DOUBLE PRECISION Tolerance that indicates when eigenvector entries are negligible w.r.t. their contribution to the residual.

*Z*Z is COMPLEX*16 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*WANTNC is LOGICAL Specifies whether NEGCNT has to be computed.

*NEGCNT*NEGCNT is 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*ZTZ is DOUBLE PRECISION The square of the 2-norm of Z.

*MINGMA*MINGMA is DOUBLE PRECISION The reciprocal of the largest (in magnitude) diagonal element of the inverse of L D L**T - sigma I.

*R*R is 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*ISUPPZ is 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*NRMINV is DOUBLE PRECISION NRMINV = 1/SQRT( ZTZ )

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

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

*WORK*WORK is DOUBLE PRECISION array, dimension (4*N)

**Author:**-
Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date:**December 2016

**Contributors:**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

Definition at line 232 of file zlar1v.f.

## Author

Generated automatically by Doxygen for LAPACK from the source code.

## Referenced By

The man page zlar1v(3) is an alias of zlar1v.f(3).