# clar1v.f - Man Page

SRC/clar1v.f

## Synopsis

### Functions/Subroutines

subroutine **clar1v** (n, b1, bn, lambda, d, l, ld, lld, pivmin, gaptol, z, wantnc, negcnt, ztz, mingma, r, isuppz, nrminv, resid, rqcorr, work)**CLAR1V** 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 clar1v (integer n, integer b1, integer bn, real lambda, real, dimension( * ) d, real, dimension( * ) l, real, dimension( * ) ld, real, dimension( * ) lld, real pivmin, real gaptol, complex, dimension( * ) z, logical wantnc, integer negcnt, real ztz, real mingma, integer r, integer, dimension( * ) isuppz, real nrminv, real resid, real rqcorr, real, dimension( * ) work)

**CLAR1V** computes the (scaled) r-th column of the inverse of the submatrix in rows b1 through bn of the tridiagonal matrix LDLT - λI.

**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.

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

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

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

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

*PIVMIN*PIVMIN is REAL The minimum pivot in the Sturm sequence.

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

*Z*Z is 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*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 REAL The square of the 2-norm of Z.

*MINGMA*MINGMA is REAL 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 REAL NRMINV = 1/SQRT( ZTZ )

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

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

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

**Author**Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**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 **227** of file **clar1v.f**.

## Author

Generated automatically by Doxygen for LAPACK from the source code.

## Referenced By

The man page clar1v(3) is an alias of clar1v.f(3).