clar1v-f(3)

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 (integerN, integerB1, integerBN, realLAMBDA, real, dimension( * )D, real, dimension( * )L, real, dimension( * )LD, real, dimension( * )LLD, realPIVMIN, realGAPTOL, complex, dimension( * )Z, logicalWANTNC, integerNEGCNT, realZTZ, realMINGMA, integerR, integer, dimension( * )ISUPPZ, realNRMINV, realRESID, realRQCORR, 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 is INTEGER
 The order of the matrix L D L**T.

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

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

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

 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 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 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.

Date:

September 2012

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 229 of file clar1v.f.

Author

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