zlar1v.f (3)

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.

subroutine zlar1v (integerN, integerB1, integerBN, double precisionLAMBDA, double precision, dimension( * )D, double precision, dimension( * )L, double precision, dimension( * )LD, double precision, dimension( * )LLD, double precisionPIVMIN, double precisionGAPTOL, complex*16, dimension( * )Z, logicalWANTNC, integerNEGCNT, double precisionZTZ, double precisionMINGMA, integerR, integer, dimension( * )ISUPPZ, double precisionNRMINV, double precisionRESID, double precisionRQCORR, 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:

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 zlar1v.f.

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