SRC/zlar1v.f(3)            Library Functions Manual            SRC/zlar1v.f(3)

NAME
       SRC/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 -
           <lambda>I.

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 -
       <lambda>I.

       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.

       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 zlar1v.f.

Author
       Generated automatically by Doxygen for LAPACK from the source code.

LAPACK                          Version 3.12.0                 SRC/zlar1v.f(3)