CLAR1V(3f)

NAME
CLAR1V - 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

SYNOPSIS

SUBROUTINE CLAR1V(
    N, B1, BN, LAMBDA, D, L, LD, LLD, PIVMIN, GAPTOL, Z, WANTNC, NEGCNT, ZTZ, MINGMA, R, ISUPPZ, NRMINV, RESID, RQCORR, WORK )

LOGICAL
    WANTNC

INTEGER
    B1, BN, N, NEGCNT, R

REAL
    GAPTOL, LAMBDA, MINGMA, NRMINV, PIVMIN, RESID, RQCORR, ZTZ

INTEGER
    ISUPPZ( * )

REAL
    D( * ), L( * ), LD( * ), LLD( * ), WORK( * )

COMPLEX
    Z( * )

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.

ARGUMENTS

N        (input) INTEGER
    The order of the matrix L D L^T.

B1       (input) INTEGER
    First index of the submatrix of L D L^T.

BN       (input) INTEGER
    Last index of the submatrix of L D L^T.

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

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

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

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

PIVMIN   (input) REAL
    The minimum pivot in the Sturm sequence.

GAPTOL   (input) REAL
    Tolerance that indicates when eigenvector entries are negligible w.r.t. their contribution to the residual.

Z        (input/output) 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   (input) LOGICAL
    Specifies whether NEGCNT has to be computed.

NEGCNT   (output) 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      (output) REAL
    The square of the 2-norm of Z.

MINGMA   (output) REAL
    The reciprocal of the largest (in magnitude) diagonal element of the inverse of L D L^T - sigma I.

R        (input/output) 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   (output) 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   (output) REAL
    NRMINV = 1/SQRT( ZTZ )

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

RQCORR   (output) REAL
    The Rayleigh Quotient correction to LAMBDA. RQCORR = MINGMA*TMP

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

FURTHER DETAILS
Based on contributions by
   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