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.

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)

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

December 2016

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