slar1v()

subroutine slar1v	(	integer	n,
		integer	b1,
		integer	bn,
		real	lambda,
		real, dimension( * )	d,
		real, dimension( * )	l,
		real, dimension( * )	ld,
		real, dimension( * )	lld,
		real	pivmin,
		real	gaptol,
		real, dimension( * )	z,
		logical	wantnc,
		integer	negcnt,
		real	ztz,
		real	mingma,
		integer	r,
		integer, dimension( * )	isuppz,
		real	nrminv,
		real	resid,
		real	rqcorr,
		real, dimension( * )	work )

SLAR1V computes the (scaled) r-th column of the inverse of the submatrix in rows b1 through bn of the tridiagonal matrix LDLT - λI.

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Purpose:

!>
!> SLAR1V 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

[in]	N	!> N is INTEGER !> The order of the matrix L D L**T. !>
[in]	B1	!> B1 is INTEGER !> First index of the submatrix of L D L**T. !>
[in]	BN	!> BN is INTEGER !> Last index of the submatrix of L D L**T. !>
[in]	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. !>
[in]	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. !>
[in]	D	!> D is REAL array, dimension (N) !> The n diagonal elements of the diagonal matrix D. !>
[in]	LD	!> LD is REAL array, dimension (N-1) !> The n-1 elements L(i)*D(i). !>
[in]	LLD	!> LLD is REAL array, dimension (N-1) !> The n-1 elements L(i)*L(i)*D(i). !>
[in]	PIVMIN	!> PIVMIN is REAL !> The minimum pivot in the Sturm sequence. !>
[in]	GAPTOL	!> GAPTOL is REAL !> Tolerance that indicates when eigenvector entries are negligible !> w.r.t. their contribution to the residual. !>
[in,out]	Z	!> Z is REAL 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. !>
[in]	WANTNC	!> WANTNC is LOGICAL !> Specifies whether NEGCNT has to be computed. !>
[out]	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. !>
[out]	ZTZ	!> ZTZ is REAL !> The square of the 2-norm of Z. !>
[out]	MINGMA	!> MINGMA is REAL !> The reciprocal of the largest (in magnitude) diagonal !> element of the inverse of L D L**T - sigma I. !>
[in,out]	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. !>
[out]	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 ). !>
[out]	NRMINV	!> NRMINV is REAL !> NRMINV = 1/SQRT( ZTZ ) !>
[out]	RESID	!> RESID is REAL !> The residual of the FP vector. !> RESID = ABS( MINGMA )/SQRT( ZTZ ) !>
[out]	RQCORR	!> RQCORR is REAL !> The Rayleigh Quotient correction to LAMBDA. !> RQCORR = MINGMA*TMP !>
[out]	WORK	!> WORK is REAL 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