subroutine clar1v (N, B1, BN, LAMBDA, D, L, LD, LLD, PIVMIN, GAPTOL, Z, WANTNC, NEGCNT, ZTZ, MINGMA, R, ISUPPZ, NRMINV, RESID, RQCORR, WORK)
CLAR1V computes the (scaled) r-th column of the inverse of the submatrix in rows b1 through bn of the tridiagonal matrix LDLT - λI.
CLAR1V computes the (scaled) r-th column of the inverse of the submatrix in rows b1 through bn of the tridiagonal matrix LDLT - λI.
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.
Parameters:
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)
Author:
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
Contributors:
Definition at line 232 of file clar1v.f.
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