CLAR1V(1) LAPACK auxiliary routine (version 3.2) CLAR1V(1)NAME
CLAR1V - computes the (scaled) r-th column of the inverse of the sumb‐
matrix 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 sumbma‐
trix 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 accu‐
rate eigenvector. Usually, r corresponds to the index where the eigen‐
vector 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 negligi‐
ble 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
LAPACK auxiliary routine (versioNovember 2008 CLAR1V(1)
