ELA, Volume 9, pp. 55-66, May 2002, abstract.

The convergence rate of the chebyshev 
semiiterative method under a perturbation 
of the foci of an elliptic domain

Xiezhang Li and Fangjun Arroyo

The Chebyshev semiiterative method (CHSIM)
is a powerful method for finding the iterative 
solution of a nonsymmetric real linear system 
Ax=b if an ellipse excluding the origin well 
fits the spectrum of A. The asymptotic rate 
of convergence of the CHSIM for solving the above
system under a perturbation of the foci of the 
optimal ellipse is studied. Several formulae to 
approximate the asymptotic rates of convergence, 
up to the first order of a perturbation, are 
derived. These generalize the results about the 
sensitivity of the asymptotic rate of convergence 
to a perturbation of a real-line segment spectrum 
by Hageman and Young, and by the first author. 
A numerical example is given to illustrate 
the theoretical results.