The convergence rate of the Chebyshev sim under a perturbation of a complex line-segment spectrum

Abstract

The Chebyshev semiiterative method ( chsim) is probably the best known and most often used method for the iterative solution of linear system x = Tx + c, where the spectrum of T is located in a complex line segment [α, β] excluding 1. The asymptotic convergence factor (ACF) of the chsim, under a perturbation of [α, β], is considered. Several formulae for the approximation to the ACFs, up to the second order of a perturbation, are derived. This generalizes the results about the sensitivity of the asymptotic rate of convergence to the estimated eigenvalues by Hageman and Young in the case that both α and β are real. Two numerical examples are given to illustrate the theoretical results.