The convergence rate of the Chebyshev SIM under a perturbation of a complex line-segment spectrum (Q1904046)

The Chebyshev semiiterative method (CHSIM) is the most often used method for the iterative solution of the linear system \(x= Tx+ c\), where the spectrum of \(T\) is located in a complex line segment \([\alpha, \beta]\) excluding 1. In practice, the exact endpoints \(\alpha\) and \(\beta\) are often not available, only estimates \(\alpha_e\) and \(\beta_e\) for \(\alpha\) and \(\beta\) are known. So, studying the asymptotic convergence factor (ACF) of the CHSIM, under a perturbation of \([\alpha, \beta]\), is very important. \textit{L. A. Hageman} and \textit{D. M. Young} [Applied iterative methods (1981; Zbl 0459.65014)] studied this problem in the case that \([\alpha, \beta]\) is located on the \(x\)-axis. This paper extends their results. Eight different perturbations of the endpoints of \([\alpha, \beta]\) and their corresponding ACFs are considered. Several formulae for the approximation to the ACFs, up to the second order of the perturbation, are derived.