The convergence rate of the Chebyshev semiiterative method under a perturbation of the foci of an elliptic domain. (English) Zbl 0997.65059

Authors’ abstract: 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 L. A. Hageman and D. M. Young [Applied iterative methods (1981; Zbl 0459.65014)], and by the first author [Linear Algebra Appl. 230, 47-60 (1995; Zbl 0839.65032)]. A numerical example is given to illustrate the theoretical results.


65F10 Iterative numerical methods for linear systems
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