Comparison between the convergence rates of the Chebyshev method and the related (2,2)‐step methods

An optimal Chebyshev method for solving A x = b , where all the eigenvalues of the real and non‐symmetric matrix A are located in the open right half plane, is dependent on an optimal ellips∂Ω^* such that the spectrum of A is contrained in Ω^*, the closed interior of the ellipse. The relationship between the convergence rates of the Chebyshev method and the closely related (2,2)‐step iterative methods are studied. (2,2)‐step iterative methods are faster than an optimal Chebyshev method under certain conditions. A numerical example illustrates such an improvement of a (2,2)‐step iterative method. Copyright © 2000 John Wiley & Sons, Ltd.