Is a Chebyshev Method Optimal for an Elliptic Region also Optimal for a Nearly Elliptic Region?

The asymptotic rate of convergence of an optimal Chebyshev semiiterative method for solving a real and nonsymmetric linear system x=Tx+c can be improved by the related (2,2)-step iterative methods under certain conditions. The condition for which a Chebyshev method asymptotically optimal for an elliptic region is also asymptotically optimal for a nearly elliptic region is presented. Thus a (2,2)-step method is asymptotically superior to the Chebyshev method asymtotically optimal for a nearly elliptic region under certain conditions. A numerical example illustrates our results