The superiority of a new type (2,2)-step iterative method over the related Chebyshev method

A new type (2,2)-step iterative method related to an optimal Chebyshev method is developed for solving real and nonsymmetric linear systems of the form Ax=b. It is an extension of the (2,2)-step iterative method introduced in [Numer. Linear Algebra Appl. 7 (2000) 169]. The superiority of the new type (2,2)-step iterative method over the optimal Chebyshev method is derived in the case where the known (2,2)-step iterative method may not improve the asymptotic rate of convergence. Two numerical examples are given to illustrate the results.