A condition for the superiority of the (2, 2)-step methods over the related Chebyshev method

The (2, 2)-step iterative methods related to an optimal Chebyshev method for solving a real and nonsymmetric linear system Ax = b are studied. A condition under which the asymptotic rate of convergence of the optimal Chebyshev method can be improved by a related (2, 2)-step method is derived. The condition depends not only on the location of the extreme eigenvalues of T but also on whether the ratio of the minor axis to the major axis of the optimal ellipse is greater than the golden ratio. Two numerical examples are given to illustrate our results.