A Condition for the Superiority of the (2,2)-step Methods Over the Related Chebyshev Method

The (2,2)-step iterative method related to the optimal Chebyshev method for solving a large linear system Ax=bAx=b for a nonsymmetric matrix AA is studied closely. A condition guaranteeing the superiority of this (2,2)-step method over the Chebyshev method is derived. Interestingly, this condition involves the golden ratio. A numerical example is given to illustrate the theoretic results