Optimal Parameters for Linear Second-Degree Stationary Iterative Methods

Abstract

In this paper we show that the optimal parameters for linear second-degree stationary iterative methods applied to nonsymmetric linear systems can be found by solving the same minimax problem used to find optimal parameters for the Chebyshev iteration. In fact, the Chebyshev iteration is asymptotically equivalent to a linear seconnd-degree stationary method. The method of finding optimal parameters for the Chebyshev iteration given in Manteuffel [Numer. Math., 28 (1977), pp. 307–327] can be used to find optimal parameters for the sitationary method as well.