One Step Closer to an Optimal Two-Parameter SOR Method: A Geometric Approach

Abstract

The well-known SOR method is obtained from a one-part splitting of the system matrix $A$, using one parameter $\omega$ for the diagonal. A strong interest in using more than one parameter for the SOR method to improve the convergence has been developed. Sisler, Niethammer, and Hadiidimos worked on the two-parameter method in the seventies. This author has generalized Sisler's method and introduced a range for the second parameter, providing a faster two-parameter method compared to the SOR method. In this paper, we go one step further by removing the hypothesis that requires the eigenvalues of the Jacobi iteration matrix to be real. The result is an optimal value for the second parameter when the eigenvalues of the SOR method are in a certain well-defined region.