Optimum first and second order extrapolations of Successive Overrelaxation type methods for certain classes of matrices

This paper deals with the iterative solution of the linear systemx=Bx+c when its Jacobi matrixB is weakly 2-cyclic consistently ordered and has a complex eigenvalue spectrum which lies on a straight-line segment. The optimization problem of the following three methods is considered and solved: i) The extrapolation of the optimum Successive Overrelaxation (SOR) ii) The second order extrapolation of a “good” SOR and iii) The second order extrapolation of the Gauss-Seidel method. In addition a variant of the second order methods considered, suitable for the solution of the system even ifB isnot necessarily weakly 2-cyclic consistently ordered, is proposed. Finally a reference to a theoretical comparison of the various optimum methods in the paper is made and their asymptotic convergence factors for selected eigenvalue spectra are illustrated in a Table in support of the theory developed.