Controlled over-relaxation method and the general extrapolation method

For solving systems of linear equations, relaxation methods have been used to accelerate the convergence of iterative methods; especially SOR methods in which, discussion of convergence and the choice of the relaxation parameter is restricted to the Gauss-Seidel splitting of various classes of matrices. This paper is concerned with a wider range of splittings of coefficient matrices. With a geometric and analytic approach, a two parameter COR method is developed. The choice of appropriate parameters are calculated so that the first parameter controls the spectral radius of the iteration matrix and guarantees convergence. The second , modifies the acceleration. This method can be useful as last resort in the numerical procedures that systems of linear equations have to be solved as an intermediate step (such as Newton's method or predictor-corrector methods for ODEs). For all splittings appropriate for COR, existence of the optimum parameter for general extrapolation method is shown and approximations for the parameters are derived.