Iterative solvers of Ax=b derived from Ode's numerical integration methods

Although iterative methods for solving linear systems has been the subject of study for a long time, the acceleration of such methods is still object of interest, research focusing in improvements of already known methods as well as on new, faster ones. In this sense we can cite among several other authors, for example, the works of Martins [8], Hadjidimos [9] and Evans [7] The purpose of this paper is twofold. First we show that the numerical integration methods for Ordinary Differential Equations, obtained by Taylor expansions, result in a B-extrapolation method [3] for iteratively solving linear algebraic systems. Second, we compare the best rates of convergence of the algorithms developed here, with the best rate of convergence of the Jacobi Over-Relaxation method, (JOR), proving that depending on the choice of the step of integration and the behavior of the spectrum of the matrix A of the original system Ax = b the third order methods derived from the Taylor expansions can be better than the JOR. The procedure used here with the splitting A = I - J can also be easely applied to other splittings, resulting in a comparison of the convergence of the present method with the SOR or iterations methods or any other linear iterative methods of degree one.