An iterative scheme to obtain multiple solutions simultaneously

Abstract

In this manuscript, we propose an iterative step that, combined with any other method, allows us to obtain an iterative scheme for approximating the simple roots of a polynomial simultaneously. We show that adding this step, the order of convergence of the new scheme is tripled respect to the original one. With this idea, we also present an iterative method that obtains multiple solutions of any nonlinear equation simultaneously, without the need to know the multiplicity of the solutions. We conclude with several numerical experiments to confirm the behaviour of the proposed methods.