Multipoint efficient iterative methods and the dynamics of Ostrowski's method

Abstract

ABSTRACTIn this work, we introduce a modification into the technique, presented in A. Cordero, J.L. Hueso, E. Martínez, and J.R. Torregrosa [Increasing the convergence order of an iterative method for nonlinear systems, Appl. Math. Lett. 25 (2012), pp. 2369–2374], that increases by two units the convergence order of an iterative method. The main idea is to compose a given iterative method of order p with a modification of Newton's method that introduces just one evaluation of the function, obtaining a new method of order p+2, avoiding the need to compute more than one derivative, so we improve the efficiency index in the scalar case. This procedure can be repeated n times, with the same approximation to the derivative, obtaining new iterative methods of order p+2n. We perform different numerical tests that confirm the theoretical results. By applying this procedure to Newton's method one obtains the well known fourth order Ostrowski's method. We finally analyse its dynamical behaviour on second and third degree real polynomials.