Generating optimal derivative free iterative methods for nonlinear equations by using polynomial interpolation

In this work we show a general procedure to obtain optimal derivative free iterative methods [4] for nonlinear equations f(x) = 0, applying polynomial interpolation to a generic optimal derivative free iterative method of lower order. Let us consider an optimal method of order q = 2n−1, v = φn(x), that uses n functional evaluations. Performing a Newton step w = v − f(v) f ′(v) one obtains a method of order 2, that is not optimal because it introduces two new functional evaluations. Instead, we approximate the derivative by using a polynomial of degree n that interpolates n + 1 already known functional values and keeps the order 2. We have applied this idea to Steffensen’s method, [5], obtaining a family of optimal derivative free iterative methods of arbitrary high order. We provide different numerical tests, that confirm the theoretical results and compare the new family with other well known family of similar characteristics. This research was supported by Ministerio de Ciencia e Innovación MTM2011-28636C02-02 and by Vicerrectorado de Investigación. Universitat Politècnica de València PAID06-2010-2285 ∗Corresponding author Email addresses: [email protected] (Alicia Cordero), [email protected] (José L. Hueso ), [email protected] (Eulalia Mart́ınez), [email protected] (Juan R. Torregrosa) Preprint submitted to Mathematical and Computer Modelling December 27, 2011