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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 836901, 15 pages
http://dx.doi.org/10.1155/2012/836901
Research Article

A Family of Derivative-Free Methods with High Order of Convergence and Its Application to Nonsmooth Equations

Instituto de Matemática Multidisciplinar and Instituto de Matemática Pura y Aplicada, Universitat Politècnica de València, 46022 Valencia, Spain

Received 4 January 2012; Accepted 14 March 2012

Academic Editor: Muhammad Aslam Noor

Copyright © 2012 Alicia Cordero et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A family of derivative-free methods of seventh-order convergence for solving nonlinear equations is suggested. In the proposed methods, several linear combinations of divided differences are used in order to get a good estimation of the derivative of the given function at the different steps of the iteration. The efficiency indices of the members of this family are equal to 1.6266. Also, numerical examples are used to show the performance of the presented methods, on smooth and nonsmooth equations, and to compare with other derivative-free methods, including some optimal fourth-order ones, in the sense of Kung-Traub’s conjecture.