Table of Contents
ISRN Applied Mathematics
Volume 2011 (2011), Article ID 693787, 12 pages
http://dx.doi.org/10.5402/2011/693787
Research Article

Eighth-Order Iterative Methods without Derivatives for Solving Nonlinear Equations

Padé Research Centre, 39 Deanswood Hill, West Yorkshire, Leeds LS17 5JS, UK

Received 29 March 2011; Accepted 17 May 2011

Academic Editor: A.-V. Phan

Copyright © 2011 R. Thukral. 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 new family of eighth-order derivative-free methods for solving nonlinear equations is presented. It is proved that these methods have the convergence order of eight. These new methods are derivative-free and only use four evaluations of the function per iteration. In fact, we have obtained the optimal order of convergence which supports the Kung and Traub conjecture. Kung and Traub conjectured that the multipoint iteration methods, without memory based on 𝑛 evaluations, could achieve optimal convergence order 2 𝑛 βˆ’ 1 . Thus, we present new derivative-free methods which agree with Kung and Traub conjecture for 𝑛 = 4 . Numerical comparisons are made to demonstrate the performance of the methods presented.