International Journal of Mathematics and Mathematical Sciences
Volume 2012 (2012), Article ID 493456, 12 pages
Research Article

New Eighth-Order Derivative-Free Methods for Solving Nonlinear Equations

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

Received 28 March 2012; Revised 16 August 2012; Accepted 30 August 2012

Academic Editor: Marianna Shubov

Copyright © 2012 Rajinder 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.


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 of . Thus, we present new derivative-free methods which agree with Kung and Traub conjecture for . Numerical comparisons are made to demonstrate the performance of the methods presented.