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The Scientific World Journal
Volume 2014, Article ID 410410, 18 pages
http://dx.doi.org/10.1155/2014/410410
Research Article

Optimal Sixteenth Order Convergent Method Based on Quasi-Hermite Interpolation for Computing Roots

1Centre for Advanced Studies in Pure and Applied Mathematics (CASPAM), Bahauddin Zakariya University, Multan 60800, Pakistan
2Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
3National University of Sciences and Technology (NUST), Islamabad 44000, Pakistan

Received 21 May 2014; Accepted 3 July 2014; Published 12 August 2014

Academic Editor: M. Mursaleen

Copyright © 2014 Fiza Zafar 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

We have given a four-step, multipoint iterative method without memory for solving nonlinear equations. The method is constructed by using quasi-Hermite interpolation and has order of convergence sixteen. As this method requires four function evaluations and one derivative evaluation at each step, it is optimal in the sense of the Kung and Traub conjecture. The comparisons are given with some other newly developed sixteenth-order methods. Interval Newton’s method is also used for finding the enough accurate initial approximations. Some figures show the enclosure of finitely many zeroes of nonlinear equations in an interval. Basins of attractions show the effectiveness of the method.