New modifications of Potra-Ptaks method with optimal fourth and eighth orders of convergence

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS

卷:234

Article

Martinez, Eulalia¹ 

[1] Univ Politecn Valencia, Inst Matemat Pura & Aplicada, Valencia 46022, Spain

关键词: Divided differences;    Linear interpolation;    Nonlinear equations;    Iterative methods;    Convergence order;    Efficiency index;   

DOI  :  10.1016/j.cam.2010.04.009

来源: Elsevier

【 摘 要 】

In this paper, we present two new iterative methods for solving nonlinear equations by using suitable Taylor and divided difference approximations. Both methods are obtained by modifying Potra-Ptaks method trying to get optimal order. We prove that the new methods reach orders of convergence four and eight with three and four functional evaluations, respectively. So, Kung and Traub's conjecture Kung and Traub (1974)[2], that establishes for an iterative method based on n evaluations an optimal order p = 2(n-1) is fulfilled, getting the highest efficiency indices for orders p = 4 and p = 8, which are 1.587 and 1.682. We also perform different numerical tests that confirm the theoretical results and allow us to compare these methods with Potra-Ptaks method from which they have been derived, and with other recently published eighth-order methods. (C) 2010 Elsevier B.V. All rights reserved.

【 授权许可】

Free