Some Improvements to a Third Order Variant of Newton's Method from Simpson's Rule

Algorithms

Diyashvir Kreetee Rajiv Babajee¹ 

[1] id="af1-algorithms-08-00552">African Network for Policy Research & Actions for Sustainability (ANPRAS), Midlands, Curepipe 52501, Mauriti

关键词: Non-linear equation;    Multi-point iterative methods;    Simpson's rule;    Efficiency Index;   

DOI  :  10.3390/a8030552

来源: mdpi

【 摘 要 】

In this paper, we present three improvements to a three-point third order variant of Newton's method derived from the Simpson rule. The first one is a fifth order method using the same number of functional evaluations as the third order method, the second one is a four-point 10th order method and the last one is a five-point 20th order method. In terms of computational point of view, our methods require four evaluations (one function and three first derivatives) to get fifth order, five evaluations (two functions and three derivatives) to get 10th order and six evaluations (three functions and three derivatives) to get 20th order. Hence, these methods have efficiency indexes of 1.495, 1.585 and 1.648, respectively which are better than the efficiency index of 1.316 of the third order method. We test the methods through some numerical experiments which show that the 20th order method is very efficient.

【 授权许可】

CC BY   
© 2015 by the authors; licensee MDPI, Basel, Switzerland.