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Algorithms 2015, 8(3), 552-561; doi:10.3390/a8030552

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

African Network for Policy Research & Actions for Sustainability (ANPRAS), Midlands, Curepipe 52501, Mauritius
Academic Editor: Alicia Cordero
Received: 11 May 2015 / Revised: 20 July 2015 / Accepted: 21 July 2015 / Published: 29 July 2015
(This article belongs to the Special Issue Numerical Algorithms for Solving Nonlinear Equations and Systems)
View Full-Text   |   Download PDF [203 KB, uploaded 29 July 2015]

Abstract

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. View Full-Text
Keywords: Non-linear equation; Multi-point iterative methods; Simpson’s rule; Efficiency Index Non-linear equation; Multi-point iterative methods; Simpson’s rule; Efficiency Index
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. (CC BY 4.0).

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Babajee, D.K.R. Some Improvements to a Third Order Variant of Newton’s Method from Simpson’s Rule. Algorithms 2015, 8, 552-561.

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