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Algorithms 2015, 8(3), 786-798; doi:10.3390/a8030786

A Family of Newton Type Iterative Methods for Solving Nonlinear Equations

1
School of Mathematics and Physics, Bohai University, Jinzhou 121013, China
2
College of Engineering, Bohai University, Jinzhou 121013, China
*
Author to whom correspondence should be addressed.
Academic Editors: Alicia Cordero, Juan R. Torregrosa and Francisco I. Chicharro
Received: 9 July 2015 / Revised: 14 September 2015 / Accepted: 15 September 2015 / Published: 22 September 2015
(This article belongs to the Special Issue Numerical Algorithms for Solving Nonlinear Equations and Systems)
View Full-Text   |   Download PDF [249 KB, uploaded 22 September 2015]   |  

Abstract

In this paper, a general family of n-point Newton type iterative methods for solving nonlinear equations is constructed by using direct Hermite interpolation. The order of convergence of the new n-point iterative methods without memory is 2n requiring the evaluations of n functions and one first-order derivative in per full iteration, which implies that this family is optimal according to Kung and Traub’s conjecture (1974). Its error equations and asymptotic convergence constants are obtained. The n-point iterative methods with memory are obtained by using a self-accelerating parameter, which achieve much faster convergence than the corresponding n-point methods without memory. The increase of convergence order is attained without any additional calculations so that the n-point Newton type iterative methods with memory possess a very high computational efficiency. Numerical examples are demonstrated to confirm theoretical results. View Full-Text
Keywords: multipoint iterative methods; nonlinear equations; R-order convergence; root-finding methods multipoint iterative methods; nonlinear equations; R-order convergence; root-finding methods
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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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Wang, X.; Qin, Y.; Qian, W.; Zhang, S.; Fan, X. A Family of Newton Type Iterative Methods for Solving Nonlinear Equations. Algorithms 2015, 8, 786-798.

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