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Article

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
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Author to whom correspondence should be addressed.
Academic Editors: Alicia Cordero, Juan R. Torregrosa and Francisco I. Chicharro
Algorithms 2015, 8(3), 786-798; https://0-doi-org.brum.beds.ac.uk/10.3390/a8030786
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)
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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MDPI and ACS Style

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. https://0-doi-org.brum.beds.ac.uk/10.3390/a8030786

AMA Style

Wang X, Qin Y, Qian W, Zhang S, Fan X. A Family of Newton Type Iterative Methods for Solving Nonlinear Equations. Algorithms. 2015; 8(3):786-798. https://0-doi-org.brum.beds.ac.uk/10.3390/a8030786

Chicago/Turabian Style

Wang, Xiaofeng, Yuping Qin, Weiyi Qian, Sheng Zhang, and Xiaodong Fan. 2015. "A Family of Newton Type Iterative Methods for Solving Nonlinear Equations" Algorithms 8, no. 3: 786-798. https://0-doi-org.brum.beds.ac.uk/10.3390/a8030786

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