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Mathematics 2016, 4(2), 22; doi:10.3390/math4020022

Higher Order Methods for Nonlinear Equations and Their Basins of Attraction

Department of Mathematics, Pondicherry Engineering College, Pondicherry 605014, India
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Author to whom correspondence should be addressed.
Academic Editor: Michel Chipot
Received: 30 December 2015 / Revised: 18 March 2016 / Accepted: 22 March 2016 / Published: 1 April 2016
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Abstract

In this paper, we have presented a family of fourth order iterative methods, which uses weight functions. This new family requires three function evaluations to get fourth order accuracy. By the Kung–Traub hypothesis this family of methods is optimal and has an efficiency index of 1.587. Furthermore, we have extended one of the methods to sixth and twelfth order methods whose efficiency indices are 1.565 and 1.644, respectively. Some numerical examples are tested to demonstrate the performance of the proposed methods, which verifies the theoretical results. Further, we discuss the extraneous fixed points and basins of attraction for a few existing methods, such as Newton’s method and the proposed family of fourth order methods. An application problem arising from Planck’s radiation law has been verified using our methods. View Full-Text
Keywords: nonlinear equation; iterative methods; basins of attraction; extraneous fixed points; efficiency index nonlinear equation; iterative methods; basins of attraction; extraneous fixed points; 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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Madhu, K.; Jayaraman, J. Higher Order Methods for Nonlinear Equations and Their Basins of Attraction. Mathematics 2016, 4, 22.

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