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Article

Higher Order Methods for Nonlinear Equations and Their Basins of Attraction

Department of Mathematics, Pondicherry Engineering College, Pondicherry 605014, India
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Academic Editor: Michel Chipot
Mathematics 2016, 4(2), 22; https://doi.org/10.3390/math4020022
Received: 30 December 2015 / Revised: 18 March 2016 / Accepted: 22 March 2016 / Published: 1 April 2016
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
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MDPI and ACS Style

Madhu, K.; Jayaraman, J. Higher Order Methods for Nonlinear Equations and Their Basins of Attraction. Mathematics 2016, 4, 22. https://doi.org/10.3390/math4020022

AMA Style

Madhu K, Jayaraman J. Higher Order Methods for Nonlinear Equations and Their Basins of Attraction. Mathematics. 2016; 4(2):22. https://doi.org/10.3390/math4020022

Chicago/Turabian Style

Madhu, Kalyanasundaram, and Jayakumar Jayaraman. 2016. "Higher Order Methods for Nonlinear Equations and Their Basins of Attraction" Mathematics 4, no. 2: 22. https://doi.org/10.3390/math4020022

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