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Mathematics 2019, 7(4), 322; https://doi.org/10.3390/math7040322

Optimal Fourth, Eighth and Sixteenth Order Methods by Using Divided Difference Techniques and Their Basins of Attraction and Its Application

1,† and 2,*,†
1
School of Computer Science and Engineering, Qujing Normal University, Qujing 655011, China
2
Department of Mathematics, Saveetha Engineering College, Chennai 602105, India
*
Author to whom correspondence should be addressed.
These authors contributed equally to this work.
Received: 26 February 2019 / Revised: 24 March 2019 / Accepted: 26 March 2019 / Published: 30 March 2019
(This article belongs to the Special Issue Iterative Methods for Solving Nonlinear Equations and Systems)
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Abstract

The principal objective of this work is to propose a fourth, eighth and sixteenth order scheme for solving a nonlinear equation. In terms of computational cost, per iteration, the fourth order method uses two evaluations of the function and one evaluation of the first derivative; the eighth order method uses three evaluations of the function and one evaluation of the first derivative; and sixteenth order method uses four evaluations of the function and one evaluation of the first derivative. So these all the methods have satisfied the Kung-Traub optimality conjecture. In addition, the theoretical convergence properties of our schemes are fully explored with the help of the main theorem that demonstrates the convergence order. The performance and effectiveness of our optimal iteration functions are compared with the existing competitors on some standard academic problems. The conjugacy maps of the presented method and other existing eighth order methods are discussed, and their basins of attraction are also given to demonstrate their dynamical behavior in the complex plane. We apply the new scheme to find the optimal launch angle in a projectile motion problem and Planck’s radiation law problem as an application. View Full-Text
Keywords: non-linear equation; basins of attraction; optimal order; higher order method; computational order of convergence non-linear equation; basins of attraction; optimal order; higher order method; computational order of convergence
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Tao, Y.; Madhu, K. Optimal Fourth, Eighth and Sixteenth Order Methods by Using Divided Difference Techniques and Their Basins of Attraction and Its Application. Mathematics 2019, 7, 322.

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