Variants of Chebyshev’s Method with Eighth-Order Convergence for Solving Nonlinear Equations
Applied and Computational Mathematics
Volume 5, Issue 6, December 2016, Pages: 247-251
Received: Oct. 26, 2016; Accepted: Jan. 7, 2017; Published: Feb. 3, 2017
Views 2319      Downloads 134
Authors
Muhamad Nizam Muhaijir, Department of Mathematics, University of Riau, Pekanbaru, Indonesia
M. Imran, Department of Mathematics, University of Riau, Pekanbaru, Indonesia
Moh Danil Hendry Gamal, Department of Mathematics, University of Riau, Pekanbaru, Indonesia
Article Tools
Follow on us
Abstract
This paper develops the variants of Chebyshev’s method by applying Lagrange interpolation and finite difference to eliminate the second derivative appearing in the Chebyshev’s method. The results of this research show that the modified eight-order method has the efficiency index 1.5157. Numerical simulations show that the effectiveness and performance of the new method in solving nonlinear equations are encouraging.
Keywords
Chebyshev’s Method, Finite Differences, Lagrange Interpolation, Nonlinear Equations, Order of Convergence
To cite this article
Muhamad Nizam Muhaijir, M. Imran, Moh Danil Hendry Gamal, Variants of Chebyshev’s Method with Eighth-Order Convergence for Solving Nonlinear Equations, Applied and Computational Mathematics. Vol. 5, No. 6, 2016, pp. 247-251. doi: 10.11648/j.acm.20160506.13
Copyright
Copyright © 2016 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
References
[1]
S. Amat, S. Busquier, J. M. Guiterres and M. A. Hernandes, On the global convergence of Chebyshev's iterative method, Journal of Computational and Applied Mathematics, 220 (2008), 17-21.
[2]
R. Behl and V. Kanwar, Variant of Chebyshev's methods with optimal order convergence, Tamsui Oxford Journal of Information and Mathematical Sciences, 29 (2013), 39-53.
[3]
V. Candela and A. Marquina, Recurance relations for rational cubic method, Computing, 45 (1990), 355-367.
[4]
H. Esmaeili and A. N. Rezaei, A uniparametric family of modifications for Chebyshev's method, Lecturas Matematicas, 33 (2012), 95-106.
[5]
J. Jayakumar and P. Jayasilan, Second derivative free modification with parameter for Chebyshev's method, International Journal of Computational Engineering Research, 03 (2013), 38-42.
[6]
V. Kanwar, A family of third order multipoint methods for solving nonlinear equations, Applied Mathematics and Computation, 176 (2006), 409-413.
[7]
J. Kou, L. Yitian and W. Xiuhua, A uniparametric Chebyshev-type method free from second derivatives, Applied Mathematics and Computation. 179 (2006), 296-300.
[8]
J. Kou, L. Yitian and W. Xiuhua, Third-order modification of Newton's method, Journal of Computational and Applied Mathematics, 205 (2007), 1-5.
[9]
M. A. Noor, W. A. Khan and A. Husain, A new modifed Halley method without second derivatives for nonlinear equation, Applied Mathematics and Computation, 189 (2007), 1268-1273.
[10]
A. Y. Ozban, Some new variants of Newton's method, Applied Mathematics Letters 13 (2004), 87-93.
[11]
S. Weerakoon and T. G. I. Fernando, A variant of Newton's methods with accelarated third order convergence, Applied Mathematics Letters, 1 (2000), 87-93.
ADDRESS
Science Publishing Group
1 Rockefeller Plaza,
10th and 11th Floors,
New York, NY 10020
U.S.A.
Tel: (001)347-983-5186