Family of Optimal Eighth-Order of Convergence for Solving Nonlinear Equations

  • Ibrahim Ahmed Al-Subaihi Department of Mathematics, Faculty of Science, Taibah University, Saudi Arabia
  • A. A. Al-Harbi Department of Mathematics, Faculty of Science, Taibah University, Saudi Arabia
Keywords: Convergence order, Efficiency index, Iterative methods, Nonlinear equations, Optimal eighth-order.

Abstract

In this paper, a new family of optimal eighth-order iterative methods are presented. The new family is developed by combining Traub-Ostrowskis fourth-order method adding Newtons method as a third step and using the forward divided difference and three real-valued functions in the third step to reduce the number of function evaluations. We employed several numerical comparisons to demonstrate the performance of the proposed method.

Downloads

Download data is not yet available.

References

I. A. Al-subaihi, A. J. Alqarni. Higher-Order Iterative Methods for Solving Nonlinear Equations, Life Science Journal, 11, 12, pp. 85-91, 2014.

W. Bi, Q. Wu, H. Ren, A new family of eighth-order iterative methods for solving nonlinear equations, Appl. Math.214, pp.236–245, 2009.

W. Bi, H. Ren, and Q.Wu, Three-step iterative methods with eighth-order convergence for solving nonlinear equations, J. Comput. Appl. Math., 225, pp.105-112, 2009.

C. Chun, B. Neta, J. Kozdon, and M. Scott, “Choosing weight functions in iterative methods for simple roots,” Applied Mathematics and Computation, 227, pp. 788–800, 2014.

Y. H. Geum and Y. I. Kim, A multi-parameter family of three-step eighth-order iterative methods locating a simple root, Appl. Math. Comput., 215, pp.3375-3382, 2010.

J. Kou, X. Wang, and Y. Li, Some eighth-order root-finding three-step methods, Commun. Nonlinear Sci. Numer. Simul, 15, pp. 536-544, 2010.

H. T. Kung and J. F. Traub, “Optimal order of one-point and multipoint iteration,” Journal of the Association for Computing Machinery, 21, pp. 643–651, 1974.

L. Liu, X. Wang, Eighth-order methods with high efficiency index for solving nonlinear equations, Applied Mathematics and Computation, 215, pp.3449-3454, 2010.

A.M. Ostrowski, Solution of Equations in Euclidean and Banach Spaces, Academic Press, New York, 1960.

M. S. Petkovic, B. Neta, L. D. Petkovic, and J. Dzunic, Multipoint Methods for Solving Nonlinear Equations, Elsevier, 2012.

Miodrag S. Petkovic, Ljiljana D. Petkovic, Families Of Optimal Multipoint Method For Solving Nonlinear Equations: A SURVEY, Appl. Anal. Discrete Math. 4, pp.1-22, 2010.

J.R.Sharma, R. Sharma, A new family of modified Ostrowski’s methods with accelerated eighth order convergence, Numerical Algorithms, 54, pp. 445-458, 2010.

A.Singh and J. P. Jaiswal, An Efficient Family of Optimal Eighth-Order Iterative Methods for Solving Nonlinear Equations and Its Dynamics, Journal of Mathematics, pp.1-14, 2014.

J. F. Traub, Iterative Methods for Solution of Equations, Chelsea Publishing, New York, NY, USA, 2nd edition, 1982.

S. Weerakoon, G. I. Fernando. A variant of Newton’s method with accelerated third-order convergence, Appl. Math. Lett., 17, 8, pp.87-93, 2000.

Published
2015-08-03
How to Cite
Al-Subaihi, I., & Al-Harbi, A. A. (2015). Family of Optimal Eighth-Order of Convergence for Solving Nonlinear Equations. Journal of Progressive Research in Mathematics, 4(4), 393-398. Retrieved from https://scitecresearch.com/journals/index.php/jprm/article/view/291
Section
Articles