A Variant of Newton Method with Eighth-Order Convergence for Solving Nonlinear Equations

Abstract:

In this paper, we present a variant of Newton method with order of convergence eight for solving nonlinear equations. The method is free from second derivatives. It requires three evaluations of the functions and two evaluations of derivatives in each step. Therefore the efficiency index of the presented method is 1.5157 which is better than that of classical Newton’s method 1.4142. Some numerical experiments illustrate that the proposed method is more efficient and performs better than classical Newton's method.

• Periodical: Advanced Materials Research (Volumes 490-495)
• Pages: 51-55
• DOI: https://doi.org/10.4028/www.scientific.net/AMR.490-495.51
• Online since: March 2012
• Authors: Liang Fang
• Keywords: Iterative Method, Newton Method, Nonlinear Equations, Order of Convergence

References

[1] J.F. Traub: Iterative Methods for Solution of Equations, Prentice-Hall, Englewood Clis, NJ(1964).

[2] A.N. Muhammad and I.N. Khalida: Modified iterative methods with cubic convergence for solving nonlinear equations, Appl. Math. Comput. 184 (2007), pp.322-325.

[3] L. Fang, G. He, Y. Hu and L. Sun: Some modified Newton-type methods with order of convergence varied from two to six under weak conditions, IEEE CSO 2009, pp.637-640.

DOI: 10.1109/cso.2009.284

[4] M. Grau and J.L. Diaz-Barrero: An improvement to Ostrowski root-finding method, Appl. Math. Comput. 173 (2006), pp.450-456.

DOI: 10.1016/j.amc.2005.04.043

[5] C. Chun: Some second-derivative-free variants of super-Halley method with fourth-order convergence, Appl. Math. Comput. 192 (2007), pp.537-541.

DOI: 10.1016/j.amc.2007.05.003

[6] J. Li: Adomian's decomposition method and homotopy perturbation method in solving nonlinear equations, Journal of Computational and Applied Mathematics. 228 (2009), pp.168-173.

DOI: 10.1016/j.cam.2008.09.007

[7] X. Wang, J. Kou and C. Gu: A new modified secant-like method for solving nonlinear equations, Computers & Mathematics with Applications, 60 (2010), pp.1633-1638.

DOI: 10.1016/j.camwa.2010.06.045

[8] L. Chen and Y. Ma: A new modified King–Werner method for solving nonlinear equations, Computers & Mathematics with Applications, 62 (2011), pp.3700-3705.

DOI: 10.1016/j.camwa.2011.09.012

[9] L. Yang, Y. Chen, X. Tong and C. Deng: A new smoothing Newton method for solving constrained nonlinear equations, Appl. Math. Comput. 217 (2011), pp.9855-9863.

DOI: 10.1016/j.amc.2011.04.045

[10] F.A. Potra, Potra-Pták: Nondiscrete induction and iterative processes, Research Notes in Mathematics, Vol. 103, Pitman, Boston, (1984).

[11] J. Kou: The improvements of modified Newton's method, Appl. Math. Comput. 189 (2007), pp.602-609.

[12] L. Fang, G. He and Z. Hu: A cubically convergent Newton-type method under weak conditions, J. Comput. Appl. Math. 220 (2008), pp.409-412.

DOI: 10.1016/j.cam.2007.08.013

[13] M.A. Noor and K.I. Noor: Fifth-order iterative methods for solving nonlinear equations, Appl. Math. Comput. 188 (2007), pp.406-410.

DOI: 10.1016/j.amc.2006.10.007

[14] L. Fang and G. He: Some modifications of Newton's method with higher-order convergence for solving nonlinear equations, J. Comput. Appl. Math. 228 (2009), pp.296-303.

DOI: 10.1016/j.cam.2008.09.023

[15] W. Gautschi: Numerical Analysis: An Introduction, Birkhauser Boston Inc. Cambridge, MA, USA , (1997).

[16] L. Fang, L. Sun and G. He: An efficient Newton-type method with fifth-order for solving nonlinear equations, Comput. Appl. Math. 27 (2008), pp.269-274.

DOI: 10.1590/s1807-03022008000300003

[17] A.N. Muhammad and A. Faizan: Fourth-order convergent iterative method for nonlinear equation, Appl. Math. Comput. 182 (2006), pp.1149-1153.

[18] C. Chun: A simply constructed third-order modifications of Newton's method, J. Comput. Appl. Math. 219(2008), pp.81-89.