A Family of Combined Iterative Methods for Solving Nonlinear Equations

Shuping Chen^1, and Youhua Qian²

¹College of Mathematics, Xiamen University of Technology, xiamen 361024, P. R. China

²College of Mathematics, Physics and Information Engineering, Zhejiang Normal University, Jinhua, 321004, P. R. China

Cite this paper

Shuping Chen and Youhua Qian. A Family of Combined Iterative Methods for Solving Nonlinear Equations. American Journal of Applied Mathematics and Statistics. 2017; 5(1):22-32. doi: 10.12691/AJAMS-5-1-5

Abstract

In this article we construct some higher-order modifications of Newton’s method for solving nonlinear equations, which is based on the undetermined coefficients. This construction can be applied to any iteration formula. It can be found that per iteration the resulting methods add only one additional function evaluation, their order of convergence can be increased by two or three units. Higher order convergence of our methods is proved and corresponding asymptotic error constants are expressed. Numerical examples, obtained using Matlab with high precision arithmetic, are shown to demonstrate the convergence and efficiency of the combined iterative methods. It is found that the combined iterative methods produce very good results on tested examples, compared to the results produced by the existing higher order schemes in the related literature.

Keywords

Newton’s method, combined iterative methods, nonlinear equations, order of convergence, computational efficiency

Copyright

This work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/

References

[1]	Zhou, X.J., “A class of Newton’s methods with third-order convergence,” Applied Mathematics Letters, 20 (9). 1026-1030. Sep.2007.
[2]	Cordero, A., Jordán, C. and Torregrosa, J.R.,”One-point Newton-type iterative methods: A unified point of view,” Journal of Computational and Applied Mathematics, 275. 366-374. Feb. 2015.
[3]	Neta, B. and Scott, M., “On a family of Halley-like methods to find simple roots of nonlinear equations,” Applied Mathematics and Computation, 219 (15). 7940-7944. Mar. 2013.
[4]	Soleymani, F., Khattri, S.K. and Vanani, S.K., “Two new classes of optimal Jarratt-type fourth-order methods,” Applied Mathematics Letters, 25 (5). 847-853. May. 2012.
[5]	Wang, X. and Liu, L.P., “Two new families of sixth-order methods for solving non-linear equations,” Applied Mathematics and Computation, 213 (1). 73-78. Jul. 2009.
[6]	Kou, J.S., Li, Y.T. and Wang, X.H., “Some modifications of Newton’s method with fifth-order convergence,” Journal of Computational and Applied Mathematics, 209 (2). 146-152. Dec. 2007.
[7]	Chun, C. and Ham, Y., “Some sixth-order variants of Ostrowski root-finding methods,” Applied Mathematics and Computation, 193 (2). 389-394. Nov. 2007.
[8]	Parhi, S.K. and Gupta, D.K., “A sixth order method for nonlinear equations,” Applied Mathematics and Computation, 203 (1). 50-55. Sep. 2008.
[9]	Weerakoon, S. and Fernando, T.G.I., “A variant of Newton's method with accelerated third-order convergence,” Applied Mathematics Letters, 13 (8). 87-93. Nov. 2000.
[10]	Petkovic, S.M., Neta B., Petkovic L.D. and Džunic, J., “Multiple methods for solving nonlinear equations: A survey,” Applied Mathematics and Computation, 226. 635-660. Jan. 2014.
[11]	Steffensen, I.F., “Remarks on iteration,” Scandinavian Actuarial Journal, 1993(1). 64-72. Dec. 2011.
[12]	Khattri, S.K. and Argyros, I.K., “How to develop fourth and seventh order iterative methods,” Novi Sad Journal of Mathematics, 40 (2). 61-67. 2010.
[13]	Traub, J.F., Iterative methods for the solution of equations, Chelsea Publishing Company, New York, 1982.
[14]	Kanwar, V. and Tomar, S.K., “Modified families of Newton, Halley and Chebyshev methods,” Applied Mathematics and Computation, 192 (1). 20-26. Sep. 2007.
[15]	Gautschi, W., Numerical Analysis: An introduction, Birkhäuser, 1997.
[16]	Özban, A. Y., “Some new variants of Newton’s method,” Applied Mathematics Letters, 17 (6). 677-682. Jun. 2004.
[17]	Jarratt, P., “Some efficient fourth order multipoint methods for solving equations,” BIT Numerical Mathematics, 9(2). 119-124. Jun. 1969.
[18]	Khattri, S.K. and Abbasbandy, S., “Optimal fourth order family of iterative methods,” Matematicki Vesnik, 63 (1). 67-72. Jan. 2011.
[19]	Cordero, A. and Torregrosa, J.R., “Variants of Newton’s method using fifth-order quadrature formulas,” Applied Mathematics and Computation, 190 (1). 686-698. Jul. 2007.
[20]	King, R.F, “A family of fourth-order methods for nonlinear equations,” SIAM Journal on Numerical Analysis, 10 (5). 876-879. Oct. 1973.
[21]	Ostrowski, A.M., Solution of equations in Euclidean and Banach spaces, Academic, New York, 1960.
[22]	Grau-Sánchez, M., “Improving order and efficiency: Composition with a modified Newton’s method,” Journal of Computational and Applied Mathematics, 231 (2). 592-597. Sep. 2009.