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Solving Nonsmooth Equations Using Derivative-Free Methods

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In this paper, a family of derivative-free methods of cubic convergence for solving nonlinear equations is suggested. In the proposed methods, several linear combinations of divided differences are used in order to get a good estimation of the derivative of the given function at the different steps of the iteration. The efficiency indices of the members of this family are equal to 1.442. The convergence and error analysis are given. Also, numerical examples are used to show the performance of the presented methods and to compare with other derivative-free methods. And, were applied these methods on smooth and nonsmooth equations.


The Bulletin of Society for Mathematical Services and Standards (Volume 3)
M.S.M. Bahgat and M.A. Hafiz, "Solving Nonsmooth Equations Using Derivative-Free Methods", The Bulletin of Society for Mathematical Services and Standards, Vol. 3, pp. 37-45, 2012
Online since:
Sep 2012

M. Frontini, E. Sormani, Modified Newtons method with third-order convergence and multiple roots, J. Comput. Appl. Math. 156(2003) 345-354.

H.H.H. Homeier, A modified Newton method for root finding with cubic convergence, J. Comput. Appl. Math. 157(2003) 227-230.

H.H.H. Homeier , On Newton-type methods with cubic convergence, J. Comput. Appl. Math. 176(2005) 425-432.

M. S. M. Bahgat, New Two-Step Iterative Methods for Solving Nonlinear Equations, J. Math. Research, 4, No. 3; June (2012) ISSN 1916-9795.

D. Kincaid, W. Cheney, Numerical Analysis, second ed., Brooks/Cole, Pacific Grove, CA, (1996).

P. Jain, Steffensen type methods for solving nonlinear equations, Applied Mathematics and Computation 194(2007) 527-533.

Q. Zheng, J. Wang, P. Zhao, L. Zhang, A Steffensen-like method and its higher-order variants, Applied Mathematics and Computation 214(2009)10-16.

M. Dehghan, M. Hajarian, Some derivative free quadratic and cubic convergence iterative formulas for solving nonlinear equations, Journal of Computational and Applied Mathematics 29(2010) 19- 30.

N. Yasmin, M. Junjua, Some Drivative Free Iterative Methods for Solving nonlinear Equations. Academic Research International, Vol. 2, No. 1, (2012), 75-82.

A. Cordero, J.L. Hueso, E. Martínez, J.R. Torregrosa, Steffensen type methods for solving nonlinear equations J. of Comput and App Math 236 (2012) 3058-3064.

J. M. Ortega, W. G. Rheinboldt, Iterative solutions of nonlinear equations in several variables, Press, New York- London, (1970).

E. Halley, Anew exact and easy method for finding the roots of equations generally and without any previous reduction, Phil. Roy. Soc. London 8(1964) 136-147.

K. Jisheng,L. Yitian, W. Xiuhua, A composite fourth-order iterative method for solving, Applied Mathematics and Computation (2006), doi: 10. 1016/j. amc. 2006. 05. 181.

R. Ezzati and F. Saleki, On the Construction of New Iterative Methods with Fourth-Order Convergence by Combining Previous Methods. International Mathematical Forum, Vol. 6, 2011, no. 27, 1319 - 1326.

F. Soleymani , V. Hosseinabadi, New Third- and Sixth-Order Derivative-Free Techniques for Nonlinear Equations. Journal of Mathematics Research 3 (2011) ISSN 1916-9809 (Online).

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