New Newton's Method with Third-order Convergence for Solving Nonlinear Equations

Authors: Osama Yusuf Ababneh

Abstract:

For the last years, the variants of the Newton-s method with cubic convergence have become popular iterative methods to find approximate solutions to the roots of non-linear equations. These methods both enjoy cubic convergence at simple roots and do not require the evaluation of second order derivatives. In this paper, we present a new Newton-s method based on contra harmonic mean with cubically convergent. Numerical examples show that the new method can compete with the classical Newton's method.

Keywords: Third-order convergence, non-linear equations, root finding, iterative method.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1080316

References:

[1] S.Weerakoon, T.G.I. Fernando, A variant of Newtons method with accelerated third-order convergence, Appl. Math. Lett. 13 (2000) 87-93.
[2] M. Frontini, E. Sormani, Some variants of Newtons method with thirdorder convergence, Appl. Math. Comput. 140 (2003) 419-426.
[3] A.Y. ¨ozban, Some new variants of Newtons method, Appl. Math. Lett. 17 (2004) 677-682.
[4] M. Frontini, E. Sormani, Modified Newtons method with third-order convergence and multiple roots, J. Comput. Appl. Math. 156 (2003) 345-354.
[5] Changbum Chun, A two-parameter third-order family of methods for solving nonlinear equations, Applied Mathematics and Computation 189 (2007) 1822-1827.
[6] Kou Jishenga, LiYitianb, Wang Xiuhuac, Third-order modification of Newtons method,Journal of Computational and Applied Mathematics 205 (2007) 1 5.
[7] Mamta, V. Kanwar , V.K. Kukreja, Sukhjit Singh, On some third-order iterative methods for solving nonlinear equations,Applied Mathematics and Computation 171 (2005) 272-280.
[8] A.M. Ostrowski, Solution of Equations in Euclidean and Banach Space, third ed., Academic Press, NewYork, 1973.