A sixth order method for nonlinear equations

Abstract

In this paper, a sixth order method is developed by extending a third order method of Weerakoon and Fernando [S. Weerakoon, T.G.I. Fernando, A variant of Newton’s method with accelerated third-order convergence, Appl. Math. Lett. 13 (2000) 87–93] for finding the real roots of nonlinear equations in R . Starting with a suitably chosen x 0 , the method generates a sequence of iterates converging to the root. The convergence analysis is provided to establish its sixth order of convergence. In terms of computational cost, it requires evaluations of only two functions and two first derivatives per iteration. This implies that efficiency index of our method is 1.565. Our method is comparable with the methods of Neta [B. Neta, A sixth order family of methods for nonlinear equations, Intern. J. Computer Math. 7 (1979) 157–161] and that of Kou and Li [Jisheng Kou, Yitian Li, An improvement of Jarratt method, Appl. Math. Comput. 189 (2007) 1816–1821]. It does not require the evaluation of the second order derivative of the given function as required in the family of Chebyshev–Halley type methods [Jisheng Kou, Xiuhua Wang, Sixth-order variants of Chebyshev–Halley methods for solving non-linear equations, Appl. Math. Comput. 190 (2007) 1839–1843; Jisheng Kou, On Chebyshev–Halley methods with sixth-order convergence for solving non-linear equations, Appl. Math. Comput. 190 (2007) 126–131]. The efficacy of the method is tested on a number of numerical examples. It is observed that our method takes less number of iterations than Newton’s method and the method of Weerakoon and Fernando. On comparison with the other sixth order methods, it behaves either similarly or better for the examples considered.