New modifications of Newton-type methods with eighthorder convergence for solving nonlinear equations

Abstract

The aims of this paper are, firstly, to define a new family of the Thukral and Petkovic type methods for finding zeros of nonlinear equations and secondly, to introduce new formulas for approximating the order of convergence of the iterative method. It is proved that these methods have the convergence order of eight requiring only four function evaluations per iteration. In fact, the optimal order of convergence which supports the Kung and Traub conjecture have been obtained. Kung and Traub conjectured that the multipoint iteration methods, without memory based on n evaluations, could achieve optimal convergence order 2^n-1.  Thus, new iterative methods which agree with the Kung and Traub conjecture for n = 4  have been presented. It is observed that our proposed methods are competitive with other similar robust methods and very effective in high precision computations.