Fourth-order iterative method without calculating the higher derivatives for nonlinear equation

Abstract

In this paper, we present firstly several one-step iterative methods including classical Newton’s method and Halley’s method for root-finding problem based on Thiele’s continued fraction. Secondly, by using approximants of the second derivative and the third derivative, we obtain an improved iterative method, which is not only two-step iterative method but also avoids calculating the higher derivatives of the function. Analysis of its convergence shows that the order of convergence of the modified iterative method is four for a simple root of the equation. Finally, to illustrate the efficiency and performance of the proposed method, we give some numerical experiments and comparison.