New Efficient Optimal Derivative-Free Method for Solving Nonlinear Equations

In this paper, we suggest a new technique which uses Lagrange polynomials to get derivative-free iterative methods for solving nonlinear equations. With the use of the proposed technique and Steffens on-like methods, a new optimal fourth-order method is derived. By using three-degree Lagrange polynomials with other two-step methods which are efficient optimal methods, eighth-order methods can be achieved. Besides, we can get sixteenth-order methods if we use other three-step methods and higher-order degree Lagrange polynomials. The error equations and asymptotic convergence constants are obtained for the proposed methods. Some numerical examples are illustrated to verify the accuracy of the proposed computational scheme.