Improved Higher Order Compositions for Nonlinear Equations

In the present study, two new compositions of convergence order six are presented for solving nonlinear equations. The first method is obtained from the third-order one given by Homeier using linear interpolation, and the second one is obtained from the third-order method given by Traub using divided differences. The first method requires three evaluations of the function and one evaluation of the first derivative, thereby enhancing the efficiency index. In the second method, the computation of a derivative is reduced by approximating it using divided differences. Various numerical experiments are performed which demonstrate the accuracy and efficacy of the proposed methods.