Efficient optimal eighth order method for solving nonlinear equations

This article presents a new iterative scheme to find the roots of nonlinear equations. We use composition technique and weight function procedure to construct new three-step eighth order method. Several iterative methods have recently been proposed for finding the roots of a nonlinear equation that requires the evaluation of the second and higher-order derivatives of the function, which increases the processing cost. We have introduced a second and higher derivative-free method, which requires three evaluations of the function and one evaluation of its first derivative. In fact, the new eighth-order method has a maximum computational efficiency of 1.682 in the sense of Kung-Traub’s optimality conjecture. We have presented an extensive theoretical convergence analysis of new eighth order method using Taylor developments. Numerical tests are presented with several examples to confirm the applicability and to justify the rapid convergence of the current technique. Finally, we confirm on the basis of obtained results that new proposed method has better absolute residual errors as compared to the other existing standard methods of the same order.