Design and Analysis of Derivative-Free Iterative Approaches to Solving Nonlinear Equations

In this manuscript, we propose a new method to solve nonlinear equations without the need for higher-order derivatives. In particular, we propose a fourth-order, derivative-free iterative scheme for approximating the roots of nonlinear functions. The convergence analysis of the proposed method is carried out in two steps: first, to show its derivative-free nature, and second, to include a proper weight function. For comparing the efficiency of the method proposed, we conduct comparative analyses with some popular methods for a group of nonlinear equations. The results indicate that our method is more efficient than other methods in all the cases. Numerical experiments also validate the efficiency and accuracy of the method, and show better performance in CPU time, residual error, reduction of error per iteration, and number of iterations to converge.