Abstract

Root-finding techniques are essential in computational science, especially for addressing nonlinear models. This paper introduces an enhanced two-step optimal iterative method for solving nonlinear equations and systems of nonlinear equations. The proposed approach achieves optimal convergence in line with the Kung-Traub conjecture, utilizing only three function evaluations per iteration for a fourth-order optimal iterative method. The method is developed by integrating two well-established third-order techniques. This new method offers a notable advancement over current optimal iterative methods. Extensive testing on various polynomial functions has shown that it delivers precise and efficient results, outperforming existing algorithms in both speed and accuracy. Additionally, the convergence of these iterative methods is illustrated graphically.