A New Perspective on the Convergence of Mean-Based Methods for Nonlinear Equations

Many problems in science, engineering, and economics require solving of nonlinear equations, often arising from attempts to model natural systems and predict their behavior. In this context, iterative methods provide an effective approach to approximate the roots of nonlinear functions. This work introduces five new parametric families of multipoint iterative methods specifically designed for solving nonlinear equations. Each family is built upon a two-step scheme: the first step applies the classical Newton method, while the second incorporates a convex mean, a weight function, and a frozen derivative (i.e., the same derivative from the previous step). The careful design of the weight function was essential to ensure fourth-order convergence while allowing arbitrary parameter values. The proposed methods are theoretically analyzed and dynamically characterized using tools such as stability surfaces, parameter planes, and dynamical planes on the Riemann sphere. These analyses reveal regions of stability and divergence, helping identify suitable parameter values that guarantee convergence to the root. Moreover, a general result proves that all the proposed optimal parametric families of iterative methods are topologically equivalent, under conjugation. Numerical experiments confirm the robustness and efficiency of the methods, often surpassing classical approaches in terms of convergence speed and accuracy. Overall, the results demonstrate that convex-mean-based parametric methods offer a flexible and stable framework for the reliable numerical solution of nonlinear equations.