Simultaneous Root Approximation: A High-Convergence Iterative Approach

This paper introduces a novel and innovative iterative methodology that not only transforms arbitrary iterative schemes into an efficient framework but also redefines the process of simultaneous root approximation for polynomials and nonlinear equations. The proposed methods are distinguished by their exceptional convergence orders, achieving up to $p+2$ for polynomial equations and $2p$ for nonlinear equations, where $p$ is the order of the base iterative scheme. In contrast to existing techniques, these methods incorporate advanced correction mechanisms, such as an arithmetic mean blending Newton's and Ehrlich-Aberth methods, to enhance stability and convergence performance. Comprehensive numerical experiments validate the robustness and efficiency of our approaches, with clear advantages in terms of convergence speed, computational cost, and error minimization. Moreover, we present a detailed analysis of convergence behavior, supported by graphical illustrations of residual errors, shedding new light on the dynamics of iterative methods. These findings not only establish the superiority of the proposed schemes but also open new avenues for applying iterative techniques to complex mathematical and engineering problems.