Expansion of Convergence Domain of a Parameter-Based Iteration Scheme for Equations in Banach Spaces

The performance of iterative schemes used to solve nonlinear operator equations is strongly influenced by the initial guess. Therefore, it is essential to accurately determine convergence radii and develop theoretical strategies to broaden the region where convergence is guaranteed in order to enhance the reliability and efficiency of these methods. A crucial tool for this purpose is local convergence analysis, which investigates behavior near the true solution to establish convergence criteria. This work is dedicated to extending the convergence region of a parameter-based iteration scheme of the fifth-order. We carry out a comprehensive local convergence study within the framework of Banach spaces and derive precise formulas for the convergence radius, error estimates, and convergence zones associated with the method. A notable advantage of our approach is that it relies solely on the first derivative and avoids the need for additional conditions, making it easier to apply and significantly expanding the convergence region relative to earlier approaches. The theoretical contributions are further validated through a series of numerical experiments applied to diverse classes of nonlinear equations. Furthermore, the examination of the basins of attraction and their symmetry provides a deeper understanding of the method’s dynamic characteristics, robustness, and effectiveness in tackling complex-valued polynomial equations.