Local convergence analysis of a three-step iterative scheme with Lagrangeinterpolation and basin of attraction

Developing efficient and robust iterative methods for solving nonlinear equations is a critical task in various scientific and engineering fields. In this study, we presented a three-step eighth-order derivative-free iterative scheme based on Lagrange interpolation. The method involves four parameters and one variable weight function, and it is specifically designed to avoid the computation of higher-order derivatives. A detailed local convergence analysis is carried out under the assumption that the method relies only on the first-order derivative, satisfying a Lipschitz condition. This analysis establishes the convergence radius, provides error estimates, and confirms the existence and uniqueness of the solution. These results support the effective selection of a suitable initial guess based on the computed convergence region. Numerical experiments are conducted to validate the theoretical findings and demonstrate that the presented method provides a larger radius of convergence compared to existing methods of the respective domain. Furthermore, the dynamic behavior of the method is examined using the basin of attraction, which illustrates improved stability and reduced chaotic behavior when applied to transcendental equations.