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

Abstract

 Developing efficient and robust iterative methods for solving non-linear 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 in volves four parameters and one variable weight function, and it is specif ically 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 Lip schitz condition. This analysis establishes the convergence radius, pro vides 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 experimentsare 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 basin of attraction, which il lustrates improved stability and reduced chaotic behavior when applied to transcendental equations