On the local convergence and stability analysis of a two-step iterative method for solving non-linear equations

作者：Maira Khalid, Saima Akram, Muhammad Ibrahim · 发表于：Kuwait Journal of Science · 年份：2025 · DOI：10.1016/j.kjs.2025.100475 · 被引用次数：1 · 研究领域：Iterative Methods for Nonlinear Equations、Matrix Theory and Algorithms、Advanced Optimization Algorithms Research

Finding efficient and accurate solutions to non-linear equations is a critical challenge in many scientific and engineering fields. Iterative methods often require a well-chosen initial point to ensure convergence, but determining this point can be difficult. In this manuscript, we develop a two-step fourth-order iterative method that addresses this issue by calculating the radius of convergence, which provides important insights into selecting an appropriate initial point. The proposed method requires three function evaluations and one derivative evaluation per step. We present general assumptions for the local convergence analysis using a first-order derivative. The local convergence of the iterative method is validated under the conditions of a first-order derivative satisfying the Lipschitz conditions. Unlike previous studies that relied on higher-order derivatives (up to fourth order or beyond) for convergence analysis, our approach avoids such computations, making it more widely applicable. The analysis focuses on determining the convergence radius, estimating approximation errors, and establishing sufficient conditions for solution uniqueness. To validate the theoretical results, we test the method on various non-linear functions, demonstrating that the proposed method achieves a larger radius compared to existing methods. Additionally, we investigate the stability of the two-step fourth-order iterative method using a second-degree polynomial and obtain the fixed and c...