Local convergence analysis of a novel derivative-free method with and without memory for solving nonlinear systems

作者：Xiaofeng Wang, Ning Shang · 发表于：International Journal of Computer Mathematics · 年份：2025 · DOI：10.1080/00207160.2025.2464701 · 被引用次数：9 · 研究领域：Iterative Methods for Nonlinear Equations、Advanced Optimization Algorithms Research、Matrix Theory and Algorithms

First, a novel derivative-free single-parameter iterative method without memory for solving nonlinear systems is designed, and the convergence order of the method without memory is proved to be fourth order by using the higher Fréchet derivative. Second, the method with memory of high order convergence is obtained by means of acceleration parameters, which are represented by different forms of difference operators. When the acceleration parameter is replaced by Kurchatov's difference operator, the convergence order of the iterative method with memory is up to sixth-order convergence. Then, to expand the application range of the iterative method, the local convergence of the iterative method is proved by using the ω-continuity condition of the first Fréchet derivative. In this way, the influence of the existence of higher derivatives on the proof of convergence is avoided, the convergence condition is weakened and the application range of the iterative method is extended. In addition, the radius of the convergent sphere and the uniqueness of its solution are given. Numerical experiments verify the theoretical results.