Journal Article

Unified Convergence Criteria of Derivative-Free Iterative Methods for Solving Nonlinear Equations

Argyros, Ioannis K., Shakhno, Stepan, Yarmola, Halyna, Regmi, Samundra

2023

Overview

A local and semi-local convergence is developed of a class of iterative methods without derivatives for solving nonlinear Banach space valued operator equations under the classical Lipschitz conditions for first-order divided differences. Special cases of this method are well-known iterative algorithms, in particular, the Secant, Kurchatov, and Steffensen methods as well as the Newton method. For the semi-local convergence analysis, we use a technique of recurrent functions and majorizing scalar sequences. First, the convergence of the scalar sequence is proved and its limit is determined. It is then shown that the sequence obtained by the proposed method is bounded by this scalar sequence. In the local convergence analysis, a computable radius of convergence is determined. Finally, the results of the numerical experiments are given that confirm obtained theoretical estimates.

• Publisher: MDPI AG
• Subject: Banach space / Banach spaces / Convergence / Convergence (Mathematics) / divided difference / error analysis / Iterative algorithms / iterative method / Iterative methods / local convergence / Mathematical analysis / Methods / Newton methods / Nonlinear equations / Nonlinear theories / Operators (mathematics) / semi-local convergence