From Quadratic Convergence to Structural Invariance: A Selje Topological Framework for Newton-Type Methods

Articles in Press

Document Type : Research Article

Authors

1 Department of Science and Humanities, Faculty in Mathematics, Dr.Mahalingam College of Engineering and Technology, Pollachi -642003

2 Clinical Research Development Unit of Rouhami Hospital, Babol University of Medical Sciences, Babol, Iran

3 Department of Science and Humanities, Faculty in Mathematics, Dr. Mahalingam College of Engineering and Technology, Pollachi-642003

4 Payame Noor University, Tehran, Iran

5 Department of Physics, Sar.C., Islamic Azad Universiry, Sari, Iran

10.22128/ansne.2026.3276.1203

Abstract

Newton-type methods are essential for solving nonlinear equations and systems, with classical metric-based analysis focusing on quadratic convergence and local error bounds. However, these results overlook the structural stability of iterations under perturbations. This paper introduces a Selje topological framework to analyze the stability of Newton-type methods beyond traditional numerical theory. We associate nonlinear operators with Selje topological structures and study the invariance and stability of iterative sequences via induced operators $\mathcal{T}_{\mathcal{R}}{(\mathbb{X})}$,$\mu_{\mathcal{R}}{(\mathbb{X})}$, $SJ_{\mathcal{R}}{(\mathbb{X})}$ . Sufficient conditions are established for preserving topological stability in Newton-type iterations, interpreting convergence as structural consistency in the Selje space. This framework yields a generalized stability characterization that complements classical convergence theory, advancing the analysis of nonlinear iterative solvers through topology.

Keywords

Main Subjects

Analytical and Numerical Solutions for Nonlinear Equations

Articles in Press, Accepted Manuscript
Available Online from 13 April 2026

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History

  • Receive Date: 20 February 2026
  • Accept Date: 15 March 2026
  • Publish Date: 13 April 2026

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