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

Document Type : Research Article

Authors

1 Assistant Professor Faculty in Mathematics, Dr. Mahalingam College of Engineering and Technology, Pollachi, India

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

3 Payame Noor University (PNU) Tehran, Iran

4 Department of Physics, Sar.C., Islamic Azad University, Sari, Iran

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.

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References

[1] S. Abbasbandy, R. Koudsieh, Stability of a regularized Newton method with two potentials, International Journal of Theoretical and Applied Mathematics, 7, 1–11, (2021).
[2] S. Ali, S. Mathew, J. N. Mordeson, H. Rashmanlou, Vertex connectivity of fuzzy graphs with applications to human trafficking, New Mathematics and Natural Computation, 14(3), 457–485, (2018).
[3] I. K. A. Alvarez, K. Sadarangani, S. Ramos, Newton-type methods for solving equations in Banach spaces: A unified approach, Symmetry, 13, 1–21, (2021).
[4] S. Amat, S. Busquier, Review of some iterative root-finding methods from a dynamical point of view, Scientific Series A: Mathematical Sciences, 10, 3–35, (2004).
[5] S. Amat, S. Busquier (Eds.), Advances in Iterative Methods for Nonlinear Equations, Springer, Cham (2016).
[6] C. Chun, M. Y. Lee, A new family of optimal eighth-order methods for solving nonlinear equations, Applied Mathematics and Computation, 217, 6598–6605, (2011).
[7] A. Cordero, C. Jordán, E. Sanabria-Codesal, J. R. Torregrosa, Design, convergence and stability of a fourth-order class of iterative methods for solving nonlinear vectorial problems, Fractal and Fractional, 5, 125, (2021).
[8] M. Grau-Sánchez, M. Noguera, S. Amat, On the computational efficiency index and some iterative methods for solving nonlinear systems, Journal of Computational and Applied Mathematics, 236, 1259–1266, (2011).
[9] G. F. Helminck, F. Twilt, Newton flows for elliptic functions I: Structural stability, characterization and genericity, arXiv preprint arXiv:1609.01267 (2016).
[10] V. Jeyanthi, N. Selva Nandhini, Comparative Analysis of Selje Topological Space with Other Topological Spaces, Journal of the Indian Academy of Mathematics, 47(1), 41–63, (2023).
[11] V. Jeyanthi, N. Selva Nandhini, Leveraging Selje Topological Space and Blockchain Technology for Optimizing Library Book Utilization and Digital Archival Preservation, Leveraging Blockchain for Future-Ready Libraries, 5(6), 337–356, (2023).
[12] V. Jeyanthi, N. Selva Nandhini, Exploring Selje Topological Space: Novel Generalized Closed Sets and their Impact on Renal Health, ShodhKosh: Journal of Visual and Performing Arts, 5(6), 2024–2032, (2024).
[13] V. Jeyanthi, N. Selva Nandhini, On Selje Topological Spaces, Indian Journal of Natural Sciences, 14(80), 64187–64193, (2023).
[14] V. Jeyanthi, N. Selva Nandhini, Optimizing Electricity Networks with Selje Topological Space TDA and R Programming, Communications on Applied Nonlinear Analysis, 25(43), 1–9.
[15] S. Kosari, Y. Rao, H. Jiang, X. Liu, P. Wu, Z. Shao, Vague graph structure with application in medical diagnosis, Symmetry, 12(10), 1582, (2020).
[16] S. Kosari, Z. Shao, Y. Rao, X. Liu, R. Cai, H. Rashmanlou, Some types of domination in vague graphs with application in medicine, Journal of Multiple-Valued Logic and Soft Computing, 41, (2023).
[17] S. Kosari, X. Qiang, J. Kacprzyk, Q. T. Ain, H. Rashmanlou, A study on topological indices in fuzzy graphs with application in decision making problems, Journal of Multiple-Valued Logic and Soft Computing, 42, (2024).
[18] S. Kosari, X. Shi, J. Kacprzyk, Z. Chen, H. Rashmanlou, A novel description of perfectly regular fuzzy graphs with application in psychological sciences, Journal of Multiple-Valued Logic and Soft Computing, 42, (2024).
[19] S. Kosari, P. Xu, J. Kacprzyk, J. Shafi, A. Khan, H. Rashmanlou, A novel decision-making method based on fuzzy graphs, Journal of Multiple-Valued Logic and Soft Computing, 46(1), (2025).
[20] J. H. Manton, A framework for generalising the Newton method and other iterative methods from Euclidean space to manifolds, arXiv preprint arXiv:1207.5087 (2012).
[21] J. M. Ortega, W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, (1970).
[22] J. M. Ortega, W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, (1970).
[23] M. S. Petkovi, B. Neta, L. D. Petkovi, J. Duni, Multipoint Methods for Solving Nonlinear Equations, Elsevier, Amsterdam (2013).
[24] H. Rashmanlou, S. Samanta, M. Pal, R. A. Borzooei, A study on vague graphs, SpringerPlus, 5(1), 1234, (2016).
[25] J. F. Traub, Iterative Methods for the Solution of Equations, Chelsea Publishing Company, New York, (1982). 
Analytical and Numerical Solutions for Nonlinear Equations
Volume 11, Issue 1
June 2026
Pages 174-185

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History

  • Receive Date: 20 February 2026
  • Accept Date: 15 March 2026
  • Publish Date: 07 June 2026

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