Two Approaches For Solving Nonlinear Equation Systems: Newton Raphson and Red Fox

Year 2025, Volume: 14 Issue: 2, 1 - 10, 27.06.2025

Bayram Köse , Bahar Demirtürk , Şükran Konca

https://doi.org/10.46810/tdfd.1570735

Abstract

In this study, the results obtained by using the Red Fox method, a new metaheuristic optimization method, and the Newton Raphson method, which is one of the numerical methods, in finding the solutions of nonlinear systems of equations, are compared and these comparative analyses are evaluated and recommendations are presented.

Keywords

Root finding , Optimization algorithms , Newton-Raphson algorithm , Red fox algorithm.

References

• Victor YP. Root-Finding and Root-Refining for a Polynomial Equation. Computer Science Technical Reports: City University of New York (CUNY); 2012.
• Millidere M, Karaman U, Uslu S, Kasnakoğlu C, Çimen T. Newton Raphson methods in aircraft trim: A comparative study. In: AIAA AVIATION 2020 Forum. American Institute of Aeronautics and Astronautics; 2020. p. 3198.
• Gower R, Lorenz DA, Winkler M. A Bregman–Kaczmarz method for nonlinear systems of equations. Comput Optim Appl. 2023;87(3):1059–1098.
• Pourrajabian A, Ebrahimi R, Mirzaei M, Shams M. Applying genetic algorithms for solving nonlinear algebraic equations. Appl Math Comput. 2013;219(24):11483–11494.
• Kotsireas IS, Pardalos PM, Semenov A, Trevena WT, Vrahatis MN. Survey of methods for solving systems of nonlinear equations, Part II: Optimization based approaches. arXiv. 2022. Available from: https://arxiv.org/abs/2208.08532
• Odan S. Metaheuristic method for solving systems of equations. arXiv. 2024. Available from: https://arxiv.org/abs/2409.16958
• Verma P, Parouha RP. Solving systems of nonlinear equations using an innovative hybrid algorithm. Iranian J Sci Technol Trans Electr Eng. 2022; 46:1005–1027.
• Chapra RP, Canale SC. Numerical Methods for Engineers. New York: McGraw-Hill Higher Education; 2002.
• Kujawa MA. Torsional buckling of restrained thin-walled bars of open bisymmetric cross-section. TASK Quarterly. Scientific Bulletin of Academic Computer Centre in Gdansk. 2012;16(3-4).
• Sharma JR, Kumar S, Cesarano C. An efficient derivative free one-point method with memory for solving nonlinear equations. Mathematics. 2019;7(7):604.
• Burden RL, Faires JD, Burden AM. Numerical Analysis. 10th ed. Canada: Cengage Learning; 2015.
• Połap D, Woźniak M. Red fox optimization algorithm. Expert Syst Appl. 2021; 166:1-21.
• Larivière S, Pasitschniak AM. Vulpes vulpes. Mammal Species. 1996; 537:1–11.
• Mohammed H, Rashid T. FOX: a FOX-inspired optimization algorithm. Appl Intell. 2023;53(1):1030–1050.
• Červený J, Begall S, Koubek P, Nováková P, Burda H. Directional preference may enhance hunting accuracy in foraging foxes. Biol Lett. 2011;7(3):355–357.
• Bektaş Y. Kızıl Tilki Optimizasyon Algoritması. Teknobilim 2023: Optimizasyon Modelleme ve Yapay Zeka Optimizasyon Algoritmaları. İstanbul: Efe Akademi; 2023.
• Köse B, Demirtürk B, Konca Ş. Application of Search Algorithms to Root-Finding Problems in Algebraic and Transcendental Contexts. 2025. (accepted in European Journal of Pure and Applied Mathematics)