Enhancing Efficiency of Newton-Raphson Method with Chaotic Mappings for Nonlinear Equation Solving

Document Type : Computer Article

Authors

1 Department of Mathematics, Payame Noor University, Tehran, Iran

2 Department of Mathematics, Chabahar Maritime University, Chabahar, Iran

Abstract

One of the most powerful methods for solving nonlinear equations is the Newton-Raphson algorithm. Although this algorithm is highly efficient, it faces two challenges: sensitivity to the starting point and the possibility of getting stuck in a loop for the solution sequence. In this paper, by adding a small disruptive term to the Newton recursive relation, we practically generate a chaotic pseudo-random sequence, thus addressing these two challenges. The effectiveness of the proposed version has been numerically demonstrated on several nonlinear equations. The results show that the improved version is not sensitive to the initial conditions and escapes from the loop due to the generation of a random sequence, while maintaining an acceptable execution time due to the use of a deterministic system in generating the random sequence.
 

Keywords

Main Subjects

References

[1] M. Kumar, K.S. Akhilesh, and S. Akanksha. “Various Newton-Type Iterative Methods for Solving Nonlinear Equations.” Journal of the Egyptian Mathematical Society 21, no. 3 (2013): 334–39.
[2] C. Chun. “Iterative Methods Improving Newton’s Method by the Decomposition Method.” Computers & Mathematics with Applications 50, no. 10–12 (2005): 1559–68.
[3] S. Abbasbandy. “Improving Newton–Raphson Method for Nonlinear Equations by Modified Adomian Decomposition Method.” Applied Mathematics and Computation 145, no. 2–3 (2003): 887–93.
[4] L. Asiedu. “A Modification of Newton Method for Solving Non-Linear Equations.” Science and Development Journal 5, no. 1 (2021): 54–54.
[5] S. Weerakoon,  and T.G.I. Fernando. “A Variant of Newton’s Method with Accelerated Third-Order Convergence.” Applied Mathematics Letters 13, no. 8 (2000): 87–93.
[6] J. Kou, L. Yitian, and W. Xiuhua. “Some Modifications of Newton’s Method with Fifth-Order Convergence.” Journal of Computational and Applied Mathematics 209, no. 2 (2007): 146–52.
[7] T.J. McDougall, and S.J. Wotherspoon. “A Simple Modification of Newton’s Method to Achieve Convergence of Order 1+2.” Applied Mathematics Letters 29 (2014): 20–25.
[8] M.A.A. Bortoloti, T.A. Fernandes, and O.P. Ferreira. “An Efficient Damped Newton-Type Algorithm with Globalization Strategy on Riemannian Manifolds.” Journal of Computational and Applied Mathematics 403 (2022): 113853.
[9] O.P. Ferreira, and G.N. Silva. “Local Convergence Analysis of Newton’s Method for Solving Strongly Regular Generalized Equations.” Journal of Mathematical Analysis and Applications 458, no. 1 (2018): 481–96.
[10] D. Yang, L. Gang, and C. Gengdong. “On the Efficiency of Chaos Optimization Algorithms for Global Optimization.” Chaos, Solitons & Fractals 34, no. 4 (2007): 1366–75.
[11] V. Jovanovic. "Chaotic descent method and fractal conjecture." International Journal for Numerical Methods in Engineering 48, no. 1 (2000): 137-152.
[12] V. Jovanovic, and K. Kazerounian. “Optimal Design Using Chaotic Descent Method.” Journal of Mechanical Design 122, no. 3 (1998): 265–70.
[13] T. Xiang, X. Liao, and K. Wong. “An Improved Particle Swarm Optimization Algorithm Combined with Piecewise Linear Chaotic Map.” Applied Mathematics and Computation 190, no. 2 (2007): 1637–45.
[14] D. Yang, Z. Liu, and J. Zhou. “Chaos Optimization Algorithms Based on Chaotic Maps with Different Probability Distribution and Search Speed for Global Optimization.” Communications in Nonlinear Science and Numerical Simulation 19, no. 4 (2014): 1229–46.
[15] D. Yang, P. Yang, and C. Zhang. “Chaotic Characteristic Analysis of Strong Earthquake Ground Motions.” International Journal of Bifurcation and Chaos 22, no. 03 (2012): 1250045.
[16] J.L. McCauley. Chaos, Dynamics, and Fractals: An Algorithmic Approach to Deterministic Chaos. Cambridge Nonlinear Science Series. Cambridge: Cambridge University Press, 1993.
[17] Ott, Edward. Chaos in Dynamical Systems. Cambridge University Press, 1993.
[18] D. Yang, G. Li, and G. Cheng. “Convergence Analysis of First Order Reliability Method Using Chaos Theory.” Computers & Structures 84, no. 8–9 (2006): 563–71.
[19] B. Li, and W-S. Jiang. “Optimizing Complex Functions by Chaos Search.” Cybern. Syst. 29, no. 4 (1998): 409–19.
[20] M.S. Tavazoei, and M. Haeri. “Comparison of Different One-Dimensional Maps as Chaotic Search Pattern in Chaos Optimization Algorithms.” Appl. Math. Comput. 187, no. 2 (2007): 1076–85.
[21] Y.Y. He, J-Z. Zhou, X-Q. Xiang, H. Chen, and H. Qin. “Comparison of Different Chaotic Maps in Particle Swarm Optimization Algorithm for Long-Term Cascaded Hydroelectric System Scheduling.” Chaos, Solitons & Fractals 42, no. 5 (2009): 3169–76.
Journal of Modeling in Engineering
Volume 23, Issue 80
In Progress
March 2025
Pages 59-73

Files

History

  • Receive Date: 09 December 2023
  • Revise Date: 06 August 2024
  • Accept Date: 26 August 2024

Share

How to cite

Statistics