Formulation and convergence analysis of an efficient higher order iterative scheme
Vol 2, Issue 1, 2024
Issue release: vol 2, No 1
VIEWS - 27 (Abstract)
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Abstract
This contribution presents a highly efficient three-step iterative scheme. The proposed scheme is different in itself by achieving seventh-order convergence. The scheme is very useful for equations of nonlinear nature having multiple roots. The Taylor series expansion is employed to rigorously analyze the convergence of the presented scheme. That the scheme is effective and robust can be fit through a variety of examples from different fields. Numerical experimentation demonstrates the scheme’s rapid and reliable convergence to the true root and comparing its performance against existing techniques in the literature. Additionally, basins of attraction are visualized to offer a clear, comparative view of how different methods perform with varying initial guesses. The results show that this new scheme consistently compete well over other methods. This makes it a powerful tool for solving complex equations.
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References
1. Schröder E. Over an infinite number of algorithms for solving the equations (German). Mathematische Annalen. 1870; 2(2): 317-365. doi: 10.1007/bf01444024
2. Ahmad N and Singh VP. New efficient four-step iterative method with 36th-order con- vergence for solving nonlinear equations. Journal of Nonlinear Modeling and Analysis. 2023; 5(2): 1-8.
3. Bala R, Kansal M, Kanwar V. An optimal class of fourth-order multiple-root finders of Chebyshev-Halley type and their basins of attraction. International Journal of Computing Science and Mathematics. 2021; 14(1): 17. doi: 10.1504/ijcsm.2021.118074
4. Geum YH, Kim YI, Neta B. A class of two-point sixth-order multiple-zero finders of modified double-Newton type and their dynamics. Applied Mathematics and Computation. 2015; 270: 387-400. doi: 10.1016/j.amc.2015.08.039
5. Geum YH, Kim YI, Neta B. A sixth-order family of three-point modified Newton-like multiple-root finders and the dynamics behind their extraneous fixed points. Applied Mathematics and Computation. 2016; 283: 120-140. doi: 10.1016/j.amc.2016.02.029
6. King RF. A secant method for multiple roots. BIT. 1977; 17(3): 321-328. doi: 10.1007/bf01932152
7. Neta B, Scott M, Chun C. Basin attractors for various methods for multiple roots. Applied Mathematics and Computation. 2012; 218(9): 5043-5066. doi: 10.1016/j.amc.2011.10.071
8. Weerakoon S and Fernando TGI. A variant of Newton’s method with accelerated third- order convergence. Applied Mathematics Letters. 2000; 13: 87-93. https://doi.org/10.1016/S0893-9659(00)00100-2
9. Nadeem GA, Aslam W, and Ali F. An optimal fourth-order second derivative free iterative method for nonlinear scientific equations. Kuwait Journal of Sciences. 2023; 50(2A): 1-15.
10. Sharifi M, Babajee DKR, Soleymani F. Finding the solution of nonlinear equations by a class of optimal methods. Computers & Mathematics with Applications. 2012; 63(4): 764-774. doi: 10.1016/j.camwa.2011.11.040
11. Sharma R, Bahl A. General Family of Third Order Methods for Multiple Roots of Nonlinear Equations and Basin Attractors for Various Methods. Advances in Numerical Analysis. 2014; 2014: 1-8. doi: 10.1155/2014/963878
12. Sharma JR, Sharma R. Modified Jarratt method for computing multiple roots. Applied Mathematics and Computation. 2010; 217(2): 878-881. doi: 10.1016/j.amc.2010.06.031
13. Sharma R, Bahl A. Optimal eighth order convergent iteration scheme based on Lagrange interpolation. Acta Mathematicae Applicatae Sinica, English Series. 2017; 33(4): 1093-1102. doi: 10.1007/s10255-017-0722-x
14. Sharma JR, Kumar D, Cattani C. An Efficient Class of Weighted-Newton Multiple Root Solvers with Seventh Order Convergence. Symmetry. 2019; 11(8): 1054. doi: 10.3390/sym11081054
15. Traub JF. Iterative Methods for the Solution of Equations. Chelsea Publishing Company, New York; 1982.
16. Kumar D, Sharma JR, Jăntschi L. A Novel Family of Efficient Weighted-Newton Multiple Root Iterations. Symmetry. 2020; 12(9): 1494. doi: 10.3390/sym12091494
17. Wolfram S. The Mathematica Book. Wolfram Research; 2003.
18. Varona JL. Graphic and numerical comparison between iterative methods. The Mathematical Intelligencer. 2002; 24(1): 37-46. doi: 10.1007/bf03025310
19. Scott M, Neta B, Chun C. Basin attractors for various methods. Applied Mathematics and Computation. 2011; 218(6): 2584-2599. doi: 10.1016/j.amc.2011.07.076
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Copyright (c) 2024 Ranjita Guglani, Ashu Bahl, Rajni Sharma
License URL: https://creativecommons.org/licenses/by/4.0/
Editor-in-Chief
Prof. Youssri Hassan Youssri
Cairo University, Egypt
Processing Speed (2023)
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- ＜7 days: submission to screening review decision
- 53 days: received to accepted
- 68 days: received to online
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