A comparative study of the Regula Falsi method, Newton's method, and the steepest descent method for solving nonlinear equations
DOI:
https://doi.org/10.15282/daam.v6i2.13044Keywords:
Numerical Methods, Regula Falsi Method, Newton’s Method, Steepest Descent Method, Nonlinear EquationAbstract
A numerical method has been introduced to help mathematicians to solve the functions. Numerical methods are powerful tools used to approximate solutions to equations that may not have exact solutions or are difficult to solve analytically. This study presents a comparative analysis of numerical methods that is the Regula Falsi method, Newton's Method, and Steepest Descent method. These methods are employed for solving nonlinear equations. All these methods will be compared and tested with eight different types of test functions including polynomials, exponentials, cubic, and trigonometric functions and also with different tolerance and initial guess. The performance of these methods was evaluated using performance profiles based on the number of iterations and CPU time. The results demonstrate that Newton's Method outperforms the other approaches, exhibiting the fastest convergence, and the least computational cost. Regula Falsi shows moderate performance, while Steepest Descent lags in efficiency due to its higher iteration count and CPU usage. The findings underscore the significance of selecting appropriate numerical techniques to optimize computational efficiency, with potential applications across diverse scientific and engineering disciplines.
References
[1] Shaikh WA. Numerical hybrid iterative technique for solving nonlinear equations in one variable. Journal of Mechanics of Continua and Mathematical Sciences. 2021;16(7).
[2] Shams M, Kausar N, Araci S, Oros G. Numerical scheme for estimating all roots of non-linear equations with applications. AIMS Mathematics. 2023;8(10):23603–20.
[3] Rasheed M, Rashid AA, Rashid T, Abed M, Manoochehri NT. Various numerical methods for solving nonlinear equations. Journal of Al-Qadisiyah for Computer Science and Mathematics. 2021;13(3).
[4] Grossan C, Abraham A. A new approach for solving nonlinear equations systems. IEEE Transactions on Systems, Man, and Cybernetics - Part A: Systems and Humans. 2008;38(3):698–714.
[5] Yash B, Vishal VM. Regula falsi method. International Journal of Advanced Research in Computer and Communication Engineering. 2022;11(1):225–9.
[6] Nguyen T. The convergence of the regula falsi method. arXiv. 2021 Sep 8.
[7] Kiusalaas J. Numerical Methods in Engineering with Python. Cambridge: Cambridge University Press; 2010.
[8] Burden RL, Faires JD. Numerical Analysis. 9th ed. Brooks/Cole Cengage Learning; 2011.
[9] Meza JC. A survey of numerical methods for nonlinear equations. SIAM Review. 2010;52(4):637–63.
[10] Atluri SN. Numerical Methods in Science and Engineering: A Practical Approach. Springer; 2009.
[11] Atluri SN, Liu CS, Kuo CL. A modified Newton method for solving non-linear algebraic equations. Journal of Marine Science and Technology. 2009;17(3).
[12] Nocedal J, Wright SJ. Numerical Optimization. 2nd ed. Springer; 2006.
[13] Boyd S, Vandenberghe L. Convex Optimization. Cambridge: Cambridge University Press; 2004.
[14] Djordjevic SS. Unconstrained optimization methods: conjugate gradient methods and trust-region methods. In: Applied Mathematics. IntechOpen; 2019.
[15] Fülöp Z. Regula falsi in lower secondary school education II. Teaching Mathematics and Computer Science. 2021;18(2):121–42.
[16] Nguyen T. The convergence of the regula falsi method. arXiv. 2021 Sep 8.
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