Formulation and comparison of mathematical models examination of root-finding methods : A review

Gulzar, Yonis; Nidhi; Kaur, Narinder; Anshika

doi:10.47974/JIM-2094

Open Access Research Article

Formulation and comparison of mathematical models examination of root-finding methods : A review

^*Yonis GulzarCorresponding authorYgulzar@kfu.edu.saDepartment of Management Information SystemsCollege of Business AdministrationKing Faisal UniversityAl-Ahsa, 31982, Saudi ArabiaView full profile → , NidhiKushwahanidhi843@gmail.comDepartment of Computer Science & EngineeringChandigarh UniversityGharuan, Punjab, IndiaView full profile → , Narinder Kaurer.narinder@gmail.comDepartment of Computer Science & EngineeringChandigarh UniversityGharuan, Punjab, IndiaView full profile → , Anshikaanshikagoel0806@gmail.comDepartment of Computer Science & EngineeringChandigarh UniversityGharuan, Punjab, IndiaView full profile →

* Corresponding author · click or hover a name for details

pp. 1719–1727Vol. 27Issue 7November 2024DOI: 10.47974/JIM-2094XML

Received:: 06 Mar 2024
Published Online:: 30 Nov 2024
Article type:: Research Article
Language:: EN
Article no.:: JIM-2094
Pages:: 1719–1727

Abstract

This article provides a review that examines the mathematical principles that underlie the three numerical approaches that are utilized the most frequently for the purpose of solving nonlinear equations. For the purpose of demonstrating the numerical estimation of the non-linear equation f(x)=x^3-2x-5, the methods of Bisection, Newton-Raphson, and Secant are utilized. Using the Python programming language, the research investigates and analyzes a number of different tactics based on their capacity to get closer to the actual root of the function with each iteration (convergence). The Newton-Raphson and Secant methods produce more precision and faster convergence with a limited number of iterations, whereas the Bisection approach takes a substantially higher number of iterations to achieve convergence. Both of these methods achieve convergence more quickly and with greater accuracy. In order to arrive at an accurate determination of the root value of 2.09455, the Bisection strategy managed to achieve convergence at the thirteenth iteration, whilst the Newton-Raphson and Secant approaches were able to achieve convergence at the seventh and fourth iterations, respectively. As a result of analyzing three different approaches, it was determined that the Secant method is the most efficient one. On the other hand, the Newton-Raphson method converges more quickly but has a more challenging extraction process.

Keywords

Mathematical modeling Ordinary differential equations A duopoly economy Market share

Subject Classifications

Primary 65A05Secondary 26C10

References

[1] W. H. Press, et al., “Numerical Recipes in C: The Art of Scientific Computing,” Cambridge University Press (1992).
[2] C. R. Harris, “Mathematical Python: Extending Python with Mathematics and Engineering,” CRC Press (2013).
[3] W. McKinney, “Python for Data Analysis: Data Wrangling with Pandas, NumPy, and IPython,” O’Reilly Media (2017).
[4] J. H. Wilkinson, “The Algebraic Eigenvalue Problem,” Oxford University Press (1965).
[5] J. Renegar, “A Mathematical View of Interior-Point Methods in Convex Optimization,” MPS/SIAM Series on Optimization (2001).
[6] L. N. Trefethen and D. Bau III, “Numerical Linear Algebra,” SIAM (1997).
[7] J. Kiusalaas, “Numerical Methods in Engineering with Python,” Cambridge University Press (2010).
[8] A. Greenbaum and T. P. Chartier, “Numerical Methods: Design, Analysis, and Computer Implementation of Algorithms,” Princeton University Press (2012).
[9] H. Li, “Finding Roots of Equations,” (2022), doi: 10.1007/978-1-4842-6797-4_3.
[10] S. Aslam, “Study of numerical methods of root-finding problems,” Nucleation and Atmospheric Aerosols (2022), doi: 10.1063/5.0095703.
[11] Biswas, Chinmay, Tanmay Biswas, and Ritam Biswas, “Relative (p, q)-φ order and relative (p, q)-φ type oriented some growth investigations of composite p-adic entire functions,” Journal of Interdisciplinary Mathematics, Vol. 25.2, pp. 233-246 (2022).
[12] Slimane, Ibrahim, and Zoubir Dahmani. “Normalized fractional inequalities for continuous random variables,” Journal of Interdisciplinary Mathematics, Vol. 25.2, pp. 335-349 (2022).

Views: 314Downloads: 116Citations: 0