A Study on the Numerical Accuracy and Efficiency of the Bisection Method in Finding nth Roots of Positive Real Numbers

Hafijur Rahman¹ , Md. Soriful Islam² , Abul Khair³ , Md. Shohag Hossain Reyad⁴

1. Dept. of Mathematics, American International University-Bangladesh, Dhaka 1229, Bangladesh.
2. Dept. of Applied Mathematics, Gono Bishwabidyalay, Dhaka 1344, Bangladesh.
3. Dept. of Mathematics, Jahangirnagar University, Dhaka 1342, Bangladesh.
4. Dept. of Mathematics, Dhaka University of Engineering and Technology, Gazipur 1707, Bangladesh.

Section:Research Paper, Product Type: Journal-Paper
Vol.10 , Issue.5 , pp.37-42, May-2024

Online published on May 31, 2024

Copyright © Hafijur Rahman, Md. Soriful Islam, Abul Khair, Md. Shohag Hossain Reyad . This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
 

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Abstract :
An important part of computational science and engineering is finding the nth roots of positive real numbers. Calculators and computer programs that operate as calculators employ the nth root function often. However, one of the most intriguing areas of numerical computing is the search for the real roots of transcendental and algebraic equations. There are various ways to do this, including the Bisection, Newton-Raphson, Iteration, and Secant methods. The Bisection method is notable for its robustness and does not need a stability condition. Although its convergence occurs gradually, convergence always occurs. In this study, we used this method to find the nth roots of some positive real numbers to calculate the root mean square error (RMSE) value through which the numerical accuracy of the method can be reckoned. In order to evaluate the efficacy of the method, we also computed the computing time and the number of iterations required to converge to an exact root with an error tolerance of 0.000001. The RMSE value found in our investigation is of the order of 10-7, demonstrating an adequate level of accuracy of the method. This precision was achieved in each instance within 23 repetitions, and the time required is a minuscule fraction of a millisecond; these demonstrate the method’s high level of efficiency. Our investigation revealed the method’s reasonable acceptance, efficiency, and robustness.

Key-Words / Index Term :
Algebraic and transcendental equations, nth root, Bisection method, Root mean square error, Numerical computation of zeros, Efficiency, Accuracy, Rate of convergence

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