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

H. Rahman¹ , K.C. Roy² , S.K. Das³ , S.A. Hossain⁴

Section:Research Paper, Product Type: Journal-Paper
Vol.10 , Issue.3 , pp.7-12, Jun-2022

Online published on Jun 30, 2022

Copyright © H. Rahman, K.C. Roy, S.K. Das, S.A. Hossain . 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 :
Calculating square roots of positive real numbers play a vital role in scientific and engineering computing. From the daily-life used calculators to the computer software used as a calculator often use the square root function. However, finding the real roots of algebraic and transcendental equations is one of the most exciting topics in numerical computation. There are several methods, such as the Bisection, Secant, Iteration, and Newton-Raphson’s methods, to do that. The speciality of the Bisection method is its robustness and needs no stability criterion. Although it convergences slowly, it always convergences. In this study, we assessed the numerical accuracy through the root mean square error (RMSE) value for this method by applying it to finding square roots of some positive real numbers. Also, we calculated the computational time and number of iterations to convergence to an exact root with the error tolerance of 0.000001 for assessing the method’s efficiency. The method’s RMSE value obtained in our study is of order 10-7, indicating its reasonably acceptable accuracy level. We got this accuracy within 23 iterations in each case, and the computational time is a tiny fraction of a millisecond; these indicate the excellent efficiency level of the method. Our inquiry has found the method reasonably acceptable, efficient, and robust.

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

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