Statistical Study of Bisection Method for Cubic Equations with Random Coefficients

Abstract

The bisection method is an iterative approach used in numerical analysis to find nonlinear equations' solutions. It is an easily understandable and simple method that assured convergence. The main purpose of this paper is to study how the parameters of a probability distribution which covers both discrete and continuous distribution that characterizes the coefficient of a cubic polynomial influence the convergence of the bisection method. Distributions covered among discrete and continuous are discrete uniform distribution, continuous uniform distribution and normal distribution.

In case of both discrete and continuous uniform distribution inputs, the parameter r indicates the distribution interval [-r, r]. In case of normal distribution input, the parameters m and sigma implies mean and standard deviation respectively. The parameter values are gradually changed. In discrete as well as continuous uniform distribution inputs, and it is found that a second degree polynomial equation can be used to predict the average iteration for a given parameter value, i.e., second-degree polynomial is the best fit. On comparison between discrete and continuous uniform distributions, even the coefficients of both the second-degree polynomial are almost same. For normal distribution input, the average iteration does not depend upon the standard deviation when the mean is fixed and the standard deviation is varying whereas second degree polynomial is again the best fit when the standard deviation is fixed and the mean is varying. This means the average iteration depends upon mean of the normal distribution. Overall, our paper concludes that:

1. In case of uniform distribution input, the average iteration does not depend on whether the distribution is discrete or continuous but rather depend on the range of the distribution which is its parameter.
2. In case of non-uniform distribution input, the average iteration depends on the mean of the distribution but not on the standard deviation. Thus, it depends on the location parameter but not on the scale parameter.

Finally, a curtain is raised in the future direction of research in which we propose to combine the bisection method with the regula falsi and Newton Raphson methods to speed up the rate of convergence.