Comparison of the Methods for Numerical Integration

Abstract

Numerical integration methods are generally helpful to determine the integral value of a function for which the integration becomes difficult or when it is impossible to find out the exact integral value. The main objective of this paper is to determine the optimum number of partitions for various numerical integration methods so that these methods can give us the best approximate result. We have used these methods to get the optimum result in such a way that after setting initial partitions, these methods will automatically continue to take more additional partitions until the difference between two successive integral values will be less than the considered tolerance limit. For each of the methods, we have recorded the number of steps to get the optimum number of partitions along with their computational error and average CPU time. Among all the considered methods, Weddle’s rule outperformed compared to other methods because it has required less optimum number of steps and less average CPU time, and also produced a small amount of computational error to obtain the approximate integral value. The composite integration method also performed well for a large number of subdivisions. But one limitation of the method is that what number of subdivisions should be taken for a good result is not fixed in advance. After that, the order of the methods which performed well is Boole's rule, Simpson’s 3/8 rule, Simpson’s rule, Simpson’s 1/3 rule, Midpoint rule and Trapezoidal rule respectively.