Developing and Analyzing Newton – C’otes Quadrature Formulae for Approximating Definite Integrals- AC++ Approach

Abstract

In this paper, different Newton – C’otes quadrature formulae for the approximation of definite integrals and their error analysis are derived. The order of convergences of the methods is also derived and of these Newton – C’otes quadrature formulae, the Simpson’s 1/3 rule have been shown to have high order of convergence. Since the functionality of these numerical integration methods is practical only if we can use computer programs and applications to produce approximate solutions with acceptable errors within short period, C++ programs for the selected methods are written. These programs are used on the comparison of the Newton – C’otes quadrature formulae and the result obtained based on the inputs and outputs of the programs for different integrands. The results of these programs show that the convergence of the methods highly depends on the number of iterations. The results of different numerical examples show that for high accuracy of the trapezoidal rule computational effort is higher and round off errors with large number of iterations limit the accuracy. The results show that the Simpson’s 1/3 rule produces much more accurate solution than other methods even within small number of iterations. This shows that the error for Simpson’s rule 1/3 converges to zero faster than the error for the trapezoidal rule as the step size decreases. It is finally observed that Simpson’s 1/3 rule is much faster than the Trapezoidal and the Simpson’s 3/8 rules according to the results of the C++ programs.