Numerical solution of improper integrals with valid implementation

Abstract

In this paper, two theorems are explained which are used in order to find the improper integral I = $\inta\inftyf(x)dx$ numerically. It has been proved in [4], one can use the Trapezoidal and Simpson rules to find the definite integral $I_m$ = $\intamf(x)dx$ numerically using the CESTAC (Control et Estimation Stochastique des Arrondis de Calculs ) method which is based on the stochastic arithmetic, [5-8,12]. These theorems are developed on the improper integrals. Then, the CESTAC method and stochastic arithmetic are used to validate the results and implement the numerical examples. By using this method, one can find the optimal integer number m $\geq$1 such that I ~ $I_m$. In the last section two examples are solved. The programs have been provided with Fortran 90.