Dynamical Control of Accuracy Using the Stochastic Arithmetic to Estimate Double and Improper Integrals

The CESTAC (Control et Estimation STochastique des Arrondis de Calculs) method is based on a probabilistic approach of the round-off error propagation which replaces the floating-point arithmetic by the stochastic arithmetic. This is an efficient method to estimate the accuracy of the results. In this paper, we present the reliable schemes using the CESTAC method to estimate the definite double integral I = \({\int_a^b}{\int_c^d}\)f(x,y)dydx and the improper integral I = \(\int_a^\infty\)f(x)dx , where a, b, c, d ∈ R, by applying the trapezoidal or Simpson's rule. For each kind of integrals, we prove a theorem to show the accuracy of the results. According to these theorems, one can find an optimal value number of the points which we can find the best approximation of I from the computer point of view. Also, we observe that by using the stochastic arithmetic, we are able to validate the results.