Valid implementation of the Romberg integration rule to evaluate a de nite integral by applying the CADNA library

One of the most important problems in numerical integration is to nd the optimal number of iteration in quadrature rules. In this paper, the Romberg integration rule is used to implement in the stochastic arithmetic. To this end, the CESTAC method and the CADNA li- brary are applied. Also, a theorem is proved which can be guaranteed the numerical results. In this theorem, we show the number of common signi cant digits between the exact and the approx-imate values is almost equal to the number of common signi cant digits between two successive approximations. The examples shows the importance of using the stochastic arithmetic in place of the common oating-point arithmetic.

Keywords:

Stochastic arithmetic , CESTAC method , CADNA library , Romberg integration rule