A Valid Scheme to Evaluate Fuzzy Definite Integrals by Applying the CADNA Library

The aim of this paper is to estimate the value of a fuzzy integral and to find the optimal step size and nodes via the stochastic arithmetic. For this purpose, the fuzzy Romberg integration rule is considered as an integration rule, then the CESTAC (Controle et Estimation Stochastique des Arrondis de Calculs) method is applied which is a method to describe the discrete stochastic arithmetic. Also, in order to implement this method, the CADNA (Control of Accuracy and Debugging for Numerical Applications) is applied which is a library to perform the CESTAC method automatically. A theorem is proved to show the accuracy of the results by means of the concept of common significant digits. Then, an algorithm is given to perform the proposed idea on sample fuzzy integrals by computing the Hausdorff distance between two fuzzy sequential results which is considered to be an informatical zero in the termination criterion. Three sample fuzzy integrals are evaluated based on the proposed algorithm to find the optimal number of points and validate the results.