Finding optimal step of fuzzy Newton-Cotes integration rules by using the CESTAC method

Abstract

The aim of this work, is to evaluate the value of a fuzzy integral by applying the Newton-Cotes integration rules via a reliable scheme. In order to perform the numerical examples, the CADNA (Control of Accuracy and Debugging for Numerical Applications) library and the CESTAC (Controle et Estimation Stochastique des Arrondis de Calculs) method are applied based on the stochastic arithmetic. By using this method, the optimal number of points in the fuzzy numerical integration rules and the optimal approximate solution are obtained. Also, the accuracy of the fuzzy quadrature rules are discussed. An algorithm is given to illustrate the implementation of the method. In this case, the termination criterion is considered as the Hausdorff distance between two sequential results to be an informatical zero. Two sample fuzzy integrals are evaluated based on the proposed algorithm to show the importance and advantage of using the stochastic arithmetic in place of the floating-point arithmetic.