An important task in solving second order linear ordinary differential equations by the finite difference is to choose a suitable stepsize h. In this paper, by using the stochastic arithmetic, the CESTAC method and the CADNA library we present a procedure to estimate the optimal stepsize hopt, the stepsize which minimizes the global error consisting of truncation and round-off error. Keywords—o...

We formulate certain numerical problems with stochastic numbers and compare algebraically obtained results with experimental results provided by the CESTAC method. Such comparisons give additional information related to the stochastic behavior of random roundings in the course of numerical computations. The good coincidence between theoretical and experimental results confirms the adequacy of o...

In this paper, the evaluation of I = ∫ 1 −1 f(x) √ 1−x2 dx is proposed by using the opened and closed Gauss Chebyshev integration rules in the stochastic arithmetic. For this purpose, a theorem is proved to show the accuracy of the Gauss-Chebyshev rules. Then, the CESTAC 1 method and the stochastic arithmetic are used to validate the results and implement the numerical example.

Numerical validation of computed results in scienti c computation is always an essential problem as well on sequential architecture as on parallel architecture. The probabilistic approach is the only one that allows to estimate the round-o error propagation of the oating point arithmetic on computers. We begin by recalling the basics of the CESTAC method (Contrôle et Estimation STochastique des...

The CESTACmethod and its implementation known as CADNA software have been created to estimate the accuracy of the solution of real life problems when these solutions are obtained from numerical methods implemented on a computer. The method takes into account uncertainties on data and round-off errors. On another hand a theoretical model for this method in which operands are gaussian variables c...

Finite precision computations may affect the stability of algorithms and the accuracy of computed solutions. In this paper we first obtain a relation for computing the number of common significant digits between the exact solution and a computed solution of a one-dimensional initial-value problem obtained by using a single-step or multi-step method. In fact, by using the approximate solutions o...

In this paper, we apply the Newton’s and He’s iteration formulas in order to solve the nonlinear algebraic equations. In this case, we use the stochastic arithmetic and the CESTAC method to validate the results. We show that the He’s iteration formula is more reliable than the Newton’s iteration formula by using the CADNA library.

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 ...

The aim of this paper is to apply the Taylor expansion method solve first and second kinds Volterra integral equations with Abel kernel. This study focuses on two main arithmetics: FPA DSA. In order DSA, we use CESTAC CADNA library. Using method, can find optimal step approximation, error, some numerical instabilities. They are novelties DSA in comparison FPA. error analysis proved. Furthermore...