Discrete Stochastic Arithmetic for Validating Results of Numerical Software

The Discrete Stochastic Arithmetic DSA is a probabilistic approach for round-off error propagation. After a brief review of the CESTAC (Controle et Estimation Stochastique des Arrondis de Calculs) method, which is the basis of DSA, the concept of the “informatical zero”, also called “computational zero”, is defined. The stochastic order relations of the DSA are presented. The DSA is the joint use of the synchronous implementation of the CESTAC method and the stochastic order relations. After having summarized the asynchronous implementation of the CESTAC method, which has been used in the Prosolver software, and which has been legitimately criticized, the synchronous implementation is presented. Then the CADNA (Control of Accuracy and Debugging for Numerical Application) library which implements the DSA arithmetic is presented. It is shown that this library is able to dynamically control the validity of the hypotheses which must hold so that results provided by CESTAC method are reliable. If the hypotheses do not hold then warnings are printed in a special file. The user is informed that numerical anomalies have been detected. Depending on these warnings the user may conclude either that the results obtained are not reliable and that they cannot be correctly computed with this computer, or he may try to debug his code. It is shown that the numerical examples that support the criticisms and which make the Prosolver software fail, do not jeopardize the CADNA library.