A stochastic arithmetic for reliable scientific computation

After having reviewed the CESTAC (Controle et Estimation STochastique des Arrondis de Calculs) method, which uses a probabilistic approach to round-off error analysis, a new arithmetic called stochastic arithmetic is presented in this paper. This new arithmetic covers a large part of the properties of exact arithmetic, properties which are lost in the usual floating-point arithmetic. The CADNA (Control of Accuracy and Debugging for Numerical Applications) library is a tool for automatic implementation of stochastic arithmetic in any fortran or ADA program. Consequently, by the use of the CADNA library, it is possible during the run of a program, to detect the numerical instabilities, to stop correctly any iterative process, to evaluate the optimal step or the optimal mesh in approximate methods, to evaluate the accuracy of any result provided by the computer. In short, the stochastic arithmetic, i.e., the use of the CADNA library, serves to validate the results provided by a computer, and to assure the user of the reliability of scientific computations.

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