J.-M. Chesneaux et J. Vignes

Sur la robustesse de la méthode CESTAC

C.R.A.S., Paris, t.307, série 1, 1988, pp.855-860.

The aim of this paper is to show that the CESTAC method is able to correctly estimate the number of exact significant digits of computed results, even if the assumptions necessary for the validity of the method, are not exactly satisfied. For estimating the error between the computed result R, resulting from a finite sequence of computations, performed with the floating point arithmetic of the computer, and the exact result r resulting from the same sequence of computations, the CESTAC method consists to add after each floating point operation a random variable h to the last bit of the mantissa, then to run the code N times to obtain N different results. The average of the N results is taken as the computed result and the number of exact significant digits is given by a formula based on the Student's test. The validity of the method has been proved if the expectation of the computed result is the exact result r and if the distribution of R is almost gaussian. We show that, for having a bias able to make the CESTAC method fail, it is necessary that the ratio of the number of arithmetic operations to the number of data is close of 1 and that the mantissa of data are badly distributed. This double condition is, in real problems, quasi never satisfied. The robustness of Student law is established for the random variable R. In particular the corrections terms for the probability of the confidence interval are calculated when the distribution is approximated by the four first terms of the Edgeworth's expansion.