Stochastic Arithmetic as a Tool to Study the Stability of Biological Models
Résumé
The theoretical study of the stability of the
numerical solution of a differential system may be complicated
or even not feasible when the system is large
and nonlinear. Here it is shown that such a study can
be experimentally done by using stochastic arithmetic and
its discrete approach known as the CESTAC method.
The CESTAC method has been first proposed since more
than forty years by M. La Porte and J. Vignes as an experimental
statistical method to estimate the accuracy on the
result of numerical program [10], [14]. Later an abstract
formalization of the theory called Stochastic Arithmetic
has been developed and many of its algebraic properties
have been studied [2], [4], [5]. Here a brief presentation
of stochastic arithmetic, of its main properties and of
the different software existing for its implementation are
given. Then it is demonstrated that the use of stochastic
arithmetic in the solver of a differential system can easily
reveal whether the computed solution is stable or not.
Moreover the stability can be studied with respect to the
coefficients of the system or with respect to the initial
conditions. At the end it is also pointed out that the same
method can be used to detect instabilities due to the used
solver. Some examples taken from the biological literature
are given [1], [6], [7].