Stochastic Arithmetic in Multiprecision

Abstract

Floating-point arithmetic precision is limited in length the IEEE single (respectively double) precision format is 32-bit (respectively 64-bit) long. Extended precision formats can be up to 128-bit long. However some problems require a longer floating-point format, because of round-off errors. Such problems are usually solved in arbitrary precision, but round-off errors still occur and must be controlled. Interval arithmetic has been implemented in arbitrary precision, for instance in the MPFI library. Interval arithmetic provides guaranteed results, but it is not well suited for the validation of huge applications. The CADNA library estimates round-off error propagation using stochastic arithmetic. CADNA has enabled the numerical validation of real-life applications, but it can be used in single precision or in double precision only. In this paper, we present a library called SAM (Stochastic Arithmetic in Multiprecision). It is a multiprecision extension of the classic CADNA library. In SAM (as in CADNA), the arithmetic and relational operators are overloaded in order to be able to deal with stochastic numbers. As a consequence, the use of SAM in a scientific code needs only few modifications. This new library SAM makes it possible to dynamically control the numerical methods used and more particularly to determine the optimal number of iterations in an iterative process. We present some applications of SAM in the numerical validation of chaotic systems modeled by the logistic map.