This paper justifies why an arbitrary precision interval arithmetic is needed. To provide accurate results, interval computations require small input intervals; this explains why bisection is so often employed in interval algorithms. The MPFI library has been built in order to fulfill this need. Indeed, no existing library met the required specifications. The main features of this library are briefly given and a comparison with a fixed-precision interval arithmetic, on a specific problem, is presented. It shows that the overhead due to the multiple precision is completely acceptable. Eventually, some applications based on MPFI are given: robotics, isolation of polynomial real roots (by an algorithm combining symbolic and numerical computations) and approximation of real roots with arbitrary accuracy.
Motivations and State of the ArtComputing with interval arithmetic [3], [25], [31] gives guarantees on a numerical result. The fundamental principle of this arithmetic consists of replacing every number by an interval enclosing it. For instance, cannot be exactly represented using a binary or decimal arithmetic, but it is certified that belongs to [314159 314160]. Measurement errors also can be taken into account. The advantages of interval arithmetic are numerous. Computed results are validated, and rounding errors are taken into account, since computer implementations perform outward rounding. Last but not least, interval arithmetic provides global information: for instance, it provides the range of a function over a whole set S, which is a crucial information for global optimization. Such properties cannot be reached without set computing: interval arithmetic computes with sets and is easily available.However, in spite of the improvements in interval analysis, the problem of overestimation, i.e. of enclosures which are far too large and thus of little practical use, seems to plague interval computations. With fixed-precision floating-point arithThis work was done while N. Revol was a member of the ANO Laboratory, University of Lille, France, on sabbatical leave within the Arenaire project. This work was done while F. Rouillier belonged to the Spaces project, LORIA and LIP6, France. Reliable Computing (2005) c Springer 2005 11: 275-290 276 NATHALIE REVOL AND FABRICE ROUILLIERmetic, results can be wide even when the input data are provided with machine precision; a remedy for this phenomenon consists of computing with a higher precision. This proposal is the core of the MPFI library (MPFI stands for Multiple Precision Floating-point Interval), a library implementing arbitrary precision interval arithmetic which is described in this paper.This quest for extra accuracy can be found primarily in the development of algorithms for interval analysis: use of centered forms and slopes instead of direct interval evaluation, preconditioners, and many other directions [5]. Real-world applications where extra accuracy is required are to be found in control theory (we have indeed received requests for a tool that would in...
