Motivations for an Arbitrary Precision Interval
Arithmetic and the MPFI Library
INRIA, Project Arenaire, LIP (CNRS/ENSL/INRIA/UCBL),
Ecole Normale Sup
erieure de Lyon,
69364 Lyon Cedex 07, France, e-mail: Nathalie.Revol@inria.fr
Salsa Project, INRIA Rocquencourt and LIP6, France, e-mail: Fabrice.Rouillier@inria.fr
(Received: 24 July 2002; accepted: 29 October 2004)
Abstract. 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
1. Motivations and State of the Art
Computing with interval arithmetic , ,  gives guarantees on a numerical
result. The fundamental principle of this arithmetic consists of replacing every num-
ber by an interval enclosing it. For instance,
cannot be exactly represented using a
binary or decimal arithmetic, but it is certiﬁed that
belongs to [3
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 over-
estimation, 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 arith-
This 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.