Multiple Precision Interval Packages: Comparing Different Approaches [chapter]

Markus Grimmer, Knut Petras, Nathalie Revol
2004 Lecture Notes in Computer Science  
We give a survey on packages for multiple precision interval arithmetic, with the main focus on three specific packages. One is within a Maple environment, intpakX, and two are C/C++ libraries, GMP-XSC and MPFI. We discuss their different features, present timing results and show several applications from various fields, where high precision intervals are fundamental. Résumé Dans cet article est effectué un tour d'horizon des outils implémentant l'arithmétique par intervalles en précision
more » ... le. Un coup de projecteur est donné sur trois de ces outils. Le premier est un paquetage développé pour Maple: intpakX, les deux autres sont des bibliothèques C/C++: GMP-XSC et MPFI. Leurs spécificités sont présentées, puis leurs performances sont données. Enfin, sont développées quelques applications, issues de domaines d'application divers, pour lesquelles l'arithmétique par intervallesà grande précision est fondamentale. Mots-clés: Arithmétique par intervalles en précision multiple, paquetages et bibliothèques, facilité d'emploi, efficacité, fiabilité. Multiple precision is a floating-point arithmetic, where the number of digits of the mantissa can be any fixed or variable value. It is usually applied to problems where it is important to have a high accuracy (e.g., many digits of π). However, for such, mostly numerical, algorithms where extra computing precision is required, the loss of accuracy can be predictable or unpredictable. If this loss is predictable, then multiple precision arithmetic perfectly fulfils the application's needs. When it is unpredictable, interval arithmetic can reveal useful to bound this loss of accuracy. Of course, this interval arithmetic must also be based on a multiple precision arithmetic. Hence, we are particularly interested in numerical problems, with a large and unpredictable loss of accuracy. Although multiple precision interval arithmetic might help, one should be aware of the fact that this often means an increase in the computational time and memory usage, cf. section 3. Before classifying softwares that implement multiple precision interval arithmetic, let us first point out the possibility to vary or not the computing precision. Sometimes multiple precision refers only to extended and fixed precision, whereas arbitrary precision is used for variable precision. In this paper, multiple precision refers to extended precision, whether it is variable or not. Arbitrary precision arithmetic offers the possibility to set precision to an arbitrary value as needed in the computations; this can be done either statically or dynamically, i.e. during the computations. Interval packages based on GNU multiple precision arithmetic or Maple arithmetic are such. But there are also approaches offering multiple precision arithmetic without the possibility to vary the precision, as the staggered multiple precision arithmetic in the XSC languages [24, 23] .
doi:10.1007/978-3-540-24738-8_4 fatcat:cn7cdo46pbbmfjevnfkgqx7tl4