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An Interval Extension Based on Occurrence Grouping

Ignacio Araya 1 Bertrand Neveu 2, 3, 4 Gilles Trombettoni 5, 6
4 imagine [Marne-la-Vallée]
CSTB - Centre Scientifique et Technique du Bâtiment, ENPC - École des Ponts ParisTech, LIGM - Laboratoire d'Informatique Gaspard-Monge
5 COPRIN - Constraints solving, optimization and robust interval analysis
CRISAM - Inria Sophia Antipolis - Méditerranée , ENPC - École des Ponts ParisTech
6 Laboratoire d'Informatique, Signaux, et Systèmes de Sophia-Antipolis (I3S) / Equipe CEP
Laboratoire I3S - MDSC - Modèles Discrets pour les Systèmes Complexes
Abstract : In interval arithmetics, special care has been brought to the definition of interval extension functions that compute narrow interval images. In particular, when a function f is monotonic w.r.t. a variable in a given domain, it is well-known that the monotonicity-based interval extension of f computes a sharper image than the natural interval extension does. This paper presents a so-called "occurrence grouping" interval extension [ f ]og of a function f . When f is not monotonic w.r.t. a variable x in a given domain, we try to transform f into a new function f og that is monotonic w.r.t. two subsets xa and xb of the occurrences of x: f og is increasing w.r.t. xa and decreasing w.r.t. xb . [ f ]og is the interval extension by monotonicity of f og and produces a sharper interval image than the natural extension does. For finding a good occurrence grouping, we propose a linear program and an algorithm that minimize a Taylor-based over-estimate of the image diameter of [ f ]og . Experiments showthe benefits of this new interval extension for solving systems of non linear equations
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Ignacio Araya, Bertrand Neveu, Gilles Trombettoni. An Interval Extension Based on Occurrence Grouping. Computing, Springer Verlag, 2012, 94 (2), pp.173-188. ⟨hal-00733855⟩

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