An interval extension based on occurrence grouping (original) (raw)

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 x a and x b of the occurrences of x: f og is increasing w.r.t. x a and decreasing w.r.t. x b. [ 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 overestimate of the image diameter of [ f ] og. Experiments show the benefits of this new interval extension for solving systems of nonlinear equations.