Experiments with Range Computations Using Extrapolation

Abstract

The natural interval extension (NIE) used widely in interval analysis has the first order convergence property, i.e., the excess width of the range enclosures obtained with the NIE goes down at least linearly with the domain width. Here, we show how range approximations of higher convergence orders can be obtained from the sequence of range enclosures generated with the NIE and uniform subdivision. We combine the well-known Richardson Extrapolation Process (Sidi, A., Practical Extrapolation Methods, Cambridge University Press, Cambridge, 2003) with Brezinski’s error control method (Brezinski, C., Error Control in Convergence Acceleration Processes, IMA J. Nunerical Analysis 3 (1983), pp. 65–80) to generate non-validated range approximations to the true range. We demonstrate the proposed method for accelerating the convergence orders on several multidimensional examples, varying from one to six dimensions. These numerical experiments also show that considerable computational savings can be obtained with the proposed procedure. However, the theoretical basis of the proposed method remains to be investigated.