Higher Order Convergence for Multidimensional Functions with a New Taylor-Bernstein Form as Inclusion Function

Abstract

Recently, Lin and Rokne (Interval Approximation of Higher Order to the Ranges of Functions, Computers Math. Applic.31(7) (1996), pp. 101-109) introduced the so-called Taylor-Bernstein (TB) form as an inclusion function form for multidimensional functions. This form was theoretically shown to have the property of higher order convergence. In this paper, we present an improvement of Lin and Rokne's TB form to make it more effective in practice. We test and compare the higher order convergence behavior of the proposed TB form with that of Lin and Rokne's TB form and also with that of the Taylor model of Berz et al. (Computation and Application of Taylor Polynomials with Interval Remainder Bounds, Reliable Computing4(1) (1998), pp. 83-97). For the testing, we consider six benchmark examples with dimensions varying from 1 to 6. In all examples, unlike with the Taylor model and Lin and Rokne's TB form, we obtain higher order convergence of orders up to 9 with the proposed TB form. Moreover, with the proposed TB form we quite easily obtain such high orders of convergence for up to 5-dim problems.