5. Remainder Differential Algebras and Their Applications

Abstract

In many practical problems in which derivatives are calculated, their basic purpose is to be used in the modeling of a functional dependence, often based on a Taylor expansion to first or higher orders. While the practical computation of such derivatives is greatly facilitated and in many cases is possible only through the use of forward or reverse computational differentiation, there is usually no direct information regarding the accuracy of the functional model based on the Taylor expansion. We show how, in parallel to the accumulation of derivatives, error bounds of all functional dependencies can be carried along the computation. The additional effort is minor, and the resulting bounds are usually rather sharp, in particular at higher orders. This Remainder Differential Algebraic Method is more straightforward and can yield tighter bounds than the mere interval bounding of the Taylor remainder’s (n +1 )st order derivative obtained via forward differentiation. The method can be applied to various numerical problems: Here we focus on global optimization, where blow-up can often be substantially reduced compared with interval methods, in particular for the cases of complicated functions or many variables. This problem is at the core of many questions of nonlinear dynamics and can help facilitate a detailed, quantitative understanding.