Computation and Application of Taylor Polynomials with Interval Remainder Bounds

Abstract

The expansion of complicated functions of many variables in Taylor polynomials is an important problem for many applications, and in practice can be performed rather conveniently (even to high orders) using polynomial algebras. An important application of these methods is the field of beam physics, where often expansions in about six variables to orders between five and ten are used.

However, often it is necessary to also know bounds for the remainder term of the Taylor formula if the arguments lie within certain intervals. In principle such bounds can be obtained by interval bounding of the (n + 1)-st derivative, which in turn can be obtained with polynomial algebra; but in practice the method is rather inefficient and susceptible to blow-up because of the need of repeated interval evaluations of the derivative. Here we present a new method that allows the computation of sharp remainder intervals in parallel with the accumulation derivatives up to order n.

The method is useful for a variety of numerical problems, including the interval inclusion of very complicated functions prone to blow-up. To this end, the function is represented by a Taylor polynomial with remainder using the above method. Since at least for high orders, the remainder terms have a tendency to be very small, the problem is reduced to an interval evaluation of the Taylor polynomial. The method is used for guaranteed global optimization of blow-up prone functions and compared with some interval-based global optimization schemes.