Reliable Computing 4: 83–97, 1998.
1998 Kluwer Academic Publishers. Printed in the Netherlands.
Computation and Application of Taylor
Polynomials with Interval Remainder Bounds
MARTIN BERZ and GEORG HOFFST
Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48824, USA,
(Received: 20 February 1996; accepted: 12 May 1997)
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
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.
The idea of veriﬁed computation is based on the rigorous estimation of the inﬂu-
ences of uncertainties on the calculation. Such uncertainties arise mainly from two
On one hand, there are computational inaccuracies based on the ﬁnite accuracy
of computational environments.
On the other hand, there are uncertainties in the variables of the model to be
These two sources of uncertainty differ in that:
in the ﬁrst case, at least initially, the inaccuracies are small and comparable to
the machine accuracy;
in the second case, however, inaccuracies can be large even from the beginning.