TITLE:
A software package for the numerical integration of ODE by means of
high-order Taylor methods

AUTHORS:
Angel Jorba (1) and Maorong Zou (2)

(1) Departament de Matematica Aplicada i Analisi,
    Universitat de Barcelona,
    Gran Via 585, 08007 Barcelona (Spain) 
    E-mail: angel@maia.ub.es
(2) Department of Mathematics,
    University of Texas at Austin,
    Austin, TX 78712 (USA).
    E-mails: mzou@math.utexas.edu

ABSTRACT:
This paper revisits the Taylor method for the numerical integration of
initial value problems of Ordinary Differential Equations (ODEs).  The
main goal is to show that the Taylor method can be competitive, both
in speed and accuracy, with the standard methods. To this end, we
present a computer program that outputs an specific numerical
integrator for a given set of ODEs. The generated code includes
adaptive selection of order and step size at run time. The package
provides support for several extended precision arithmetics, including
user-defined types.

The paper discusses the performance of the resulting integrator in
some examples, showing that it is a very competitive method in many
situations. This is specially true for integrations that require
extended precision arithmetic. The main drawback is that the Taylor
method is an explicit method, so it has all the limitations of these
kind of schemes. For instance, it is not suitable for stiff systems.