IJPAM: Volume 4, No. 4 (2003)


Kyoko Makino$^1$, Martin Berz$^2$
$^1$Department of Physics
University of Illinois at Urbana-Champaign
1110 W. Green Street, Urbana IL 61801-3080, USA
$^2$Department of Physics and Astronomy
Michigan State University
East Lansing, MI 48824, USA
e-mail: berz@msu.edu

Abstract.A detailed comparison between Taylor model methods and other tools for validated computations is provided. Basic elements of the Taylor model (TM) methods are reviewed, beginning with the arithmetic for elementary operations and intrinsic functions. We discuss some of the fundamental properties, including high approximation order and the ability to control the dependency problem, and pointers to many of the more advanced TM tools are provided. Aspects of the current implementation, and in particular the issue of floating point error control, are discussed.

For the purpose of providing range enclosures, we compare with modern versions of centered forms and mean value forms, as well as the direct computation of remainder bounds by high-order interval automatic differentiation and show the advantages of the TM methods.

We also compare with the so-called boundary arithmetic (BA) of Lanford, Eckmann, Wittwer, Koch et al, which was developed to prove existence of fixed points in several comparatively small systems, and the ultra-arithmetic (UA) developed by Kaucher, Miranker et al which was developed for the treatment of single variable ODEs and boundary value problems as well as implicit equations. Both of these are not Taylor methods and do not provide high-order enclosures, and they do not support intrinsics and advanced tools for range bounding and ODE integration.

A summary of the comparison of the various methods including a table as well as an extensive list of references to relevant papers are given.

Received: January 7, 2003

AMS Subject Classification: 65L20, 65L06

Key Words and Phrases: Taylor model methods, high approximation order, dependency problem, centered forms, mean value forms, boundary arithmetic, ultra-arithmetic

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2003
Volume: 4
Issue: 4