Reliable Computing 9: 43–79, 2003.
2003 Kluwer Academic Publishers. Printed in the Netherlands.
Taylor Forms—Use and Limits
ur Mathematik, Universit
at Wien, Strudlhofgasse 4, A-1090 Wien, Austria,
e-mail: Arnold.Neumaier@univie.ac.at, WWW: http://www.mat.univie.ac.at/
(Received: 16 May 2002; accepted: 12 October 2002)
Abstract. This review is a response to recent discussions on the reliable computing mailing list,
and to continuing uncertainties about the properties and merits of Taylor forms, multivariate higher
degree generalizations of centered forms. They were invented around 1980 by Lanford, documented
in detail in 1984 by Eckmann, Koch, and Wittwer, and independently studied and popularized since
1996 by Berz, Makino, and Hoefkens. A highlight is their application to the veriﬁed integration of
asteroid dynamics in the solar system in 2001.
Apart from summarizing what Taylor forms are and do, this review puts them into the perspective
of more traditional methods, in particular centered forms, discusses the major applications, and
analyzes someof their elementary properties. Particular emphasis is given tooverestimation properties
and the wrapping effect. A deliberate attempt has been made to offer value statements with appropriate
justiﬁcations; but all opinions given are my own and might be controversial.
PART 1: PROPERTIES AND HISTORY OF TAYLOR FORMS
Taylor forms are higher degree generalizations of centered forms. They compute
recursively a high order polynomial approximation to a multivariate Taylor expan-
sion, with a remainder term that rigorously bounds the approximation error. Storage
is proportional to
n + d
n + d
) for an approximation of
degree d in n variables. The work is proportional to this number and to the number
of arithmetic operations. Both counts may be much less for sparse problems, or
when it is known that the function has low degree in some variables.
Rigorous multivariate Taylor arithmetic with remainder, for the four elementary
operations and composition exists at least since 1984 (Eckmann et al. , ).
Independently, a slightly different version of Taylor arithmetic with remainder has
been made popular since 1996 under the name Taylor models by Martin Berz and
his group; their papers –, –, ,  on Taylor models and their
applications can be found at:
An efﬁcient implementation is freely available within the framework of the COSY
INFINITY package (designed for beam physics applications) of Martin Berz at: