Reliable Computing 7: 379–398, 2001.
2001 Kluwer Academic Publishers. Printed in the Netherlands.
Veriﬁed High-Order Inversion of Functional
Dependencies and Interval Newton Methods
MARTIN BERZ and JENS HOEFKENS
Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory,
Michigan State University, East Lansing, MI 48824, USA, e-mail: firstname.lastname@example.org,
(Received: 25 February 2000; accepted: 28 August 2000)
Abstract. A new method for computing veriﬁed enclosures of the inverses of given functions over
large domains is presented. The approach is based on Taylor Model methods, and the sharpness of the
enclosures scales with a high order of the domain. These methods have applications in the solution of
implicit equations and the Taylor Model based integration of Differential Algebraic Equations (DAE)
as well as other tasks where obtaining veriﬁed high-order models of inverse functions is required.
The accuracy of Taylor model methods has been shown to scale with the (n + 1)-st order of the
underlying domain, and as a consequence, they are particularly well suited to model functions over
relatively large domains. Moreover, since Taylor models can control the cancellation and dependency
problems (see Makino, K. and Berz, M.: Efﬁcient Control of the Dependency Problem Based on Taylor
Model Methods, Reliable Computing 5 (1) (1999)) that often affect regular interval techniques, the
new method can successfully deal with complicated multidimensional problems. As an application
of these new methods, a high-order extension of the standard Interval Newton method that converges
approximately with the (n + 1)-st order of the underlying domain is developed.
Several examples showing various aspects of the practical behavior of the methods are given.
In the following we will develop methods for inclusion of inverses of general
functions. As a prerequisite it is necessary to prove invertibility over the domain
in question, and there are a variety of methods for this purpose. Recently, Taylor
model methods based on high-order multivariate ﬂoating point polynomials with
interval remainder bounds have been derived that allow the efﬁcient and accurate
determination of whether a given function can be guaranteed to be invertible . The
Taylor model methods allow veriﬁcation of invertibility over domains that are often
larger than the ones over which other methods succeed. Moreover, Taylor models
are signiﬁcantly less susceptible to blow up due to the complexity of the problem and
the linear algebra in the inversion. In general, Taylor model methods provide sharp
guaranteed inclusion of functional dependencies with an accuracy that scales with
the (n +1)-st order of the domain over which the functional dependence is evaluated
, , , and it has been shown  that Taylor models can often substantially
alleviate the following problems inherent in naive interval arithmetic:
Sharpness for large domain intervals,