Reliable Computing 8: 67–82, 2002.
2002 Kluwer Academic Publishers. Printed in the Netherlands.
Veriﬁcation of Invertibility of Complicated
Functions over Large Domains
JENS HOEFKENS and MARTIN BERZ
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: 8 September 2000; accepted: 30 January 2001)
Abstract. A new method to decide the invertibility of a given high-dimensional function over a
domain is presented. The problem arises in the ﬁeld of veriﬁed solution of differential algebraic
equations (DAEs) related to the need to perform projections of certain constraint manifolds over large
domains. The question of invertibility is reduced to a veriﬁed linear algebra problem involving ﬁrst
partials of the function under consideration. Different from conventional approaches, the elements of
the resulting matrices are Taylor models for the derivatives of the functions.
The linear algebra problem is solved based on Taylor model methods, and it will be shown the
method is able to decide invertibility with a conciseness that often goes substantially beyond what can
be obtained with other interval methods. The theory of the approach is presented. Comparisons with
three other interval-based methods are performed for practical examples, illustrating the applicability
of the new method.
In , , a method involving high order Taylor polynomials with remainder
bound has been presented that allows veriﬁed computations while avoiding some
difﬁculties inherent in normal interval arithmetic. This Taylor model approach
guarantees inclusion of functional dependencies with an accuracy that scales with
the (n + 1)-st order of the domain over which the functions are evaluated.
In particular, as shown in , this method can often substantially alleviate the
following problems inherent in naive interval arithmetic:
Sharpness of the Result,
The method has recently been used for a variety of applications, including
veriﬁed bounding of highly complex functions , , solution of ODEs under
avoidance of the wrapping effect for practical purposes , and high-dimensional
veriﬁed quadrature .
In this paper we will combine these techniques with a fresh look at the mathe-
matics of invertibility to