Reliable Computing 8: 267–282, 2002.
2002 Kluwer Academic Publishers. Printed in the Netherlands.
On Existence and Uniqueness Veriﬁcation for
R. BAKER KEARFOTT
Department of Mathematics, University of Louisiana at Lafayette, Lafayette, LA 70504, USA,
(Received: 14 December 2001; accepted: 21 March 2002)
Abstract. Given an approximate solution to a nonlinear system of equations at which the Jacobi
matrix is nonsingular, and given that the Jacobi matrix is continuous in a region about this approximate
solution, a small box can be constructed about the approximate solution in which interval Newton
methods can verify existence and uniqueness of an actual solution. Recently, we have shown how
to verify existence and uniqueness, up to multiplicity, for solutions at which the Jacobi matrix is
singular. We do this by efﬁcient computation of the topological index over a small box containing
the approximate solution. Since the topological index is deﬁned and computable when the Jacobi
matrix is not even deﬁned at the solution, one may speculate that efﬁcient algorithms can be devised
for veriﬁcation in this case, too. In this note, however, we discuss, through examples, key techniques
underlying our simpliﬁcation of the calculations that cannot necessarily be used when the function
is non-smooth. We also present those parts of the theory that are valid in the non-smooth case, and
suggest when degree computations involving non-smooth functions may be practical.
As a bonus, the examples lead to additional understanding of previously published work on
veriﬁcation involving the topological degree.
Given a system of nonlinear equations F(x) = 0, numerical methods produce
x to a solution x
. It is then sometimes desirable to compute
x is the center of
, and such that
is guaranteed to contain a solution x
to F(x) = 0. This leads to the problem
Given F :
x → R
x ∈ IR
, rigorously verify:
there exists a x
such that F(x
represents the set of n-dimensional vectors, as
, whose components are
In this introduction, we give a brief overview of our approaches to (1.1). For a
fuller understanding of the theory and techniques, see the references cited here.