Reliable Computing 11: 77–85
On the Proofs of Some Statements
Concerning the Theorems
of Kantorovich, Moore, and Miranda
ur Angewandte Mathematik, University of Karlsruhe, D–76128 Karlsruhe, Germany,
(Received: 13 August 2003; accepted: 21 May 2004)
Abstract. In this paper we discuss the proofs of some results comparing existence statements for the
solutions of nonlinear equations. In one important case we correct a proof already published.
Thetheorems of Kantorovich, Moore, and Miranda are important results proving
the existence of solutions of nonlinear equations. In recent years several results
have been published comparing the assumptions which have to be made for these
An interesting comparison between the theorems of Kantorovich and Moore
has been given by Rall in , see Theorem 4.1 below. Neumaier and Shen 
improved Rall’s result by using a different slope enclosure. They used the same
additional assumption (4.1) as Rall in , see Theorem 4.2 below. In , Shen and
Wolfe stated a similar result for the afﬁne invariant formulation of Kantorovich’s
theorem (see ) without the assumption (4.1).
The purpose of this paper is twofold. First, we show that the proof in  is
incorrect and that it appears that it cannot be corrected without assuming (4.1).
Therefore, we doubt that the result really holds. This is discussed in some detail
in Section 4. Secondly, we give a correct proof of Theorem 5.1. This theorem was
originally stated in  with a reference to  indicating that the proof proceeds as
speciﬁed there. However, the proof was also incorrect, as was already mentioned
Recently, the discussion has been extended by Alefeld, Frommer, Heindl, and
Mayer , leading to a comparison between Kantorovich’s theorem, a generaliza-
tion of Miranda’s theorem and Borsuk’s theorem. A survey of the whole topic is
given byFrommer .
Thepaper is organized as follows. In the next section we recall the theorems
of Kantorovich, Moore, and Miranda. In Section 3, well-known possibilities for
choosing slope matrices and slope enclosures are introduced. Section 4 leads to