Reliable Computing 4: 1-5, 1998 A NOTE ON EPSILON-INFLATION

The epsilon-inflation proved to be useful and necessary in many verification algorithms. Different definitions of an epsilon-inflation are possible, depending on the context. Recently, certain theoretical justifications and optimality results were proved for an epsilon-inflation without absolute term. In this note we show that in currently used interval iterations the epsilon-inflation without absolute term does not serve the purpose it is defined for. A new epsilon-inflation is proposed. Many verification algorithms for calculating an inclusion of the solution of a given problem use Banach’s or Brouwer’s Fixed Point theorem. The main point of those algorithms is to verify that a certain interval is mapped into itself or into its interior. We assume the reader is familiar with the fact that this self-mapping is the central part of many verification algorithms for systems of linear or nonlinear equations, algebraic eigenproblems, polynomial zeros and others. References include [2], [9], [12], [13] and many more. For an overview see e.g. [7], commercial implementations include [1], [3], [8], [16]. If this self-mapping cannot be verified for the initial test interval, an interval iteration is started. To the author’s knowledge, it was first noted by Caprani and Madsen [5] that it is useful to enlarge the computed iterates prior to the next iteration in order to increase chances for a self-mapping. The term epsilon-inflation was introduced in [13]. For a real interval X the original definition is [13, Definition 2.6], X ◦ ε := { X + d(X) · [−ε, ε] for d(X) 6= 0 X + [−η, +η] otherwise, where d denotes the diameter and η denotes the smallest representable positive machine number. In later papers an analysis of the benefits of the epsilon-inflation was given (cf. [14]). These results can be summarized as follows. Let Z, X ∈ IK be interval vectors, and let C ∈ IMn(K) be an n× n interval matrix for K ∈ {R,C}. Define the interval iteration Y k := X ◦ ε and X := Z + C · Y k for k ≥ 0 . (1.1) Using the simplified definition X ◦ ε := X + d(X) · [−ε, ε] + [−η, +η] (1.2) of the epsilon-inflation in (1.1), the following is true (|C| is the matrix of entrywise absolute values of C; ρ denotes the spectral radius): I) If interval operations are used in the iteration (1.1) and ρ(|C|) < 1/(1 + 2ε), then the inclusion