INTERVAL ANALYSIS AND COMPLEX CENTERED FORMS

Abstract

In this paper we first discuss, briefly, the problem of approximating the real numbers with floating point computer representable numbers (a continuous space approximated by a discrete space). This approximation leads to uncertainties in numerical calculations. A tool for estimating and controlling the errors in numerical calculations in a discrete space is interval analysis. Interval analysis is therefore introduced and some basic properties are given. The merits and demerits of interval analysis are then discussed in some detail. As examples of interval analysis tools and algorithms, the natural extension idea as well as Newton's method in one dimension are discussed. The computation of inclusions for the range of functions is furthermore discussed placing particular emphasis on centered forms. We then turn to the definition of a complex interval arithmetic as well as natural extensions in this arithmetic. Here we present a number of results for polynomials and rational functions showing in particular that centered circular complex forms have some nice properties (explicit formulas, convergence, comparisons). Some numerical results are also given. A final brief discussion is given for the problem of subdividing a circle for the purpose of obtaining improved inclusions.