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.
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