Disregarding all of these difficulties, we shall carry on. It is clear that we will be able to find an algorithm that can correctly graph many common equations. In fact; for any finite set of equations, we know that there exists a program that will generate the correct graph for every equation in that set.
We start by defining a novel set of numbers, along with an arithmetic over these numbers, so that we may compute without worrying excessively about numerical round-off. This arithmetic is a generalization of interval arithmetic, so we will refer to it as ``generalized interval arithmetic''. With interval arithmetic, lower and upper bounds on computed result are kept. With our previous example, of graphing y = n(x),
The true strength of interval arithmetic is revealed by sampling with intervals, rather than points. Consider graphing y=m(x),
A detailed explanation of these techniques form the bulk of this thesis. The techniques are general and may be expanded to grapple other difficult problems.
Jeff Tupper | March 1996 |