1.6 Perseverance

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 sampling computation, for x = 1, would proceed as follows:

where in each pair, the first element, a, is a lower bound, while the second element, b, is an upper bound. We have limited ourselves to three digits of precision, as before. Similarly computing samples for x = -1,0,2,3 allows us to create a reliable subgraph of y = n(x).

As our interval arithmetic was correctly carried out, we may place complete confidence in our produced subgraph: the actual samples for x=-1,0,1,2,3 lie within our computed samples.

The true strength of interval arithmetic is revealed by sampling with intervals, rather than points. Consider graphing y=m(x),

using interval arithmetic. We may sample the interval by computing , as follows:

Similarly computing samples for allows us to create a reliable graph of y = m(x).

Again, we have complete confidence in our computed graph: the true graph lies within our computed graph.

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