We begin by formalizing interval arithmetic, after a brief formal review of some standard number systems. A variety of interval arithmetics are developed, which will allow us to cope with ``badly behaved'' equations. Much of the generalizations are novel, and developed by the author.
In the third chapter, a detailed exposition of the arithmetic of generalized intervals is presented. A general approach is taken, so that a similar set of rules may be followed when computing in any one of the myriad of interval arithmetics presented.
We will then precisely define what a graph is, to bring the mathematical idealization into the realm of Computer Science. This will allow for strong results, as we will then have a concrete, realizable goal. Results using several different interval arithmetics will be presented and briefly analysed.
| Jeff Tupper | March 1996 |