Generalized interval techniques may be explored further. Knowledge of theoretical results concerning the efficiency of generalized interval evaluation would be reassuring; complete knowledge is not possible, as the exact determination of the value of a function is, in general, not computable.
In practice, a system that mechanically produces generalized interval
arithmetic libraries would be appreciated.
With such a system, human effort is applied to a more
abstract system, allowing for the deployment of
a variety of efficient interval arithmetic libraries
with a reduced likelihood of implementation error.
Another implementation approach is, for example,
to implement 
in place of 
;
a thorough evaluation of this approach may be informative.
The interval arithmetic presented may be generalized further. Other data types and operations may be considered, as in [52]; probabilistic arithmetic is another possibility. Integrating automatic differentiation [61] into this framework is another possible pursuit.
An ongoing challenge is to integrate generalized interval arithmetic into a wide variety of applications. As with graphing, the free variables provided by a generalized interval arithmetic should be gainfully exploited by the benefitting application.
Graphing may be explored further. Algorithms for accurately rendering differential equations, integral equations, and iterated function systems may be explored. Higher dimensions are intriguing; the implementation of an interval-based renderer which models the interaction of light within a scene is a tempting challenge. Such algorithms would return reliable bounds on the colour assigned to a pixel; a quantized colour system would provide a natural stopping criteria.
| Jeff Tupper | March 1996 |