Next: 3.2.1 Constant Functions Up: 3 Arithmetic Previous: 3.1.6 Basic Methods

# 3.2 Constant Interval Arithmetic

Let denote a constant interval number system, built from an underlying number system :

Some candidates for , which will allow machine implementations of , are the fixed-point, floating-point, fixed-slash, and floating-slash number systems [51]. The previous section discussed implementation details when , although the comments made are relevant for the other possibilities of . We no longer consider details pertaining to the choice of .

A general methodology for constructing constant interval models of real functions will be presented in this section. We will assume that an order-preserving mapping exists:

which allows us to focus on the case . We identify the number with the extended real number . This mapping need not be the obvious one. The construction of from will determine the construction of from :

The endpoints and are procedures which evaluate g at point(s) and return an endpoint based on those evaluations. In the first case, where is computed, and ; the evaluations of g are invocations of . In the second case, where is computed, and ; the evaluations of g are invocations of or . The same algorithm may be used for (and ) in both cases.

Throughout this section we may treat members of as constant functions, to ease the upcoming transition to linear interval arithmetic. Rather than describe the procedures and in a formal language, we will discuss evaluations of with examples. It is understood that much of the examination of g occurs while is being implemented, rather than during execution. Of course, such examination is possible during execution, and may be useful for complicated functions; interval arithmetic may be used to help perform such examinations. Complicated functions may be handled without direct analysis; the interval inclusion property allows such functions to be treated as compositions of simpler functions.

Knowledge of basic vector calculus is assumed; see [48] for reference. See, for example, [19, 27] for other approaches to the implementation of constant interval arithmetic.

Next: 3.2.1 Constant Functions Up: 3 Arithmetic Previous: 3.1.6 Basic Methods
 Jeff Tupper March 1996