Let 
denote a constant interval number system, built
from an underlying number system 
:
, 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:
.
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
:
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.
| Jeff Tupper | March 1996 |