Allowing functions with more descriptive power as an interval bounds is the obvious way to generalize interval arithmetic.
For any particular function 
there is an interval arithmetic number system
with an abstract model
. I will assume that
the function has k ``coefficients'', and n ``parameters''.
Each interval bound would then be specified by k
extended real numbers, and would vary over an n-dimensional domain:
states that the k
coefficients of f are filled in by the k elements of 
.
An interval description is valid
if the described interval does not collapse:
An interval model 
of an m-ary function g is
valid if 
has the inclusion property.
The model has the inclusion property if
The interval extension of g is also defined as before:
Different choices of f lead to differing complexity
in the implementation of the operator models
(such as 
, 
,
and 
) and the demotion operators
and 
. The choice of
f will affect how well the intervals can track the
underlying stream of real computations as well as
how useful the computed results will be.
| Jeff Tupper | March 1996 |