2.10 Generalized Floating Point Interval Arithmetic

Actual implementations of interval arithmetic use floating point numbers to describe interval bounds. I will only discuss directly since a re-reading with an appropriate fixed choice of f will provide a discussion of , , or .

I will assume the bound description function f takes n parameters and has k floating point coefficients:

As before, an interval description is valid if the described interval is non-collapsing. An interval i is a member of if and only if a description of i is valid:

An interval model of an m-ary function g has the interval inclusion property if

The interval extension of g is derived from the real interval extension :

Deriving the floating point extension from the real extension means that the really hard decision of how to bound an arbitrary function is made once. It also lends credence to the concept that is an abstract model of . Deriving the bounds in this way will lead to suboptimal bounds since the demotion does not take into account the granularity of . The difference between the two stage demotion and the direct demotion will only be on the order of machine precision, however.

There are two differences between and . One is the two-stage demotion used in the interval extension. The other is the floating point evaluation of interval bounds.

A bound f is a function from to . Implementations will have to evaluate f using floating point numbers. Formally, this is stated as a straight forward application of a demotion from to . A floating point number x is a member of interval for parameter value if

which promoted x to an extended real. This is not what an implementation would do. An implementation would round the bounds outward, so that a floating point number x is a member of for parameter value if

Jeff Tupper

March 1996