As extended numbers are useful when discussing floating point numbers, real intervals are a useful abstract model of floating point intervals. The set of real intervals is denoted by . Each real interval is specified by a lower and upper endpoint, both of which are extended real numbers.
The syntax for intervals is used for all forms of interval arithmetic, and will be used for abstract models of interval arithmetic as well. The ensuing development of interval arithmetic will flesh out the concepts introduced by floating point interval arithmetic.
Interval arithmetic is used to model computations with reals. Operators are defined over the reals and then modelled with interval operators. The interval inclusion property gives interval methods their rigor. An n-ary function is a valid interval representation of the n-ary function if satisfies the interval inclusion property. The function satisfies the inclusion property if
The judgement of model quality can again be guided by the interval extension of a real function . The interval extension is defined as before:
Real intervals behave much like the floating point intervals they abstract. The abstraction allows one to ignore the effects of rounding, which can simplify discussion and analysis.
|Jeff Tupper||March 1996|