Stable Numerics Subroutine Library
Programming Reference Manual
Version 1.1 DD-00006-011

3.1 Real Interval Arithmetic IR

Interval arithmetic consists of a tuple of two numbers representing an infinimum and supremum of an interval. By definition an interval is in R2   or Z2   we will however for distinction represent interval numbers as IR  (interval real space) and IZ  (interval integer space) as they are used in a distinct manner differing from standard R2   or Z2   arithmetic. The definition of an interval is:

      -   -       -
IR = {[x,x]|x,x∈ R,x≤ x}

The two elements in an interval x∈IR  or  2
Z   are called infinimum x and a supremum -
x where for a normalized interval    -
x ≤x . The two latter are thereby representing the respective infinimum and supremum of an interval hence by definition an interval X ∈ IR  actually defines a set of values:

X ={∀a∈ R:x≤ a≤ x}

This makes it distinct from R2   where a number in the latter would correspond to a fixed defined point.

3.1.1 Singleton Interval
3.1.2 Empty Interval
3.1.3 Sign of an Interval
3.1.4 Bounded and Whole Interval
3.1.5 Normalized Interval