Real Interval Arithmetic IR

Stable Numerics Subroutine Library
Programming Reference Manual
Version 1.0 DD-00006-010

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 $2 R$ 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

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