# Complete Interval Arithmetic and Its Implementation on the Computer

• Ulrich W. Kulisch
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5492)

## Abstract

Let $$I\textit{I \kern-.55em R}$$ be the set of closed and bounded intervals of real numbers. Arithmetic in $$I\textit{I \kern-.55em R}$$ can be defined via the power set $$\textit{I \kern-.54em P}\textit{I \kern-.55em R}$$ of real numbers. If divisors containing zero are excluded, arithmetic in $$I\textit{I \kern-.55em R}$$ is an algebraically closed subset of the arithmetic in $$\textit{I \kern-.54em P}\textit{I \kern-.55em R}$$, i.e., an operation in $$I\textit{I \kern-.55em R}$$ performed in $$\textit{I \kern-.54em P}\textit{I \kern-.55em R}$$ gives a result that is in $$I\textit{I \kern-.55em R}$$. Arithmetic in $$\textit{I \kern-.54em P}\textit{I \kern-.55em R}$$ also allows division by an interval that contains zero. Such division results in closed intervals of real numbers which, however, are no longer bounded. The union of the set $$I\textit{I \kern-.55em R}$$ with these new intervals is denoted by $$(I\textit{I \kern-.55em R})$$. This paper shows that arithmetic operations can be extended to all elements of the set $$(I\textit{I \kern-.55em R})$$.

Let $$F \subset \textit{I \kern-.55em R}$$ denote the set of floating-point numbers. On the computer, arithmetic in $$(I\textit{I \kern-.55em R})$$ is approximated by arithmetic in the subset (IF) of closed intervals with floating-point bounds. The usual exceptions of floating-point arithmetic like underflow, overflow, division by zero, or invalid operation do not occur in (IF).

### Keywords

computer arithmetic floating-point arithmetic interval arithmetic arithmetic standards

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