Complete Interval Arithmetic and Its Implementation on the Computer

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).

Authors and Affiliations

1. Institut für Angewandte und Numerische Mathematik, Universität Karlsruhe, Germany

   Ulrich W. Kulisch

Editors and Affiliations

1. Department of Mathematics and Computer Science, Universiteit Antwerpen, Middelheimlaan 1, 2020, Antwerpen, Belgium

   Annie Cuyt

2. Wissenschaftliches Rechnen/Softwaretechnologie, Bergische Universität Wuppertal, Gausstrasse 20, 42097, Wuppertal, Germany

   Walter Krämer

3. Fakultät Ingenieurwissenschaften, Abteilung Informatik und Angewandte Kognitionswissenschaft, Universität Duisburg-Essen, 47048, Duisburg, Germany

   Wolfram Luther

4. 160 Redland Road, CA 94062, Woodside, USA

   Peter Markstein

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