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Complete Interval Arithmetic and Its Implementation on the Computer

2009; Springer Science+Business Media; Linguagem: Inglês

10.1007/978-3-642-01591-5_2

1611-3349

Ulrich Kulisch,

Digital Filter Design and Implementation

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).