Complete Interval Arithmetic and its Implementation

: Let IIR be the set of closed and bounded intervals of real numbers. Arithmetic in IIR can be defined via the power set IPIR of real numbers. If divisors containing zero are excluded, arithmetic in IIR is an algebraically closed subset of the arithmetic in IPIR, i.e., an operation in IIR performed in IPIR gives a result that is in IIR. Arithmetic in IPIR 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 IIR with these new intervals is denoted by (IIR). This paper shows that arithmetic operations can be extended to all elements of the set (IIR). Let F IR denote the set of floating-point numbers. On the computer, arithmetic in (IIR) is approximated by arithmetic in the subset (IF) of closed intervals with floatingpoint bounds. The usual exceptions of floating-point arithmetic like underflow, overflow, division by zero, or invalid operation do not occur in (IF).

• Type: Conference
• Year: 2008
• Where: DAGSTUHL
• Authors: Ulrich W. Kulisch
• Keywords: Arithmetic, DAGSTUHL 2008, Real Numbers, Set Iir, Software Engineering