Mathematics and Speed for Interval Arithmetic

Abstract

After a short introduction, the article begins with an axiomatic definition of rounded arithmetic. The concepts of rounding and of rounded arithmetic operations are defined in an axiomatic manner fully independent of special data formats and encodings. Basic properties of floating-point and interval arithmetic can directly be derived from this abstract mathematical model. Interval operations are defined as set operations for elements of the set ¯IR of closed and connected sets of real numbers. As such, they form an algebraically closed subset of the powerset of the real numbers. This property leads to explicit formulas for the arithmetic operations of floating-point intervals of ¯IF, which are executable on the computer. Arithmetic for intervals of ¯IF forms an exception free calculus, i.e., arithmetic operations for intervals of ¯IF always lead to intervals of ¯IF again. Later sections are concerned with programming support and hardware for interval arithmetic. Both are a must and absolutely necessary to move interval arithmetic more into the center of scientific computing. With some minor hardware additions, interval operations can be made as fast as simple floating-point operations. In vector and matrix spaces for real, complex, and interval data, the dot product is a fundamental arithmetic operation. Computing the dot product of two vectors with floating-point components exactly substantially speeds up floating-point and interval arithmetic as well as the accuracy of the computed result. Hardware needed for the exact dot product is very modest. The exact dot product is essential for long real and long interval arithmetic. Section 9 illustrates that interval arithmetic as developed in this article already has a long tradition. Products based on these ideas have been available since 1980. Implementing what the article advocates would have a profound effect on mathematical software. Modern processor architecture from Intel, for example, comes quite close to what is requested in this article.