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.

In electronic computers the elementary-arithmetic operations are these days generally approximated by floating-point operations of highest accuracy. Vector processors and parallel computers often provide additional operations like “multiply and add”, “accumulate” or “multiply and accumulate”. Also these operations shall always deliver the correct answer whatever the data are. The user should not be oblighed to execute an error analysis for operations predefined by the manufacturer. In the first part of this paper we discuss circuits which allow a fast and correct computation of sums and scalar products making use of a matrix shaped arrangement of adders and pipeline technology. In the second part a variant is discussed which permits a drastic reduction in the number of adders required. The methods discussed in this paper can also be used to build a fast arithmetic unit for micro computers in VLSI-technology.

During recent years a number of papers concerning a mathematical foundation of computer arithmetic have been written. Some of these papers are still unpublished. The papers consider the spaces which occur in numerical computations on computers depending on a properly defined computer arithmetic. The following treatment gives a summary of the main ideas of these papers. Many of the proofs had to be sketched or completely omitted. In such cases the full information can be found in the references.

During the last years a number of papers concerning a mathematical foundation of computer arithmetic have been written. Some of these papers are still unpublished. The papers consider the spaces which occur in numerical computations on computers in dependence of a properly defined computer arithmetic. The following treatment gives a summary of the main ideas of these papers. Many of the proofs had to be sketched or completely omitted. In such cases the full information can be found in the references.