An Axiomatic Approach to Rounded Computations*

The present paper is intended to give an axiomatic approach to rounded computations. A rounding is defined as a monotone mapping of an ordered set into a subset, which in general is called a lower respectively an upper screen. The first chapter deals with roundings in ordered sets. In the second chapter further properties of roundings in linearly ordered sets are studied. The third chapter deals with the two most important applications, the approximation of the real arithmetic on a finite screen and the approximation of the real interval arithmetic on an upper screen. Beyond these examples various further applications are possible. 0. Introduction I t is a matter of fact that there hardly exist two computers of different companies which are using the same rounding procedure. The most frequently used programming languages like ALGOL-60, FORTRAN or PL/t therefore do not make any assumption of special properties of the rounding procedure for floating point numbers. Numerical methods however, which are trying to get control of the rounding errors, have to make use of clearly defined rounding procedures. Therefore an axiomatic approach to rounded computations seems to be necessary. There are already former attempts to define the approximation of the real arithmetic in a finite subset in an axiomatic way [6]. Further, more unconscious ideas in this direction can be found in [t-5]. We will t ry here to capture all the essential properties connected with rounded computations by setting up a system of axioms for more general problems, which permit a lot of applications. There are many further problems in applied mathematics which nearly show the same behavior. Let for instance M be the set of all real n • n matrices and (M) the power set of M. Then it is well known that M is an ordered set and we can define the set I (M) of intervals over M. If we define a multiplication for elements A, BEI(M) by: A • B : = { a • b[aEA ̂ bEB}, then it is easy to see, that A • B in general is an element of IP (M) and not of I (M). To get again an element of I (M), we have to round the result A • B from ~ (M) into I (M). This process is very similar to that of approximating the real arithmetic in a finite subsystem. Therefore we shall define a rounding as an approximation of a set M with an order relation defined on M in a subset T of M or, more generally as an approximation of an algebraic structure {M, ,} (*: M • M-->M) in a subset T with an * Sponsored by the Mathematics Research Center, Madison, Wisconsin, under Contract No.: DA-31-124-ARO-D-462. 1 Numer. Math., Bd. 18