A new arithmetic for scientific computation with exact evaluation of expressions

Abstract

The paper summarizes an extensive research activity in computer arithmetic and scientific computation that went on during the last fifteen years. We also discuss the experience gained through various implementations of a new approach to arithmetic on diverse processors including microprocessors.

We begin with a complete listing of the spaces that occur in numerical computations. This leads to a new and general definition of computer arithmetic.

Then we discuss aspects of traditional computer arithmetic such as the definition of the basic arithmetic operations, the definition of the operations in product spaces and some consequences of these defintions for error analysis of numerical algorithms.

In contrast to this we then give the new definition of computer arithmetic. The arithmetic operations are defined by a general mapping principle which is called a semimorphism. We discuss the properties of semimorphisms, show briefly how they can be obtained and mention the most important feartures of their implementation on computers.

Then we show that the new operations can not be properly addressed by existing programming languages. Correcting this limitation led to extensions of PASCAL and FORTRAN.

A demonstration of a computer that has been systematically equipped with the new arithmetic will follow. The new arithmetic turns out to be a key property for an automatic error control in numerical analysis. By means of a large number of examples we show that guaranteed bounds for the solution with maximum accuracy can be obtained. The computer even proves the existence and uniqueness of the solution within the calculated bounds. If there is no unique solution (e.g. in case of a singular matrix) the computer recognizes it. Toward the end of the paper we sketch how expressions or program parts can be evaluated with high accuracy.