Computer arithmetic and programming languages

Abstract

For historical reasons, the early programming languages lacked a precise definition of the arithmetic operations and the roundings which are to be employed. The paper summarizes an extensive research activity on computer arithmetic that went on during the last fifteen years as well as the experience gained through various implementations. 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. The arithmetic operations are defined by a general mapping property which is called a semimorphism. We discuss the properties of semimorphisms, show briefly how they can be obtained and mention the most important features of their implementation on computers. The properties of a semimorphism should be used for an axiomatic definition of the arithmetic operations and the roundings within each programming language. Then we show that the new, semimorphic operations can not be properly addressed by existing programming languages. Correcting this limitation led to extensions of PASCAL [4] and FORTRAN [3]. APL is much better suited for adoptation of the new arithmetic operations. A demonstration of a computer that has been systematically equipped with the new arithmetic will follow the talk. 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.