The Arithmetic of the Digital Computer: A New Approach

Abstract

A new approach to the arithmetic of the digital computer is surveyed. The methodology for defining and implementing floating-point arithmetic is described. Shortcomings of elementary floating-point arithmetic are revealed through sample problems. The development of automatic computation with emphasis on the user control of errors is reviewed. The limitations of conventional rule-of-thumb procedures for error control in scientific computation are demonstrated by means of examples. Computer arithmetic is extended so that the arithmetic operations in the linear spaces and their interval correspondents which are most commonly used in computation can be performed with maximum accuracy on digital computers. A new fundamental computer operation, the scalar product, is introduced to develop this advanced computer arithmetic. A process of automatic error control called validation which delivers high accuracy with guarantees for scientific computations is described. Validation of computations for a large class of numerical problems is made possible by advanced computer arithmetic. High accuracy is furnished by coupling the scalar product with the process of-defect correction. Guarantees and error bounds are obtained by interval techniques. This whole process establishes certain numerical algorithms such as the evaluation of rational expressions as additional higher order arithmetic operations. The development of some programming languages in the context of computer arithmetic is reviewed. A collection of constructs in terms of which a source language may accommodate the methodology of computer arithmetic in a user-friendly mode is described. Finally the current state of implementation of the ideas discussed here is reviewed.