Computer Arithmetic in Theory and Practice

Computer Arithmetic in Theory and Practice

1st Edition - March 28, 1981

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  • Authors: Ulrich W. Kulisch, Willard L. Miranker
  • eBook ISBN: 9781483264738

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Computer Arithmetic in Theory and Practice deals with computer arithmetic and the various implementations of the entire arithmetic package on diverse processors, including microprocessors. It illustrates the importance of theoretical development in the sound implementation of arithmetic on computers, and argues that such an implementation requires the establishment of various isomorphisms between different definitions of arithmetic operations. Comprised of seven chapters, this volume begins with an introduction to the theory of computer arithmetic by giving an axiomatic characterization of the essential properties of sets and subsets; complete lattices and complete subnets; screens and roundings; and arithmetic operations. The discussion then turns to the concepts of a ringoid and a vectoid as well as those of ordered or weakly ordered ringoids and vectoids; interval arithmetic; and floating-point arithmetic. The operations in interval spaces are defined by means of semimorphisms. The final chapter shows how to embed the five basic data types (integer, real, complex, real interval, and complex interval) together with the arithmetic operations that are defined for all of these types into existing higher programming languages. This book will be helpful to students and practitioners in the fields of computer science and applied mathematics.

Table of Contents

  • Preface

    Introduction and Preliminary Definition of Computer Arithmetic

    Part 1. Theory of Computer Arithmetic

    Chapter 1. First Concepts

    1. Ordered Sets

    2. Complete Lattices and Complete Subnets

    3. Screens and Roundings

    4. Arithmetic Operations and Roundings

    Chapter 2. Ringoids and Vectoids

    1. Ringoids

    2. Vectoids

    3. Completely Ordered Ringoids and Vectoids

    Chapter 3. Definition of Computer Arithmetic

    1. Introduction

    2. Preliminaries

    3. The Vertical Definition of Computer Arithmetic

    4. The Horizontal Definition of Computer Arithmetic by Semimorphisms

    5. A Remark about Roundings

    Chapter 4. Interval Arithmetic

    1. Interval Sets and Arithmetic

    2. Interval Arithmetic over a Linearly Ordered Set

    3. Interval Matrices

    4. Interval Vectors

    5. Interval Arithmetic on a Screen

    6. Interval Matrices and Interval Vectors on a Screen

    7. Complex Interval Arithmetic

    8. Complex Interval Matrices and Vectors

    Part 2. Implementation of Arithmetic on Computers

    Chapter 5. Floating-Point Arithmetic

    1. Definition and Properties of the Real Numbers

    2. Floating-Point Numbers and Roundings

    3. Floating-Point Operations

    4. Extended Floating-Point Arithmetic

    Chapter 6. Implementation of Floating-Point Arithmetic on a Computer

    1. The Level 1 Routines

    2. Introductory Remarks about the Level 2 Routines

    3. Addition and Subtraction

    4. Normalization

    5. Rounding

    6. Multiplication

    7. Division

    8. Algorithms Using the Short Accumulator

    9. Underflow and Overflow Treatment

    10. Computation with Maximum Accuracy of the Sum of n Floating-Point Numbers

    11. An Alternative Algorithm for Computing the Sum of n Floating-Point Numbers

    12. The Level 3 Routines

    Chapter 7. Computer Arithmetic and Programming Languages

    1. Introduction

    2. General Description of the Language Extension

    3. Arithmetic Standard Functions of the Language Extension

    4. Syntax of the Language Extension

    5. Syntax Variables Used in the Diagrams


    Glossary of Symbols and Formulas


Product details

  • No. of pages: 268
  • Language: English
  • Copyright: © Academic Press 1981
  • Published: March 28, 1981
  • Imprint: Academic Press
  • eBook ISBN: 9781483264738

About the Authors

Ulrich W. Kulisch

Willard L. Miranker

About the Editor

Werner Rheinboldt

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