## Book description

This is the revised and extended second edition of the successful basic book on computer arithmetic. It is consistent with the newest recent standard developments in the field. The book shows how the arithmetic and mathematical capability of the digital computer can be enhanced in a quite natural way. The work is motivated by the desire and the need to improve the accuracy of numerical computing and to control the quality of the computed results (validity). The accuracy requirements for the elementary floating-point operations are extended to the customary product spaces of computations including interval spaces. The mathematical properties of these models are extracted into an axiomatic approach which leads to a general theory of computer arithmetic. Detailed methods and circuits for the implementation of this advanced computer arithmetic on digital computers are developed in part two of the book. Part three then illustrates by a number of sample applications how this extended computer arithmetic can be used to compute highly accurate and mathematically verified results. The book can be used as a high-level undergraduate textbook but also as reference work for research in computer arithmetic and applied mathematics.

## Table of contents

1. Foreword to the second edition
2. Preface (1/3)
3. Preface (2/3)
4. Preface (3/3)
5. Introduction (1/2)
6. Introduction (2/2)
7. I Theory of computer arithmetic
1. 1 First concepts
2. 2 Ringoids and vectoids
3. 3 Definition of computer arithmetic
4. 4 Interval arithmetic
8. II Implementation of arithmetic on computers
1. 5 Floating-point arithmetic
2. 6 Implementation of floating-point arithmetic on a computer
3. 7 Hardware support for interval arithmetic
1. 7.1 Introduction
2. 7.2 Arithmetic interval operations
3. 7.3 Circuitry for the arithmetic interval operations
4. 7.4 Comparisons and lattice operations
5. 7.5 Alternative circuitry for interval operations and comparisons
4. 8 Scalar products and complete arithmetic
1. 8.1 Introduction and motivation
2. 8.2 Historical remarks
3. 8.3 The ubiquity of the scalar product in numerical analysis
4. 8.4 Implementation principles (1/2)
5. 8.4 Implementation principles (2/2)
6. 8.5 Informal sketch for computing an exact dot product
7. 8.6 Scalar product computation units (SPUs)
8. 8.7 Comments
9. 8.8 The data format complete and complete arithmetic (1/2)
10. 8.8 The data format complete and complete arithmetic (2/2)
11. 8.9 Top speed scalar product units (1/3)
12. 8.9 Top speed scalar product units (2/3)
13. 8.9 Top speed scalar product units (3/3)
14. 8.10 Hardware complete register window
9. III Principles of verified computing
1. 9 Sample applications
1. 9.1 Basic properties of interval mathematics (1/3)
2. 9.1 Basic properties of interval mathematics (2/3)
3. 9.1 Basic properties of interval mathematics (3/3)
4. 9.2 Differentiation arithmetic, enclosures of derivatives (1/2)
5. 9.2 Differentiation arithmetic, enclosures of derivatives (2/2)
6. 9.3 The interval Newton method
7. 9.4 The extended interval Newton method
8. 9.5 Verified solution of systems of linear equations (1/2)
9. 9.5 Verified solution of systems of linear equations (2/2)
10. 9.6 Accurate evaluation of arithmetic expressions (1/2)
11. 9.6 Accurate evaluation of arithmetic expressions (2/2)
12. 9.7 Multiple precision arithmetics (1/3)
13. 9.7 Multiple precision arithmetics (2/3)
14. 9.7 Multiple precision arithmetics (3/3)
15. 9.8 Remarks on Kaucher arithmetic (1/2)
16. 9.8 Remarks on Kaucher arithmetic (2/2)
10. A Frequently used symbols
11. B On homomorphism
12. Bibliography (1/10)
13. Bibliography (2/10)
14. Bibliography (3/10)
15. Bibliography (4/10)
16. Bibliography (5/10)
17. Bibliography (6/10)
18. Bibliography (7/10)
19. Bibliography (8/10)
20. Bibliography (9/10)
21. Bibliography (10/10)
22. List of figures
23. List of tables
24. Index (1/2)
25. Index (2/2)

## Product information

• Title: Computer Arithmetic and Validity
• Author(s): Ulrich Kulisch
• Release date: April 2013
• Publisher(s): De Gruyter
• ISBN: 9783110301793