Numerical Computing with IEEE Floating Point Arithmetic

121 Pages·2001·6.31 MB·English

by  Michael L. Overton

Numerical
Computing
with IEEE Floating
Point Arithmetic
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Numerical
Computing
with IEEE Floating
Point Arithmetic
Including One Theorem, One Rule of Thumb,
and One Hundred and One Exercises
Michael L. Overton
Courant Institute of Mathematical Sciences
New York University
New York, New York
siam.
Society for Industrial and Applied Mathematics
Philadelphia
Copyright © 2001 by the Society for Industrial and Applied Mathematics.
1098765432
All rights reserved. Printed in the United States of America. No part of this book may be
reproduced, stored, or transmitted in any manner without the written permission of the
publisher. For information, write to the Society for Industrial and Applied Mathematics,
3600 University City Science Center, Philadelphia, PA 19104-2688.
Library of Congress Cataloging-in-Publication Data
Overton, Michael L
Numerical computing with IEEE floating point arithmetic / Michael L Overton.
p. cm.
Includes bibliographical references and index.
ISBN 0-89871-571-7
I. Computer arithmetic. 2. Floating-point arithmetic. 3. Numerical calculations.
I. Title.
QA76.9.M35O94200I
O04'.0l'5l--dc2l
00-067941
SlcLJTL is a registered trademark.
Dedicated to girls who like math
especially my daughter
Eleuthera Overton Sa
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Contents
Preface ix
Acknowledgments xi
1 Introduction 1
2 The Real Numbers 5
3 Computer Representation of Numbers 9
4 IEEE Floating Point Representation 17
5 Rounding 25
6 Correctly Rounded Floating Point Operations 31
7 Exceptions 41
8 The Intel Microprocessors 49
9 Programming Languages 55
10 Floating Point in C 59
11 Cancellation 71
12 Conditioning of Problems 77
13 Stability of Algorithms 83
14 Conclusion 97
Bibliography 101
vii
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Preface
Numerical computing is a vital part of the modern scientific infrastructure. Almost
all numerical computing uses floating point arithmetic, and almost every modern
computer implements the IEEE1 binary floating point standard, published in 1985.
This standard is arguably the most important in the computer industry, the result of an
unprecedented cooperation between academic computer scientists and the cutting edge
of industry. Nonetheless, many years after its publication, the key ideas of the IEEE
standard remain poorly understood by many students and computer professionals.
Perhaps this is because an easily accessible yet reasonably detailed discussion of the
standard has not been available—hence, the evolution of this short book. Although it
is intended primarily for computer science or mathematics students, as a supplement
to a more traditional textbook for a course in scientific computing, numerical analysis,
or computer architecture, it also aims to reach a broader audience. As well as the IEEE
standard, topics include the floating point architecture of the Intel microprocessors,
a discussion of programming language support for the standard, and an introduction
to the key concepts of cancellation, conditioning, and stability. The book should be
accessible to any reader with an interest in computers and mathematics. Some basic
knowledge of calculus and programming is assumed in the second half. The style is
not that of a traditional textbook. There is enough variety of content that all but the
most expert readers will find something of interest here.
A web page for the book is maintained at
http://www.cs.nyu.edu/cs/faculty/overton/book/
Refer to this page for corrections to the text, to download programs from the book,
and to link to the web pages mentioned in the bibliography, which will be updated as
necessary.
MICHAEL L. OVERTON
1 Institute for Electrical and Electronics Engineers. IEEE is pronounced "I triple E.
IX