REAL Computing Made REAL Preventing Errors in Scientific and Engineersg Calculations Princeton University Press Princeton, New Jersey Copyright @ 1996 by Princeton University Press Published by Prince ton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, Chichester, West Sussex All Rights Reserved Library of Congress Cataloging-in-Publication Data Acton, Forman S. REAL Computing made real : preventing errors in scientific and engineering calculations / Forman S. Acton. cm. P. Includes Index. ISBN 0-691-03663-2 (alk. paper) 1. Error analysis (Mathematics) 2. Numerical analysis-Data processing. I. Title. QA275.A25 1996 519.4'0285'5 1 d c 2 0 95- 10 606 CIP This book has been composed in Stempel Schneidler, Flintstone and Lucida Sans The publisher would like to acknowledge Forman Acton for providing the camera-ready copy from which this book was printed Text Design by C. Alvarez Princeton University Press books are printed on acid-free paper and meet the guidelines for permanence and durability of the Committee on Production Guidelines for Book Longevity of the Council on Library Resources Frontispiece. Courtesy of the Freer Gallery of Art, Washington, D.C. 98.423: An Actor in Character by Katsukawa Shunko (1743-1812) Japan, Edo period. Detail from a painting; color, lacquer black on silk: 58.7 x 26.8 cm. All other photographs courtesy of Phisit Phisitthakorn Printed in the United States of America by Princeton Academic Press CONTENTS AN EXHORTATION Make fewer errors! (Errors are hard to find.) We need help - but much software obscures! - A kind of Kyrie. VI Contents Chapter 0 TOOLS OF W E TRADE A brief collection of those useful numerical aids that may by now have faded slightly in your memory. A WORKSHOP FOR PRACTICE IN SKETCHING FUNCTIONS .... to refresh your skills at getting realistic pic- tures of your equations (and their parts). Sketch- ing an equation before an algorithm is chosen will prevent more computational disasters than any other effort. Try some! (We give our own at the end of this chapter.) GLOOMY MUSINGS 63 .. . . Correctness, Efficiency and Responsibility How can we get into the habit of thinking care- fully about our computations before writing our programs+ How can we be sensitized to pe- venting errors without becoming paralyzed into inaction+ Can we ever learn to distrust s o b a r e even while using itt - a Dies Irae! Chapter 1 W I N E A R EQUATIONS We seek rearrangements of the equation to per- mit iterations that approach roots from one side. A picture will usually lead to an algorithm that does not degenerate into futile repetitions. (It may also show that you are about to seek a so- lution that you really don't want!) Contents MI Chapter 2 PRESERVING SIGMFICAW DIGITS - When significant digits disappear, pay heed for your algorithm is often about to break down! Searching for these places will lead you to trou- ble spots and allow you to fix them before they rise to bite you. This error-prevention dividend is far more important than the digits that could be lost. We illustrate various techniques, and include a real-life problem. Chapter 3 QUADRATURES A field where it often pays to alter the problem to fit the algorithm: How to remove singularities that can spawn monumental algorithmic ineffi- ciencies. Chapter 4 RECURRENCE RELAIONS A hands-on introduction to a delightfully simple and efficient class of algorithms - that don't always work! A world where molehills really do sometimes become mountains. Chapter 5 CHOOSING AM) 'TUMNG AN ALGORITHM We follow a straightforward small problem thru our search for suitable algorithms - and are per- haps surprised to find the process rather convo- luted. ANSWERS TO MOST OF THE EXERCISES 205 - Dona Nobis Pacem! ACKNOWLEDGMENTS Altho A.P.C. long served to cure headaches, the present team of Ac- ton, Phisit, and Carmina have here delivered a book that may produce them! Let me therefore quickly declare that the designer, Carmina Alvarez, is responsible for wonderfully turning the author's penchant for oriental statues into a coherent visual style that uses some of the seventy-three photographs taken, on very short notice, by my friend Phisit Phisitthakorn, architect, of Bangkok. But neither of them can be blamed for the verbal content, which was created entirely by your author.* I would also like to thank Jane Low who led the production team at the Princeton University Press and Trevor Lipscombe, the Physical Sciences Editor, who gallantly sponsored the book even after I con- cluded that a few Shavian simplified spellings were too ingrained by my fifty-five years of use to be expunged now. * Of course the text is error-free - but you report any you find to acton@princeton edu