17 Bertil Gustafsson Scientifi c Computing A Historical Perspective Editorial Board T. J.Barth M.Griebel D.E.Keyes R.M.Nieminen D.Roose T.Schlick Texts in Computational Science and Engineering 17 Editors Timothy J. Barth Michael Griebel David E. Keyes Risto M. Nieminen Dirk Roose Tamar Schlick More information about this series at http://www.springer.com/series/5151 Bertil Gustafsson Scientific Computing A Historical Perspective 123 Bertil Gustafsson Department of Information Technology Uppsala University Uppsala, Sweden ISSN 1611-0994 ISSN 2197-179X (electronic) Texts in Computational Science and Engineering ISBN 978-3-319-69846-5 ISBN 978-3-319-69847-2 (eBook) https://doi.org/10.1007/978-3-319-69847-2 Library of Congress Control Number: 2018953059 Mathematics Subject Classification (2010): 65-XX, 68-XX, 01-XX © Springer International Publishing AG, part of Springer Nature 2018 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Cover illustration: Figure 2.1 showing the Babylonian clay tablet. (The clay tablet YBC7289. From the Yale Babylonian Collection, with the assistance and permission of William Hallo, Curator, and Ulla Kasten, Associate Curator. Photo by Bill Casselman, http://www.math.ubc.ca/_cass/Euclid/ybc/ybc. html.) This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Preface Science and technology are traditionally based on theory and experiments, but now- adays scientific computing is an established third branch of strategic importance. It is based on mathematical models, some of which were developed many centuries ago. Most of these models are in the form of different kinds of equations that must be solved, and in the ideal case, one can find the solutions by analytic means in the form of an explicit mathematical form, as for example a function f(x,y) that can be easily evaluated for any value of the independent variables x and y.However,in most realistic cases this is not possible, and numerical methods must be used to find accurate approximations of the solutions to the mathematical model. The concept of scientific computing usually means the whole procedure including analysis of the mathematical model, development and analysis of a numerical method, programming the resulting algorithm, and finally running it on a computer. Today, there are many systems that include the whole chain and require only specifying the physical parameters for the problem to be solved numerically. However, there are always new challenges with new types of mathematical models that require new numerical methods. This book is about some of the most significant problem areas and the history of the process leading to efficient numerical methods. Mathematics in the old days was very closely connected to astronomy, which was always the major source for new mathematical problems up to the nineteenth century. According to Galileo Galilei (1564–1642), “the book of nature is written in the language of mathematics,” and this was particularly true for astronomy. As we shall see, many of the most famous mathematicians in the past could as well have been called astronomers. For example, perhaps the most famous mathematician of all time, Carl Friedrich Gauss (1777–1855), was the director of the German astronomical observatory Göttingen Observatory for 48 years. Today, there is a clear distinction between pure mathematics and applied mathematics with scientific computing quite far from pure mathematics. This was not the case in the old days. Mathematicians found new models for various physical problems, but they also carried out the necessary computations to find the unknown v vi Preface numbers they were looking for. A typical example was to predict the future location of a certain planet based on a few available observations. The content of this book is divided into four parts. The first is about the very early mathematical/numerical achievements made by the Babylonians and the Greeks. After that not much happened until the seventeenth century when Newton and others developed new mathematics, and the second part is about the development during the centuries until the Second World War. Just at the end of the war, scientific computing took a giant step forward with the construction of electronic computers. Now numerical methods that earlier led to computations of impossible size could be implemented on these computers with results obtained after a few hours. In this new situation, the development of new methods became much more interesting, and the existence of electronic computers provided a strong boost for numerical analysis and scientific computing. The third part of the book is about this postwar period until the end of the 1950s. Around that time, scientific computing became a third scientific method in addition to the traditional branches in theory and experiments. The fourth part of the book covers the period until the present time. The major numerical methods are traced back to their origin and to the people who invented them, as well as to the origin of important techniques for analysis of the methods. There is also short presentations of some of the mathematicians who played a key role in this process. There is certainly not a complete description of the whole story behind all methods, but rather an attempt to catch the key steps without going into a complete mathematical derivation. There is also a very brief presentation of the development of electronic computers, particularly the early ones. Differential equations are the dominating type of mathematical models for almost every branch of science and engineering, and there are several different principles that are used when developing numerical methods for the solution of these problems. Therefore, differential equations are given some extra space in this book. One difficulty when going back in time is that the original articles are not always easy to read and understand. During the active period from the sixteenth to the nineteenth century, Latin was a common language for mathematical texts. Another major difficulty was the way mathematics was described. Many symbols used today were not introduced at that time, and numerical methods were described in words in quite a lengthy and complicated way. In fact, in some cases there is still a certain uncertainty about the exact meaning of the description, and a consequence of this may be that the real inventor of a certain method is not known for sure. We are aware of the fact that the whole area of scientific computing is not covered in this book. There are many techniques and methods that are left out or just briefly mentioned, and there are many mathematicians who could have been included but are not. The book is intended to give the big picture and the historical development behind the rise of scientific computing as a new scientific branch. The content of this book is based on a large number of sources in addition to the original books and articles in the reference list. It is impossible to give all of these sources, in particular for some of the notes concerning important mathematicians, but as far as possible, correctness has been checked. Preface vii Hopefully, the book will be a valuable resource for all students and professionals interested in the history of numerical analysis and computing and for a broader readership alike. Uppsala, Sweden Bertil Gustafsson Acknowledgments This book was written after my retirement from the chair at the Division of Scientific Computing at Uppsala University. The Department of Information Technology has still provided full access to the necessary infrastructure, which is much appreciated. Martin Peters and Ruth Allewelt at Springer have been helpful, in particular when it comes to copyright issues. Ann Kostant corrected language deficiencies and misprints, and I am impressed by her ability to find so many errors and language details that required correction. Finally, I would like to thank my wife Margareta for accepting and supporting my full engagement in writing still another book. ix Contents 1 Scientific Computing: An Introduction.................................... 1 2 Computation Far Back in Time ............................................ 5 2.1 The Babylonians ....................................................... 6 2.2 Archimedes and Iterative Methods