Table Of ContentLectures on Numerical Mathematics
Heinz Rutishauser
Lectures on Numerical Mathematics
Edited by
Martin Gutknecht
with the Assistance of
Peter Henrici
Peter Uiuchli
Hans-Rudolf Schwarz
Translated by
Walter Gautschi
With 67 Figures
1990 Birkhauser
Boston' Basel' Berlin
Translator
Walter Gautschi
Department of Computer Sciences
Purdue University
West Lafayette, IN 47907
USA
Library of Congress Cataloging-in-Publication Data
Rutishauser, Heinz, 1918-1970.
[Vorlesungen liber numerische Mathematik. English]
Lectures on numerical mathematics/Heinz Rutishauser; edited by
Martin Gutknecht, with the assistance of Peter Henrici, Peter
Uiuchli, Hans-Rudolf Schwarz; translated by Walter Gautschi.
p. cm.
Translation of: Vorlesungen liber numerische Mathematik; German
ed. issued in 2 vols.
Includes bibliographical references.
ISBN-\3:978-1-4612-8035-4 e-ISBN-13:978-1-4612-3468-5
DOl: 10.1007/978-1-4612-3468-5
I. Numerical analysis. 1. Gutknecht, Martin. II. Tille.
QA297.R8713 1990
512' .7-dc20 90-58
Printed on acid-free paper.
© Birkhauser Boston, 1990
Softcover reprint of the hardcover 1st edition 1990
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or
transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise,
without prior permission of the copyright owner.
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CCC, 21 Congress Street, Salem, MA 01970, USA. Special requests should be addressed directly to
Birkhauser Boston, Inc., 675 Massachusetts Avenue, Cambridge, MA 02139, USA.
Camera-ready copy supplied by the translator using troff (UNIX).
9 8 7 6 5 4 3 2 I
Editor's Foreword
The present book is an edition of the manuscripts to the courses
"Numerical Methods I" and "Numerical Mathematics I and II" which
Professor H. Rutishauser held at the E.T.H. in Zurich. The first-named
course was newly conceived in the spring semester of 1970, and intended
for beginners, while the two others were given repeatedly as elective
courses in the sixties. For an understanding of most chapters the funda
mentals of linear algebra and calculus suffice. In some places a little
complex variable theory is used in addition. However, the reader can get
by without any knowledge of functional analysis.
The first seven chapters discuss the direct solution of systems of
linear equations, the solution of nonlinear systems, least squares prob
lems, interpolation by polynomials, numerical quadrature, and approxima
tion by Chebyshev series and by Remez' algorithm. The remaining
chapters include the treatment of ordinary and partial differential equa
tions, the iterative solution of linear equations, and a discussion of eigen
value problems. In addition, there is an appendix dealing with the qd
algorithm and with an axiomatic treatment of computer arithmetic.
For a few algorithms, also problems of programming are discussed
and fragments of ALGOL-programs are given. It should be pointed out
that a number of complete and safe procedures to the methods described
here are to be published (also with Birkhiiuser, as Vol. 33 of the Interna
tional Series of Numerical Mathematics) by W. Gander, L. Molinari and
H. §vecova under the title "Numerische Prozeduren aus Nachlass und
Lehre von Prof. Heinz Rutishauser".
When Professor H. Rutishauser died on November 10, 1970, at the
age of 52, he left behind, among other things, the manuscripts to the
vi Editor's Forword
courses mentioned above. These notes, which Mrs. M. Rutishauser
kindly made available to us, have in the course of the years been repeat
edly revised and updated to reflect progress in research. His desire to
publish them later as a text book was known; some sections were already
finished in an almost ready-to-print form. Unfortunately, however, he did
not live to see the book completed. In view of the quality of these
manuscripts, and given the world-wide reputation of the author, we
deemed it more than justified that these manuscripts be published, even
though in form and volume they certainly did not yet fully come up to the
high demands of their author. From the beginning it was our intention to
change the text as little as possible and to make no extensive reorganiza
tions or additions. We were equally intent on not altering the character of
the work and, for example, on preserving the occasionally pictorial
language, which facilitates reading and understanding. Nevertheless,
much remained to be done, especially in those parts which were only
drafted by the author. Also, individual chapters which in part were avail
able in several versions had to be merged as smoothly as possible into a
seamless whole. The numerical examples were almost all newly recom
puted. Finally, the figures needed to be prepared; the drawing of
mathematically defined curves was generally done by a plotter.
Throughout the editing of the text, I was assisted above all by the
coeditors Profs. P. Henrici, P. Uiuchli and H.R. Schwarz, who read the
entire manuscript and who consulted with me during many hours on ques
tions of principles and details. Many further colleagues also helped me
through their criticism; to be mentioned are particularly Prof. R. Jeltsch,
Dr. R. Bloch and Dr. J. Waldvogel. Thanks go also to Miss G. Bonzli
and Mrs. L. Gutknecht, who typed large parts of the text, and to Dr. V.
Sechovcov, who drew the figures in ink. I am also very pleased that Mr.
Stutz and others agreed to help me with the correction of the galley
proofs.
My editing work for the most part was financed by the Swiss
National Science Foundation. Finally, I wish to thank the publisher for
the very careful and speedy printing.
Vancouver, B.C., February 1976 M.GUTKNECHT
Preface
Heinz Rutishauser is one of the pioneers of modem numerical
mathematics. Educated originally as a function theorist, he in 1950 joined
as a collaborator the Institute of Applied Mathematics, founded shortly
before at the Federal Institute of Technology. There, his extraordinary
algorithmic talent soon became evident. With concisely written publica
tions he introduced methods and directions of research into numerical
mathematics which later on proved to be fundamental. The stability
theory in the numerical solution of ordinary differential equations,
"economization" of power series by the use of Chebyshev polynomials,
the quotient-difference algorithm,. the LR-method, the exact justification
of the Romberg algorithm, and many other contributions all go back to
Rutishauser. He was also one of the first to recognize that the computer
itself could be used for the preparation of computer programs, and he
played a leading role in the development of the programming language
ALGOL. In the last years of his life, Rutishauser concerned himself with
the axiomatization of numerical computation and as a result gave perhaps
the most satisfactory treatment, from a theoretical point of view, of the
propagation of rounding errors. His health-related aversion to travel and,
no doubt, a touch of introversion, prevented all these achievements from
becoming known and appreciated as they deserved to be.
After Rutishauser's death in 1970, his widow, Mrs. Margrit
Rutishauser, asked the undersigned to sift through his unpublished
scientific notes. It became immediately clear to us that Rutishauser's lec
tures on numerical mathematics constituted an important part of these
notes. The lectures, which in quality and originality far excel the average
presentations in this area, were already intended for publication by
viii Preface
Rutishauser himself, but have only partly been prepared in detail for pub
lication. It so happened that Dr. Martin Gutknecht, who still heard these
lectures as a student, and who also has the necessary technical knowledge,
could be prevailed upon to successfully complete the preparation for pub
lication. Commendably, the work of Dr. Gutknecht has been supported
by the Swiss National Science Foundation. We are pleased, thanks to the
cooperation of the Birkhauser publishing house, to be able to present the
outcome to the public.
Zurich, February 1976 P. HENRICI
P. LAUCHLI
H.R. SCHWARZ
Translator's Preface
Rutishauser's Vorlesungen iiber numerische Mathematik appeared in
1976 in two volumes. Even though more than twelve years have elapsed
since the work was first published, it has retained much of its freshness
and timeliness. The material treated, though no longer entirely up-to-date
in some areas, still provides a sound and stimulating introduction to the
field of scientific computing. It was felt desirable, therefore, to make the
work accessible to a wider audience by providing an English translation.
The undersigned was happy to undertake this task, as he has known
Rutishauser personally and has great admiration for his scientific achieve
ments. In preparing the translation, he has combined the original two
volumes into a single volume. He has refrained from making any major
changes to the text itself, other than correcting a fair number of typo
graphical errors. However, following a suggestion already made by G.W.
Stewart in his review of the German original (cf. Bull. Amer. Math. Soc.,
v. 84, 1978, pp. 660--663), he has supplemented each chapter with notes
designed to make the reader aware of significant developments in compu
tational techniques that occurred since the original volumes have
appeared, and to direct him to appropriate sources for further study. The
preparation of these notes took considerably longer than anticipated, and
in fact would never have been completed, were it not for the invaluable
assistance he has received from a number of colleagues. John K. Reid
helped with the notes to Chapters 2 and 3, Florian A. Potra and Hermann
Brunner with those to Chapters 4 and 5, and Chapters 8 and 9, respec
tively. The notes to Chapters 10 and 11 were contributed entirely by Lars
B. Wahlbin, those to Chapters 12 and 13 in large part by Beresford N.
Parlett. Comments from Carl de Boor pertaining to the notes for Chapters
x Translator's Preface
6 and 7, and from Hans J. Stetter on the notes for Chapter 8, were also
incorporated. The help of all these colleagues is herewith gratefully ac
knowledged.
Thanks are also due to Ms. Connie Heer, who capably and unremit
tingly prepared the photo-ready copy of the manuscript on a computer of
the Department of Computer Sciences at Purdue University, using
UNIX's troft system. Finally, we thank the publisher for patiently wait
ing for the completion of this project and for assisting us in the produc
tion of this volume.
West Lafayette, Ind., November 1989 WALTER GAUTSCHI
Contents
Editor's Foreword
Preface
Translator's Preface
Chapter 1. An Outline of the Problems .......... ............. ..... ...... ............ 1
§ 1.1. Reliability of programs ........ ........................................................ 1
§ 1.2. The evolution of a program ........ ............... ... ........ ....... ...... .......... 2
§ 1.3. Difficulties ................................................................................... 3
Notes to Chapter 1 ...................................................................... 8
Chapter 2. Linear Equations and Inequalities .. ........ ... .......... ............ 10
§ 2.1. The classical algorithm of Gauss ................................ ................ 12
§ 2.2. The triangular decomposition ...................................................... 16
§ 2.3. Iterative refinement ............. ......................................................... 21
§ 2.4. Pivoting strategies ............... .......................... .............................. 23
§ 2.5. Questions of programming .......................................................... 27
§ 2.6. The exchange algorithm .............................................................. 32
§ 2.7. Questions of programming .......................................................... 41
§ 2.8. Linear inequalities (optimization) ............................................... 43
Notes to Chapter 2 ...................................................................... 49
Chapter 3. Systems of Equations With Positive Definite Symmetric
Coefficient Matrix .......... ............. ... ........ ............. ......... ...... 53
§ 3.1. Positive definite matrices .... ........................................................ 53
§ 3.2. Criteria for positive definiteness ................. ................................. 55
§ 3.3. The Cholesky decomposition ...................................................... 59
§ 3.4. Programming the Cholesky decomposition ................................. 63
§ 3.5. Solution of a linear system .......................................................... 66
