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Numerical Linear Algebra Texts in Applied Mathematics 55 Grégoire Allaire 55 Sidi Mahmoud Kaber TEXTS IN APPLIED MATHEMATICS Numerical Linear Algebra Texts in Applied Mathematics 55 Editors J.E. Marsden L. Sirovich S.S. Antman Advisors G. Iooss P. Holmes D. Barkley M. Dellnitz P. Newton Texts in Applied Mathematics 1. Sirovich: Introduction to Applied Mathematics. 2. Wiggins: Introduction to Applied Nonlinear Dynamical Systems and Chaos. 3. Hale/Koc¸ak: Dynamics and Bifurcations. 4. Chorin/Marsden: A Mathematical Introduction to Fluid Mechanics, 3rd ed. 5. Hubbard/West: Differential Equations: A Dynamical Systems Approach: Ordinary Differential Equations. 6. Sontag: Mathematical Control Theory: Deterministic Finite Dimensional Systems, 2nd ed. 7. Perko: Differential Equations and Dynamical Systems, 3rd ed. 8. Seaborn: Hypergeometric Functions and Their Applications. 9. Pipkin: A Course on Integral Equations. 10. Hoppensteadt/Peskin: Modeling and Simulation in Medicine and the Life Sciences, 2nd ed. 11. Braun: Differential Equations and Their Applications, 4th ed. 12. Stoer/Bulirsch: Introduction to Numerical Analysis, 3rd ed. 13. Renardy/Rogers: An Introduction to Partial Differential Equations. 14. Banks: Growth and Diffusion Phenomena: Mathematical Frameworks and Applications. 15. Brenner/Scott: The Mathematical Theory of Finite Element Methods, 2nd ed. 16. Van de Velde: Concurrent Scientific Computing. 17. Marsden/Ratiu: Introduction to Mechanics and Symmetry, 2nd ed. 18. Hubbard/West: Differential Equations: A Dynamical Systems Approach: Higher-Dimensional Systems. 19. Kaplan/Glass: Understanding Nonlinear Dynamics. 20. Holmes: Introduction to Perturbation Methods. 21. Curtain/Zwart: An Introduction to Infinite-Dimensional Linear Systems Theory. 22. Thomas: Numerical Partial Differential Equations: Finite Difference Methods. 23. Taylor: Partial Differential Equations: Basic Theory. 24. Merkin: Introduction to the Theory of Stability of Motion. 25. Naber: Topology, Geometry, and Gauge Fields: Foundations. 26. Polderman/Willems: Introduction to Mathematical Systems Theory: A Behavioral Approach. 27. Reddy: Introductory Functional Analysis with Applications to Boundary Value Problems and Finite Elements. 28. Gustafson/Wilcox: Analytical and Computational Methods of Advanced Engineering Mathematics. 29. Tveito/Winther: Introduction to Partial Differential Equations: A Computational Approach (continued after index) Gregoire´ Allaire Sidi Mahmoud Kaber Numerical Linear Algebra Translated by Karim Trabelsi ABC Gregoire´ Allaire Sidi Mahmoud Kaber CMAP Ecole Polytechnique Universite´ Pierre et Marie Curie 91128 Palaiseau 75252 Paris France France [email protected] [email protected] Series Editors J.E. Marsden L. Sirovich Control and Dynamic Systems, 107-81 Division of Applied Mathematics California Institute of Technology Brown University Pasadena, CA 91125 Providence, RI 02912 USA USA S.S. Antman Department of Mathematics and Institute for Physical Science and Technology University of Maryland College Park, MD 20742-4015 USA [email protected] ISBN 978-0-387-34159-0 e-ISBN 978-0-387-68918-0 DOI: 10.1007/978-0-387-68918-0 Library of Congress Control Number: 2007939527 Mathematics Subject Classification (2000): 15-01, 54F05, 65F10, 65F15, 65F35 Original French edition: Introduction a` Scilab-exercices pratiques corriges´ d’algebre lineaire´ & Algebre lineaire´ numerique.´ Cours et exercices, 2002 °c 2008 Springer Science+Business Media, LLC Introduction a` Scilab-exercices pratiques corriges´ d’algebre lineaire´ & Algebre lineaire´ numerique.´ Cours et exercices ¡ published by Ellipses-Edition Marketing S.A. All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper 9 8 7 6 5 4 3 2 1 springer.com Series Preface Mathematics is playing an ever more important role in the physical and biolog- ical sciences, provoking a blurring of boundaries between scienti¯c disciplines and a resurgence of interest in the modern as well as the classical techniques of applied mathematics. This renewal of interest, both in research and teach- ing, has led to the establishment of the series Texts in Applied Mathematics (TAM). The development of new courses is a natural consequence of a high level of excitement on the research frontier as newer techniques, such as numerical and symbolic computer systems, dynamical systems, and chaos, mix with and reinforce the traditional methods of applied mathematics. Thus, the purpose of this textbook series is to meet the current and future needs of these advances and to encourage the teaching of new courses. TAM will publish textbooks suitable for use in advanced undergraduate and beginning graduate courses, and will complement the Applied Mathe- matical Sciences (AMS) series, which will focus on advanced textbooks and research-level monographs. Pasadena, California J.E. Marsden Providence, Rhode Island L. Sirovich College Park, Maryland S.S. Antman Preface The origin of this textbook is a course on numerical linear algebra that we taught to third-year undergraduate students at Universit´e Pierre et Marie Curie (Paris 6 University). Numerical linear algebra is the intersection of numerical analysis and linear algebra and, more precisely, focuses on practical algorithms for solving on a computer problems of linear algebra. Indeed, most numerical computations in all fields of applications, such as physics, mechanics, chemistry, and finance. involve numerical linear algebra. All in all, these numerical simulations boil down to a series of matrix com- putations. There are mainly two types of matrix computations: solving linear systems of equations and computing eigenvalues and eigenvectors of matrices. Of course, there are other important problems in linear algebra, but these two are predominant and will be studied in great detail in this book. From a theoretical point of view, these two questions are by now com- pletely understood and solved. Necessary and sufficient conditions for the existence and/or uniqueness of solutions to linear systems are well known, as well as criteria for diagonalizing matrices. However, the steady and impres- sive progress of computer power has changed those theoretical questions into practical issues. An applied mathematician cannot be satisfied by a mere exis- tence theorem and rather asks for an algorithm, i.e., a method for computing unknown solutions. Such an algorithm must be efficient: it must not take too long to run and too much memory on a computer. It must also be stable, that is, small errors in the data should produce similarly small errors in the output. Recall that errors cannot be avoided, because of rounding off in the computer. These two requirements, efficiency and stability, are key issues in numerical analysis. Many apparently simple algorithms are rejected because of them. This book is intended for advanced undergraduate students who have al- ready been exposed to linear algebra (for instance, [9], [10], [16]). Nevertheless, to be as self-contained as possible, its second chapter recalls the necessary de- finitions and results of linear algebra that will be used in the sequel. On the other hand, our purpose is to be introductory concerning numerical analysis, VIII Preface for which we do not ask for any prerequisite. Therefore, we do not pretend to be exhaustive nor to systematically give the most efficient or recent algorithms if they are too complicated. We leave this task to other books at the graduate level, such as [2], [7], [11], [12], [14], [17], [18]. For pedagogical reasons we satisfy ourselves in giving the simplest and most illustrative algorithms. Since the inception of computers and, all the more, the development of simple and user-friendly software such as Maple, Mathematica, Matlab, Octave, and Scilab, mathematics has become a truly experimental science like physics or mechanics. It is now possible and very easy to perform numerical experiments on a computer that help in increasing intuition, checking conjec- tures or theorems, and quantifying the effectiveness of a method. One original feature of this book is to follow an experimental approach in all exercises. The reader should use Matlab for solving these exercises, which are given at the end of each chapter. For some of them, marked by a (∗), complete solutions, including Matlab scripts, are given in the last chapter. The solutions of the other exercises are available in a solution manual available for teachers and professors on request to Springer. The original french version of this book (see our web page http://www.ann.jussieu.fr/numalgebra) used Scilab, which is probably less popular than Matlab but has the advantage of being free software (see http://www.scilab.org). Finally we thank Karim Trabelsi for translating a large part of this book from a French previous version of it. We hope the reader will enjoy more mathematics by seeing it “in practice.” G.A., S.M.K.
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