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numas - Numerical Methods Course Notes Version 0.11(UCSD...

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Numerical Methods Course Notes Version 0.11 (UCSD Math 174, Fall 2004) Steven E. Pav 1 October 13, 2005 1 Department of Mathematics, MC0112, University of California at San Diego, La Jolla, CA 92093-0112. <[email protected]> This document is Copyright c 2004 Steven E. Pav. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front- Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled ”GNU Free Documentation License”.
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Preface These notes were originally prepared during Fall quarter 2003 for UCSD Math 174, Numerical Methods. In writing these notes, it was not my intention to add to the glut of Numerical Analysis texts; they were designed to complement the course text, Numerical Mathematics and Computing, Fourth edition, by Cheney and Kincaid [7]. As such, these notes follow the conventions of that text fairly closely. If you are at all serious about pursuing study of Numerical Analysis, you should consider acquiring that text, or any one of a number of other fine texts by e.g. , Epperson, Hamming, etc. [3, 4, 5]. 3.1 3.2 8.4 3.3 1.4 3.4 1.1 4.2 7.1 10.1 4.3 5.1 5.2 7.2 8.3 8.2 9.1 9.2 9.3 10.2 10.3 Figure 1: The chapter dependency of this text, though some dependencies are weak. Special thanks go to the students of Math 174, 2003–2004, who suffered through early versions of these notes, which were riddled with (more) errors. Revision History 0.0 Transcription of course notes for Math 174, Fall 2003. 0.1 As used in Math 174, Fall 2004. 0.11 Added material on functional analysis and Orthogonal Least Squares. Todo More homework questions and example problems. Chapter on optimization. Chapters on basic finite difference and finite element methods? Section on root finding for functions of more than one variable. i
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Contents Preface i 1 Introduction 1 1.1 Taylor’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Loss of Significance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Vector Spaces, Inner Products, Norms . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3.1 Vector Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3.2 Inner Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3.3 Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.4 Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.4.1 Matrix Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2 A “Crash” Course in octave/Matlab 13 2.1 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2 Useful Commands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.3 Programming and Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.3.1 Logical Forks and Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.4 Plotting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3 Solving Linear Systems 25 3.1 Gaussian Elimination with Na¨ ıve Pivoting . . . . . . . . . . . . . . . . . . . . . . . . 25 3.1.1 Elementary Row Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.1.2 Algorithm Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.1.3 Algorithm Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.2 Pivoting Strategies for Gaussian Elimination . . . . . . . . . . . . . . . . . . . . . . 29 3.2.1 Scaled Partial Pivoting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.2.2 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.2.3 Another Example and A Real Algorithm . . . . . . . . . . . . . . . . . . . . . 32 3.3 LU Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.3.1 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.3.2 Using LU Factorizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.3.3 Some Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.3.4 Computing Inverses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.4 Iterative Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.4.1 An Operation Count for Gaussian Elimination . . . . . . . . . . . . . . . . . 37 iii
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iv CONTENTS 3.4.2 Dividing by Multiplying . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.4.3 Impossible Iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.4.4 Richardson Iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.4.5 Jacobi Iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.4.6 Gauss Seidel Iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.4.7 Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.4.8 A Free Lunch? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 4 Finding Roots 49 4.1 Bisection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 4.1.1 Modifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 4.1.2 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 4.2 Newton’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 4.2.1
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