Analisis numerico

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Numerical Analysis
L. Ridgway Scott

Numerical Analysis

Numerical Analysis

L. Ridgway Scott

PRINCETON UNIVERSITY PRESS PRINCETON AND OXFORD

Copyright c 2011 by Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, 6 Oxford Street, Woodstock, Oxfordshire OX20 1TWpress.princeton.edu All Rights Reserved Library of Congress Control Number: 2010943322 ISBN: 978-0-691-14686-7 British Library Cataloging-in-Publication Data is available The publisher would like to acknowledge the author of this volume for typeA setting this book using L TEX and Dr. Janet Englund and Peter Scott for providing the cover photograph Printed on acid-free paper ∞ Printed in the UnitedStates of America 10 9 8 7 6 5 4 3 2 1

Dedication
To the memory of Ed Conway1 who, along with his colleagues at Tulane University, provided a stable, adaptive, and inspirational starting point for my career.

1 Edward Daire Conway, III (1937–1985) was a student of Eberhard Friedrich Ferdinand Hopf at the University of Indiana. Hopf was a student of Erhard Schmidt and Issai Schur. Contents

Preface Chapter 1. Numerical Algorithms 1.1 1.2 1.3 1.4 1.5 1.6 1.7 Finding roots Analyzing Heron’s algorithm Where to start An unstable algorithm General roots: effects of floating-point Exercises Solutions

xi 1 2 5 6 8 9 11 13 15 16 20 25 26 27 27 30 35 36 38 42 44 47 47 50 51 51 56 58 60 60 63

Chapter 2. Nonlinear Equations 2.1 2.2 2.3 2.4 2.5 2.6 2.7 Fixed-point iteration Particularmethods Complex roots Error propagation More reading Exercises Solutions

Chapter 3. Linear Systems 3.1 3.2 3.3 3.4 3.5 3.6 3.7 Gaussian elimination Factorization Triangular matrices Pivoting More reading Exercises Solutions

Chapter 4. Direct Solvers 4.1 4.2 4.3 4.4 4.5 4.6 Direct factorization Caution about factorization Banded matrices More reading Exercises Solutions

viii
Chapter 5.Vector Spaces 5.1 5.2 5.3 5.4 5.5 5.6 5.7 Normed vector spaces Proving the triangle inequality Relations between norms Inner-product spaces More reading Exercises Solutions

CONTENTS

65 66 69 71 72 76 77 79 81 82 84 89 89 92 95 97 98 103 108 110 111 111 114 115 116 117 123 128 128 131 133 133 137 141 147 147 148 149

Chapter 6. Operators 6.1 6.2 6.3 6.4 6.5 6.6 Operators Schurdecomposition Convergent matrices Powers of matrices Exercises Solutions

Chapter 7. Nonlinear Systems 7.1 7.2 7.3 7.4 7.5 7.6 7.7 Functional iteration for systems Newton’s method Limiting behavior of Newton’s method Mixing solvers More reading Exercises Solutions

Chapter 8. Iterative Methods 8.1 8.2 8.3 8.4 8.5 8.6 Stationary iterative methods General splittings Necessary conditions for convergenceMore reading Exercises Solutions

Chapter 9. Conjugate Gradients 9.1 9.2 9.3 9.4 9.5 9.6 9.7 Minimization methods Conjugate Gradient iteration Optimal approximation of CG Comparing iterative solvers More reading Exercises Solutions

CONTENTS

ix
151 151 152 157 160 160 163 167 167 170 171 173 177 178 180 183 183 187 191 193 195 196 196 199 203 203 209 212 219 221 221 224 225 225 227 232 233234 237 238 240

Chapter 10. Polynomial Interpolation 10.1 10.2 10.3 10.4 10.5 10.6 Local approximation: Taylor’s theorem Distributed approximation: interpolation Norms in infinite-dimensional spaces More reading Exercises Solutions

Chapter 11. Chebyshev and Hermite Interpolation 11.1 11.2 11.3 11.4 11.5 11.6 11.7 Error term ω Chebyshev basis functions Lebesgue function Generalizedinterpolation More reading Exercises Solutions

Chapter 12. Approximation Theory 12.1 12.2 12.3 12.4 12.5 12.6 12.7 12.8 Best approximation by polynomials Weierstrass and Bernstein Least squares Piecewise polynomial approximation Adaptive approximation More reading Exercises Solutions

Chapter 13. Numerical Quadrature 13.1 13.2 13.3 13.4 13.5 13.6 13.7 Interpolatory quadrature Peano kernel theorem...
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