Numerical Analysis - Department of Computer Science

341 Pages·2011·2.31 MB·English

Numerical Analysis

Numerical Analysis
L. Ridgway Scott
PRINCETON UNIVERSITY PRESS
PRINCETON AND OXFORD
Copyright c 2011 by Princeton University Press
(cid:13)
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 1TW
press.princeton.edu
All Rights Reserved
Library of Congress Control Number: 2010943322
ISBN: 978-0-691-14686-7
British Library Cataloging-in-PublicationData is available
The publisher wouldliketo acknowledgethe authorofthis volume fortype-
setting this book using LATEX and Dr. Janet Englund and Peter Scott for
providing the cover photograph
Printed on acid-free paper
∞
Printed in the United States 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.
1EdwardDaireConway,III(1937–1985)wasastudentofEberhardFriedrichFerdinand
HopfattheUniversityofIndiana. HopfwasastudentofErhardSchmidtandIssaiSchur.

Contents
Preface xi
Chapter1. Numerical Algorithms 1
1.1 Finding roots 2
1.2 AnalyzingHeron’s algorithm 5
1.3 Whereto start 6
1.4 An unstablealgorithm 8
1.5 General roots: effects of floating-point 9
1.6 Exercises 11
1.7 Solutions 13
Chapter2. Nonlinear Equations 15
2.1 Fixed-point iteration 16
2.2 Particular methods 20
2.3 Complex roots 25
2.4 Error propagation 26
2.5 More reading 27
2.6 Exercises 27
2.7 Solutions 30
Chapter3. Linear Systems 35
3.1 Gaussian elimination 36
3.2 Factorization 38
3.3 Triangular matrices 42
3.4 Pivoting 44
3.5 More reading 47
3.6 Exercises 47
3.7 Solutions 50
Chapter4. Direct Solvers 51
4.1 Direct factorization 51
4.2 Caution about factorization 56
4.3 Banded matrices 58
4.4 More reading 60
4.5 Exercises 60
4.6 Solutions 63
viii CONTENTS
Chapter5. Vector Spaces 65
5.1 Normed vector spaces 66
5.2 Proving thetriangle inequality 69
5.3 Relations between norms 71
5.4 Inner-productspaces 72
5.5 More reading 76
5.6 Exercises 77
5.7 Solutions 79
Chapter6. Operators 81
6.1 Operators 82
6.2 Schurdecomposition 84
6.3 Convergent matrices 89
6.4 Powers of matrices 89
6.5 Exercises 92
6.6 Solutions 95
Chapter7. Nonlinear Systems 97
7.1 Functionaliteration for systems 98
7.2 Newton’s method 103
7.3 Limiting behavior of Newton’s method 108
7.4 Mixing solvers 110
7.5 More reading 111
7.6 Exercises 111
7.7 Solutions 114
Chapter8. Iterative Methods 115
8.1 Stationary iterative methods 116
8.2 General splittings 117
8.3 Necessary conditions for convergence 123
8.4 More reading 128
8.5 Exercises 128
8.6 Solutions 131
Chapter9. Conjugate Gradients 133
9.1 Minimization methods 133
9.2 Conjugate Gradient iteration 137
9.3 Optimal approximation of CG 141
9.4 Comparing iterative solvers 147
9.5 More reading 147
9.6 Exercises 148
9.7 Solutions 149
CONTENTS ix
Chapter10. Polynomial Interpolation 151
10.1 Local approximation: Taylor’s theorem 151
10.2 Distributed approximation: interpolation 152
10.3 Norms in infinite-dimensional spaces 157
10.4 More reading 160
10.5 Exercises 160
10.6 Solutions 163
Chapter11. Chebyshev and Hermite Interpolation 167
11.1 Error term ω 167
11.2 Chebyshevbasis functions 170
11.3 Lebesgue function 171
11.4 Generalized interpolation 173
11.5 More reading 177
11.6 Exercises 178
11.7 Solutions 180
Chapter12. Approximation Theory 183
12.1 Best approximation by polynomials 183
12.2 Weierstrass and Bernstein 187
12.3 Least squares 191
12.4 Piecewise polynomial approximation 193
12.5 Adaptiveapproximation 195
12.6 More reading 196
12.7 Exercises 196
12.8 Solutions 199
Chapter13. Numerical Quadrature 203
13.1 Interpolatory quadrature 203
13.2 Peano kerneltheorem 209
13.3 Gregorie-Euler-Maclaurin formulas 212
13.4 Otherquadraturerules 219
13.5 More reading 221
13.6 Exercises 221
13.7 Solutions 224
Chapter14. Eigenvalue Problems 225
14.1 Eigenvalue examples 225
14.2 Gershgorin’s theorem 227
14.3 Solving separately 232
14.4 How not to eigen 233
14.5 Reduction to Hessenberg form 234
14.6 More reading 237
14.7 Exercises 238
14.8 Solutions 240
x CONTENTS
Chapter15. Eigenvalue Algorithms 241
15.1 Power method 241
15.2 Inverseiteration 250
15.3 Singular valuedecomposition 252
15.4 Comparing factorizations 253
15.5 More reading 254
15.6 Exercises 254
15.7 Solutions 256
Chapter16. OrdinaryDifferential Equations 257
16.1 Basic theory of ODEs 257
16.2 Existence and uniquenessof solutions 258
16.3 Basic discretization methods 262
16.4 Convergence of discretization methods 266
16.5 More reading 269
16.6 Exercises 269
16.7 Solutions 271
Chapter17. Higher-order ODEDiscretization Methods 275
17.1 Higher-orderdiscretization 276
17.2 Convergence conditions 281
17.3 Backward differentiation formulas 287
17.4 More reading 288
17.5 Exercises 289
17.6 Solutions 291
Chapter18. Floating Point 293
18.1 Floating-point arithmetic 293
18.2 Errors in solving systems 301
18.3 More reading 305
18.4 Exercises 305
18.5 Solutions 308
Chapter19. Notation 309
Bibliography 311
Index 323