Numerical+Methods.pdf
Numerical Analysis for almost four decades at the Indian Institute of Technology New Delhi. and co-author of their earlier books on Numerical Analysis.
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An Introduction to Numerical Methods and Analysis
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What is numerical analysis?
- Introduction Numerical analysis is a branch of Mathematics that deals with devising e?cient methods for obtaining numerical solutions to di?cult Mathematical problems. Most of the Mathematical problems that arise in science and engineering are very hard and sometime impossible to solve exactly.
What are the three parts of numerical analysis?
- Numerical analysis include three parts. The ?rst part of the subject is about the development of a method to a problem. The second part deals with the analysis of the method, which includes the error analysis and the e?ciency analysis.
Who invented the equivalence class of rational numbers?
- in which each real number is defined to be an equivalence class of Cauchy sequences of rational numbers (see Chap. 3), is carried out in Sec. 5 of the book by Hewitt and Stromberg. The cuts in Q which we used here were invented by Dedekind. The construction of R from Q by means of Cauchy sequences is due to Cantor.
Numerical Analysis
Numerical Analysis
L. Ridgway Scott
PRINCETON UNIVERSITY PRESS
PRINCETON AND OXFORD
Copyrightc?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 OxfordStreet,Woodstock, Oxfordshire OX20 1TW
press.princeton.eduAll 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 type- setting this book using LATEX and Dr. Janet Englund and Peter Scott for
providing the cover photographPrinted 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 Conway
1who, along with his colleagues at Tulane
University, provided a stable, adaptive, and inspirational starting point for my career.1Edward 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
Prefacexi
Chapter 1. Numerical Algorithms1
1.1 Finding roots2
1.2 Analyzing Heron"s algorithm5
1.3 Where to start6
1.4 An unstable algorithm8
1.5 General roots: effects of floating-point9
1.6 Exercises11
1.7 Solutions13
Chapter 2. Nonlinear Equations15
2.1 Fixed-point iteration16
2.2 Particular methods20
2.3 Complex roots25
2.4 Error propagation26
2.5 More reading27
2.6 Exercises27
2.7 Solutions30
Chapter 3. Linear Systems35
3.1 Gaussian elimination36
3.2 Factorization38
3.3 Triangular matrices42
3.4 Pivoting44
3.5 More reading47
3.6 Exercises47
3.7 Solutions50
Chapter 4. Direct Solvers51
4.1 Direct factorization51
4.2 Caution about factorization56
4.3 Banded matrices58
4.4 More reading60
4.5 Exercises60
4.6 Solutions63
viiiCONTENTSChapter 5. Vector Spaces65
5.1 Normed vector spaces66
5.2 Proving the triangle inequality69
5.3 Relations between norms71
5.4 Inner-product spaces72
5.5 More reading76
5.6 Exercises77
5.7 Solutions79
Chapter 6. Operators81
6.1 Operators82
6.2 Schur decomposition84
6.3 Convergent matrices89
6.4 Powers of matrices89
6.5 Exercises92
6.6 Solutions95
Chapter 7. Nonlinear Systems97
7.1 Functional iteration for systems98
7.2 Newton"s method103
7.3 Limiting behavior of Newton"s method108
7.4 Mixing solvers110
7.5 More reading111
7.6 Exercises111
7.7 Solutions114
Chapter 8. Iterative Methods115
8.1 Stationary iterative methods116
8.2 General splittings117
8.3 Necessary conditions for convergence123
8.4 More reading128
8.5 Exercises128
8.6 Solutions131
Chapter 9. Conjugate Gradients133
9.1 Minimization methods133
9.2 Conjugate Gradient iteration137
9.3 Optimal approximation of CG141
9.4 Comparing iterative solvers147
9.5 More reading147
9.6 Exercises148
9.7 Solutions149
CONTENTSix
Chapter 10. Polynomial Interpolation151
10.1 Local approximation: Taylor"s theorem151
10.2 Distributed approximation: interpolation152
10.3 Norms in infinite-dimensional spaces157
10.4 More reading160
10.5 Exercises160
10.6 Solutions163
Chapter 11. Chebyshev and Hermite Interpolation 16711.1 Error termω167
11.2 Chebyshev basis functions170
11.3 Lebesgue function171
11.4 Generalized interpolation173
11.5 More reading177
11.6 Exercises178
11.7 Solutions180
Chapter 12. Approximation Theory183
12.1 Best approximation by polynomials183
12.2 Weierstrass and Bernstein187
12.3 Least squares191
12.4 Piecewise polynomial approximation193
12.5 Adaptive approximation195
12.6 More reading196
12.7 Exercises196
12.8 Solutions199
Chapter 13. Numerical Quadrature203
13.1 Interpolatory quadrature203
13.2 Peano kernel theorem209
13.3 Gregorie-Euler-Maclaurin formulas212
13.4 Other quadrature rules219
13.5 More reading221
13.6 Exercises221
13.7 Solutions224
Chapter 14. Eigenvalue Problems225
14.1 Eigenvalue examples225
14.2 Gershgorin"s theorem227
14.3 Solving separately232
14.4 How not to eigen233
14.5 Reduction to Hessenberg form234
14.6 More reading237
14.7 Exercises238
14.8 Solutions240
xCONTENTSChapter 15. Eigenvalue Algorithms241
15.1 Power method241
15.2 Inverse iteration250
15.3 Singular value decomposition252
15.4 Comparing factorizations253
15.5 More reading254
15.6 Exercises254
15.7 Solutions256
Chapter 16. Ordinary Differential Equations 257
16.1 Basic theory of ODEs257
16.2 Existence and uniqueness of solutions258
16.3 Basic discretization methods262
16.4 Convergence of discretization methods266
16.5 More reading269
16.6 Exercises269
16.7 Solutions271
Chapter 17. Higher-order ODE Discretization Methods 27517.1 Higher-order discretization276
17.2 Convergence conditions281
17.3 Backward differentiation formulas287
17.4 More reading288
17.5 Exercises289
17.6 Solutions291
Chapter 18. Floating Point293
18.1 Floating-point arithmetic293
18.2 Errors in solving systems301
18.3 More reading305
18.4 Exercises305
18.5 Solutions308
Chapter 19. Notation309
Bibliography311
Index323
Preface
"...by faith and faith alone, embrace, believing where we cannot prove," fromIn Memoriamby Alfred Lord Ten- nyson, a memorial to Arthur Hallum. Numerical analysis provides the foundations for a major paradigm shift in what we understand as an acceptable "answer" to a scientific or techni- cal question. In classical calculus we look for answers like⎷ sinx, that is, answers composed of combinations of names of functions thatare familiar. This presumes we can evaluate such an expression as needed, and indeed numerical analysis has enabled the development of pocket calculators and computer software to make this routine. But numerical analysis has done much more than this. We will see that far more complex functions, defined, e.g., only implicitly, can be evaluated just as easily and with the same tech- nology. This makes the search for answers in classical calculus obsolete in many cases. This new paradigm comes at a cost: developing stable, con- vergent algorithms to evaluate functions is often more difficult than more classical analysis of these functions. For this reason, thesubject is still be- ing actively developed. However, it is possible to present many important ideas at an elementary level, as is done here. Today there are many good books on numerical analysis at the graduate level, including general texts [47, 134] as well as more specialized texts. We reference many of the latter at the ends of chapters where we suggest fur- ther reading in particular areas. At a more introductory level, the recent trend has been to provide texts accessible to a wide audience. The book by Burden and Faires [28] has been extremely successful. It is a tribute to the importance of the field of numerical analysis that such books and others [131] are so popular. However, such books intentionally diminish the role of advanced mathematics in the subject of numerical analysis. As a result, numerical analysis is frequently presented as an elementary subject. As a corollary, most students miss exposure to numerical analysis as a mathemat- ical subject. We hope to provide an alternative. Several books written some decades ago addressed specifically a mathe-quotesdbs_dbs20.pdfusesText_26[PDF] numerical analysis handwritten notes pdf
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