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Publisher:

Princeton University Press

Publication Date:

2011

Number of Pages:

325

Format:

Hardcover

Price:

65.00

ISBN:

9780691146867

Category:

Textbook

The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by , on ]

William J. Satzer

08/29/2011

There can’t be many numerical analysis textbooks that advertise themselves as excellent follow-up books to Rudin’s *Principles of Mathematical Analysis*. This is one. The author’s intentions are clear from the beginning. In his preface, he writes of introductory books on numerical analysis that are accessible to a wide audience:

... 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 mathematical subject. We hope to provide an alternative.

That alternative is, in part, “an invitation to study more deeply advanced topics in mathematics”, and in doing so, “the general style of a course using this book will be to prove theorems.” The author proposes a modified “Moore method” whereby he provides a sequence of steps to be verified as exercises. The stated prerequisites are not too extensive: a “sophisticated understanding of real numbers”, the background to follow arguments using compactness, and basic linear algebra. A good bit of mathematical maturity wouldn’t hurt either.

Most of the topics treated here are common to virtually all numerical analysis books: general solution methods for linear and nonlinear equations, iterative methods and conjugate gradient optimization, interpolation (polynomial, Chebyshev, and Hermite), numerical integration, eigenvalue problems and numerical solution methods for differential equations. The level of rigor, however, is significantly higher than most numerical analysis books currently available.

The author does not address computer programming or software development as they pertain to numerical analysis. He allows the value of programming experience for students, and argues that the discipline of software development gives a “useful model in making complex mathematical arguments understandable to others”. While the value of a rigorous treatment of numerical analysis is clear, there is a certain indifference here to practical matters. That is the author’s prerogative, but most students of numerical analysis probably feel otherwise. It does seem odd to read a book about numerical analysis that has very few actual numbers in it.

Who are the prospective readers of this book? They could be, as the author suggests, students taking a second course in analysis, one prior to graduate-level real analysis. There is some lovely classical analysis here, clearly worthy of study. Another group might be those looking for a deeper understanding, proofs, or rigorous statements of the theorems of numerical analysis. While *Numerical Recipes* has been the technical equivalent of a runaway best-seller, it does not pretend to be more than a good general sourcebook of numerical methods and algorithms. Anyone looking to go deeper needs other sources. My favorite introductory text, Acton’s *Numerical Methods That Work* (with the sly addition in faint letters of *“Usually”* to the title on the book’s cover) does very well at conveying concepts and dispensing wisdom, but has no aspirations to rigor. However, the good sense it does offer to students is worth a lot. One might hope that readers of Scott’s book also spend a little time with Acton’s.

Bill Satzer (wjsatzer@mmm.com) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.

Preface xi

Chapter 1. Numerical Algorithms 1

1.1 Finding roots 2

1.2 Analyzing Heron's algorithm 5

1.3 Where to start 6

1.4 An unstable algorithm 8

1.5 General roots: effects of floating-point 9

1.6 Exercises 11

1.7 Solutions 13

Chapter 2. 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

Chapter 3. 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

Chapter 4. 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

Chapter 5. Vector Spaces 65

5.1 Normed vector spaces 66

5.2 Proving the triangle inequality 69

5.3 Relations between norms 71

5.4 Inner-product spaces 72

5.5 More reading 76

5.6 Exercises 77

5.7 Solutions 79

Chapter 6. Operators 81

6.1 Operators 82

6.2 Schur decomposition 84

6.3 Convergent matrices 89

6.4 Powers of matrices 89

6.5 Exercises 92

6.6 Solutions 95

Chapter 7. Nonlinear Systems 97

7.1 Functional iteration 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

Chapter 8. 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

Chapter 9. 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

Chapter 10. 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

Chapter 11. Chebyshev and Hermite Interpolation 167

11.1 Error term ! 167

11.2 Chebyshev basis 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

Chapter 12. 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 Adaptive approximation 195

12.6 More reading 196

12.7 Exercises 196

12.8 Solutions 199

Chapter 13. Numerical Quadrature 203

13.1 Interpolatory quadrature 203

13.2 Peano kernel theorem 209

13.3 Gregorie-Euler-Maclaurin formulas 212

13.4 Other quadrature rules 219

13.5 More reading 221

13.6 Exercises 221

13.7 Solutions 224

Chapter 14. 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

Chapter 15. Eigenvalue Algorithms 241

15.1 Power method 241

15.2 Inverse iteration 250

15.3 Singular value decomposition 252

15.4 Comparing factorizations 253

15.5 More reading 254

15.6 Exercises 254

15.7 Solutions 256

Chapter 16. Ordinary Differential Equations 257

16.1 Basic theory of ODEs 257

16.2 Existence and uniqueness of 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

Chapter 17. Higher-order ODE Discretization Methods 275

17.1 Higher-order discretization 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

Chapter 18. 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

Chapter 19. Notation 309

Bibliography 311

Index 323

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