Numerical Analysis (ebook)

Argumento de Numerical Analysis (ebook)

Computational science is fundamentally changing how technological questions are addressed. The design of aircraft, automobiles, and even racing sailboats is now done by computational simulation. The mathematical foundation of this new approach is numerical analysis, which studies algorithms for computing expressions defined with real numbers. Emphasizing the theory behind the computation, this book provides a rigorous and self-contained introduction to numerical analysis and presents the advanced mathematics that underpin industrial software, including complete details that are missing from most textbooks.

Using an inquiry-based learning approach, Numerical Analysis is written in a narrative style, provides historical background, and includes many of the proofs and technical details in exercises. Students will be able to go beyond an elementary understanding of numerical simulation and develop deep insights into the foundations of the subject. They will no longer have to accept the mathematical gaps that exist in current textbooks. For example, both necessary and sufficient conditions for convergence of basic iterative methods are covered, and proofs are given in full generality, not just based on special cases.

The book is accessible to undergraduate mathematics majors as well as computational scientists wanting to learn the foundations of the subject.

• Presents the mathematical foundations of numerical analysis
• Explains the mathematical details behind simulation software
• Introduces many advanced concepts in modern analysis
• Self-contained and mathematically rigorous
• Contains problems and solutions in each chapter
• Excellent follow-up course to Principles of Mathematical Analysis by Rudin

"Students will be able to go beyond an elementary understanding of numerical simulation and develop deep insights into the foundations of the subject. They will no longer have to accept the mathematical gaps that exist in current textbooks."--World Book Industry0Preface 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