Numerical analysis is an increasingly important link between pure mathematics and its application in science and technology. This textbook provides an introduction to the justification and development of constructive methods that provide sufficiently accurate approximations to the solution of numerical problems, and the analysis of the influence that errors in data, finite-precision calculations, and approximation formulas have on results, problem formulation and the choice of method. It also serves as an introduction to scientific programming in MATLAB, including many simple and difficult, theoretical and computational exercises. A unique feature of this book is the consequent development of interval analysis as a tool for rigorous computation and computer assisted proofs, along with the traditional material.
Author(s): Arnold Neumaier
Publisher: CUP
Year: 2001
Language: English
Pages: 365
Cover......Page 1
Title Page......Page 4
Copyright......Page 5
Contents......Page 6
Preface......Page 8
1. The Numerical Evaluation of Expressions......Page 10
1.1 Arithmetic Expressions and Automatic Differentiation......Page 11
1.2 Numbers, Operations, and Elementary Functions......Page 23
1.3 Numerical Stability......Page 32
1.4 Error Propagation and Condition......Page 42
1.5 Interval Arithmetic......Page 47
1.6 Exercises......Page 62
2. Linear Systems of Equations......Page 70
2.1 Gaussian Elimination......Page 72
2.2 Variations on a Theme......Page 82
2.3 Rounding Errors, Equilibration, and Pivot Search......Page 91
2.4 Vector and Matrix Norms......Page 103
2.5 Condition Numbers and Data Perturbations......Page 108
2.6 Iterative Refinement......Page 115
2.7 Error Bounds for Solutions of Linear Systems......Page 121
2.8 Exercises......Page 128
3. Interpolation and Numerical Differentiation......Page 139
3.1 Interpolation by Polynomials......Page 140
3.2 Extrapolation and Numerical Differentiation......Page 154
3.3 Cubic Splines......Page 162
3.4 Approximation by Splines......Page 174
3.5 Radial Basis Functions......Page 179
3.6 Exercises......Page 191
4.1 The Accuracy of Quadrature Formulas......Page 188
4.2 Gaussian Quadrature Formulas......Page 196
4.3 The Trapezoidal Rule......Page 205
4.4 Adaptive Integration......Page 212
4.5 Solving Ordinary Differential Equations......Page 219
4.6 Step Size and Order Control......Page 228
4.7 Exercises......Page 234
5. Univariate Nonlinear Equations......Page 242
5.1 The Secant Method......Page 243
5.2 Bisection Methods......Page 250
5.3 Spectral Bisection Methods for Eigenvalues......Page 259
5.4 Convergence Order......Page 265
5.5 Error Analysis......Page 274
5.6 Complex Zeros......Page 282
5.7 Methods Using Derivative Information......Page 290
5.8 Exercises......Page 302
6. Systems of Nonlinear Equations......Page 310
6.1 Preliminaries......Page 311
6.2 Newton's Method and Its Variants......Page 320
6.3 Error Analysis......Page 331
6.4 Further Techniques for Nonlinear Systems......Page 338
6.5 Exercises......Page 349
References......Page 354
Index......Page 360
