Concise numerical mathematics / Robert Plato ; translated by Richard Le Borne, Sabine Le Borne.

Format:

Book

Author/Creator:

Plato, Robert, 1962-

Series:

Graduate studies in mathematics 1065-7339 ; v. 57.

Graduate studies in mathematics, 1065-7339 ; v. 57

Standardized Title:

Numerische Mathematik kompakte. English

Language:

English

German

Subjects (All):

Numerical analysis.

Physical Description:

xiv, 453 pages : illustrations ; 26 cm.

Place of Publication:

Providence, R.I. : American Mathematical Society, [2003]

Contents:

Chapter 1. Interpolation by Polynomials 1

1.1. General prerequisites and Landau symbols 1

1.2. Existence and uniqueness of an interpolating polynomial 3

1.3. Neville's algorithm 6

1.4. Newton's interpolation formula, divided differences 8

1.5. The interpolation error 11

1.6. Chebyshev polynomials 14

Chapter 2. Spline Functions 23

2.2. Interpolating linear spline functions 24

2.3. Minimality properties of cubic spline functions 25

2.4. The calculation of interpolating cubic spline functions 27

2.5. Error estimates for interpolating cubic splines 33

Chapter 3. The Discrete Fourier Transform and Its Applications 41

3.1. Discrete Fourier transform 41

3.2. Applications of the discrete Fourier transform 43

3.3. Fast Fourier transform (FFT) 49

Chapter 4. Solution of Linear Systems of Equations 59

4.1. Triangular systems 59

4.2. Gaussian elimination 61

4.3. The factorization PA = LR 66

4.4. LR factorization 74

4.5. Cholesky factorization for positive definite matrices 76

4.6. Banded matrices 79

4.7. Norms and error estimates 81

4.8. The factorization A = QS 91

Chapter 5. Nonlinear Systems of Equations 105

5.2. The one-dimensional case (N = 1) 107

5.3. Banach's fixed point theorem 109

5.4. Newton's method 112

Chapter 6. The Numerical Integration of Functions 123

6.1. Quadrature by interpolation formulas 124

6.2. Special quadrature by interpolation formulas 125

6.3. The error due to quadrature by interpolation 129

6.4. Degree of exactness for the closed Newton-Cotes formulas, n even 132

6.5. Composite Newton-Cotes formulas 137

6.6. Asymptotic form of the composite trapezoidal rule 141

6.7. Extrapolation methods 142

6.8. Gaussian quadrature 146

6.9. Appendix: Proof of the asymptotic form for the composite trapezoidal rule 155

Chapter 7. Explicit One-Step Methods for Initial Value Problems in Ordinary Differential Equations 161

7.1. An existence and uniqueness theorem 162

7.2. Theory of one-step methods 163

7.3. One-Step methods 166

7.4. Analysis of round-off error 170

7.5. Asymptotic expansion of the approximations 172

7.6. Extrapolation methods for one-step methods 178

7.7. Step size control 182

Chapter 8. Multistep Methods for Initial Value Problems of Ordinary Differential Equations 189

8.2. The global discretization error for multistep methods 192

8.3. Specific linear multistep methods - preparations 201

8.4. Adams method 204

8.5. Nystrom and Milne-Simpson methods 210

8.6. BDF method 214

8.7. Predictor-corrector methods 216

8.8. Linear homogeneous difference equations 222

8.9. Stiff differential equations 232

Chapter 9. Boundary Value Problems for Ordinary Differential Equations 247

9.1. Problem setting, existence, uniqueness 247

9.2. Difference methods 250

9.3. Galerkin methods 260

9.4. Simple shooting methods 274

Chapter 10. Jacobi, Gauss-Seidel and Relaxation Methods for the Solution of Linear Systems of Equations 281

10.1. Iteration methods for the solution of linear systems of equations 281

10.2. Linear fixed point iteration 282

10.3. Some special classes of matrices and their properties 287

10.4. The Jacobi method 289

10.5. The Gauss-Seidel method 292

10.6. The relaxation method and first convergence results 295

10.7. The relaxation method for consistently ordered matrices 300

Chapter 11. The Conjugate Gradient and GMRES Methods 311

11.1. Prerequisites 311

11.2. The orthogonal residual approach (11.2) for positive definite matrices 313

11.3. The CG method for positive definite matrices 316

11.4. The convergence rate of the CG method 319

11.5. The CG method for the normal equations 323

11.6. Arnoldi process 324

11.7. Realization of GMRES on the basis of the Arnoldi process 328

11.8. Convergence rate of the GMRES method 333

11.9. Appendix 1: Krylov subspaces 334

11.10. Appendix 2: Interactive program systems with multifunctionality 335

Chapter 12. Eigenvalue Problems 339

12.2. Perturbation theory for eigenvalue problems 339

12.3. Localization of eigenvalues 343

12.4. Variational formulation for eigenvalues of symmetric matrices 346

12.5. Perturbation results for the eigenvalues of symmetric matrices 349

12.6. Appendix: Factorization of matrices 350

Chapter 13. Numerical Methods for Eigenvalue Problems 355

13.2. Transformation to Hessenberg form 357

13.3. Newton's method for the calculation of the eigenvalues of Hessenberg matrices 362

13.4. The Jacobi method for the off-diagonal element reduction for symmetric matrices 366

13.5. The QR algorithm 373

13.6. The LR algorithm 386

13.7. The vector iteration 387

Chapter 14. Peano's Error Representation 393

14.2. Peano kernels 394

14.3. Applications 397

Chapter 15. Approximation Theory 401

15.2. Existence of a best approximation 402

15.3. Uniqueness of a best approximation 404

15.4. Approximation theory in spaces with a scalar product 408

15.5. Uniform approximation of continuous functions by polynomials of maximum degree n - 1 411

15.6. Applications of the alternation theorem 415

15.7. Haar spaces, Chebyshev systems 417

Chapter 16. Computer Arithmetic 423

16.1. Number representations 423

16.2. General floating point number systems 424

16.3. Floating point number systems in practical applications 429

16.4. Rounding, truncating 432

16.5. Arithmetic in floating point number systems 436.

Notes:

Includes bibliographical references (pages 443-447) and index.

ISBN:

082182953X

0821834142

OCLC:

51266040