| Preface to the English Edition | p. xi |
| Preface to the German Edition | p. xiii |
| Interpolation by Polynomials | p. 1 |
| General prerequisites and Landau symbols | p. 1 |
| Existence and uniqueness of an interpolating polynomial | p. 3 |
| Neville's algorithm | p. 6 |
| Newton's interpolation formula, divided differences | p. 8 |
| The interpolation error | p. 11 |
| Chebyshev polynomials | p. 14 |
| Additional topics and literature | p. 18 |
| Exercises | p. 19 |
| Spline Functions | p. 23 |
| Introductory remarks | p. 23 |
| Interpolating linear spline functions | p. 24 |
| Minimality properties of cubic spline functions | p. 25 |
| The calculation of interpolating cubic spline functions | p. 27 |
| Error estimates for interpolating cubic splines | p. 33 |
| Additional topics and literature | p. 38 |
| Exercises | p. 38 |
| The Discrete Fourier Transform and Its Applications | p. 41 |
| Discrete Fourier transform | p. 41 |
| Applications of the discrete Fourier transform | p. 43 |
| Fast Fourier transform (FFT) | p. 49 |
| Additional topics and literature | p. 56 |
| Exercises | p. 57 |
| Solution of Linear Systems of Equations | p. 59 |
| Triangular systems | p. 59 |
| Gaussian elimination | p. 61 |
| The factorization PA = LR | p. 66 |
| LR factorization | p. 74 |
| Cholesky factorization for positive definite matrices | p. 76 |
| Banded matrices | p. 79 |
| Norms and error estimates | p. 81 |
| The factorization A = QS | p. 91 |
| Additional topics and literature | p. 100 |
| Exercises | p. 100 |
| Nonlinear Systems of Equations | p. 105 |
| Preliminary remarks | p. 105 |
| The one-dimensional case (N = 1) | p. 107 |
| Banach's fixed point theorem | p. 109 |
| Newton's method | p. 112 |
| Additional topics and literature | p. 121 |
| Exercises | p. 121 |
| The Numerical Integration of Functions | p. 123 |
| Quadrature by interpolation formulas | p. 124 |
| Special quadrature by interpolation formulas | p. 125 |
| The error due to quadrature by interpolation | p. 129 |
| Degree of exactness for the closed Newton-Cotes formulas, n even | p. 132 |
| Composite Newton-Cotes formulas | p. 137 |
| Asymptotic form of the composite trapezoidal rule | p. 141 |
| Extrapolation methods | p. 142 |
| Gaussian quadrature | p. 146 |
| Appendix: Proof of the asymptotic form for the composite trapezoidal rule | p. 155 |
| Additional topics and literature | p. 159 |
| Exercises | p. 159 |
| Explicit One-Step Methods for Initial Value Problems in Ordinary Differential Equations | p. 161 |
| An existence and uniqueness theorem | p. 162 |
| Theory of one-step methods | p. 163 |
| One-Step methods | p. 166 |
| Analysis of round-off error | p. 170 |
| Asymptotic expansion of the approximations | p. 172 |
| Extrapolation methods for one-step methods | p. 178 |
| Step size control | p. 182 |
| Additional topics and literature | p. 186 |
| Exercises | p. 186 |
| Multistep Methods for Initial Value Problems of Ordinary Differential Equations | p. 189 |
| Fundamental terms | p. 189 |
| The global discretization error for multistep methods | p. 192 |
| Specific linear multistep methods - preparations | p. 201 |
| Adams method | p. 204 |
| Nystrom and Milne-Simpson methods | p. 210 |
| BDF method | p. 214 |
| Predictor-corrector methods | p. 216 |
| Linear homogeneous difference equations | p. 222 |
| Stiff differential equations | p. 232 |
| Additional topics and literature | p. 241 |
| Exercises | p. 242 |
| Boundary Value Problems for Ordinary Differential Equations | p. 247 |
| Problem setting, existence, uniqueness | p. 247 |
| Difference methods | p. 250 |
| Galerkin methods | p. 260 |
| Simple shooting methods | p. 274 |
| Additional topics and literature | p. 276 |
| Exercises | p. 277 |
| Jacobi, Gauss-Seidel and Relaxation Methods for the Solution of Linear Systems of Equations | p. 281 |
| Iteration methods for the solution of linear systems of equations | p. 281 |
| Linear fixed point iteration | p. 282 |
| Some special classes of matrices and their properties | p. 287 |
| The Jacobi method | p. 289 |
| The Gauss-Seidel method | p. 292 |
| The relaxation method and first convergence results | p. 295 |
| The relaxation method for consistently ordered matrices | p. 300 |
| Additional topics and literature | p. 305 |
| Exercises | p. 305 |
| The Conjugate Gradient and GMRES Methods | p. 311 |
| Prerequisites | p. 311 |
| The orthogonal residual approach (11.2) for positive definite matrices | p. 313 |
| The CG method for positive definite matrices | p. 316 |
| The convergence rate of the CG method | p. 319 |
| The CG method for the normal equations | p. 323 |
| Arnoldi process | p. 324 |
| Realization of GMRES on the basis of the Arnoldi process | p. 328 |
| Convergence rate of the GMRES method | p. 333 |
| Appendix 1: Krylov subspaces | p. 334 |
| Appendix 2: Interactive program systems with multifunctionality | p. 335 |
| Additional topics and literature | p. 336 |
| Exercises | p. 337 |
| Eigenvalue Problems | p. 339 |
| Introduction | p. 339 |
| Perturbation theory for eigenvalue problems | p. 339 |
| Localization of eigenvalues | p. 343 |
| Variational formulation for eigenvalues of symmetric matrices | p. 346 |
| Perturbation results for the eigenvalues of symmetric matrices | p. 349 |
| Appendix: Factorization of matrices | p. 350 |
| Additional topics and literature | p. 351 |
| Exercises | p. 351 |
| Numerical Methods for Eigenvalue Problems | p. 355 |
| Introductory remarks | p. 355 |
| Transformation to Hessenberg form | p. 357 |
| Newton's method for the calculation of the eigenvalues of Hessenberg matrices | p. 362 |
| The Jacobi method for the off-diagonal element reduction for symmetric matrices | p. 366 |
| The QR algorithm | p. 373 |
| The LR algorithm | p. 386 |
| The vector iteration | p. 387 |
| Additional topics and literature | p. 389 |
| Exercises | p. 390 |
| Peano's Error Representation | p. 393 |
| Introductory remarks | p. 393 |
| Peano kernels | p. 394 |
| Applications | p. 397 |
| Additional topics and literature | p. 398 |
| Exercises | p. 398 |
| Approximation Theory | p. 401 |
| Introductory remarks | p. 401 |
| Existence of a best approximation | p. 402 |
| Uniqueness of a best approximation | p. 404 |
| Approximation theory in spaces with a scalar product | p. 408 |
| Uniform approximation of continuous functions by polynomials of maximum degree n - 1 | p. 411 |
| Applications of the alternation theorem | p. 415 |
| Haar spaces, Chebyshev systems | p. 417 |
| Additional topics and literature | p. 420 |
| Exercises | p. 420 |
| Computer Arithmetic | p. 423 |
| Number representations | p. 423 |
| General floating point number systems | p. 424 |
| Floating point number systems in practical applications | p. 429 |
| Rounding, truncating | p. 432 |
| Arithmetic in floating point number systems | p. 436 |
| Additional topics and literature | p. 441 |
| Bibliography | p. 443 |
| Index | p. 449 |
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