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Preface to the Second Edition | |

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Preface to the First Edition | |

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Error Analysis | |

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Representation of Numbers | |

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Roundoff Errors and Floating-Point Arithmetic | |

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Error Propagation | |

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Examples | |

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Interval Arithmetic; Statistical Roundoff Estimation | |

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Interpolation | |

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Interpolation by Polynomials | |

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Theoretical Foundation: The Interpolation Formula of Lagrange | |

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Neville's Algorithm | |

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Newton's Interpolation Formula: Divided Differences | |

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The Error in Polynomial Interpolation | |

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Hermite Interpolation | |

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Interpolation by Rational Functions | |

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General Properties of Rational Interpolation | |

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Inverse and Reciprocal Differences. Thiele's Continued Fraction | |

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Algorithms of the Neville Type | |

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Comparing Rational and Polynomial Interpolations | |

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Trigonometric Interpolation | |

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Basic Facts | |

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Fast Fourier Transforms | |

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The Algorithms of Goertzel and Reinsch | |

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The Calculation of Fourier Coefficients. Attenuation Factors | |

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Interpolation by Spline Functions | |

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Theoretical Foundations | |

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Determining Interpolating Cubic Spline Functions | |

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Convergence Properties of Cubic Spline Functions | |

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B-Splines | |

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The Computation of B-Splines | |

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Topics in Integration | |

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The Integration Formulas of Newton and Cotes | |

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Peano's Error Representation | |

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The Euler-Maclaurin Summation Formula | |

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Integrating by Extrapolation | |

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About Extrapolation Methods | |

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Gaussian Integration Methods | |

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Integrals with Singularities | |

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Systems of Linear Equations | |

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Gaussian Elimination. The Triangular Decomposition of a Matrix | |

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The Gauss-Jordan Algorithm | |

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The Cholesky Decomposition | |

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Error Bounds | |

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Roundoff-Error Analysis for Gaussian Elimination | |

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Roundoff Errors in Solving Triangular Systems | |

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Orthogonalization Techniques of Householder and Gram-Schmidt | |

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Data Fitting | |

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Linear Least Squares. The Normal Equations | |

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The Use of Orthogonalization in Solving Linear Least-Squares Problems | |

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The Condition of the Linear Least-Squares Problem | |

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Nonlinear Least-Squares Problems | |

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The Pseudoinverse of a Matrix | |

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Modification Techniques for Matrix Decompositions | |

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The Simplex Method | |

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Phase One of the Simplex Method | |

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Appendix to Chapter 4 | |

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Elimination Methods for Sparse Matrices | |

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Finding Zeros and Minimum Points by Iterative Methods | |

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The Development of Iterative Methods | |

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General Convergence Theorems | |

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The Convergence of Newton's Method in Several Variables | |

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A Modified Newton Method | |

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On the Convergence of Minimization Methods | |

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Application of the Convergence Criteria to the Modified Newton Method | |

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Suggestions for a Practical Implementation of the Modified Newton Method. A Rank-One Method Due to Broyden | |

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Roots of Polynomials. Application of Newton's Method | |

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Sturm Sequences and Bisection Methods | |

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Bairstow's Method | |

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The Sensitivity of Polynomial Roots | |

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Interpolation Methods for Determining Roots | |

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The [Delta][superscript 2]-Method of Aitken | |

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Minimization Problems without Constraints | |

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Eigenvalue Problems | |

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Basic Facts on Eigenvalues | |

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The Jordan Normal Form of a Matrix | |

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The Frobenius Norma] Form of a Matrix | |

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The Schur Normal Form of a Matrix; Hermitian and Normal Matrices; Singular Values of Matrices | |

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Reduction of Matrices to Simpler Form | |

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Reduction of a Hermitian Matrix to Tridiagonal Form: The Method of Householder | |

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Reduction of a Hermitian Matrix to Tridiagonal or Diagonal Form: The Methods of Givens and Jacobi | |

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Reduction ofa Hermitian Matrix to Tridiagonal Form: The Method of Lanczos | |

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Reduction to Hessenberg Form | |

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Methods for Determining the Eigenvalues and Eigenvectors | |

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Computation of the Eigenvalues of a Hermitian Tridiagonal Matrix | |

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Computation of the Eigenvalues of a Hessenberg Matrix. The Method of Hyman | |

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Simple Vector Iteration and Inverse Iteration of Wielandt | |

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The LR and QR Methods | |

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The Practical Implementation of the QR Method | |

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Computation of the Singular Values of a Matrix | |

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Generalized Eigenvalue Problems | |

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Estimation of Eigenvalues | |

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Ordinary Differential Equations | |

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Some Theorems from the Theory of Ordinary Differential Equations | |

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Initial-Value Problems | |

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One-Step Methods: Basic Concepts | |

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Convergence of One-Step Methods | |

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Asymptotic Expansions for the Global Discretization Error of One-Step Methods | |

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The Influence of Rounding Errors in One-Step Methods | |

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Practical Implementation of One-Step Methods | |

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Multistep Methods: Examples | |

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General Multistep Methods | |

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An Example of Divergence | |

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Linear Difference Equations | |

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Convergence of Multistep Methods | |

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Linear Multistep Methods | |

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Asymptotic Expansions of the Global Discretization Error for Linear Multistep Methods | |

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Practical Implementation of Multistep Methods | |

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Extrapolation Methods for the Solution of the Initial-Value Problem | |

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Comparison of Methods for Solving Initial-Value Problems | |

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Stiff Differential Equations | |

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Implicit Differential Equations. Differential-Algebraic Equations | |

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Boundary-Value Problems | |

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Introduction | |

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The Simple Shooting Method | |

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The Simple Shooting Method for Linear Boundary-Value Problems | |

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An Existence and Uniqueness Theorem for the Solution of Boundary-Value Problems | |

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Difficulties in the Execution of the Simple Shooting Method | |

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The Multiple Shooting Method | |

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Hints for the Practical Implementation of the Multiple Shooting Method | |

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An Example: Optimal Control Program for a Lifting Reentry Space Vehicle | |

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The Limiting Case m [actual symbol not reproducible] of the Multiple Shooting Method (General Newton's Method, Quasilinearization) | |

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Difference Methods | |

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Variational Methods | |

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Comparison of the Methods for Solving Boundary-Value Problems for Ordinary Differential Equations | |

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Variational Methods for Partial Differential Equations. The Finite-Element Method | |

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Iterative Methods for the Solution of Large Systems of Linear Equations. Some Further Methods | |

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General Procedures for the Construction of Iterative Methods | |

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Convergence Theorems | |

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Relaxation Methods | |

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Applications to Difference Methods - An Example | |

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Block Iterative Methods | |

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The ADI-Method of Peaceman and Rachford | |

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The Conjugate-Gradient Method of Hestenes and Stiefel | |

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The Algorithm of Buneman for the Solution of the Discretized Poisson Equation | |

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Multigrid Methods | |

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Comparison of Iterative Methods | |

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General Literature on Numerical Methods | |

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Index | |