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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 | |