Introduction to numerical analysis / J. Stoer, R. Bulirsch ; translated by R. Bartels, W. Gautschi, and C. Witzgall.
Idioma: Inglés Lenguaje original: Alemán Editor: New York : Springer-Verlag, c1980Descripción: ix, 609 p. : il. ; 25 cmISBN: 0387904204; 3540904204Títulos uniformes: Einführung in die Numerische Mathematik. Inglés Tema(s): Numerical analysisOtra clasificación: 65-011 Error Analysis [1] 1.1 Representation of Numbers [2] 1.2 Roundoff Errors and Floating-Point Arithmetic [5] 1.3 Error Propagation [9] 1.4 Examples [20] 1.5 Interval Arithmetic; Statistical Roundoff Estimation [27] Exercises for Chapter 1 [33] References for Chapter 1 [36] 2 Interpolation [37] 2.1 Interpolation by Polynomials [38] 2.1.1 Theoretical Foundation: The Interpolation Formula of Lagrange [38] 2.1.2 Neville’s Algorithm [40] 2.1.3 Newton’s Interpolation Formula: Divided Differences [43] 2.1.4 The Error in Polynomial Interpolation [49] 2.1.5 Hermite Interpolation [52] 2.2 Interpolation by Rational Functions [58] 2.2.1 General Properties of Rational Interpolation [58] 2.2.2 Inverse and Reciprocal Differences. Thiele’s Continued Fraction [63] 2.2.3 Algorithms of the Neville Type [67] 2.2.4 Comparing Rational and Polynomial Interpolations [71] 2.3 Trigonometric Interpolation [72] 2.3.1 Basic Facts [72] 2.3.2 Fast Fourier Transforms [78] 2.3.3 The Algorithms of Goertzel and Reinsch [84] 2.3.4 The Calculation of Fourier Coefficients. Attenuation Factors [88] Interpolation by Spline Functions [93] Theoretical Foundations [93] Determining Interpolating Spline Functions [97] Convergence Properties of Spline Functions [102] Exercises for Chapter 2 [107] References for Chapter 2 [115] Topics in Integration [117] The Integration Formulas of Newton and Cotes [118] Peano’s Error Representation [123] The Euler-Maclaurin Summation Formula [127] Integrating by Extrapolation [131] About Extrapolation Methods [136] Gaussian Integration Methods [142] Integrals with Singularities [152] Exercises for Chapter 3 [154] References for Chapter 3 [158] Systems of Linear Equations [159] Gaussian Elimination. The Triangular Decomposition of a Matrix 159 The Gauss-Jordan Algorithm [169] The Cholesky Decomposition [172] Error Bounds [175] Roundoff-Error Analysis for Gaussian Elimination [183] Roundoff Errors in Solving Triangular Systems [188] Orthogonalization Techniques of Householder and Gram-Schmidt 190 Data Fitting [197] Linear Least Squares. The Normal Equations [199] The Use of Orthogonalization in Solving Linear Least-Squares Problems [201] The Condition of the Linear Least-Squares Problem [202] Nonlinear Least-Squares Problems [209] The Pseudoinverse of a Matrix [210] Modification Techniques for Matrix Decompositions [213] The Simplex Method [222] Phase One of the Simplex Method [233] Exercises for Chapter 4 [237] References for Chapter 4 [242] Finding Zeros and Minimum Points by Iterative Methods [244] The Development of Iterative Methods [245] General Convergence Theorems [248] The Convergence of Newton’s Method in Several Variables [253] A Modified Newton Method [256] 5.4.1 On the Convergence of Minimization Methods [257] 5.4.2 Application of the Convergence Criteria to the Modified Newton Method [262] 5.4.3 Suggestions for a Practical Implementation of the Modified Newton Method. A Rank-One Method Due to Broyden [266] 5.5 Roots of Polynomials. Application of Newton’s Method [270] 5.6 Sturm Sequences and Bisection Methods [281] 5.7 Bairstow’s Method [285] 5.8 The Sensitivity of Polynomial Roots [287] 5.9 Interpolation Methods for Determining Roots [290] 5.10 The Δ2-Method of Aitken [296] 5.11 Minimization Problems without Constraints [300] Exercises for Chapter 5 [309] References for Chapter 5 [312] 6 Eigenvalue Problems [314] 6.0 Introduction [314] 6.1 Basic Facts on Eigenvalues [316] 6.2 The Jordan Normal Form of a Matrix [319] 6.3 The Frobenius Normal Form of a Matrix [324] 6.4 The Schur Normal Form of a Matrix; Hermitian and Normal Matrices; Singular Values of Matrices [328] 6.5 Reduction of Matrices to Simpler Form [334] 6.5.1 Reduction of a Hermitian Matrix to Tridiagonal Form. The Method of Householder [336] 6.5.2 Reduction of a Hermitian Matrix to Tridiagonal or Diagonal Form: The Methods of Givens and Jacobi [341] 6.5.3 Reduction to Frobenius Form [345] 6.5.4 Reduction to Hessenberg Form [347] 6.6 Methods for Determining the Eigenvalues and Eigenvectors [351] 6.6.1 Computation of the Eigenvalues of a Hermitian Tridiagonal Matrix [351] 6.6.2 Computation of the Eigenvalues of a Hessenberg Matrix. The Method of Hyman [353] 6.6.3 Simple Vector Iteration and Inverse Iteration of Wielandt [354] 6.6.4 The LR Method [361] 6.6.5 Practical Implementation of the LR Method [368] 6.6.6 The QR Method [370] 6.7 Computation of the Singular Values of a Matrix [377] 6.8 Generalized Eigenvalue Problems [382] 6.9 Estimation of Eigenvalues [383] Exercises for Chapter 6 [396] References for Chapter 6 [402] 7 Ordinary Differential Equations [404] 7.0 Introduction [404] 7.1 Some Theorems from the Theory of Ordinary Differential Equations [406] 7.2 Initial-Value Problems [410] 7.2.1 One-Step Methods: Basic Concepts [410] 7.2.2 Convergence of One-Step Methods [415] 7.2.3 Asymptotic Expansions for the Global Discretization Error of One-Step Methods [419] 7.2.4 The Influence of Rounding Errors in One-Step Methods [420] 7.2.5 Practical Implementation of One-Step Methods [423] 7.2.6 Multistep Methods: Examples [429] 7.2.7 General Multistep Methods [432] 7.2.8 An Example of Divergence [435] 7.2.9 Linear Difference Equations [438] 7.2.10 Convergence of Multistep Methods [441] 7.2.11 Linear Multistep Methods [445] 7.2.12 Asymptotic Expansions of the Global Discretization Error for Linear Multistep Methods [450] 7.2.13 Practical Implementation of Multistep Methods [455] 7.2.14 Extrapolation Methods for the Solution of the Initial-Value Problem [458] 7.2.15 Comparison of Methods for Solving Initial-Value Problems [461] 7.2.16 Stiff* Differential Equations [462] 7.3 Boundary-Value Problems [466] 7.3.0 Introduction [466] 7.3.1 The Simple Shooting Method [469] 7.3.2 The Simple Shooting Method for Linear Boundary-Value Problems [474] 7.3.3 An Existence and Uniqueness Theorem for the Solution of Boundary-Value Problems [476] 7.3.4 Difficulties in the Execution of the Simple Shooting Method [478] 7.3.5 The Multiple Shooting Method [483] 7.3.6 Hints for the Practical Implementation of the Multiple Shooting Method [487] 7.3.7 An Example: Optimal Control Program for a Lifting Reentry Space Vehicle [491] 7.3.8 The Limiting Case m -> ∞ of the Multiple Shooting Method (General Newton’s Method, Quasilinearization) [498] 7.4 Difference Methods [502] 7.5 Variational Methods [507] 7.6 Comparison of the Methods for Solving Boundary-Value Problems for Ordinary Differential Equations [515] 7.7 Variational Methods for Partial Differential Equations. The Finite-Element Method [519] Exercises for Chapter 7 [526] References for Chapter 7 [532] 8 Iterative Methods for the Solution of Large Systems of Linear Equations. Some Further Methods [536] 8.0 Introduction [536] 8.1’ General Procedures for the Construction of Iterative Methods [537] 8.2 Convergence Theorems [540] 8.3 Relaxation Methods [545] 8.4 Applications to Difference Methods—An Example [554] 8.5 Block Iterative Methods [560] 8.6 The ADI-Method of Peaceman and Rachford [563] 8.7 The Conjugate-Gradient Method of Hestenes and Stiefel [572] 8.8 The Algorithm of Buneman for the Solution of the Discretized Poisson Equation [576] 8.9 Comparison of Iterative Methods 584 Exercises for Chapter 8 588 References for Chapter 8 [595] General Literature on Numerical Methods [597] Index [599]
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Instituto de Matemática, CONICET-UNS | Libros ordenados por tema | 65 St872 (Browse shelf) | Checked out | 2025-02-10 | A-5132 |
Traducción de: Einführung in die Numerische Mathematik.
Incluye índice.
Incluye referencias bibliográficas.
1 Error Analysis [1] --
1.1 Representation of Numbers [2] --
1.2 Roundoff Errors and Floating-Point Arithmetic [5] --
1.3 Error Propagation [9] --
1.4 Examples [20] --
1.5 Interval Arithmetic; Statistical Roundoff Estimation [27] --
Exercises for Chapter 1 [33] --
References for Chapter 1 [36] --
2 Interpolation [37] --
2.1 Interpolation by Polynomials [38] --
2.1.1 Theoretical Foundation: The Interpolation Formula of Lagrange [38] --
2.1.2 Neville’s Algorithm [40] --
2.1.3 Newton’s Interpolation Formula: Divided Differences [43] --
2.1.4 The Error in Polynomial Interpolation [49] --
2.1.5 Hermite Interpolation [52] --
2.2 Interpolation by Rational Functions [58] --
2.2.1 General Properties of Rational Interpolation [58] --
2.2.2 Inverse and Reciprocal Differences. Thiele’s Continued Fraction [63] --
2.2.3 Algorithms of the Neville Type [67] --
2.2.4 Comparing Rational and Polynomial Interpolations [71] --
2.3 Trigonometric Interpolation [72] --
2.3.1 Basic Facts [72] --
2.3.2 Fast Fourier Transforms [78] --
2.3.3 The Algorithms of Goertzel and Reinsch [84] --
2.3.4 The Calculation of Fourier Coefficients. Attenuation Factors [88] --
Interpolation by Spline Functions [93] --
Theoretical Foundations [93] --
Determining Interpolating Spline Functions [97] --
Convergence Properties of Spline Functions [102] --
Exercises for Chapter 2 [107] --
References for Chapter 2 [115] --
Topics in Integration [117] --
The Integration Formulas of Newton and Cotes [118] --
Peano’s Error Representation [123] --
The Euler-Maclaurin Summation Formula [127] --
Integrating by Extrapolation [131] --
About Extrapolation Methods [136] --
Gaussian Integration Methods [142] --
Integrals with Singularities [152] --
Exercises for Chapter 3 [154] --
References for Chapter 3 [158] --
Systems of Linear Equations [159] --
Gaussian Elimination. The Triangular Decomposition of a Matrix 159 The Gauss-Jordan Algorithm [169] --
The Cholesky Decomposition [172] --
Error Bounds [175] --
Roundoff-Error Analysis for Gaussian Elimination [183] --
Roundoff Errors in Solving Triangular Systems [188] --
Orthogonalization Techniques of Householder and Gram-Schmidt 190 Data Fitting [197] --
Linear Least Squares. The Normal Equations [199] --
The Use of Orthogonalization in Solving Linear Least-Squares Problems [201] --
The Condition of the Linear Least-Squares Problem [202] --
Nonlinear Least-Squares Problems [209] --
The Pseudoinverse of a Matrix [210] --
Modification Techniques for Matrix Decompositions [213] --
The Simplex Method [222] --
Phase One of the Simplex Method [233] --
Exercises for Chapter 4 [237] --
References for Chapter 4 [242] --
Finding Zeros and Minimum Points by Iterative --
Methods [244] --
The Development of Iterative Methods [245] --
General Convergence Theorems [248] --
The Convergence of Newton’s Method in Several Variables [253] --
A Modified Newton Method [256] --
5.4.1 On the Convergence of Minimization Methods [257] --
5.4.2 Application of the Convergence Criteria to the Modified Newton Method [262] --
5.4.3 Suggestions for a Practical Implementation of the Modified Newton Method. A Rank-One Method Due to Broyden [266] --
5.5 Roots of Polynomials. Application of Newton’s Method [270] --
5.6 Sturm Sequences and Bisection Methods [281] --
5.7 Bairstow’s Method [285] --
5.8 The Sensitivity of Polynomial Roots [287] --
5.9 Interpolation Methods for Determining Roots [290] --
5.10 The Δ2-Method of Aitken [296] --
5.11 Minimization Problems without Constraints [300] --
Exercises for Chapter 5 [309] --
References for Chapter 5 [312] --
6 Eigenvalue Problems [314] --
6.0 Introduction [314] --
6.1 Basic Facts on Eigenvalues [316] --
6.2 The Jordan Normal Form of a Matrix [319] --
6.3 The Frobenius Normal Form of a Matrix [324] --
6.4 The Schur Normal Form of a Matrix; Hermitian and Normal Matrices; Singular Values of Matrices [328] --
6.5 Reduction of Matrices to Simpler Form [334] --
6.5.1 Reduction of a Hermitian Matrix to Tridiagonal Form. The Method of Householder [336] --
6.5.2 Reduction of a Hermitian Matrix to Tridiagonal or Diagonal Form: The Methods of Givens and Jacobi [341] --
6.5.3 Reduction to Frobenius Form [345] --
6.5.4 Reduction to Hessenberg Form [347] --
6.6 Methods for Determining the Eigenvalues and Eigenvectors [351] --
6.6.1 Computation of the Eigenvalues of a Hermitian Tridiagonal Matrix [351] --
6.6.2 Computation of the Eigenvalues of a Hessenberg Matrix. The Method of Hyman [353] --
6.6.3 Simple Vector Iteration and Inverse Iteration of Wielandt [354] --
6.6.4 The LR Method [361] --
6.6.5 Practical Implementation of the LR Method [368] --
6.6.6 The QR Method [370] --
6.7 Computation of the Singular Values of a Matrix [377] --
6.8 Generalized Eigenvalue Problems [382] --
6.9 Estimation of Eigenvalues [383] --
Exercises for Chapter 6 [396] --
References for Chapter 6 [402] --
7 Ordinary Differential Equations [404] --
7.0 Introduction [404] --
7.1 Some Theorems from the Theory of Ordinary Differential Equations [406] --
7.2 Initial-Value Problems [410] --
7.2.1 One-Step Methods: Basic Concepts [410] --
7.2.2 Convergence of One-Step Methods [415] --
7.2.3 Asymptotic Expansions for the Global Discretization Error of One-Step Methods [419] --
7.2.4 The Influence of Rounding Errors in One-Step Methods [420] --
7.2.5 Practical Implementation of One-Step Methods [423] --
7.2.6 Multistep Methods: Examples [429] --
7.2.7 General Multistep Methods [432] --
7.2.8 An Example of Divergence [435] --
7.2.9 Linear Difference Equations [438] --
7.2.10 Convergence of Multistep Methods [441] --
7.2.11 Linear Multistep Methods [445] --
7.2.12 Asymptotic Expansions of the Global Discretization Error for Linear Multistep Methods [450] --
7.2.13 Practical Implementation of Multistep Methods [455] --
7.2.14 Extrapolation Methods for the Solution of the Initial-Value Problem [458] --
7.2.15 Comparison of Methods for Solving Initial-Value Problems [461] --
7.2.16 Stiff* Differential Equations [462] --
7.3 Boundary-Value Problems [466] --
7.3.0 Introduction [466] --
7.3.1 The Simple Shooting Method [469] --
7.3.2 The Simple Shooting Method for Linear Boundary-Value Problems [474] --
7.3.3 An Existence and Uniqueness Theorem for the Solution of Boundary-Value Problems [476] --
7.3.4 Difficulties in the Execution of the Simple Shooting Method [478] --
7.3.5 The Multiple Shooting Method [483] --
7.3.6 Hints for the Practical Implementation of the Multiple Shooting Method [487] --
7.3.7 An Example: Optimal Control Program for a Lifting Reentry Space Vehicle [491] --
7.3.8 The Limiting Case m -> ∞ of the Multiple Shooting Method (General Newton’s Method, Quasilinearization) [498] --
7.4 Difference Methods [502] --
7.5 Variational Methods [507] --
7.6 Comparison of the Methods for Solving Boundary-Value Problems for Ordinary Differential Equations [515] --
7.7 Variational Methods for Partial Differential Equations. The Finite-Element Method [519] --
Exercises for Chapter 7 [526] --
References for Chapter 7 [532] --
8 Iterative Methods for the Solution of Large Systems of Linear Equations. Some Further Methods [536] --
8.0 Introduction [536] --
8.1’ General Procedures for the Construction of Iterative Methods [537] --
8.2 Convergence Theorems [540] --
8.3 Relaxation Methods [545] --
8.4 Applications to Difference Methods—An Example [554] --
8.5 Block Iterative Methods [560] --
8.6 The ADI-Method of Peaceman and Rachford [563] --
8.7 The Conjugate-Gradient Method of Hestenes and Stiefel [572] --
8.8 The Algorithm of Buneman for the Solution of the Discretized Poisson Equation [576] --
8.9 Comparison of Iterative Methods 584 Exercises for Chapter 8 588 References for Chapter 8 [595] --
General Literature on Numerical Methods [597] --
Index [599] --
MR, 83d:65003
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