Introduction to numerical analysis / J. Stoer, R. Bulirsch ; translated by R. Bartels, W. Gautschi, and C. Witzgall.

Por: Stoer, JosefColaborador(es): Bulirsch, RolandIdioma: 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: 65-01
Contenidos:
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]
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Traducción de: Einführung in die Numerische Mathematik.

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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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