Numerical methods / Germund Dahlquist, Åke Björck ; translated by Ned Anderson.
Idioma: Inglés Lenguaje original: Sueco Series Prentice-Hall series in automatic computationEditor: Englewood Cliffs, N.J. : Prentice-Hall, c1974Descripción: xviii, 573 p. : il. ; 24 cmISBN: 0136273157Tema(s): Numerical analysis -- Data processingOtra clasificación: 65-021 SOME GENERAL PRINCIPLES OF NUMERICAL [1] CALCULATION [1] 1.1. Introduction [1] 1.2. Some Common Ideas and Concepts in Numerical Methods [2] 1.3. Numerical Problems and Algorithms [13] 1.3.1. Definitions [13] 1.3.2. Recursive Formulas; Homer’s Rule [14] 1.3.3. An Example of Numerical Instability [16] 2 HOW TO OBTAIN AND ESTIMATE ACCURACY IN NUMERICAL CALCULATIONS [21] 2.1. Basic Concepts in Error Estimation [21] 2.1.1. Introduction [21] 2.1.2. Sources of Error [22] 2.1.3. Absolute and Relative Errors [23] 2.1.4. Rounding and Chopping [24] 2.2 Propagation of Errors [26] 2.2.1. Simple Examples of Error Analysis [26] 2.2.2. The General Formula for Error Propagation; Maximum Error and Standard Error [29] 2.2.3. On the Practical Application of Error Estimation [34] 2.2.4. The Use of Experimental Perturbations [35] 2.2.5. Automatic Control of Accuracy [37] 2.3. Number Systems; Floating and Fixed Representation [42] 2.3.1. The Position System [42] 2.3.2. Floating and Fixed Representation [43] 2.3.3. Floating Decimal Point [44] 2.3.4. Fixed Decimal Point [46] 2.3.5. Round-off Errors in Computation with Floating Arithmetic Operations [46] 2.4. Backward Error Analysis*; Condition Numbers [51] 2.4.1. Backward Error Analysis [51] 2.4.2. Condition Numbers for Problems and Algorithms [53] 2.4.3. Geometrical Illustration of Error Analysis [56] 3 NUMERICAL USES OF SERIES [60] 3.1. Elementary Uses of Series [60] 3.1.1. Simple Examples [60] 3.1.2. Estimating the Remainder [62] 3.1.3. Power Series [65] 3.2. Acceleration of Convergence [71] 3.2.1. Slowly Converging Alternating Series [71] 3.2.2. Slowly Converging Series with Positive Terms [73] 3.2.3. Other Simple Ways to Accelerate Convergence [74] 3.2.4. Ill-Conditioned Series [75] 3.2.5. Numerical Use of Divergent Series [77] 4 APPROXIMATION OF FUNCTIONS [81] 4.1. Basic Concepts in Approximation [81] 4.1.1. Introduction [81] 4.1.2. The Idea of a Function Space [84] 4.1.3. Norms and Seminorms [85] 4.1.4. Approximation of Functions as a Geometric Problem in Function Space [87] 4.2. The Approximation of Functions by the Method of Least Squares [88] 4.2.1. Statement of the Problems [88] 4.2.2. Orthogonal Systems [89] 4.2.3. Solution of the Approximation Problem [92] 4.3. Polynomials [97] 4.3.1. Basic Terminology; the Weierstrass Approximation Theorem [97] 4.3.2, Triangle Families of Polynomials [98] 2.2 Propagation of Errors [26] 2.2.1. Simple Examples of Error Analysis [26] 2.2.2. The General Formula for Error Propagation; Maximum Error and Standard Error [29] 2.2.3. On the Practical Application of Error Estimation [34] 2.2.4. The Use of Experimental Perturbations [35] 2.2.5. Automatic Control of Accuracy [37] 2.3. Number Systems; Floating and Fixed Representation [42] 2.3.1. The Position System [42] 2.3.2. Floating and Fixed Representation [43] 2.3.3. Floating Decimal Point [44] 2.3.4. Fixed Decimal Point [46] 2.3.5. Round-off Errors in Computation with Floating Arithmetic Operations [46] 2.4. Backward Error Analysis*; Condition Numbers [51] 2.4.1. Backward Error Analysis [51] 2.4.2. Condition Numbers for Problems and Algorithms [53] 2.4.3. Geometrical Illustration of Error Analysis [56] 3 NUMERICAL USES OF SERIES [60] 3.1. Elementary Uses of Series [60] 3.1.1. Simple Examples [60] 3.1.2. Estimating the Remainder [62] 3.1.3. Power Series [65] 3.2. Acceleration of Convergence [71] 3.2.1. Slowly Converging Alternating Series [71] 3.2.2. Slowly Converging Series with Positive Terms [73] 3.2.3. Other Simple Ways to Accelerate Convergence [74] 3.2.4. Ill-Conditioned Series [75] 3.2.5. Numerical Use of Divergent Series [77] 4 APPROXIMATION OF FUNCTIONS [81] 4.1. Basic Concepts in Approximation [81] 4.1.1. Introduction [81] 4.1.2. The Idea of a Function Space [84] 4.1.3. Norms and Seminorms [85] 4.1.4. Approximation of Functions as a Geometric Problem in Function Space [87] 4.2. The Approximation of Functions by the Method of Least Squares [88] 4.2.1. Statement of the Problems [88] 4.2.2. Orthogonal Systems [89] 4.2.3. Solution of the Approximation Problem [92] 4.3. Polynomials [97] 4.3.1. Basic Terminology; the Weierstrass Approximation Theorem [97] 4.3.2, Triangle Families of Polynomials [98] 4.3.3. A Triangle Family and Its Application to Interpolation [99] 4.3.4. Equidistant Interpolation and the Runge Phenomenon [101] 4.4. Orthogonal Polynomials and Applications [104] 4.4.1. Tchebycheff Polynomials [104] 4.4.2. Tchebycheff Interpolation and Smoothing [106] 4.4.3. General Theory of Orthogonal Polynomials [108] 4.4.4. Legendre Polynomials and Gram Polynomials [113] 4.5. Complementary Observations on Polynomial Approximation [117] 4.5.1. Summary of the Use of Polynomials [117] 4.5.2. Some Inequalities for En(f) with Applications, to the Computation of Linear Functionals [120] 4.5.3. Approximation in the Maximum Norm [124] 4.5.4. Economization of Power Series; Standard Functions [125] 4.5.5. Some Statistical Aspects of the Method of Least Squares [126] 4.6. Spline Functions [131] 5 NUMERICAL LINEAR ALGEBRA [137] 5.1. Introduction [137] 5.2. Basic Concepts of Linear Algebra [138] 5.2.1. Fundamental Definitions [138] 5.2.2. Partitioned Matrices [140] 5.2.3. Linear Vector Spaces [141] 5.2.4. Eigenvalues and Similarity Transformations [142] 5.2.5. Singular-Value Decomposition and Pseudo-Inverse [143] 5.3. Direct Methods for Solving Systems of Linear Equations [146] 5.3.1. Triangular Systems [146] 5.3.2. Gaussian Elimination [147] 5.3.3. Pivoting Strategies [150] 5.3.4. LU-Decomposition [152] 5.3.5. Compact Schemes for Gaussian Elimination [157] 5.3.6. Inverse Matrices [159] 5.4. Special Matrices [162] 5.4.1. Symmetric Positive-Definite Matrices [162] 5.4.2. Band Matrices [165] 5.4.3. Large-Scale Linear Systems [168] 5.4.4. Other Sparse Matrices [169] 5.5. Error Analysis for Linear Systems [174] 5.5.1. An Ill-Conditioned Example [174] 5.5.2. Vector and Matrix Norms [175] 5.5.3. Perturbation Analysis [176] 5.5.4. Rounding Errors in Gaussian Elimination [177] 5.5.5. Scaling of Linear Systems [181] 5.5.6. Iterative Improvement of a Solution [183] 5.6. Iterative Methods [188] 5.7. Overdetermined Linear Systems [196] 5.7.1. The Normal Equations [197] 5.7.2. Orthogonalization Methods [201] 5.7.3. Improvement of Least-Squares Solution [204] 5.7.4. Least-Squares Problems with Linear Constraints [205] 5.8. Computation of Eigenvalues and Eigenvectors [208] 5.8.1. The Power Method [209] 5.8.2. Methods Based on Similarity Transformations [211] 5.8.3. Eigenvalues by Equation Solving [215] 5.8.4. The QR-Algorithm [216]
6 NONLINEAR EQUATIONS [218] 6.1. Introduction [218] 6.2. Initial Approximations; Starting Methods [219] 6.2.1. Introduction [219] 6.2.2. The Bisection Method [220] 6.3. Newton-Raphson’s Method [222] 6.4. The Secant Method [227] 6.4.1. Description of the Method [227] 6.4.2. Error Analysis for the Secant Method [228] 6.4.3. Regula Falsi [230] 6.4.4. Other Related Methods [230] 6.5. General Theory of Iteration Methods [233] 6.6. Error Estimation and Attainable Accuracy in Iteration Methods [238] 6.6.1. Error Estimation [238] 6.6.2. Attainable Accuracy; Termination Criteria [240] 6.7. Multiple Roots [242] 6.8. Algebraic Equations [243] 6.8.1. Introduction [243] 6.8.2. Deflation [245] 6.8.3. Ill-Conditioned Algebraic Equations [246] 6.9. Systems of Nonlinear Equations [248] 6.9.1. Iteration [249] 6.9.2. Newton-Raphson’s Method and Some Modifications [249] 6.9.3. Other Methods [251] 7 FINITE DIFFERENCES WITH APPLICATIONS TO NUMERICAL INTEGRATION, DIFFERENTIATION, AND INTERPOLATION [255] 7.1. Difference Operators and Their Simplest Properties [255] 7.2. Simple Methods for Deriving Approximation Formulas and Error Estimates [263] 7.2.1. Statement of the Problems and Some Typical Examples [263] 7.2.2. Repeated Richardson Extrapolation [269] 7.3. Interpolation [275] 7.3.1. Introduction [275] 7.3.2. When is Linear Interpolation Sufficient? [276] 7.3.3. Newton’s General Interpolation Formula [277] 7.3.4. Formulas for Equidistant Interpolation [279] 7.3.5. Complementary Remarks on Interpolation [282] 7.3.6. Lagrange’s Interpolation Formula [284] 7.3.7. Hermite Interpolation [285] 7.3.8. Inverse Interpolation [286] 7.4. Numerical Integration [290] 7.4.1. The Rectangle Rule, Trapezoidal Rule, and Romberg’s Method [291] 7.4.2. The Truncation Error of the Trapezoidal Rule [293] 7.4.3. Some Difficulties and Possibilities in Numerical Integration [294] 7.4.4. The Euler-Maclaurin Summation Formula [297] 7.4.5. Uses of the Euler-Maclaurin Formula [300] 7.4.6. Other Methods for Numerical Integration [302] 7.5. Numerical Differentiation [307] 7.6. The Calculus of Operators [311] 7.6.1. Operator Algebra [311] 7.6.2. Operator Series with Applications [312] 7.7. Functions of Several Variables [318] 7.7.1. Working with One Variable at a Time [319] 7.7.2. Rectangular Grids [319] 7.7.3. Irregular Triangular Grids [322] 8 DIFFERENTIAL EQUATIONS [330] 8.1. Theoretical Background [330] 8.1.1. Initial-Value Problems for Ordinary Differential Equations [330] 8.1.2. Error Propagation [333] 8.1.3. Other Differential Equation Problems [337] 8.2. Euler’s Method, with Repeated Richardson Extrapolation [338] 8.3. Other Methods for Initial-Value Problems in Ordinary Differential Equations [342] 8.3.1. The Modified Midpoint Method [342] 8.3.2. The Power-Series Method [345] 8.3.3. Runge-Kutta Methods [346] 8.3.4. Implicit Methods [347] 8.3.5. Stiff Problems [349] 8.3.6. Control of Step Size [350] 8.3.7. A Finite-Difference Method for a Second-Order Equation [352] 8.4. Orientation on Boundary and Eigenvalue Problems for Ordinary Differential Equations [359] 8.4.1. Introduction [359] 8.4.2. The Shooting Method [359] 8.4.3. The Band Matrix Method [361] 8.4.4. Numerical Example of an Eigenvalue Problem [363] 8.5. Difference Equations [367] 8.5.1. Homogeneous Linear Difference Equations with Constant Coefficients [368] 8.5.2. General Linear Difference Equations [370] 8.5.3. Analysis of a Numerical Method with the Help of a Test Problem [372] 8.5.4. Linear Multistep Methods [375] 8.6. Partial Differential Equations [383] 8.6.1. Introduction [383] 8.6.2. An Example of an Initial-Value Problem [384] 8.6.3. An Example of a Boundary-Value Problem [389] 8.6.4. Methods of Undetermined Coefficients and Variational Methods [392] 8.6.5. Finite-Element Methods [395] 8.6.6. Integral Equations [397] 9 FOURIER METHODS [405] 9.1. Introduction [405] 9.2. Basic Formulas and Theorems in Fourier Analysis [406] 9.2.1. Functions of One Variable [406] 9.2.2. Functions of Several Variables [411] 9.3. Fast Fourier Analysis [413] 9.3.1. An Important Special Case [413] 9.3.2. Fast Fourier Analysis, General Case [414] 9.4. Periodic Continuation of a Nonperiodic Function [417] 9.5. The Fourier Integral Theorem [419] 10 OPTIMIZATION [422] 10.1. Statement of the Problem, Definitions, and Normal Form [422] 10.2. The Simplex Method [426] 10.3. Duality [435] 10.4. The Transportation Problem and Some Other Optimization Problems [436] 10.5. Nonlinear Optimization Problems [438] 10.5.1. Basic Concepts and Introductory Examples [438] 10.5.2. Line Search [440] 10.5.3. Algorithms for Unconstrained Optimization [441] 10.5.4. Overdetermined Nonlinear Systems [443] 10.5.5. Constrained Optimization [444] 11 THE MONTE CARLO METHOD AND SIMULATION [448] 11.1. Introduction [448] 11.2. Random Digits and Random Numbers [449] 11.3. Applications; Reduction of Variance [455] 11.4. Pseudorandom Numbers [463] 12 SOLUTIONS TO PROBLEMS [465] 13 BIBLIOGRAPHY AND PUBLISHED ALGORITHMS [536] 13.1. Introduction [536] 13.2. General Literature in Numerical Analysis [536] 13.3. Tables, Collections of Formulas, and Problems [539] 13.4. Error Analysis and Approximation of Functions [540] 13.5. Linear Algebra and Nonlinear Systems of Equations [541] 13.6. Interpolation, Numerical Integration, and Numerical Treatment of Differential Equations [543] 13.7. Optimization; Simulation [545] 13.8. Reviews, Abstracts and Other Periodicals [547] 13.9. Survey of Published Algorithms [548] Index by Subject to Algorithms, 1960-1970 [548] APPENDIX TABLES [563] INDEX [565]
| Item type | Home library | Shelving location | Call number | Materials specified | Copy number | Status | Date due | Barcode | Course reserves |
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Instituto de Matemática, CONICET-UNS | Libros ordenados por tema | 65 D131 (Browse shelf) | Available | A-5785 | ||||
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Instituto de Matemática, CONICET-UNS | Libros ordenados por tema | 65 D131 (Browse shelf) | Ej. 2 | Available | A-6402 | |||
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Instituto de Matemática, CONICET-UNS | Libros ordenados por tema | 65 D131 (Browse shelf) | Ej. 3 | Available | A-6676 | |||
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Instituto de Matemática, CONICET-UNS | Libros ordenados por tema | 65 D131 (Browse shelf) | Ej. 4 | Available | A-6677 |
"An extended and updated translation of": Numeriska metoder / by Å. Björck and G. Dahlquist. Lund, Suecia : Gleerup, 1969.
"Bibliography and published algorithms": p. 536-562.
1 SOME GENERAL PRINCIPLES OF NUMERICAL [1] --
CALCULATION [1] --
1.1. Introduction [1] --
1.2. Some Common Ideas and Concepts in Numerical Methods [2] --
1.3. Numerical Problems and Algorithms [13] --
1.3.1. Definitions [13] --
1.3.2. Recursive Formulas; Homer’s Rule [14] --
1.3.3. An Example of Numerical Instability [16] --
2 HOW TO OBTAIN AND ESTIMATE ACCURACY IN NUMERICAL CALCULATIONS [21] --
2.1. Basic Concepts in Error Estimation [21] --
2.1.1. Introduction [21] --
2.1.2. Sources of Error [22] --
2.1.3. Absolute and Relative Errors [23] --
2.1.4. Rounding and Chopping [24] --
2.2 Propagation of Errors [26] --
2.2.1. Simple Examples of Error Analysis [26] --
2.2.2. The General Formula for Error Propagation; Maximum Error and Standard Error [29] --
2.2.3. On the Practical Application of Error Estimation [34] --
2.2.4. The Use of Experimental Perturbations [35] --
2.2.5. Automatic Control of Accuracy [37] --
2.3. Number Systems; Floating and Fixed Representation [42] --
2.3.1. The Position System [42] --
2.3.2. Floating and Fixed Representation [43] --
2.3.3. Floating Decimal Point [44] --
2.3.4. Fixed Decimal Point [46] --
2.3.5. Round-off Errors in Computation with Floating --
Arithmetic Operations [46] --
2.4. Backward Error Analysis*; Condition Numbers [51] --
2.4.1. Backward Error Analysis [51] --
2.4.2. Condition Numbers for Problems and Algorithms [53] --
2.4.3. Geometrical Illustration of Error Analysis [56] --
3 NUMERICAL USES OF SERIES [60] --
3.1. Elementary Uses of Series [60] --
3.1.1. Simple Examples [60] --
3.1.2. Estimating the Remainder [62] --
3.1.3. Power Series [65] --
3.2. Acceleration of Convergence [71] --
3.2.1. Slowly Converging Alternating Series [71] --
3.2.2. Slowly Converging Series with Positive Terms [73] --
3.2.3. Other Simple Ways to Accelerate Convergence [74] --
3.2.4. Ill-Conditioned Series [75] --
3.2.5. Numerical Use of Divergent Series [77] --
4 APPROXIMATION OF FUNCTIONS [81] --
4.1. Basic Concepts in Approximation [81] --
4.1.1. Introduction [81] --
4.1.2. The Idea of a Function Space [84] --
4.1.3. Norms and Seminorms [85] --
4.1.4. Approximation of Functions as a Geometric Problem in Function Space [87] --
4.2. The Approximation of Functions by the Method of Least Squares [88] --
4.2.1. Statement of the Problems [88] --
4.2.2. Orthogonal Systems [89] --
4.2.3. Solution of the Approximation Problem [92] --
4.3. Polynomials [97] --
4.3.1. Basic Terminology; the Weierstrass Approximation Theorem [97] --
4.3.2, Triangle Families of Polynomials [98] --
2.2 Propagation of Errors [26] --
2.2.1. Simple Examples of Error Analysis [26] --
2.2.2. The General Formula for Error Propagation; Maximum Error and Standard Error [29] --
2.2.3. On the Practical Application of Error Estimation [34] --
2.2.4. The Use of Experimental Perturbations [35] --
2.2.5. Automatic Control of Accuracy [37] --
2.3. Number Systems; Floating and Fixed Representation [42] --
2.3.1. The Position System [42] --
2.3.2. Floating and Fixed Representation [43] --
2.3.3. Floating Decimal Point [44] --
2.3.4. Fixed Decimal Point [46] --
2.3.5. Round-off Errors in Computation with Floating Arithmetic Operations [46] --
2.4. Backward Error Analysis*; Condition Numbers [51] --
2.4.1. Backward Error Analysis [51] --
2.4.2. Condition Numbers for Problems and Algorithms [53] --
2.4.3. Geometrical Illustration of Error Analysis [56] --
3 NUMERICAL USES OF SERIES [60] --
3.1. Elementary Uses of Series [60] --
3.1.1. Simple Examples [60] --
3.1.2. Estimating the Remainder [62] --
3.1.3. Power Series [65] --
3.2. Acceleration of Convergence [71] --
3.2.1. Slowly Converging Alternating Series [71] --
3.2.2. Slowly Converging Series with Positive Terms [73] --
3.2.3. Other Simple Ways to Accelerate Convergence [74] --
3.2.4. Ill-Conditioned Series [75] --
3.2.5. Numerical Use of Divergent Series [77] --
4 APPROXIMATION OF FUNCTIONS [81] --
4.1. Basic Concepts in Approximation [81] --
4.1.1. Introduction [81] --
4.1.2. The Idea of a Function Space [84] --
4.1.3. Norms and Seminorms [85] --
4.1.4. Approximation of Functions as a Geometric Problem in Function Space [87] --
4.2. The Approximation of Functions by the Method of Least Squares [88] --
4.2.1. Statement of the Problems [88] --
4.2.2. Orthogonal Systems [89] --
4.2.3. Solution of the Approximation Problem [92] --
4.3. Polynomials [97] --
4.3.1. Basic Terminology; the Weierstrass Approximation Theorem [97] --
4.3.2, Triangle Families of Polynomials [98] --
4.3.3. A Triangle Family and Its Application to Interpolation [99] --
4.3.4. Equidistant Interpolation and the Runge Phenomenon [101] --
4.4. Orthogonal Polynomials and Applications [104] --
4.4.1. Tchebycheff Polynomials [104] --
4.4.2. Tchebycheff Interpolation and Smoothing [106] --
4.4.3. General Theory of Orthogonal Polynomials [108] --
4.4.4. Legendre Polynomials and Gram Polynomials [113] --
4.5. Complementary Observations on Polynomial Approximation [117] --
4.5.1. Summary of the Use of Polynomials [117] --
4.5.2. Some Inequalities for En(f) with Applications, to the Computation of Linear Functionals [120] --
4.5.3. Approximation in the Maximum Norm [124] --
4.5.4. Economization of Power Series; Standard Functions [125] --
4.5.5. Some Statistical Aspects of the Method of Least Squares [126] --
4.6. Spline Functions [131] --
5 NUMERICAL LINEAR ALGEBRA [137] --
5.1. Introduction [137] --
5.2. Basic Concepts of Linear Algebra [138] --
5.2.1. Fundamental Definitions [138] --
5.2.2. Partitioned Matrices [140] --
5.2.3. Linear Vector Spaces [141] --
5.2.4. Eigenvalues and Similarity Transformations [142] --
5.2.5. Singular-Value Decomposition and Pseudo-Inverse [143] --
5.3. Direct Methods for Solving Systems of Linear Equations [146] --
5.3.1. Triangular Systems [146] --
5.3.2. Gaussian Elimination [147] --
5.3.3. Pivoting Strategies [150] --
5.3.4. LU-Decomposition [152] --
5.3.5. Compact Schemes for Gaussian Elimination [157] --
5.3.6. Inverse Matrices [159] --
5.4. Special Matrices [162] --
5.4.1. Symmetric Positive-Definite Matrices [162] --
5.4.2. Band Matrices [165] --
5.4.3. Large-Scale Linear Systems [168] --
5.4.4. Other Sparse Matrices [169] --
5.5. Error Analysis for Linear Systems [174] --
5.5.1. An Ill-Conditioned Example [174] --
5.5.2. Vector and Matrix Norms [175] --
5.5.3. Perturbation Analysis [176] --
5.5.4. Rounding Errors in Gaussian Elimination [177] --
5.5.5. Scaling of Linear Systems [181] --
5.5.6. Iterative Improvement of a Solution [183] --
5.6. Iterative Methods [188] --
5.7. Overdetermined Linear Systems [196] --
5.7.1. The Normal Equations [197] --
5.7.2. Orthogonalization Methods [201] --
5.7.3. Improvement of Least-Squares Solution [204] --
5.7.4. Least-Squares Problems with Linear Constraints [205] --
5.8. Computation of Eigenvalues and Eigenvectors [208] --
5.8.1. The Power Method [209] --
5.8.2. Methods Based on Similarity Transformations [211] --
5.8.3. Eigenvalues by Equation Solving [215] --
5.8.4. The QR-Algorithm [216] --
6 NONLINEAR EQUATIONS [218] --
6.1. Introduction [218] --
6.2. Initial Approximations; Starting Methods [219] --
6.2.1. Introduction [219] --
6.2.2. The Bisection Method [220] --
6.3. Newton-Raphson’s Method [222] --
6.4. The Secant Method [227] --
6.4.1. Description of the Method [227] --
6.4.2. Error Analysis for the Secant Method [228] --
6.4.3. Regula Falsi [230] --
6.4.4. Other Related Methods [230] --
6.5. General Theory of Iteration Methods [233] --
6.6. Error Estimation and Attainable Accuracy in Iteration Methods [238] --
6.6.1. Error Estimation [238] --
6.6.2. Attainable Accuracy; Termination Criteria [240] --
6.7. Multiple Roots [242] --
6.8. Algebraic Equations [243] --
6.8.1. Introduction [243] --
6.8.2. Deflation [245] --
6.8.3. Ill-Conditioned Algebraic Equations [246] --
6.9. Systems of Nonlinear Equations [248] --
6.9.1. Iteration [249] --
6.9.2. Newton-Raphson’s Method and Some Modifications [249] --
6.9.3. Other Methods [251] --
7 FINITE DIFFERENCES WITH APPLICATIONS TO NUMERICAL INTEGRATION, DIFFERENTIATION, AND INTERPOLATION [255] --
7.1. Difference Operators and Their Simplest Properties [255] --
7.2. Simple Methods for Deriving Approximation Formulas and Error Estimates [263] --
7.2.1. Statement of the Problems and Some Typical Examples [263] --
7.2.2. Repeated Richardson Extrapolation [269] --
7.3. Interpolation [275] --
7.3.1. Introduction [275] --
7.3.2. When is Linear Interpolation Sufficient? [276] --
7.3.3. Newton’s General Interpolation Formula [277] --
7.3.4. Formulas for Equidistant Interpolation [279] --
7.3.5. Complementary Remarks on Interpolation [282] --
7.3.6. Lagrange’s Interpolation Formula [284] --
7.3.7. Hermite Interpolation [285] --
7.3.8. Inverse Interpolation [286] --
7.4. Numerical Integration [290] --
7.4.1. The Rectangle Rule, Trapezoidal Rule, and Romberg’s Method [291] --
7.4.2. The Truncation Error of the Trapezoidal Rule [293] --
7.4.3. Some Difficulties and Possibilities in Numerical Integration [294] --
7.4.4. The Euler-Maclaurin Summation Formula [297] --
7.4.5. Uses of the Euler-Maclaurin Formula [300] --
7.4.6. Other Methods for Numerical Integration [302] --
7.5. Numerical Differentiation [307] --
7.6. The Calculus of Operators [311] --
7.6.1. Operator Algebra [311] --
7.6.2. Operator Series with Applications [312] --
7.7. Functions of Several Variables [318] --
7.7.1. Working with One Variable at a Time [319] --
7.7.2. Rectangular Grids [319] --
7.7.3. Irregular Triangular Grids [322] --
8 DIFFERENTIAL EQUATIONS [330] --
8.1. Theoretical Background [330] --
8.1.1. Initial-Value Problems for Ordinary Differential Equations [330] --
8.1.2. Error Propagation [333] --
8.1.3. Other Differential Equation Problems [337] --
8.2. Euler’s Method, with Repeated Richardson Extrapolation [338] --
8.3. Other Methods for Initial-Value Problems in Ordinary Differential Equations [342] --
8.3.1. The Modified Midpoint Method [342] --
8.3.2. The Power-Series Method [345] --
8.3.3. Runge-Kutta Methods [346] --
8.3.4. Implicit Methods [347] --
8.3.5. Stiff Problems [349] --
8.3.6. Control of Step Size [350] --
8.3.7. A Finite-Difference Method for a Second-Order Equation [352] --
8.4. Orientation on Boundary and Eigenvalue Problems for Ordinary Differential Equations [359] --
8.4.1. Introduction [359] --
8.4.2. The Shooting Method [359] --
8.4.3. The Band Matrix Method [361] --
8.4.4. Numerical Example of an Eigenvalue Problem [363] --
8.5. Difference Equations [367] --
8.5.1. Homogeneous Linear Difference Equations with Constant Coefficients [368] --
8.5.2. General Linear Difference Equations [370] --
8.5.3. Analysis of a Numerical Method with the Help of a Test Problem [372] --
8.5.4. Linear Multistep Methods [375] --
8.6. Partial Differential Equations [383] --
8.6.1. Introduction [383] --
8.6.2. An Example of an Initial-Value Problem [384] --
8.6.3. An Example of a Boundary-Value Problem [389] --
8.6.4. Methods of Undetermined Coefficients and Variational Methods [392] --
8.6.5. Finite-Element Methods [395] --
8.6.6. Integral Equations [397] --
9 FOURIER METHODS [405] --
9.1. Introduction [405] --
9.2. Basic Formulas and Theorems in Fourier Analysis [406] --
9.2.1. Functions of One Variable [406] --
9.2.2. Functions of Several Variables [411] --
9.3. Fast Fourier Analysis [413] --
9.3.1. An Important Special Case [413] --
9.3.2. Fast Fourier Analysis, General Case [414] --
9.4. Periodic Continuation of a Nonperiodic Function [417] --
9.5. The Fourier Integral Theorem [419] --
10 OPTIMIZATION [422] --
10.1. Statement of the Problem, Definitions, and Normal Form [422] --
10.2. The Simplex Method [426] --
10.3. Duality [435] --
10.4. The Transportation Problem and Some Other Optimization Problems [436] --
10.5. Nonlinear Optimization Problems [438] --
10.5.1. Basic Concepts and Introductory Examples [438] --
10.5.2. Line Search [440] --
10.5.3. Algorithms for Unconstrained Optimization [441] --
10.5.4. Overdetermined Nonlinear Systems [443] --
10.5.5. Constrained Optimization [444] --
11 THE MONTE CARLO METHOD AND SIMULATION [448] --
11.1. Introduction [448] --
11.2. Random Digits and Random Numbers [449] --
11.3. Applications; Reduction of Variance [455] --
11.4. Pseudorandom Numbers [463] --
12 SOLUTIONS TO PROBLEMS [465] --
13 BIBLIOGRAPHY AND PUBLISHED ALGORITHMS [536] --
13.1. Introduction [536] --
13.2. General Literature in Numerical Analysis [536] --
13.3. Tables, Collections of Formulas, and Problems [539] --
13.4. Error Analysis and Approximation of Functions [540] --
13.5. Linear Algebra and Nonlinear Systems of Equations [541] --
13.6. Interpolation, Numerical Integration, and Numerical --
Treatment of Differential Equations [543] --
13.7. Optimization; Simulation [545] --
13.8. Reviews, Abstracts and Other Periodicals [547] --
13.9. Survey of Published Algorithms [548] --
Index by Subject to Algorithms, 1960-1970 [548] --
APPENDIX TABLES [563] --
INDEX [565] --
MR, 51 #4620
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