Computing Methods

Foreword

Preface

Chapter 6 The Solution of Sets of Linear Algebraic Equations

§ 1. Classification of Methods

§ 2. Elimination

1. The Gauss Method with Selection of the Pivotal Element (Pivotal Condensation)

2. Compact Gauss Method

3. Inversion of Matrices

4. Calculation of Determinants

5. Jordan Method

6. The Method without Back-Substitution

§ 3. The Square-root Method

§ 4. Orthogonalization

§ 5. Conjugate Gradients

§ 6. Partitioning into Sub-Matrices

§ 7. Linear Operators. Operator "Norms"

1. Finite Dimensional Linear Normalized Spaces

2. Linear Operators in Finite Dimensional Linear Normalized Space and their Link with Matrices

3. The Convergence of Sequences of Matrices and Matrix Series

§ 8. Methods of Successive Approximation

§ 9. Linear First-order Full-step Methods

1. The Convergence of Linear First-order Full-step Methods Simple Iteration

2. Eichardson's Method

3. The Inversion of Matrices by the Method of Successive Approximation

§ 10. Linear First-order Single-step Methods

1. The Seidel Method

2. Convergence of the Seidel Method

3. The Relaxation Method

§ 11. The Method of Steepest Descent

Exercises

References

Chapter 7 Numerical Solution of High Degree Algebraic Equations and Transcendental Equations

§ 1. Introduction

§ 2. Isolation of Roots

1. General Remarks

2. Bounds of the Roots of Algebraic Equations

3. The Number of Real Roots in an Algebraic Equation

4. Isolation of the Real Roots of an Algebraic Equation

5. Isolation of the Complex Roots of Algebraic Equations

§ 3. The Lobachevskii Solution of Algebraic Equations (Graeffe's Root-squaring Method)

1. Root-Squaring. Real Roots of Different Absolute Magnitude

2. The Root-Squaring Method. Complex Roots

3. Root-Squaring Method. Close or Equal Roots

4. The Error in the Lobachevskii Root-Squaring Method

5. Lehmer's Modification of the Lobachevskii Root-Squaring Method

§ 4. Iterative Methods of Solving Algebraic and Transcendental Equations

1. Compressed Transformations and their Application to the Proof of Convergence in Iteration

2. Simple Iteration: The Rule of False Position (Secants) and The Newton-Raphson (Tangents) Method

3. The Chebyshev Method for Higher-Order Iteration

4. König's Theorem and High-Order Iterations

5. Aitken's Method of Finding Higher-Order Iterations

6. Example

§ 5. The Solution of Sets of Equations

1. The Iterative Method of Solving Sets of a Special Kind

2. The Newton Method

3. The Method of Steepest Descent

§ 6. Finding the Roots of Algebraic Equations by Factorization

1. Lin's Method of Factorization

2. Friedman's Method

3. Hitchcock's Method of Isolating Quadratic Factors

Exercises

References

Chapter 8 The Evaluation of Eigenvalues and Eigenvectors of Matrices

§ 1. Introduction

§ 2. Krylov's Method

1. Eigenvalues

2. Eigenvectors

§ 3. Lanczos' Method

1. Eigenvalues

2. Eigenvectors

§ 4. Danilevskii's Method

1. Modification of the Danilevskii Method

§ 5. Other Methods of Finding the Characteristic Polynomial

1. Leverrier's Method

2. Bordering

3. The Escalator Method

4. Samuelson's Method

5. Iteration

§ 6. Defining the Bounds of Eigenvalues

1. The Symmetric Matrix

2. Non-Symmetric Matrices

§ 7. Iterative Methods of Finding Eigenvalues and Eigenvectors

1. The Absolutely Largest, Real Eigenvalue of a Simple Matrix. The Case of a Symmetric Matrix

2. Finding Other Eigenvalues and Corresponding Eigenvectors for Symmetric Matrices

3. The Eigenvalues and Eigenvectors of Simple Non-Symmetric Matrices

4. A Few Remarks on the Eigenvalues and Eigenvectors of General Matrices

§ 8. Acceleration of Convergence in Iterative Processes for the Solution of Problems in Linear Algebra

1. General Remarks

2. Gavurin's Method

3. Lyusternik's Method

4. Aitken's δ2-process

5. Improving the Convergence of Iterative Processes for Finding Eigenvalues

§ 9. The Irremovable Error in the Numerical Solution of Sets of Linear Algebraic Equations

Exercises

References

Chapter 9 Approximate Methods of Solving Ordinary Differential Equations

§ 1. Introduction

§ 2. Chaplygin's Method

1. Theorems of Differential Inequalities

2. Chaplygin's Method of Improving Approximations

3. Another Method of Improving Approximations

4. Chaplygin's Approximate Method for Linear Second-Order Differential Equations

§ 3. The Method of Small Parameters (Poincaré's Theorem)

§ 4. The Runge-Kutta Method

1. The Runge-Kutta Method for First-Order Differential Equations

2. The Runge-Kutta Method for Sets of First-Order Differential Equations

3. The Runge-Kutta Method for Second-Order Equations

§ 5. Difference Methods for Ordinary First-order Differential Equations

1. Certain Extrapolation Formulae for Integrating First-order Differential Equations

2. Examples of Interpolation Formulae

3. The Method of Undetermined Coefficients for the Deduction of Difference Formulae

4. Krylov's Method of Finding the Initial Values of a Solution

5. Examples

§ 6. Difference Methods of Solving Ordinary Differential Equations of Higher Orders

§ 7. Estimating the Error, Convergence and Stability of the Difference Methods of Solving Ordinary Differential Equations

1. Linear Difference Equations

2. The Difference Equation for the Error in the Approximate Solution

3. Estimates of the Error in the Solutions Found by the Adams' Formulae

4. The Stability of the Difference Methods of Solving Differential Equations

5. Estimating Error and Convergence in the Stable Difference Methods of Solving Differential Equations

§ 8. The Solution of the Boundary Value Problem for Ordinary Differential Equations by the Method of Finite Differences

1. The Method of Finite Differences. Its Application to the Boundary Value Problem for Linear Second-order Differential Equations

2. The Method of Finite Differences in the Solution of the Boundary Value Problem for Non-Linear Second-order Differential Equations

§ 9. The Gel'fand-Lokutsiyevskii Method of "Chasing"

§ 10. The Solution of the Boundary Value Problem for Ordinary Differential Equations by Variational Methods

1. Variational Methods of Solving Operator Equations in Hubert Space

2. Ritz's Method of Solving Variational Problems

3. Galerkin's Method

Exercises

References

Chapter 10 Approximate Methods of Solving Partial Differential and Integral Equations

§ 1. Introduction

§ 2. The Mesh Method of Solving Boundary Value Problems in Differential Equations of the Elliptic Type

1. The Main Idea

2. The Approximation of Differential Equations by Difference Equations

3. The Approximation of Boundary Conditions

4. The Solvability of Difference Equations and Methods of Solution

5. Estimating Error and Convergence in the Mesh Method

§ 3. The Mesh Method of Solving Linear Differential Equations of the Hyperbolic Type

1. Solution of the Cauchy Problem

2. Estimating the Error and Convergence of the Mesh Method of Solving Inhomogeneous Wave Equations

3. The Mesh Method of Solving Mixed Problems

4. Other Difference Methods

§ 4. The Method of Characteristics for the Numerical Solution of Hyperbolic Sets of Partial Quasi-linear Differential Equations

1. Characteristic Equations for a Set of Second-order Quasilinear Differential Equations

2. Examples: Characteristic Equations for Certain Sets of Differential Equations in Gas Dynamics

3. Characteristic Equations for Second-order Quasi-Linear Hyperbolic Differential Equations

4. The Numerical Solution of Quasi-Linear Hyperbolic Sets of Two First-order Differential Equations by Masseau's Method

5. The Numerical Solution of a Hyperbolic Set of Three First-order Differential Equations by Masseau's Method

6. Masseau's Numerical Method of Solving Quasi-Linear Hyperbolic Second-order Equations

7. The Fundamental Problems Involved in the Study of a Planar, Eddyless, Supersonic and Steady Flow of Ideal Gas

§ 5. The Mesh Method of Solving Linear Differential Equations of the Parabolic Type

1. The Mesh Method of Solving the Cauchy Problem

2. The Mesh Method of Solving Mixed Problems. The Stability of Difference Methods

§ 6. The Gerfand-Lokutsiyevskii "Chasing" Method of Solving Boundary Value Problems in Partial Differential Equations

1. The Equation of Heat Conduction

2. Poisson's Equation

§ 7. The Convergence and Stability of Difference Methods

1. The Difference Approximation of a Differential Equation and the Boundary Conditions

2. The "Correctness" and Stability of Difference Schemes

3. The Connexion Between the Convergence and "Correctness" of Difference Schemes

4. Various Methods of Investigating the Stability of Difference Schemes

5. General Remarks

§ 8. Straight-line Methods of Solving Boundary Value Problems in Partial Differential Equations

1. Essential Features

2. The Straight-Line Method of Solving the Dirichlet Problem in Poisson's Equation

3. The Straight-Line Method of Solving the Mixed Problem in the Equation of Vibration of a String

4. The Straight-Line Method of Solving the Mixed Problem in the Equation of Thermal Conductivity

§ 9. Variational Methods of Solving Boundary Value Problems in the Differential Equations of Mathematical Physics

1. The Ritz Method of Solving Operator Equations and Finding the Eigenvalues of Operators in Hubert Space

2. The Ritz Method of Finding the Approximate Solution of Boundary Value Problems in Second-Order Partial Differential Equations of the Elliptic Type

3. Other Variational Methods

4. The Ritz Method of Solving Eigenvalue Problems

5. Galerkin's Method of Solving Boundary Value Problems

§ 10. Approximate Methods of Solving Integral Equations

1. The Solution of Fredholm Equations by Substituting a Finite Sum for the Integral

2. The Solution of Fredholm Integral Equations of the Second Kind by Substituting a Degenerated Kernel

3. Method of Moments

4. The Method of Least Squares

5. Trial and Error

6. Approximate Solution of Volterra Equations

Exercises

References

Index