A THEORETICAL INTRODUCTION TO NUMERICAL ANALYSIS

Transcription

1 A THEORETICAL INTRODUCTION TO NUMERICAL ANALYSIS Victor S. Ryaben'kii Semyon V. Tsynkov Chapman &. Hall/CRC Taylor & Francis Group Boca Raton London New York Chapman & Hall/CRC is an imprint of the Taylor & Francis Group, an informa business

2 Contents Preface Acknowledgments xi xiii 1 Introduction Discretization 4 Exercises Conditioning 6 Exercises a Error ' Unavoidable Error Error of the Method Round-off Error 10 Exercises On Methods of Computation Accuracy Operation Count Stability Loss of Significant Digits Convergence General Comments. 18 Exercises 19 1 Interpolation of Functions. Quadratures 21 2 Algebraic Interpolation Existence and Uniqueness of Interpolating Polynomial The Lagrange Form of Interpolating Polynomial The Newton Form of Interpolating Polynomial. Divided Differences Comparison of the Lagrange and Newton Forms Conditioning of the Interpolating Polynomial On Poor Convergence of Interpolation with Equidistant Nodes 33 Exercises Classical Piecewise Polynomial Interpolation i Definition of Piecewise Polynomial Interpolation 35 in

3 IV Formula for the Interpolation Error Approximation of Derivatives for a Grid Function Estimate of the Unavoidable Error and the Choice of Degree for Piecewise Polynomial Interpolation Saturation of Piecewise Polynomial Interpolation 42 Exercises Smooth Piecewise Polynomial Interpolation (Splines) Local Interpolation of Smoothness s and Its Properties Nonlocal Smooth Piecewise Polynomial Interpolation Proof of Theorem Exercises Interpolation of Functions of Two Variables Structured Grids Unstructured Grids 59 Exercises 60 Trigonometric Interpolation Interpolation of Periodic Functions An Important Particular Choice of Interpolation Nodes Sensitivity of the Interpolating Polynomial to Perturbations of the Function Values Estimate of Interpolation Error An Alternative Choice of Interpolation Nodes Interpolation of Functions on an Interval. Relation between Algebraic and Trigonometric Interpolation Periodization Trigonometric Interpolation Chebyshev Polynomials. Relation between Algebraic and Trigonometric Interpolation : Properties of Algebraic Interpolation with Roots of the Chebyshev Polynomial T n+ \(x) as Nodes An Algorithm for Evaluating the Interpolating Polynomial Algebraic Interpolation with Extrema of the Chebyshev Polynomial T n (x) as Nodes More on the Lebesgue Constants and Convergence of Interpolants 80 Exercises 89 Computation of Definite Integrals. Quadratures Trapezoidal Rule, Simpson's Formula, and the Like General Construction of Quadrature Formulae Trapezoidal Rule Simpson's Formula 98 Exercises Quadrature Formulae with No Saturation. Gaussian Quadratures.. 102

4 Exercises Improper Integrals. Combination of Numerical and Analytical Methods 108 Exercises ".' Multiple Integrals Repeated Integrals and Quadrature Formulae Ill The Use of Coordinate Transformations The Notion of Monte Carlo Methods 113 II Systems of Scalar Equations Systems of Linear Algebraic Equations: Direct Methods Different Forms of Consistent Linear Systems Canonical Form of a Linear System Operator Form Finite-Difference Dirichlet Problem for the Poisson Equation? 121 Exercises Linear Spaces, Norms, and Operators Normed Spaces Norm of a Linear Operator 129 Exercises Conditioning of Linear Systems Condition Number Characterization of a Linear System by Means of Its Condition Number 136 Exercises Gaussian Elimination and Its Tri-Diagonal Version Standard Gaussian Elimination Tri-Diagonal Elimination Cyclic Tri-Diagonal Elimination Matrix Interpretation of the Gaussian Elimination. LU Factorization Cholesky Factorization Gaussian Elimination with Pivoting An Algorithm with a Guaranteed Error Estimate 155 Exercises Minimization of Quadratic Functions and Its Relation to Linear Systems 157 Exercises The Method of Conjugate Gradients Construction of the Method Flexibility in Specifying the Operator A Computational Complexity 163 Exercises 163

5 VI 5.7 Finite Fourier Series Fourier Series for Grid Functions Representation of Solution as a Finite Fourier Series Fast Fourier Transform 169 Exercises 171 Iterative Methods for Solving Linear Systems Richardson Iterations and the Like General Iteration Scheme A Necessary and Sufficient Condition for Convergence The Richardson Method for A = A* > Preconditioning Scaling 192 Exercises Chebyshev Iterations and Conjugate Gradients Chebyshev Iterations Conjugate Gradients 196 Exercises Krylov Subspace Iterations Definition of Krylov Subspaces GMRES 201 Exercises Multigrid Iterations Idea of the Method Description of the Algorithm Bibliography Comments 210 Exercises 210 Overdetermined Linear Systems. The Method of Least Squares Examples of Problems that Result in Overdetermined Systems Processing of Experimental Data. Empirical Formulae Improving the Accuracy of Experimental Results by Increasing the Number of Measurements Weak Solutions of Full Rank Systems. QR Factorization Existence and Uniqueness of Weak Solutions Computation of Weak Solutions. QR Factorization Geometric Interpretation of the Method of Least Squares Overdetermined Systems in the Operator Form 221 Exercises Rank Deficient Systems. Singular Value Decomposition Singular Value Decomposition and Moore-Penrose Pseudoinverse Minimum Norm Weak Solution 227 Exercises 229

6 Vll 8 Numerical Solution of Nonlinear Equations and Systems Commonly Used Methods of Rootfinding The Bisection Method The Chord Method The Secant Method Newton's Method Fixed Point Iterations The Case of One Scalar Equation The Case of a System of Equations 240 Exercises Newton's Method Newton's Linearization for One Scalar Equation Newton's Linearization for Systems Modified Newton's Methods 246 Exercises 247 III The Method of Finite Differences for the Numerical Solution of Differential Equations Numerical Solution of Ordinary Differential Equations Examples of Finite-Difference Schemes. Convergence Examples of Difference Schemes Convergent Difference Schemes Verification of Convergence for a Difference Scheme Approximation of Continuous Problem by a Difference Scheme. Consistency Truncation Error <5/ (/i) Evaluation of the Truncation Error 5/W Accuracy of Order h k. \ Examples.' Replacement of Derivatives by Difference Quotients Other Approaches to Constructing Difference Schemes Exercises Stability of Finite-Difference Schemes Definition of Stability The Relation between Consistency, Stability, and Convergence Convergent Scheme for an Integral Equation The Effect of Rounding General Comments. A-stability 280 Exercises The Runge-Kutta Methods The Runge-Kutta Schemes Extension to Systems 286 Exercises 288

7 Vlll 9.5 Solution of Boundary Value Problems The Shooting Method Tri-Diagonal Elimination Newton's Method 291 Exercises Saturation of Finite-Difference Methods by Smoothness 293 Exercises The Notion of Spectral Methods 301 Exercises Finite-Difference Schemes for Partial Differential Equations Key Definitions and Illustrating Examples Definition of Convergence Definition of Consistency Definition of Stability The Courant, Friedrichs, and Lewy Condition The Mechanism of Instability The Kantorovich Theorem On the Efficacy of Finite-Difference Schemes Bibliography Comments 323 Exercises Construction of Consistent Difference Schemes Replacement of Derivatives by Difference Quotients The Method of Undetermined Coefficients Other Methods. Phase Error Predictor-Corrector Schemes 344 Exercises Spectral Stability Criterion for Finite-Difference Cauchy Problems Stability with Respect to Initial Data A Necessary Spectral Condition for Stability Examples Stability in C Sufficiency of the Spectral Stability Condition in l Scalar Equations vs. Systems 365 Exercises Stability for Problems with Variable Coefficients The Principle of Frozen Coefficients Dissipation of Finite-Difference Schemes 372 Exercises Stability for Initial Boundary Value Problems The Babenko-Gelfand Criterion Spectra of the Families of Operators. The Godunov- Ryaben'kii Criterion The Energy Method 402

8 IX A Necessary and Sufficient Condition of Stability. The Kreiss Criterion 409 Exercises Maximum Principle for the Heat Equation An Explicit Scheme An Implicit Scheme 425 Exercises Discontinuous Solutions and Methods of Their Computation Differential Form of an Integral Conservation Law Differential Equation in the Case of Smooth Solutions The Mechanism of Formation of Discontinuities Condition at the Discontinuity Generalized Solution of a Differential Problem The Riemann Problem 434 Exercises c Construction of Difference Schemes Artificial Viscosity The Method of Characteristics Conservative Schemes. The Godunov Scheme 439 Exercises Discrete Methods for Elliptic Problems A Simple Finite-Difference Scheme. The Maximum Principle Consistency Maximum Principle and Stability Variable Coefficients 451 Exercises The Notion of Finite Elements. Ritz and Galerkin Approximations Variational Problem The Ritz Method The Galerkin Method An Example of Finite Element Discretization Convergence of Finite Element Approximations 466 Exercises 469 IV The Methods of Boundary Equations for the Numerical Solution of Boundary Value Problems Boundary Integral Equations and the Method of Boundary Elements Reduction of Boundary Value Problems to Integral Equations Discretization of Integral Equations and Boundary Elements The Range of Applicability for Boundary Elements 480

9 14 Boundary Equations with Projections and the Method of Difference Potentials Formulation of Model Problems Interior Boundary Value Problem Exterior Boundary Value Problem Problem of Artificial Boundary Conditions Problem of Two Subdomains Problem of Active Shielding Difference Potentials Auxiliary Difference Problem The Potential u + = P + v r Difference Potential u~ =P~v y Cauchy Type Difference Potential w ± = P ± v r Analogy with Classical Cauchy Type Integral Solution of Model Problems Interior Boundary Value Problem Exterior Boundary Value Problem Problem of Artificial Boundary Conditions Problem of Two Subdomains Problem of Active Shielding General Remarks Bibliography Comments 506 List of Figures 507 Referenced Books 509 Referenced Journal Articles 517 Index 521