# Numerical Analysis An Introduction

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1 Walter Gautschi Numerical Analysis An Introduction 1997 Birkhauser Boston Basel Berlin

2 CONTENTS PREFACE xi CHAPTER 0. PROLOGUE Overview Numerical analysis software Textbooks and monographs Journals 9 CHAPTER 1. MACHINE ARITHMETIC AND RELATED MATTERS Real Numbers, Machine Numbers, and Rounding Real numbers Machine numbers Rounding Machine Arithmetic A model of machine arithmetic Error propagation in arithmetic operations; cancellation error The Condition of a Problem Condition numbers Examples The Condition of an Algorithm Computer Solution of a Problem; Overall Error 37 Notes to Chapter 1 39 Exercises and Machine Assignments to Chapter 1 42 CHAPTER 2. APPROXIMATION AND INTERPOLATION Least Squares Approximation Inner products The normal equations Least squares error; convergence Examples of orthogonal systems Polynomial Interpolation Lagrange interpolation formula; interpolation operator Interpolation error 79

3 vi Contents 2.3. Convergence Chebyshev polynomials and nodes Barycentric formula Newton's formula Hermite interpolation Inverse interpolation Approximation and Interpolation by Spline Functions Interpolation by piecewise linear functions A basis for S?(A) Least squares approximation Interpolation by cubic splines Minimality properties of cubic spline interpolants Notes to Chapter Exercises and Machine Assignments to Chapter CHAPTER 3. NUMERICAL DIFFERENTIATION AND INTEGRATION Numerical Differentiation A general differentiation formula for unequally spaced points Examples Numerical differentiation with perturbed data Numerical Integration The composite trapezoidal and Simpson's rules (Weighted) Newton-Cotes and Gauss formulae Properties of Gaussian quadrature rules Some applications of the Gauss quadrature rule Approximation of linear functionals: method of interpolation vs. method of undetermined coefficients Peano representation of linear functionals Extrapolation methods 179 Notes to Chapter Exercises and Machine Assignments to Chapter 3 191

4 Contents vii CHAPTER 4. NONLINEAR EQUATIONS Examples 210 l.l.-a transcendental equation A two-point boundary value problem A nonlinear integral equation s-orthogonal polynomials Iteration, Convergence, and Efficiency The Methods of Bisection and Sturm Sequences Bisection method Method of Sturm sequences Method of False Position Secant Method Newton's Method Fixed Point Iteration Algebraic Equations Newton's method applied to an algebraic equation An accelerated Newton method for equations with real roots Systems of Nonlinear Equations Contraction mapping principle Newton's method for systems of equations 242 Notes to Chapter Exercises and Machine Assignments to Chapter CHAPTER 5. INITIAL VALUE PROBLEMS FOR ODEs ONE- STEP METHODS Examples Types of differential equations Existence and uniqueness Numerical methods Local Description of One-Step Methods Examples of One-Step Methods Euler's method Method of Taylor expansion Improved Euler methods 276

5 viii Contents 2.4. Second-order two-stage methods Runge-Kutta methods Global Description of One-Step Methods Stability Convergence Asymptotics of global error Error Monitoring and Step Control Estimation of global error Truncation error estimates Step control Stiff Problems A-stability Pade approximation Examples of A-stable one-step methods Regions of absolute stability 312 Notes to Chapter Exercises and Machine Assignments to Chapter CHAPTER 6. INITIAL VALUE PROBLEMS FOR ODEs MULTI- STEP METHODS Local Description of Multistep Methods Explicit and implicit methods Local accuracy Polynomial degree vs. order Examples of Multistep 'Methods Adams-Bashforth method Adams-Moulton method Predictor-corrector methods Global Description of Multistep Methods Linear difference equations Stability and root condition Convergence Asymptotics of global error Estimation of global error Analytic Theory of Order and Stability 366

6 Contents ix 4.1. Analytic characterization of order Stable methods of maximum order Applications Stiff Problems A-stability A(a)-stability 388 Notes to Chapter Exercises and Machine Assignments to Chapter CHAPTER 7. TWO-POINT BOUNDARY VALUE PROBLEMS FOR ODEs Existence and Uniqueness Examples A scalar boundary value problem General linear and nonlinear systems Initial Value Techniques Shooting method for a scalar boundary value problem Linear and nonlinear systems Parallel shooting Finite Difference Methods " Linear second-order equations Nonlinear second-order equations Variational Methods Variational formulation The extremal problem Approximate solution of the extremal problem 436 Notes to Chapter Exercises and Machine Assignments to Chapter References 451 Subject Index 482

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