## Numerical Analysis'The book reads like an unfolding story... Topics are motivated with great care and ingenuity that might be given to establishing the drive behind characters in a good novel... Clarity is never sacrificed for elegance. Above all, the pace is always lively and brisk, the writing concise, and the author never lets the exposition bog down... Both the theoretical problems and machine assignments are a great resource... This is a stylish, lucid, and engaging book... [It] successfully conveys the author's interest and experience in the subject to the reader. ' -- SIAM Review, 1998 'This book is well written... Every teacher should look at this textbook to see how this material has been presented by a numerical analyst with many years of teaching experience and a high reputation.' -- Computing Reviews (March 1998) The term 'Numerical Analysis,' in this text, means the branch of mathematics that develops and analyzes computational methods dealing with problems arising in classical analysis, approximation theory, the theory of equations, and ordinary differential equations. The topics included in the book are presented with a view toward stressing basic principles and maintaining simplicity and teachability as far as possible. In this sense, the text is an introduction. Topics that, even though important, require a level of technicality that goes beyond the standards of simplicity imposed, are referenced in detailed bibliographic notes at the end of each chapter. In this way, the reader is given guidance and an opportunity to pursue advanced modern topics in more depth. Contrary to tradition, the text does not include numerical linear algebra, which is felt by the author to have matured into an autonomous discipline having an identity of its own and therefore deserving treatment in separate books and separate courses on the graduate level. For similar reasons, the numerical solution of partial differential equations is not covered either. The text is geared to a one- or two-semester graduate course in numerical analysis for students who have a good background in calculus and advanced calculus and some knowledge of linear algebra, complex analysis, and differential equations. Previous exposure to numerical methods in an undergraduate course is desirable but not absolutely necessary. A significant feature of the book is a large collection of exercises, both the kind that deal with theoretical and practical aspects of the subject and those requiring machine computation and the use of mathematical software. A list of corrections can be obtained at http://www.cs.purdue.edu/homes/wxg/CS514/text.html the author's website. Contents PREFACE CHAPTER 0. PROLOGUE 0.1 Overview 0.2 Numerical analysis software 0.3 Textbooks and monographs 0.4 Journals CHAPTER 1. MACHINE ARITHMETIC AND RELATED MATTERS 1. Real Numbers, Machine Numbers, and Rounding 1.1 Real numbers 1.2 Machine numbers 1.3 Rounding 2. Machine Arithmetic 2.1 A model of machine arithmetic 2.2 Error propagation in arithmetric operations; cancellation error 3. The Condition of a Problem 3.1 Condition numbers 3.2 Examples 4. The Condition of an Algorithm 5. Computer Solution of a Problem; Overall Error Notes to Chapter 1 Exercises and Machine Assignments to Chapter 1 CHAPTER 2. APPROXIMATION AND INTERPOLATION 1. Least Squares Approximation 1.1 Inner products 1.2 The normal equations 1.3 Least squares error; convergence 1.4 Examples of orthogonal systems 2. Polynomial Interpolation 2.1 Lagrange interpolation formula; interpolation operator 2.2 Interpolation error 2.3 Convergence 2.4 Chebyshev polynomials and nodes 2.5 Barycentric formula 2.6 Newton's formula 2.7 Hermite interpolation 2.8 Inverse interpolation 3. Approximation and Interpolation by Spline Functions 3.1 Interpolation by piecewise linear functions 3.2 A basis for S 0 1(Delta) 3.3 Least squares approximation 3.4 Interpolation by cubic splines 3.5 Minimality properties of cubic spline interpolants Notes to Chapter 2 Exercises and Machine Assignments to Chapter 2 CHAPTER 3. NUMERICAL DIFFERENTIATION AND INTEGRATION 1. Numerical Differentiation 1.1 A general differentiation formula for unequally spaced points 1.2 Examples 1.3 Numerical differentiation with perturbed data 2. Numerical Integration 2.1 The composite trapezoidal and Simpson's rules 2.2 (Weighted) Newton-Cotes and Gauss formulae 2.3 Properties of Gaussian quadrature rules 2.4 Some applications of the Gauss quadrature rule 2.5 Approximation of linear functionals: method of interpolation vs. method of undermined coefficients 2.6 Peano representation of linear functionals 2.7 Extrapolation methods Notes to Chapter 3 Exercises and Machine Assignments to Chapter 3 CHAPTER 4. NONLINEAR EQUATIONS 1. Examples 1.1 A transcendental equation 1.2 A two-point boundary value problem 1.3 A nonlinear integral equation 1.4 s- Orthogonal polynomials 2. Iteration, Convergence, and Efficiency 3. The Methods of Bisection and Sturm Sequences 3.1 Bisection method 3.2 Method of Sturm sequences 4. Method of False Position 5. Secant Method 6. Newton's Method 7. Fixed Iteration 8. Algebraic Equations 8.1 Newton's method applied to an algebraic equation 8.2 An accelerated Newton method for equations with real roots 9. Systems of Nonlinear Equations 9.1 Contraction mapping principle 9.2 Newton's method for systems of equations Notes to Chapter 4 Exercises and Machine Assignments to Chapter 4 CHAPTER 5. INITIAL VALUE PROBLEMS FOR ODEs - ONE-STEP METHODS 0.1 Examples 0.2 Types of diffential equations 0.3 Existence and uniqueness 0.4 Numerical Methods 1. Local Description of One-Step Methods 2. Examples of One-Step Methods 2.1 Euler's method 2.2 Method of Taylor expansion 2.3 Improved Euler methods 2.4 Second-order two-stage methods 2.5 Runge-Kutta methods 3. Global Description of One-Step Methods 3.1 Stability 3.2 Convergence 3.3 Asymptotics of global error 4. Error Monitoring and Step Control 4.1 Estimation of global error 4.2 Truncation error estimates 4.3 Step control 5. Stiff Problems 5.1 A-stability 5.2 Pade approximation 5.3 Examples of A-stable one-step methods 5.4 Regions of absolute stability Notes to Chapter 5 Exercises and Machine Assignments to Chapter 5 CHAPTER 6. INITIAL VALUE PROBLEMS FOR ODEs - MULTISTEP METHODS 1. Local Description of Multistep Methods 1.1 Explicit and implicit methods 1.2 Local accuracy 1.3 Polynomial degree vs. order 2. Examples of Multistep Methods 2.1 Adams-Bashforth method 2.2 Adams-Moulton method 2.3 Predictor-corrector methods 3. Global Description of Multistep Methods 3.1 Linear difference equations 3.2 Stability and root condition 3.3 Convergence 3.4 Asymptotics of global error 3.5 Estimation of global error 4. Analytic Theory of Order and Stability 4.1 Analytic characterization of order 4.2 Stable methods of maximum order 4.3 Applications 5. Stiff Problems 5.1 A-stability 5.2 A(alpha)-stability Notes to Chapter 6 Exercises and Machine Assignments to Chapter 6 CHAPTER 7. TWO-POINT BOUNDARY VALUE PROBLEMS FOR ODEs 1. Existence and Uniqueness 1.1 Examples 1.2 A scalar boundary value problem 1.3 General linear and nonlinear systems 2. Initial Value Techniques 2.1 Shooting method for a scalar boundary value problem 2.2 Linear and nonlinear systems 2.3 Parallel shooting 3. Finite Difference Methods 3.1 Linear second-order equations 3.2 Nonlinear second-order equations 4. Variational Methods 4.1 Variational formualtion 4.2 The extremal problem 4.3 Approximate solution of the extremal problem Notes to Chapter 7 Exercises and Machine Assignments to Chapter 7 References Subject Index |

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### Contents

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### Common terms and phrases

A-stable algebraic algorithm analysis applied arithmetic assume boundary value problem Chebyshev Chebyshev polynomials coefficients compute condition number Consider convergence defined definite degree of exactness denote derivative Determine differential equations divided differences eigenvalues endpoints error function estimate Euler Euler's method evaluation exact solution example floating-point follows Gauss Gautschi given grid function hence Hint:Use implicit initial value problem integral interpolation polynomial interval Lagrange least squares approximation linearly Lipschitz mathematical matrix method of order multistep method Newton's method nodes nonlinear equations norm obtain one-step method orthogonal polynomials polynomial of degree positive principal error function proof of Theorem quadratic quadrature formula quadrature rule recursion result root condition Runge-Kutta method satisfies secant method Show solving spline stability starting value step theory trapezoidal rule truncation error un+k unique solution variables vector weight function xn+i y(xn zero

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Page 456 - interpolation par la methode des moindres carres', Mem. Acad. Impe'r. Sci. St. Pe'tersbourg (7) 1, no. 15, pp.