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nabook - A Concise Introduction to Numerical Analysis...

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A Concise Introduction to Numerical Analysis Douglas N. Arnold School of Mathematics, University of Minnesota, Minneapolis, MN 55455 E-mail address : [email protected] URL : http://umn.edu/~arnold/
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1991 Mathematics Subject Classification. Primary 65-01 c 1999, 2000, 2001 by Douglas N. Arnold. All rights reserved. Not to be disseminated without explicit permission of the author.
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Preface These notes were prepared for use in teaching a one-year graduate level introductory course on numerical analysis at Penn State University. The author taught the course during the 1998–1999 academic year (the first offering of the course), and then again during the 2000–2001 academic year. They were never put into final form, and cannot be used without express permission of the author. Douglas N. Arnold iii
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Contents Preface iii Chapter 1. Approximation and Interpolation 1 1. Introduction and Preliminaries 1 2. Minimax Polynomial Approximation 4 3. Lagrange Interpolation 14 4. Least Squares Polynomial Approximation 22 5. Piecewise polynomial approximation and interpolation 26 6. Piecewise polynomials in more than one dimension 34 7. The Fast Fourier Transform 44 Exercises 48 Bibliography 53 Chapter 2. Numerical Quadrature 55 1. Basic quadrature 55 2. The Peano Kernel Theorem 57 3. Richardson Extrapolation 60 4. Asymptotic error expansions 61 5. Romberg Integration 65 6. Gaussian Quadrature 66 7. Adaptive quadrature 70 Exercises 74 Chapter 3. Direct Methods of Numerical Linear Algebra 77 1. Introduction 77 2. Triangular systems 78 3. Gaussian elimination and LU decomposition 79 4. Pivoting 82 5. Backward error analysis 83 6. Conditioning 87 Exercises 88 Chapter 4. Numerical solution of nonlinear systems and optimization 89 1. Introduction and Preliminaries 89 2. One-point iteration 90 3. Newton’s method 92 4. Quasi-Newton methods 94 5. Broyden’s method 95 v
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vi CONTENTS 6. Unconstrained minimization 99 7. Newton’s method 99 8. Line search methods 100 9. Conjugate gradients 105 Exercises 112 Chapter 5. Numerical Solution of Ordinary Differential Equations 115 1. Introduction 115 2. Euler’s Method 117 3. Linear multistep methods 123 4. One step methods 134 5. Error estimation and adaptivity 138 6. Stiffness 141 Exercises 148 Chapter 6. Numerical Solution of Partial Differential Equations 151 1. BVPs for 2nd order elliptic PDEs 151 2. The five-point discretization of the Laplacian 153 3. Finite element methods 162 4. Difference methods for the heat equation 177 5. Difference methods for hyperbolic equations 183 6. Hyperbolic conservation laws 189 Exercises 190 Chapter 7. Some Iterative Methods of Numerical Linear Algebra 193 1. Introduction 193 2. Classical iterations 194 3. Multigrid methods 198 Exercises 204 Bibliography 205
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CHAPTER 1 Approximation and Interpolation 1. Introduction and Preliminaries The problem we deal with in this chapter is the approximation of a given function by a simpler function. This has many possible uses. In the simplest case, we might want to evaluate the given function at a number of points, and an algorithm for this, we construct and evaluate the simpler function. More commonly the approximation problem is only the first step towards developing an algorithm to solve some other problem. For example, an
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