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Unformatted text preview: A Concise Introduction to Numerical Analysis Douglas N. Arnold School of Mathematics, University of Minnesota, Minneapolis, MN 55455 Email address : arnold@umn.edu URL : http://umn.edu/~arnold/ 1991 Mathematics Subject Classication. Primary 6501 c 1999, 2000, 2001 by Douglas N. Arnold. All rights reserved. Not to be disseminated without explicit permission of the author. Preface These notes were prepared for use in teaching a oneyear graduate level introductory course on numerical analysis at Penn State University. The author taught the course during the 19981999 academic year (the rst oering of the course), and then again during the 20002001 academic year. They were never put into nal form, and cannot be used without express permission of the author. Douglas N. Arnold iii 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. Onepoint iteration 90 3. Newtons method 92 4. QuasiNewton methods 94 5. Broydens method 95 v vi CONTENTS 6. Unconstrained minimization 99 7. Newtons method 99 8. Line search methods 100 9. Conjugate gradients 105 Exercises 112 Chapter 5. Numerical Solution of Ordinary Dierential Equations 115 1. Introduction 115 2. Eulers Method 117 3. Linear multistep methods 123 4. One step methods 134 5. Error estimation and adaptivity 138 6. Stiness 141 Exercises 148 Chapter 6. Numerical Solution of Partial Dierential Equations 151 1. BVPs for 2nd order elliptic PDEs 151 2. The vepoint discretization of the Laplacian 153 3. Finite element methods 162 4. Dierence methods for the heat equation 177 5. Dierence 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 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 a simpler function....
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This note was uploaded on 11/12/2011 for the course STA 3032 taught by Professor Kyung during the Summer '08 term at University of Florida.
 Summer '08
 Kyung
 Statistics, The Land, Sula

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