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# Introduction to Numerical Analysis

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Introduction to Numerical Analysis Doron Levy Department of Mathematics and Center for Scientific Computation and Mathematical Modeling CSCAMM University of Maryland September 21 2010 D Levy Preface i D Levy CONTENTS Contents Preface i 1 Introduction 1 2 Methods for Solving Nonlinear Problems 2 1 Preliminary Discussion 2 1 1 Are there any roots anywhere 2 1 2 Examples of root finding methods 2 2 Iterative Methods 2 3 The Bisection Method 2 4 Newton s Method 2 5 The Secant Method 2 2 3 5 6 8 11 15 3 Interpolation 3 1 What is Interpolation 3 2 The Interpolation Problem 3 3 Newton s Form of the Interpolation Polynomial 3 4 The Interpolation Problem and the Vandermonde Determinant 3 5 The Lagrange Form of the Interpolation Polynomial 3 6 Divided Differences 3 7 The Error in Polynomial Interpolation 3 8 Interpolation at the Chebyshev Points 3 9 Hermite Interpolation 3 9 1 Divided differences with repetitions 3 9 2 The Lagrange form of the Hermite interpolant 3 10 Spline Interpolation 3 10 1 Cubic splines 3 10 2 What is natural about the natural spline 19 19 20 22 23 25 28 31 33 40 42 44 47 49 53 4 Approximations 4 1 Background 4 2 The Minimax Approximation Problem 4 2 1 Existence of the minimax polynomial 4 2 2 Bounds on the minimax error 4 2 3 Characterization of the minimax polynomial 4 2 4 Uniqueness of the minimax polynomial 4 2 5 The near minimax polynomial 4 2 6 Construction of the minimax polynomial 4 3 Least squares Approximations 4 3 1 The least squares approximation problem 4 3 2 Solving the least squares problem a direct method 56 56 61 62 64 65 65 66 67 69 69 69 iii CONTENTS 4 3 3 4 3 4 4 3 5 4 3 6 4 3 7 D Levy Solving the least squares problem with orthogonal polynomials The weighted least squares problem Orthogonal polynomials Another approach to the least squares problem Properties of orthogonal polynomials 5 Numerical Differentiation 5 1 Basic Concepts 5 2 Differentiation Via Interpolation 5 3 The Method of Undetermined Coefficients 5 4 Richardson s

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