First Semester in Numerical Analysis with Julia, 2020a

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CHAPTER 4. NUMERICAL QUADRATURE AND DIFFERENTIATION 173 f(x 0 + h) =f(x 0 )+hf ′ (x 0 )+ h2 2 f ′′ (x 0 )+ h3 6 f (3) (x 0 )+ h4 24 f (4) (ξ + ) f(x 0 − h) =f(x 0 ) − hf ′ (x 0 )+ h2 2 f ′′ (x 0 ) − h3 6 f (3) (x 0 )+ h4 24 f (4) (ξ − ) where ξ + is between x 0 and x 0 + h, and ξ − is between x 0 and x 0 − h. Add the equations to get f(x 0 + h)+f(x 0 − h) =2f(x 0 )+h 2 f ′′ (x 0 )+ h4 [ f (4) (ξ + )+f (4) (ξ − ) ] . 24 Solving for f ′′ (x 0 ) gives f ′′ (x 0 )= f(x 0 + h) − 2f(x 0 )+f(x 0 − h) h 2 − h2 [ f (4) (ξ + )+f (4) (ξ − ) ] . 24 Note that f (4) (ξ + )+f (4) (ξ − ) 2 is a number between f (4) (ξ + ) and f (4) (ξ − ), so from the Intermediate Value Theorem 6, we can conclude there exists some ξ between ξ − and ξ + so that Then the above formula simplifies as f (4) (ξ) = f (4) (ξ + )+f (4) (ξ − ) . 2 f ′′ (x 0 )= f(x 0 + h) − 2f(x 0 )+f(x 0 − h) h 2 for some ξ between x 0 − h and x 0 + h. − h2 12 f (4) (ξ)
CHAPTER 4. NUMERICAL QUADRATURE AND DIFFERENTIATION 174 Numerical differentiation and roundoff error Arya and the mysterious black box College life is full of mysteries, and Arya faces one in an engineering class: a black box! What is a black box? It is a computer program, or some device, which produces an output when an input is provided. We do not know the inner workings of the system, hence comes the name black box. Let’s think of the black box as a function f, and represent the input and output as x, f(x). Of course, we do not have a formula for f. What Arya’s engineering classmates want to do is compute the derivative information of the black box, that is, f ′ (x), whenx =2. (The input to this black box can be any real number.) Students want to use the three-point midpoint formula to estimate f ′ (2): f ′ (2) ≈ 1 [f(2 + h) − f(2 − h)] . 2h They argue how to pick h in this formula. One of them says they should make h as small as possible, like 10 −8 . Arya is skeptical. She mutters to herself, "I know I slept through some of my numerical analysis lectures, but not all!"
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First Semester in Numerical Analysi
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First Semester in Numerical Analysi
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Contents 1 Introduction 5 1.1 Revie
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Preface This book is based on my le
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Chapter 1 Introduction 1.1 Review o
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CHAPTER 1. INTRODUCTION 8 Solution.
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CHAPTER 1. INTRODUCTION 10 Let’s
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CHAPTER 1. INTRODUCTION 24 -0.62988
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CHAPTER 1. INTRODUCTION 28 Sometime
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CHAPTER 1. INTRODUCTION 42 The abso
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CHAPTER 1. INTRODUCTION 48 Exercise
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CHAPTER 1. INTRODUCTION 50 Sources
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CHAPTER 2. SOLUTIONS OF EQUATIONS:
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CHAPTER 3. INTERPOLATION 92 basis f
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CHAPTER 3. INTERPOLATION 96 Summary
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CHAPTER 3. INTERPOLATION 98 and thu
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CHAPTER 3. INTERPOLATION 100 With t
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CHAPTER 3. INTERPOLATION 102 Exampl
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CHAPTER 3. INTERPOLATION 104 Exerci
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CHAPTER 3. INTERPOLATION 108 increa
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CHAPTER 3. INTERPOLATION 114 Figure
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CHAPTER 3. INTERPOLATION 120 Exerci
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Page 221 and 222: Index Absolute error, 38 Beasley-Sp
Page 223 and 224: INDEX 220 Chebyshev, 203 Julia code
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First Semester in Numerical Analysis with Julia, 2020a

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