First Semester in Numerical Analysis with Julia, 2020a

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CHAPTER 5. APPROXIMATION THEORY 197 Exercise 5.1-1: Find the function of the form y = ae x + b sin(4x) that best fits the data below in the least squares sense. x 1 2 3 4 5 y -4 6 -1 5 20 Plot the function and the data together. Exercise 5.1-2: Power-law type relationships are observed in many empirical data. Two variables y, x are said to be related via a power-law if y = kx α , where k, α are some constants. The following data 2 lists the top 10 family names in the order of occurrence according to Census 2000. Investigate whether relative frequency of occurrences and the rank of the name are related via a power-law, by a) Let y be the relative frequencies (number of occurrences divided by the total number of occurrences), and x be the rank, that is, 1 through 10. b) Use least squares to find a function of the form y = kx α . Use linearization. c) Plot the data together with the best fitting function found in part (b). 2 https://www.census.gov/topics/population/genealogy/data/2000_surnames.html
CHAPTER 5. APPROXIMATION THEORY 198 Name Number of Occurrences Smith 2,376,206 Johnson 1,857,160 Williams 1,534,042 Brown 1,380,145 Jones 1,362,755 Miller 1,127,803 Davis 1,072,335 Garcia 858,289 Rodriguez 804,240 Wilson 783,051 5.2 Continuous least squares In discrete least squares, our starting point was a set of data points. Here we will start with a continuous function f on [a, b] and answer the following question: how can we find the "best" polynomial P n (x) = ∑ n j=0 a jx j of degree at most n, that approximates f on [a, b]? As before, "best" polynomial will mean the polynomial that minimizes the least squares error: E = ∫ b a ( f(x) − Compare this expression with that of the discrete least squares: E = To minimize E in (5.10) weset ∂E ∂a k ∂E ∂a k = ⎛ ∂ ⎝ ∂a k = −2 ∫ b a ∫ b ( m∑ y i − i=1 ) 2 n∑ a j x j dx. (5.10) j=0 ) 2 n∑ a j x j i . j=0 =0, for k =0, 1, ..., n, and observe ∫ ( b n∑ ) f 2 (x)dx − 2 f(x) a j x j dx + a a j=0 n∑ ∫ b f(x)x k dx +2 a j x j+k dx =0, j=0 a ∫ b a ( n∑ j=0 ⎞ ) 2 a j x j dx⎠
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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 12 Out[8]:
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CHAPTER 1. INTRODUCTION 14 Press ]
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CHAPTER 1. INTRODUCTION 16 Matrix o
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CHAPTER 1. INTRODUCTION 18 stdlib/v
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CHAPTER 1. INTRODUCTION 20 In [37]:
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CHAPTER 1. INTRODUCTION 24 -0.62988
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CHAPTER 1. INTRODUCTION 26 5 7 9 He
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CHAPTER 1. INTRODUCTION 28 Sometime
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CHAPTER 1. INTRODUCTION 30 where s
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CHAPTER 1. INTRODUCTION 32 Notice t
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CHAPTER 1. INTRODUCTION 34 and real
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CHAPTER 1. INTRODUCTION 38 Chopping
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CHAPTER 1. INTRODUCTION 40 Machine
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CHAPTER 1. INTRODUCTION 42 The abso
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CHAPTER 1. INTRODUCTION 44 Next wil
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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 In this cha
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CHAPTER 3. INTERPOLATION 90 Here is
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CHAPTER 3. INTERPOLATION 92 basis f
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CHAPTER 3. INTERPOLATION 94 therefo
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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 106 end en
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CHAPTER 3. INTERPOLATION 108 increa
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CHAPTER 3. INTERPOLATION 110 The ne
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CHAPTER 3. INTERPOLATION 112 there
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CHAPTER 3. INTERPOLATION 114 Figure
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CHAPTER 3. INTERPOLATION 116 Then t
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CHAPTER 3. INTERPOLATION 120 Exerci
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CHAPTER 3. INTERPOLATION 122 yi yi+
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CHAPTER 3. INTERPOLATION 124 • Cl
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CHAPTER 3. INTERPOLATION 126 In [2]
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CHAPTER 3. INTERPOLATION 128 end fo
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CHAPTER 3. INTERPOLATION 132 end u=
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CHAPTER 3. INTERPOLATION 134 Especi
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CHAPTER 3. INTERPOLATION 136 Arya a
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CHAPTER 3. INTERPOLATION 138 This l
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CHAPTER 3. INTERPOLATION 140 Recrea
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CHAPTER 4. NUMERICAL QUADRATURE AND
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CHAPTER 4. NUMERICAL QUADRATURE AND
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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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