First Semester in Numerical Analysis with Julia, 2020a

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CHAPTER 5. APPROXIMATION THEORY 193 A[1,4]=sum(t->cos(2*pi*t/365),time) A[2,2]=sum(t->t^2,time) A[2,3]=sum(t->t*sin(2*pi*t/365),time) A[2,4]=sum(t->t*cos(2*pi*t/365),time) A[3,3]=sum(t->(sin(2*pi*t/365))^2,time) A[3,4]=sum(t->(sin(2*pi*t/365)*cos(2*pi*t/365)),time) A[4,4]=sum(t->(cos(2*pi*t/365))^2,time) for i=2:4 for j=1:i A[i,j]=A[j,i] end end In [10]: A Out[10]: 4×4 Array{Float64,2}: 1056.0 558096.0 12.2957 -36.2433 558096.0 3.93086e8 -50458.1 -38477.3 12.2957 -50458.1 542.339 10.9944 -36.2433 -38477.3 10.9944 513.661 Now we define the vector r. The function dot(x,y) takes the dot product of the arrays x, y. For example, dot([1, 2, 3], [4, 5, 6]) = 1 × 4+2× 5+3× 6=32. In [11]: r=zeros(4,1) r[1]=sum(temp) r[2]=dot(temp,time) r[3]=dot(temp,map(t->sin(2*pi*t/365),time)) r[4]=dot(temp,map(t->cos(2*pi*t/365),time)); In [12]: r Out[12]: 4×1 Array{Float64,2}: 21861.499999999996 1.1380310200000009e7 1742.0770653857649 2787.7612743690415
CHAPTER 5. APPROXIMATION THEORY 194 We can solve the matrix equation now. In [13]: A\r Out[13]: 4×1 Array{Float64,2}: 20.28975634590378 0.0011677302061853312 2.7211617643953634 6.88808560736693 Recall that these constants are the values of a, b, c, d in the definition of f(t). Here is the best fitting function to the data: In [14]: f(t)=20.2898+0.00116773*t+2.72116*sin(2*pi*t/365)+ 6.88809*cos(2*pi*t/365) Out[14]: f (generic function with 1 method) We next plot the data together with f(t): In [15]: xaxis=1:1:1056 yvals=map(t->f(t),xaxis) plot(xaxis,yvals,label="Least squares approximation") xlabel("time (t)") ylabel("Temperature (T)") plot(temp,linestyle="-",alpha=0.5,label="Data") legend(loc="upper center");
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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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Page 181 and 182: Chapter 5 Approximation Theory 5.1
Page 183 and 184: CHAPTER 5. APPROXIMATION THEORY 180
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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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