First Semester in Numerical Analysis with Julia, 2020a

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CHAPTER 5. APPROXIMATION THEORY 211 Computing Chebyshev polynomials The following function implements the recursion Chebyshev polynomials satisfy: T n+1 (x) =2xT n (x) − T n−1 (x), for n =1, 2, ..., with T 0 (x) =1and T 1 (x) =x. Note that in the code the index is shifted from n +1to n. In [11]: function cheb(x,n) n==0 && return 1 n==1 && return x 2x*cheb(x,n-1)-cheb(x,n-2) end Out[11]: cheb (generic function with 1 method) Here is a plot of the first five Chebyshev polynomials: In [12]: xaxis=-1:1/100:1 chebzero=map(x->cheb(x,0),xaxis) chebone=map(x->cheb(x,1),xaxis) chebtwo=map(x->cheb(x,2),xaxis) chebthree=map(x->cheb(x,3),xaxis) chebfour=map(x->cheb(x,4),xaxis) plot(xaxis,chebzero,label=L"T_0(x)") plot(xaxis,chebone,label=L"T_1(x)") plot(xaxis,chebtwo,label=L"T_2(x)") plot(xaxis,chebthree,label=L"T_3(x)") plot(xaxis,chebfour,label=L"T_4(x)") legend(loc="lower right");
CHAPTER 5. APPROXIMATION THEORY 212 Least squares using Chebyshev polynomials In Example 94, we computed the least squares approximation to e x using Chebyshev polynomials. The inner products, after a transformation, were computed using the composite Simpson rule. Below is the Julia code for the composite Simpson rule we discussed in the previous chapter. In [13]: function compsimpson(f::Function,a,b,n) h=(b-a)/n nodes=Array{Float64}(undef,n+1) for i in 1:n+1 nodes[i]=a+(i-1)h end sum=f(a)+f(b) for i in 3:2:n-1 sum=sum+2*f(nodes[i]) end for i in 2:2:n sum=sum+4*f(nodes[i]) end return(sum*h/3) end
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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 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 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 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 140 Recrea
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CHAPTER 4. NUMERICAL QUADRATURE AND
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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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