First Semester in Numerical Analysis with Julia, 2020a

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CHAPTER 5. APPROXIMATION THEORY 213 Out[13]: compsimpson (generic function with 1 method) The integrals in Example 94 were computed using the composite Simpson rule with n =20. For example, the second integral 〈e x ,T 1 〉 = ∫ π 0 ecos θ cos θdθ is computed as: In [14]: compsimpson(x->exp(cos(x))*cos(x),0,pi,20) Out[14]: 1.7754996892121808 Next we write two functions, polyChebCoeff(f::Function,n) and polyCheb(x,n). The first function computes the coefficients 〈f,T j〉 of the least 〈f,T j 〉 α j squares polynomial P n (x) = ∑ n j=0 T j (x),j =0, 1, ..., n, for any f and n, where T j are the Chebyshev polynomials. Note that the indices are shifted in the code so that j starts at 1 not 0. The coefficients are stored in the global array A. The integral 〈f,T j 〉 is transformed to the integral ∫ π f(cos θ)cos(jθ)dθ, similar to 0 the derivation in Example 94, and then the transformed integral is computed using the composite Simpson’s rule by polyChebCoeff. In [15]: function polyChebCoeff(f::Function,n) global A=Array{Float64}(undef,n+1) A[1]=compsimpson(x->f(cos(x)),0,pi,20)/pi for j in 2:n+1 A[j]=compsimpson(x->f(cos(x))*cos((j-1)*x),0,pi,20)*2/pi end end Out[15]: polyChebCoeff (generic function with 1 method) In [16]: function polyCheb(x,n) sum=0.; for j in 1:n+1 sum=sum+A[j]*cheb(x,(j-1)) end return(sum) end Out[16]: polyCheb (generic function with 1 method) α j
CHAPTER 5. APPROXIMATION THEORY 214 Next we plot y = e x together with polynomial approximations of degree two and three using Chebyshev basis polynomials. In [17]: xaxis=-1:1/100:1 polyChebCoeff(x->e^x,2) deg2=map(x->polyCheb(x,2),xaxis) polyChebCoeff(x->e^x,3) deg3=map(x->polyCheb(x,3),xaxis) plot(xaxis,map(x->e^x,xaxis),label=L"e^x") plot(xaxis,deg2,label="Chebyshev least squares poly of degree 2") plot(xaxis,deg3,label="Chebyshev least squares poly of degree 3") legend(loc="upper left"); The cubic Legendre and Chebyshev approximations are difficult to distinguish from the function itself. Let’s compare the quadratic approximations obtained by Legendre and Chebyshev polynomials. Below, you can see visually that Chebyshev does a better approximation at the end points of the interval. Is this expected? In [18]: xaxis=-1:1/100:1 polyChebCoeff(x->e^x,2) cheb2=map(x->polyCheb(x,2),xaxis) polyLegCoeff(x->e^x,2) leg2=map(x->polyLeg(x,2),xaxis)
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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 110 The ne
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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 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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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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