First Semester in Numerical Analysis with Julia, 2020a

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CHAPTER 2. SOLUTIONS OF EQUATIONS: ROOT-FINDING 63 In [3]: x=range(-2,2,length=1000) y=map(x->x^5+2*x^3-5*x-2,x) ax = gca() ax.spines["bottom"].set_position("center") ax.spines["left"].set_position("center") ax.spines["top"].set_position("center") ax.spines["right"].set_position("center") ylim([-40,40]) plot(x,y); The derivative is f ′ =5x 4 +6x 2 − 5, we set pin =1,eps= ɛ =10 −4 ,andN =20, in the code. In [4]: newton(x -> x^5+2x^3-5x-2,x->5x^4+6x^2-5,1,10^(-4),20) p is 1.3196411672093726 and the iteration number is 6 Recall that the bisection method required 16 iterations to approximate the root in [0, 2] as p =1.31967. (However, the stopping criterion used in bisection and Newton’s methods are slightly different.) 1.3196 is the rightmost root in the plot. But there are other roots of the function. Let’s run the code with pin = 0.
CHAPTER 2. SOLUTIONS OF EQUATIONS: ROOT-FINDING 64 In [5]: newton(x -> x^5+2x^3-5x-2,x->5x^4+6x^2-5,0,10^(-4),20) p is -0.43641313299799755 and the iteration number is 4 Nowweusepin=−2.0 which will give the leftmost root. In [6]: newton(x -> x^5+2x^3-5x-2,x->5x^4+6x^2-5,-2,10^(-4),20) p is -1.0000000001014682 and the iteration number is 7 Theorem 32. Let f ∈ C 2 [a, b] and assume f(p) =0,f ′ (p) ≠0for p ∈ (a, b). If p 0 is chosen sufficiently close to p, then Newton’s method generates a sequence that converges to p with lim n→∞ p − p n+1 (p − p n ) = − f ′′ (p) 2 2f ′ (p) . Proof. Since f ′ is continuous and f ′ (p) ≠0, there exists an interval I =[p − ɛ, p + ɛ] on which f ′ ≠0. Let M = max x∈I |f ′′ (x)| 2 min x∈I |f ′ (x)| . Pick p 0 from the interval I (which means |p − p 0 | ≤ ɛ), sufficiently close to p so that M|p − p 0 | < 1. From Equation (2.5) wehave: |p − p 1 | = |p − p 0||p − p 0 | 2 ∣ f ′′ (ξ(p)) f ′ (p 0 ) ∣ < |p − p 0||p − p 0 |M
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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 3. INTERPOLATION 114 Figure
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CHAPTER 3. INTERPOLATION 116 Then t
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CHAPTER 3. INTERPOLATION 118 end en
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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 130 end en
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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 5 Approximation Theory 5.1
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CHAPTER 5. APPROXIMATION THEORY 180
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Index Absolute error, 38 Beasley-Sp
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INDEX 220 Chebyshev, 203 Julia code
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First Semester in Numerical Analysis with Julia, 2020a

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