First Semester in Numerical Analysis with Julia, 2020a

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CHAPTER 2. SOLUTIONS OF EQUATIONS: ROOT-FINDING 85 0.28707160940672327 0.291222000031716 0.2924065231373124 0.2927509058227538 0.292851556555365 0.2928810179561428 0.2928896454165901 0.29289217219391755 In [9]: plot(list); Figure 2.6: Fixed-point iteration The graph suggests the limit of linear convergence. p n− √ 2 p n−1 − √ 2 exists and it is around 0.295, supporting 2.7 High-order fixed-point iteration In the proof of Theorem 44, weshowed |p n+1 − p| lim n→∞ |p n − p| = |g ′ (p)| which implied that the fixed-point iteration has linear convergence, if g ′ (p) ≠0.
CHAPTER 2. SOLUTIONS OF EQUATIONS: ROOT-FINDING 86 If this limit were zero, then we would have |p n+1 − p| lim n→∞ |p n − p| which means the denominator is growing at a larger rate than the numerator. We could then ask if |p n+1 − p| lim = nonzero constant n→∞ |p n − p| α for some α>1. Theorem 46. Assume p is a solution of g(x) =x where g ∈ C α (I) for some interval I that contains p, and for some α ≥ 2. Furthermore assume =0, g ′ (p) =g ′′ (p) =... = g (α−1) (p) =0, and g (α) (p) ≠0. Then if the initial guess p 0 is sufficiently close to p, the fixed-point iteration p n = g(p n−1 ),n≥ 1, will have order of convergence of α, and Proof. From Taylor’s theorem, lim n→∞ p n+1 − p (p n − p) = g(α) (p) . α α! p n+1 = g(p n )=g(p)+(p n − p)g ′ (p)+... + (p n − p) α−1 g (α−1) (p)+ (p n − p) α g (α) (ξ n ) (α − 1)! α! where ξ n isanumberbetweenp n and p, and all numbers are in I. From the hypothesis, this simplifies as p n+1 = p + (p n − p) α g (α) (ξ n ) ⇒ p n+1 − p α! (p n − p) = g(α) (ξ n ) . α α! From Theorem 44, ifp 0 is chosen sufficiently close to p, then lim n→∞ p n = p. The order of convergence is α with |p n+1 − p| lim n→∞ |p n − p| = lim |g (α) (ξ n )| α n→∞ α! = |g(α) (p)| α! ≠0.
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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 16 Matrix o
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CHAPTER 1. INTRODUCTION 18 stdlib/v
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CHAPTER 1. INTRODUCTION 24 -0.62988
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CHAPTER 1. INTRODUCTION 28 Sometime
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CHAPTER 1. INTRODUCTION 32 Notice t
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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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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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