First Semester in Numerical Analysis with Julia, 2020a

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CHAPTER 2. SOLUTIONS OF EQUATIONS: ROOT-FINDING 79 4 y = x 4 3 3 y = g(x) 2 2 1 1 0 p 0 Figure 2.4: Fixed-point iteration: |g ′ (p)| < 1. 4 y = x 4 3 3 2 2 1 1 y = g(x) 0 p 0 Figure 2.5: Fixed-point iteration: |g ′ (p)| > 1. Example 43. Consider the root-finding problem x 3 − 2x 2 − 1=0on [1, 3]. 1. Write the problem as a fixed-point problem, g(x) =x, for some g. Verify that the hypothesis of Theorem 40 (or Remark 41) is satisfied so that the fixed-point iteration converges. 2. Let p 0 =1. Use Corollary 42 to find n that ensures an estimate to p accurate to within 10 −4 . Solution. 1. There are several ways we can write this problem as g(x) =x : (a) Let f(x) =x 3 − 2x 2 − 1, andp be its root, that is, f(p) =0. Ifweletg(x) = x − f(x), theng(p) =p − f(p) =p, so p is a fixed-point of g. However, this choice
CHAPTER 2. SOLUTIONS OF EQUATIONS: ROOT-FINDING 80 for g will not be helpful, since g does not satisfy the first condition of Theorem 40: g(x) /∈ [1, 3] for all x ∈ [1, 3] (g(3) = −5 /∈ [1, 3]). (b) Since p is a root for f, we have p 3 =2p 2 +1,orp =(2p 2 +1) 1/3 . Therefore, p is the solution to the fixed-point problem g(x) =x where g(x) =(2x 2 +1) 1/3 . • g is increasing on [1, 3] and g(1) = 1.44,g(3) = 2.67, thusg(x) ∈ [1, 3] for all x ∈ [1, 3]. Therefore, g satisfies the first condition of Theorem 40. • g ′ 4x (x) = and g ′ (1) = 0.64,g ′ (3) = 0.56 and g ′ is decreasing on [1, 3]. 3(2x 2 +1) 2/3 Therefore g satisfies the condition in Remark 41 with λ =0.64. Then, from Theorem 40 and Remark 41, the fixed-point iteration converges if g(x) =(2x 2 +1) 1/3 . 2. Take λ = k =0.64 in Corollary 42 and use bound (4): |p − p n |≤(0.64) n max{1 − 1, 3 − 1} =2(0.64 n ). We want 2(0.64 n ) < 10 −4 , which implies n log 0.64 < −4 log 10 − log 2, or n > −4 log 10−log 2 log 0.64 ≈ 22.19. Therefore n =23is the smallest number of iterations that ensures an absolute error of 10 −4 . Julia code for fixed-point iteration The following code starts with the initial guess p 0 (pzero in the code), computes p 1 = g(p 0 ), and checks if the stopping criterion |p 1 − p 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 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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