First Semester in Numerical Analysis with Julia, 2020a

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CHAPTER 2. SOLUTIONS OF EQUATIONS: ROOT-FINDING 57 a new interval in the following step, we will simply call the new interval [a, b], overwriting the old one. Similarly, we will call the midpoint p, and update it at each step. In [1]: function bisection(f::Function,a,b,eps,N) n=1 p=0. # to ensure the value of p carries out of the while loop while n
CHAPTER 2. SOLUTIONS OF EQUATIONS: ROOT-FINDING 58 In [3]: x=1.319671630859375; x^5+2x^3-5x-2 Out[3]: 0.000627945623044468 Let’s see what happens if N is set too small and the method does not converge. In [4]: bisection(x -> x^5+2x^3-5x-2,0,2,10^(-4.),5) Method did not converge. The last iteration gives 1.3125 with function value -0.14562511444091797 Theorem 28. Suppose that f ∈ C 0 [a, b] and f(a)f(b) < 0. The bisection method generates a sequence {p n } approximating a zero p of f(x) with |p n − p| ≤ b − a ,n≥ 1. 2n Proof. Let the sequences {a n } and {b n } denote the left-end and right-end points of the subintervals generated by the bisection method. Since at each step the interval is halved, we have b n − a n = 1 2 (b n−1 − a n−1 ). By mathematical induction, we get b n − a n = 1 2 (b n−1 − a n−1 )= 1 2 2 (b n−2 − a n−2 )=... = 1 2 n−1 (b 1 − a 1 ). Therefore b n − a n = 1 2 n−1 (b − a). Observe that and thus |p n − p| →0 as n →∞. |p n − p| ≤ 1 2 (b n − a n )= 1 (b − a) (2.3) 2n Corollary 29. The bisection method has linear convergence. Proof. The bisection method does not satisfy (2.1) for any C
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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 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 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 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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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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