First Semester in Numerical Analysis with Julia, 2020a

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CHAPTER 2. SOLUTIONS OF EQUATIONS: ROOT-FINDING 61 by This is called Newton’s method. p n = p n−1 − f(p n−1) ,n≥ 1. (2.6) f ′ (p n−1 ) Graphical interpretation: Start with p 0 . Draw the tangent line at (p 0 ,f(p 0 )) and approximate p by the intercept p 1 of the line: f ′ (p 0 )= 0 − f(p 0) p 1 − p 0 ⇒ p 1 − p 0 = − f(p 0) f ′ (p 0 ) ⇒ p 1 = p 0 − f(p 0) f ′ (p 0 ) . Now draw the tangent at (p 1 ,f(p 1 )) and continue. Remark 31. 1. Clearly Newton’s method will fail if f ′ (p n )=0for some n. Graphically this means the tangent line is parallel to the x-axis so we cannot get the x-intercept. 2. Newton’s method may fail to converge if the initial guess p 0 is not close to p. In Figure (2.3), either choice for p 0 results in a sequence that oscillates between two points. 1 p0 p0 0 Figure 2.3: Non-converging behavior for Newton’s method 3. Newton’s method requires f ′ (x) is known explicitly. Exercise 2.3-1: f(x) =0? Sketch the graph for f(x) =x 2 −1. What are the roots of the equation
CHAPTER 2. SOLUTIONS OF EQUATIONS: ROOT-FINDING 62 1. Let p 0 =1/2 and find the first two iterations p 1 ,p 2 of Newton’s method by hand. Mark the iterates on the graph of f you sketched. Do you think the iterates will converge to a zero of f? 2. Let p 0 =0and find p 1 . What are your conclusions about the convergence of the iterates? Julia code for Newton’s method The Julia code below is based on Equation (2.6). The variable pin in the code corresponds to p n−1 ,andp corresponds to p n . The code overwrites these variables as the iteration continues. Also notice that the code has two functions as inputs; f and fprime (the derivative f ′ ). In [1]: function newton(f::Function,fprime::Function,pin,eps,N) n=1 p=0. # to ensure the value of p carries out of the while loop while n
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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 3. INTERPOLATION 120 Exerci
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CHAPTER 3. INTERPOLATION 126 In [2]
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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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