First Semester in Numerical Analysis with Julia, 2020a

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CHAPTER 2. SOLUTIONS OF EQUATIONS: ROOT-FINDING 83 In [1]: function fixedpt(g::Function,pzero,eps,N) n=1 while nx-(x^2-2)/4,2,10^(-5),15) p is 1.414214788550556 and the iteration number is 10 Let’s try p 0 = −5. Note that this is not only outside the interval of convergence I, but g ′ (−5) = 3.5 > 1, so we do not expect convergence.
CHAPTER 2. SOLUTIONS OF EQUATIONS: ROOT-FINDING 84 In [5]: fixedpt(x->x-(x^2-2)/4,-5.,10^(-5),15) Did not converge. The last estimate is p=-Inf. Let’s verify the linear convergence of the fixed-point iteration numerically in this example. We write another version of the fixed-point code, fixedpt2, and we compute pn−√ 2 p n−1 − √ for each 2 n. The code uses some functions we have not seen before: • global list defines the variable list as a global variable, which allows us to access list outside the function definition, where we plot it; • Float64[ ] creates an empty array of dimension 1, with a float type; • append!(list,x) appends x to the list. In [6]: using PyPlot In [7]: function fixedpt2(g::Function,pzero,eps,N) n=1 global list list=Float64[] error=1. while n10^-5. pone=g(pzero) error=abs(pone-pzero) append!(list,(pone-2^0.5)/(pzero-2^0.5)) pzero=pone n=n+1 end return(list) end Out[7]: fixedpt2 (generic function with 1 method) In [8]: fixedpt2(x->x-(x^2-2)/4,1.5,10^(-7),15) Out[8]: 9-element Array{Float64,1}: 0.27144660940672544
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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 14 Press ]
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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 20 In [37]:
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CHAPTER 1. INTRODUCTION 24 -0.62988
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CHAPTER 1. INTRODUCTION 26 5 7 9 He
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CHAPTER 1. INTRODUCTION 28 Sometime
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CHAPTER 1. INTRODUCTION 30 where s
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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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