First Semester in Numerical Analysis with Julia, 2020a

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CHAPTER 4. NUMERICAL QUADRATURE AND DIFFERENTIATION 167 For each sample size, we compute the relative error, and then plot the results. In [6]: error=[relerror(n) for n in samples]; In [7]: plot(samples,error) xlabel("Sample size (n)") ylabel("Relative error"); Figure 4.4: Monte Carlo relative error for the integral (4.9) 4.5 Improper integrals The quadrature rules we have learned so far cannot be applied (or applied with a poor performance) to integrals such as ∫ b f(x)dx if a, b = ±∞ or if a, b are finite but f is not continuous a at one or both of the endpoints: recall that both Newton-Cotes and Gauss-Legendre error bound theorems require the integrand to have a number of continuous derivatives on the closed interval [a, b]. For example, an integral in the form ∫ 1 −1 f(x) √ 1 − x 2 dx clearly cannot be approximated using the trapezoidal or Simpson’s rule without any modifications, since both rules require the values of the integrand at the end points which do not exist. One could try using the Gauss-Legendre rule, but the fact that the integrand does
CHAPTER 4. NUMERICAL QUADRATURE AND DIFFERENTIATION 168 not satisfy the smoothness conditions required by the Gauss-Legendre error bound means the error of the approximation might be large. A simple remedy to the problem of improper integrals is to change the variable of integration and transform the integral, if possible, to one that behaves well. Example 82. Consider the previous integral ∫ 1 f(x) √ −1 1−x 2 dx. Try the transformation θ = cos −1 x. Then dθ = −dx/ √ 1 − x 2 and ∫ 1 −1 ∫ f(x) 0 √ dx = − f(cos θ)dθ = 1 − x 2 π ∫ π 0 f(cos θ)dθ. The latter integral can be evaluated using, for example, Simpson’s rule, provided f is smooth on [0,π]. If the interval of integration is infinite, another approach that might work is truncation of the interval to a finite one. The success of this approach depends on whether we can estimate the resulting error. Example 83. Consider the improper integral ∫ ∞ 0 e −x2 dx. Write the integral as ∫ ∞ 0 e −x2 dx = ∫ t 0 e −x2 dx + ∫ ∞ t e −x2 dx, where t is the "level of truncation" to determine. We can estimate the first integral on the right-hand side using a quadrature rule. The second integral on the right-hand side is the error due to approximating ∫ ∞ e −x2 dx by ∫ t 0 0 e−x2 dx. An upper bound for the error can be found easily for this example: note that when x ≥ t, x 2 = xx ≥ tx, thus ∫ ∞ t e −x2 dx ≤ ∫ ∞ t e −tx dx = e −t2 /t. When t =5, e −t2 /t ≈ 10 −12 , therefore, approximating the integral ∫ ∞ e −x2 dx by ∫ 5 0 0 e−x2 dx will be accurate within 10 −12 . Additional error will come from estimating the latter integral by numerical quadrature. 4.6 Numerical differentiation The derivative of f at x 0 is f ′ f(x 0 + h) − f(x 0 ) (x 0 ) = lim . h→0 h
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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 1. INTRODUCTION 32 Notice t
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CHAPTER 1. INTRODUCTION 34 and real
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CHAPTER 1. INTRODUCTION 38 Chopping
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CHAPTER 1. INTRODUCTION 40 Machine
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CHAPTER 1. INTRODUCTION 42 The abso
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CHAPTER 1. INTRODUCTION 48 Exercise
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CHAPTER 1. INTRODUCTION 50 Sources
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CHAPTER 2. SOLUTIONS OF EQUATIONS:
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Chapter 3 Interpolation In this cha
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CHAPTER 3. INTERPOLATION 90 Here is
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CHAPTER 3. INTERPOLATION 92 basis f
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CHAPTER 3. INTERPOLATION 94 therefo
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CHAPTER 3. INTERPOLATION 96 Summary
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CHAPTER 3. INTERPOLATION 98 and thu
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CHAPTER 3. INTERPOLATION 100 With t
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CHAPTER 3. INTERPOLATION 102 Exampl
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CHAPTER 3. INTERPOLATION 104 Exerci
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CHAPTER 3. INTERPOLATION 106 end en
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CHAPTER 3. INTERPOLATION 108 increa
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CHAPTER 3. INTERPOLATION 110 The ne
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CHAPTER 3. INTERPOLATION 112 there
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CHAPTER 3. INTERPOLATION 114 Figure
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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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