First Semester in Numerical Analysis with Julia, 2020a

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CHAPTER 5. APPROXIMATION THEORY 187 The corresponding error is: In [13]: leastsqerror(a,xd,yd) Out[13]: 1.2664285714285708 The next polynomial is of degree four: In [14]: a=leastsqfit(xd,yd,4) xaxis=1:1/100:6 yvals=map(x->poly(x,a),xaxis) plot(xaxis,yvals) scatter(xd,yd); The least squares error is: In [15]: leastsqerror(a,xd,yd) Out[15]: 0.7232142857142789 Finally, we try a fifth degree polynomial. Recall that the normal equations have a unique solution when x i are distinct, and n ≤ m − 1. Since m =6in this example, n =5is the largest degree with guaranteed unique solution.
CHAPTER 5. APPROXIMATION THEORY 188 In [16]: a=leastsqfit(xd,yd,5) xaxis=1:1/100:6 yvals=map(x->poly(x,a),xaxis) plot(xaxis,yvals) scatter(xd,yd); The approximating polynomial of degree five is the interpolating polynomial! What is the least squares error? Least squares with non-polynomials The method of least squares is not only for polynomials. For example, suppose we want to find the function f(t) =a + bt + c sin(2πt/365) + d cos(2πt/365) (5.3) that has the best fit to some data (t 1 ,T 1 ), ..., (t m ,T m ) in the least-squares sense. This function is used in modeling weather temperature data, where t denotes time, and T denotes the temperature. The following figure plots the daily maximum temperature during a period of 1,056 days, from 2016 until November 21, 2018, as measured by a weather station at Melbourne airport, Australia 1 . 1 http://www.bom.gov.au/climate/data/
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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 44 Next wil
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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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CHAPTER 3. INTERPOLATION 116 Then t
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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 128 end fo
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CHAPTER 3. INTERPOLATION 132 end u=
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CHAPTER 3. INTERPOLATION 134 Especi
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Page 221 and 222: Index Absolute error, 38 Beasley-Sp
Page 223 and 224: INDEX 220 Chebyshev, 203 Julia code
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First Semester in Numerical Analysis with Julia, 2020a

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