First Semester in Numerical Analysis with Julia, 2020a

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CHAPTER 5. APPROXIMATION THEORY 181 which are called the normal equations. The solution to this system of equations is a = m ∑ m i=1 x iy i − ∑ m i=1 x ∑ m i i=1 y i m ( ∑ m i=1 x2 i ) − (∑ m i=1 x i) 2 ,b= Let’s consider a slightly more general question. Given data • Data: (x 1 ,y 1 ), (x 2 ,y 2 ), ..., (x m ,y m ) can we find the best polynomial approximation ∑ m i=1 x2 i ∑ m i=1 y i − ∑ m i=1 x iy i ∑ m i=1 x i m ( ∑ m i=1 x2 i ) − (∑ m i=1 x i) 2 . • Polynomial approximation: P n (x) =a n x n + a n−1 x n−1 + ... + a 0 where m will be usually much larger than n. Similar to the above discussion, we want to minimize ( ) 2 m∑ m∑ n∑ E = (y i − P n (x i )) 2 = y i − a j x j i i=1 with respect to the parameters a n ,a n−1 , ..., a 0 . For a minimum to occur, the necessary conditions are ( ∂E m∑ n∑ m ) ∑ =0⇒− y i x k i + a j x k+j i =0 ∂a k i=1 for k =0, 1, ..., n. (we are skipping some algebra here!) The normal equations for polynomial approximation are ( n∑ m ) ∑ m∑ a j = y i x k i (5.1) j=0 i=1 x k+j i for k =0, 1, ..., n. This is a system of (n +1) equations and (n +1) unknowns. We can write this system as a matrix equation i=1 j=0 i=1 i=1 j=0 Aa = b (5.2) where a is the unknown vector we are trying to find, and b is the constant vector ⎡ ⎤ ⎡ ∑ a m 0 i=1 y ⎤ i ∑ a a = 1 ⎢ ⎥ ⎣ . ⎦ ,b= m i=1 y ix i ⎢ ⎥ ⎣ . ⎦ ∑ a m n i=1 y ix n i and A is an (n +1)by (n +1)symmetric matrix with (kj)th entry A kj ,k =1, ..., n +1,j =
CHAPTER 5. APPROXIMATION THEORY 182 1, 2, ..., n +1given by A kj = m∑ i=1 x k+j−2 i . The equation Aa = b has a unique solution if the x i are distinct, and n ≤ m − 1. Solving this equation by computing the inverse matrix A −1 is not advisable, since there could be significant roundoff error. Next, we will write a Julia code for least squares approximation, and use the Julia function A\b to solve the matrix equation Aa = b for a. The\ operation in Julia uses numerically optimized matrix factorizations to solve the matrix equation. More details on this topic can be found in Heath [10] (Chapter 3). Julia code for least squares approximation In [1]: using PyPlot The function leastsqfit takes the x and y-coordinates of the data, and the degree of the polynomial we want to use, n, as inputs. It solves the matrix Equation (5.2). In [2]: function leastsqfit(x::Array,y::Array,n) m=length(x) # number of data points d=n+1 # number of coefficients to determine A=zeros(d,d) b=zeros(d,1) # the linear system we want to solve is Ax=b p=Array{Float64}(undef,2*n+1) for k in 1:d sum=0 for i in 1:m sum=sum+y[i]*x[i]^(k-1) end b[k]=sum end # p[i] below is the sum of the (i-1)th power of the x # coordinates p[1]=m for i in 2:2*n+1 sum=0
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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 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 128 end fo
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Page 181 and 182: Chapter 5 Approximation Theory 5.1
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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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