First Semester in Numerical Analysis with Julia, 2020a

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CHAPTER 3. INTERPOLATION 125 There are eight equations and eight unknowns. However, a 0 = c 0 =0, so that reduces the number of equations and unknowns to six. We rewrite the equations below, substituting a 0 = c 0 =0, and simplifying when possible: b 0 + d 0 =1 a 1 + b 1 + c 1 + d 1 =1 a 1 +2b 1 +4c 1 +8d 1 =0 b 0 +3d 0 = b 1 +2c 1 +3d 1 3d 0 = c 1 +3d 1 c 1 +6d 1 =0 We will use Julia to solve this system of equations. To do that, we first rewrite the system of equations as a matrix equation Ax = v where ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 1 1 0 0 0 0 b 0 1 0 0 1 1 1 1 d 0 1 0 0 1 2 4 8 a 1 0 A = ,x= ,v = . 1 3 0 −1 −2 −3 b 1 0 ⎢ ⎣0 3 0 0 −1 −3 ⎥ ⎢ ⎦ ⎣c ⎥ ⎢ 1 ⎦ ⎣0 ⎥ ⎦ 0 0 0 0 1 6 d 1 0 We enter the matrices A, v in Julia and solve the equation Ax = v using the command A\v. In [1]: A=[1 1 0 0 0 0 ; 0 0 1 1 1 1; 0 0 1 2 4 8; 1 3 0 -1 -2 -3; 0 3 0 0 -1 -3; 0 0 0 0 1 6] Out[1]: 6×6 Array{Int64,2}: 1 1 0 0 0 0 0 0 1 1 1 1 0 0 1 2 4 8 1 3 0 -1 -2 -3 0 3 0 0 -1 -3 0 0 0 0 1 6
CHAPTER 3. INTERPOLATION 126 In [2]: v=[1 1 0 0 0 0]' Out[2]: 6×1 Array{Int64,2}: 1 1 0 0 0 0 In [3]: A\v Out[3]: 6×1 Array{Float64,2}: 1.5 -0.5 -1.0 4.5 -3.0 0.5 Therefore, the polynomials are: S 0 (x) =1.5x − 0.5x 3 S 1 (x) =−1+4.5x − 3x 2 +0.5x 3 Solving the equations of a spline even for three data points can be tedious. Fortunately, there is a general approach to solving the equations for natural and clamped splines, for any number of data points. We will use this approach when we write Julia codes for splines next. Exercise 3.4-1: Find the natural cubic spline interpolant for the following data: x -1 0 1 y 1 2 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 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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