First Semester in Numerical Analysis with Julia, 2020a

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CHAPTER 3. INTERPOLATION 113 We seek a polynomial that fits the y and y ′ values, that is, we seek a polynomial H(x) such that H(x i )=y i and H ′ (x i )=y i, ′ i =0, 1, ..., n. This makes 2n +2equations, and if we let H(x) =a 0 + a 1 x + ... + a 2n+1 x 2n+1 , then there are 2n +2 unknowns, a 0 , ..., a 2n+1 , to solve for. The following theorem shows that there is a unique solution to this system of equations; a proof can be found in Burden, Faires, Burden [4]. Theorem 61. If f ∈ C 1 [a, b] and x 0 , ..., x n ∈ [a, b] are distinct, then there is a unique polynomial H 2n+1 (x), of degree at most 2n +1, agreeing with f and f ′ at x 0 , ..., x n . The polynomial can be written as: H 2n+1 (x) = n∑ y i h i (x)+ i=0 n∑ y i ′ ˜h i (x) i=0 where h i (x) =(1− 2(x − x i )l ′ i(x i )) (l i (x)) 2 ˜hi (x) =(x − x i )(l i (x)) 2 . Here l i (x) is the ith Lagrange basis function for the nodes x 0 , ..., x n , and l i(x) ′ is its derivative. H 2n+1 (x) is called the Hermite interpolating polynomial. The only difference between Hermite interpolation and polynomial interpolation is that in the former, we have the derivative information, which can go a long way in capturing the shape of the underlying function. Example 62. We want to interpolate the following data: x-coordinates : −1.5, 1.6, 4.7 y-coordinates :0.071, −0.029, −0.012. The underlying function the data comes from is cos x, but we pretend we do not know this. Figure (3.2) plots the underlying function, the data, and the polynomial interpolant for the data. Clearly, the polynomial interpolant does not come close to giving a good approximation to the underlying function cos x.
CHAPTER 3. INTERPOLATION 114 Figure 3.2: Polynomial interpolation Now let’s assume we know the derivative of the underlying function at these nodes: x-coordinates : −1.5, 1.6, 4.7 y-coordinates :0.071, −0.029, −0.012 y ′ -values :1, −1, 1. We then construct the Hermite interpolating polynomial, incorporating the derivative information. Figure (3.3) plots the Hermite interpolating polynomial, together with the polynomial interpolant, and the underlying function. It is visually difficult to separate the Hermite interpolating polynomial from the underlying function cos x in Figure (3.3). Going from polynomial interpolation to Hermite interpolation results in rather dramatic improvement in approximating the underlying function.
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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 16 Matrix o
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CHAPTER 1. INTRODUCTION 18 stdlib/v
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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 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 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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