First Semester in Numerical Analysis with Julia, 2020a

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CHAPTER 4. NUMERICAL QUADRATURE AND DIFFERENTIATION 155 interpolant for P (x), using the nodes as x 1 , ..., x n P (x) = n∑ P (x i )l i (x) = i=1 n∑ n∏ i=1 j=1,j≠i x − x j x i − x j P (x i ). There is no error term above because the error term depends on the nth derivative of P (x), but P (x) is a polynomial of degree less than n, so that derivative is zero. Integrate both sides to get ∫ 1 −1 P (x)dx = ∫ 1 −1 [ n∑ i=1 n∏ j=1,j≠i ] x − x j P (x i ) dx = x i − x j = = = ∫ 1 [ n∑ P (x i ) −1 i=1 [ n∑ ∫ 1 P (x i ) i=1 −1 [ n∑ ∫ 1 i=1 n∏ −1 j=1,j≠i n∑ w i P (x i ). i=1 n∏ j=1,j≠i n∏ j=1,j≠i x − x j x i − x j dx ] x − x j dx x i − x j ] x − x j dx x i − x j ] P (x i ) Therefore the theorem is correct for polynomials of degree less than n. Now let P (x) be a polynomial of degree greater than or equal to n, but less than or equal to 2n − 1. Divide P (x) by the Legendre polynomial L n (x) to get P (x) =Q(x)L n (x)+R(x). Note that P (x i )=Q(x i )L n (x i )+R(x i )=R(x i ) since L n (x i )=0, for i =1, 2, ..., n. Also observe the following facts: 1. ∫ 1 −1 Q(x)L n(x)dx =0, since Q(x) is a polynomial of degree less than n, and from the second property of Legendre polynomials. 2. ∫ 1 −1 R(x)dx = ∑ n i=1 w iR(x i ), since R(x) is a polynomial of degree less than n, and from the first part of the proof. Putting these facts together we get: ∫ 1 P (x)dx = ∫ 1 [Q(x)L n (x)+R(x)] dx = ∫ 1 −1 −1 −1 R(x)dx = n∑ w i R(x i )= i=1 n∑ w i P (x i ). i=1
CHAPTER 4. NUMERICAL QUADRATURE AND DIFFERENTIATION 156 Table 4.1 displays the roots of the Legendre polynomials L 2 ,L 3 ,L 4 ,L 5 and the corresponding weights. n Roots Weights 2 √3 1 =0.5773502692 1 − √ 1 3 = −0.5773502692 1 3 −( 3 5 )1/2 5 = −0.7745966692 =0.5555555556 9 8 0.0 =0.8888888889 9 ( 3 5 )1/2 5 =0.7745966692 =0.5555555556 9 4 0.8611363116 0.3478548451 0.3399810436 0.6521451549 -0.3399810436 0.6521451549 -0.8611363116 0.3478548451 5 0.9061798459 0.2369268850 0.5384693101 0.4786286705 0.0 0.5688888889 -0.5384693101 0.4786286705 -0.9061798459 0.2369268850 Table 4.1: Roots of Legendre polynomials L 2 through L 5
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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 5. APPROXIMATION THEORY 206
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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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