First Semester in Numerical Analysis with Julia, 2020a

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INDEX 219 stopping criteria, 53 Julia abs, 75 Base.MathConstants, 208 bitstring, 32 Complex, 75 Distributions, 68 dot, 193 factorial, 35 global, 84 JuliaDB, 190 LatexStrings, 108 LinearAlgebra, 191 reverse, 105 standard normal distribution, 68 Lagrange interpolation, 91 Least squares, 178 continuous, 198 discrete, 178 Julia code Chebyshev, 212 discrete, 182 Legendre, 208 linearizing, 195 non-polynomials, 188 normal equations, continuous, 199 normal equations, discrete, 181 orthogonal polynomials, 200 Legendre polynomials, 202 Julia code, 207 Linear convergence, 54 Machine epsilon, 40 alternative definition, 40 Mean value theorem, 7 Midpoint rule, 145 Monte Carlo integration, 163 Muller’s method, 73 convergence rate, 74 Julia code, 75 Multiple integrals, 159 Newton interpolation Julia code, 104 Newton’s method, 60 error theorem, 64 Julia code, 62 quadratic convergence, 65 Newton-Cotes, 141 closed, 144 Julia code, 149 open, 145 Normal equations continuous, 199 discrete, 181 Numerical differentiation, 168 three-point endpoint, 171 three-point midpoint, 171 backward-difference, 170 forward-difference, 170 noisy data, 172 roundoff, 173 second derivative, 172 Numerical quadrature, 141 midpoint rule, 145 Monte Carlo, 163 multiple integrals, 159 Newton-Cotes, 141 Simpson’s rule, 143 trapezoidal rule, 142 Orthogonal functions, 200 Orthogonal polynomials, 200
INDEX 220 Chebyshev, 203 Julia code, 207 Legendre, 202 Overflow, 33 Polynomial interpolation, 89 error theorem, 97 Existence and uniqueness, 96 high degree, 107 Lagrange basis functions, 91 monomial basis functions, 90 Newton basis functions, 94 Polynomials nested form, 47 standard form, 48 Power-law, 197 Propagation of error, 41 adding numbers, 44 alternating sum, 45 cancellation of leading digits, 41 division by a small number, 42 quadratic formula, 43 sample variance, 45 Quadratic convergence, 54 Significant digits, 38 Simpson’s rule, 143 Spline interpolation, 120 clamped cubic, 124 cubic, 122 Julia code, 127 linear, 120 natural cubic, 124 quadratic, 121 Runge’s function, 131 Stirling’s formula, 158 Subnormal numbers, 32 Superlinear convergence, 54 Taylor’s theorem, 7 Trapezoidal rule, 142 Two’s complement, 33 Underflow, 33 van der Monde matrix, 91 Weight function, 201 Weighted mean value theorem for integrals, 142 Relative error, 38 Representation of integers, 33 Rhind papyrus, 50 Rolle’s theorem, 112 generalized, 112 Root-finding, 50 Rounding, 38 Runge’s function, 107 Secant method, 70 error theorem, 71 Julia code, 71
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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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CHAPTER 3. INTERPOLATION 136 Arya a
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CHAPTER 3. INTERPOLATION 138 This l
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CHAPTER 3. INTERPOLATION 140 Recrea
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CHAPTER 4. NUMERICAL QUADRATURE AND
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Page 221: Index Absolute error, 38 Beasley-Sp
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First Semester in Numerical Analysis with Julia, 2020a

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