First Semester in Numerical Analysis with Julia, 2020a

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CHAPTER 3. INTERPOLATION 127 Exercise 3.4-2: The following is a clamped cubic spline for a function f defined on [1, 3]: ⎧ ⎨s 0 (x) =(x − 1) + (x − 1) 2 − (x − 1) 3 , if 1 ≤ x
CHAPTER 3. INTERPOLATION 128 end for i in 2:n u[i]=3*(a[i+1]-a[i])/h[i]-3*(a[i]-a[i-1])/h[i-1] end s=Array{Float64}(undef,m) z=Array{Float64}(undef,m) t=Array{Float64}(undef,n) s[1]=1 z[1]=0 t[1]=0 for i in 2:n s[i]=2*(x[i+1]-x[i-1])-h[i-1]*t[i-1] t[i]=h[i]/s[i] z[i]=(u[i]-h[i-1]*z[i-1])/s[i] end s[m]=1 z[m]=0 c[m]=0 for i in reverse(1:n) c[i]=z[i]-t[i]*c[i+1] b[i]=(a[i+1]-a[i])/h[i]-h[i]*(c[i+1]+2*c[i])/3 d[i]=(c[i+1]-c[i])/(3*h[i]) end Out[2]: CubicNatural (generic function with 1 method) Once the matrix equation is solved, and the coefficients of the cubic polynomials are computed by CubicNatural, the next step is to evaluate the spline at a given value. This is done by the following function CubicNaturalEval. The inputs are the value at which the spline is evaluated, w, and the x-coordinates of the data. The function first finds the interval [x i ,x i+1 ],i =1, ..., m − 1, wbelongs to, and then evaluates the spline at w using the corresponding cubic polynomial. Note that in the previous section the data and the intervals are counted starting at 0, so the formulas used in the Julia code do not exactly match the formulas given earlier. In [3]: function CubicNaturalEval(w,x::Array)
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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 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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