First Semester in Numerical Analysis with Julia, 2020a

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CHAPTER 2. SOLUTIONS OF EQUATIONS: ROOT-FINDING 75 3. Muller’s method converges for a variety of starting values even though pathological examples that do not yield convergence can be found (for example, when the three starting values fall on a line). Julia code for Muller’s method The following Julia code takes initial guesses p 0 ,p 1 ,p 2 (written as pzero, pone, ptwo in the code), computes the coefficients a, b, c from Equation (2.14), and sets the root p 3 to p. Itthen updates the three initial guesses as the last three iterates, and continues until the stopping criterion is satisfied. We need to compute the square root, and the absolute value, of possibly complex numbers in Equations (2.15) and(2.16). The Julia function for the square root of a possibly complex number z is Complex(z) 0.5 , and its absolute value is abs(z). In [1]: function muller(f::Function,pzero,pone,ptwo,eps,N) n=1 p=0 while n
CHAPTER 2. SOLUTIONS OF EQUATIONS: ROOT-FINDING 76 end is $n") end pzero=pone pone=ptwo ptwo=p n=n+1 end y=f(p) println("Method did not converge. The last iteration gives $p with function value $y") Out[1]: muller (generic function with 1 method) The polynomial x 5 +2x 3 − 5x − 2 has three real roots, and two complex roots that are conjugates. Let’s find them all, by experimenting with various initial guesses. In [2]: muller(x->x^5+2x^3-5x-2,0.5,1.0,1.5,10^(-5.),10) p is 1.3196411677283386 + 0.0im and the iteration number is 4 In [3]: muller(x->x^5+2x^3-5x-2,0.5,0,-0.1,10^(-5.),10) p is -0.43641313299908585 + 0.0im and the iteration number is 5 In [4]: muller(x->x^5+2x^3-5x-2,0,-0.1,-1,10^(-5.),10) p is -1.0 + 0.0im and the iteration number is 1 In [5]: muller(x->x^5+2x^3-5x-2,5,10,15,10^(-5.),20) p is 0.05838598289491982 - 1.8626227582154478im and the iteration number is 18
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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 3. INTERPOLATION 126 In [2]
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CHAPTER 3. INTERPOLATION 140 Recrea
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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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