First Semester in Numerical Analysis with Julia, 2020a

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CHAPTER 2. SOLUTIONS OF EQUATIONS: ROOT-FINDING 77 2.6 Fixed-point iteration Many root-finding methods are based on the so-called fixed-point iteration; a method we discuss in this section. Definition 37. Anumberp is a fixed-point for a function g(x) if g(p) =p. We have two problems that are related to each other: • Fixed-point problem: Find p such that g(p) =p. • Root-finding problem: Find p such that f(p) =0. We can formulate a root-finding problem as a fixed-point problem, and vice versa. For example, assume we want to solve the root finding problem, f(p) =0. Define g(x) =x−f(x), and observe that if p is a fixed-point of g(x), that is, g(p) =p − f(p) =p, then p is a root of f(x). Here the function g is not unique: there are many ways one can represent the root-finding problem f(p) =0as a fixed-point problem, and as we will learn later, not all will be useful to us in developing fixed-point iteration algorithms. The next theorem answers the following questions: When does a function g have a fixedpoint? If it has a fixed-point, is it unique? Theorem 38. 1. If g is a continuous function on [a, b] and g(x) ∈ [a, b] for all x ∈ [a, b], then g has at least one fixed-point in [a, b]. 2. If, in addition, |g(x) − g(y)| ≤λ|x − y| for all x, y ∈ [a, b] where 0
CHAPTER 2. SOLUTIONS OF EQUATIONS: ROOT-FINDING 78 for all x, y ∈ [a, b]. The following theorem describes how we can find a fixed point. Theorem 40. If g is a continuous function on [a, b] satisfying the conditions 1. g(x) ∈ [a, b] for all x ∈ [a, b], 2. |g(x) − g(y)| ≤λ|x − y|, for x, y ∈ [a, b] where 0
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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 4. NUMERICAL QUADRATURE AND
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Chapter 5 Approximation Theory 5.1
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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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