Chapter 6 I Nonlinear Equations

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Chapter 6 (I)Nonlinear Equations
Prof. Chuan-Ming Liu MCSE Lab, NTUT TAWIAN
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Outline

• The Bisection Method
• Newtons Method
• Secant Method
• Fixed-point Iteration

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Nonlinear Equations

• Root finding Problem
• For an equation f(x) 0, the root-finding
  problem is to find values of the variables x that
  satisfies f(x) 0.
• A solution to this problem is called a zero of f
  or a root of f(x) 0.

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The Bisection Method

• Intermediate Value Theorem

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Bisection Method
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Bisection Method
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Stopping Condition
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Bisection Method
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Proof
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Example
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Newtons Method
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Newtons Method
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Newtons Method
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Newtons Method
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Example
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Newtons Method
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Newtons Method
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Secant Method
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Secant Method
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Error Analysis for Iterative Methods
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Error Analysis for Iterative Methods

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Fixed-Point Iteration

• Determine the solution to some function g of the
  form g(x) x.
• An solution to such an equation is said to be a
  fixed-point of the function g.

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Fixed-Point Iteration

• Note
• If a fixed point can be found for any given g,
  then every root finding problem could also be
  solved.
• Consider f(x) 0, then x-f(x)x. Let
  g(x)x-f(x). The solutions to f(x)0. Correspond
  to the fixed point s of g(x)x when g(x)x-f(x).
• g(x) x ltgt f(x)0

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Example
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Theorem
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Proof of part A
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Proof of part A
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Proof of part B
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Proof of part B
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Note