Newton’s Method

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• Misc
  • Math colloquium
    • Wednesday, Nov. 5, 2:30 PM
    • Newton 201
    • Tony Macula, Geneseo
    • “An Example of Using Maple in Abstract Algebra”
    • Extra credit for going and writing summary
• Questions?
• Newton’s Method
  • Video lecture summary
    • How to find x such that f(x) = 0
    • Initial guess
      • Could come from plot
      • Or from knowledge of function
      • etc.
    • Process
      • x₁ = x₀ - f(x₀)/f′(x₀)
      • |ε_a| = 100 | (x₁ - x₀)/x₁|
      • Check |ε_a| ≤ |ε_s|
      • Repeat if not until |ε_a| ≤ |ε_s|
    • ε_a, ε_s are error estimate and tolerance, based on idea that small changes in x mean algorithm is getting close to a zero.
      • But that’s not necessarily true:
        
      • You can also run Newton’s method until the f(x) values are very small, although that isn’t guaranteed either.
      • Moral: Newton’s method works very well in most cases, but you need to be sensitive to how you detect convergence and it might not work at all for some starting values on some functions
  • Try it by hand, e.g., on f(x) = x² - 1
    • f′(x) = 2x
    • Guess x₀ = 0.5
    • x₁ = 0.5 - -0.75/1 = 1.25
    • x₂ = x₁ - f(x₁)/f′(x₁) = 1.25 - 0.5625/2.5 = 1.025
• Next
  • Refine pseudocode for Newton a little
  • Talk about numerical differentiation
  • Start coding Newton

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