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08 interpolation lagrange

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Transcript

  • 1. Interpolation Lagrange Interpolation Polynomial
  • 2. Lagrange Method
    • First, we learned that a polynomial can pass by the points by using a simple polynomial with (n-1) terms.
    • Then, we learned a way that “looks like” the Taylor expansion (Newton’s method)
  • 3. Lagrange Method (cont’d)
    • Now, we will use polynomials that are zero at all points except the one we are evaluating at but in an easier form!
  • 4. For the two points
  • 5. Two lines added!
  • 6. Homework
    • Show that the polynomial obtained by solving a set of equations is equivalent to that obtained by Lagrange method
  • 7. For the three points
  • 8. Similarly
  • 9. Finally
  • 10. Three parabolas added!!!
  • 11. Example
    • Find a 3 rd order polynomial to interpolate the function described by the given points using Lagrange’s method
    16 2 5 1 2 0 1 -1 Y x
  • 12. Solution
  • 13. Solution
  • 14. Solution
  • 15. Solution
  • 16. Homework #6
    • Solve the example presented in the previous lecture (Newton’s method) using Lagrange method
    • Chapter 18, pp. 505-506, numbers: 18.6, 18.7.