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07 interpolation

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Transcript

  • 1. Interpolation/Curve Fitting
  • 2. Objectives
    • Understanding the difference between regression and interpolation
    • Knowing how to “best fit” a polynomial into a set of data
    • Knowing how to use a polynomial to interpolate data
  • 3. Measured Data
  • 4. Polynomial Fit!
  • 5. Line Fit!
  • 6. Which is better?
  • 7. Curve Fitting
    • If the data measured is of high accuracy and it is required to estimate the values of the function between the given points, then, polynomial interpolation is the best choice.
    • If the measurements are expected to be of low accuracy , or the number of measured points is too large, regression would be the best choice.
  • 8. Interpolation
  • 9. Why Interpolation?
    • When the accuracy of your measurements are ensured
    • When you have discrete values for a function (numerical solutions, digital systems, etc …)
  • 10. Acquired Data
  • 11. But, how to get the equation of a function that passes by all the data you have!
  • 12. Equation of a Line: Revision If you have two points
  • 13. Solving for the constants!
  • 14. What if I have more than two points?
    • We may fit a polynomial of order one less that the number of points we have. i.e. four points give third order polynomial.
  • 15. Third-Order Polynomial For the four points
  • 16. In Matrix Form Solve the above equation for the constants of the polynomial.
  • 17. Newton's Interpolation Polynomial
  • 18. Newton’s Method
    • In the previous procedure, we needed to solve a system of linear equations for the unknown constants.
    • This method suggests that we may just proceed with the values of x & y we have to get the constants without setting a set of equations
    • The method is similar to Taylor’s expansion without differentiation!
  • 19. Equation of a Line: Revision If you have two points
  • 20. For the two points
  • 21. For the three points
  • 22. Using a table y3 x3 y2 x2 y1 x1 yi xi
  • 23. In General
    • Newton’s Interpolation is performed for an n th order polynomial as follows
  • 24. Example
    • Find a 3 rd order polynomial to interpolate the function described by the given points
    16 2 5 1 2 0 1 -1 Y x
  • 25. Solution: System of equations
    • A third order polynomial is given by:
  • 26. In matrix form
  • 27. Newton’s Method
    • Newton’s methods defines the polynomial in the form:
  • 28. Newton’s Method 16 2 11 5 1 4 3 2 0 1 1 1 1 -1 Y x
  • 29. Newton’s Method
    • Finally:
  • 30. Advantage of Newton’s Method
    • The main advantage of Newton’s method is that you do not need to invert a matrix!
  • 31. Homework #6
    • Chapter 18, pp. 505-506, numbers: 18.1, 18.2, 18.3, 18.5.