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EE 3561 : Computational Methods Unit 5 : Interpolation

EE 3561 : Computational Methods Unit 5 : Interpolation. Dr. Mujahed AlDhaifallah (Term 342) Read Chapter 18, Sections 1-5. Introduction to Interpolation. Introduction Interpolation problem Existence and uniqueness Linear and Quadratic interpolation

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EE 3561 : Computational Methods Unit 5 : Interpolation

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  1. EE 3561 : Computational MethodsUnit 5:Interpolation Dr. Mujahed AlDhaifallah (Term 342) Read Chapter 18, Sections 1-5 (c)AL-DHAIFALLAH1435

  2. Introduction to Interpolation Introduction Interpolation problem Existence and uniqueness Linear and Quadratic interpolation Newton’s Divided Difference method Properties of Divided Differences (c)AL-DHAIFALLAH1435

  3. Introduction Interpolation was used for long time to provide an estimate of a tabulated function at values that are not available in the table. What is sin (0.15)? Using Linear Interpolation sin (0.15) ≈ 0.1493 True value( 4 decimal digits) sin (0.15)= 0.1494 (c)AL-DHAIFALLAH1435

  4. The interpolation Problem Given a set of n+1 points, find an nth order polynomial that passes through all points (c)AL-DHAIFALLAH1435

  5. Example An experiment is used to determine the viscosity of water as a function of temperature. The following table is generated Problem: Estimate the viscosity when the temperature is 8 degrees. (c)AL-DHAIFALLAH1435

  6. Interpolation Problem Find a polynomial that fit the data points exactly. Linear Interpolation V(T)= 1.73 − 0.0422 T V(8)= 1.3924 (c)AL-DHAIFALLAH1435

  7. Existence and Uniqueness Given a set of n+1 points Assumption are distinct Theorem: There is a unique polynomial fn(x) of order ≤ n such that (c)AL-DHAIFALLAH1435

  8. Linear Interpolation Given any two points, there is one polynomial of order ≤ 1 that passes through the two points. Quadratic Interpolation Given any three points there is one polynomial of order ≤ 2 that passes through the three points. Examples of Polynomial Interpolation (c)AL-DHAIFALLAH1435

  9. Linear Interpolation Given any two points, The line that interpolates the two points is Example : find a polynomial that interpolates (1,2) and (2,4) (c)AL-DHAIFALLAH1435

  10. Given any three points, The polynomial that interpolates the three points is Quadratic Interpolation (c)AL-DHAIFALLAH1435

  11. Given any n+1 points, The polynomial that interpolates all points is General nth order Interpolation (c)AL-DHAIFALLAH1435

  12. Divided Differences (c)AL-DHAIFALLAH1435

  13. Divided Difference Table (c)AL-DHAIFALLAH1435

  14. Divided Difference Table Entries of the divided difference table are obtained from the data table using simple operations (c)AL-DHAIFALLAH1435

  15. Divided Difference Table The first two column of the table are the data columns Third column : first order differences Fourth column: Second order differences (c)AL-DHAIFALLAH1435

  16. Divided Difference Table (c)AL-DHAIFALLAH1435

  17. Divided Difference Table (c)AL-DHAIFALLAH1435

  18. Divided Difference Table (c)AL-DHAIFALLAH1435

  19. Divided Difference Table f2(x)= F[x0]+F[x0,x1] (x-x0)+F[x0,x1,x2] (x-x0)(x-x1) (c)AL-DHAIFALLAH1435

  20. Two Examples Obtain the interpolating polynomials for the two examples What do you observe? (c)AL-DHAIFALLAH1435

  21. Two Examples Ordering the points should not affect the interpolating polynomial (c)AL-DHAIFALLAH1435

  22. Properties of divided Difference Ordering the points should not affect the divided difference (c)AL-DHAIFALLAH1435

  23. Example • Find a polynomial to interpolate the data. (c)AL-DHAIFALLAH1435

  24. Example (c)AL-DHAIFALLAH1435

  25. Summary (c)AL-DHAIFALLAH1435

  26. Lagrange Interpolation Dr. Mujahed Al-Dhaifallah (Term 342) (c)AL-DHAIFALLAH1435

  27. The interpolation Problem Given a set of n+1 points, find an nth order polynomial that passes through all points (c)AL-DHAIFALLAH1435

  28. Lagrange Interpolation Problem: Given Find the polynomial of least order such that Lagrange Interpolation Formula (c)AL-DHAIFALLAH1435

  29. Lagrange Interpolation (c)AL-DHAIFALLAH1435

  30. Lagrange Interpolation Example (c)AL-DHAIFALLAH1435

  31. Example find a polynomial to interpolate both divided difference interpolation method and Lagrange interpolation method must give the same answer (c)AL-DHAIFALLAH1435

  32. (c)AL-DHAIFALLAH1435

  33. Interpolating Polynomial (c)AL-DHAIFALLAH1435

  34. Interpolating Polynomial Using Lagrange Interpolation method (c)AL-DHAIFALLAH1435

  35. Inverse interpolation Error in polynomial interpolation Dr. Mujahed Al-Dhaifallah (Term 342) (c)AL-DHAIFALLAH1435

  36. Inverse Interpolation One approach: Use polynomial interpolation to obtain fn(x) to interpolate the data then use Newton’s method to find a solution to (c)AL-DHAIFALLAH1435

  37. Inverse InterpolationAlternative Approach Inverse interpolation: 1. Exchange the roles of x and y 2. Perform polynomial Interpolation on the new table. 3. Evaluate (c)AL-DHAIFALLAH1435

  38. Inverse Interpolation x y x y (c)AL-DHAIFALLAH1435

  39. Inverse Interpolation Question: What is the limitation of inverse interpolation • The original function has an inverse • y1,y2,…,yn must be distinct. (c)AL-DHAIFALLAH1435

  40. Inverse Interpolation Example (c)AL-DHAIFALLAH1435

  41. Errors in polynomial Interpolation • Polynomial interpolations may lead to large error (especially when high order polynomials are used) BE CAREFUL • When an nth order interpolating polynomial is used, the error is related to the (n+1) th order derivative (c)AL-DHAIFALLAH1435

  42. 10 th order Polynomial Interpolation (c)AL-DHAIFALLAH1435

  43. Error in polynomial interpolation Dr. Mujahed Al-Dhaifallah (Term 342) (c)AL-DHAIFALLAH1435

  44. Errors in polynomial InterpolationTheorem (c)AL-DHAIFALLAH1435

  45. Example 1 (c)AL-DHAIFALLAH1435

  46. Interpolation Error Not useful. Why? (c)AL-DHAIFALLAH1435

  47. Approximation of the Interpolation Error (c)AL-DHAIFALLAH1435

  48. Example • Given the following table of data (c)AL-DHAIFALLAH1435

  49. (c)AL-DHAIFALLAH1435

  50. Divided Difference Theorem (c)AL-DHAIFALLAH1435

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