1. EE3561_Unit 5(c)AL-DHAIFALLAH14361 EE 3561 : Computational Methods Unit 5 : Interpolation Dr. Mujahed AlDhaifallah (Term 351) Read Chapter 18, Sections 1-5

2. EE3561_Unit 5(c)AL-DHAIFALLAH14362 Introduction to Interpolation Introduction Interpolation problem Existence and uniqueness Linear and Quadratic interpolation Newton’s Divided Difference method Properties of Divided Differences

3. EE3561_Unit 5(c)AL-DHAIFALLAH14363 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)? xsin(x) 00.0000 0.10.0998 0.20.1987 0.30.2955 0.40.3894 Using Linear Interpolation sin (0.15) ≈ 0.1493 True value( 4 decimal digits) sin (0.15)= 0.1494

4. EE3561_Unit 5(c)AL-DHAIFALLAH14364 The interpolation Problem Given a set of n+1 points, find an n th order polynomial that passes through all points

5. EE3561_Unit 5(c)AL-DHAIFALLAH14365 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. Temperature (degree) Viscosity 01.792 51.519 101.308 151.140

6. EE3561_Unit 5(c)AL-DHAIFALLAH14366 Interpolation Problem Find a polynomial that fit the data points exactly. Linear Interpolation V(T)= 1.73 − 0.0422 T V(8)= 1.3924

7. EE3561_Unit 5(c)AL-DHAIFALLAH14367 Existence and Uniqueness Given a set of n+1 points Assumption are distinct Theorem: There is a unique polynomial f n (x) of order ≤ n such that

8. EE3561_Unit 5(c)AL-DHAIFALLAH14368 Examples of Polynomial Interpolation 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.

9. EE3561_Unit 5(c)AL-DHAIFALLAH14369 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)

10. EE3561_Unit 5(c)AL-DHAIFALLAH143610 Quadratic Interpolation  Given any three points,  The polynomial that interpolates the three points is

11. EE3561_Unit 5(c)AL-DHAIFALLAH143611 General n th order Interpolation Given any n+1 points, The polynomial that interpolates all points is

12. EE3561_Unit 5(c)AL-DHAIFALLAH143612 Divided Differences

13. EE3561_Unit 5(c)AL-DHAIFALLAH143613 Divided Difference Table xF[ ]F[, ]F[,, ]F[,,,] x0x0 F[x0]F[x 0,x 1 ]F[x 0,x 1, x 2 ]F[x 0,x 1,x 2,x 3 ] x1x1 F[x 1 ]F[x 1,x 2 ]F[x 1,x 2,x 3 ] x2x2 F[x 2 ]F[x 2,x 3 ] x3x3 F[x 3 ]

14. EE3561_Unit 5(c)AL-DHAIFALLAH143614 Divided Difference Table f(x i ) 0-5 1-3 -15 xF[ ]F[, ]F[,, ] 0-52-4 1-36 -15 Entries of the divided difference table are obtained from the data table using simple operations

15. EE3561_Unit 5(c)AL-DHAIFALLAH143615 Divided Difference Table f(x i ) 0-5 1-3 -15 xF[ ]F[, ]F[,, ] 0-52-4 1-36 -15 The first two column of the table are the data columns Third column : first order differences Fourth column: Second order differences

16. EE3561_Unit 5(c)AL-DHAIFALLAH143616 Divided Difference Table 0-5 1-3 -15 xF[ ]F[, ]F[,, ] 0-52-4 1-36 -15

17. EE3561_Unit 5(c)AL-DHAIFALLAH143617 Divided Difference Table 0-5 1-3 -15 xF[ ]F[, ]F[,, ] 0-52-4 1-36 -15

18. EE3561_Unit 5(c)AL-DHAIFALLAH143618 Divided Difference Table 0-5 1-3 -15 xF[ ]F[, ]F[,, ] 0-52-4 1-36 -15

19. EE3561_Unit 5(c)AL-DHAIFALLAH143619 Divided Difference Table 0-5 1-3 -15 xF[ ]F[, ]F[,, ] 0-52-4 1-36 -15 f 2 (x)= F[x 0 ]+F[x 0,x 1 ] (x-x 0 )+F[x 0,x 1,x 2 ] (x-x 0 )(x-x 1 )

20. EE3561_Unit 5(c)AL-DHAIFALLAH143620 Two Examples xy 10 23 38 Obtain the interpolating polynomials for the two examples xy 23 10 38 What do you observe?

21. EE3561_Unit 5(c)AL-DHAIFALLAH143621 Two Examples xY 1031 235 38 xY 2331 104 38 Ordering the points should not affect the interpolating polynomial

22. EE3561_Unit 5(c)AL-DHAIFALLAH143622 Properties of divided Difference Ordering the points should not affect the divided difference

23. EE3561_Unit 5(c)AL-DHAIFALLAH143623 Example  Find a polynomial to interpolate the data. xf(x) 23 45 51 66 79

24. EE3561_Unit 5(c)AL-DHAIFALLAH143624 Example xf(x)f[, ]f[,, ]f[,,, ]f[,,,, ] 23 1-1.6667 1.5417-0.6750 45-4 4.5-1.8333 51 5 66 3 79

25. EE3561_Unit 5(c)AL-DHAIFALLAH143625 Summary

26. EE3561_Unit 5(c)AL-DHAIFALLAH143626 Inverse interpolation Error in polynomial interpolation Dr. Mujahed Al-Dhaifallah (Term 351)

27. EE3561_Unit 5(c)AL-DHAIFALLAH143627 Inverse Interpolation …. One approach: Use polynomial interpolation to obtain f n (x) to interpolate the data then use Newton’s method to find a solution to

28. EE3561_Unit 5(c)AL-DHAIFALLAH143628 Inverse Interpolation Alternative Approach …. Inverse interpolation: 1. Exchange the roles of x and y 2. Perform polynomial Interpolation on the new table. 3. Evaluate ….

29. EE3561_Unit 5(c)AL-DHAIFALLAH143629 Inverse Interpolation x y y x

30. EE3561_Unit 5(c)AL-DHAIFALLAH143630 Inverse Interpolation Question: What is the limitation of inverse interpolation The original function has an inverse y 1,y 2,…,y n must be distinct.

31. EE3561_Unit 5(c)AL-DHAIFALLAH143631 Inverse Interpolation Example x123 y3.22.0 1.6 3.21-.83331.0416 2.02-2.5 1.63

32. EE3561_Unit 5(c)AL-DHAIFALLAH143632 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

33. EE3561_Unit 5(c)AL-DHAIFALLAH143633 10 th order Polynomial Interpolation

34. EE3561_Unit 5(c)AL-DHAIFALLAH143634 Error in polynomial interpolation Dr. Mujahed Al-Dhaifallah (Term 351)

35. EE3561_Unit 5(c)AL-DHAIFALLAH143635 Errors in polynomial Interpolation Theorem

36. EE3561_Unit 5(c)AL-DHAIFALLAH143636 Example 1

37. EE3561_Unit 5(c)AL-DHAIFALLAH143637 Interpolation Error Not useful. Why?

38. EE3561_Unit 5(c)AL-DHAIFALLAH143638 Approximation of the Interpolation Error

39. Example  Given the following table of data EE3561_Unit 5(c)AL-DHAIFALLAH143639

40. EE3561_Unit 5(c)AL-DHAIFALLAH143640

41. EE3561_Unit 5(c)AL-DHAIFALLAH143641 Divided Difference Theorem 14100 2510 361 47

42. EE3561_Unit 5(c)AL-DHAIFALLAH143642 Summary  The interpolating polynomial is unique  Different methods can be used to obtain it Newton’s Divided difference Lagrange interpolation others  Polynomial interpolation can be sensitive to data.  BE CAREFUL when high order polynomials are used