1. Interpolation

2. What is Interpolation ?
Given (x0,y0), (x1,y1), …… (xn,yn), find the value of ‘y’ at a value of ‘x’ that is not given.
Figure 1 Interpolation of discrete.

3. Interpolants Evaluate Differentiate, and Integrate
Polynomials are the most common choice of interpolants because they are easy to:
Evaluate
Differentiate, and
Integrate

4. 1. Direct method

5. Direct Method Given ‘n+1’ data points (x0,y0), (x1,y1),………….. (xn,yn),
pass a polynomial of order ‘n’ through the data as given
below:
where a0, a1,………………. an are real constants.
Set up ‘n+1’ equations to find ‘n+1’ constants.
To find the value ‘y’ at a given value of ‘x’, simply substitute the value of ‘x’ in the above polynomial.

6. Example 1
The upward velocity of a rocket is given as a function of time in Table 1.
Find the velocity at t=16 seconds using the direct method for linear interpolation.
Table 1 Velocity as a function of time. 10
227.04 15
362.78 20
517.35
22.5
602.97 30
901.67
Figure 2 Velocity vs. time data for the rocket example

7. Linear Interpolation Solving the above two equations gives, Hence
Figure 3 Linear interpolation.
Hence

8. Example 2
The upward velocity of a rocket is given as a function of time in Table 2.
Find the velocity at t=16 seconds using the direct method for quadratic interpolation.
Table 2 Velocity as a function of time. 10
227.04 15
362.78 20
517.35
22.5
602.97 30
901.67
Figure 5 Velocity vs. time data for the rocket example

9. Quadratic Interpolation
Figure 6 Quadratic interpolation.
Solving the above three equations gives

10. Quadratic Interpolation (cont.)
The absolute relative approximate error obtained between the results from the first and second order polynomial is

11. Example 3
The upward velocity of a rocket is given as a function of time in Table 3.
Find the velocity at t=16 seconds using the direct method for cubic interpolation.
Table 3 Velocity as a function of time. 10
227.04 15
362.78 20
517.35
22.5
602.97 30
901.67
Figure 6 Velocity vs. time data for the rocket example

12. Cubic Interpolation Figure 7 Cubic interpolation.

13. Cubic Interpolation (contd)
The absolute percentage relative approximate error between second and third order polynomial is

14. Comparison Table
Table 4 Comparison of different orders of the polynomial.

15. Distance from Velocity Profile
Find the distance covered by the rocket from t=11s to t=16s ?

16. Acceleration from Velocity Profile
Find the acceleration of the rocket at t=16s given that

17. 2. Spline Method

18. Why Splines ?

19. Figure : Higher order polynomial interpolation is a bad idea
Why Splines ?
Figure : Higher order polynomial interpolation is a bad idea

20. Linear Interpolation

21. Linear Interpolation (contd)

22. Figure. Velocity vs. time data
Example
The upward velocity of a rocket is given as a function of time in Table 1. Find the velocity at t=16 seconds using linear splines.
Table Velocity as a
function of time
(s)
(m/s) 10
227.04 15
362.78 20
517.35
22.5
602.97 30
901.67
Figure. Velocity vs. time data
for the rocket example

23. Linear Interpolation

24. Quadratic Interpolation

25. Quadratic Interpolation (contd)

26. Quadratic Splines (contd)

27. Quadratic Splines (contd)

28. Quadratic Splines (contd)

29. Quadratic Spline Example
The upward velocity of a rocket is given as a function of time. Using quadratic splines
Find the velocity at t=16 seconds
Find the acceleration at t=16 seconds
Find the distance covered between t=11 and t=16 seconds
Table Velocity as a
function of time
(s)
(m/s) 10
227.04 15
362.78 20
517.35
22.5
602.97 30
901.67
Figure. Velocity vs. time data
for the rocket example 29

30. Solution Let us set up the equations 30

31. Each Spline Goes Through Two Consecutive Data Points 31

32. Each Spline Goes Through Two Consecutive Data Points
v(t) s
m/s 10
227.04 15
362.78 20
517.35
22.5
602.97 30
901.67 32

33. Derivatives are Continuous at Interior Data Points 33

34. Derivatives are continuous at Interior Data Points
At t=10
At t=15
At t=20
At t=22.5 34

35. Last Equation

36. Final Set of Equations 36

37. Coefficients of Splineai bi ci 1
22.704 2
0.8888
4.928
88.88 3
−0.1356
35.66
−141.61 4
1.6048
−33.956
554.55 5
28.86
−152.13 37

38. Final Solution 38

39. Velocity at a Particular Point
a) Velocity at t=16

40. Acceleration from Velocity Profile
b) The quadratic spline valid at t=16 is given by

41. Distance from Velocity Profile
c) Find the distance covered by the rocket from t=11s to t=16s. 41

42. 3. Newton’s Divided Differences

43. Newton’s Divided Difference Method
Linear interpolation: Given pass a linear interpolant through the data
where

44. Figure 2: Velocity vs. time data
Example
The upward velocity of a rocket is given as a function of time in Table 1. Find the velocity at t=16 seconds using the Newton Divided Difference method for linear interpolation. t
v(t) s
m/s 10
227.04 15
362.78 20
517.35
22.5
602.97 30
901.67
Figure 2: Velocity vs. time data
for the rocket example
Table 1: Velocity as a
function of time

45. Linear Interpolation

46. Linear Interpolation (contd)

47. Quadratic Interpolation

48. Figure 2: Velocity vs. time data
Example
The upward velocity of a rocket is given as a function of time in Table 1. Find the velocity at t=16 seconds using the Newton Divided Difference method for quadratic interpolation. t
v(t) s
m/s 10
227.04 15
362.78 20
517.35
22.5
602.97 30
901.67
Figure 2: Velocity vs. time data
for the rocket example
Table 1: Velocity as a
function of time

49. Quadratic Interpolation (contd)

50. Quadratic Interpolation (contd)

51. Quadratic Interpolation (contd)

52. General Form
where
Rewriting

53. General Form

54. General form

55. Figure 2: Velocity vs. time data
Example
The upward velocity of a rocket is given as a function of time in Table 1. Find the velocity at t=16 seconds using the Newton Divided Difference method for cubic interpolation. t
v(t) s
m/s 10
227.04 15
362.78 20
517.35
22.5
602.97 30
901.67
Figure 2: Velocity vs. time data
for the rocket example
Table 1: Velocity as a
function of time

56. Example The velocity profile is chosen as
we need to choose four data points that are closest to

57. Example

58. Example

59. Comparison Table

60. Distance from Velocity Profile
Find the distance covered by the rocket from t=11s to
t=16s ?

61. Acceleration from Velocity Profile
Find the acceleration of the rocket at t=16s given that