1 Interpolation

2 What is Interpolation ? Given (x 0,y 0 ), (x 1,y 1 ), …… (x n,y n ), find the value of ‘y’ at a value of ‘x’ that is not given. Figure 1 Interpolation of discrete.

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

1. Direct method 4

5 Direct Method Given ‘n+1’ data points (x 0,y 0 ), (x 1,y 1 ),………….. (x n,y n ), pass a polynomial of order ‘n’ through the data as given below: where a 0, a 1,………………. a n 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. Figure 2 Velocity vs. time data for the rocket example

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

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. Figure 5 Velocity vs. time data for the rocket example

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

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. 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

2. Spline Method 17

18 Why Splines ?

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

20 Linear Interpolation

21 Linear Interpolation (contd)

22 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 Figure. Velocity vs. time data for the rocket example (s) (m/s)

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 a) Find the velocity at t=16 seconds b) Find the acceleration at t=16 seconds c) Find the distance covered between t=11 and t=16 seconds Table Velocity as a function of time Figure. Velocity vs. time data for the rocket example (s) (m/s)

30 Solution Let us set up the equations

31 Each Spline Goes Through Two Consecutive Data Points

32 tv(t) sm/s Each Spline Goes Through Two Consecutive Data Points

33 Derivatives are Continuous at Interior Data Points

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

35 Last Equation

36 Final Set of Equations

37 Coefficients of Spline iaiai bibi cici − − − −152.13

38 Final Solution

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.

3. Newton’s Divided Differences 42

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

44 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. tv(t) sm/s Table 1: Velocity as a function of time Figure 2: Velocity vs. time data for the rocket example

45 Linear Interpolation

46 Linear Interpolation (contd)

47 Quadratic Interpolation

48 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. tv(t) sm/s Table 1: Velocity as a function of time Figure 2: Velocity vs. time data for the rocket example

49 Quadratic Interpolation (contd)

50 Quadratic Interpolation (contd)

51 Quadratic Interpolation (contd)

52 General Form where Rewriting

53 General Form

54 General form

55 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. tv(t) sm/s Table 1: Velocity as a function of time Figure 2: Velocity vs. time data for the rocket example

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