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INTERPOLASI

INTERPOLASI. Direct Method of Interpolation. 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. Interpolants.

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INTERPOLASI

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  1. INTERPOLASI

  2. Direct Method of Interpolation

  3. 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.

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

  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. Figure 2 Velocity vs. time data for the rocket example

  7. Linear Interpolation Solving the above two equations gives, 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. Figure 5 Velocity vs. time data for the rocket example

  9. Quadratic Interpolation 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. 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. Lagrange Method of Interpolation

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

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

  20. Lagrangian Interpolation

  21. 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 Lagrangian method for linear interpolation. Table Velocity as a function of time Figure. Velocity vs. time data for the rocket example

  22. Linear Interpolation

  23. Linear Interpolation (contd)

  24. Quadratic Interpolation

  25. 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 Lagrangian method for quadratic interpolation. Table Velocity as a function of time Figure. Velocity vs. time data for the rocket example

  26. Quadratic Interpolation (contd)

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

  28. Cubic Interpolation

  29. 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 Lagrangian method for cubic interpolation. Table Velocity as a function of time Figure. Velocity vs. time data for the rocket example

  30. Cubic Interpolation (contd)

  31. Cubic Interpolation (contd) The absolute relative approximate error obtained between the results from the first and second order polynomial is

  32. Comparison Table

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

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

  35. Newton’s Divided Difference Method of Interpolation

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

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

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

  39. 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. Table. Velocity as a function of time Figure. Velocity vs. time data for the rocket example

  40. Linear Interpolation

  41. Linear Interpolation (contd)

  42. Quadratic Interpolation

  43. 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. Table. Velocity as a function of time Figure. Velocity vs. time data for the rocket example

  44. Quadratic Interpolation (contd)

  45. Quadratic Interpolation (contd)

  46. Quadratic Interpolation (contd)

  47. General Form where Rewriting

  48. General Form

  49. General form

  50. 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. Table. Velocity as a function of time Figure. Velocity vs. time data for the rocket example

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