Interpolation Topic: Spline Interpolation Method Major: General
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 ...
Interpolants <ul><li>Polynomials are the most common choice of interpolants because they are easy to: </li></ul><ul><ul><l...
Why Splines ?
Why Splines ? Figure : Higher order polynomial interpolation is a bad idea
Linear Interpolation
Linear Interpolation (contd)
Example <ul><li>The upward velocity of a rocket is given as a function of time in Table 1. Find the velocity at t=16 secon...
Linear Interpolation
Quadratic Interpolation
Quadratic Interpolation (contd)
Quadratic Splines (contd)
Quadratic Splines (contd)
Quadratic Splines (contd)
Example <ul><li>The upward velocity of a rocket is given as a function of time in Table 1. Find the velocity at t=16 secon...
Solution
Solution (contd)
Solution (contd)
Solution (contd)
Solution (contd)
Solution (contd)
Distance from Velocity Profile <ul><li>Find the distance covered by the rocket from t=11s to </li></ul><ul><li>t=16s </li>...
Acceleration from Velocity Profile , Find the acceleration of the rocket at t=16s. The quadratic spline valid at t=16 is g...
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Spline Interpolation

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Basic spline interpolation

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Spline Interpolation

  1. 1. Interpolation Topic: Spline Interpolation Method Major: General
  2. 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.
  3. 3. Interpolants <ul><li>Polynomials are the most common choice of interpolants because they are easy to: </li></ul><ul><ul><li>Evaluate </li></ul></ul><ul><ul><li>Differentiate, and </li></ul></ul><ul><ul><li>Integrate . </li></ul></ul>
  4. 4. Why Splines ?
  5. 5. Why Splines ? Figure : Higher order polynomial interpolation is a bad idea
  6. 6. Linear Interpolation
  7. 7. Linear Interpolation (contd)
  8. 8. Example <ul><li>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. </li></ul>Table : Velocity as a function of time Figure : Velocity vs. time data for the rocket example 901.67 30 602.97 22.5 517.35 20 362.78 15 227.04 10 0 0 m/s s v(t) t
  9. 9. Linear Interpolation
  10. 10. Quadratic Interpolation
  11. 11. Quadratic Interpolation (contd)
  12. 12. Quadratic Splines (contd)
  13. 13. Quadratic Splines (contd)
  14. 14. Quadratic Splines (contd)
  15. 15. Example <ul><li>The upward velocity of a rocket is given as a function of time in Table 1. Find the velocity at t=16 seconds using quadratic splines. </li></ul>Table : Velocity as a function of time Figure : Velocity vs. time data for the rocket example 901.67 30 602.97 22.5 517.35 20 362.78 15 227.04 10 0 0 m/s s v(t) t
  16. 16. Solution
  17. 17. Solution (contd)
  18. 18. Solution (contd)
  19. 19. Solution (contd)
  20. 20. Solution (contd)
  21. 21. Solution (contd)
  22. 22. Distance from Velocity Profile <ul><li>Find the distance covered by the rocket from t=11s to </li></ul><ul><li>t=16s </li></ul>
  23. 23. Acceleration from Velocity Profile , Find the acceleration of the rocket at t=16s. The quadratic spline valid at t=16 is given by

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