1. http://numericalmethods.eng.usf.edu 1 Spline Interpolation Method Major: All Engineering Majors Authors: Autar Kaw, Jai Paul http://numericalmethods.eng.usf.edu Transforming Numerical Methods Education for STEM Undergraduates

2. Spline Method of Interpolation http://numericalmethods.eng.usf.edu http://numericalmethods.eng.usf.edu

3. 3 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.

4. http://numericalmethods.eng.usf.edu4 Interpolants Polynomials are the most common choice of interpolants because they are easy to: Evaluate Differentiate, and Integrate.

5. http://numericalmethods.eng.usf.edu5 Why Splines ?

6. http://numericalmethods.eng.usf.edu6 Why Splines ? Figure : Higher order polynomial interpolation is a bad idea

7. http://numericalmethods.eng.usf.edu7 Linear Interpolation

8. http://numericalmethods.eng.usf.edu8 Linear Interpolation (contd)

9. http://numericalmethods.eng.usf.edu9 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) 00 10227.04 15362.78 20517.35 22.5602.97 30901.67

10. http://numericalmethods.eng.usf.edu10 Linear Interpolation

11. http://numericalmethods.eng.usf.edu11 Quadratic Interpolation

12. http://numericalmethods.eng.usf.edu12 Quadratic Interpolation (contd)

13. http://numericalmethods.eng.usf.edu13 Quadratic Splines (contd)

14. http://numericalmethods.eng.usf.edu14 Quadratic Splines (contd)

15. http://numericalmethods.eng.usf.edu15 Quadratic Splines (contd)

16. http://numericalmethods.eng.usf.edu16 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) 00 10227.04 15362.78 20517.35 22.5602.97 30901.67

17. http://numericalmethods.eng.usf.edu17 Solution Let us set up the equations

18. http://numericalmethods.eng.usf.edu18 Each Spline Goes Through Two Consecutive Data Points

19. http://numericalmethods.eng.usf.edu19 tv(t) sm/s 00 10227.04 15362.78 20517.35 22.5602.97 30901.67 Each Spline Goes Through Two Consecutive Data Points

20. http://numericalmethods.eng.usf.edu20 Derivatives are Continuous at Interior Data Points

21. http://numericalmethods.eng.usf.edu21 Derivatives are continuous at Interior Data Points At t=10 At t=15 At t=20 At t=22.5

22. http://numericalmethods.eng.usf.edu22 Last Equation

23. http://numericalmethods.eng.usf.edu23 Final Set of Equations

24. http://numericalmethods.eng.usf.edu24 Coefficients of Spline iaiai bibi cici 1022.7040 20.88884.92888.88 3−0.135635.66−141.61 41.6048−33.956554.55 50.2088928.86−152.13

25. http://numericalmethods.eng.usf.edu25 Quadratic Spline Interpolation Part 2 of 2 http://numericalmethods.eng.usf.edu http://numericalmethods.eng.usf.edu

26. 26 Final Solution

27. http://numericalmethods.eng.usf.edu27 Velocity at a Particular Point a) Velocity at t=16

28. http://numericalmethods.eng.usf.edu28 Acceleration from Velocity Profile b) The quadratic spline valid at t=16 is given by

29. http://numericalmethods.eng.usf.edu29 Distance from Velocity Profile c) Find the distance covered by the rocket from t=11s to t=16s.

30. Additional Resources For all resources on this topic such as digital audiovisual lectures, primers, textbook chapters, multiple-choice tests, worksheets in MATLAB, MATHEMATICA, MathCad and MAPLE, blogs, related physical problems, please visit http://numericalmethods.eng.usf.edu/topics/spline_met hod.html

31. THE END http://numericalmethods.eng.usf.edu