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Spline Interpolation Method

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

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Spline Interpolation Method

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  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 http://numericalmethods.eng.usf.edu

  2. Spline Method of Interpolationhttp://numericalmethods.eng.usf.edu

  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. http://numericalmethods.eng.usf.edu

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

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

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

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

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

  9. 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 http://numericalmethods.eng.usf.edu

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

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

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

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

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

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

  16. 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 Figure. Velocity vs. time data for the rocket example http://numericalmethods.eng.usf.edu

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

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

  19. Each Spline Goes Through Two Consecutive Data Points http://numericalmethods.eng.usf.edu

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

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

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

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

  24. Coefficients of Spline http://numericalmethods.eng.usf.edu

  25. Quadratic Spline InterpolationPart 2 of 2http://numericalmethods.eng.usf.edu http://numericalmethods.eng.usf.edu

  26. Final Solution http://numericalmethods.eng.usf.edu

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

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

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

  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_method.html

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

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