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Interpolation

Interpolation. Reading Between the Lines. 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 Interpolation of discrete data. APPLIED PROBLEMS. Fly Rocket Fly, Fly Rocket Fly.

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Interpolation

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  1. Interpolation Reading Between the Lines http://nm.mathforcollege.com

  2. 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 Interpolation of discrete data. http://nm.mathforcollege.com

  3. APPLIED PROBLEMS http://nm.mathforcollege.com

  4. Fly Rocket Fly, Fly Rocket Fly The upward velocity of a rocket is given as a function of time in table below. Find the velocity and acceleration at t=16 seconds. Table Velocity as a function of time. Velocity vs. time data for the rocket example http://nm.mathforcollege.com

  5. Specific Heat of Carbon A graphite block needs to be pyrolized by heating it up from room temperature of 300K to 1800K. How much heat is required to do so? http://nm.mathforcollege.com

  6. THE END http://nm.mathforcollege.com

  7. 5.01 BACKGROUND OF INTERPOLATION http://nm.mathforcollege.com

  8. The number of different polynomials that can go though two fixed points (x1,y1) and (x2,y2) is • 0 • 1 • 2 • infinite http://nm.mathforcollege.com

  9. Given n+1data points, a unique polynomial of degree _______ passes through the n+1data points • n+1 • n+1 or less • n • n or less http://nm.mathforcollege.com

  10. If a polynomial of degree n has more than n zeros, then the polynomial is • oscillatory • zero everywhere • quadratic • not defined http://nm.mathforcollege.com

  11. THE END http://nm.mathforcollege.com

  12. 5.02 DIRECT METHOD http://nm.mathforcollege.com

  13. The following velocity vs time data is given. To find the velocity at t=14.9s, the three time data points you would choose for second order polynomial interpolation are • 0, 15, 18 • 15, 18, 22 • 0, 15, 22 • 0, 18, 24 http://nm.mathforcollege.com

  14. The following x-y data is given • -1.048 • 0.1433 • 4.333 • 24.00 A first order polynomial is chosen as an interpolant for the first two data points as The value of b1 is most nearly http://nm.mathforcollege.com

  15. The polynomial that passes through the following x-y data • 0.2500 • 8.125 • 24.00 • not obtainable with the information given is given by The corresponding polynomial using Newton’s divided difference polynomial method is given by The value of b2is http://nm.mathforcollege.com

  16. THE END http://nm.mathforcollege.com

  17. 5.04 SPLINE INTERPOLATION http://nm.mathforcollege.com

  18. Given n data points of y vs x for conducting quadratic spline interpolation, the x-data needs to be • equally spaced • in ascending or descending order • integers • positive http://nm.mathforcollege.com

  19. Given n+1 data points (xo,y0),(x1,y1),…,(xn-1,yn-1), (xn,yn), and assume you pass a function f(x) through all the data points. If now the value of the function f(x) is required to be found outside the range of given x-data, the procedure is called • extrapolation • interpolation • guessing • regression http://nm.mathforcollege.com

  20. In quadratic spline interpolation, • the first derivatives of the splines are continuous at the interior data points • the second derivatives of the splines are continuous at the interior data points • the first or the second derivatives of the splines are continuous at the interior data points • the first and second derivatives are continuous at the interior data points http://nm.mathforcollege.com

  21. In cubic spline interpolation, • the first derivatives of the splines are continuous at the interior data points • the second derivatives of the splines are continuous at the interior data points • the first and the second derivatives of the splines are continuous at the interior data points • the first or the second derivatives of the splines are continuous at the interior data points http://nm.mathforcollege.com

  22. A robot needs to follow a path that passes through six points as shown in the figure. To find the shortest path that is also smooth you would recommend • Pass a 5th order polynomial through the data • Pass linear splines through the data • Pass quadratic splines through the data • Regress the data to a 2nd order polynomial http://nm.mathforcollege.com

  23. A robot path on an x-y plane is found by interpolating 3 data points given below. The interpolant is The length of the path from x=4to x=7is http://nm.mathforcollege.com

  24. THE END http://nm.mathforcollege.com

  25. EXTRA PROBLEMS http://nm.mathforcollege.com

  26. The following data of the velocity of a body is given as a function of time • 6.26 • 6.29 • 6.44 • cannot be found Using quadratic interpolation, the interpolant approximates the velocity of the body from t=4 to t=7 s. From this information, at what time in seconds is the velocity of the body 20 m/s http://nm.mathforcollege.com

  27. The following type of functions can be used for interpolation • polynomial • exponential • trigonometric • all of the above http://nm.mathforcollege.com

  28. Polynomials are most commonly used functions for interpolation because they are easy to • evaluate • differentiate • integrate • all of the above http://nm.mathforcollege.com

  29. The data of velocity vs time is given. The velocity in m/s at t=16s using linear interpolation is • 27.867 • 28.333 • 30.429 • 43.000 http://nm.mathforcollege.com

  30. The length of a straight line path from (1, 2.2) to (4, 6.2) is • 3.0 • 4.0 • 5.0 • 25.0 http://nm.mathforcollege.com

  31. The following incomplete y vs. x data is given • 2bx + c = 50x - 303 • 12b + c = -3 • 36b + 6c + d = 10 • 36x 2 + 6x + d = 25x 2-303x+928 The data is fit by quadratic spline interpolants given by At x=6, the first derivative is continuous gives the equation http://nm.mathforcollege.com

  32. The following incomplete y vs. x data is given The data is fit by quadratic spline interpolants given by • 23.50 • 25.67 • 26.42 • 28.00 where a, b, c, d, e, f, g are constants. What is the value of http://nm.mathforcollege.com

  33. Given three data points (1,6), (3,28), (10,231), it is found that the function y=2x 2+3x+1 passes through the three data points. Your estimate of y at x=2 is most nearly • 6 • 15 • 17 • 28 http://nm.mathforcollege.com

  34. The following incomplete y vs. x data is given The data is fit by quadratic spline interpolants given by • -144.5 • -4.000 • 3.600 • 12.20 where a, b, c, d, e, f, g are constants. The value of df/dx at x=2.6 most nearly is http://nm.mathforcollege.com

  35. Thermistor Calibration Thermistors are based on change in resistance of a material with temperature. A manufacturer of thermistors makes the following observations on a thermistor. Determine the calibration curve for thermistor. http://nm.mathforcollege.com

  36. 4 3 5 2 6 Y 1 X 7 Following the Cam A curve needs to be fit through the given points to fabricate the cam. http://nm.mathforcollege.com

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