1 / 38

Chem 302 - Math 252

Chem 302 - Math 252. Chapter 3 Interpolation / Extrapolation. Interpolation / Extrapolation. Experimental data at discrete points Need to know the dependent variable at a value of the independent variable that was not measured

rowens
Download Presentation

Chem 302 - Math 252

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Chem 302 - Math 252 Chapter 3Interpolation / Extrapolation

  2. Interpolation / Extrapolation • Experimental data at discrete points • Need to know the dependent variable at a value of the independent variable that was not measured • Need to know what value of the independent variable gives a particular value of the dependent variable • Point is within range of experimental data then called interpolation • Point is outside range of experimental data then call extrapolation • Same techniques • Extrapolation more risky

  3. Linear Interpolation • Assume data varies linearly between 2 points • Connect-the-dots

  4. Linear Interpolation – Viscosity of Water Find  at 25 °C Exp 25 =8.937 mP

  5. Linear Interpolation – Viscosity of Water ln() vs 1/T nearly linear

  6. Linear Interpolation – Viscosity of Water

  7. Linear Interpolation – Viscosity of Water Find  at 25 °C Exp 25 =8.937 mP

  8. Linear Interpolation – Heat Capacity of Benzene Find C at 220, 250 & 270 K Find C at 20 K Exp C20 = 8.4 J/K

  9. Quadratic Interpolation • Assume data is quadratic between 3 points

  10. Quadratic Interpolation Do it !

  11. Quadratic Interpolation

  12. Quadratic Interpolation – Viscosity of Water Find  at 25 °C Exp 25 =8.937 mP

  13. Quadratic Interpolation – Viscosity of Water Find  at 25 °C Using points 2,3,4 Using points 3,4,5 Exp 25 =8.937 mP

  14. Quadratic Interpolation – Heat Capacity of Benzene Find C at 20, 220, 250 & 270 K

  15. Lagrangian Interpolation • Generalization of linear & quadratic interpolations • Uses nth order polynomial & n+1 points Unique solution

  16. Lagrangian Interpolation

  17. Lagrangian Interpolation

  18. Other Interpolation Functions • Does not have to be a power series • Methods are same as Lagrangian Interpolation • Usually 2nd order (quadratic) or 3rd order (cubic) Lagrangian interpolation is sufficient

  19. Lorentzian Interpolation • Uses Lorentzian lineshape Peak height – A Peak Position – x0 Full Width at Half Height (FWHH) – 2/B 3 three points (usually three at top of peak)

  20. Lorentzian Interpolation

  21. Lorentzian Interpolation Quadratic interpolation on 1/y

  22. Magnitude-Lorentzian Interpolation Uses square root of Lorentzian lineshape Peak height – A1/2 Peak Position – x0 Full Width at Half Height (FWHH) – 3 three points (usually three at top of peak)

  23. Magnitude-Lorentzian Interpolation Quadratic interpolation on 1/y

  24. KCe Interpolation Based on Lorentzian & Magnitude-Lorentzian e = 1 – quadratic e = -1 – Lorentzian e = -1/2 – Magnitude-Lorentzian Optimized e for different lineshapes (mostly used in FTICR-MS) Keefe, Comisarow, App. Spectrosc. 44, 600 (1990)

  25. Magnitude-Lorentzian Interpolation Quadratic interpolation on y-e

  26. Gaussian Interpolation Based on Gaussian lineshape Peak height – A Peak Position – x0 Full Width at Half Height (FWHH) –

  27. Gaussian Interpolation Can be converted to form Quadratic interpolation on lny

  28. Find Peak Position & Height

  29. Find Peak Position & Height • Use various interpolation functions to find peak position and height • Determine interpolation function (top 3, 5 or 7 points) • Differentiate interpolation function and find root (i.e. find location max) - position • Evaluate interpolation function at peak position (height) 1034.6 0.73050 1035.1 0.75487 1035.5 0.76894 1036.0 0.77183 1036.5 0.75979 1037.0 0.73585 1037.5 0.69860 Seven Highest points

  30. Find Peak Position & Height Quadratic Interpolation

  31. Find Peak Position & Height Comparison of Interpolation Methods

  32. Spline Interpolation • So far methods have used a moving window of subset of data • May be discontinuous at edges of windows • Causes jagged plots • Spline interpolation forces slopes (and in some cases higher derivatives) to match at edges of windows • Creates smooth plots

  33. Cubic Spline with Slope Matching Value of x such that x2 < x < x3 p(x) forced to pass through (x2,y2) & (x3,y3) p(x) forced to match slopes at (x2,y2) & (x3,y3)

  34. Cubic Spline with Slope Matching Approximate slopes

  35. Cubic Spline with Slope Matching • Between 1st and last pair of points • Can set slopes = 0 • Natural spline • Good if data is flat at extremes • Can set • Useful if slope is basically constant • Can extrapolate using closest region • Can set

More Related