1. Chem 302 - Math 252 Chapter 3 Interpolation / 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 T / °C  / mP 017.921 1013.077 2010.050 308.007 406.560 505.494 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. 1/T × K  ln(  / mP) 0.0036612.88597 0.0035322.57086 0.0034112.30757 0.0032992.08032 0.0031931.88099 0.0030951.70366 Find  at 25 °C Exp  25 =8.937 mP

8. Linear Interpolation – Heat Capacity of Benzene T / KC / (J K -1 ) 200.0083.7 240.00104.1 260.00116.1 278.69128.7 Find C at 220, 250 & 270 K Find C at 20 K Exp C 20 = 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 T / °C  / mP 017.921 1013.077 2010.050 308.007 406.560 505.494 Find  at 25 °C Exp  25 =8.937 mP

13. Quadratic Interpolation – Viscosity of Water 1/T × K  ln(  / mP) 0.0036612.88597 0.0035322.57086 0.0034112.30757 0.0032992.08032 0.0031931.88099 0.0030951.70366 Find  at 25 °C Exp  25 =8.937 mP Using points 2,3,4 Using points 3,4,5

14. Quadratic Interpolation – Heat Capacity of Benzene T / KC / (J K -1 ) 200.0083.7 240.00104.1 260.00116.1 278.69128.7 Find C at 20, 220, 250 & 270 K

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

16. Lagrangian Interpolation

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

19. Lorentzian Interpolation Uses Lorentzian lineshape Peak height – A Peak Position – x 0 Full Width at Half Height (FWHH) – 2/B 3 three points (usually three at top of peak)

20. Lorentzian Interpolation

21. Quadratic interpolation on 1/y

22. Magnitude-Lorentzian Interpolation Uses square root of Lorentzian lineshape Peak height – A 1/2 Peak Position – x 0 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 – x 0 Full Width at Half Height (FWHH) –

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

28. Find Peak Position & Height

29. Seven Highest points 1034.60.73050 1035.10.75487 1035.50.76894 1036.00.77183 1036.50.75979 1037.00.73585 1037.50.69860 Use various interpolation functions to find peak position and height 1. Determine interpolation function (top 3, 5 or 7 points) 2. Differentiate interpolation function and find root (i.e. find location max) - position 3.Evaluate interpolation function at peak position (height)

30. Find Peak Position & Height Quadratic Interpolation

31. Find Peak Position & Height Comparison of Interpolation Methods Interpolation MethodPeak Position /cm -1 Peak Height /(cm 3 mol -1 ) No Interpolation1036.00.77183 Quadratic1035.8470.77253 Lorentzian1035.8460.77255 Magnitude-Lorentzian1035.8460.77255 Gaussian1035.8460.77254 KC21035.8470.77254 KC41035.8460.77254 5-point Lagrangian1035.8400.77263 7-point Lagrangian1035.8340.77270

32. 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 Spline Interpolation

33. Cubic Spline with Slope Matching Value of x such that x 2 < x < x 3 p(x) forced to pass through (x 2,y 2 ) & (x 3,y 3 ) p(x) forced to match slopes at (x 2,y 2 ) & (x 3,y 3 )

34. Cubic Spline with Slope Matching Approximate slopes

35. Cubic Spline with Slope Matching Between 1 st 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