Chem 302 Math 252

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Chem 302 - Math 252

• Chapter 3Interpolation / Extrapolation

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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

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Linear Interpolation

• Assume data varies linearly between 2 points
• Connect-the-dots

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Linear Interpolation Viscosity of Water
Find ? at 25 C
Exp ?25 8.937 mP
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Linear Interpolation Viscosity of Water
ln(?) vs 1/T nearly linear
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Linear Interpolation Viscosity of Water
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Linear Interpolation Viscosity of Water
Find ? at 25 C
Exp ?25 8.937 mP
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Linear Interpolation Heat Capacity of Benzene
Find C at 220, 250 270 K
Find C at 20 K
Exp C20 8.4 J/K
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Quadratic Interpolation

• Assume data is quadratic between 3 points

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Quadratic Interpolation
Do it !
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Quadratic Interpolation
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Quadratic Interpolation Viscosity of Water
Find ? at 25 C
Exp ?25 8.937 mP
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Quadratic Interpolation Viscosity of Water
Find ? at 25 C
Using points 2,3,4
Using points 3,4,5
Exp ?25 8.937 mP
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Quadratic Interpolation Heat Capacity of Benzene
Find C at 20, 220, 250 270 K
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Lagrangian Interpolation

• Generalization of linear quadratic
  interpolations
• Uses nth order polynomial n1 points

Unique solution
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Lagrangian Interpolation
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Lagrangian Interpolation
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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

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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)
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Lorentzian Interpolation
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Lorentzian Interpolation
Quadratic interpolation on 1/y
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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)
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Magnitude-Lorentzian Interpolation
Quadratic interpolation on 1/y
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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)
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Magnitude-Lorentzian Interpolation
Quadratic interpolation on y-e
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Gaussian Interpolation

• Based on Gaussian lineshape

Peak height A Peak Position x0 Full Width at
Half Height (FWHH)
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Gaussian Interpolation
Can be converted to form
Quadratic interpolation on lny
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Find Peak Position Height
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Find Peak Position Height

• Seven Highest points

• 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 0.6
9860
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Find Peak Position Height
Quadratic Interpolation
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Find Peak Position Height
Comparison of Interpolation Methods
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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

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Cubic Spline with Slope Matching

• Value of x such that x2 lt x lt x3

p(x) forced to pass through (x2,y2)
(x3,y3) p?(x) forced to match slopes at (x2,y2)
(x3,y3)
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Cubic Spline with Slope Matching
Approximate slopes
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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

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