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MathGroup Archive 2011

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Interpolation problems

  • To: mathgroup at smc.vnet.net
  • Subject: [mg120229] Interpolation problems
  • From: James Womack <james.c.womack at gmail.com>
  • Date: Thu, 14 Jul 2011 05:21:51 -0400 (EDT)

Hello all,

I am having difficulty with interpolating a data set. My data is a 
function of one coordinate -- each data point consists of 2 numbers, one 
denoting position and one denoting the value of the function. If you are 
interested, the data represents a potential energy function.

The best way to demonstrate my issue is to link to some images of plots.

First, here is the result of:
ListPlot[data, PlotRange -> {{0.4, 0.6}, {-350, 350}}, Joined -> True]
http://jcwx86.dyndns.org/public/listplot.jpeg

Second, here is the result of:
Plot[Interpolation[data][x], {x, 0.4, 0.6}, PlotRange -> {-350, 350}]
http://jcwx86.dyndns.org/public/interpolationfunction.jpeg

These plots focus only on the relevant region of the data. The data 
spans x = 0.000857412 to x = 25.6297. The peak you see in the plots is 
cut off with a maximum value of the function at about y = 4000.

As you can see, the interpolation results in two small additional minima 
at the base of the peak. This is problematic, since the function is a 
potential energy function and minima represent stable-states. Additional 
minima present in the data therefore result in problems with subsequent 
calculations using the InterpolationFunction of the data.

I suspect that the issue is to do with the Interpolation command trying 
to fit the function to every datapoint. The rapid change in gradient at 
the base of the peak means that in order to fit splines to the function, 
these minima are necessary.

Ideally, I would like an InterpolationFunction of the data that is (i) 
smooth and (ii) lacks these spurious minima. I have considered deleting 
data points from the base of the peak, as this might enable the splines 
to better fit the peak. Does anyone know of a way that I can modify the 
Interpolation command that enables a fit which better represents the 
data? Or perhaps you know of an alternative method of creating 
interpolated functions which might suit my needs better? I have already 
tried increasing the interpolation order -- this seems only to result in 
more "wobbles" at the base of the peak. Decreasing the interpolation 
order to 1 removes the minima, but also means the function is no longer 
"smooth".

If you want to play around with the data yourself, I will post it below 
in full.

Thanks in advance for your thoughts,

James

data = {{0.000857412, 14838.7}, {0.000952607, 14879.2}, {0.0010546,
   14922.6}, {0.00116363, 14967.6}, {0.00127992, 15012.4}, {0.00140372,
    15055.}, {0.00153526, 15092.5}, {0.00167477, 15121.8}, {0.00182249,
    15139.7}, {0.00197865, 15142.7}, {0.00214349,
   15127.9}, {0.00231725, 15092.8}, {0.00250016, 15036.}, {0.00269245,
   14957.1}, {0.00289436, 14856.4}, {0.00310612, 14735.2}, {0.00332798,
    14594.7}, {0.00356017, 14436.}, {0.00380292, 14259.1}, {0.00405647,
    14062.8}, {0.00432105, 13844.5}, {0.00459691,
   13601.1}, {0.00488428, 13329.}, {0.00518339, 13025.6}, {0.00549449,
   12689.5}, {0.0058178, 12320.9}, {0.00615357, 11922.}, {0.00650203,
   11496.5}, {0.00686342, 11049.1}, {0.00723798, 10585.3}, {0.00762594,
    10110.9}, {0.00802755, 9631.17}, {0.00844304,
   9151.38}, {0.00887265, 8675.96}, {0.00931661, 8208.59}, {0.00977517,
    7752.08}, {0.0102486, 7308.4}, {0.010737, 6878.79}, {0.0112408,
   6463.95}, {0.0117602, 6064.19}, {0.0122953, 5679.67}, {0.0128465,
   5310.55}, {0.0134139, 4957.09}, {0.0139978, 4619.7}, {0.0145986,
   4298.88}, {0.0152163, 3995.17}, {0.0158512, 3709.05}, {0.0165036,
   3440.87}, {0.0171737, 3190.75}, {0.0178618, 2958.54}, {0.0185681,
   2743.84}, {0.0192929, 2545.97}, {0.0200363, 2364.08}, {0.0207987,
   2197.12}, {0.0215803, 2043.97}, {0.0223812, 1903.44}, {0.0232019,
   1774.37}, {0.0240425, 1655.66}, {0.0249032, 1546.25}, {0.0257843,
   1445.24}, {0.0266861, 1351.8}, {0.0276088, 1265.26}, {0.0285526,
   1185.03}, {0.0295178, 1110.63}, {0.0305046, 1041.65}, {0.0315134,
   977.742}, {0.0325443, 918.601}, {0.0335975, 863.929}, {0.0346735,
   813.433}, {0.0357722, 766.816}, {0.0368942, 723.771}, {0.0380395,
   683.979}, {0.0392085, 647.117}, {0.0404014, 612.862}, {0.0416184,
   580.898}, {0.0428598, 550.931}, {0.0441259, 522.687}, {0.0454169,
   495.929}, {0.046733, 470.455}, {0.0480746, 446.101}, {0.0494419,
   422.747}, {0.0508351, 400.309}, {0.0522545, 378.739}, {0.0537004,
   358.02}, {0.055173, 338.159}, {0.0566725, 319.178}, {0.0581993,
   301.111}, {0.0597536, 283.995}, {0.0613357, 267.864}, {0.0629458,
   252.743}, {0.0645842, 238.642}, {0.0662511, 225.56}, {0.0679469,
   213.476}, {0.0696718, 202.352}, {0.071426, 192.138}, {0.0732098,
   182.767}, {0.0750235, 174.163}, {0.0768674, 166.246}, {0.0787418,
   158.929}, {0.0806468, 152.129}, {0.0825829, 145.764}, {0.0845501,
   139.761}, {0.0865489, 134.055}, {0.0885796, 128.59}, {0.0906423,
   123.323}, {0.0927373, 118.223}, {0.094865, 113.266}, {0.0970257,
   108.441}, {0.0992195, 103.746}, {0.101447, 99.1832}, {0.103708,
   94.7614}, {0.106003, 90.492}, {0.108333, 86.3876}, {0.110697,
   82.4605}, {0.113096, 78.7209}, {0.11553, 75.1764}, {0.118,
   71.8308}, {0.120505, 68.6837}, {0.123047, 65.731}, {0.125625,
   62.9647}, {0.128239, 60.3737}, {0.130891, 57.9446}, {0.13358,
   55.6622}, {0.136306, 53.511}, {0.13907, 51.4754}, {0.141873,
   49.541}, {0.144714, 47.6948}, {0.147594, 45.9259}, {0.150512,
   44.2252}, {0.153471, 42.5863}, {0.156469, 41.0045}, {0.159507,
   39.477}, {0.162585, 38.0023}, {0.165704, 36.5801}, {0.168864,
   35.2105}, {0.172065, 33.894}, {0.175308, 32.6307}, {0.178593,
   31.4207}, {0.18192, 30.2633}, {0.185289, 29.1574}, {0.188702,
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{20.1565, -0.00955576}, {22.1739, -0.00998324}, {25.6297, -0.0104947}}


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