Image:Rungesphenomenon.png

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Download high-resolution version (700x664, 46 KB)

Illustration of Runge's phenomenon at higher resolution

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Below is the Maple code to reproduce the diagram:

> restart; with(CurveFitting);
> f := 1/(1+25*x^2);
> sample_interval := -1+(i-1)*2/n;
> fifth_order := [
> eval(sample_interval, [i=1, n=5]), 
> eval(sample_interval, [i=2, n=5]), 
> eval(sample_interval, [i=3, n=5]), 
> eval(sample_interval, [i=4, n=5]), 
> eval(sample_interval, [i=5, n=5]), 
> eval(sample_interval, [i=6, n=5])];
> ninth_order := [
> eval(sample_interval, [i=1, n=9]), 
> eval(sample_interval, [i=2, n=9]), 
> eval(sample_interval, [i=3, n=9]), 
> eval(sample_interval, [i=4, n=9]), 
> eval(sample_interval, [i=5, n=9]), 
> eval(sample_interval, [i=6, n=9]), 
> eval(sample_interval, [i=7, n=9]), 
> eval(sample_interval, [i=8, n=9]), 
> eval(sample_interval, [i=9, n=9]), 
> eval(sample_interval, [i=10, n=9])];
> fifth_val := [
> eval(f, x=eval(sample_interval, [i=1, n=5])), 
> eval(f, x=eval(sample_interval, [i=2, n=5])), 
> eval(f, x=eval(sample_interval, [i=3, n=5])), 
> eval(f, x=eval(sample_interval, [i=4, n=5])), 
> eval(f, x=eval(sample_interval, [i=5, n=5])), 
> eval(f, x=eval(sample_interval, [i=6, n=5]))];
> ninth_val := [
> eval(f, x=eval(sample_interval, [i=1, n=9])), 
> eval(f, x=eval(sample_interval, [i=2, n=9])), 
> eval(f, x=eval(sample_interval, [i=3, n=9])), 
> eval(f, x=eval(sample_interval, [i=4, n=9])), 
> eval(f, x=eval(sample_interval, [i=5, n=9])), 
> eval(f, x=eval(sample_interval, [i=6, n=9])), 
> eval(f, x=eval(sample_interval, [i=7, n=9])), 
> eval(f, x=eval(sample_interval, [i=8, n=9])), 
> eval(f, x=eval(sample_interval, [i=9, n=9])), 
> eval(f, x=eval(sample_interval, [i=10, n=9]))];
> fifth_approx := PolynomialInterpolation(fifth_order, fifth_val, x);
> ninth_approx := PolynomialInterpolation(ninth_order, ninth_val, x);
> plotsetup(ps, plotoutput="plot.ps", 
> plotoptions="color=rgb, width=600, height=600");
> plot([f, fifth_approx, ninth_approx], x=-1..1, y=-0.4..1.2,
> resolution=600, numpoints=600,
> color=[red, blue, green], axes=box, labels=["", ""], thickness=2);

File history

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• (del) (cur) 01:52, 24 October 2004 . . Madoka (Talk | contribs) . . 700×664 (47,897 bytes) (Illustration of Runge's phenomenon at higher resolution {GFDL})

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• Runge's phenomenon