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Runge Function

  1. Sep 13, 2007 #1
    My teacher asked a very interesting question. so given a runge function 1/(1+x^2) and i interpolate it on uniformly spaced point in the inteval -1 and 1 by p_n(x)
    How does the pole -i and i contribute to the oscillation of p_n(x)? I never thought pole would come into play.
     
  2. jcsd
  3. Sep 13, 2007 #2

    HallsofIvy

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    What exactly do you mean by "interpolate it ... by p_n(x)". A polynomial at n points?

    Slightly different but you might think about this: The Taylor's series for 1/(1+x2), around x=0, has "radius of convergence" equal to 1 precisely because it has poles at i and -i. In the complex plane, the radius of convergence really is a "radius". It can't go beyond i or -i, both at distance 1 from 0, because they are poles.
     
  4. Sep 13, 2007 #3
    so i have x0 = {-1, -0.9, -0.8,....., 0.9, 1} equidistant point. Then by lagrange polynomial there exists a polynomial, degree <= cardinal[x0] - 1, that will interpolate the ordered pair (x0, f(x0)}.

    more specifially my teacher showed the comparison between runge(x) and exp(-10x^2) on the same set of point. The polynomial has very large oscillation near the end point of the interval. I can see how taylor diverges but now we are interpolating n+1 point instead of its derivative.. so i dont know.
     
    Last edited: Sep 13, 2007
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