Analyzing Pole Effects on Interpolated Runge Function

[leon1127]

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.

 

[HallsofIvy]

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.

 

[leon1127]

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 don't know.