I'm trying to interpolate x*atan(x) on [-5, 5] using (n+1) equidistant nodes. The oscillations on the ends seem to be caused by the same reasons Runge phenomenon is. The question is why exactly are they occuring and if they will diminish provided n tends to infinity (the nodes should still be equidistant).
2026-03-25 04:35:38.1774413338
Interpolation error for x*atan(x)
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It depends on what you mean by the same as the Runge phenomenon. The Runge phenomenon comes about because sines and cosines do not have discontinuities, so have trouble fitting functions that have them. They can get there pointwise, but it takes a lot of terms and you have the overshoot because the high order terms are trying to compensate for badly fitting low order terms. Polynomials do not like having equally spaced roots, so you tend to get overshoot at the ends of the interval. Sines and cosines have equally spaced roots and are a much better choice if you are using equally spaced nodes.