Runge's Phenomenon - Problem

Problem

Consider the function:

Runge found that if this function is interpolated at equidistant points xi between −1 and 1 such that:

with a polynomial Pn(x) of degree ≤ n, the resulting interpolation oscillates toward the end of the interval, i.e. close to −1 and 1. It can even be proven that the interpolation error tends toward infinity when the degree of the polynomial increases:

This shows that high-degree polynomial interpolation at equidistant points can be troublesome.

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