Runge's Phenomenon
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To my understanding, the Runge’s phenomenon is a kind of abnormality that occurs in polynomial interpolation. Theoretically, for every continuous function f(x) defined on an interval [a,b], there exists a set of polynomial functions Pn(x) for n=0, 1, 2, …,n. The more n is, the better approximation we will have, so when we set n to infinity, we will get uniform convergence over [a,b]. The abnormality of Runge’s phenomenon is that, as n increases, the polynomial functions Pn(x) would actually deviate from f(x) in a way of oscillation. This usually happens in the endpoints close to the interpolation point, when using equidistant data points for high-degree polynomial interpolation.
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