Runge's phenomenon

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In the mathematical field of numerical analysis, Runge's phenomenon (German: [ˈʁʊŋə]) is a problem of oscillation at the edges of an interval that occurs when using polynomial interpolation with polynomials of high degree over a set of equispaced interpolation points. It was discovered by Carl Runge when exploring the behavior of errors when using polynomial interpolation to approximate certain functions. The discovery shows that going to higher degrees does not always improve accuracy. The phenomenon is similar to the Gibbs phenomenon in Fourier series approximations. The Weierstrass approximation theorem states that 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,…, each of degree at most n, that approximates f(x) with uniform convergence over [a,b] as n tends to infinity. This can be expressed as limn→∞(supa≤x≤b|f(x)−Pn(x)|)=0. Consider the case where one desires to interpolate through n+1 equispaced points...

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