Runge phenomenon

Wild, growing oscillation near the ends of the interval when a smooth function is interpolated at equally spaced nodes of increasing degree -- more nodes make it worse, not better.

The Runge phenomenon is the failure of equally spaced polynomial interpolation to converge as the degree $n$ increases, for some perfectly smooth functions -- most famously $f(x)=1/(1+25x^2)$ on $[-1,1]$ . Rather than the interpolant settling down toward $f$ everywhere as $n$ grows, the error near the endpoints of the interval grows without bound, even though the error near the center stays small. It is a property of the node placement, not of any flaw in the interpolation formula itself.

Examples

Interpolating $f(x)=1/(1+25x^2)$ at 20 equally spaced nodes on $[-1,1]$ oscillates so wildly near $x=\pm1$ that the interpolant is a WORSE fit than a much lower-degree one.