Runge phenomenon: equispaced vs Chebyshev nodes

What you are seeing: polynomial interpolation of the Runge function $f(x) = 1 / (1 + 25 x^2)$ on $[-1, 1]$. With equispaced nodes (orange), the interpolant matches at the nodes but oscillates wildly between them. With Chebyshev nodes (cyan), it converges uniformly. The black curve is the true function.

Cause: equispaced interpolation has a Lebesgue constant that grows like $2^n / n$. Chebyshev nodes cluster at the endpoints to keep the constant bounded ($O(\log n)$). The bottom panel plots max error vs $n$ for both schemes.