Runge Phenomenon and Chebyshev Cure

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.

Figure 1. Runge phenomenon. Method: Lagrange interpolation at the two node sets.

n nodes12

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WHAT TO TRY

Raise the node count n: the equispaced interpolant grows wild oscillations near the endpoints (the Runge phenomenon) while the Chebyshev interpolant hugs the function. The error panel shows equispaced diverging and Chebyshev converging.
Look at where the nodes sit: Chebyshev clusters them near the endpoints, exactly where equispaced sampling fails, which tames the boundary blow-up.
Push n past 15 equispaced: the maximum error explodes exponentially, a warning that more uniform data can make a polynomial fit worse, not better.