The Runge phenomenon, where more points give a worse fit

Adam Zettel · September 21, 2026 · written with an agent

f (x) = \frac{1}{1 + 25 x ^{2}}, ∥ f - p_{n} ∥_{\infty} \to \infty

Interpolate $1/ (1 + 25 x^{2})$ by a polynomial through equally spaced points and the error grows as you add points. Move the same number of points to $cos (j π / n)$ and it falls geometrically, at a rate you can work out from where the function's poles are.

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The function has poles at $x = \pm i /5$ . Polynomial interpolation in Chebyshev points converges like $ρ^{- n}$ , where $ρ$ is the size of the largest ellipse with foci $\pm 1$ that the function is analytic inside. The ellipse through $\pm i /5$ has $ρ = \frac{1}{5} + 1 + \frac{1}{25} = 1.2198$ . Both interpolants use the barycentric formula, which is stable in Chebyshev points. With equally spaced points the problem is ill-conditioned (its Lebesgue constant grows exponentially with $n$ ), and this formula adds rounding error of its own.

Printed

equi_10: 1.91561

equi_20: 59.8221

cheb_20: 0.0177266

cheb_40: 0.000338655

rate: 1.21883

predicted_rate: 1.2198

equally spacedChebyshev

Tests 3 of 3 pass

with equally spaced points, doubling them makes the error more than thirty times larger (passes)
with Chebyshev points the same 21 are good to two percent (passes)
and the error falls like rho^-n, with rho set by the poles at +-i/5 (passes)