Runge's phenomenon

Common sense says: to fit a curve better, sample it at more points and use a higher-degree polynomial. Runge discovered, alarmingly, that this can backfire spectacularly. Interpolating a perfectly nice, smooth function at MANY equally spaced points with a single high-degree polynomial can produce a curve that oscillates more and more violently near the ends — and the error can grow without bound as you add points.

The classic example is the bell-shaped function f(x) = 1 / (1 + 25 x^2) on the interval [-1, 1]. Interpolate it at equispaced nodes: with a few points the fit looks fine, but as the degree climbs the interpolant develops huge wiggles that swing far above and below the true curve near x = +/-1, while staying close in the middle. Doubling the points makes the end oscillations WORSE, not better. The interpolation error formula explains it: the node product (x - x_0)...(x - x_n) is enormously larger near the endpoints than in the centre for equispaced nodes, and for the Runge function the high derivatives grow fast enough that this end-blowup overwhelms the (n+1)! in the denominator — so the polynomial diverges there.

The lesson reshaped numerical practice: do not trust a single high-degree polynomial on equispaced data. There are two standard cures. First, change the NODES: cluster them toward the ends as Chebyshev nodes do, which tames the node product and restores convergence (combine with the barycentric formula for speed and stability). Second, change the STRATEGY: abandon one global polynomial for piecewise low-degree pieces — splines — which never oscillate this way. Runge's phenomenon is the headline reason high-order interpolation must be done thoughtfully, not blindly.