Runge's Phenomenon
High-degree polynomial interpolation on equally spaced nodes oscillates violently near the interval ends, a failure cured by better node placement or splines.
When more points make things worse
Intuition suggests that interpolating with more points should improve accuracy. Runge's phenomenon shows this is false for a single high-degree polynomial on equally spaced nodes: the interpolant develops large oscillations near the ends of the interval that grow, not shrink, as the number of points increases.
The classic example is Runge's function 1/(1 + 25 x^2) on [-1, 1]. With 11 or more equally spaced nodes, the degree-n interpolant swings far above and below the smooth true curve near x = plus or minus 1, and the maximum error diverges as n grows.
Why it happens
The interpolation error contains the product of distances (x - x_i) over all nodes. For equally spaced nodes this product is far larger near the endpoints than in the middle, and it grows faster than the (n+1)th derivative shrinks. The mismatch produces the exploding end oscillations.
Cures
- Use Chebyshev nodes, clustered toward the endpoints, which flatten the error product and give near-optimal interpolation.
- Use piecewise low-degree interpolation such as cubic splines, keeping each piece's degree small.
- Fit a lower-degree polynomial in the least-squares sense instead of interpolating exactly.
- Increase resolution by adding pieces, not by raising a single polynomial's degree.
With Chebyshev nodes, polynomial interpolation converges rapidly for smooth functions and is the backbone of spectral methods. Equally spaced high-degree interpolation, by contrast, is almost always a mistake in practice.
This lesson guides how tabulated data and spatial fields are reconstructed in the breeder Hyperion codes: local splines and clustered nodes are used instead of one global high-degree polynomial.