Rungeverschijnsel

Rungeverschijnsel, or Runge phenomenon, is a problem in polynomial interpolation where high-degree polynomials fitted to a function at equally spaced nodes on a finite interval exhibit large oscillations near the interval ends. Although the function is smooth, the interpolants can become wildly inaccurate at the boundaries as the degree increases, even while the approximation improves in the interior. The phenomenon was demonstrated by Carl Runge in the early 20th century and is a fundamental example of numerical instability in interpolation.

The underlying cause is linked to the growth of the Lagrange basis polynomials for many nodes on a finite interval. With equidistant nodes, the interpolation error and the oscillations near the endpoints grow with the polynomial degree, leading to poor convergence at the boundaries despite good interior behavior. The effect is most pronounced for functions with non-polynomial behavior that cannot be captured well by a single global polynomial.

A classic illustration uses the Runge function f(x) = 1/(1+x^2) on the interval [-1, 1]. Interpolating f with polynomials of increasing degree at equally spaced nodes produces increasingly large oscillations near x = ±1, showing that more nodes do not guarantee a better approximation.

Mitigations include using nonuniform node distributions such as Chebyshev or Chebyshev-Gauss-Lobatto nodes, which minimize the maximum interpolation error. Alternative approaches include piecewise polynomial interpolation (splines), rational interpolation, or spectral methods with appropriate basis functions. These strategies reduce or eliminate Rungeverschijnsel and are standard in numerical analysis and scientific computing.