Rungeefekti

Rungeefekti, or Runge phenomenon, describes a problem in polynomial interpolation where high-degree polynomials fitted to a function at equally spaced nodes on an interval exhibit large oscillations near the interval endpoints. This leads to poor approximations even for smooth functions and serves as a cautionary example in numerical analysis of the limits of polynomial interpolation with uniform node spacing. The phenomenon is named after Carl Runge, who highlighted the issue in the early 20th century.

The core cause is the growth of the interpolation operator’s error with the degree of the polynomial when nodes are evenly spaced. The resulting oscillations are most pronounced at the ends of the interval, and the associated Lebesgue constant grows rapidly with the degree, amplifying any small function or rounding errors.

A classic demonstration uses the Runge function f(x) = 1/(1+25x^2) on [-1,1]. Interpolating this function with high-degree polynomials at equidistant nodes produces large overshoots near x = ±1. Replacing equidistant nodes with nonuniform distributions, such as Chebyshev or Chebyshev–Lobatto nodes, substantially reduces or eliminates the oscillations. Alternative approaches include using piecewise polynomial interpolation (splines) or spectral methods with appropriate basis functions.

In practice, Rungeefekti informs the choice of interpolation strategy: avoid high-degree polynomials on equally spaced nodes for smooth functions, and prefer splines, Chebyshev-based methods, or other stabilization techniques. See also Runge function, Runge’s phenomenon, Chebyshev polynomials, and spline interpolation.