Chemistry - Runge's phenomenon

In the mathematical subfield of numerical analysis Runge's phenomenon is a problem which occurs when using polynomial interpolation with polynomials of high degree. It was discovered by Carle David Tolmé Runge when exploring the behaviour of errors when using polynomial interpolation to approximate certain functions.

Problem

Consider the function:

$f(x) = \frac{1}{1+25x^2}$

Runge found that if you interpolate this function at equidistant points between -1 and 1 such that:

$x_i = -1 + (i-1)\frac{2}{n}, i \in \left\{ 1, 2, \cdots n+1 \right\}$

with a polynomial $P n (x)$ which has a degree $\leq n$ , the resulting interpolation would oscillate toward the end of the interval, i.e. close to -1 and 1. It can even be proven that the interpolation error tends toward infinity when the degree of the polynomial increases:

$\lim_{n \rightarrow \infty} \left( \max_{-1 \leq x \leq 1} | f(x) -P_n(x)| \right) = \infty$

Solutions to the problem of Runge's phenomenon

The oscillation can be minimized by using Chebyshev nodes instead of equidistant nodes. In this case the maximum error is guaranteed to diminish with increasing polynomial order. The phenomenon demonstrates that high degree polynomials are generally unsuitable for interpolation. The problem can be avoided by using spline curves which are piecewise polynomials. When trying to decrease the interpolation error one can increase the number of polynomial pieces which are used to construct the spline instead of increasing the degree of the polynomials used.

Runge's phenomenon

Problem

Solutions to the problem of Runge's phenomenon

See also