Rungephenomena

The Runge phenomenon, also known as the Runge's phenomenon, is a problem that occurs in numerical analysis and approximation theory when using polynomial interpolation. It was named after the German mathematician Carl David Tolmé Runge, who first described it in 1901. The phenomenon demonstrates that polynomial interpolation can produce large errors even when using high-degree polynomials, especially at the endpoints of the interval.

The Runge phenomenon typically occurs when interpolating functions with certain characteristics, such as the Runge function, which is defined as \( f(x) = \frac{1}{1 + 25x^2} \) on the interval \([-1, 1]\). When attempting to interpolate this function using high-degree polynomials at equally spaced points, the resulting polynomial exhibits large oscillations, particularly near the endpoints of the interval. These oscillations can lead to significant errors in the approximation, despite the polynomial's high degree.

The cause of the Runge phenomenon is related to the condition number of the Vandermonde matrix, which becomes increasingly ill-conditioned as the number of interpolation points increases. This ill-conditioning leads to large coefficients in the interpolating polynomial, which in turn result in the observed oscillations. To mitigate the Runge phenomenon, various techniques can be employed, such as using non-uniform interpolation points, employing Chebyshev nodes, or using alternative approximation methods like spline interpolation or least squares approximation. These methods help to reduce the oscillations and improve the accuracy of the interpolation.