Why do polynomials for higher degrees have large oscillations near the edge of the interval?

Question

In regards to Polynomial Interpolation, especially Lagrange Interpolation, I noticed that near the edges of the interval there are huge oscillations. My question is: Why do polynomials of higher degrees have big oscillations near the end points? And how do we avoid running into this scenario?

Answer 1

This is known as the Runge phenomenon. Also, the interpolation with many nodes is numerically instable. Therefore, in practice , either splines are used (in particular cubic splines) or Tchebycheff-interpolation also avoiding this annoying phenomenon.

The reason is that the interpolation polynomial has moderate values in a large range , hence the polynomial usually has many minima and maxima. If they have large absolute values, we have the oscillation.

Answer 2

This happens if you do interpolation on equally spaced points. It does not happen if you use Chebyshev points. There's a nice treatment of these matters in Trefethen, Approximation Theory and Approximation Practice