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?
Why do polynomials for higher degrees have large oscillations near the edge of the interval
interpolationlagrange-interpolationlinear algebrapolynomials
Best Answer
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.