Question

Question about Runge's phenomenon

For higher order interpolating polynomial why oscillation is not occur at mid-point and why it becomes severe near end-point?

Most texts say just it is because of Runge's Phenomenon , and the derivative of polynomial is large as n increases so the graph oscillates, but I cannot understand these.

Answer 1

it can be proven that as the degree of polynomial increases,the error of interpolation also increases.Runges problem is a consequence of the 2 properties of the problem

1.magnitude of derivatives of functions grows quickly as the value of n increases

2.the equidistance between points lead to LEBESGUE constant that increases quickly as n increases,LEBESGYE constant varies exponentially rather than linearly ,so oscillations are severe towards ends,not at midpoint