Runge's Phenomenon - Problem

Problem

Consider the function:

Runge found that if this function is interpolated at equidistant points xi between −1 and 1 such that:

with a polynomial Pn(x) of degree ≤ n, the resulting interpolation oscillates 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:

This shows that high-degree polynomial interpolation at equidistant points can be troublesome.

Read more about this topic:  Runge's Phenomenon

Famous quotes containing the word problem:

    “Every child is an artist. The problem is how to remain an artist once he grows up.”
    —Pablo Picasso (1881–1973)

    “My problem lies in reconciling my gross habits with my net income.”
    —Errol Flynn (1909–1959)

    “Any solution to a problem changes the problem.”
    —R.W. (Richard William)