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:
“The problem of induction is not a problem of demonstration but a problem of defining the difference between valid and invalid
predictions.”
—Nelson Goodman (1906)
“The writer operates at a peculiar crossroads where time and place and eternity somehow meet. His problem is to find that location.”
—Flannery OConnor (19251964)
“The family environment in which your children are growing up is different from that in which you grew up. The decisions our parents made and the strategies they used were developed in a different context from what we face today, even if the content of the problem is the same. It is a mistake to think that our own experience as children and adolescents will give us all we need to help our children. The rules of the game have changed.”
—Lawrence Kutner (20th century)