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:
“It is part of the educators responsibility to see equally to two things: First, that the problem grows out of the conditions of the experience being had in the present, and that it is within the range of the capacity of students; and, secondly, that it is such that it arouses in the learner an active quest for information and for production of new ideas. The new facts and new ideas thus obtained become the ground for further experiences in which new problems are presented.”
—John Dewey (18591952)
“I tell you, sir, the only safeguard of order and discipline in the modern world is a standardized worker with interchangeable parts. That would solve the entire problem of management.”
—Jean Giraudoux (18821944)
“Hypocrisy is the essence of snobbery, but all snobbery is about the problem of belonging.”
—Alexander Theroux (b. 1940)