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:
“One thing in any case is certain: man is neither the oldest nor the most constant problem that has been posed for human knowledge.”
—Michel Foucault (1926–1984)
“The problem of the novelist who wishes to write about a man’s encounter with God is how he shall make the experience—which is both natural and supernatural—understandable, and credible, to his reader. In any age this would be a problem, but in our own, it is a well- nigh insurmountable one. Today’s audience is one in which religious feeling has become, if not atrophied, at least vaporous and sentimental.”
—Flannery O’Connor (1925–1964)
“Every child is an artist. The problem is how to remain an artist once he grows up.”
—Pablo Picasso (1881–1973)