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:
“From cradle to grave this problem of running order through chaos, direction through space, discipline through freedom, unity through multiplicity, has always been, and must always be, the task of education, as it is the moral of religion, philosophy, science, art, politics and economy; but a boy’s will is his life, and he dies when it is broken, as the colt dies in harness, taking a new nature in becoming tame.”
—Henry Brooks Adams (1838–1918)
“But a problem occurs about nothing. For that from which something is made is a cause of the thing made from it; and, necessarily, every cause contributes some assistance to the effect’s existence.”
—Anselm of Canterbury (1033–1109)
“And just as there are no words for the surface, that is,
No words to say what it really is, that it is not
Superficial but a visible core, then there is
No way out of the problem of pathos vs. experience.”
—John Ashbery (b. 1927)