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 disesteem into which moralists have fallen is due at bottom to their failure to see that in an age like this one the function of the moralist is not to exhort men to be good but to elucidate what the good is. The problem of sanctions is secondary.”
—Walter Lippmann (1889–1974)
“What had really caused the women’s movement was the additional years of human life. At the turn of the century women’s life expectancy was forty-six; now it was nearly eighty. Our groping sense that we couldn’t live all those years in terms of motherhood alone was “the problem that had no name.” Realizing that it was not some freakish personal fault but our common problem as women had enabled us to take the first steps to change our lives.”
—Betty Friedan (20th century)
“The general public is easy. You don’t have to answer to anyone; and as long as you follow the rules of your profession, you needn’t worry about the consequences. But the problem with the powerful and rich is that when they are sick, they really want their doctors to cure them.”
—Molière [Jean Baptiste Poquelin] (1622–1673)