Runge's Phenomenon - Problem

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:

    “In the nineteenth century the problem was that God is dead; in the twentieth century the problem is that man is dead.”
    —Erich Fromm (1900–1980)

    “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 (1882–1944)

    “You are a problem and rune,
    you are mystery;
    writ on a stone.”
    —Hilda Doolittle (1886–1961)