Fixed Points
If f(x) = x for some x in X, then x is called a fixed point of the iterated sequence. The set of fixed points is often denoted as Fix(f). There exist a number of fixed-point theorems that guarantee the existence of fixed points in various situations, including the Banach fixed point theorem and the Brouwer fixed point theorem.
There are several techniques for convergence acceleration of the sequences produced by fixed point iteration. For example, the Aitken method applied to an iterated fixed point is known as Steffensen's method, and produces quadratic convergence.
Read more about this topic: Iterated Function
Famous quotes containing the words fixed and/or points:
“The word “obligation” is very seldom in my thoughts, and consequently very seldom is it uttered by my tongue. For I am satisfied that whoever hath the word “obligation” continually in his mouth hath the love of tyranny steadily fixed in his heart.”
—Sarah Fielding (1710–1768)
“Sometimes apparent resemblances of character will bring two men together and for a certain time unite them. But their mistake gradually becomes evident, and they are astonished to find themselves not only far apart, but even repelled, in some sort, at all their points of contact.”
—Sébastien-Roch Nicolas De Chamfort (1741–1794)