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 firelight of a long, blind, dreaming story
Lingers upon your lips; and I have seen
Firm, fixed forever in your closing eyes,
The Corn King beckoning to his Spring Queen.”
—Randall Jarrell (1914–1965)
“Wonderful “Force of Public Opinion!” We must act and walk in all points as it prescribes; follow the traffic it bids us, realise the sum of money, the degree of “influence” it expects of us, or we shall be lightly esteemed; certain mouthfuls of articulate wind will be blown at us, and this what mortal courage can front?”
—Thomas Carlyle (1795–1881)