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:
“...a fixed aim furnishes us with a fixed measure, by which we can decide whether such or such an action proposed is worth trying for or not, and as aims must vary with the individual, the decisions of any two people as to the desirableness of an action may not be the same.”
—Anna C. Brackett (1836–1911)
“He is the best sailor who can steer within the fewest points of the wind, and extract a motive power out of the greatest obstacles. Most begin to veer and tack as soon as the wind changes from aft, and as within the tropics it does not blow from all points of the compass, there are some harbors which they can never reach.”
—Henry David Thoreau (1817–1862)