Editing Iterated function (section)

==Fixed points==

If {{math|1='' ''x'' = f''(''x'')}} for some {{mvar|x}} in {{mvar|X}} (that is, the period of the orbit of {{mvar|x}} is {{math|1}}), then {{mvar|x}} is called a '''[[fixed point (mathematics)|fixed point]]''' of the iterated sequence.  The set of fixed points is often denoted as {{math|'''Fix'''(''f'')}}. There exist a number of [[fixed-point theorem]]s 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]].<ref>{{Cite book| last1=Carleson|first1=L.| last2=Gamelin|first2=T. D. W.| title=Complex dynamics|series=Universitext: Tracts in Mathematics| publisher=Springer-Verlag| year=1993| isbn=0-387-97942-5| url-access=registration| url=https://archive.org/details/complexdynamics0000carl}}</ref> For example, the [[Aitken method]] applied to an iterated fixed point is known as [[Steffensen's method]], and produces quadratic convergence.