Content deleted Content added
No edit summary |
No edit summary |
||
Line 4:
[[File:An iterated direct similarity yields spirals.svg|thumb|upright=1.8|[[Function composition|Composed]] with itself '''repeatedly''', [[Similarity (geometry)|similarity]] {{math|''F''}} of [[Similarity (geometry)#Similarity_with_a_center|center]] ''S ''enlarges the smallest [[Regular polygon|'''regular''']] [[pentagon]] into successive [[Concentric objects|concentric]] pentagons, in manner that the outline of each one passes through all vertices of the previous pentagon, of which it is the [[Image_(mathematics)#Image_of_an_element|image]] under {{math|''F''}}. If [[Transformation (function)|transformation]] {{math|''F''}} is '''iterated''' indefinitely, then ''A ''and ''K'' are the starting points of two infinite [[spiral]]s.]]
In [[mathematics]], an '''iterated function''' is a function {{math|''X → X''}} (a function from some [[Set (mathematics)|set]] {{mvar|X}} to itself) that is obtained by [[function composition|composing]] another function {{math|''f'' : ''X'' → ''X''}} with itself a certain number of times. The process of repeatedly applying the same function is called [[iteration]]. In this process, starting from some initial object, the result of applying a given function is fed again in the function as input, and this process is repeated.
:{{nobr|1=''L'' = <math>\mathit{F}\,</math>( ''K ''), ''M '' = <math>\mathit{F}\,\circ \mathit{F}\,</math>( ''K '') = <math>\mathit{F}\;^{2}\,</math>( ''K ''),}}<br />with the circle‑shaped symbol of function composition.▼
For example on the image on the right:
Iterated functions are objects of study in [[computer science]], [[fractals]], [[dynamical system]]s, mathematics and [[renormalization group]] physics.▼
▲:{{nobr|1=''L'' = <math>\mathit{F}\,</math>( ''K ''), ''M '' = <math>\mathit{F}\,\circ \mathit{F}\,</math>( ''K '') = <math>\mathit{F}\;^{2}\,</math>( ''K ''),}}
▲Iterated functions are
==Definition==
| |||