Content deleted Content added
mNo edit summary |
m Grammar error Tags: Visual edit Mobile edit Mobile web edit Newcomer task Newcomer task: copyedit |
||
Line 20:
== Contractive functions ==
A Lipschitz-continuous function with constant ''L'' is called '''[[contractive]]''' if ''L'' < 1; it is called '''[[weakly-contractive]]''' if ''L ≤'' 1. Every contractive function satisfying Brouwer's conditions has a ''unique'' fixed point. Moreover, fixed-point computation for contractive functions is easier than for general functions.
[[File:Fixed point anime.gif|alt=computing a fixed point using function iteration|thumb|Computing a fixed point using function iteration]]
The first algorithm for fixed-point computation was the '''[[fixed-point iteration]]''' algorithm of Banach. [[Banach fixed point theorem|Banach's fixed-point theorem]] implies that, when fixed-point iteration is applied to a contraction mapping, the error after ''t'' iterations is in <math>O(L^t)</math>. Therefore, the number of evaluations required for a ''δ''-relative fixed-point is approximately <math>\log_L(\delta) = \log(\delta)/\log(L) = \log(1/\delta)/\log(1/L) </math>. Sikorski and Wozniakowski<ref name=":5">{{cite journal |last1=Sikorski |first1=K |last2=Woźniakowski |first2=H |title=Complexity of fixed points, I |journal=Journal of Complexity |date=December 1987 |volume=3 |issue=4 |pages=388–405 |doi=10.1016/0885-064X(87)90008-2 |doi-access=free }}</ref> showed that Banach's algorithm is optimal when the dimension is large. Specifically, when <math>d\geq \log(1/\delta)/\log(1/L) </math>, the number of required evaluations of ''any'' algorithm for ''δ''-relative fixed-point is larger than 50% the number of evaluations required by the iteration algorithm. Note that when ''L'' approaches 1, the number of evaluations approaches infinity. In fact, no finite algorithm can compute a ''δ''-absolute fixed point for all functions with L=1.<ref name=":4">{{cite book |last1=Sikorski |first1=Krzysztof A. |title=Optimal Solution of Nonlinear Equations |date=2001 |publisher=Oxford University Press |isbn=978-0-19-510690-9 }}{{page needed|date=April 2023}}</ref>
| |||