Fixed-point computation: Difference between revisions

Revision as of 22:43, 25 July 2024

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A Lipschitz-continuous function with constant <math>L</math> is called '''[[contractive]]''' if <math>L<1</math>; it is called '''[[weakly-contractive]]''' if <math>L\le 1</math>. 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 <math>t</math> iterations is in <math>O(L^t)</math>. Therefore, the number of evaluations required for a <math>\delta</math>-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 <math>\delta</math>-relative fixed-point is larger than 50% the number of evaluations required by the iteration algorithm. Note that when <math>L</math> approaches 1, the number of evaluations approaches infinity. No finite algorithm can compute a <math>\delta</math>-absolute fixed point for all functions with <math>L=1</math>.<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>

When <math>L</math> < 1 and ''d'' = 1, the optimal algorithm is the '''Fixed Point Envelope''' (FPE) algorithm of Sikorski and Wozniakowski.<ref name=":5" /> It finds a ''δ''-relative fixed point using <math>O(\log(1/\delta) + \log \log(1/(1-L))) </math> queries, and a ''δ''-absolute fixed point using <math>O(\log(1/\delta)) </math> queries. This is faster than the fixed-point iteration algorithm.<ref>{{cite book |doi=10.1007/978-1-4615-9552-6_4 |chapter=Fast Algorithms for the Computation of Fixed Points |title=Robustness in Identification and Control |year=1989 |last1=Sikorski |first1=K. |pages=49–58 |isbn=978-1-4615-9554-0 }}</ref>