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{{Short description|Computing the fixed point of a function}}
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'''Fixed-point computation''' refers to the process of computing an exact or approximate [[Fixed point (mathematics)|fixed point]] of a given function.<ref name=":1">{{cite book |doi=10.1007/978-3-642-50327-6 |title=The Computation of Fixed Points and Applications |series=Lecture Notes in Economics and Mathematical Systems |year=1976 |volume=124 |isbn=978-3-540-07685-8 }}</ref> In its most common form, the given function <math>f</math> satisfies the condition to the [[Brouwer fixed-point theorem]]: that is, <math>f</math> is continuous and maps the unit [[N-cube|''d''-cube]] to itself. The [[Brouwer fixed-point theorem]] guarantees that <math>f</math> has a fixed point, but the proof is not [[Constructive proof|constructive]]. Various algorithms have been devised for computing an approximate fixed point. Such algorithms are used in
* [[ * [[ * [[ == Definitions ==
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For Lipschitz-continuous functions, the absolute criterion is stronger than the residual criterion: If <math>f</math> is Lipschitz-continuous with constant <math>L</math>, then <math>|x-x_0|\leq \delta</math> implies <math>|f(x)-f(x_0)|\leq L\cdot \delta</math>. Since <math>x_0</math> is a fixed-point of <math>f</math>, this implies <math>|f(x)-x_0|\leq L\cdot \delta</math>, so <math>|f(x)-x|\leq (1+L)\cdot \delta</math>. Therefore, a δ-absolute fixed-point is also an {{mvar|ε}}-residual fixed-point with <math>\varepsilon = (1+L)\cdot \delta</math>.
The most basic step of a fixed-point computation algorithm is a '''value query''': given any <math>x</math> in <math>E^d</math>, the
The function <math>f</math> is accessible via '''evaluation''' queries: for any <math>x</math>, the algorithm can evaluate <math>f(x)</math>. The run-time complexity of an algorithm is usually given by the number of required evaluations.
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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
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>
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* A later algorithm by [[Harold W. Kuhn|Harold Kuhn]]<ref>{{Cite journal |last=Kuhn |first=Harold W. |date=1968 |title=Simplicial Approximation of Fixed Points |jstor=58762 |journal=Proceedings of the National Academy of Sciences of the United States of America |volume=61 |issue=4 |pages=1238–1242 |doi=10.1073/pnas.61.4.1238 |pmid=16591723 |pmc=225246 |doi-access=free }}</ref> used simplices and simplicial partitions instead of primitive sets.
* Developing the simplicial approach further, Orin Harrison Merrill<ref>{{cite thesis |last1=Merrill |first1=Orin Harrison |date=1972 |title=Applications and Extensions of an Algorithm that Computes Fixed Points of Certain Upper Semi-continuous Point to Set Mappings |id={{NAID|10006142329}} |oclc=570461463 |url=https://www.proquest.com/openview/9bd010ff744833cb3a23ef521046adcb/1 }}</ref> presented the ''restart algorithm''.
* B. Curtis Eaves<ref>{{cite journal |last1=Eaves |first1=B. Curtis |title=Homotopies for computation of fixed points |journal=Mathematical Programming |date=December 1972 |volume=3-3 |issue=1 |pages=1–22 |doi=10.1007/BF01584975 |s2cid=39504380 }}</ref> presented the
* A book by Michael Todd<ref name=":1" /> surveys various algorithms developed until 1976. * [[David Gale]]<ref>{{cite journal |first1=David |last1=Gale |year=1979 |title=The Game of Hex and Brouwer Fixed-Point Theorem |journal=The American Mathematical Monthly |volume=86 |issue=10 |pages=818–827 |doi=10.2307/2320146 |jstor=2320146 }}</ref> showed that computing a fixed point of an ''n''-dimensional function (on the unit ''d''-dimensional cube) is equivalent to deciding who is the winner in a ''d''-dimensional game of [[Hex (board game)|Hex]] (a game with ''d'' players, each of whom needs to connect two opposite faces of a ''d''-cube). Given the desired accuracy ''{{mvar|ε}}''
** Construct a Hex board of size ''kd'', where <math>k > 1/\varepsilon</math>. Each vertex ''z'' corresponds to a point ''z''/''k'' in the unit ''n''-cube.
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