Fixed point (mathematics) - Revision history

Jochen Burghardt: /* top */ suggest to use image with known definition and known fixpoint values

2024-09-18T10:23:13Z

top: suggest to use image with known definition and known fixpoint values

← Previous revision		Revision as of 10:23, 18 September 2024
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	{{Short description\|Element mapped to itself by a mathematical function}}		{{Short description\|Element mapped to itself by a mathematical function}}
	{{hatnote\|1=Fixed points in mathematics are not to be confused with [[Fixed point (disambiguation)\|other uses of "fixed point"]], or [[stationary point]]s where <math> f'(x) = 0</math>.}}		{{hatnote\|1=Fixed points in mathematics are not to be confused with [[Fixed point (disambiguation)\|other uses of "fixed point"]], or [[stationary point]]s where <math> f'(x) = 0</math>.}}
	[[File:~~Fixed~~ ~~point example~~.svg\|thumb\|~~right~~\|A function ~~with~~ ~~three~~ fixed points]]		[[File:Fixpoint012 svg.svg\|thumb\|300px\|The function <math>f(x)=x^3 - 3x^2 + 3x</math> (shown in red) has the fixed points 0, 1, and 2.]]

	In [[mathematics]], a '''fixed point''' (sometimes shortened to '''fixpoint'''), also known as an '''invariant point''', is a value that does not change under a given [[transformation (mathematics)\|transformation]]. Specifically, for [[function (mathematics)\|functions]], a fixed point is an element that is mapped to itself by the function. Any set of fixed points of a transformation is also an [[invariant set]].		In [[mathematics]], a '''fixed point''' (sometimes shortened to '''fixpoint'''), also known as an '''invariant point''', is a value that does not change under a given [[transformation (mathematics)\|transformation]]. Specifically, for [[function (mathematics)\|functions]], a fixed point is an element that is mapped to itself by the function. Any set of fixed points of a transformation is also an [[invariant set]].

96.244.78.198 at 16:20, 7 September 2024

2024-09-07T16:20:41Z

← Previous revision		Revision as of 16:20, 7 September 2024
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	{{Short description\|Element mapped to itself by a mathematical function}}		{{Short description\|Element mapped to itself by a mathematical function}}
	{{hatnote\|1=Fixed points in mathematics are not to be confused with [[Fixed point (disambiguation)\|other uses of "fixed point"]], or [[stationary point]]s where {{math~~\|1=''~~f'''(''x'') = 0}}.}}		{{hatnote\|1=Fixed points in mathematics are not to be confused with [[Fixed point (disambiguation)\|other uses of "fixed point"]], or [[stationary point]]s where <math> f'(x) = 0</math>.}}
	[[File:Fixed point example.svg\|thumb\|right\|A function with three fixed points]]		[[File:Fixed point example.svg\|thumb\|right\|A function with three fixed points]]

Mathnerd314159: what is a "mathematical point"?

2024-08-31T15:49:56Z

what is a "mathematical point"?

← Previous revision		Revision as of 15:49, 31 August 2024
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	[[File:Fixed point example.svg\|thumb\|right\|A function with three fixed points]]		[[File:Fixed point example.svg\|thumb\|right\|A function with three fixed points]]

	In [[mathematics]], a '''fixed point''' (sometimes shortened to '''fixpoint''') ~~of a [[Map (mathematics)\|map]]~~, also known as an '''invariant point''', is a ~~[[mathematical~~ ~~point]],~~ ~~such~~ as a [[~~Value~~ (mathematics)\|~~value~~]] ~~or a [[Coordinate system\|coordinate point]], that does not change when applying the given map~~. Specifically, for ~~maps that are~~ [[function (mathematics)\|functions]] ~~or [[Partial function\|partial functions]] of sets~~, a fixed point ~~of the map~~ is ~~a [[Element (mathematics)\|set~~ element]] that is mapped to itself by the ~~function or partial~~ function. Any set of fixed points of a ~~map~~ is also an [[invariant set]].		In [[mathematics]], a '''fixed point''' (sometimes shortened to '''fixpoint'''), also known as an '''invariant point''', is a value that does not change under a given [[transformation (mathematics)\|transformation]]. Specifically, for [[function (mathematics)\|functions]], a fixed point is an element that is mapped to itself by the function. Any set of fixed points of a transformation is also an [[invariant set]].

	== Fixed point of a function ==		== Fixed point of a function ==
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	{{Main\|Fixed-point iteration}}		{{Main\|Fixed-point iteration}}

	In [[numerical analysis]], ''fixed-point iteration'' is a method of computing fixed points of a function. Specifically, given a function <math>f</math> with the same domain and codomain, ~~also called a [[Transformation (function)\|transformation]], and~~ a point <math>x_0</math> in the domain of <math>f</math>, the fixed-point iteration is		In [[numerical analysis]], ''fixed-point iteration'' is a method of computing fixed points of a function. Specifically, given a function <math>f</math> with the same domain and codomain, a point <math>x_0</math> in the domain of <math>f</math>, the fixed-point iteration is

	<math display="block">x_{n+1}=f(x_n), \, n=0, 1, 2, \dots</math>		<math display="block">x_{n+1}=f(x_n), \, n=0, 1, 2, \dots</math>

RowanElder: /* Fixed point iteration */ Found a place to add a link to "transformation (function)" so that page doesn't lose its inbound link from this page where it fits.

2024-08-31T14:31:35Z

Fixed point iteration: Found a place to add a link to "transformation (function)" so that page doesn't lose its inbound link from this page where it fits.

← Previous revision		Revision as of 14:31, 31 August 2024
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	{{Main\|Fixed-point iteration}}		{{Main\|Fixed-point iteration}}

	In [[numerical analysis]], ''fixed-point iteration'' is a method of computing fixed points of a function. Specifically, given a function <math>f</math> with the same domain and codomain, a point <math>x_0</math> in the domain of <math>f</math>, the fixed-point iteration is		In [[numerical analysis]], ''fixed-point iteration'' is a method of computing fixed points of a function. Specifically, given a function <math>f</math> with the same domain and codomain, also called a [[Transformation (function)\|transformation]], and a point <math>x_0</math> in the domain of <math>f</math>, the fixed-point iteration is

	<math display="block">x_{n+1}=f(x_n), \, n=0, 1, 2, \dots</math>		<math display="block">x_{n+1}=f(x_n), \, n=0, 1, 2, \dots</math>

RowanElder: Another go at the problem described in the talk page, this time favoring correctness over consistency with the article's lead as I found it.

2024-08-31T14:23:00Z

Another go at the problem described in the talk page, this time favoring correctness over consistency with the article's lead as I found it.

← Previous revision		Revision as of 14:23, 31 August 2024
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	[[File:Fixed point example.svg\|thumb\|right\|A function with three fixed points]]		[[File:Fixed point example.svg\|thumb\|right\|A function with three fixed points]]

	In [[mathematics]], a '''fixed point''' (sometimes shortened to '''fixpoint'''), also known as an '''invariant point''', is a ~~value~~ ~~that~~ ~~does~~ ~~not change under~~ a ~~given~~ [[~~transformation~~ (mathematics)\|~~transformation~~]]. Specifically, for [[function (mathematics)\|functions]], a fixed point is an element that is mapped to itself by the function. Any set of fixed points of a ~~transformation~~ is also an [[invariant set]].		In [[mathematics]], a '''fixed point''' (sometimes shortened to '''fixpoint''') of a [[Map (mathematics)\|map]], also known as an '''invariant point''', is a [[mathematical point]], such as a [[Value (mathematics)\|value]] or a [[Coordinate system\|coordinate point]], that does not change when applying the given map. Specifically, for maps that are [[function (mathematics)\|functions]] or [[Partial function\|partial functions]] of sets, a fixed point of the map is a [[Element (mathematics)\|set element]] that is mapped to itself by the function or partial function. Any set of fixed points of a map is also an [[invariant set]].

	== Fixed point of a function ==		== Fixed point of a function ==

Jochen Burghardt: /* Fixed point of a function */ mention non-disjointness of dom,codom; mv graphical interpretation up, before examples; try to fix it without introduction of "identity function"; refer to picture; inline example parabola

2024-08-31T06:34:50Z

Fixed point of a function: mention non-disjointness of dom,codom; mv graphical interpretation up, before examples; try to fix it without introduction of "identity function"; refer to picture; inline example parabola

← Previous revision		Revision as of 06:34, 31 August 2024
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	Formally, {{mvar\|c}} is a fixed point of a function {{mvar\|f}} if {{mvar\|c}} belongs to both the [[domain of a function\|domain]] and the [[codomain]] of {{mvar\|f}}, and {{math\|1=''f''(''c'') = ''c''}}.		Formally, {{mvar\|c}} is a fixed point of a function {{mvar\|f}} if {{mvar\|c}} belongs to both the [[domain of a function\|domain]] and the [[codomain]] of {{mvar\|f}}, and {{math\|1=''f''(''c'') = ''c''}}.
			In particular, {{mvar\|f}} cannot have any fixed point if its domain is disjoint from its codomain.
			If {{math\|''f''}} is defined on the [[real number]]s, it corresponds, in graphical terms, to a [[curve]] in the [[Euclidean plane]], and each fixed-point {{math\|''c''}} corresponds to an intersection of the curve with the line {{math\|1=''y'' = ''x''}}, cf. picture.

	For example, if {{math\|''f''}} is defined on the [[real number]]s by		For example, if {{math\|''f''}} is defined on the [[real number]]s by
	<math ~~display="block"~~> f(x) = x^2 - 3 x + 4,</math>		<math> f(x) = x^2 - 3 x + 4,</math>
	then 2 is a fixed point of {{math\|''f''}}, because {{math\|1=''f''(2) = 2}}.		then 2 is a fixed point of {{math\|''f''}}, because {{math\|1=''f''(2) = 2}}.

	Not all functions have fixed points: for example, {{math\|1=''f''(''x'') = ''x'' + 1}} has no fixed points because {{math\|''x''}} is never equal to {{math\|''x'' + 1}} for any real number. In graphical terms, a fixed-point {{math\|''x''}} means the point {{math\|(''x'', ''f''(''x''))}} is on the line {{math\|1=''y'' = ''x''}}, or in other words the [[graph of a function\|graph]] of {{math\|''f''}} has a point in common with that line.		Not all functions have fixed points: for example, {{math\|1=''f''(''x'') = ''x'' + 1}} has no fixed points because {{math\|''x''}} is never equal to {{math\|''x'' + 1}} for any real number.

	== Fixed point iteration ==		== Fixed point iteration ==

Jochen Burghardt: Partly undid revision 1243150976 by RowanElder (talk): rm unneccessary restriction dom(f)=ran(f); convexity can't be defined in topological spaces in general, afaik

2024-08-31T06:13:45Z

Partly undid revision 1243150976 by RowanElder (talk): rm unneccessary restriction dom(f)=ran(f); convexity can't be defined in topological spaces in general, afaik

← Previous revision		Revision as of 06:13, 31 August 2024
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	[[File:Fixed point example.svg\|thumb\|right\|A function with three fixed points]]		[[File:Fixed point example.svg\|thumb\|right\|A function with three fixed points]]

	In [[mathematics]], a '''fixed point''' ~~of a [[Transformation (function)\|transformation]]~~ (sometimes shortened to '''fixpoint'''), also known as an '''invariant point''', is a value that does not change under ~~that~~ given transformation. ~~More~~ ~~explicitly~~, for [[function (mathematics)\|~~mathematical~~ functions~~]] with a common [[Domain of a function\|domain]] and [[codomain~~]], a fixed point ~~of the function~~ is ~~any~~ element ~~of the domain~~ that is mapped to itself by the function. Any set of fixed points of a transformation is also ~~fixed by the transformation and is called~~ an ~~'''~~[[invariant set]]~~'''~~.		In [[mathematics]], a '''fixed point''' (sometimes shortened to '''fixpoint'''), also known as an '''invariant point''', is a value that does not change under a given [[transformation (mathematics)\|transformation]]. Specifically, for [[function (mathematics)\|functions]], a fixed point is an element that is mapped to itself by the function. Any set of fixed points of a transformation is also an [[invariant set]].

	== Fixed point of a function ==		== Fixed point of a function ==
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	then 2 is a fixed point of {{math\|''f''}}, because {{math\|1=''f''(2) = 2}}.		then 2 is a fixed point of {{math\|''f''}}, because {{math\|1=''f''(2) = 2}}.

	Not all functions have fixed points: for example {{math\|1=''f''(''x'') = ''x'' + 1}} has no fixed points because {{math\|''x''}} is never equal to {{math\|''x'' + 1}} for any real number. In graphical terms, a fixed-point {{math\|''x''}} ~~of a function of one variable is an x such that~~ the point {{math\|(''x'', ''f''(''x''))}} is on the line {{math\|1=''y'' = ''x''}}, or in other words the [[graph of a function\|graph]] of {{math\|''f''}} has a point in common with ~~the~~ ~~graph of the identity function {{math\|1=''i''(''x'') = ''x''}}~~.		Not all functions have fixed points: for example, {{math\|1=''f''(''x'') = ''x'' + 1}} has no fixed points because {{math\|''x''}} is never equal to {{math\|''x'' + 1}} for any real number. In graphical terms, a fixed-point {{math\|''x''}} means the point {{math\|(''x'', ''f''(''x''))}} is on the line {{math\|1=''y'' = ''x''}}, or in other words the [[graph of a function\|graph]] of {{math\|''f''}} has a point in common with that line.

	== Fixed point iteration ==		== Fixed point iteration ==
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	The FPP is a [[topological invariant]], i.e., it is preserved by any [[homeomorphism]]. The FPP is also preserved by any [[Retraction (topology)\|retraction]].		The FPP is a [[topological invariant]], i.e., it is preserved by any [[homeomorphism]]. The FPP is also preserved by any [[Retraction (topology)\|retraction]].

	According to the [[Brouwer fixed-point theorem]], every [[compact space\|compact]] and [[convex set\|convex]] [[subset]] of a [[Euclidean space]] has the FPP. Compactness alone does not imply the FPP, and convexity is not a ~~topologically invariant~~ property, so it makes sense to ask how to topologically characterize the FPP. In 1932 [[Karol Borsuk\|Borsuk]] asked whether compactness together with [[contractible space\|contractibility]] could be a necessary and sufficient condition for the FPP to hold. The problem was open for 20 years until the conjecture was disproved by ~~Shin'ichi~~ Kinoshita, who found an example of a compact contractible space without the FPP.<ref>{{cite journal \|last=Kinoshita \|first=Shin'ichi \|year=1953 \|title=On Some Contractible Continua without Fixed Point Property \|journal=[[Fundamenta Mathematicae\|Fund. Math.]] \|volume=40 \|issue=1 \|pages=96–98 \|doi=10.4064/fm-40-1-96-98 \|issn=0016-2736 \|doi-access=free}}</ref>		According to the [[Brouwer fixed-point theorem]], every [[compact space\|compact]] and [[convex set\|convex]] [[subset]] of a [[Euclidean space]] has the FPP. Compactness alone does not imply the FPP, and convexity is not even a topological property, so it makes sense to ask how to topologically characterize the FPP. In 1932 [[Karol Borsuk\|Borsuk]] asked whether compactness together with [[contractible space\|contractibility]] could be a necessary and sufficient condition for the FPP to hold. The problem was open for 20 years until the conjecture was disproved by Kinoshita, who found an example of a compact contractible space without the FPP.<ref>{{cite journal \|last=Kinoshita \|first=Shin'ichi \|year=1953 \|title=On Some Contractible Continua without Fixed Point Property \|journal=[[Fundamenta Mathematicae\|Fund. Math.]] \|volume=40 \|issue=1 \|pages=96–98 \|doi=10.4064/fm-40-1-96-98 \|issn=0016-2736 \|doi-access=free}}</ref>

	==Fixed points of partial orders ==		==Fixed points of partial orders ==

RowanElder: General copyediting, problems remain

2024-08-30T20:25:27Z

General copyediting, problems remain

Jochen Burghardt at 18:39, 26 March 2024

2024-03-26T18:39:48Z

← Previous revision		Revision as of 18:39, 26 March 2024
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	For example, the [[Banach fixed-point theorem]] (1922) gives a general criterion guaranteeing that, if it is satisfied, [[fixed-point iteration]] will always converge to a fixed point.		For example, the [[Banach fixed-point theorem]] (1922) gives a general criterion guaranteeing that, if it is satisfied, [[fixed-point iteration]] will always converge to a fixed point.

	The [[Brouwer fixed-point theorem]] (1911) says that any [[continuous function]] from the closed [[unit ball]] in ''n''-dimensional [[~~Euclidean spaces\|~~Euclidean space]] to itself must have a fixed point, but it doesn't describe how to find the fixed point.		The [[Brouwer fixed-point theorem]] (1911) says that any [[continuous function]] from the closed [[unit ball]] in ''n''-dimensional [[Euclidean space]] to itself must have a fixed point, but it doesn't describe how to find the fixed point.

	The [[Lefschetz fixed-point theorem]] (and the [[Nielsen theory\|Nielsen fixed-point theorem]]) from [[algebraic topology]] give a way to count fixed points.		The [[Lefschetz fixed-point theorem]] (and the [[Nielsen theory\|Nielsen fixed-point theorem]]) from [[algebraic topology]] give a way to count fixed points.
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	* The stationary distribution of a [[Markov chain]] is the fixed point of the one step transition probability function.		* The stationary distribution of a [[Markov chain]] is the fixed point of the one step transition probability function.
	* Fixed points are used to finding formulas for [[Iterated function\|iterated functions]].		* Fixed points are used to finding formulas for [[Iterated function\|iterated functions]].
	* [[doi:10.1016/j.cam.2018.04.057\|Algorithms for zeros of two accretive operators for solving convex minimization problems and its application to image restoration problems]]
	*Fixed Point Algorithms for Optimization ([[List of algorithms\|see more algorithms]])
	*[https://tacs-coe.com/highlight-research/ Splitting algorithms for Convex Minimization Problems and Image Restoration Problems]

	==See also==		==See also==
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	*[[Invariant (mathematics)]]		*[[Invariant (mathematics)]]

	*[[ Image restoration problems]]
	{{div col end}}		{{div col end}}

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	==External links==		==External links==
	* [http://www.osaka-ue.ac.jp/zemi/nishiyama/math2010/fixedpoint.pdf An Elegant Solution for Drawing a Fixed Point]		* [http://www.osaka-ue.ac.jp/zemi/nishiyama/math2010/fixedpoint.pdf An Elegant Solution for Drawing a Fixed Point]
	* https://fixedpointkmutt.wordpress.com/
	* https://www.routledge.com/authors/i17016-poom-kumam#

	[[Category:Fixed points (mathematics)\| ]]		[[Category:Fixed points (mathematics)\| ]]

PKMoobin: /* Applications */

2024-03-26T07:33:00Z

Applications

← Previous revision		Revision as of 07:33, 26 March 2024
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	* Fixed points are used to finding formulas for [[Iterated function\|iterated functions]].		* Fixed points are used to finding formulas for [[Iterated function\|iterated functions]].
	* [[doi:10.1016/j.cam.2018.04.057\|Algorithms for zeros of two accretive operators for solving convex minimization problems and its application to image restoration problems]]		* [[doi:10.1016/j.cam.2018.04.057\|Algorithms for zeros of two accretive operators for solving convex minimization problems and its application to image restoration problems]]
			*Fixed Point Algorithms for Optimization ([[List of algorithms\|see more algorithms]])
	*
			*[https://tacs-coe.com/highlight-research/ Splitting algorithms for Convex Minimization Problems and Image Restoration Problems]

	==See also==		==See also==

← Previous revision		Revision as of 10:23, 18 September 2024
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	{{Short description\|Element mapped to itself by a mathematical function}}		{{Short description\|Element mapped to itself by a mathematical function}}
	{{hatnote\|1=Fixed points in mathematics are not to be confused with [[Fixed point (disambiguation)\|other uses of "fixed point"]], or [[stationary point]]s where <math> f'(x) = 0</math>.}}		{{hatnote\|1=Fixed points in mathematics are not to be confused with [[Fixed point (disambiguation)\|other uses of "fixed point"]], or [[stationary point]]s where <math> f'(x) = 0</math>.}}
	[[File:~~Fixed~~ ~~point example~~.svg\|thumb\|~~right~~\|A function ~~with~~ ~~three~~ fixed points]]		[[File:Fixpoint012 svg.svg\|thumb\|300px\|The function <math>f(x)=x^3 - 3x^2 + 3x</math> (shown in red) has the fixed points 0, 1, and 2.]]

	In [[mathematics]], a '''fixed point''' (sometimes shortened to '''fixpoint'''), also known as an '''invariant point''', is a value that does not change under a given [[transformation (mathematics)\|transformation]]. Specifically, for [[function (mathematics)\|functions]], a fixed point is an element that is mapped to itself by the function. Any set of fixed points of a transformation is also an [[invariant set]].		In [[mathematics]], a '''fixed point''' (sometimes shortened to '''fixpoint'''), also known as an '''invariant point''', is a value that does not change under a given [[transformation (mathematics)\|transformation]]. Specifically, for [[function (mathematics)\|functions]], a fixed point is an element that is mapped to itself by the function. Any set of fixed points of a transformation is also an [[invariant set]].

← Previous revision		Revision as of 16:20, 7 September 2024
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	{{Short description\|Element mapped to itself by a mathematical function}}		{{Short description\|Element mapped to itself by a mathematical function}}
	{{hatnote\|1=Fixed points in mathematics are not to be confused with [[Fixed point (disambiguation)\|other uses of "fixed point"]], or [[stationary point]]s where {{math~~\|1=''~~f'''(''x'') = 0}}.}}		{{hatnote\|1=Fixed points in mathematics are not to be confused with [[Fixed point (disambiguation)\|other uses of "fixed point"]], or [[stationary point]]s where <math> f'(x) = 0</math>.}}
	[[File:Fixed point example.svg\|thumb\|right\|A function with three fixed points]]		[[File:Fixed point example.svg\|thumb\|right\|A function with three fixed points]]

← Previous revision		Revision as of 06:34, 31 August 2024
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	Formally, {{mvar\|c}} is a fixed point of a function {{mvar\|f}} if {{mvar\|c}} belongs to both the [[domain of a function\|domain]] and the [[codomain]] of {{mvar\|f}}, and {{math\|1=''f''(''c'') = ''c''}}.		Formally, {{mvar\|c}} is a fixed point of a function {{mvar\|f}} if {{mvar\|c}} belongs to both the [[domain of a function\|domain]] and the [[codomain]] of {{mvar\|f}}, and {{math\|1=''f''(''c'') = ''c''}}.
			In particular, {{mvar\|f}} cannot have any fixed point if its domain is disjoint from its codomain.
			If {{math\|''f''}} is defined on the [[real number]]s, it corresponds, in graphical terms, to a [[curve]] in the [[Euclidean plane]], and each fixed-point {{math\|''c''}} corresponds to an intersection of the curve with the line {{math\|1=''y'' = ''x''}}, cf. picture.

	For example, if {{math\|''f''}} is defined on the [[real number]]s by		For example, if {{math\|''f''}} is defined on the [[real number]]s by
	<math ~~display="block"~~> f(x) = x^2 - 3 x + 4,</math>		<math> f(x) = x^2 - 3 x + 4,</math>
	then 2 is a fixed point of {{math\|''f''}}, because {{math\|1=''f''(2) = 2}}.		then 2 is a fixed point of {{math\|''f''}}, because {{math\|1=''f''(2) = 2}}.

	Not all functions have fixed points: for example, {{math\|1=''f''(''x'') = ''x'' + 1}} has no fixed points because {{math\|''x''}} is never equal to {{math\|''x'' + 1}} for any real number. In graphical terms, a fixed-point {{math\|''x''}} means the point {{math\|(''x'', ''f''(''x''))}} is on the line {{math\|1=''y'' = ''x''}}, or in other words the [[graph of a function\|graph]] of {{math\|''f''}} has a point in common with that line.		Not all functions have fixed points: for example, {{math\|1=''f''(''x'') = ''x'' + 1}} has no fixed points because {{math\|''x''}} is never equal to {{math\|''x'' + 1}} for any real number.

	== Fixed point iteration ==		== Fixed point iteration ==

← Previous revision		Revision as of 20:25, 30 August 2024
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	[[File:Fixed point example.svg\|thumb\|right\|A function with three fixed points]]		[[File:Fixed point example.svg\|thumb\|right\|A function with three fixed points]]

	In [[mathematics]], a '''fixed point''' (sometimes shortened to '''fixpoint'''), also known as an '''invariant point''', is a value that does not change under a given [[transformation ~~(mathematics)\|transformation]].~~ ~~Specifically~~, for [[function (mathematics)\|functions]], a fixed point is an element that is mapped to itself by the function.		In [[mathematics]], a '''fixed point''' of a [[Transformation (function)\|transformation]] (sometimes shortened to '''fixpoint'''), also known as an '''invariant point''', is a value that does not change under that given transformation. More explicitly, for [[function (mathematics)\|mathematical functions]] with a common [[Domain of a function\|domain]] and [[codomain]], a fixed point of the function is any element of the domain that is mapped to itself by the function. Any set of fixed points of a transformation is also fixed by the transformation and is called an '''[[invariant set]]'''.

	== Fixed point of a function ==		== Fixed point of a function ==
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	then 2 is a fixed point of {{math\|''f''}}, because {{math\|1=''f''(2) = 2}}.		then 2 is a fixed point of {{math\|''f''}}, because {{math\|1=''f''(2) = 2}}.

	Not all functions have fixed points: for example, {{math\|1=''f''(''x'') = ''x'' + 1}}, has no fixed points, ~~since~~ {{math\|''x''}} is never equal to {{math\|''x'' + 1}} for any real number. In graphical terms, a fixed-point {{math\|''x''}} ~~means~~ the point {{math\|(''x'', ''f''(''x''))}} is on the line {{math\|1=''y'' = ''x''}}, or in other words the [[graph of a function\|graph]] of {{math\|''f''}} has a point in common with ~~that~~ ~~line~~.		Not all functions have fixed points: for example {{math\|1=''f''(''x'') = ''x'' + 1}} has no fixed points because {{math\|''x''}} is never equal to {{math\|''x'' + 1}} for any real number. In graphical terms, a fixed-point {{math\|''x''}} of a function of one variable is an x such that the point {{math\|(''x'', ''f''(''x''))}} is on the line {{math\|1=''y'' = ''x''}}, or in other words the [[graph of a function\|graph]] of {{math\|''f''}} has a point in common with the graph of the identity function {{math\|1=''i''(''x'') = ''x''}}.

	== Fixed point iteration ==		== Fixed point iteration ==
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	there exists <math>x \in X</math> such that <math>f(x)=x</math>.		there exists <math>x \in X</math> such that <math>f(x)=x</math>.

	The FPP is a [[topological invariant]], i.e. is preserved by any [[homeomorphism]]. The FPP is also preserved by any [[Retraction (topology)\|retraction]].		The FPP is a [[topological invariant]], i.e., it is preserved by any [[homeomorphism]]. The FPP is also preserved by any [[Retraction (topology)\|retraction]].

	According to the [[Brouwer fixed-point theorem]], every [[compact space\|compact]] and [[convex set\|convex]] [[subset]] of a [[Euclidean space]] has the FPP. Compactness alone does not imply the FPP, and convexity is not ~~even~~ a ~~topological~~ property, so it makes sense to ask how to topologically characterize the FPP. In 1932 [[Karol Borsuk\|Borsuk]] asked whether compactness together with [[contractible space\|contractibility]] could be a necessary and sufficient condition for the FPP to hold. The problem was open for 20 years until the conjecture was disproved by Kinoshita who found an example of a compact contractible space without the FPP.<ref>{{cite journal \|last=Kinoshita \|first=S. \|year=1953 \|title=On Some Contractible Continua without Fixed Point Property \|journal=[[Fundamenta Mathematicae\|Fund. Math.]] \|volume=40 \|issue=1 \|pages=96–98 \|doi=10.4064/fm-40-1-96-98 \|issn=0016-2736 \|doi-access=free}}</ref>		According to the [[Brouwer fixed-point theorem]], every [[compact space\|compact]] and [[convex set\|convex]] [[subset]] of a [[Euclidean space]] has the FPP. Compactness alone does not imply the FPP, and convexity is not a topologically invariant property, so it makes sense to ask how to topologically characterize the FPP. In 1932 [[Karol Borsuk\|Borsuk]] asked whether compactness together with [[contractible space\|contractibility]] could be a necessary and sufficient condition for the FPP to hold. The problem was open for 20 years until the conjecture was disproved by Shin'ichi Kinoshita, who found an example of a compact contractible space without the FPP.<ref>{{cite journal \|last=Kinoshita \|first=Shin'ichi \|year=1953 \|title=On Some Contractible Continua without Fixed Point Property \|journal=[[Fundamenta Mathematicae\|Fund. Math.]] \|volume=40 \|issue=1 \|pages=96–98 \|doi=10.4064/fm-40-1-96-98 \|issn=0016-2736 \|doi-access=free}}</ref>

	==Fixed points of partial orders ==		==Fixed points of partial orders ==
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	In [[order theory]], the [[least fixed point]] of a [[function (mathematics)\|function]] from a [[partially ordered set]] (poset) to itself is the fixed point which is less than each other fixed point, according to the order of the poset. A function need not have a least fixed point, but if it does then the least fixed point is unique.		In [[order theory]], the [[least fixed point]] of a [[function (mathematics)\|function]] from a [[partially ordered set]] (poset) to itself is the fixed point which is less than each other fixed point, according to the order of the poset. A function need not have a least fixed point, but if it does then the least fixed point is unique.

	One way to express the [[Knaster–Tarski theorem]] is to say that a [[monotone function]] on a [[complete lattice]] has a [[least ~~fixpoint~~]] that coincides with its least prefixpoint (and similarly its greatest ~~fixpoint~~ coincides with its greatest postfixpoint).<ref>Yde Venema (2008) [http://staff.science.uva.nl/~yde/teaching/ml/mu/mu2008.pdf Lectures on the Modal μ-calculus] {{webarchive\|url=https://web.archive.org/web/20120321162526/http://staff.science.uva.nl/~yde/teaching/ml/mu/mu2008.pdf\|date=March 21, 2012}}</ref>		One way to express the [[Knaster–Tarski theorem]] is to say that a [[monotone function]] on a [[complete lattice]] has a [[least fixed point]] that coincides with its least prefixpoint (and similarly its greatest fixed point coincides with its greatest postfixpoint).<ref>Yde Venema (2008) [http://staff.science.uva.nl/~yde/teaching/ml/mu/mu2008.pdf Lectures on the Modal μ-calculus] {{webarchive\|url=https://web.archive.org/web/20120321162526/http://staff.science.uva.nl/~yde/teaching/ml/mu/mu2008.pdf\|date=March 21, 2012}}</ref>

	==Fixed-point combinator==		==Fixed-point combinator==