Fixed Point (mathematics)

Information about Fixed Point (mathematics)

In mathematics, a fixed point (sometimes shortened to fixpoint) of a function is a point that is mapped to itself by the function. That is to say, $\text{[math]}$ is a fixed point of the function $\text{[math]}$ if and only if $\text{[math]}$ . For example, if $\text{[math]}$ is defined on the real numbers by

$\text{[math]}$

then 2 is a fixed point of $\text{[math]}$ , because $\text{[math]}$ .

Not all functions have fixed points: for example, if $\text{[math]}$ is a function defined on the real numbers as $\text{[math]}$ , then it has no fixed points, since $\text{[math]}$ is never equal to $\text{[math]}$ for any real number. In graphical terms, a fixed point means the point $\text{[math]}$ is on the line $\text{[math]}$ , or in other words the graph of $\text{[math]}$ has a point in common with that line. The example is a case where the graph and the line are a pair of parallel lines.

Points which come back to the same value after a finite number of iterations of the function are known as periodic points; a fixed point is a periodic point with period equal to one.

Attractive fixed points

An attractive fixed point of a function f is a fixed point x₀ of f such that for any value of x in the domain that is close enough to x₀, the iterated function sequence

$\text{[math]}$

converges to x₀. How close is "close enough" is sometimes a subtle question.

The natural cosine function ("natural" means in radians, not degrees or other units) has exactly one fixed point, which is attractive. In this case, "close enough" is not a stringent criterion at all -- to demonstrate this, start with any real number and repeatedly press the cos key on a calculator. It quickly converges to about 0.73908513, which is a fixed point. That is where the graph of the cosine function intersects the line $\text{[math]}$ .

Not all fixed points are attractive: for example, $\text{[math]}$ is a fixed point of the function $\text{[math]}$ , but iteration of this function for any value other than zero rapidly diverges.

Attractive fixed points are a special case of a wider mathematical concept of attractors.

An attractive fixed point is said to be a stable fixed point if it is also Lyapunov stable.

A fixed point is said to be a neutrally stable fixed point if it is Lyapunov stable but not attracting. The center of a second order homogeneous linear differential equation is an example of a neutrally stable fixed point.

Theorems guaranteeing fixed points

There are numerous theorems in different parts of mathematics that guarantee that functions, under certain circumstances, must have one or more fixed points. These are amongst the most basic qualitative results available: such fixed-point theorems that apply in generality are very valuable insights.

Applications

In many fields, equilibria or stability are fundamental concepts that can be described in terms of fixed points. For example, in economics, a Nash equilibrium of a game is a fixed point of the game's best response correspondence.

In compilers, fixed point computations are used for whole program analysis, which are often required to do code optimization. The vector of PageRank values of all web pages is the fixed point of a linear transformation derived from the World Wide Web's link structure.

Logician Saul Kripke makes use of fixed points in his influential theory of truth. He shows how one can generate a partially defined truth predicate (one which remains undefined for problematic sentences like "This sentence is not true"), by recursively defining "truth" starting from the segment of a language which contains no occurrences of the word, and continuing until the process ceases to yield any newly well-defined sentences. (This will take a denumerable infinity of steps.) That is, for a language L, let L-prime be the language generated by adding to L, for each sentence S in L, the sentence "S is true." A fixed point is reached when L-prime is L; at this point sentences like "This sentence is not true" remain undefined, so, according to Kripke, the theory is suitable for a natural language which contains its own truth predicate.

External links

Animations for Fixed Point Iteration

Mathematics (colloquially, maths or math) is the body of knowledge centered on such concepts as quantity, structure, space, and change, and also the academic discipline that studies them. Benjamin Peirce called it "the science that draws necessary conclusions".
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function expresses dependence between two quantities, one of which is given (the independent variable, argument of the function, or its "input") and the other produced (the dependent variable, value of the function, or "output").
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“Iff” redirects here. For other uses, see IFF.

If and only if, in logic and fields that rely on it such as mathematics and philosophy, is a logical connective between statements which means that the truth of either one of the statements
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In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2.4871773339…. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as π and
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graph of a function f is the collection of all ordered pairs (x,f(x)). In particular, graph means the graphical representation of this collection, in the form of a curve or surface, together with axes, etc.
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Parallel is a term in geometry and in everyday life that refers to a property in Euclidean space of two or more lines or planes, or a combination of these. The existence and properties of parallel lines are the basis of Euclid's parallel postulate.
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In mathematics, iterated functions are the objects of deep study in fractals and dynamical systems. An iterated function is a function which is composed with itself, repeatedly, a process called iteration.
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periodic point of a function is a point which returns to itself after a certain number of function iterations or a certain amount of time.

Iterated functions

Given an endomorphism f on a set X

a point x in X
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Convergence is the approach toward a definite value, a definite point, a common view or opinion, or toward a fixed or equilibrium state.

Convergence or convergent may also refer to:

Convergence (Mexico), a political party in Mexico

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trigonometric functions (also called circular functions) are functions of an angle. They are important in the study of triangles and modeling periodic phenomena, among many other applications.
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radian, in mathematics, is a unit of plane angle, equal to 180/π degrees, or about 57.2958 degrees. It is represented by the symbol "rad" or, more rarely, by the superscript c (for "circular measure"). For example, an angle of 1.2 radians would be written as "1.
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An attractor is a set to which a dynamical system evolves after a long enough time. That is, points that get close enough to the attractor remain close even if slightly disturbed.
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In mathematics, the notion of Lyapunov stability occurs in the study of dynamical systems. In simple terms, if all solutions of the dynamical system that start out near an equilibrium point stay near forever, then is Lyapunov stable.
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In mathematics, a fixed-point theorem is a result saying that a function F will have at least one fixed point (a point x for which F(x) = x), under some conditions on F that can be stated in general terms.
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Stability can refer to:

Aircraft flight Stability (aircraft)
Atmospheric stability, a measure of the turbulence in the ambient atmosphere
BIBO stability (Bounded Input, Bounded Output stability), in signal processing and control theory, part of electrical

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Economics is the social science that studies the production, distribution, and consumption of goods and services. The term economics comes from the Greek for oikos (house) and nomos (custom or law), hence "rules of the house(hold).
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In game theory, the Nash equilibrium (named after John Forbes Nash, who proposed it) is a solution concept of a game involving two or more players, in which no player has anything to gain by changing only his or her own strategy unilaterally.
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Game theory is a branch of applied mathematics that is often used in the context of economics. It studies strategic interactions between agents. In strategic games, agents choose strategies which will maximize their return, given the strategies the other agents choose.
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In game theory, the best response is the strategy (or strategies) which produces the most favorable immediate outcome for the current player, taking other players' strategies as given (; ).
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compiler is a computer program (or set of programs) that translates text written in a computer language (the source language) into another computer language (the target language).
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In computing, optimization is the process of modifying a system to make some aspect of it work more efficiently or use fewer resources. For instance, a computer program may be optimized so that it executes more rapidly, or is capable of operating within a reduced amount of memory
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PageRank is a link analysis algorithm that assigns a numerical weighting to each element of a hyperlinked set of documents, such as the World Wide Web, with the purpose of "measuring" its relative importance within the set.
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In mathematics, a linear map (also called a linear transformation or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication.
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World Wide Web (commonly shortened to the Web) is a system of interlinked, hypertext documents accessed via the Internet. With a web browser, a user views web pages that may contain text, images, videos, and other multimedia and navigates between them using hyperlinks.
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Saul Aaron Kripke (born in November 13, 1940 in Bay Shore, New York) is an American philosopher and logician now emeritus from Princeton and teaches as distinguished professor of philosophy at CUNY Graduate Center.
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eigenvector of the transformation and the blue vector is not. Since the red vector was neither stretched nor compressed, its eigenvalue is 1. All vectors with the same vertical direction - i.e., parallel to this vector - are also eigenvectors, with the same eigenvalue.
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In mathematics, the point is an equilibrium point for the differential equation

if for all .

Similarly, the point is an equilibrium point (or fixed point) for the difference equation

if for .
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