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In mathematics, a fixed point (sometimes shortened to fixpoint) of a function is a point that is mapped to itself by the function. For example, if f is defined on the real numbers by Euclid, a famous Greek mathematician known as the father of geometry, is shown here in detail from The School of Athens by Raphael. ...
Partial plot of a function f. ...
In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ...
- f(x) = x2 − 3x + 4,
then 2 is a fixed point of f, because f(2) = 2.
Not all functions have fixed points: for example, the function f(x) = x + 1 has no fixed point on the reals, since x is never equal to x + 1 for any real number. In graphical terms, a fixed point means the point (x, f(x)) is on the line y = x, or in other words the graph of f 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. 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. ...
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 In mathematics, iterated functions are the objects of study in fractals and dynamical systems. ...
 converges to x0. How close is "close enough" is sometimes a subtle question. Converge denotes Converge PL a programming language developed by Laurence Tratt Converge, a metalcore band from Massachusetts For the mathematical meaning of this term see Convergence. ...
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 (provided, of course, it is set to find cosines in radians), the fixed point. That is where the graph of the cosine function intersects the line y = x, and this is no coincidence. In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ...
In mathematics and physics, the radian is a unit of angle measure. ...
The radian (symbol: rad, or a superscript c ( half circle)) is the SI unit of plane angle. ...
Not all fixed points are attractive: for example, x = 0 is a fixed point of the function f(x) = 2x, 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. In dynamical systems, an attractor is a set to which the system evolves after a long enough time. ...
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. 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. ...
Applications In many fields, equilibrium 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. Look up equilibrium in Wiktionary, the free dictionary. ...
The word stability has a number of technical meanings, all related to the common meaning of the word. ...
Buyers bargain for good prices while sellers put forth their best front in Chichicastenango Market, Guatemala. ...
In game theory, the Nash equilibrium (named after John Nash, who proposed it) is a kind of solution concepts of a game involving two or more players, where no player has anything to gain by changing only his or her own strategy. ...
Game theory is a branch of applied mathematics that studies strategic situations where players choose different actions in an attempt to maximize their returns. ...
In game theory, the best response is the strategy in a single period that creates the most favorable immediate outcome for the current player, taking other players strategies as given. ...
In compilers, fixed point computations are used for whole program analysis, which is 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. A compiler is a computer program that translates a computer program written in one computer language (called the source language) into an equivalent program written in another computer language (called the output or the target language). ...
It has been suggested that Loop optimization be merged into this article or section. ...
PageRank is a patented method (an algorithm) to assign 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. ...
In mathematics, a linear transformation (also called linear map or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ...
The World Wide Web (WWW or simply the Web) is a global information space which people can read-from and write-to via a large number of different Internet-connected devices (e. ...
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. Saul Aaron Kripke (born in November, 1940, Omaha, Nebraska) is an American philosopher and logician now emeritus from Princeton and professor of philosophy at CUNY Graduate Center. ...
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