Fixed point (mathematics)

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In mathematics, a fixed point 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

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 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.

Table of contents

1 Attractive fixed points

2 Theorems guaranteeing fixed points

3 Applications

4 See also

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 sequence

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, the fixed point. That's where the graph of the cosine function intersects the line y = x, and this is no coincidence.

Not all fixed points are attractive: for example, x=0 is a fixed point of the function x -> x²+x, 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.

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 compilers, fixed point computations are used for whole program analysis, which is often required to do code optimization. Fixed point is also used in finding PageRank of a webpage.

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