Fixed point (mathematics)

In mathematics we mean by a fixed point a point that is mapped by a given figure on itself. The fixed points of a mirror image, the points of the mirror axis. A point reflection has only one fixed point, namely the center.

Definition

Be a quantity and a function. Then is a point fixed point, if it satisfies the equation.

Comments

  • Is a linear map on the vector space, then it is called the fixed points of also Fixvektoren. In particular Fixvektoren eigenvectors are therefore of respect to the eigenvalue 1
  • Since each equation into a fixed point form can deal with convert, fixed-point equations are a prototype of nonlinear equations.

Fixed points in the numerical

In addition, the following applies: The fixed point is stable or unstable if, the amount of the derivative of the function under consideration, at the intersection or is. This clearly means that you can even apply the function to the point, without changing it, with a little disorder (or a lot) is amended so as it leads to the fixed point (or leads away from the fixed point ).

Related to the fixed point problem is the problem of " iterated images", which is important in numerical analysis and chaos research. With a given initial value starting, jumping here according to the scheme like stairs between the function and the diagonal back and forth, and indeed to the fixed point towards or away from it, depending on whether the fixed point is stable or unstable. Details include the below book by HG Schuster refer to.

Examples

  • The parabolic function is given by, the two fixing points 0 and 1
  • Be a vector space and the identical picture, so the picture with, then all fixed points.
  • Be the Schwartz space and the continuous Fourier transform. For the density function of the - dimensional normal distribution. Therefore, the density function of the normal distribution is a fixed point of the Fourier transform.

Space with fixed-point property

Definition

A topological space has the fixed point property if every continuous map has a fixed point.

Examples

  • The sphere has the fixed point property is not, because the point reflection at the center has no fixed point.
  • A solid sphere has the fixed point property. This implies the fixed point theorem of Brouwer.

Fixed point theorems

The existence of fixed points is the subject of some important mathematical theorems. The Banach'sche fixed point theorem states that a contraction of a complete metric space has exactly one fixed point. If a self-map is only continuous, the fixed point need not be unique and other fixed point theorems then show only the existence. They usually provide stronger conditions on the space on which the function is defined. For example, the fixed point theorem of Schauder showing the existence of a fixed point in a compact, convex subset of a Banach space. This theorem is a generalization of the fixed point theorem of Brouwer, which states that every continuous mapping of the closed unit ball has a fixed point in itself. In contrast to the other two sets, however, this is only valid in finite spaces, ie, in or in.

The fixed point theorem of Banach also provides convergence and an error estimate of the fixed point iteration in the room considered. This theorem thus gives a concrete numerical method for the computation of fixed points.

Reflection (mathematics) Probability density function Fixed-point theorem Schauder fixed point theorem Fixed-point iteration
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