Wikipedia / Fixed-point iteration
Wikipedia / Fixed-point iteration
In numerical analysis, fixed-point iteration is a method of computing fixed points of a function.
defined on the real numbers in the domain which gives rise to the sequence of iterated function which is hoped to converge
can be defined on any metric space with values in that same space.
Examples
A first simple and useful example is the Babylonian method for computing the square root of , i.e. the mean value of and ). This is a special case of Newton's method quoted below.
This example does satisfy (at the latest after the first iteration step) the assumptions of the Banach fixed-point theorem for some constant , we say that we have linear convergence. The Banach fixed-point theorem allows one to obtain fixed-point iterations with linear convergence.
The requirement that . However, 0 is not as this function is not, and in fact has no fixed points.
Attracting fixed points
An attracting fixed point of a function is a fixed point of with a neighborhood of "close enough" points around such that for any value of in is contained in and converges to . The basin of attraction of is the largest such neighborhood .
The natural cosine function ("natural" means in radians, not degrees or other units) has exactly one fixed point, and that fixed point is attracting. 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 (checking first that the calculator is in "radians" mode). It eventually converges to the Dottie number.
Not all fixed points are attracting. For example, 0 is a fixed point of the function is repelling.
An attracting 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 linear homogeneous differential equation of the second order is an example of a neutrally stable fixed point.
Multiple attracting points can be collected in an attracting fixed set.
Banach fixed-point theorem
The Banach fixed-point theorem gives a sufficient condition for the existence of attracting fixed points. A contraction mapping defined on a complete metric space is defined on the real line with real values and is Lipschitz continuous, and (2) the function is continuously differentiable in an open neighbourhood of a fixed point .
Although there are other fixed-point theorems, this one in particular is very useful because not all fixed-points are attractive. When constructing a fixed-point iteration, it is very important to make sure it converges to the fixed point. We can usually use the Banach fixed-point theorem to show that the fixed point is attractive.
Attractors
Attracting fixed points are a special case of a wider mathematical concept of attractors. Fixed-point iterations are a discrete dynamical system on one variable. Bifurcation theory studies dynamical systems and classifies various behaviors such as attracting fixed points, periodic orbits, or strange attractors. An example system is the logistic map.
Iterative methods
In computational mathematics, an iterative method is a mathematical procedure that uses an initial value to generate a sequence of improving approximate solutions for a class of problems, in which the n-th approximation is derived from the previous ones. Convergent fixed-point iterations are mathematically rigorous formalizations of iterative methods.
Iterative method examples
Convergence acceleration
The speed of convergence of the iteration sequence can be increased by using a convergence acceleration method such as Anderson acceleration and Aitken's delta-squared process. The application of Aitken's method to fixed-point iteration is known as Steffensen's method, and it can be shown that Steffensen's method yields a rate of convergence that is at least quadratic.
Chaos game
The term chaos game refers to a method of generating the fixed point of any iterated function system (IFS). Starting with any point , successive iterations are formed as , where is a member of the given IFS randomly selected for each iteration. Hence the chaos game is a randomized fixed-point iteration. The chaos game allows plotting the general shape of a fractal such as the Sierpinski triangle by repeating the iterative process a large number of times. More mathematically, the iterations converge to the fixed point of the IFS. Whenever belongs to the attractor of the IFS, all iterations stay inside the attractor and, with probability 1, form a dense set in the latter.
See also
References
Further reading
External links