Fixed-point iteration

This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. Find sources: "Fixed-point iteration" – news · newspapers · books · scholar · JSTOR (May 2010) (Learn how and when to remove this message)

In numerical analysis, fixed-point iteration is a method of computing fixed points of a function.

More specifically, given a function ${\displaystyle f}$ defined on the real numbers with real values and given a point ${\displaystyle x_{0}}$ in the domain of ${\displaystyle f}$, the fixed-point iteration is $${\displaystyle x_{n+1}=f(x_{n}),\,n=0,1,2,\dots }$$ which gives rise to the sequence ${\displaystyle x_{0},x_{1},x_{2},\dots }$ of iterated function applications ${\displaystyle x_{0},f(x_{0}),f(f(x_{0})),\dots }$ which is hoped to converge to a point ${\displaystyle x_{\text{fix}}}$. If ${\displaystyle f}$ is continuous, then one can prove that the obtained ${\displaystyle x_{\text{fix}}}$ is a fixed point of ${\displaystyle f}$, i.e., $${\displaystyle f(x_{\text{fix}})=x_{\text{fix}}.}$$

More generally, the function ${\displaystyle f}$ can be defined on any metric space with values in that same space.