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Banach Fixed-Point Theorem

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z overview

If $ g$ is a contraction of a non-empty closed set $ D\subset\mathbb{R}^n$, i.e., if with $ c<1$, then $ g$ has a unique fixed point $ x_*=g(x_*)\in D$. Starting with any point $ x_0\in D$, $ x_\ast$ can be approximated by the sequence

$\displaystyle x_0,\, x_1 = g(x_0),\, x_2=g(x_1),\,\ldots

The error satisfies

$\displaystyle \Vert x_*-x_k\Vert\le \frac{c^k}{1-c}\, \Vert x_1-x_0\Vert

i.e., the iteration converges linearly.

More generally, the fixed point theorem holds in complete metric spaces. Since the proof neither requires translation invariance nor homogeneity of the norm, $ \Vert x-y\Vert$ can be replaced by a general distance function $ d(x,y)$.


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  automatically generated 9/22/2016