Contraction (math)

In analysis and related areas of mathematics, a contraction is a representation of a set in itself, which reduces the distance between any two points by at least as much as a centric stretch with a fixed stretching factor , that is, the set "contracts itself" when used multiple times “( Contracted ). It clearly appears clear that by continuing to use such a contraction, the initial set is gradually mapped to an "arbitrarily small" subset and finally (could only be mapped infinitely often) contracts to one point. That this intuitive assumption applies in a more precise sense in very general cases can be proven mathematically. Sentences that make statements about the existence of the "limit point" towards which the contraction tends, its calculation and the approximation error after a finite number of steps ( iterations ) of this approximation are called contraction sentences or fixed point sentences . ${\ displaystyle M}$ ${\ displaystyle M}$ ${\ displaystyle \ lambda <1}$

definition

${\ displaystyle (M, d)}$ be a metric space . A figure is called contraction if there is a number with which applies to all : ${\ displaystyle \ varphi \ colon M \ to M}$ ${\ displaystyle \ lambda \ in [0,1)}$ ${\ displaystyle x, y \ in M}$

{\ displaystyle d \ left (\ varphi (x), \ varphi (y) \ right) \ leq \ lambda \ cdot d (x, y)}

.

Then called the picture even twitch or even contractionary on . ${\ displaystyle M}$

In other words: the mapping is a contraction if and only if it ${\ displaystyle \ varphi}$

depicts the crowd in itself and ${\ displaystyle M}$
satisfies a Lipschitz condition with a Lipschitz constant . ${\ displaystyle \ lambda <1}$

Application: Real contraction theorem

A contracting self-mapping of an interval has exactly one fixed point . This can be calculated using the iteration sequence with any starting value . The error estimate applies to the terms of the iteration sequence . ${\ displaystyle f}$ ${\ displaystyle I = [a, b]}$ ${\ displaystyle \ xi}$ ${\ displaystyle x_ {n + 1}: = f (x_ {n})}$ ${\ displaystyle x_ {0} \ in I}$ ${\ displaystyle | x_ {n} - \ xi | \ leq {\ frac {\ lambda ^ {n}} {1- \ lambda}} | x_ {1} -x_ {0} |}$

A generalization of this theorem is Banach's Fixed Point Theorem .

Examples

Let and be a real-valued function which satisfies the Lipschitz condition with . When it comes to the starting point of an interval are, on the , then the function is a contracting self-image of . A fixed point in can be calculated using the recursion sequence from the real contraction theorem (see above). ${\ displaystyle X \ subseteq \ mathbb {R}}$ ${\ displaystyle f}$ ${\ displaystyle X}$ ${\ displaystyle X}$ ${\ displaystyle \ lambda <1}$ ${\ displaystyle x_ {0} \ in X}$ ${\ displaystyle I = [x_ {0} -r, x_ {0} + r] \ subseteq X}$ ${\ displaystyle | f (x_ {0}) - x_ {0} | <| (1- \ lambda) r |}$ ${\ displaystyle f}$ ${\ displaystyle I}$ ${\ displaystyle I}$

A well-known application of the real contraction theorem is Heron's method for determining the square root of an integer . Instead of the equation submitted for the solution, the equation is solved , i.e. a fixed point of the function is determined . This function is contracting on the interval , where is set. The contraction constant can be selected. ${\ displaystyle a> 1}$ ${\ displaystyle x ^ {2} = a}$ ${\ displaystyle x = {\ frac {x} {2}} + {\ frac {a} {2x}}}$ ${\ displaystyle f (x) = {\ frac {x} {2}} + {\ frac {a} {2x}}}$ ${\ displaystyle I = [w, w + 1]}$ ${\ displaystyle w: = \ max \ {w \ in \ mathbb {N} \ mid w ^ {2} <a \}}$ ${\ displaystyle \ lambda = {\ tfrac {1} {2}}}$

literature

Harro Heuser: Textbook of Analysis Part 1 . 5th edition. Teubner-Verlag, 1988, ISBN 3-519-42221-2