Fixed-point iteration

In <a href="/facts/Numerical_analysis/QT9cYa9z">numerical analysis</a>, fixed-point iteration is a method of computing <a href="/facts/Fixed_point_(mathematics)/vVVddTKc">fixed points</a> of a function.
More specifically, given a function 
 
 
 
 f
 
 
 {\displaystyle f}
 
 defined on the <a href="/facts/Real_number/R02gw5Pb">real numbers</a> with real values and given a point 
 
 
 
 
 x
 
 0
 
 
 
 
 {\displaystyle x_{0}}
 
 in the <a href="/facts/Domain_of_a_function/FQvoeUpX">domain</a> of 
 
 
 
 f
 
 
 {\displaystyle f}
 
, the fixed-point iteration is

x
          
            n
            +
            1
          
        
        =
        f
        (
        
          x
          
            n
          
        
        )
        ,
        
        n
        =
        0
        ,
        1
        ,
        2
        ,
        …
      
    
    {\displaystyle x_{n+1}=f(x_{n}),\,n=0,1,2,\dots }

which gives rise to the <a href="/facts/Sequence/gV2S4uqp">sequence</a> 
 
 
 
 
 x
 
 0
 
 
 ,
 
 x
 
 1
 
 
 ,
 
 x
 
 2
 
 
 ,
 …
 
 
 {\displaystyle x_{0},x_{1},x_{2},\dots }
 
 of <a href="/facts/Iterated_function/Vzsx64An">iterated function</a> applications 
 
 
 
 
 x
 
 0
 
 
 ,
 f
 (
 
 x
 
 0
 
 
 )
 ,
 f
 (
 f
 (
 
 x
 
 0
 
 
 )
 )
 ,
 …
 
 
 {\displaystyle x_{0},f(x_{0}),f(f(x_{0})),\dots }
 
 which is hoped to <a href="/facts/Limit_(mathematics)/j8JBhj4e">converge</a> to a point 
 
 
 
 
 x
 
 fix
 
 
 
 
 {\displaystyle x_{\text{fix}}}
 
. If 
 
 
 
 f
 
 
 {\displaystyle f}
 
 is continuous, then one can prove that the obtained 
 
 
 
 
 x
 
 fix
 
 
 
 
 {\displaystyle x_{\text{fix}}}
 
 is a fixed point of 
 
 
 
 f
 
 
 {\displaystyle f}
 
, i.e.,

f
        (
        
          x
          
            fix
          
        
        )
        =
        
          x
          
            fix
          
        
        .
      
    
    {\displaystyle f(x_{\text{fix}})=x_{\text{fix}}.}

More generally, the function 
 
 
 
 f
 
 
 {\displaystyle f}
 
 can be defined on any <a href="/facts/Metric_space/gnBBRXtc">metric space</a> with values in that same space.

Fixed-point iteration open-in-new

Fixed-point iteration