Fixed-point iteration

In numerical analysis, fixed-point iteration is a method of computing fixed points of a function. More specifically, given a function f {\displaystyle f} defined on the real numbers with real values and given a point x 0 {\displaystyle x_{0}} in the domain 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 sequence x 0 , x 1 , x 2 , … {\displaystyle x_{0},x_{1},x_{2},\dots } of iterated function 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 converge 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 metric space with values in that same space.