Fixed-point iteration

In numerical analysis, fixed-point iteration is a method of computing fixed points of iterated functions. In numerical analysis, fixed-point iteration is a method of computing fixed points of iterated functions. 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 which gives rise to the sequence x 0 , x 1 , x 2 , … {displaystyle x_{0},x_{1},x_{2},dots } which is hoped to converge to a point x {displaystyle x} . If f {displaystyle f} is continuous, then one can prove that the obtained x {displaystyle x} is a fixed point of f {displaystyle f} , i.e., More generally, the function f {displaystyle f} can be defined on any metric space with values in that same space. converges to 0 for all values of x 0 {displaystyle x_{0}} .However, 0 is not a fixed point of the function as this function is not continuous at x = 0 {displaystyle x=0} , and in fact has no fixed points. If a function f {displaystyle f} defined on the real line with real values is Lipschitz continuous with Lipschitz constant L < 1 {displaystyle L<1} , then this function has precisely one fixed point, and the fixed-point iteration converges towards that fixed point for any initial guess x 0 . {displaystyle x_{0}.} This theorem can be generalized to any metric space.