Fixed-point iteration

Fixed-point iteration is a method in numerical analysis for computing the fixed points of a function, meaning the points where the function returns its own input. Given a function f defined on the real numbers with real values, and a starting point x0 in its domain, the method repeatedly applies the function to produce the sequence x0, f(x0), f(f(x0)) and so on, using the rule that each term is f of the one before. The hope is that this sequence converges to a point x-fix. If f is continuous, any such limit is provably a fixed point, satisfying f(x-fix) = x-fix. The idea is not limited to the real line, since the function can be defined on any metric space with values in that same space, which is why it appears throughout numerical methods.