# Numerical Stability of Fixed-Point Interation

The fixed-point iteration $$x_{n+1} = phi(x_n)$$ for some Lipschitz-continuous function $$phi$$ with Lipschitz-constant $$L<1$$ is one of the methods in numerical analysis to obtain a solution $$x^*$$ of the nonlinear system $$y = f(x^*)$$. How would I determine whether the fixed-point iteration is numerically stable? Or, in other words, how would I determine the stability in the sense of forward analysis, that is, the smallest number $$sigma$$ which satisfies $$Vert tilde g(x) – tilde g(tilde x)Vertleq kappasigmaepsilon$$, where $$tilde g$$ is an algorithm that solves the problem (i.e. in that case the fixed-point iteration), $$kappa$$ the relative condition of the problem (i.e. $$frac{Vert x^*Vert}{Vert f(x^*)Vert}Vertmathrm D f(x^*)^{-1}Vert$$), and $$epsilon$$ is the machine precision.

Unfortunately, I am completely lost. One of the major confusions arises from the fact that I find very differently looking definitions of numerical stability…