Summary

1. Express $F(x) = 0$ in the form $x = G(x)$. The fixed-point iteration scheme is
   
   $$x_{n+1} = G(x_{n}).$$
   
   

2. Fixed-point iteration generates a first order scheme with
   
   $$\epsilon_{n+1} \approx G^{\prime}(r)\epsilon_{n}.$$
   
   

3. Fixed-point iteration with converge to a root at $x = r$ if
   
   $$\vert G^{\prime}(r)\vert < 1,$$
   
   
   and will not converge if
   
   $$\vert G^{\prime}(r)\vert >1.$$
   
   
   These conditions are local conditions in the sense that they apply if the intial estimate $x_{0}$ is sufficiently close to the root.