To find a fixed point numerically, fixed-point iteration repeatedly applies the function:

$$x_{n+1}=g(x_n).$$

Starting from an initial guess $x_0$, the method asks whether the sequence approaches a fixed point $x_$. If $x_n\to x_$ and $g$ is continuous near the limit, then

$$x_=g(x_).$$

The crucial question is whether applying $g$ shrinks or amplifies nearby errors. In one dimension, linearizing around a fixed point gives approximately

$$x_{n+1}-x_\approx g'(x_)(x_n-x_*).$$

A common local convergence condition is therefore

$$|g'(x_*)|<1.$$

Then sufficiently small errors tend to contract from one iteration to the next. If $|g'(x_*)|>1$, nearby errors tend to grow instead.

The same equation can often be rearranged into several forms $x=g(x)$, and those rearrangements need not have the same convergence behavior. Fixed-point iteration is therefore not just repeated substitution: choosing a useful iteration function and analyzing its stability are part of the numerical method.