Fixed-point iteration

The idea

Rearrange $f(x) = 0$ into $x = g(x)$, then iterate $x_{n+1} = g(x_n)$. If the values settle, they approach a root. On a graph of $y = x$ and $y = g(x)$, the steps form a staircase or a cobweb.

1. Start with the given $x_1$.
2. Substitute into $g$ to get $x_2$, then $x_3$, keeping full calculator accuracy.
3. An iteration converges only if $|g'(x)| < 1$ near the root.

Watch out Not every rearrangement converges: some move away from the root.