Glossary

Fixed point

A value the iteration function leaves unchanged: $g(\alpha)=\alpha$ .

Definition

Rewriting $f(x)=0$ as $x=g(x)$ , solving the equation is equivalent to finding the fixed point of $g$ . The iteration $x_{k+1}=g(x_k)$ converges locally if $|g'(\alpha)|<1$ , and the smaller that derivative, the faster the convergence.

How it is used

The same equation admits many $g$ 's: choosing one with small $|g'|$ near the root separates a useful iteration from a divergent one. Newton is a fixed point with $g'(\alpha)=0$ .