Frozen Newton correction for a fixed-point equation
= Frozen Newton correction for a fixed-point equation
{title2=$A(p)=p-J(\Phi(p)-p)$}
To solve $\Phi(p)=p$, let $J$ be a bounded approximation to $(D\Phi(p_0)-I)^{-1}$ and set $A(p)=p-J(\Phi(p)-p)$. Its <derivative> is $DA(p)=I-J(D\Phi(p)-I)$. Every <fixed point> of $\Phi$ is fixed by $A$; the converse holds when $J$ is injective. A good inverse approximation makes $A$ a <contraction mapping> even if $\Phi$ itself has an expanding direction.
// Target: analysis.bigb