Frozen Newton correction for a fixed-point equation ($A(p)=p-J(\Phi (p)-p)$)

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