Newton iteration in a Banach space ($N(p)=p-[D\Phi (p){]}^{-1}\Phi (p)$)

For a $C^2$ map $\Phi$ on a Banach space, define $N(p)=p-[D\Phi (p){]}^{-1}\Phi (p)$ wherever the bounded derivative is invertible. It solves the local affine approximation to $\Phi =0$. Every zero is a fixed point of $N$. Near a zero with invertible derivative, bounded inverses and a locally Lipschitz derivative give the quadratic Newton error bound in a Banach space.

// Target: numerical-analysis.bigb

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

Let $\Phi (\bar{p})=0$, $\Vert [D\Phi (p){]}^{-1}\Vert \le M$ and $\Vert D\Phi (p)-D\Phi (q)\Vert \le L\Vert p-q\Vert$ near $\bar{p}$. The integral Taylor remainder givesA small ball with $(ML/2)r<1$ is invariant under $N$. Iteration there converges at least quadratically, with $C{e}_{n+k}\le (C{e}_n{)}^{2^k}$ for $C=ML/2$.

// Target: numerical-analysis.bigb