Quadratic Newton error bound in a Banach space
= Quadratic Newton error bound in a Banach space
Let $\Phi(\bar p)=0$, $\|[D\Phi(p)]^{-1}\|\leq M$ and $\|D\Phi(p)-D\Phi(q)\|\leq L\|p-q\|$ near $\bar p$. The integral <Taylor remainder> gives
$$
\|N(p)-\bar p\|\leq\frac{ML}{2}\|p-\bar p\|^2.
$$
A small ball with $(ML/2)r<1$ is invariant under $N$. Iteration there converges at least quadratically, with $Ce_{n+k}\leq(Ce_n)^{2^k}$ for $C=ML/2$.
// Target: numerical-analysis.bigb