= Newton iteration in a Banach space
{c}
{title2=$N(p)=p-[D\Phi(p)]^{-1}\Phi(p)$}
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
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