A Note on the Convergence of Newton’s Method

In a Banach space X, let P be an operator with Lipschitz continuous derivative $P'$, and $x^ * \in X$ such that $P(x^ * ) = 0$ and $[P'(x^ * )]^{ - 1} $ exists. It is shown that an open ball $\Omega ^ * $ with center $x^ * $ and specified radius exists such that the theorem of L. V. Kantorovic guarantees rapid convergence of Newton’s method to $x^ * $ starting from any $x_0 \in \Omega ^ * $. The value given for the radius of $\Omega ^ * $ is shown to be best possible.