A Note on the Convergence of Newton's Method
Abstract¶
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.
Summary¶
Provides refined conditions and tighter bounds on the local quadratic convergence of Newton's method for solving nonlinear equations in Banach spaces, extending classical Kantorovich-type analyses.
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