Convergence and complexity of interpolatory—newton iteration in a Banach space

The class of interpolatory—Newton iterations is defined and analyzed for the computation of a simple zero of a non-linear operator in a Banach space of finite or infinite dimension. Convergence of the class is established.

The concepts of “informationally optimal class of algorithms” and “optimal algorithm” are formalized. For the multivariate case, the optimality of Newton iteration is established in the class of one-point iterations under an “equal cost assumption”.