Optimal Iterative Methods for Nonlinear Equations

Abstract

We prove an existence and uniqueness theorem for equations with regularly smooth operators acting between Banach spaces. It allows us to formulate a novel concept of optimality of iterative methods for solving nonlinear equations based on the entropy of the solution's position within the existence and uniqueness set guaranteed by the theorem. Using this concept in the special case of scalar equations (which is not trivial), we show that it is feasible to get a method that needs the same information as Newton's (one value of the operator and one of its derivative) per iteration but exhibits more than quadratic rate of convergence.