Newton-Type Methods for Solving Equations in Banach Spaces: A Unified Approach

Abstract

A plethora of quantum physics problems are related to symmetry principles. Moreover, by using symmetry theory and mathematical modeling, these problems reduce to solving iteratively finite differences and systems of nonlinear equations. In particular, Newton-type methods are introduced to generate sequences approximating simple solutions of nonlinear equations in the setting of Banach spaces. Specializations of these methods include the modified Newton method, Newton’s method, and other single-step methods. The convergence of these methods is established with similar conditions. However, the convergence region is not large in general. That is why a unified semilocal convergence analysis is developed that can be used to handle these methods under even weaker conditions that are not previously considered. The approach leads to the extension of the applicability of these methods in cases not covered before but without new conditions. The idea is to replace the Lipschitz parameters or other parameters used by smaller ones to force convergence in cases not possible before. It turns out that the error analysis is also extended. Moreover, the new idea does not depend on the method. That is why it can also be applied to other methods to also extend their applicability. Numerical applications illustrate and test the convergence conditions.

Keywords

• convergence
• Banach spaces
• Fréchet derivative
• iterative method