A Convergence Criterion of Newton’s Method Based on the Heisenberg Uncertainty Principle

The objective in this article is to extend the applicability of Newton’s method for solving Banach space valued nonlinear equations. In particular, a new semi-local convergence criterion for Newton’s method (NM) based on Kantorovich theorem in Banach space is developed by application of the Heisenberg Uncertainty Principle (HUP). The convergence region given by this theorem is small in general limiting the applicability of NM. But, using HUP and the Fourier transform of the operator involved, we show that it is possible to extend the applicability of NM without additional hypotheses. This is done by enlarging the convergence region of NM and using the concept of epsilon-concentrated operator. Numerical experiments further validate our theoretical results by solving equations in case not covered before by the Newton–Kantorovich theorem.