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

Review (2022)

Authors

S. Kouhkani Islamic Azad University branch of Shabestar
H. Koppelaar Interactive Intelligence - EEMCS
O. Taghipour Birgani Iran University of Science and Technology
I. K. Argyros Cameron University
S. Radenović University of Belgrade
Research Group
Interactive Intelligence (EEMCS) (TU Delft)
Copyright: © 2022 S. Kouhkani, H. Koppelaar, O. Taghipour Birgani, I. K. Argyros, S. Radenović
DOI
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https://doi.org/10.1007/s40819-021-01214-z
Iterative method Banach space Heisenberg uncertainty principle Kantorovich theorem Newton’s method
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Published Date

2022

Language

English

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Faculty

Electrical Engineering, Mathematics and Computer Science

Department

Intelligent Systems

Research Group

Interactive Intelligence

Abstract

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.