An Adaptive Newton-Method Based on a Dynamical Systems Approach

TL;DR

This paper presents an adaptive Newton method based on a dynamical systems perspective, aiming to improve stability and efficiency while maintaining quadratic convergence near solutions.

Contribution

It introduces a novel adaptive step size control for the Newton method using a dynamical systems framework, reducing chaotic behavior without sacrificing convergence.

Findings

Enhanced stability of Newton method demonstrated

Maintains quadratic convergence near roots

Effective in algebraic and differential equations

Abstract

The traditional Newton method for solving nonlinear operator equations in Banach spaces is discussed within the context of the continuous Newton method. This setting makes it possible to interpret the Newton method as a discrete dynamical system and thereby to cast it in the framework of an adaptive step size control procedure. In so doing, our goal is to reduce the chaotic behavior of the original method without losing its quadratic convergence property close to the roots. The performance of the modified scheme is illustrated with various examples from algebraic and differential equations.