Newton's method's basins of attraction revisited

TL;DR

This paper explores how a modified Newton's method's convergence regions are affected by a parameter, revealing complex behaviors in the basins of attraction through polynomiography.

Contribution

It introduces a parameter-dependent modification to Newton's method and analyzes the resulting complex basin structures using visualizations.

Findings

Convergent regions vary significantly with the parameter.

The basin boundaries can be highly intricate and sensitive.

Modified Newton's method exhibits complex convergence behaviors.

Abstract

In this paper, we revisit the chaotic number of iterations needed by Newton's method to converge to a root. Here, we consider a simple modified Newton method depending on a parameter. It is demonstrated using polynomiography that even in the simple algorithm the presence and the position of the convergent regions, i.e. regions where the method converges nicely to a root, can be complicatedly a function of the parameter.