A new tool to study real dynamics: The Convergence Plane

TL;DR

The paper introduces the Convergence Plane, a novel visualization tool for analyzing the real dynamics of parameter-dependent iterative methods, aiding in selecting optimal elements and understanding basin changes.

Contribution

It presents the Convergence Plane as a new, compact method for studying iterative dynamics, demonstrated through the Damped Newton's method example.

Findings

Effective visualization of convergence properties.

Ability to identify good and bad elements in a family.

Insights into basin of attraction variations.

Abstract

In this paper, the author presents a new tool, called The Convergence Plane, that allows to study the real dynamics of iterative methods whose iterations depends on one parameter in an easy and compact way. This tool can be used, inter alia, to find the elements of a family that have good convergence properties and discard the bad ones or to see how the basins of attraction changes along the elements of the family. To show the applicability of the tool an example of the dynamics of the Damped Newton's method applied to a cubic polynomial is presented.