# Comparing the geometry of the basins of attraction, the speed and the efficiency of several numerical methods

## TL;DR

This paper systematically compares the basins of attraction, convergence speed, and efficiency of various numerical methods applied to complex equations, revealing similarities and influences of root complexity.

## Contribution

It provides a comprehensive comparison of multiple numerical methods' convergence properties and efficiency based on basin structures and iteration distributions.

## Key findings

- Most iterative schemes exhibit similar convergence structures.
- The complexity of roots influences the convergence characteristics.
- Certain methods demonstrate superior speed and efficiency under specific conditions.

## Abstract

We use simple equations in order to compare the basins of attraction on the complex plane, corresponding to a large collection of numerical methods, of several order. Two cases are considered, regarding the total number of the roots, which act as numerical attractors. For both cases we use the iterative schemes for performing a thorough and systematic classification of the nodes on the complex plane. The distributions of the required iterations as well as the probability and their correlations with the corresponding basins of convergence are also discussed. Our numerical calculations suggest that most of the iterative schemes provide relatively similar convergence structures on the complex plane. In addition, several aspects of the numerical methods are compared in an attempt to obtain general conclusions regarding their speed and efficiency. Moreover, we try to determine how the complexity of the each case influences the main characteristics of the numerical methods.