# Global dynamics of the real secant method

## TL;DR

This paper analyzes the secant method for root finding as a dynamical system, exploring basin structures, stability, and extending the method to a topological space to improve understanding and efficiency.

## Contribution

It provides a detailed dynamical systems perspective on the secant method, including basin analysis and an extension to the punctured torus for better root-finding insights.

## Key findings

- Basin shapes and distributions for roots are characterized.
- Existence of stable dynamics affecting algorithm efficiency is shown.
- Extension to the punctured torus enhances understanding of behavior near infinity.

## Abstract

We investigate the root finding algorithm given by the secant method applied to a real polynomial $p$ as a discrete dynamical system defined on $\mathbb R^2$. We study the shape and distribution of the basins of attraction associated to the roots of $p$, and we also show the existence of other stable dynamics that might affect the efficiency of the algorithm. Finally we extend the secant map to the punctured torus $\mathbb T^2_{\infty}$ which allow us to better understand the dynamics of the secant method near $\infty$ and facilitate the use of the secant map as a method to find all roots of a polynomial.