On the basins of attraction of root-finding algorithms

Abstract

Root-finding algorithms have historically been employed to solve numerically nonlinear equations of the form $f(x)=0$. Newton's method, one of the most well-known techniques, started being analyzed as a dynamical system in the complex plane during the late 19th century. This thesis explores the dynamics of damped Traub's methods $T_{p, \delta}$ when applied to polynomials. These methods encompass a range from Newton's method $(\delta=0)$ to Traub's method $(\delta=1)$. Our focus lies in investigating various topological properties of the basins of attraction, particularly their simple connectivity and unboundedness, which are crucial in identifying a universal set of initial conditions that ensure convergence to all roots of $p$. While the former topological properties are already proven for Newton's method $(\delta=0)$, they remain open for $\delta \neq 0$. We present results that contribute to addressing this gap, including a proof for cases where $\delta$ is close to 0 and for the polynomial family $p_d(z)=z\left(z^d-1\right)$.