In this paper we study the dynamics of damped Traub's methods $T_\delta$ when applied to polynomials. The family of damped Traub's methods consists of root finding algorithms which contain both Newton's ($\delta=0$) and Traub's method ($\delta=1$). Our goal is to obtain several topological properties of the basins of attraction of the roots of a polynomial $p$ under $T_1$, which are used to determine a (universal) set of initial conditions for which convergence to all roots of $p$ can be guaranteed. We also numerically explore the global properties of the dynamical plane for $T_\delta$ to better understand the connection between Newton's method and Traub's method.

翻译：本文研究了阻尼Traub法$T_\delta$应用于多项式时的动力学特性。阻尼Traub法族包含牛顿法（$\delta=0$）与Traub法（$\delta=1$）两类求根算法。我们的目标是获得多项式$p$在$T_1$映射下各根吸引域的若干拓扑性质，这些性质可用于确定保证收敛到$p$所有根的（普适）初始条件集。我们还通过数值方法探究$T_\delta$动力平面的全局性质，以深入理解牛顿法与Traub法之间的关联。