Kneisl Kyle
CB #3250, Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, North Carolina 27599.
Chaos. 2001 Jun;11(2):359-370. doi: 10.1063/1.1368137.
We study numerically and dynamically three cubically convergent iterative root-finding algorithms, namely Cauchy's method, the super-Newton method, and Halley's method. Using the concept of a universal Julia set (motivated by the results of McMullen), we establish that these algorithms converge when applied to any quadratic with distinct roots. We give examples showing the existence of attracting periodic orbits not associated to a root for the super-Newton method and Halley's method applied to cubic polynomials. We include computer plots showing the dynamic structure for each algorithm applied to a variety of polynomials. (c) 2001 American Institute of Physics.
我们对三种三次收敛的迭代求根算法进行了数值和动态研究，即柯西方法、超牛顿方法和哈雷方法。利用通用朱利亚集的概念（受麦克马伦结果的启发），我们证明了这些算法应用于任何具有不同根的二次函数时都会收敛。我们给出了一些例子，表明对于应用于三次多项式的超牛顿方法和哈雷方法，存在与根无关的吸引周期轨道。我们还展示了应用于各种多项式的每种算法的动态结构的计算机绘图。(c) 2001美国物理研究所。
