重根牛顿变换的Julia集

Julia Sets of Newton Method for Multiple Roots

中文关键词:  标准牛顿变换  松弛牛顿变换  重根牛顿变换  Julia集  不动点  吸引域.

英文关键词:standard Newton method  relax Newton method  multiple roots Newton method  Julia set  fixed point  attraction region.

中文摘要:

      作者分析了重根牛顿变换的Julia集理论,并利用迭代法构造了标准牛顿变换、松弛牛顿变换和重根牛顿变换的Julia集. 采用实验数学方法,作者得出如下结论: (1) 函数$f(z)=z^\alpha(z^\beta-1)$的三种牛顿变换Julia集的中心为原点且具有$\beta$倍的旋转对称性; (2) 三种牛顿变换Julia集的重根吸引域对$\alpha$具有敏感的依赖性; (3) 由于的零点是松弛牛顿变换的中性或斥性不动点,故松弛牛顿变换的Julia集中不存在单根吸引域; (4) 由于$\infty$点不是重根牛顿变换的不动点,故重根牛顿变换的Julia集中多为重根和单根吸引域; (5) 重根牛顿法受计算误差影响最小,松弛牛顿法次之,标准牛顿法最大.

英文摘要:

      In this paper we analyze the theory of Julia sets of Newton method for multiple roots, construct the standard, relax and multiple root's Julia sets. Utilizing the method of experimental mathematics, the authors obtain the following conclusion: (1) The Julia sets of above methods for $f(z)=z^\alpha(z^\beta-1)$ has $\beta$ rotational symmetry and its center is the origin; (2) The multiple root attraction region of these kinds of Julia sets are sensitive to $\alpha$; (3) There is not simple root attraction region in the relax method, because $z^*$, the root of $f(z)$, is neutral or repelling fixed point of the relax Newton transform $F(z)$; (4) $\infty$ is not the fixed point of multiple roots Newton transform $F(z)$, so multiple root's Julia set is multiple and simple roots attraction region; (5) The experimental errors and truncation of coefficients cause the minimal effect to the multiple root method, the more to the relax and the maximal to the standard Newton' method.