比较若干数值方法的吸引域几何、速度与效率
Comparing the geometry of the basins of attraction, the speed and the efficiency of several numerical methods
AI总结：
本文使用简单方程比较复平面上多种阶数数值方法的吸引域，对节点进行系统分类，分析迭代分布与收敛域相关性，并探讨方法的收敛结构、速度、效率及问题复杂性对方法特征的影响。
摘要：
我们使用简单方程来比较复平面上对应于大量不同阶数值方法的吸引域。根据作为数值吸引子的根的总数，考虑了两种情形。对于这两种情形，我们使用迭代格式对复平面上的节点进行了全面且系统的分类。所需迭代的分布、概率及其与相应收敛域的相关性也得到了讨论。我们的数值计算表明，大多数迭代格式在复平面上提供了相对相似的收敛结构。此外，我们比较了数值方法的若干方面，试图得出关于其速度和效率的一般性结论。而且，我们试图确定每种情形的复杂性如何影响数值方法的主要特征。
英文摘要：
We use simple equations in order to compare the basins of attraction on the complex plane, corresponding to a large collection of numerical methods, of several order. Two cases are considered, regarding the total number of the roots, which act as numerical attractors. For both cases we use the iterative schemes for performing a thorough and systematic classification of the nodes on the complex plane. The distributions of the required iterations as well as the probability and their correlations with the corresponding basins of convergence are also discussed. Our numerical calculations suggest that most of the iterative schemes provide relatively similar convergence structures on the complex plane. In addition, several aspects of the numerical methods are compared in an attempt to obtain general conclusions regarding their speed and efficiency. Moreover, we try to determine how the complexity of the each case influences the main characteristics of the numerical methods.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
