Durand-Kerner算法的全局化*

本文利用对称多项式与一元多项式之间的关系,结合连续同伦思想,构造了一条概率为1的正则同论曲线.然后,对这条同论路径,进行离散化跟踪,导出了一类带有步长参数的Durand-Kerner算法,我们证明了这类算法的整体收敛性,从而在理论上解决了人们关于Durand-Kerner算法具有整体性的推测.本文还深入讨论了步长参数的选择问题.最后,我们以足够的数值例子,检验了理论的正确性.

The Globalization of Durand-Kerner Algorithm

Making use of the theory of continuous homotopy and the relation between symmetric polynomial and polynomial in one variable the authors devoted this article to constructing a regularly homotopic curve with probability one. Discrete tracing along this homotopic curve leads to a class of Durand-Kerner algorithm with step parameters. The convergence of this class of algorithms is given, which solves the conjecture about the global property of Durand-Kerner algorithm.The problem for steplength selection is thoroughly discussed. Finaily,sufficient numerical examples are used to verify our theory.