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. Finally, sufficient numerical examples are used to verify our theory.