On the determination of the safe initial approximation for the Durand- Kerner algorithm (Q1184142)

The Durand-Kerner method is a well-known algorithm for solving polynomial equations. Although it has the property of parallel computing and convergence rate of order two applications are restricted due to the difficulty for getting safe initial values. Using majorant functions the authors establish a new proof of the convergence of the algorithm and obtain estimates for the attractive radius of the solution. These estimates turn out to be an improvement of known results. Then the authors focus their interest on the problem of finding safe initial guesses. For this purpose they combine the Kuhn algorithm which makes intensive use of completely labelled simplices. The resulting discriminant condition is easy to check in the computing process. Finally the authors discuss the complexity of the presented algorithm by comparison with known circular iterative methods.