The Durand-Kerner polynomials roots-finding method in case of multiple roots

The convergence of the Durand-Kerner algorithm is quadratic in case of simple roots but only linear in case of multiple roots. This paper shows that, at each step, the mean of the components converging to the same root approaches it with an error proportional to the square of the error at the previous step. Since it is also shown that it is possible to estimate the multiplicity order of the roots during the algorithm, a modification of the Durand-Kerner iteration is proposed to preserve a quadratic-like convergence even in case of multiple zeros.This work is supported in part by the Research Program C3 of the French CNRS and MEN, and by the Direction des Recherches et Etudes Techniques (DGA).