Nearest multivariate system with given root multiplicities

We present a symbolic-numeric technique to find the closest multivariate polynomial system to a given one which has roots with prescribed multiplicity structure. Our method generalizes the “Weierstrass iteration”, defined by Ruatta, to the case when the input system is not exact, i.e. when it is near to a system with multiple roots, but itself might not have multiple roots. First, using interpolation techniques, we define the “generalized Weierstrass map”, a map from the set of possible roots to the set of systems which have these roots with the given multiplicity structure. Minimizing the 2-norm of this map formulates the problem as an optimization problem over all possible roots. We use Gauss-Newton iteration to compute the closest system to the input with given root multiplicity together with its roots. We give explicitly an iteration function which computes this minimum. These results extends previous results of Zhi and Wu and results of Zeng from the univariate case to the multivariate case. Finally, we give a simplified version of the iteration function analogously to the classical Weierstrass iteration, which allows a componentwise expression, and thus reduces the computational cost of each iteration. We provide numerical experiments that demonstrate the effectiveness of our method.