Verified error bounds for multiple roots of systems of nonlinear equations

摘要

It is well known that it is an ill-posed problem to decide whether a function has a multiple root. Even for a univariate polynomial an arbitrary small perturbation of a polynomial coefficient may change the answer from yes to no. Let a system of nonlinear equations be given. In this paper we describe an algorithm for computing verified and narrow error bounds with the property that a slightly perturbed system is proved to have a double root within the computed bounds. For a univariate nonlinear function f we give a similar method also for a multiple root. A narrow error bound for the perturbation is computed as well. Computational results for systems with up to 1000 unknowns demonstrate the performance of the methods.