Verified Error Bounds for Isolated Singular Solutions of Polynomial   Systems: Case of Breadth One
  In this paper we describe how to improve the performance of the symbolic-numeric method in (Li and Zhi,2009, 2011) for computing the multiplicity structure and refining approximate isolated singular solutions in the breadth one case. By introducing a parameterized and deflated system with smoothing parameters, we generalize the algorithm in (Rump and Graillat, 2009) to compute verified error bounds such that a slightly perturbed polynomial system is guaranteed to have a breadth-one multiple root within the computed bounds. 
Nan Li, Lihong Zhi
math.NA