Weak Singularities and the Continuous Newton Method

Some issues related to the determination of the singular roots of a nonlinear vector function f : RnªRn are addressed in this paper. It is usually assumed that Newton-like fields are not defined at singularities; thus a particular treatment for these points is necessary. Nevertheless, in dimension 1 and in several higher dimensional instances it is possible to make a smooth extension of the field to singular points; when this is the case for a singular root, it can be treated in a way similar to that of regular ones. Necessary and sufficient conditions for this extension to be possible are given, under some structural assumptions, through the concept of weak singularity. The actual setting for this result is a general class of quotient functions which includes, in particular, the Newton field. For the specific case of the continuous-time Newton method, we enlarge some previous results concerning the relation between singular roots and equilibrium points of the extended field, as well as their asymptotic stability. Finally, a computational tool obtained from the extended continuous Newton method by means of the cell mapping technique is shown to be well behaved for the location of these singular roots.