Starlike domains of convergence for Newton's method at singularities

Abstract

Given a solutionx * of a system of nonlinear equationsf with singular Jacobian ?f(x *) we construct an open starlike domainR of initial points, from which Newton's method converges linearly tox *. Under certain conditions the union of those straight lines throughx *, that do not intersect withR is shown to form a closed set of measure zero, which is necessarily disjoint from any starlike domain of convergence. The results apply to first and higher order singularities.