http://mathlab.mathlab.sunysb.edu/~scott/Papers/Newton/Published.pdf This paper is quite interesting. Given any polynomial P(z) of degree=D such that we know all its roots lie within the unit circle, they describe a "universal" set of 2.22 * D * (lnD)^2 points such that Newton's method starting from those points is guaranteed to converge to all the roots of P. By "universal" I mean that this point set does not know or care what P is (aside from knowing its degree and knowing al its roots lie in the unit disk). The point set is: Let N=ceiling(8.32547*D*lnD) and M=ceiling(0.26632*lnD). Equispace N points around the perimeters of each of the following M circles centered at z=0: radius = (1+sqrt(2)) * ((D-1)/D)^((2*v-1)/(4*M)) for each v=1,2,3,...,M. That's all.