Robust approximate zeros

Smale's notion of an approximate zero of an analytic function f : C -> C is extended to take into account the errors incurred in the evaluation of the Newton operator. Call this stronger notion a robust approximate zero. We develop a corresponding robust point estimate for such zeros: we prove that if z(0) is an element of C satisfies alpha(f, z(0)) < 0.02 then z(0) is a robust approximate zero, with the associated zero z* lying in the closed disc (B) over bar (z(0), 0.07/gamma(f,z0)). Here alpha(f, z), gamma(f, z) are standard functions in point estimates.