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 z₀ ∈ ℂ satisfies α(f, z ₀) < 0.02 then z₀ is a robust approximate zero, with the associated zero z* lying in the closed disc B̄(z₀, 0.07/γ(f, z₀). Here α(f, z), γ(f, z) are standard functions in point estimates. Suppose f(z) is an L-bit integer square-free polynomial of degree d. Using our new algorithm, we can compute an n-bit absolute approximation of z* ∈ IR starting from a bigfloat Z ₀, in time O[dM(n + d²(L + lg d) lg(n + L))], where M(n) is the complexity of multiplying n-bit integers.