Optimal Unbounded Search Strategies

We present here strategies for searching the (unique) zero of a real function, or its n-th derivative; we assume no a priori bound on the value x of this zero. The proposed strategy performs logry + llogry+ ... +1 + log*ry evaluations of f to determine x = ɛy with error less than ɛ (here r depends only on n). An argument of slowly converning integrals shows that these strategies are essentially optimal.