  Consider the problem: given a real number $x$ and an error bound $\epsilon$,
find an interval such that it contains the $\sqrt[n]{x}$ and its width is less
than $\epsilon$. One way to solve the problem is to start with an initial
interval and to repeatedly update it by applying an interval refinement map on
it until it becomes narrow enough. In this paper, we prove that the well known
Secant-Newton map is optimal among a certain family of natural generalizations.
