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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 lessthan $\epsilon$. One way to solve the problem is to start with an initialinterval and to repeatedly update it by applying an interval refinement map onit until it becomes narrow enough. In this paper, we prove that the well knownSecant-Newton map is optimal among a certain family of natural generalizations.

The Secant-Newton Map is Optimal Among Contracting $n^{th}$ Degree Maps for $n^{th}$ Root Computation