The Secant-Newton Map is Optimal Among Contracting Quadratic Maps for Square Root Computation

Consider the problem: given a real number x and an error bound ε, find an interval such that it contains √ x and its width is less than ε. One way to solve the problem is to start with an initial interval and repeatedly to 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 the optimal among a certain family of natural generalizations.