Quadratic Interval Refinement for Real Roots

We present a new algorithm for refining a real interval containing a single real root: the new method combines characteristics of the classical Bisection algorithm and Newtons Iteration. Our method exhibits quadratic convergence when refining isolating intervals of simple roots of polynomials (and other well-behaved functions). We assume the use of arbitrary precision rational arithmetic. Unlike Newtons Iteration our method does not need to evaluate the derivative.