Halving interval guaranteed for Dekker and Brent root finding methods

Hybrid methods are widely used in many areas of applied mathematics. One of the simplest and most common problems in this field is root finding, for which various methods exist. Some of the most efficient approaches combine two or more techniques into hybrid methods. Among these are the Dekker and Brent methods, for which we propose a modification to ensure that the search interval is halved in each iteration. We apply this modification to two examples: a transcendental equation and a cubic equation of state. The results demonstrate that the proposed modifications guarantee at least interval halving and offer a slight improvement in the efficiency of the root-finding process.