Applying root-finding techniques to extend Quantized-State-Systems-based solvers

In this work we propose the usage of root isolation algorithms to extend Quantized-State-Systems-based methods for integrating systems of Ordinary Differential Equations to higher orders. QSS methods of order n, at their inner loop, neet to compute the minimum positive root of a n-degree polynomial. The lack of analytical expressions for the roots of polynomial of degree greater than four limits the QSS methods to fourth order or less. We make an observation which, combined with the usage of root-finding techniques, allows the generalization to QSS of any order. Moreover, we show experimentally that, considering our algorithmic improvements, higher order methods do require considerably fewer iterations than lower order ones.