An improvement of Gargantini's simultaneous inclusion method for polynomial roots by Schro¨der's correction

Abstract The interval version of the complex third-order method of Maehly, Borsch-Supan, Ehrlich, and Aberth is the most efficient method for simultaneous inclusion of simple polynomial roots [14]. In this note, Gargantini's generalization of this third-order interval method for multiple roots is accelerated using Schroder's modification of Newton's corrections and modifying the required interval inversions. The underlying idea is that the iteration of the midpoints of the interval method should be similar to Nourein's acceleration of the above-mentioned complex third-order method improving the convergence of the midpoints. Since the convergence of the radii and the midpoints are coupled it can be proved that the R-orders of convergence of the radii of the newly presented Schroder-like interval methods are asymptotically greater than 3.5. Hence two of these methods are more efficient than the most efficient one known before. Numerical results and an analysis of computational efficiency are included.