Convergence of Super-Halley method for simultaneous approximation of polynomial zeros

Abstract

Recently, Ivanov (Mathematics. 8: 1599, 2020) has proven two local convergence results with error estimates about the well-known Super-Halley method for individual approximation of multiple polynomial roots. In this talk, we study the Super-Halley method as a method in n-dimensional vector space, that is, as a method for extracting all the zeros of a complex polynomial at once. By this means, we obtain a local convergence theorem which specifies precise domains of initial vectors to assure the Q-cubic convergence of this algorithm. Furtheron, using this theorem, we obtain a semilocal convergence result which is of great practical importance due to its computable initial condition and error estimate. We demonstrate the practical applicability of our semilocal theorem via two numerical experiments, where we apply it to two significant real-world problems.