Fully computable convergence analysis of a fourth order iterative method for finding polynomial roots

Abstract This paper investigates the convergence of a fourth-order iterative method for simultaneously finding all roots of a complex polynomial. The method was introduced by Kyurkchiev in 1983 and, in a completely different form, by Zheng and Sun in 1999. To date, only local convergence results are available for this method. In this study, we provide a comprehensive semilocal convergence analysis for the Kyurkchiev-Zheng-Sun method. The resulting theorem includes fully computable initial conditions, a computable a posteriori error estimate, and a localization for each root of the polynomial. The paper concludes with numerical experiments that demonstrate the effectiveness and applicability of the proposed theorem.