A convergence analysis of a fourth-order method for computing all zeros of a polynomial simultaneously

In 2011, Petković, Rančić and Milošević (Petković et al., 2011) introduced and studied a new fourth-order iterative method for finding all zeros of a polynomial simultaneously. They obtained a semilocal convergence theorem for their method with computationally verifiable initial conditions, which is of practical importance. In this paper, we provide new local as well as semilocal convergence results for this method over an algebraically closed normed field. Our semilocal results improve and complement the result of Petković, Rančić and Milošević in several directions. The main advantage of the new semilocal results are: weaker sufficient convergence conditions, computationally verifiable a posteriori error estimates, and computationally verifiable sufficient conditions for all zeros of a polynomial to be simple. Furthermore, several numerical examples are provided to show some practical applications of our semilocal results.