On the guaranteed convergence of the square-root iteration method

The construction of initial conditions which guarantee the convergence of the applied iterative method, is one of the most important problems in solving nonlinear equations which attracted the great attention for many years. In this paper, we give a precise convergent analysis of the Ostrowski-like method of the fourth order for the simultaneous determination of polynomial zeros. Using a procedure based on Smaleʹs point estimation theory and some recent results related to the localization of complex polynomial zeros, we state initial conditions which enable both the guaranteed and fast convergence of this method. These conditions are computationally verifiable since they depend only on polynomial coefficients, its degree and initial approximations, which is of practical importance.