The guaranteed convergence of Laguerre-like method

The construction of initial conditions that provide a guaranteed convergence of zero-finding methods has attracted a great deal of attention for many years. In this paper, we consider convergent properties of the Laguerre-like method of the fourth order for the simultaneous approximation of polynomial zeros. Using a procedure based on Smaleʹs point estimation theory and some recent results concerned with 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 initial approximations, polynomial coefficients, and polynomial degree, which is of practical importance.