The convergence of a family of parallelzero-finding methods

One of the crucial problems in solving polynomial equations is to state some kind of quantitative conditions that provide both safe and fast convergence. In the spirit of Smaleʹs point estimation theory introduced in 1981, we construct in this paper computationally verifiable initial conditions which guarantee the convergence of the recently proposed one parameter family of methods of the fourth order for the simultaneous finding polynomial zeros. These conditions are computationally verifiable since they depend only on initial approximations, polynomial coefficients, and polynomial degree, which is of significant practical importance.