||The construction of initial conditions which guarantee the safe
convergence of iterative processes is one of the most important problems in solving
nonlinear equations. In this paper we give initial conditions for a one parameter
family of square-root iteration methods for the simultaneous approximation of all
simple zeros of a polynomial. This family, based on the Hansen-Patrick third
order method, has also the cubic convergence and generates some new methods.
The presented initial conditions are of practical importance since depend only
of available data: coefficients of a polynomial and initial approximations to the
wanted zeros of a polynomial.