Empirical versus asymptotic rate of convergence of a class of methods for solving a polynomial equation

Given alternative methods with identical order of convergence for solving the polynomial equation -(z) = 0, the method with the smaller asymptotic error constant might be assumed to be superior in terms of the number of iterations required for convergence. We present empirical evidence for a parameterized class of methods of second order showing that a parameter choice which does not correspond to the minimal asymptotic error constant may nevertheless be superior in practice.