x[i+1] := RootOf( F(x[i])-p(z), z );

Selecting which root of the polynomial to use is a crucial point.  If we want to insure convergence, we must select the root which is closest to the iteration value.

Since this selection is necessary for convergence, we will use it in every step of the algorithm.  Furthermore, many iterative methods for solving polynomials can profit significantly from starting their iterators at an approximation of the root; the correct root could come almost for free. 

For example, for the solution of the equation

x[i+1] := RootOf( z^5-6*z^4+5*z^3+6*z-5 = 7*x[i]^2*ln(x[i]), z );

If we start with x₀ = 1, then the rounded roots are

The root which is closest to x₀=1 is 0.758845...  Hence we set x₁=0.758845...  For the second iteration the polynomial becomes z⁵-6z⁴+5z³+6z-5 = -1.11236...  The roots of this polynomial are

and the root closest to x₁ is 0.586566...  which becomes x₂.  This process converges quite quickly,