On the cost of approximating all roots of a complex polynomial

This work examines the computational complexity of a homotopy algorithm in approximating all roots of a complex polynomialf. It is shown that, probabilistically, monotonic convergence to each of the roots occurs after a determined number of steps. Moreover, in all subsequent steps, each rootz is approximated by a complex numberx, where ifx₀ =x, x_j =x_j–1 –f(x_j–1)/f(x_j–1),j = 1, 2,, then |x_j –z| < (1/|x₀ –z|)|x_j–1–z|².