Robustness

The method also seems quite robust. The cancellation problem when computing $k_0$ is avoided by using the proper expression. Another cancellation proble occurs while computing $R-\sqrt{Q^3+ R^2}$ when $R \gg Q$. However, this has no consequence as it affects only the $T$ term which in this case is much smaller than $S$ and only $S+T$ is used to compute the solution. As the algorithm converges, all the coefficients also converge ($k$, $\alpha$, $\beta$, $\gamma$, $\delta$, $a_1$, $a_2$, $a_3$, $Q$, $R$, $S$, $T$). There is no numerical explosion anywhere.