Before revision:  Even though the presented method is mathematically equivalent to the latter algorithm, we show that exploiting the rank structure allows to guarantee a small backward error on the polynomial, up to an explicitly computable amplification factor \hat\gamma_1(p), which depends on the polynomial under consideration.
After revision:  Even though the presented method is mathematically equivalent to the unbalanced QR algorithm, we show that exploiting the rank structure allows to guarantee a small backward error on the polynomial, up to an explicitly computable amplification factor \hat\gamma_1(p), which depends on the polynomial under consideration.