Methods to compute a Sturm sequence

Question

Are there any known methods of computing a Sturm sequence of a polynomial $p$ other than the standard algorithm of applying Euclid's algorithm to $p$ and $p'$?

I am asking this question because I noticed that Euclid's algorithm is numerically unstable when implemented with floating point numbers. (See also my related question on the Computer Graphics StackExchange)

Answer 1

Gleyse (1997) used the sequence $$ \begin{align*} p_0&=\sum_{i=0}^n a_iT_i\\ p_1&=\sum_{i=1}^n a_iU_{i-1}\\ p_2&=T_1p_1-p_0\\ p_{k+1}&=U_1p_k-p_{k-1},\quad k\geq 2 \end{align*} $$ instead of Euclid algorithm.