Let $(W,S)$ be an arbitrary Coxeter system, $y\in S^*$. We describe an algorithm which will compute, directly from $y$ and the Coxeter matrix of $W$, the interval from the identity to $y$ in the Bruhat ordering, together with the (partially defined) left and right actions of the generators. This provides us with exactly the data that are needed to compute the Kazhdan-Lusztig polynomials $P_{x,z}$, $x\leq z\leq y$. The correctness proof ot the algorithm is based on a remarkable theorem due to Matthew Dyer.