The stable roommates problem with $n$ agents has worst case complexity $O(n^2)$ in time and space. Random instances can be solved faster and with less memory, however. We introduce an algorithm that has average time and space complexity $O(n^\frac{3}{2})$ for random instances. We use this algorithm to simulate large instances of the stable roommates problem and to measure the probabilty $p_n$ that a random instance of size $n$ admits a stable matching. Our data supports the conjecture that $p_n = Θ(n^{-1/4})$.