Computing rational points on rank 1 elliptic curves via $L$-series and canonical heights

Let $E/\mathbb{Q}$ be an elliptic curve of rank 1. We describe an algorithm which uses the value of $L'(E,1)$ and the theory of canonical heghts to efficiently search for points in $E(\mathbb{Q})$ and $E(\mathbb{Z}_{S})$. For rank 1 elliptic curves $E/\mathbb{Q}$ of moderately large conductor (say on the order of $10^{7}$ to $10^{10}$) and with a generator having moderately large canonical height (say between 13 and 50), our algorithm is the first practical general purpose method for determining if the set $E(\mathbb{Z}_{S})$ contains non-torsion points.