We describe a numerical method for calculating hydrodynamic interactions between spherical particles efficiently and accurately, both for particles immersed in an infinite liquid and for systems with periodic boundary conditions. Our method is based on a multipole expansion in Cartesian tensors. We then show how to solve the equations of motion for translational and rotational motion of suspended particles at large Peclet numbers. As an example we study the sedimentation of an array of spheres with and without periodic boundary conditions. We also study the effect of perturbations on the stability of the trajectories.