Jacobi techniques for computing the symmetric eigenvalue and singular value decompositions have achieved recent prominence because of interest in parallel computation. They are ideally suited for certain multiprocessor systems having processors that are connected in nearest neighbor fashion. If the processors are reasonably powerful and have significant local memory, then block Jacobi procedures are attractive because they render a more favorable computation to communication ratio. This paper examines some of the practical details associated with two block Jacobi methods for the singular value decomposition. The methods differ in how the 2-by-2 subproblems are solved.