The backpropagation algorithm for error gradient calculations in multilayer, feed-forward neural networks is derived in matrix form involving inner and outer products. It is demonstrated that these calculations can be carried out efficiently using systolic processing techniques , particularly using the SPRINT, a 64-element systolic processor developed at Lawrence Livermore National Laboratory. This machine contains one million synapses, and forward-propagates 12 million connections per second, using 100 watts of power. When executing the algorithm, each SPRINT processor performs useful work 97% of the time. The theory and applications are confirmed by some nontrivial examples involving seismic signal recognition.