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The Davidson method is extensively used in quantum chemistry and atomic physics for finding a few extreme eigenpairs of a large, sparse, symmetric matrix. It can be viewed as a preconditioned version of the Lanczos method which reduces the number of iterations at the expense of a more complicated step. Frequently, the problem sizes involved demand the use of large multicomputers with hundreds or thousands of processors. The difficulties occurring in parallelizing the Davidson step are dealt with and results on a smaller scale machine are reported. The new version improves the parallel characteristics of the Davidson algorithm and holds promise for a large number of processors. Its stability and reliability is similar to that of the original method.