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Many engineering applications require the computation of the q algebraically largest eigenvalues and a corresponding eigenspace of a large, sparse, real, symmetric matrix. An iterative, block version of the symmetric Lanczos algorithm has been developed for this computation. There are no restrictions on the sparsity pattern within the matrix or on the distribution of the eigenvalues of the matrix. Zero eigenvalues, eigenvalues equal in magnitude but opposite in sign, and multiple eigenvalues can all be handled directly by the procedure.