The paper deals with the solution of the real symmetric tridiagonal eigenvalue problem on a multiprocessor. A multiprocessor algorithm, TREPS, is descibed to find few or all eigenvalues and corresponding eigenvectors of large tridiagonal symmetric matrices. The method consists of isolation, extraction of a cluster by bisection or by the Zeroin method, computation of the eigenvectors of the cluster by inverse iteration, grouping of close eigenvalues (computation of Sturm sequences is described) and orthogonalization of the corresponding groups of vectors by the modified Gram-Schmidt process. The algorithm TREPS is implemented on the multiprocessors Alliant FX/8 (8 processors) and CRAY X-MP/48 (4 processors). Time and speed-up for computing the eigenvalues and eigenvectors are shown in tables and compared with those for the BISECT, TINVIT and TQL2 routines.