A method for determining all eigenvalues of large real symmetric tridiagonal matrices on multiprocessor systems with vector facilities is presented. The method is based on the Sturm sequence and uses bisection for isolating and extracting the eigenvalues. For the extraction bisection is accelerated by a superlinear convergent zero finder, the Pegasus method. Some new ideas are presented for weighting the values of the characteristic polynomial to avoid under- and overflow. The experiments are realized on SUPRENUM (with 16 processors) and CRAY Y- MP8/832 (with 8 processors).