We propuse a new and simple way to parallelise the QR algorithm for computing eigenvalues and eigenvectors of real symetric tridiagonal matrices. This approach is suitable for all prallel computers, ranging from multiprocessor supercomputers with shared memory to massively parallel computer with local memory. We report on numerical experiments we have completed on a cray-Y-MP, an alliant FX-80 a sequent symmetry S81b a nCUBE 2, a thinking machenes CM200 and a cluter of Sun SPARCstations.