Parallel QR algorithm for the complete eigensystem of symmetric matrices

We propose a parallel organization of the QR algorithm for computing the complete eigensystem of symmetric matrices. We developed Occam versions of standard sequential implementations of the QR algorithm: the procedure qr1 which computes only eigenvalues and qr2 for the computation of all eigenvalues and eigenvectors. The Occam procedure parqr2 is a parallel implementation of qr2 and was tested on a pipeline of 16 transputers. Although parqr2 could be used to compute the eigenvalues and eigenvectors of a symmetric tridiagonal matrix, it is best suited to be used in conjunction with a parallel algorithm for the reduction of a dense symmetric matrix to tridiagonal form where the orthogonal transformations are accumulated in an explicit way. In the practical tests parqr2 has proved to be efficient and we have carried out a simple analyses that appears to indicate that it is possible to use efficiently a number p of processors of the same order of magnitude of the size n of the matrix (p⩽n/6). This is an interesting result from the point of view of the scalability of our parallel algorithm