The Parallel Solution Of Eigenproblems On Multiprocessor Systems
P. J. Eberlein · Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE · 1988
We describe here 'one-sided' Jacobi methods for solving eigenproblems on parallel architectures. These methods are modifications of the procedure introduced by Hestenesl, which uses only columns of the matrix. Thus, only 'local' information is needed in each processor. The Jacobi angle is determined from columns of the factored matrix and columns of the matrix which ultimately becomes the eigenvector matrix. The singular value decomposition may be considered as a special case. The eigenvalue decompositions for symmetric, real normal, and, orthogonal matrices may be regarded as generalizations. Extensions to the complex field are easily derived. We also describe a new formation of Jacobi rotation sets. These require only one send and one receive per set, and a complete set is found in the minimal n-1 steps. We use the ordering of the diagonal elements of the matrix so that, in general, one may obtain only a few of the eigenvalues (and vectors) as required. Implementations of these algorithms may be made on various kinds of parallel architectures, including the hypercube.