Dynamic ordering for a parallel block-Jacobi SVD algorithm

A new approach for the parallel computation of singular value decomposition (SVD) of matrix A ∈ Cm×n is proposed. Contrary to the known algorithms that use a static cyclic ordering of subproblems simultaneously solved in one iteration step, the proposed implementation of the two-sided block-Jacobi method uses a dynamic ordering of subproblems. The dynamic ordering takes into account the actual status of matrix A. In each iteration step, a set of the off-diagonal blocks is determined that reduces the Frobenius norm of the off-diagonal elements of A as much as possible and, at the same time, can be annihilated concurrently. The solution of this task is equivalent to the solution of the maximum-weight perfect matching problem. The greedy algorithm for the efficient solution of this problem is presented. The computational experiments with both types of ordering, incorporated into the two-sided block-Jacobi method, were performed on an SGI - Cray Origin 2000 parallel computer using the Message Passing Interface (MPI). The results confirm that the dynamic ordering is much more efficient with regard to the amount of work required for the computation of SVD of a given accuracy than the static cyclic ordering.