Generalization of the Dynamic Ordering for the One-Sided Block Jacobi SVD Algorithm: I. Analysis and Design

The efficiency of the one-sided parallel block-Jacobi algorithm for computation of the singular value decomposition (SVD) of a general matrix A ∈ Rm×n, m ≥ n, depends–besides some numerical tricks that speed-up the convergence–crucially on the parallel ordering of subproblems, which are to be solved in each parallel iteration step. We discuss in detail possible generalizations of the so-called dynamic ordering of subproblems that was originally designed for the two-sided parallel block-Jacobi SVD algorithm. It turns out that the straightforward generalization leads to the algorithm that requires too much communication in each parallel iteration step, so that the Jacobi method would spend too much time in communication and its efficiency due to the clever usage of some numerical tricks will be lost. Therefore, a modification of the dynamic ordering is proposed based on the use of the maximum principal angle between two subspaces of the same dimension. This approach can be described as a heuristics. Its efficiency should be tested with using a wide set of random matrices and compared with the (row or column) cyclic ordering.