New Dynamic Orderings for the Parallel One–Sided Block-Jacobi SVD Algorithm

Abstract

Five variants of a new dynamic ordering are presented for the parallel one-sided block Jacobi SVD algorithm. Similarly to the two-sided algorithm, the dynamic ordering takes into account the actual status of a matrix—this time of its block columns with respect to their mutual orthogonality. Variants differ in the computational and communication complexities and in proposed global and local stopping criteria. Their performance is tested on a square random matrix of order 8192 with a random distribution of singular values using [Formula: see text], 32, 64, 96 and 128 processors. All variants of dynamic ordering are compared with a parallel cyclic ordering, two-sided block-Jacobi method with dynamic ordering and the ScaLAPACK routine PDGESVD with respect to the number of parallel iteration steps needed for the convergence and total parallel execution time. Moreover, the relative errors in the orthogonality of computed left singular vectors and in the matrix assembled from computed singular triplets are also discussed. It turns out that the variant 3, for which a local optimality in some precisely defined sense can be proved, and its combination with variant 2, are the most efficient ones. For relatively small blocking factors [Formula: see text], they outperform the ScaLAPACK procedure PDGESVD and are about 2 times faster.