Dynamic Ordering for the Parallel One-sided Block-jacobi Svd Algorithm

The serial Jacobi algorithm (either one-sided or two-sided) for the computation of a singular value decomposition (SVD) of a general matrix has excellent numerical properties and parallelization potential, but it is considered to be the slowest method for computing the SVD. Even its parallelization with some parallel cyclic (static) ordering of subproblems does not lead to much improvement when comparing with parallel methods based on the matrix bi-diagonalization principle. However, in last 10 years some progress has been achieved in increasing the efficiency of the parallel block-Jacobi SVD method by using two new ideas: i) the new parallel dynamic ordering of subproblems, and, ii) the matrix pre-processing by QR iterations. For the parallel two-sided block-Jacobi algorithm, these ideas were already thoroughly tested on various parallel platforms, and our implementation can be faster than the ScaLAPACK routine PDGESVD for some distributions of singular values. With respect to the one-sided variant, the new parallel dynamic ordering, when compared to a parallel cyclic ordering, can substantially decrease the number of parallel iteration steps needed for the convergence. However, its more scalable implementation is desirable because currently it occupies a relatively high portion of the total parallel execution time.