Preconditioned Parallel Block-jacobi Svd Algorithm

We show experimentally, that the QR factorization with the complete column pivoting, optionally followed by the LQ factorization of the Rfactor, can lead to a substantial decrease of the number of outer parallel iteration steps in the parallel block-Jacobi SVD algorithm, whereby the details depend on the condition number and on the shape of spectrum, including the multiplicity of singular values. Best results were achieved for well-conditioned matrices with a multiple minimal singular value, where the number of parallel iteration steps has been reduced by two orders of magnitude. However, the gain in speed, as measured by the total parallel execution time, depends decisively on how efficient is the implementation of the distributed QR and LQ factorizations on a given parallel architecture. In general, the reduction of the total parallel execution time up to one order of magnitude has been achieved.