On Data Layout in the Parallel Block-jacobi Svd Algorithm with Pre–processing

An efficient version of the parallel two-sided block-Jacobi algorithm for the singular value decomposition of an m × n matrix A includes the pre-processing step, which consists of the QR factorization of A with column pivoting followed by the optional LQ factorization of the Rfactor. Then the iterative two-sided block-Jacobi algorithm is applied in parallel to the R-factor (or L-factor). Having p processors, these iterations are efficiently computed with two block-columns stored in each processor; hence the blocking factor is l = 2p and the process grid 1 × p is used. However, this process grid is not well suited for the pre-processing step due to the (block) column oriented approach in the QR (or LQ) factorization with (or without) column pivoting implemented in the ScaLAPACK. Instead, some matrix block cyclic distribution on a process grid r × c with p = r × c, r, c > 1, and block size nb × nb is required so that all processors remain busy during the whole parallel QR (or LQ) factorization. Optimal values for parameters r, c and nb are estimated experimentally using matrices of order from n = 2000 to n = 8000 and the number of processors form p = 4 to p = 16. It turns out that the optimal values are about nb = 100 and r ≤ c with both r, c near to √ p. It is shown that using optimal parameters in the pre-processing step, the parallel two-sided block-Jacobi SVD algorithm becomes competitive with the ScaLAPACK routine PDGESVD for matrices with a multiple maximal/minimal singular value regardless to the condition number.