New Clique-Based Parallel Orderings for the Block-Jacobi EVD/SVD Algorithm

We propose a new method for finding a parallel ordering needed in the parallel two-sided block-Jacobi EVD/SVD method. For a given matrix A, partitioned into block columns and block rows, such an ordering defines the subproblems that are solved in parallel in each parallel iteration step. Our approach is based on modeling the matrix block partition as a complete, edge-weighted graph, where the weight of edge (i, j) is defined as the sum of squares of Frobenius norms of the off-diagonal blocks Aij and Aji. The distinction between the physical and logical blocking factors enables to compose the SVD subproblems of varying size by using the contexts of processors under the Message Passing Interface paradigm. We show that finding the ordering that maximalizes the off-diagonal Frobenius norm of covered matrix blocks is equivalent to finding the partition of a complete graph into disjunct cliques of a given size where the total sum of all weights through all cliques is maximized. Since this task belongs to the class of NP-hard, we have designed and implemented a serial genetic algorithm for solving this problem approximately. We report first numerical results using 12 processors and well-conditioned matrices with a multiple minimal singular value of orders from 1000 to 10000.