Obtaining triangular diagonal blocks in sparse matrices using cutsets

The paper proposes an algorithm which computes a (2 × 2) block partition of an irreducible, sparse matrix with a zero-free diagonal so that one of the diagonal blocks is triangular. The algorithm works by computing a cutset of the directed graph associated with the off-diagonal part of the matrix and is linear in the number of nonzeros therein. The proposed algorithm is then extended to reducible, sparse matrices with a zero-free diagonal. Experiments on benchmark unsymmetric matrices show that, in many cases the order of the triangular block can be increased by using another algorithm for computing cutsets from the literature, which is based on a greedy randomized adaptive search procedure; however, this is not efficient timewise unless the matrix is relatively small. A block iterative solver based on the partition returned by the proposed algorithm is compared with an industrial strength direct solver for time, space, and accuracy. Results indicate that there are cases in which it is advantageous to compute and use block partitions based on cutsets.