The stability of the partitioned inverse approach to parallel sparse triangular

Abstract

Several authors have recently considered a parallel method for solving sparse triangular systems with many right-hand sides. The method employs a partition into sparse factors of the product form of the inverse of the coefficient matrix. It is shown here that the method can be unstable, stability is guaranteed if a certain scalar that depends on the matrix and the partition is small, and that this scalar is small when the matrix is well-conditioned. Moreover, when the partition is chosen so that the factors have the same sparsity structure as the coefficient matrix, the backward error matrix can be taken to be sparse.