Stability of the Partitioned Inverse Method for Parallel Solution of Sparse Triangular Systems

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 while 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. Key words, sparse matrix, triangular system, substitution algorithm, parallel algorithm, rounding error analysis, matrix inverse AMS subject classifications, primary 65F05, 65F50, 65G05