Nj. Higham et A. Pothen, STABILITY OF THE PARTITIONED INVERSE METHOD FOR PARALLEL SOLUTION OF SPARSE TRIANGULAR SYSTEMS, SIAM journal on scientific computing, 15(1), 1994, pp. 139-148
Several authors have recently considered a parallel method for solving
sparse triangular systems with many right-hand sides. The method empl
oys 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 smal
l when the matrix is well conditioned. Moreover, when the partition is
chosen so that the factors have the same sparsity structure as the co
efficient matrix, the backward error matrix can be taken to be sparse.