On some parallel banded system solvers

We consider algorithms for solving narrow-banded diagonally dominant linear systems which are suitable for multiprocessors. We describe a direct solver similar to that in [12] for tridiagonal systems, and in [9] for solving a banded system on a linearly connected set of processors. We will also provide and analyze a parallel implementation of the partitioning algorithm and the matrix decomposition which we refer to as a hybrid solver (direct and iterative) which is superior to the direct solver especially for strongly diagonally dominant systems. When the interconnection network is not sufficiently powerful, a bottleneck develops in one of the stages of the direct solver in which the cost of the computation is proportional to the number of processors. This inefficiency may be alleviated by using an iterative scheme in this stage that takes full advantage of the parallelism offered even by a linear array of p processors. A similar approach is also used to handle the positive-definite system that arises from the standard five-point finite-difference discretization of the Helmholtz equation. This problem arises frequently in situations where fast solvers are of primary importance. In this paper we consider the matrix decomposition solver that has been described in several papers, e.g. [1,2,6,10,11].