A Comparison of Parallel Solvers for Diagonally Dominant and General Narrow-Banded Linear Systems

We investigate and compare stable parallel algorithms for solving diagonally dominant and general narrow-banded linear systems of equations. Narrow-banded means that the bandwidth is very small compared with the matrix order and is typically between 1 and 100. The solvers compared are the banded system solvers of ScaLAPACK [11] and those investigated by Arbenz and Hegland [3, 6]. For the diagonally dominant case, the algorithms are analogs of the well-known tridiagonal cyclic reduction algorithm, while the inspiration for the general case is the lesser-known bidiagonal cyclic reduction, which allows a clean parallel implementation of partial pivoting. These divide-and-conquer type algorithms complement ne-grained algorithms which perform well only for wide-banded matrices, with each family of algorithms having a range of problem sizes for which it is superior. We present theoretical analyses as well as numerical experiments conducted on the Intel Paragon.