A comparison of parallel solvers for diagonally dominant and general narrow-banded linear systems

  • Authors:

  • Peter Arbenz;Andrew Cleary;Jack Dongarra;Markus Hegland

  • Affiliations:

  • Institute of Scientific Computing, Swiss Federal Institute of Technology (ETH), 8092 Zurich, Switzerland;Center for Applied Scientific Computing, Lawrence Livermore National Laboratory, P.O. Box 808, L-561, Livermore CA;Department of Computer Science, University of Tennessee, Knoxville TN;Computer Sciences Laboratory, RSISE, Australian National University, Canberra ACT 0200, Australia

  • Venue:
  • Parallel numerical linear algebra
  • Year:
  • 2001

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Abstract

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 [12] and those investigated by Arbenz and Hegland [4, 8]. 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 fine-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.