The Use of Vector and Parallel Computers in the Solution of Large Sparse Linear Equations

  • Iain S. Duff
Part of the Progress in Scientific Computing book series (PSC, volume 7)


We discuss three main approaches that are used in the direct solution of sparse unsymmetric linear equations and indicate how they perform on computers with vector or parallel architecture. The principal methods which we consider are general solution schemes, frontal methods, and multifrontal techniques. In each case, we illustrate the approach by reference to a package in the Harwell Subroutine Library. We consider the implementation of the various approaches on machines with vector architecture (like the CRAY-1) and on parallel architectures, both with shared memory and with local memory and message passing.


Gaussian Elimination Elimination Tree Innermost Loop Frontal Method Nest Dissection 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Alaghband, G. and Jordan, H. F. (1983). Parallelization of the MA28 sparse matrix package for the HEP. Report CSDG-83–3, Department of Electrical and Computer Engineering, University of Colorado, Boulder, Colorado.Google Scholar
  2. Benner, R. E. (1986). Shared memory, cache, and frontwidth considerations in multifrontal algorithm development. Report SAND85–2752, Fluid and Thermal Sciences Department, Sandia National Laboratories, Albuquerque, New Mexico.Google Scholar
  3. Berger, P., Dayde, M., and Fraboul, C. (1985). Experience in parallelizing numerical algorithms for MIMD architectures use of asynchronous methods. La Recherche Aerospatiale 5, 325–340.MathSciNetGoogle Scholar
  4. Dave, A. K. and Duff, I. S. (1986). Sparse matrix calculations on the CRAY-2. Report CSS 197, Computer Science and Systems Division, AERE Harwell. In Proceedings International Conference on Vector and Parallel Computing, Loen, Norway, June 2–6, 1986. Parallel Computing (To appear).Google Scholar
  5. Dongarra, J. J. and Duff, I.S. (1986). Performance of vector computers for direct and indirect addressing in Fortran. Harwell Report. (To appear).Google Scholar
  6. Duff, I.S. (1977). MA28 — a set of Fortran subroutines for sparse unsymmetric linear equations. AERE R8730, HMSO, London.Google Scholar
  7. Duff, I. S. (1981). MA32 — A package for solving sparse unsymmetric systems using the frontal method. AERE R10079, HMSO, London.Google Scholar
  8. Duff, I.S. (1983). Enhancements to the MA32 package for solving sparse unsymmetric equations. AERE R11009, HMSO, London.Google Scholar
  9. Duff, I.S. (1984a). The solution of sparse linear systems on the CRAY-1. In Kowalik (1984), 293–309.Google Scholar
  10. Duff, I.S. (1984b). Design features of a frontal code for solving sparse unsymmetric linear systems out-of-core. SIAM J. Sci. Stat. Comput. 5, 270–280.MathSciNetMATHCrossRefGoogle Scholar
  11. Duff, I.S. (1986a). Parallel implementation of multifrontal schemes. Parallel Computing 3, 193–204.MathSciNetMATHCrossRefGoogle Scholar
  12. Duff, I. S. (1986b). The parallel solution of sparse linear equations. In Händler, Haupt, Jeltsch, Juling, and Lange (1986), 18–24.Google Scholar
  13. Duff, I.S. and Johnsson, S. L. (1986). Node orderings and concurrency in sparse problems: an experimental investigation. Proceedings International Conference on Vector and Parallel Computing, Loen, Norway, June 2–6, 1986. Harwell Report. (To appear).Google Scholar
  14. Duff, I.S. and Reid, J. K. (1983)1 The multifrontal solution of indefinite sparse symmetric linear systems. ACM Trans. Math. Softw. 9, 302–325.MathSciNetMATHCrossRefGoogle Scholar
  15. Duff, I. S. and Reid, J. K. (1984). The multifrontal solution of unsymmetric sets of linear systems. SIAM J. Sci. Stat. Comput. 5, 633–641.MathSciNetMATHCrossRefGoogle Scholar
  16. Duff, I. S., Erisman, A. M., and Reid, J. K. (1986). Direct methods for sparse matrices. Oxford University Press, London.MATHGoogle Scholar
  17. George, A. and Liu, J. W. H. (1981). Computer solution of large sparse positive-definite systems. Prentice-Hall, New Jersey.MATHGoogle Scholar
  18. George, A. and Ng, E. (1984). Symbolic factorization for sparse Gaussian elimination with partial pivoting. CS-84–43, Department of Computer Science, University of Waterloo, Ontario, Canada.Google Scholar
  19. George, A. and Ng, E. (1985). An implementation of Gaussian elimination with partial pivoting for sparse systems. SIAM J. Sci. Stat. Comput. 6, 390–409.MathSciNetMATHCrossRefGoogle Scholar
  20. George, A., Heath, M., Liu, J., and Ng, E. (1986). Sparse Cholesky factorization on a local-memory multiprocessor. Report CS-86–01. Department of Computer Science, York University, Ontario, Canada.Google Scholar
  21. Händler, W., Haupt, D., Jeltsch, R., Juling, W., and Lange, O. (Eds.) (1986). CONPAR 86. Lecture Notes in Computer Science 237, Springer-Verlag, Berlin, Heidelberg, New York, and Tokyo.Google Scholar
  22. Hockney, R. W. and Jesshope, C. R. (1981). Parallel computers. Adam Hilger Ltd., Bristol.MATHGoogle Scholar
  23. Hood, P. (1976). Frontal solution program for unsymmetric matrices. Int. J. Numer. Meth. Engng. 10, 379–400.MATHCrossRefGoogle Scholar
  24. Irons, B. M. (1970). A frontal solution program for finite-element analysis. Int. J. Numer. Meth. Engng. 2, 5–32.MATHCrossRefGoogle Scholar
  25. Knuth, D. E. (1973). The art of computer programming. Second edition. Volume 1. Fundamental algorithms. Addison-Wesley, Massachusetts, Palo Alto, and London.Google Scholar
  26. Kowalik, J.S. (Ed.) (1984). High-speed computation. NATO ASI Series. Vol. F.7. Springer-Verlag, Berlin, Heidelberg, New York, and Tokyo.MATHGoogle Scholar
  27. Kung, S.-Y., Arun, K., Bhuskerio, D., and Ho, Y. (1981a). A matrix data flow language/architecture for parallel matrix operations based on computational wave concept. In Kung, Sproull, and Steele (1981b).Google Scholar
  28. Kung, H., Sproull, R., and Steele, G. (Eds.) (1981b). VLSI systems and computations. Computer Science Press, Rockville, Maryland.Google Scholar
  29. Lewis, J. G. and Simon, H. D. (1986). The impact of hardware gather/scatter on sparse Gaussian elimination. Super computing Forum, Boeing Computer Services 1(2), 9–11.Google Scholar
  30. Liu, J. W. H. (1985). Computational models and task scheduling for parallel sparse Cholesky factorization. Report CS-85–01. Department of Computer Science, York University, Ontario, Canada.Google Scholar
  31. Vavasis, S. (1986). Parallel Gaussian elimination. Report CS 367A, Department of Computer Science, Stanford University, Stanford, California.Google Scholar

Copyright information

© Birkhäuser Boston 1987

Authors and Affiliations

  • Iain S. Duff

There are no affiliations available

Personalised recommendations