Skip to main content

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

  • Chapter
Large Scale Scientific Computing

Part of the book series: Progress in Scientific Computing ((PSC,volume 7))

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  • 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 

  • 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 

  • 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.

    MathSciNet  Google Scholar 

  • 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 

  • 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 

  • Duff, I.S. (1977). MA28 — a set of Fortran subroutines for sparse unsymmetric linear equations. AERE R8730, HMSO, London.

    Google Scholar 

  • Duff, I. S. (1981). MA32 — A package for solving sparse unsymmetric systems using the frontal method. AERE R10079, HMSO, London.

    Google Scholar 

  • Duff, I.S. (1983). Enhancements to the MA32 package for solving sparse unsymmetric equations. AERE R11009, HMSO, London.

    Google Scholar 

  • Duff, I.S. (1984a). The solution of sparse linear systems on the CRAY-1. In Kowalik (1984), 293–309.

    Google Scholar 

  • 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.

    Article  MathSciNet  MATH  Google Scholar 

  • Duff, I.S. (1986a). Parallel implementation of multifrontal schemes. Parallel Computing 3, 193–204.

    Article  MathSciNet  MATH  Google Scholar 

  • Duff, I. S. (1986b). The parallel solution of sparse linear equations. In Händler, Haupt, Jeltsch, Juling, and Lange (1986), 18–24.

    Google Scholar 

  • 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 

  • 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.

    Article  MathSciNet  MATH  Google Scholar 

  • 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.

    Article  MathSciNet  MATH  Google Scholar 

  • Duff, I. S., Erisman, A. M., and Reid, J. K. (1986). Direct methods for sparse matrices. Oxford University Press, London.

    MATH  Google Scholar 

  • George, A. and Liu, J. W. H. (1981). Computer solution of large sparse positive-definite systems. Prentice-Hall, New Jersey.

    MATH  Google Scholar 

  • 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 

  • 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.

    Article  MathSciNet  MATH  Google Scholar 

  • 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 

  • 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 

  • Hockney, R. W. and Jesshope, C. R. (1981). Parallel computers. Adam Hilger Ltd., Bristol.

    MATH  Google Scholar 

  • Hood, P. (1976). Frontal solution program for unsymmetric matrices. Int. J. Numer. Meth. Engng. 10, 379–400.

    Article  MATH  Google Scholar 

  • Irons, B. M. (1970). A frontal solution program for finite-element analysis. Int. J. Numer. Meth. Engng. 2, 5–32.

    Article  MATH  Google Scholar 

  • Knuth, D. E. (1973). The art of computer programming. Second edition. Volume 1. Fundamental algorithms. Addison-Wesley, Massachusetts, Palo Alto, and London.

    Google Scholar 

  • Kowalik, J.S. (Ed.) (1984). High-speed computation. NATO ASI Series. Vol. F.7. Springer-Verlag, Berlin, Heidelberg, New York, and Tokyo.

    MATH  Google Scholar 

  • 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 

  • Kung, H., Sproull, R., and Steele, G. (Eds.) (1981b). VLSI systems and computations. Computer Science Press, Rockville, Maryland.

    Google Scholar 

  • 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 

  • 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 

  • Vavasis, S. (1986). Parallel Gaussian elimination. Report CS 367A, Department of Computer Science, Stanford University, Stanford, California.

    Google Scholar 

Download references

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 1987 Birkhäuser Boston

About this chapter

Cite this chapter

Duff, I.S. (1987). The Use of Vector and Parallel Computers in the Solution of Large Sparse Linear Equations. In: Deuflhard, P., Engquist, B. (eds) Large Scale Scientific Computing. Progress in Scientific Computing, vol 7. Birkhäuser Boston. https://doi.org/10.1007/978-1-4684-6754-3_20

Download citation

  • DOI: https://doi.org/10.1007/978-1-4684-6754-3_20

  • Publisher Name: Birkhäuser Boston

  • Print ISBN: 978-1-4684-6756-7

  • Online ISBN: 978-1-4684-6754-3

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics