Advertisement

A multifrontal approach for solving sparse linear equations

  • Iain Duff
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 1005)

Abstract

A principal characteristic of multifrontal schemes for solving sparse sets of linear equations is their use of full submatrices during the elimination and thus, in the symbolic phases, much work and storage can be saved by only storing index lists rather than the full submatrices.

In this paper, we indicate how such a scheme can be efficiently implemented and show how it need not be restricted to systems which are symmetric and positive definite.

We illustrate our remarks by examining the performance of a Harwell code based on this approach. In particular, we show that the simple inner loop of such codes performs well on machines capable of vectorization.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Bunch, J.R. and Parlett, B.N. (1971). Direct methods for solving symmetric indefinite systems of linear equations. SIAM J. Numer. Anal. 8, 639–655.MathSciNetCrossRefzbMATHGoogle Scholar
  2. Duff, I.S. (1981). MA32 — a package for solving sparse unsymmetric systems using the frontal method. Harwell Report AERE R. 10079, HMSO, London.Google Scholar
  3. Duff, I.S. (1982). The design and use of a frontal scheme for solving sparse unsymmetric equations. In Numerical Analysis, Proceedings of the third IIMAS Workshop held at Cocoyoc, Mexico, January 1981. J.P. Hennart (Ed). Lecture Notes in Mathematics 909, Springer-Verlag. pp 240–247.Google Scholar
  4. Duff, I.S., Grimes, R.G., Lewis, J.G. and Poole, W.G. (1982). Sparse matrix test problems. Poster session at Sparse Matrix Symposium, Fairfield Glade, Tennessee, October 24–27, 1982.Google Scholar
  5. Duff, I.S. and Reid, J.K. (1982a). The multifrontal solution of indefinite sparse symmetric linear systems. Harwell Report CSS 122.Google Scholar
  6. Duff, I.S. and Reid, J.K. (1982b). MA27 — A set of Fortran subroutines for solving sparse symmetric sets of linear equations. AERE Harwell Report R. 10533, HMSO, London.Google Scholar
  7. Duff, I.S., Reid, J.K., Munksgaard, N and Nielsen, H.B. (1979). Direct solution of sets of linear equations whose matrix is sparse, symmetric and indefinite. J. Inst. Maths. Applics. 23, 235–250.MathSciNetCrossRefzbMATHGoogle Scholar
  8. Eisenstat, S.C., Gursky, M.C., Schultz, M.H. and Sherman, A.H. (1982). The Yale Sparse Matrix Package I: The symmetric codes. Int. J. Numer. Meth. Engng. 18 pp 1145–1151.CrossRefzbMATHGoogle Scholar
  9. George, A., Liu, J.W. and Ng, E. (1980). User guide for SPARSPAK: Waterloo Sparse Linear Equations Package. Research Report CS-78-30 (revised Jan. 1980). Dept. Comp. Sci. University of Waterloo.Google Scholar
  10. George, A. and Liu, J.W. (1981). Computer Solution of Large Sparse Positive Definite Systems. Prentice-Hall, New Jersey.zbMATHGoogle Scholar
  11. Hood, P. (1976). Frontal solution program for unsymmetric matrices. Int. J. Numer. Meth. Engng. 10, 379–400.CrossRefzbMATHGoogle Scholar
  12. Irons, B.M. (1970). A frontal solution program for finite element analysis. Int. J. Numer. Meth. Engng. 2, 5–32.CrossRefzbMATHGoogle Scholar
  13. Munksgaard, N. (1977). Fortran subroutines for direct solution of sets of sparse and symmetric linear equations. Report 77.05. Numerical Inst., Lyngby, Denmark.Google Scholar

Copyright information

© Springer-Verlag 1983

Authors and Affiliations

  • Iain Duff
    • 1
  1. 1.Computer Science and Systems DivisionAERE, HarwellDidcotEngland

Personalised recommendations