Abstract
This paper discusses the direct solution of sparse symmetric positive definite linear systems of equations and describes three parallel algorithms for the Cholesky factorization step. An empirical comparison of the communication costs of the three algorithms on a partitioned-memory multiprocessor is made. It is shown that the scheme used to partition the matrix among the processors, and the existence of a multicasting capability in the communication network, are important factors in the trade-off between the algorithms. Performance results on a BBN Butterfly TC2000 multiprocessor are reported.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
C. Ashcraft, S. Eisenstat, and J. W.-H. Liu, “A fan-in algorithm for distributed sparse numerical factorization,” SIAM J. Sci. Statist Comput., Vol. 11, pp. 593–599, 1990.
BBN Advanced Computers Inc., TC2000 Technical Product Summary, 1989.
I. Duff, A. Erisman, and J. Reid, Direct Methods for Sparse Matrices, Oxford University Press, Oxford, U.K., 1987.
I. Duff and J. Reid, “The multifrontal solution of indefinite sparse symmetric linear equations,” ACM Trans. Math. Software, Vol. 9, pp. 302–325, 1983.
K. Eswar, P. Sadayappan, and V. Visvanathan, “Multifrontal factorization of sparse matrices on shared-memory multiprocessors,” Proceedings of the Twentieth International Conference on Parallel Processing, St. Charles, IL, Vol. III, pp. 159–166, 1991.
A. George and J. W.-H. Liu, “An automatic nested dissection algorithm for irregular finite element problems,” SIAM J. Numer. Anal, Vol. 15, pp. 1053–1069, 1978.
A. George and J. W.-H. Liu, Computer Solution of Large Sparse Positive Definite Systems, Prentice-Hall, Englewood Cliffs, NJ, 1981.
A. George and J. W.-H. Liu, “The evolution of the minimum degree ordering algorithm,” SIAM Rev., Vol. 31, pp. 1–19, 1989.
A. George, M. Heath, J. W.-H. Liu, and E. G.-Y. Ng, “Sparse Cholesky factorization on a local-memory multiprocessor,” SIAM J. Sci. Statist. Comput., Vol. 9, pp. 327–340, 1988.
J. W.-H. Liu, “The multifrontal method for sparse matrix solution: theory and practice,” Tech. Report CS-90-04, Dept. of Computer Science, York University, North York, Ontario, 1990.
R. Lucas, T. Blank, and J. Tiemann, “A parallel solution method for large sparse systems of equations,” IEEE Trans, on Computer-Aided Design, Vol. CAD-6, pp. 981–991, 1987.
J. Ortega, Introduction to Parallel and Vector Solution of Linear Systems, Plenum Press, New York, 1988.
E. Rothberg and A. Gupta, “Efficient sparse matrix factorization on high performance workstations — Exploiting the memory hierarchy,” ACM Trans. Math. Software, Vol. 17, pp. 313–334, 1991.
R. Schreiber, “A new implementation of sparse Gaussian elimination,” ACM Trans. Math. Software, Vol. 8, pp. 256–276, 1982.
M. Yannakakis, “Computing the minimum fill-in is NP-complete,” SIAM J. Algebraic Discrete Methods, Vol. 2, pp. 77–79, 1981.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1993 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Eswar, K., Sadayappan, P., Visvanathan, V. (1993). Parallel Direct Solution of Sparse Linear Systems. In: Özgüner, F., Erçal, F. (eds) Parallel Computing on Distributed Memory Multiprocessors. NATO ASI Series, vol 103. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-58066-6_6
Download citation
DOI: https://doi.org/10.1007/978-3-642-58066-6_6
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-63460-4
Online ISBN: 978-3-642-58066-6
eBook Packages: Springer Book Archive