Hostname: page-component-7c8c6479df-hgkh8 Total loading time: 0 Render date: 2024-03-29T04:38:59.250Z Has data issue: false hasContentIssue false

AN APPROXIMATE MATRIX INVERSION PROCEDURE BY PARALLELIZATION OF THE SHERMAN–MORRISON FORMULA

Published online by Cambridge University Press:  09 March 2010

KENTARO MORIYA*
Affiliation:
Head Office, Nikon System Inc., Japan (email: kmoriya@nikon-sys.co.jp)
LINJIE ZHANG
Affiliation:
College of Mathematical Sciences, Ocean University of China, Japan (email: zhanglinjie@hotmail.com)
TAKASHI NODERA
Affiliation:
Department of Mathematics, Faculty of Science and Technology, Keio University, Japan (email: nodera@math.keio.ac.jp)
*
For correspondence; e-mail: kmoriya@nikon-sys.co.jp
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The Sherman–Morrison formula is one scheme for computing the approximate inverse preconditioner of a large linear system of equations. However, parallelizing a preconditioning approach is not straightforward as it is necessary to include a sequential process in the matrix factorization. In this paper, we propose a formula that improves the performance of the Sherman–Morrison preconditioner by partially parallelizing the matrix factorization. This study shows that our parallel technique implemented on a PC cluster system of eight processing elements significantly reduces the computational time for the matrix factorization compared with the time taken by a single processor. Our study has also verified that the Sherman–Morrison preconditioner performs better than ILU or MR preconditioners.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 2010

References

[1]Bru, R., Credán, J., Marín, J. and Mas, J., “Preconditioning sparse nonsymmetric linear systems with the Sherman–Morrison formula”, SIAM J. Sci. Comput. 25 (2003) 701715.CrossRefGoogle Scholar
[2]Grote, M. and Huckel, T., “Parallel preconditioning with sparse approximate inverses”, SIAM J. Sci. Comput. 18 (1997) 838853.CrossRefGoogle Scholar
[3]Saad, Y. and Schultz, M. K., “GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems”, SIAM J. Sci. Stat. Comput. 7 (1986) 856869.CrossRefGoogle Scholar
[4]Simoncini, Y., “On the convergence of restarted Krylov subspace method”, SIAM J. Matrix Anal. Appl. 22 (2000) 430452.CrossRefGoogle Scholar
[5] “University of Florida sparse matrix collection”, online http://www.cise.ufl.edu/research/sparse/matrices.Google Scholar