Abstract
In this paper we present a parallel method for finding several eigenvalues and eigenvectors of a generalized eigenvalue problem A x = λB x, where A and B are large sparse matrices. A moment-based method by which to find all of the eigenvalues that lie inside a given domain is used. In this method, a small matrix pencil that has only the desired eigenvalues is derived by solving large sparse systems of linear equations constructed from A and B. Since these equations can be solved independently, we solve them on remote hosts in parallel. This approach is suitable for master-worker programming models. We have implemented and tested the proposed method in a grid environment using a grid RPC (remote procedure call) system called OmniRPC. The performance of the method on PC clusters that were used over a wide-area network was evaluated.
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
Arnoldi, W.E.: The principle ofminimized iteration in the solution of the matrix eigenproblem. Quarterly of Appl. Math. 9, 17–29 (1951)
Bai, Z.: Krylov subspace techniques for reduced-order modeling of large-scale dynamical systems. Appl. Numer. Math. 43, 9–44 (2002)
Golub, G.H., Milanfar, P., Varah, J.: A stable numerical method for inverting shape from moments. SIAM J. Sci. Comp. 21(4), 1222–1243 (1999)
Hyodo, S.: Meso-scale fusion: A method for molecular electronic state calculation in inhomogeneous materials. In: Proc. the 15th Toyota Conference, Mikkabi (2001) in Special issue of J. Comput. Appl. Math. 149, 101–118 (2002)
Inadomi, Y., Nakano, T., Kitaura, K., Nagashima, U.: Definition of molecular orbitals in fragment molecular orbital method. Chemical Physics Letters 364, 139–143 (2002)
Kravanja, P., Sakurai, T., Van Barel, M.: On locating clusters of zeros of analytic functions. BIT 39, 646–682 (1999)
Kravanja, P., Sakurai, T., Sugiura, H., Van Barel, M.: A perturbation result for generalized eigenvalue problems and its application to error estimation in a quadrature method for computing zeros of analytic functions. J. Comput. Appl. Math. 161, 339–347 (2003)
OmniRPC, http://www.omni.hpcc.jp/OmniRPC
Ruhe, A.: Rational Krylov algorithms for nonsymmetric eigenvalue problems II: matrix pairs. Linear Algevr. Appl. 197, 283–295 (1984)
Saad, Y.: Iterative Methods for Large Eigenvalue Problems. Manchester University Press, Manchester (1992)
Sakurai, T., Kravanja, P., Sugiura, H., Van Barel, M.: An error analysis of two related quadrature methods for computing zeros of analytic functions. J. Comput. Appl. Math. 152, 467–480 (2003)
Sakurai, T., Sugiura, H.: A projection method for generalized eigenvalue problems. J. Comput. Appl. Math. 159, 119–128 (2003)
Sato, M., Boku, T., Takahashi, D.: OmniRPC: a Grid RPC System for parallel programming in cluster and grid environment. In: Proc. CCGrid 2003, pp. 206–213 (2003)
van der Vorst, H.A., Melissen, J.B.M.: A Petrov-Galerkin type method for solving Ax = b, where A is a symmetric complex matrix. IEEE Trans. on Magnetics 26(2), 706–708 (1990)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Sakurai, T., Hayakawa, K., Sato, M., Takahashi, D. (2006). A Parallel Method for Large Sparse Generalized Eigenvalue Problems by OmniRPC in a Grid Environment. In: Dongarra, J., Madsen, K., Waśniewski, J. (eds) Applied Parallel Computing. State of the Art in Scientific Computing. PARA 2004. Lecture Notes in Computer Science, vol 3732. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11558958_138
Download citation
DOI: https://doi.org/10.1007/11558958_138
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-29067-4
Online ISBN: 978-3-540-33498-9
eBook Packages: Computer ScienceComputer Science (R0)