Abstract
This work studies the eigenproblem of large, sparse and symmetric matrices through algorithms implemented in distributed memory multiprocessor architectures. The implemented parallel algorithm operates in three stages: structuring input matrix (Lanczos Method), computing eigenvalues (Sturm Sequence) and computing eigenvectors (Inverse Iteration). Parallel implementation has been carried out using a SPMD programming model and the PVM standard library. Algorithms have been tested in a multiprocessor system Cray T3E. Speed-up, load balance, cache faults and profile are discussed. From this study, it follows that for large input matrices our parallel implementations perceptibly improve the management of the memory hierarchy.
This work was supported by the Ministry of Education and Culture of Spain (CICYT TIC96-1125-C03-03)
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
Basermann, A.; Weidner, P. A parallel algorithm for determining all eigenvalues of large real symmetric tridiagonal matrices. Parallel Computing 18, pag 1129–1141. 1992.
Bernstein, H.J. An Accelerated Bisection Method for the Calculation of Eigenvalues of a Symmetric Tridiagonal Matrix. Numer. Math. 43, pag. 153–160. 1984.
Berry, M.W. Large-Scale Sparse Singular Value Computations. The International Journal of Supercomputer. Vol 6, pg. 171–186. 1992.
Cullum, J.K. and Willoughby, R.A. Lanczos Algorithms for Large Symmetric Eigenvalue Computations. Vol. I Theory. Birkhauser Boston, Inc., 1985.
Duff, I.S.; Grimes, R.G. and Lewis, J.G. User’s Guide for the Harwell-Boeing Sparse Matrix Collection. Research and Technology Division, Boeing Computer Services. 1992.
García, I; Garzón, E.M.; Cabaleiro, J.C.; Carazo, J.M. and Zapata, E.L. Parallel Tridiagonalization of Symmetric Matrices Based on Lanczos Method. Parallel Computing and Transputer Applications. Vol I, 236–245, 1992.
Garzón, E.M. and García, I. Parallel Implementation of the Lanczos Method for Sparse Matrices: Analysis of the Data Distributions. Proceedings ICS’96. p. 294–300. Philadelphia, 1996.
Garzón, E.M. and García, I. Evaluation of the Work Load Balance in Irregular Problems Using Value Based Data Distributions. Proceedings of Euro-PDS’97. p. 137–143. 1997.
Golub, G.H. and Van Loan C. F. Matrix Computation. The Johns Hopkins University Press., Baltimore and London., 1993.
Simon, H.D. Analysis of the symmetric Lanczos Algorithm with Reorthogonalization Methods. Linear Algebra and its Applications. Vol 61, pg. 101–131. 1984.
Sy-Shin Lo, Bernard Philippe and Ahmed Samed. A multiprocessor Algorithm for the Symmetric Tridiagonal Eigenvalue Problem. SIAM J. Sci. Stat. Comput. vol-8, No. 2, March 1987.
Wilkinson, J.H. The Algebraic Eigenvalue Problem. Clarendon Press. Oxford, 1965.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1999 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Garzón, E.M., García, I. (1999). A Parallel Implementation of the Eigenproblem for Large, Symmetric and Sparse Matrices. In: Dongarra, J., Luque, E., Margalef, T. (eds) Recent Advances in Parallel Virtual Machine and Message Passing Interface. EuroPVM/MPI 1999. Lecture Notes in Computer Science, vol 1697. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-48158-3_47
Download citation
DOI: https://doi.org/10.1007/3-540-48158-3_47
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-66549-6
Online ISBN: 978-3-540-48158-4
eBook Packages: Springer Book Archive