Efficient Jacobi algorithms on multicomputers
In this paper, we study the parallelization of the Jacobi method to solve the symmetric eigenvalue problem on distributed-memory multiprocessors. To obtain a theoretical efficiency of 100% when solving this problem, it is necessary to exploit the symmetry of the matrix. The only previous algorithm we know exploiting the symmetry on multicomputers is that in , but that algorithm uses a storage scheme appropriate for a logical ring of processors, thus having a low scalability. In this paper we show how matrix symmetry can be exploited on a logical mesh of processors obtaining a higher scalability than that obtained with the algorithm in . Algorithms for ring and mesh logical topologies are compared experimentally on the PARSYS SN-1040 and iPSC/860 multicomputers.
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- 1.R. P. Brent and F. T. Luk. A systolic architecture for almost linear-time solution of the symmetric eigenvalue problem. Technical Report TR-CS-82-10, Department of Computer Science, Australian National University, Camberra, August 1982.Google Scholar
- 2.P. J. Eberlein and H. Park. Efficient implementation of Jacobi algorithms and Jacobi sets on distributed memory architectures. Journal of Parallel and Distributed Computing, 8:358–366, 1990.Google Scholar
- 3.A. Edelman. Large dense linear algebra in 1993: The parallel computing influence. The International Journal of Supercomputer Applications, 7(2):113–128, 1993.Google Scholar
- 4.D. Giménez, V. Hernández, R. van de Geijn and A. M. Vidal. A jacobi method by blocks to solve the symmetric eigenvalue problem on a mesh of processors. ILAS Conference, 1994.Google Scholar
- 5.G. H. Golub and C. F. Van Loan. Matrix Computations. The Johns Hopkins University Press, 1989.Google Scholar
- 6.I. N. Levine. Molecular Spectroscopy. John Wiley and Sons, 1975.Google Scholar
- 7.M. Pourzandi and B. Tourancheau. A Parallel Performance Study of Jacobi-like Eigenvalue Solution. Technical report, March 1994.Google Scholar
- 8.G. H. Stewart. A Jacobi-like algorithm for computing the Schur decomposition of a nonhermitian matrix. SIAM J. Sci. Stat. Comput., 4:853–864, 1985.Google Scholar
- 9.V. Strumpen and P. Arbenz. Improving Scalability by Communication Latency Hiding. In D. H. Bailey, P. E. Bjørstad, J. R. Gilbert, M. V. Mascagni, R. S. Schreiber, H. D. Simon, V. J. Torczon and L. T. Watson, editor, Proceedings of the Seventh SIAM Conference on Parallel Processing for Scientific Computing, pages 778–779. SIAM, 1995.Google Scholar
- 10.R. A. van de Geijn. Storage schemes for Parallel Eigenvalue Algorithms. In G. H. Golub and P. Van Dooren, editor, Numerical Linear Algebra. Digital Signal Processing and Parallel Algorithms, volume 70 of NATO ASI Series. Springer-Verlag, 1991.Google Scholar