Toward an efficient parallel eigensolver for dense symmetric matrices

SIAM Journal on Scientific Computing

Volumn 20, Issue 3, 1998, Pages 1132-1154

  Hendrickson, Bruce ^a,b   Jessup, Elizabeth ^a,b   Smith, Christopher ^a,b,c  

^a SANDIA NATIONAL LABORATORIES   (United States)

^b University of Colorado at Boulder   (United States)

^c Tower Automotive   (United States)

Author keywords

Backtransformation; Dense matrix; Eigensolver; Householder tridiagonalization; Parallel algorithm; Torus wrap mapping

Indexed keywords

BACKTRANSFORMATION; DENSE MATRIX; EIGENSOLVER; HOUSEHOLDER TRIDIAGONALIZATION; TORUS WRAP MAPPING;

DATA STORAGE EQUIPMENT; MATRIX ALGEBRA; PARALLEL ALGORITHMS; PROBLEM SOLVING;

EIGENVALUES AND EIGENFUNCTIONS;

EID: 0032226427     PISSN: 10648275     EISSN: None     Source Type: Journal    
DOI: 10.1137/S1064827596300681     Document Type: Article

References (46)

1. E. ANDERSON, Z. BAI, C. BISCHOF, J. DEMMEL, J. DONGARRA, J. DuCnoz, A. GREENBAUM, S. HAMMARLING, A. MCKENNEY, S. OSTROUCHOV, AND D. SORENSEN, LAPACK User's Guide, 2nd ed., SIAM, Philadelphia, 1995.
2. R. H. BARLOW AND D. EVANS, A parallel organization of the bisection algorithm, Comput. J., 22 (1977), pp. 267-269.
3. R. H. BARLOW, D. J. EVANS, AND J. SHANEHCHI, Parallel multisection applied to the eigenvalue problem, Comput. J., 26 (1983), pp. 6-9.
4. H. J. BERNSTEIN AND M. GOLDSTEIN, Parallel implementation of bisection for the calculation of eigenvalues of tridiagonal symmetric matrices, Computing, 37 (19S6), pp. 85-91.
5. H. J. BERNSTEIN AND M. GOLDSTEIN, Optimizing Givens' algorithm for multiprocessors, SIAM J. Sei. Statist. Comput., 9 (1988), pp. 601-602.
6. C. BISCHOF, B. LANG, AND X. SUN, A Framework for Symmetric Band Reduction, Tech. report ANL/MCS-P586-Q496, preprint, Math. & Comput. Science Division, Argonne National Laboratory, Argonne, IL, 1996.
7. C. BISCHOF, B. LANG, AND X. SUN, The SDR Toolbox-Software for Successive Band Reduction, Tech. report ANL/MCS-P587-0496, preprint, Math. & Comput. Science Division, Argonne National Laboratory, Argonne, IL, 1996.
8. R. BlSSELlNG, Personal communication, 1992.
9. H. Y. CHANG, S. UTKU, M. SALAMA, AND D. RAPP, A parallel Householder tridiagonalization stratagem using scattered square decomposition, Parallel Comput., 6 (1988), pp. 297-311.
10. H. Y. CHANG, S. UTKU, M. SALAMA, AND D. RAPP, A parallel Householder tridiagonalization strategem using scattered row decomposition, Internat. J. Numer. Methods Engrg., 26 (1988), pp. 857-873.
11. J. CHOI, J. DONGARRA, AND D. WALKER, The design of a parallel, dense linear algebra software library: Reduction to Hessenberg, tridiagonal, and bidiagonal form, Numer. Algorithms, 10 (1995), pp. 379-400.
12. J. CULLUM AND R. A. WlLLOUGHBY, Lanczos and the computation in specified intervals of the spectrum of large, sparse, real symmetric matrices, in Sparse Matrix Proceedings, SIAM, Philadelphia, 1978, pp. 220-225.
13. 2) Algorithm for the Symmetric Tridiagonal Eigenvalue/Eigenvector Problem, Ph.D. thesis, University of California, Berkeley, 1997.
14. J. DONGARRA, J. DuCROZ, S. ÜAMMARLING, AND R. HANSON, An extended set of Fortran basic linear algebra subprograms: Model implementation and test programs, ACM Trans. Math. Software, 14 (1988), pp. 18-32.
15. J. DONGARRA, J. DUCROZ, I. DUFF, AND S. HAMMARLING, A set of level 3 basic linear algebra subprograms, ACM Trans. Math. Software, 16 (1990), pp. 1-17.
16. J. DONGARRA, J. DuCROZ, S. HAMMARLING, AND R. HANSON, An extended set of Fortran basic linear algebra subprograms, ACM Trans. Math. Software, 14 (1988), pp. 1-17.
17. J. DONGARRA, S. HAMMARLING, AND D. SORENSEN, Block reduction of matrices to condensed form for eigenvalue computations, J. Comput. Appl. Math., 27 (1989), pp. 215-227.
18. J. DONGARRA AND R. VAN DE GEIJN, Reduction to Condensed Form for the Eigenvalue Problem on Distributed Memory Architectures, LAPACK Working Note 30, Dept. of Computer Science, University of Tennessee, Knoxville, TN, 1991.
19. J. J. DONGARRA AND D. C. SORENSEN, A fully parallel algorithm for the symmetric eigenvalue problem, SIAM J. Sei. Statist. Comput., 8 (1987), pp. sl39-s154.
20. G. I. FANN AND R. J. LITTLEFIELD, Parallel inverse iteration with reorthogonalization, in Proc. 6th Conference Parallel Processing Sei. Comput., SIAM, Philadelphia, 1993, pp. 409-413.
21. G. C. Fox, M. A. JOHNSON, G. A. LYZENGA, S. W. OTTO, J. K. SALMON, AND D. W. WALKER, Solving Problems on Concurrent Processors: Volume 1, Prentice-Hall, Englewood Cliffs, NJ, 1988.
22. G. H. GOLUB AND C. F. VAN LOAN, Matrix Computations, 2nd ed., The Johns Hopkins Press, Baltimore, MD, 1989.
23. M. HEGLAND, A distribution independent algorithm for the reduction to tridiagonal form using one-sided rotations, in Proc. 1st IEEE International Conference on Algorithms and Architectures for Parallel Processing, Brisbane, Australia, IEEE Computer Society Press, Piscataway, NJ, 1995, pp. 286-289.
24. B. HENDRICKSON, E. JESSUP, AND C. SMITH, A Parallel Eigensolver for Dense Symmetric Matrices, Tech. report SAND96-0822, Sandia National Laboratories, Albuquerque, NM, 1996.
25. B. HENDRICKSON, R. LELAND, AND S. PLIMPTON, An efficient parallel algorithm for matrixvector multiplication, Internat. J. High Speed Comput., 7 (1995), pp. 73-88.
26. B. HENDRICKSON AND D. WOMBLE, The torus-wrap mapping for dense matrix calculations on massively parallel computers, SIAM J. Sei. Comput., 15 (1994), pp. 1201-1226.
27. H. HUANG, A Parallel Algorithm for Symmetric Tridiagonal Eigenvalue Problems, CAC Document 109, Center for Advanced Computation, University of Illinois, Urbanna, IL, 1974.
28. S. HUSS-LEDERMAN, A. TSAO, AND G. ZHANG, A parallel implementation of the invariant subspace decomposition algorithm for dense symmetric matrices, in Proc. 6th SIAM Conf. Parallel Proc. Sei. Comput., SIAM, Philadelphia, 1993, pp. 367-374.
29. I. C. F. IPSEN AND E. R. JESSUP, Solving the symmetric tridiagonal eigenvalue problem on the hypcrcube, SIAM J. Sei. Statist. Comput., 11 (1990), pp. 203-229.
30. E. R. JESSUP, Parallel Solution of the Symmetric Tridiagonal Eigenproblem, Ph.D. thesis, Dept. of Computer Science, Yale University, New Haven, CT, 1989.
31. E. R. JESSUP AND I. C. F. IPSEN, Improving the accuracy of inverse iteration, SIAM J. Sei. Statist. Comput., 13 (1992), pp. 550-572.
32. L. KAUFMAN, A parallel QR algorithm for the symmetric tridiagonal eigenvalue problem, J. Parallel Distrib. Comput., 23 (1994), pp. 429-434.
33. L. KAUFMAN, Computing the MDMT decomposition, ACM Trans. Math. Software, 21 (1995), pp. 479-487.
34. J. R. KUTTLER AND V. G. SIGILLITO, Eigenvalues of the Laplacian in two dimensions, SIAM Rev., 26 (1984), pp. 163-193.
35. B. LANG, A parallel algorithm for reducing symmetric banded matrices to tridiagonal form, SIAM J. Sei. Comput., 14 (1993), pp. 1320-1338.
36. T. LI, H. ZHANG, AND X. SUN, Parallel homotopy algorithm for symmetric tridiagonal eigenvalue problem, SIAM J. Sei. Statist. Comput., 12 (1991), pp. 469-487.
37. R. J. LITTLEFIELD AND K. J. MASCHHOFF, Investigating the performance of parallel eigensolvers for large processor counts, Theoretica Chimica Acta, 84 (1993), pp. 457-473.
38. S. Lo, B. PHILLIPE, AND A. SAMEH, A multiprocessor algorithm for the symmetric tridiagonal eigenvalue problem, SIAM J. Sei. Statist. Comput., 8 (1987), pp. s155-s165.
39. MESSAGE PASSING INTERFACE FORUM, MPI: A message passing interface standard, Internat. J. Supcrcomput. Appl., 8 (1994), pp. 159-416.
40. B. T. SMITH, J. M. BOYLE, J. J. DONGARRA, B. S. GARBOW, Y. IKEDE, V. C. KLEMA, AND C. B. MOLER, Matrix Eigensystem Routines-EISPACK Guide, 2nd ed., Lecture Notes in Comput. Sei. 6, Springer-Verlag, New York, 1976.
41. J. SPEISER AND H. WHITEHOUSE, Parallel processing algorithms and architectures for real-time signal processing, in Proc. SPIE Real Time Signal Processing IV, SPIE, 1981, pp. 2-9.
42. K. STANLEY, Execution time of Symmetric Eigensolvers, Ph.D. thesis, Dept. Computer Science, University of California, Berkeley, 1997.
43. P. E. STRAZDINS, Matrix factorization using distributed panels on the Fujitsu AP1000, in Proc. IEEE 1st International Conference on Algorithms and Architectures for Parallel Processing, Brisbane, Australia, IEEE Computer Society Press, Piscataway, NJ, 1995.
44. R. VAN DE GEIJN, Deferred shifting schemes for parallel QR methods, SIAM J. Matrix Anal. Appl., 14 (1993), pp. 180-194.
45. R. A. VAN DE GEIJN, Efficient global combine operations, in Proc. 6th Distributed Memory Computing Conference, Portland, OR, IEEE Computer Society Press, 1991, pp. 291-294.
46. R. A. VAN DE GEIJN, Personal communication, 1992.