A parallelizable eigensolver for real diagonalizable matrices with real eigenvalues

SIAM Journal on Scientific Computing

Volumn 18, Issue 3, 1997, Pages 869-885

  Huss Lederman, Steven ^a   Tsao, Anna ^a   Turnbull, Thomas ^a  

^a Center for Computing Sciences   (United States)

Author keywords

Divide and conquer algorithm; Eigenvalues; Invariant subspaces; Parallel algorithm

Indexed keywords

DIAGONALIZABLE MATRICES; EIGENVECTORS; INVARIANT SUBSPACES;

COMPUTER WORKSTATIONS; MATRIX ALGEBRA; PARALLEL ALGORITHMS; VECTORS;

EIGENVALUES AND EIGENFUNCTIONS;

EID: 0031129191     PISSN: 10648275     EISSN: None     Source Type: Journal    
DOI: 10.1137/S1064827592228833     Document Type: Article

References (32)

1. E. ANDERSON, Z. BAI, C. BISCHOF, J. DEMMEL, J. DONGARRA, J. Du CROZ, A. GREENBAUM, S. HAMMARLING, A. MCKENNEY, AND D. SORENSEN, LAPACK: A portable linear algebra library for high-performance computers, in Proc. Supercomputing '90, IEEE Computer Society Press, Los Alamitos, CA, 1990, pp. 2-11.
2. L. AUSLANDER AND A. TSAO, On parallelizable eigensolvers, Adv. Appl. Math., 13 (1992), pp. 253-261.
3. Z. BAI AND J. DEMMEL, On a block implementation of Hessenberg multishift QR iteration, Internat. J. High Speed Comput., 1 (1989), pp. 97-112.
4. Z. BAI AND J. DEMMEL, Design of Parallel Nonsymmetric Eigenroutine Toolbox, Part I, Research report 92-09, University of Kentucky, Lexington, KY, December 1992.
5. Z. BAI, J. DEMMEL, AND A. MCKENNEY, On the Conditioning the Nonsymmetric Eigenproblem: Theory and Software, LAPACK Working note 13, Courant Institute, New York, 1989.
6. A. N. BEAVERS, JR. AND E. D. DENMAN, A computational method for eigenvalues and eigenvectors of a matrix with real eigenvalues, Numer. Math., 21 (1973), pp. 389-396.
7. M. BERRY AND A. SAMEH, Parallel algorithms for the singular value and dense symmetric eigenvalue problem, J. Comput. Appl. Math., 27 (1989), pp. 191-213.
8. C. BISCHOF, A parallel QR factorization with controlled local pivoting, SIAM J. Sci. Statist. Comput., 12 (1991), pp. 36-57.
9. J. J. M. CUPPEN. A divide and conquer method for the symmetric tridiagonal eigenproblem, Numer. Math., 36 (1981), pp. 177-195.
10. J. DEMMEL AND K. VESELIC, Jacobi's method is more accurate than QR, SIAM J. Matrix Anal. Appl., 13 (1992), pp. 1204-1245.
11. E. D. DENMAN AND A. N. BEAVERS, JR., The matrix sign function and computations in systems, Appl. Math. Comput., 2 (1976), pp. 63-94.
12. E. D. DENMAN AND J. LEYVA-RAMOS, Spectral decomposition of a matrix using the generalised sign matrix, Appl. Math. Comput., 8 (1981), pp. 237-250.
13. J. DONGARRA, C. B. MOLER, J. R. BUNCH, AND G. W. STEWART, LINPACK User's Guide, SIAM, Philadelphia, PA, 1979.
14. J. DONGARRA AND D. SORENSEN, A fully parallel algorithm for the symmetric eigenvalue problem, SIAM J. Sci. Statist. Comput., 8 (1987), pp. 139-154.
15. G. GOLUB, V. KLEMA, AND G. W. STEWART, Rank Degeneracy and Least Squares Problems, Tech. report TR-456, University of Maryland, College Park, MD, 1976.
16. G. GOLUB AND C. F. VAN LOAN, Matrix Computations, 2nd ed., The Johns Hopkins University Press, Baltimore, MD, 1989.
17. R. W. HAMMING, Digital Filters, 2nd ed., Prentice-Hall, Englewood Cliffs, NJ, 1983.
18. K. HUFFMAN AND R. KUNZE, Linear Algebra, Prentice-Hall, Englewood Cliffs, NJ, 1971.
19. J. L. HOWLAND, The sign matrix and the separation of matrix eigenvalues, Linear Algebra Appl., 49 (1983), pp. 221-232.
20. Y. HUO AND R. SCHREIBER, Efficient, massively parallel eigenvalue computation, Internat. J. Supercomput. Appl., 7 (1993), pp. 292-303.
21. I. IPSEN AND E. JESSUP, Solving the symmetric tridiagonal eigenvalue problem on the hypercube, Tech. report RR-548, Yale University, New Haven, CT, 1987.
22. I. IPSEN AND E. JESSUP, Improving the accuracy of inverse iteration, SIAM J. Sci. Statist. Comput., 13 (1992), pp. 550-572.
23. C. KENNEY AND A. J. LAUB, Rational iterative methods for the matrix sign function, SIAM J. Matrix Anal. Appl., 12 (1990), pp. 273-291.
24. T. Y. LI, Z. ZENG, AND L. CONG, Solving eigenvalue problems of real nonsymmetric matrices with real homotopies, SIAM J. Numer. Anal., 29 (1992), pp. 229-248.
25. T.-Y. LI, H. ZHANG, AND X.-H. SUN, Parallel homotopy algorithm for symmetric tridiagonal eigenvalue problems, SIAM J. Sci. Statist. Comput., 12 (1991), pp. 469-487.
26. C.-C. LIN AND E. ZMIJEWSKI, A Parallel Algorithm for Computing the Eigenvalues of an Unsymmetric Matrix on a SIMD Mesh of Processors, Tech. report TRCS 91-15, Department of Computer Science, University of California, Santa Barbara, CA, 1991.
27. V. PAN AND R. SCHREIBER, An improved Newton iteration for the generalized inverse of a matrix, with applications, SIAM J. Sci. Statist. Comput., 12 (1991), pp. 1109-1129.
28. J. RUTTER, A Serial Implementation of Cuppen's Divide and Conquer Algorithm for the Symmetric Eigenvalue Problem, Tech. report UCB/CSD 94/799, University of California, Berkeley, California, 1994.
29. R. SCHREIBER, Solving eigenvalue and singular value problems on an undersized systolic array, SIAM J. Sci. Statist. Comput., 7 (1986), pp. 441-451.
30. G. SHROFF AND R. SCHREIBER, On the convergence of the cyclic Jacobi method for parallel block orderings, SIAM J. Sci. Statist. Comput., 10 (1989), pp. 326-346.
31. G. W. STEWART, A Jacobi-like algorithm, for computing the Schur decomposition of a nonhermitian matrix, SIAM J. Sci. Statist. Comput., 6 (1985), pp. 853-864.
32. R. A. VAN DE GEIJN, Deferred shifting schemes for parallel QR methods, SIAM J. Matrix Anal. Appl., 14 (1993), pp. 180-194.