Inexact newton preconditioning techniques for large symmetric eigenvalue problems

Electronic Transactions on Numerical Analysis

Volumn 7, Issue , 1998, Pages 202-214

  Wu, Kesheng ^a   Saad, Yousef ^b   Stathopoulos, Andreas ^c  

^a LAWRENCE BERKELEY NATIONAL LABORATORY   (United States)

^b University of Minnesota   (United States)

^c The College of William and Mary   (United States)

Author keywords

Newton method; Preconditioning for eigenvalue method; Sparse matrix eigenvalue problem

Indexed keywords

EID: 0001034392     PISSN: 10689613     EISSN: 10689613     Source Type: Journal    
DOI: None     Document Type: Article

References (40)

1. W. E. ARNOLDI,. The principle of minimized iteration in the solution of the matrix eigenvalue problem, Quarter. Appl. Math., 9 (1951), pp. 17-29.
2. O. AXELSSON AND P. S. VASSILEVSKI, A survey of multilevel preconditioned iterative methods, BIT, 29(4), (1989), pp. 769-793.
3. _, A block generalized Conjugate Gradient solver with inner iterations and variable step preconditioning, SIAM J. Matrix Anal. Appl., 12, (1991), pp. 625-644.
4. OWE AXELSSON, A survey of preconditioned iterative methods for linear systems of equations, BIT, 25 (1985), pp. 166-187.
5. M. BENZI, C. D. MEYER AND M. TUMA, A sparse approximate inverse preconditioner for the conjugate gradient method, SIAM J. Sci. Comput., 17(5) (1996), pp. :135-1149.
6. F. CHATELIN, Eigenvalues of Matrices, Wiley, New York, NY, 1993.
7. E. CHOW AND Y. SAAD, Approximate inverse preconditioners via sparse-sparse iterations, SIAM J. Sci. Comput., 18(6) (1997), pp. 1657-1675.
8. M. CROUZEIX, B. PHILIPPE AND M. SADKANE, The Davidson method, SIAM J. Sci. Comput., 15 (1994), pp. 62-76.
9. ERNEST R. DAVIDSON, The iterative calculation of a few of the lowest eigenvalues and corresponding eigenvectors of large real-symmetric matrices, J. Comput. Phys., 17 (1974), pp. 87-94.
10. , Super-matrix methods, Comput. Phys. Comm., 53 (1989), pp. 49-60.
11. JAMES W. DEMMEL, Applied Numerical Linear Algebra, SIAM, Philadelphia, PA, 1997.
12. J. J. DONGARRA, C. B. MOLER AND J. H. WILKINSON, Improving the accuracy of computed eigenvalues and eigenvectors, SIAM J. Numer. Anal., 20 (1983), pp. 23-45.
13. I. S. DUFF, R. G. GRIMES AND J. G. LEWIS, Sparse matrix test problems, ACM Trans. Math. Software, 15 (1989), pp. 1-14.
14. T. ERICSSON AND A. RUHE, The spectral transformation Lanczos method for numerical solution of large sparse generalized symmetric eigenvalue problems, Math. Comp., 35 (1980), pp. 1251-1268.
15. ALBERT T. GALICK, Efficient solution of large sparse eigenvalue problems in microelectronic simulation, Ph.D. thesis, University of Illinois at Urbana-Champaign, 1993.
16. G. H. GOLUB AND C. F. VAN LOAN, Matrix Computations, 3rd Ed., The Johns Hopkins University Press, Baltimore, MD, 1996.
17. GENE H. GOLUB, ZHENYUE ZHANG AND HONGYUAN ZHA, Large sparse symmetric eigenvalue problems with homogeneous linear constraints: The Lanczos process with inner-outer iterations, Technical Report CSE-96-050, Dept. of Computer Science and Engineering, The Pennsylvania State University, 1996.
18. R. G. GRIMES, J. G. LEWIS AND H. D. SIMON, A shifted block Lanczos algorithm for solving sparse symmetric generalized eigenproblems, SIAM J. Matrix Anal. Appl., 15(1) (1994), pp. 228-272.
19. M. J. GROTE AND T. HUCKLE, Parallel preconditioning with sparse approximate inverses, SIAM J. Sci. Comput., 18(3) (1997), pp. 838-853.
20. L. YU. KOLOTILINA AND A. YU. YEREMIN, Factorized sparse approximate inverse preconditionings I. theory, SIAM J. Matrix Anal. Appl., 14(1) (1993), pp. 45-58.
21. CORNELIUS LANCZOS, An iterative method for the solution of the eigenvalue problem of linear differential and integral operators, J. Res. National Bureau Standards, 45 (1950), pp. 255-282.
22. R. B. LEHOUCQ AND K. MEERBERGEN, Using generalized Cayley transformations within an inexact rational Krylov sequence method, SIAM J. Matrix Anal. Appl., 20(1), (1999), pp. 131-148.
23. K. MEERBERGEN AND A. SPENCE, Implicitly restarted Arnoldi with purification for the shift-invert transformation, Math. Comp., 66(218) (1997), pp. 667-689.
24. J. A. MEIJERINK AND H. A. VAN DER VORST, An iterative solution method for linear systems of which the coefficient matrix is a symmetric M-matrix, Math. Comp., 31 (1977), pp. 148-162.
25. R. B. MORGAN AND D. S. SCOTT, Generalizations of Davidson's method for computing eigenvalues of sparse symmetric matrices, SIAM J. Sci. Statist. Comput., 7 (1986), pp. 817-825.
26. J. OLSEN, P. JORGENSEN AND J. SIMONS, Passing the one-billion limit in full configuration- interaction (FCI) calculations, Chem. Phys. Letters, 169 (1990), pp. 463-472.
27. BERESFORD N. PARLETT, The symmetric eigenvalue problem. Classics in Applied Mathematics. SIAM, Philadelphia, PA, 1998.
28. G. PETERS AND J. H. WILKINSON, Inverse iteration, ill-conditioned equations and Newton's method, SIAM Rev., 21(3) (1979), pp. 339-360.
29. A. RUHE, Rational Krylov sequence methods for eigenvalue computation, Linear Algebra Appl., 58 (1984), pp. 391-405.
30. YOUSEF SAAD, A flexible inner-outer preconditioned GMRES algorithm, SIAM J. Sci. Comput., 14(2) (1993), pp. 461-469.
31. _, Numerical Methods for Large Eigenvalue Problems. Manchester University Press, 1993.
32. _, Highly parallel preconditioners for general sparse matrices, G. Golub, A. Greenbaum, and M. Luskin, eds., Recent advances in Iterative Methods, volume 60 of IMA volumes in mathematics and its applications, Springer-Verlag (1994), pp. 165-199.
33. G. L. G. SLEIJPEN, A. G. L. BOOTEN, D. R. FOKKEMA AND H. A. VAN DER VORST, Jacobi-Davidson type methods for generalized eigenproblems and polynomial eigenproblems, BIT 36(3) (1996), pp. 595-633.
34. G. L. G. SLEIJPEN AND H. A. VAN DER VORST, The Jacobi-Davidson method for eigenvalue problems as an accelerated inexact Newton scheme, In Iterative methods in linear algebra II : proceedings of the ImACS International Symposium on Iterative Methods in Linear Algebra, June 17-20, 1995, Blagoevgrad, pages 377-389. Elsevier, Amsterdam; New York, March 1996.
35. _, Jacobi-Davidson iteration method for linear eigenvalue problems, SIAM J. Matrix Anal. Appl., 17(2), (1996), pp. 401-425.
36. A. STATHOPOULOS, Y SAAD, AND C. F. FISCHER, Robust preconditioning of large, sparse, symmetric eigenvalue problems, J. Comput. Appl. Math., 64 (1995), pp. 197-215.
37. A. STATHOPOULOS, Y. SAAD AND K. WU, Dynamic thick restarting of the Davidson and the implicitly restarted Arnoldi methods, SIAM J. Sci. Comput., 19(1) (1998), pp. 227-245.
38. I. A. TAPI A, Newton's method for optimization problems with equality constraints, SIAM J. Numer. Anal., 11 (1974), pp. 874-886.
39. KESHENG WU, An experimental study of Krylov subspace accelerators, Technical Report UMSI 96/20, Minnesota Supercomputer Institute, University of Minnesota, 1996.
40. _, Preconditioned Techniques for Large Eigenvalue Problems, Ph.D. thesis, University of Minnesota, 1997.