An inverse free preconditioned Krylov subspace method for symmetric generalized eigenvalue problems

SIAM Journal on Scientific Computing

Volumn 24, Issue 1, 2003, Pages 312-334

  Golub, Gene H ^a   Ye, Qiang ^b  

^a Stanford University   (United States)

^b UNIVERSITY OF KENTUCKY   (United States)

Author keywords

Eigenvalue problems; Krylov subspace; Preconditioning

Indexed keywords

CONVERGENCE OF NUMERICAL METHODS; EIGENVALUES AND EIGENFUNCTIONS; ITERATIVE METHODS; MATHEMATICAL TRANSFORMATIONS; MATRIX ALGEBRA; PROBLEM SOLVING;

GENERALIZED EIGENVALUE PROBLEM; INVERSE FREE KRYLOV SUBSPACE METHOD;

INVERSE PROBLEMS;

EID: 0037252748     PISSN: 10648275     EISSN: None     Source Type: Journal    
DOI: 10.1137/S1064827500382579     Document Type: Article

References (41)

1. Z. Bai, J. Demmel, J. Dongarra, A. Ruhe, and H. van der Vorst, eds., Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide, SIAM, Philadelphia, 2000.
2. L. Bergamaschi, G. Pini, and F. Sartoretto, Approximate inverse preconditioning in the parallel solution of sparse eigenproblems, Numer. Linear Algebra Appl., 7 (2000), pp. 99-116.
3. A. Bouras and V. Fraysse, A Relaxation Strategy for the Arnoldi Method in Eigenproblems, Technical report TR/PA/00/16, CERFACS, Toulouse, France, 2000.
4. J. Bramble, J. Pasciak, and A. Knyazev, A subspace preconditioning algorithm for eigenvector/eigenvalue computation, Adv. Comput. Math., 6 (1996), pp. 150-189.
5. M. Crouzeix, B. Philippe, and M. Sadkane, The Davidson method, SIAM J. Sci. Comput, 15 (1994), pp. 62-76.
6. J. Demmel, Applied Numerical Linear Algebra, SIAM, Philadelphia, 1997.
7. E. G. D'yakonov and M. Y. Orekhov, Minimization of the computational labor in determining the first eigenvalue of differential operators, Math. Notes, 27 (1980), pp. 382-391.
8. D. R. Fokkema, G. L. G. Sleijpen, and H.A. Van der Vorst, Jacobi-Davidson style QR and QZ algorithms for the reduction of matrix pencils, SIAM J. Sci. Comput., 20 (1998), pp. 94-125.
9. G.H. Golub and C.F. Van Loan, Matrix Computations, 2nd ed., The Johns Hopkins University Press, Baltimore, MD, 1989.
10. G.H. Golub and Q. Ye, Inexact inverse iterations for the generalized eigenvalue problems, BIT, 40 (2000), pp. 672-684.
11. G.H. Golub, Z. Zhang, and H. Zha, Large sparse symmetric eigenvalue problems with homogeneous linear constraints: The Lanczos process with inner-outer iterations, Linear Algebra Appl., 309 (2000), pp. 289-306.
12. L. V. Kantorovich and G.P. Akilov, Functional Analysis in Normed Spaces, MacMillan, New York, 1964.
13. A. V. Knyazev, Convergence rate estimates for iterative methods for a mesh symmetric eigenvalue problem, Soviet J. Numer. Anal. Math. Modelling, 2 (1987), pp. 371-396.
14. A. V. Knyazev, Preconditioned eigensolvers - An oxymoron?, Electron. Trans. Numer. Anal., 7 (1998), pp. 104-123.
15. A. V. Knyazev, Toward the optimal preconditioned eigensolver: Locally optimal block preconditioned conjugate gradient method, SIAM J. Sci. Comput., 23 (2001) pp. 517-541.
16. Y. Lai, K. Lin, and W. Lin, An inexact inverse iteration for large sparse eigenvalue problems, Numer. Linear Algebra Appl., 4 (1997), pp. 425-437.
17. R. B. Lehoucq and K. Meerbergen, Using generalized Cayley transformations within an inexact rational Krylov sequence method, SIAM J. Matrix Anal. Appl., 20 (1998), pp. 131-148.
18. R. Lehoucq, D. Sorenson, and C. Yang, ARPACK Users' Guides, Solution of Large Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods, SIAM, Philadelphia, 1998.
19. K. Meerbergen and D. Roose, The restarted Arnoldi method applied to iterative linear solvers for the computation of rightmost eigenvalues, SIAM J. Matrix Anal. Appl., 18 (1997), pp. 1-20.
20. G. Meurant, Computer Solution of Large Linear Systems, Elsevier, Armsterdam, 1999.
21. 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.
22. R. Morgan and D. Scott, Preconditioning the Lanczos algorithm for sparse symmetric eigenvalue problems, SIAM J. Sci. Comput., 14 (1993), pp. 585-593.
23. K. Neymeyr, A geometric theory for preconditioned inverse iteration, I: Extrema of the Rayleigh quotient, Linear Algebra Appl., 322 (2001), pp. 61-85.
24. S. Oliveira, On the convergence rate of a preconditioned subspace eigensolver, Computing, 63 (1999), pp. 219-231.
25. E. Ovtchinnikov and L. Xanthis, Successive Eigenvalue Relaxation: A New Method for the Generalized Eigenvalue Problem and Convergence Estimate, preprint, University of Westminster, London, UK.
26. B.N. Parlett, The Symmetric Eigenvalue Problem, Prentice-Hall, Englewood Cliffs, NJ, 1980.
27. Y. Saad, Numerical Methods for Large Eigenvalue Problems, Manchester University Press, Manchester, UK, 1992.
28. Y. Saad, Iterative Methods for Sparse Linear Systems, PWS, Boston, 1996.
29. H.R. Schwarz, Eigenvalue Problems and Preconditioning, Internat. Ser. Numer. Math. vol. 96, Birkhauser, Basel, 1991, pp. 191-28.
30. D.S. Scott, Solving sparse symmetric generalized eigenvalue problems without factorization, SIAM J. Numer. Anal., 18 (1981), pp. 102-110.
31. G.L.G. Sleijpen and H.A. van der Vorst, A Jacobi-Davidson iteration method for linear eigenvalue problems, SIAM J. Matrix Anal. Appl., 17 (1996), pp. 401-425.
32. G.L.G. Sleijpen, J. Booten, D.R. Fokkema, and H.A. van der Vorst, Jacobi-Davidson type methods for generalized eigenproblems and polynomial eigenproblems, BIT, 36 (1996), pp. 593-633.
33. D.C. Sorenson and C. Yang, A truncated RQ iteration for large scale eigenvalue calculations, SIAM J. Matrix Anal. Appl., 19 (1998), pp. 1045-1073.
34. P. Smit and M. Paardekooper, The effects of inexact solvers in algorithms for symmetric eigenvalue problems, Linear Algebra Appl., 287 (1998), pp. 337-357.
35. A. Stathopoulos, Y. Saad, and C. Fisher, Robust preconditioning of large sparse symmetric eigenvalue problems, J. Comput. Appl. Math., 64 (1995), pp. 197-215.
36. A. Stathopoulos and Y. Saad, Restarting techniques for the (Jacobi-) Davidson symmetric eigenvalue method, Electron. Trans. Numer. Anal., 7 (1998), pp. 163-181.
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 (1998), pp. 227-245.
38. G.W. Stewart, Matrix Algorithms Volume II: Eigensystems; SIAM, Philadelphia, 2001.
39. H.A. van der Vorst, A generalized Lanczos scheme, Math. Comp., 39 (1982), pp. 559-561.
40. K. Wu, Y. Saad, and A. Stathopoulos, Inexact Newton preconditioning techniques for large symmetric eigenvalue problems, Electron. Trans. Numer. Anal., 7 (1998), pp. 202-214.
41. C. Yang, Convergence analysis of an inexact truncated RQ iterations, Electron. Trans. Numer. Anal., 7 (1998), pp. 40-55.