Preconditioning eigenvalues and some comparison of solvers

Journal of Computational and Applied Mathematics

Volumn 123, Issue 1-2, 2000, Pages 101-115

  Morgan, Ronald B ^a  

^a BAYLOR UNIVERSITY   (United States)

Author keywords

15A18; 65F15; Davidson's method; Eigenvalues; Jacobi Davidson; Preconditioning

Indexed keywords

ITERATIVE METHODS; LINEAR EQUATIONS; MATRIX ALGEBRA; PROBLEM SOLVING; VECTORS;

DAVIDSONS METHOD; PRECONDITIONING EIGENVALUES; RAYLEIGH QUOTIENT ITERATION; SOLVERS;

EIGENVALUES AND EIGENFUNCTIONS;

EID: 0034321493     PISSN: 03770427     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0377-0427(00)00395-2     Document Type: Article

References (40)

1. Arnoldi W.E. The principle of minimized iterations in the solution of the matrix eigenvalue problem. Quart. Appl. Math. 9:1951;17-29.
2. Barrett R., Berry M., Chan T., Demmel J., Donato J., Dongarra J., Eijkhout V., Pozo R., Romine C., van der Vorst H. Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods. 1994;SIAM, Philadelphia, PA.
3. Crouzeix M., Philippe B., Sadkane M. The Davidson method. SIAM J. Sci. Comput. 15:1994;62-76.
4. Davidson E.R. The iterative calculation of a few of the lowest eigenvalues and corresponding eigenvectors of large real-symmetric matrices. J. Comput. Phys. 17:1975;87-94.
5. Duff I.S., Grimes R.G., Lewis J.G. Sparse matrix test problems. ACM Trans. Math. Software. 15:1989;1-14.
6. Eisenstat S.C., Walker H.F. Choosing the forcing terms in an inexact Newton method. SIAM J. Sci. Comput. 17:1996;16-32.
7. Freund R.W., Nachtigal N.M. QMR: a Quasi-minimal residual method for non-Hermitian linear systems. Numer. Math. 60:1991;315-339.
8. A. Knyazev, Preconditioning eigensolvers, in: Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide, SIAM, 2000, to appear.
9. Lanczos C. An iterative method for the solution of the eigenvalue problem of linear differential and integral operators. J. Res. Nat. Bur. Standards. 45:1950;255-282.
10. Lehoucq R.B., Meerbergen K. Using generalized Cayley transformations within an inexact rational Krylov sequence method. SIAM J. Matrix Anal. Appl. 20:1998;131-148.
11. J.G. Lewis, Algorithms for Sparse Matrix Eigenvalue Problems, Ph.D. Thesis, Stanford University, Stanford, CA, 1977.
12. K. Meerbergen, R.B. Morgan, Inexact methods, in: Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide. SIAM, 2000, to appear.
13. Meerbergen K., Roose D. The restarted Arnoldi method applied to iterative linear system solvers for the computation of rightmost eigenvalues. SIAM J. Matrix Anal. Appl. 18:1997;1-20.
14. Meijerink J.A., van der Vorst H.A. An iterative solution method for linear systems of which the coefficient matrix is a symmetric M-matrix. Math. Comput. 31:1977;148-162.
15. R.B. Morgan, Preconditioning Eigenvalue Problems, Ph.D. Thesis, University of Texas, Austin, TX, 1986.
16. Morgan R.B. Davidson's method and preconditioning for generalized eigenvalue problems. J. Comput. Phys. 89:1990;241-245.
17. R.B. Morgan, Theory for preconditioning eigenvalue problems, Proceedings of the Copper Mountain Conference on Iterative Methods, 1990.
18. Morgan R.B. Generalizations of Davidson's method for computing eigenvalues of large nonsymmetric matrices. J. Comput. Phys. 101:1992;287-291.
19. Morgan R.B., Scott D.S. Generalizations of Davidson's method for computing eigenvalues of sparse symmetric matrices. SIAM J. Sci. Statist. Comput. 7:1986;817-825.
20. Morgan R.B., Scott D.S. Preconditioning the Lanczos algorithm for sparse symmetric eigenvalue problems. SIAM J. Sci. Comput. 14:1993;585-593.
21. Morgan R.B., Zeng M. Harmonic projection methods for large non-symmetric eigenvalue problems. Numer. Linear. Algebra. Appl. 5:1998;33-55.
22. Parlett B.N. The Symmetric Eigenvalue Problem. 1980;Prentice-Hall, Englewood Cliffs, NJ.
23. Ruhe A. SOR-methods for the eigenvalue problem with large sparse matrices. Math. Comput. 28:1974;695-710.
24. Ruhe A. Iterative eigenvalue algorithms based on convergent splittings. J. Comput. Phys. 19:1975;110-120.
25. Ruhe A., Wiberg T. The method of conjugate gradients used in inverse iteration. BIT. 12:1972;543-554.
26. Saad Y. Numerical Methods for Large Eigenvalue Problems. 1992;Halsted Press, New York, NY.
27. Saad Y. Iterative Methods for Sparse Linear Systems. 1996;PWS Publishing, Boston, MA.
28. Saad Y., Schultz M.H. GMRES: a generalized minimum residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Statist. Comput. 7:1986;856-869.
29. G. Sleijpen, H. van der Vorst, Jacobi-Davidson methods, in: Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide, SIAM, 2000, to appear.
30. Sleijpen G.L.G., van der Sluis A. Further results on the convergence behavior of conjugate-gradients and Ritz values. Linear Algebra Appl. 246:1996;233-278.
31. Sleijpen G.L.G., van der Vorst H.A. A Jacobi-Davidson iteration method for linear eigenvalue problems. SIAM J. Matrix Anal. Appl. 17:1996;401-425.
32. Sleijpen G.L.G., van der Vorst H.A. Preconditioning for eigenvalue correction equations. Technical Report in preparation. 1999;Utrecht University, Utrecht, The Netherlands.
33. Sleijpen G.L.G., van der Vorst H.A., Meijerink E. Efficient expansion of subspaces in the Jacobi-Davidson method for standard and generalized eigenvalue problems. ETNA. 7:1998;75-89.
34. Smith B.T., Boyle J.M., Ikebe Y., Klema B.C., Moler C.B. Matrix Eigensystems Routines: EISPACK Guide. 1970;Springer, New York, NY.
35. Sorensen D.C. Implicit application of polynomial filters in a k-step Arnoldi method. SIAM J. Matrix Anal. Appl. 13:1992;357-385.
36. Sorensen D.C., Yang C. A truncated RQ iteration for large scale eigenvalue calculations. SIAM J. Matrix Anal. Appl. 19:1998;1045-1073.
37. Stathopoulos A., Saad Y., Fischer C. Robust preconditioning of large, sparse, symmetric eigenvalue problems. J. Comput. Appl. Math. 64:1995;197-215.
38. Stathopoulos A., Saad Y., Wu K. Dynamic thick restarting of the Davidson, and the implicitly restarted Arnoldi methods. SIAM J. Sci. Comput. 19:1998;227-245.
39. Szyld D.B. Criteria for combining inverse and Rayleigh quotient iteration. SIAM J. Numer. Anal. 6:1988;1369-1375.
40. van der Vorst H.A. Bi-CGSTAB: A fast and smoothly converging variant of Bi-CG for the solution of non-symmetric linear systems. SIAM J. Sci. Statist. Comput. 12:1992;631-644.