Skip to main content
×
×
Home

A Filtered-Davidson Method for Large Symmetric Eigenvalue Problems

  • Cun-Qiang Miao (a1)
Abstract
Abstract

For symmetric eigenvalue problems, we constructed a three-term recurrence polynomial filter by means of Chebyshev polynomials. The new filtering technique does not need to solve linear systems and only needs matrix-vector products. It is a memory conserving filtering technique for its three-term recurrence relation. As an application, we use this filtering strategy to the Davidson method and propose the filtered-Davidson method. Through choosing suitable shifts, this method can gain cubic convergence rate locally. Theory and numerical experiments show the efficiency of the new filtering technique.

Copyright
Corresponding author
*Corresponding author. Email address: miaocunqiang@lsec.cc.ac.cn (C.-Q. Miao)
References
Hide All
[1] 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), pp. 8794.
[2] Fang H.R. and Saad Y., A filtered Lanczos procedure for extreme and interior eigenvalue problems, SIAM J. Sci. Comput., 34(2012), pp. A2220A2246.
[3] Golub G.H. and Ye Q., Inexact inverse iteration for generalized eigenvalue problems, BIT, 40(2000), pp. 671684.
[4] Jian S., A block preconditioned steepest descent method for symmetric eigenvalue problems, Appl. Math. Comput., 219(2013), pp. 1019810217.
[5] Knyazev A.V., Toward the optimal preconditioned eigensolver: locally optimal block preconditioned conjugate gradient method, SIAM J. Sci. Comput., 23(2001), pp. 517541.
[6] Lai Y.-L., Lin K.-Y. and Lin W.-W., An inexact inverse iteration for large sparse eigenvalue problems, Numer. Linear Algebra Appl., 4(1997), pp. 425437.
[7] Morgan R.B., Generalizations of Davidson's method for computing eigenvalues of large nonsymmetric matrices, J. Comput. Phys., 101(1992), pp. 287291.
[8] Morgan R.B. and Scott D.S., Generalizations of Davidson's method for computing eigenvalues of sparse symmetric matrices, SIAM J. Sci. Statisst. Copmut., 7(1986), pp. 817825.
[9] Notay Y., Convergence analysis of inexact Rayleigh quotient iteration, SIAM J. Matrix Anal. Appl., 24(2003), pp. 627644.
[10] Ovtchinnikov E., Cluster robustness of preconditioned gradient subspace iteration eigensolvers, Linear Algebra Appl., 415(2006), pp. 140166.
[11] Ovtchinnikov E.E., Sharp convergence estimates for the preconditioned steepest descent method for Hermitian eigenvalue problems, SIAM J. Numer. Anal., 43(2006), pp. 26682689.
[12] Parlett B.N., The Symmetric Eigenvalue Problems, SIAM, Philadelphia, PA, 1998.
[13] Saad Y., Chebyshev acceleration techniques for solving nonsymmetric eigenvalue problems, Math. Comp., 42(1984), pp. 567588.
[14] Saad Y., Numerical Methods for Large Eigenvalue Problems, Second Edition, SIAM, Philadelphia, PA, 2011.
[15] Saad Y., On the rates of convergence of the Lanczos and the Block-Lanczos methods, SIAM J. Numer. Anal., 17(1980), pp. 687706.
[16] Sleijpen G.L.G., Booten A.G.L., Fokkema D.R. and Van Der Vorst H.A., Jacobi-Davidson type methods for generalized eigenproblems and polynomial eigenproblems, BIT, 36(1996), pp. 595633.
[17] Sleijpen G.L.G. and Van Der Vorst H.A., A Jacobi-Davidson iteration method for linear eigenvalue problems, SIAM J. Matrix Anal. Appl., 17(1996), pp. 401425.
[18] Sorensen D.C., Implicit application of polynomial filters in a k-step Arnoldi method, SIAM J. Matrix Anal. Appl., 13(1992), pp. 357385.
[19] Van Den Eshof J., The convergence of Jacobi-Davidson iterations for Hermitian eigenproblems, Numer. Linear Algebra Appl., 9(2002), pp. 163179.
[20] Xue F. and H.Elman C., Convergence analysis of iterative solvers in inexact Rayleigh quotient iteration, SIAM J. Matrix Anal. Appl., 31(2009), pp. 877899.
[21] Zhou Y.-K. and Saad Y., A Chebyshev-Davidson algorithm for large symmetric eigenproblems, SIAM J. Matrix Anal. Appl., 29(2007), pp. 954971.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

East Asian Journal on Applied Mathematics
  • ISSN: 2079-7362
  • EISSN: 2079-7370
  • URL: /core/journals/east-asian-journal-on-applied-mathematics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Keywords:

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 52 *
Loading metrics...

Abstract views

Total abstract views: 235 *
Loading metrics...

* Views captured on Cambridge Core between 31st January 2017 - 22nd January 2018. This data will be updated every 24 hours.