Hostname: page-component-76c49bb84f-zv2rg Total loading time: 0 Render date: 2025-07-08T11:49:07.166Z Has data issue: false hasContentIssue false
  • English
  • Français

A Filtered-Davidson Method for Large Symmetric Eigenvalue Problems

Part of: Basic linear algebra Partial differential equations, boundary value problems Ordinary differential operators Numerical linear algebra

Published online by Cambridge University Press:  31 January 2017

Cun-Qiang Miao
Cun-Qiang Miao*
Affiliation:
Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
*
*Corresponding author. Email address:miaocunqiang@lsec.cc.ac.cn (C.-Q. Miao)
Get access

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.

Keywords

Symmetric eigenproblemfiltering techniqueChebyshev polynomialsKrylov subspaceDavidson-type method

MSC classification

Secondary: 15A18: Eigenvalues, singular values, and eigenvectors 65F15: Eigenvalues, eigenvectors 65F50: Sparse matrices 34L15: Eigenvalues, estimation of eigenvalues, upper and lower bounds 65N25: Eigenvalue problems

Information

Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 7 , Issue 1 , February 2017 , pp. 21 - 37
Copyright
Copyright © Global-Science Press 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

[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. 87–94.CrossRefGoogle Scholar
[2] Fang, H.R. and Saad, Y., A filtered Lanczos procedure for extreme and interior eigenvalue problems, SIAM J. Sci. Comput., 34(2012), pp. A2220–A2246.Google Scholar
[3] Golub, G.H. and Ye, Q., Inexact inverse iteration for generalized eigenvalue problems, BIT, 40(2000), pp. 671–684.Google Scholar
[4] Jian, S., A block preconditioned steepest descent method for symmetric eigenvalue problems, Appl. Math. Comput., 219(2013), pp. 10198–10217.Google Scholar
[5] Knyazev, A.V., Toward the optimal preconditioned eigensolver: locally optimal block preconditioned conjugate gradient method, SIAM J. Sci. Comput., 23(2001), pp. 517–541.CrossRefGoogle Scholar
[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. 425–437.3.0.CO;2-G>CrossRefGoogle Scholar
[7] Morgan, R.B., Generalizations of Davidson's method for computing eigenvalues of large nonsymmetric matrices, J. Comput. Phys., 101(1992), pp. 287–291.Google Scholar
[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. 817–825.Google Scholar
[9] Notay, Y., Convergence analysis of inexact Rayleigh quotient iteration, SIAM J. Matrix Anal. Appl., 24(2003), pp. 627–644.CrossRefGoogle Scholar
[10] Ovtchinnikov, E., Cluster robustness of preconditioned gradient subspace iteration eigensolvers, Linear Algebra Appl., 415(2006), pp. 140–166.CrossRefGoogle Scholar
[11] Ovtchinnikov, E.E., Sharp convergence estimates for the preconditioned steepest descent method for Hermitian eigenvalue problems, SIAM J. Numer. Anal., 43(2006), pp. 2668–2689.Google Scholar
[12] Parlett, B.N., The Symmetric Eigenvalue Problems, SIAM, Philadelphia, PA, 1998.CrossRefGoogle Scholar
[13] Saad, Y., Chebyshev acceleration techniques for solving nonsymmetric eigenvalue problems, Math. Comp., 42(1984), pp. 567–588.CrossRefGoogle Scholar
[14] Saad, Y., Numerical Methods for Large Eigenvalue Problems, Second Edition, SIAM, Philadelphia, PA, 2011.Google Scholar
[15] Saad, Y., On the rates of convergence of the Lanczos and the Block-Lanczos methods, SIAM J. Numer. Anal., 17(1980), pp. 687–706.Google Scholar
[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. 595–633.CrossRefGoogle Scholar
[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. 401–425.Google Scholar
[18] Sorensen, D.C., Implicit application of polynomial filters in a k-step Arnoldi method, SIAM J. Matrix Anal. Appl., 13(1992), pp. 357–385.Google Scholar
[19] Van Den Eshof, J., The convergence of Jacobi-Davidson iterations for Hermitian eigenproblems, Numer. Linear Algebra Appl., 9(2002), pp. 163–179.Google Scholar
[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. 877–899.CrossRefGoogle Scholar
[21] Zhou, Y.-K. and Saad, Y., A Chebyshev-Davidson algorithm for large symmetric eigenproblems, SIAM J. Matrix Anal. Appl., 29(2007), pp. 954–971.Google Scholar