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A Filtered-Davidson Method for Large Symmetric Eigenvalue Problems

  • Cun-Qiang Miao (a1)

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.

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*Corresponding author. Email address: (C.-Q. Miao)
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[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.
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East Asian Journal on Applied Mathematics
  • ISSN: 2079-7362
  • EISSN: 2079-7370
  • URL: /core/journals/east-asian-journal-on-applied-mathematics
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