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Preconditioned inexact Newton-like method for large nonsymmetric eigenvalue problems

  • * Corresponding author: Li Wang

    * Corresponding author: Li Wang

The authors are supported by the National Natural Science Foundation of China under grant 41571380 and 10971102, Major project 16KJA110001 of the Natural Science Foundation of the Jiangsu Higher Education Institution

Abstract Full Text(HTML) Figure(0) / Table(6) Related Papers Cited by
  • An efficiently preconditioned Newton-like method for the computation of the eigenpairs of large and sparse nonsymmetric matrices is proposed. A sequence of preconditioners based on the Broyden-type rank-one update formula are constructed for the solution of the linearized Newton system. The properties of the preconditioned matrix are investigated. Numerical results are given which reveal that the new proposed algorithms are efficient.

    Mathematics Subject Classification: Primary: 65F15, 65N25; Secondary: 65F10.

    Citation:

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  • Table 1.  The results for the largest real eigenvalue of $ A $

    Algorithm eigenvalue CPU time (s)
    GMRES-INewton 7.9817 9.8430
    PGMRES-INewton 7.9817 6.9330
     | Show Table
    DownLoad: CSV

    Table 2.  The results of GMRES-INewton and PGMRES-INewton method

    GMRES-INewton PGMRES-INewton
    k $ ||r_{k}|| $ $ |\lambda_{k}-\lambda^{*}| $ $ cond(\mathcal{A}) $ $ ||r_{k}|| $ $ |\lambda_{k}-\lambda^{*}| $ $ cond(P^{-1}\mathcal{A}) $
    1 2.8670e+03 1.4750e+00 3.1275e+09 2.8670e+03 1.4750e+00 3.1275e+09
    2 5.6282e+01 7.2040e-01 2.8736e+05 2.4063e+01 4.8650e-02 4.0326e+04
    3 1.2415e+00 3.8120e-02 6.1342e+04 3.4120e-03 6.7520e-03 3.3108e+04
    4 1.3586e-04 3.3470e-05 3.1625e+04 5.4758e-05 1.2050e-05 3.0213e+03
     | Show Table
    DownLoad: CSV

    Table 3.  The results for the largest real eigenvalue of $ A $

    Algorithm eigenvalue CPU time (s)
    GMRES-INewton 0.4338 12.5310
    PGMRES-INewton 0.4338 10.4990
     | Show Table
    DownLoad: CSV

    Table 4.  The results of GMRES-INewton and PGMRES-INewton method

    GMRES-INewton PGMRES-INewton
    k $ ||r_{k}|| $ $ |\lambda_{k}-\lambda^{*}| $ $ cond(\mathcal{A}) $ $ ||r_{k}|| $ $ |\lambda_{k}-\lambda^{*}| $ $ cond(P^{-1}\mathcal{A}) $
    1 2.6740e+02 5.8630e+01 1.2640e+10 2.6740e+02 5.8630e+01 1.2640e+10
    2 4.3572e+01 1.0270e+01 6.0304e+07 7.6230e+00 9.6810e-02 5.6210e+05
    3 3.0450e+00 7.8140e-01 2.4639e+05 3.2540e-02 4.2830e-04 3.2790e+04
    4 6.3580e-02 4.6720e-02 5.1852e+04 4.9130e-05 1.4270e-06 4.5870e+04
    5 2.3710e-04 1.0390e-03 1.3089e+05
     | Show Table
    DownLoad: CSV

    Table 5.  The results for the largest real eigenvalue of a random matrix

    Algorithm eigenvalue CPU time (s)
    GMRES-INewton 298.2441 15.7350
    PGMRES-INewton 298.2441 9.1470
     | Show Table
    DownLoad: CSV

    Table 6.  The results for the largest imaginary eigenvalue of a random matrix

    Algorithm eigenvalue CPU time (s)
    GMRES-INewton -0.0001+0.0054$ i $ 16.2370
    PGMRES-INewton -0.0001+0.0054$ i $ 10.4790
     | Show Table
    DownLoad: CSV
  • [1] Z. Z. Bai and G. H. Golub, Accelerated Hermitian and skew-Hermitian splitting methods for saddle-point problems, SIAM J. Numer. Anal., 27 (2007), 1-23.  doi: 10.1093/imanum/drl017.
    [2] Z. Z. Bai and C. Q. Miao, On local quadratic convergence of inexact simplified Jacobi-Davidson method, Linear Algebra Appl., 520 (2017), 215-241.  doi: 10.1016/j.laa.2017.01.018.
    [3] K. J. Bathe, Finite Element Procedures in Engineering Analysis, Prentice-Hall, Englewood Cliffs, 1982.
    [4] M. BenziG. H. Golub and J. Liesen, Numerical solution of saddle point problems, Acta Numer., 14 (2005), 1-137.  doi: 10.1017/S0962492904000212.
    [5] L. BergamaschiR. BruA. Martinez and M. Putti, Quasi-Newton preconditioners for the inexact Newton method, Electronic Transactions on Numerical Analysis, 23 (2006), 76-87. 
    [6] L. Bergamaschi and A. Martinez, Efficiently preconditioned inexact Newton methods for large symmetric eigenvalue problems, arXiv: 1312.1553v1, [math.NA], 5 Dec, 2013. doi: 10.1080/10556788.2014.908878.
    [7] D. R. FokkemaG. L. G. Sleijpen and H. A. Van der vorst, Acclerated inexact Newton schemes for large systems of nolinear equations, SIAM Journal on Scientific Computing, 19 (1997), 657-674.  doi: 10.1137/S1064827595296148.
    [8] M. A. Freitag and A. Spence, Rayleigh quotient iteration and simplified Jacobi-Davidson method with preconditioned iterative solves, Linear Algebra Appl., 428 (2008), 2049-2060.  doi: 10.1016/j.laa.2007.11.013.
    [9] G. Hagerty, Finding a Few Eigenvalues of Large Sparse Nonsymmetric Matrices, Ph.D Thesis, Washington State University, 2001.
    [10] M. E. Hochstenbach and Y. Notay, The Jacobi-Davidson method, GAMM-Mitteilungen, 29 (2006), 368-382.  doi: 10.1002/gamm.201490038.
    [11] J. Pestana and A. J. Wathen, Natural preconditioning and iterative methods for saddle point problems, SIAM Rev., 57 (2015), 71-91.  doi: 10.1137/130934921.
    [12] Y. Saad, Iterative Methods for Aparse Linear Systems, PWS publishing, Boston, 1996. doi: 10.1137/1.9780898718003.
    [13] G. L. Sleijpen and H. A. Van Der Vorst, A Jacobi-Davidson method for linear eigenvalue problems, SIAM J. Matrix Anal., 17 (1996), 401-425.  doi: 10.1137/S0895479894270427.
    [14] G. L. Sleljpen and H. A. Van Der Vorst, Inexact Rayleigh quotient-type methods for eigenvalue computations, BIT., 42 (2002), 159-182.  doi: 10.1023/A:1021930421106.
    [15] G. Wu and L. Zhang, On expansion of search subspaces for large non-Hermitian eigenproblems, J. Linear Algebra and Applications, 454 (2014), 107-129.  doi: 10.1016/j.laa.2014.04.021.
    [16] K. WuY. Saad and A. Stathopoulos, Inexact Newton preconditioning techniques for large symmetric eigenvalue problems, J. Electronic Transactions on Numerical Analysis, 7 (1998), 202-214. 
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