Skip to main content
×
Home
    • Aa
    • Aa

Implicitly Restarted Refined Partially Orthogonal Projection Method with Deflation

  • Wei Wei (a1) and Hua Dai (a1)
Abstract
Abstract

In this paper we consider the computation of some eigenpairs with smallest eigenvalues in modulus of large-scale polynomial eigenvalue problem. Recently, a partially orthogonal projection method and its refinement scheme were presented for solving the polynomial eigenvalue problem. The methods preserve the structures and properties of the original polynomial eigenvalue problem. Implicitly updating the starting vector and constructing better projection subspace, we develop an implicitly restarted version of the partially orthogonal projection method. Combining the implicit restarting strategy with the refinement scheme, we present an implicitly restarted refined partially orthogonal projection method. In order to avoid the situation that the converged eigenvalues converge repeatedly in the later iterations, we propose a novel explicit non-equivalence low-rank deflation technique. Finally some numerical experiments show that the implicitly restarted refined partially orthogonal projection method with the explicit non-equivalence low-rank deflation technique is efficient and robust.

Copyright
Corresponding author
*Corresponding author. Email addresses:w.wei@nuaa.edu.cn (W. Wei), hdai@nuaa.edu.cn (H. Dai)
Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[1] S. Adhikari and B. Pascual , Eigenvalues of linear viscoelastic systems, J. Sound Vib. 325, 10001011 (2009).

[2] L. Bao , Y.-Q. Lin and Y.-M. Wei , Restarted generalized Krylov subspace methods for solving large-scale polynomial eigenvalue problems, Numer. Algorithms 50, 1732 (2009).

[3] T. Betcke , N.J. Higham , V. Mehrmann , C. Schroder and F. Tisseur , NLEVP: A collection of non-linear eigenvalue problems, ACM Trans. Math. Software 39, 728 (2013).

[4] E.K.-W. Chu , Perturbation of eigenvalues for matrix polynomials via the Bauer-Fike theorems, SIAM J. Matrix Anal. Appl. 25, 551573 (2003).

[5] J. Dedieu and F. Tisseur , Perturbation theory for homogeneous polynomial eigenvalue problems, Linear Algebra Appl. 358, 7194 (2003).

[6] I.S. Duff , R.G. Grimes and J.G. Lewis , Sparse matrix test problems, ACM Trans. Math. Software 15, 114 (1989).

[7] I. Gohberg , P. Lancaster and L. Rodman , Perturbation theory for divisors of operator polynomials, SIAM J. Math. Anal. 10, 11611183 (1979).

[9] K.K. Gupta , On a finite dynamic element method for free vibration analysis of structures, Comput. Methods Appl. Mech. Eng. 9, 105120 (1976).

[10] N.J. Higham , R.-C. Li and F. Tisseur , Backward error of polynomial eigenproblems solved by linearization, SIAM J. Matrix Anal. Appl. 29, 12181241 (2007).

[11] N.J. Higham , D.S. Mackey and F. Tisseur , The conditioning of linearizations of matrix polynomials, SIAM J. Matrix Anal. Appl. 28, 10051028 (2006).

[13] N.J. Higham and F. Tisseur , Bounds for eigenvalues of matrix polynomials, Linear Algebra Appl. 358, 522 (2003).

[14] L. Hoffnung , R.-C. Li and Q. Ye , Krylov type subspace methods for matrix polynomials, Linear Algebra Appl. 415, 5281 (2006).

[15] W.-Q. Huang , T.-X. Li , Y.-T. Li and W.-W. Lin , A semiorthogonal generalized Arnoldi method and its variations for quadratic eigenvalue problems, Numer. Linear Algebra Appl. 20, 259280 (2013).

[16] T.-M. Hwang , W.-W. Lin , J.-L. Liu and W. Wang , Jacobi-Davidson methods for cubic eigenvalue problems, Numer. Linear Algebra Appl. 12, 605624 (2005).

[17] T.-M. Hwang , W.-W. Lin , W.-C. Wang and W. Wang , Numerical simulation of three dimensional pyramid quantum dot, J. Comput. Phys. 196, 208232 (2004).

[18] F.-N. Hwang , Z.-H. Wei , T.-M. Hwang and W. Wang , A parallel additive Schwarz preconditioned Jacobi-Davidson algorithm for polynomial eigenvalue problems in quantum dot simulation, J. Comput. Phys. 229, 29322947 (2010).

[19] Z. Jia , Refined iterative algorithms based on Arnoldi's process for large unsymmetric eigenproblems, Linear Algebra Appl. 259, 123 (1997).

[20] P.W. Lawrence and R.M. Corless , Backward error of polynomial eigenvalue problems solved by linearization of Lagrange interpolants, SIAM J. Matrix Anal. Appl. 36, 14251442 (2015).

[21] D.S. Mackey , N. Mackey , C. Mehl and V. Mehrmann , Vector spaces of linearizations for matrix polynomials, SIAM J. Matrix Anal. Appl. 28, 9711004 (2006).

[22] D.S. Mackey , N. Mackey , C. Mehl and V. Mehrmann , Structured polynomial eigenvalue problems: good vibrations from good linearization, SIAM J. Matrix Anal. Appl. 28, 10291051 (2006).

[23] C.B. Moler and G.W. Stewart , An algorithm for generalized matrix eigenvalue problems, SIAM J. Numer. Anal. 10, 241256 (1973).

[25] G.L.G. Sleijpen , A.G.L. Booten , D.R. Fokkema and H.A. van der Vorst , Jacobi-Davidson type methods for generalized eigenproblems and polynomial eigenproblems, BIT 36, 595633 (1996).

[27] F. Tisseur , Backward error and condition of polynomial eigenvalue problems, Linear Algebra Appl. 309, 339361 (2000).

[28] F. Tisseur and K. Meerbergen , The quadratic eigenvalue problem, SIAM Rev. 43, 235286 (2001).

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: 29 *
Loading metrics...

Abstract views

Total abstract views: 105 *
Loading metrics...

* Views captured on Cambridge Core between 31st January 2017 - 25th April 2017. This data will be updated every 24 hours.