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Implicitly Restarted Refined Partially Orthogonal Projection Method with Deflation

Part of: Numerical linear algebra

Published online by Cambridge University Press:  31 January 2017

Wei Wei  and
Hua Dai
Wei Wei*
Affiliation:
Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China
Hua Dai*
Affiliation:
Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China
*
*Corresponding author. Email addresses:[email protected] (W. Wei), [email protected] (H. Dai)
*Corresponding author. Email addresses:[email protected] (W. Wei), [email protected] (H. Dai)
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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.

Keywords

Polynomial eigenvalue problempartially orthogonal projection methodrefinementimplicitly restartingnon-equivalence low-rank deflation

MSC classification

Secondary: 65F15: Eigenvalues, eigenvectors
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 7 , Issue 1 , February 2017 , pp. 1 - 20
Copyright
Copyright © Global-Science Press 2017 

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References

[1] Adhikari, S. and Pascual, B., Eigenvalues of linear viscoelastic systems, J. Sound Vib. 325, 1000–1011 (2009).Google Scholar
[2] Bao, L., Lin, Y.-Q. and Wei, Y.-M., Restarted generalized Krylov subspace methods for solving large-scale polynomial eigenvalue problems, Numer. Algorithms 50, 17–32 (2009).Google Scholar
[3] Betcke, T., Higham, N.J., Mehrmann, V., Schroder, C. and Tisseur, F., NLEVP: A collection of non-linear eigenvalue problems, ACM Trans. Math. Software 39, 7–28 (2013).Google Scholar
[4] Chu, E.K.-W., Perturbation of eigenvalues for matrix polynomials via the Bauer-Fike theorems, SIAM J. Matrix Anal. Appl. 25, 551–573 (2003).Google Scholar
[5] Dedieu, J. and Tisseur, F., Perturbation theory for homogeneous polynomial eigenvalue problems, Linear Algebra Appl. 358, 71–94 (2003).Google Scholar
[6] Duff, I.S., Grimes, R.G. and Lewis, J.G., Sparse matrix test problems, ACM Trans. Math. Software 15, 1–14 (1989).Google Scholar
[7] Gohberg, I., Lancaster, P. and Rodman, L., Perturbation theory for divisors of operator polynomials, SIAM J. Math. Anal. 10, 1161–1183 (1979).Google Scholar
[8] Gohberg, I., Lancaster, P. and Rodman, L., Matrix Polynomials, Academic Press, New York (1982).Google Scholar
[9] Gupta, K.K., On a finite dynamic element method for free vibration analysis of structures, Comput. Methods Appl. Mech. Eng. 9, 105–120 (1976).Google Scholar
[10] Higham, N.J., Li, R.-C. and Tisseur, F., Backward error of polynomial eigenproblems solved by linearization, SIAM J. Matrix Anal. Appl. 29, 1218–1241 (2007).Google Scholar
[11] Higham, N.J., Mackey, D.S. and Tisseur, F., The conditioning of linearizations of matrix polynomials, SIAM J. Matrix Anal. Appl. 28, 1005–1028 (2006).Google Scholar
[12] Higham, N.J. and Tisseur, F., More on pseudospectra for polynomial eigenvalue problems and applications in control theory, Linear Algebra Appl. 351, 435–453 (2002).Google Scholar
[13] Higham, N.J. and Tisseur, F., Bounds for eigenvalues of matrix polynomials, Linear Algebra Appl. 358, 5–22 (2003).Google Scholar
[14] Hoffnung, L., Li, R.-C. and Ye, Q., Krylov type subspace methods for matrix polynomials, Linear Algebra Appl. 415, 52–81 (2006).Google Scholar
[15] Huang, W.-Q., Li, T.-X., Li, Y.-T. and Lin, W.-W., A semiorthogonal generalized Arnoldi method and its variations for quadratic eigenvalue problems, Numer. Linear Algebra Appl. 20, 259–280 (2013).Google Scholar
[16] Hwang, T.-M., Lin, W.-W., Liu, J.-L. and Wang, W., Jacobi-Davidson methods for cubic eigenvalue problems, Numer. Linear Algebra Appl. 12, 605–624 (2005).Google Scholar
[17] Hwang, T.-M., Lin, W.-W., Wang, W.-C. and Wang, W., Numerical simulation of three dimensional pyramid quantum dot, J. Comput. Phys. 196, 208–232 (2004).Google Scholar
[18] Hwang, F.-N., Wei, Z.-H., Hwang, T.-M. and Wang, W., A parallel additive Schwarz preconditioned Jacobi-Davidson algorithm for polynomial eigenvalue problems in quantum dot simulation, J. Comput. Phys. 229, 2932–2947 (2010).Google Scholar
[19] Jia, Z., Refined iterative algorithms based on Arnoldi's process for large unsymmetric eigenproblems, Linear Algebra Appl. 259, 1–23 (1997).Google Scholar
[20] Lawrence, P.W. and Corless, R.M., Backward error of polynomial eigenvalue problems solved by linearization of Lagrange interpolants, SIAM J. Matrix Anal. Appl. 36, 1425–1442 (2015).Google Scholar
[21] Mackey, D.S., Mackey, N., Mehl, C. and Mehrmann, V., Vector spaces of linearizations for matrix polynomials, SIAM J. Matrix Anal. Appl. 28, 971–1004 (2006).Google Scholar
[22] Mackey, D.S., Mackey, N., Mehl, C. and Mehrmann, V., Structured polynomial eigenvalue problems: good vibrations from good linearization, SIAM J. Matrix Anal. Appl. 28, 1029–1051 (2006).Google Scholar
[23] Moler, C.B. and Stewart, G.W., An algorithm for generalized matrix eigenvalue problems, SIAM J. Numer. Anal. 10, 241–256 (1973).Google Scholar
[24] Saad, Y., Numerical Methods for Large Eigenvalue Problems, Halsted Press, New York (1992).Google Scholar
[25] 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, 595–633 (1996).Google Scholar
[26] Sorensen, D.C., Truncated QZ methods for large scale generalized eigenvalue problems, Electron. Trans. Numer. Anal. 7, 141–162 (1998).Google Scholar
[27] Tisseur, F., Backward error and condition of polynomial eigenvalue problems, Linear Algebra Appl. 309, 339–361 (2000).Google Scholar
[28] Tisseur, F. and Meerbergen, K., The quadratic eigenvalue problem, SIAM Rev. 43, 235–286 (2001).Google Scholar
[29] Wei, W. and Dai, H., Partially orthogonal projection method and its variations for solving polynomial eigenvalue problem (in Chinese), Numer. Math. - J. Chinese Universities 38, 116–133 (2016).Google Scholar