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A Jacobi–Davidson type method for computing real eigenvalues of the quadratic eigenvalue problem

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Abstract

This paper presents a new Jacobi–Davidson type method to compute several real eigenvalues of the Hermitian quadratic eigenvalue problem. This method uses a simple index to sort the eigenvalues of the projected quadratic eigenvalue problem and extracts the approximate eigenvectors for the quadratic eigenvalue problem with the eigenvectors of the projected quadratic eigenvalue problem corresponding to the eigenvalues with the smallest indices. Numerical examples show that our method is effective and efficient to compute real eigenvalues of the Hermitian quadratic eigenvalue problem.

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References

  1. Bai, Z., Demmel, J., Dongarra, J., Ruhe, A., Vander Vorst, H.: Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide. SIAM, Philadelphia (2000)

    Book  MATH  Google Scholar 

  2. Bai, Z., Su, Y.F.: SOAR: a second-order Arnoldi method for the solution of the quadratic eigenvalue problem. SIAM J. Matrix Anal. Appl. 26(3), 64–659 (2003)

    MathSciNet  Google Scholar 

  3. Betcke, T., Voss, H.: A Jacobi–Davidson-type projection method for nonlinear eigenvalue problems. Future Gener. Comput. Syst. 20, 363–372 (2004)

    Article  Google Scholar 

  4. Chu, M., Huang, T.M., Lin, W.W.: A novel deflation technique for solving quadratic eigenvalue problems. Bull. Inst. Math. Acad. Sinica 9(1), 57–84 (2014)

    MathSciNet  MATH  Google Scholar 

  5. Esseni, D., Palestri, P., Selmi, L.: Nanoscale MOS Transistors. Cambridge University Press, Cambridge (2011)

    Book  Google Scholar 

  6. Gohberg, I., Lancaster, P., Rodman, L.: Spectral analysis of selfadjoint matrix polynomials. Ann. Math. 112, 33–71 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  7. Gu, K., Kharitonov, V.L., Chen, J.: Stability of Time-delay Systems. Birkhäser, Boston (2003)

    Book  MATH  Google Scholar 

  8. Guo, J.S., Lin, W.W., Wang, C.S.: Numerical solutions for large sparse quadratic eigenvalue problems. Linear Algebra Appl. 225, 57–89 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  9. Higham, N.J., Tisseur, F.: Bounds for eigenvalues of matrix polynomials. Linear Algebra Appl. 358, 5–22 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  10. Hochstenbach, M.E., Sleijpen, G.L.G.: Harmonic and refined Rayleigh–Ritz for the polynomial eigenvalue problem. Linear Algebra Appl. 15(1), 35–54 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  11. Huitfeldt, J., Ruhe, A.: A new algorithm for numerical path following applied to an example from hydrodynamical flow. SIAM J. Sci. Stat. Comput. 11, 1181–1192 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  12. Jarlebring, E.: Critical delays and polynomial eigenvalue problems. J. Comput. Appl. Math. 224, 296–306 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  13. Jarlebring, E., Hochstenbach, M.E.: Polynomial two-parameter eigenvalue problems and matrix pencil methods for stability of delay-differential equations. Linear Algebra Appl. 431, 369–380 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. Lancaster, P., Ye, Q.: Variational and numerical methods for symmetric matrix pencils. Bull. Aust. Math. Soc. 43, 1–17 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  15. Li, N., Saad, Y., Chow, E.: Crout versions of ilu for general sparse matrices. SIAM J. Sci. Comput. 25(2), 716–728 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  16. Meerbergen, K.: Locking and restarting quadratic eigenvalue solvers. SIAM J. Sci. Comput. 22, 1814–1839 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  17. Neumaier, J.: Residual inverse iteration for the nonlinear eigenvalue problem. SIAM J. Numer. Anal. 22, 914–923 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  18. Paige, C.C., Saunders, M.A.: Solution of sparse indefinite systems of linear equations. SIAM J. Numer. Anal. 12, 617–629 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  19. Saad, Y., Schultz, M.H.: GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 7, 856–869 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  20. Sleijpen, G.L.G., van der Vorst, H.: Jacobi–Davidson iteration method for linear eigenproblems. SIAM Rev. 42, 267–293 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  21. Sleijpen, G.L.G., Booten, G., Fokkema, D., van der Vorst, H.: Jacobi–Davidson type methods for generalized eigenproblems and polynomial eigenproblems. BIT Numer. Math. 36, 595–633 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  22. Sleijpen, G.L.G., van der Vorst, H.: A Jacobi–Davidson iteration method for linear eigenvalue problems. SIAM J. Matrix Anal. Appl. 17, 401–425 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  23. Tisseur, F., Meerbergen, K.: The quadratic eigenvalue problem. SIAM Rev. 43, 235–286 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  24. van der Corput, J.G.: On the method of steepest descent. Journal d’Analyse Mathématique 8, 159–183 (1960)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. LMAM and School of Mathematical Sciences, Peking University, Beijing, 100871, China

    Hao Li & Yunfeng Cai

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  1. Hao Li
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  2. Yunfeng Cai
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Correspondence to Hao Li.

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Li, H., Cai, Y. A Jacobi–Davidson type method for computing real eigenvalues of the quadratic eigenvalue problem. Calcolo 53, 737–749 (2016). https://doi.org/10.1007/s10092-015-0171-y

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  • DOI: https://doi.org/10.1007/s10092-015-0171-y

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