Skip to main content
SpringerLink
Log in

Convergence proof of the Harmonic Ritz pairs of iterative projection methods with restart strategies for symmetric eigenvalue problems

  • Original Paper
  • Published:
Japan Journal of Industrial and Applied Mathematics Aims and scope Submit manuscript

Abstract

We consider numerical methods for computing eigenvalues located in the interior part of the spectrum of a large symmetric matrix. For such difficult eigenvalue problems, an effective solution is to use the Harmonic Ritz pairs in projection methods because the error bounds on the Harmonic Ritz pairs are well studied. In this paper, we prove global convergence of the iterative projection methods with the Harmonic Ritz pairs in an abstract form, where the standard restart strategy is employed. To this end, we reformulate the existing convergence proof of the Ritz pairs to be successfully applied to the Harmonic Ritz pairs with the inexact linear system solvers. Our main theorem obtained by the above convergence analysis shows important features concerning the global convergence of the Harmonic Ritz pairs.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. \(|\beta _{i}^{(\ell )}|\le \Vert A\Vert \ (i=1,\ldots ,k, \ell =0,1,\ldots )\) are due to \(|\beta _{i}^{(\ell )}|\le \Vert \varPi _{i}^{(\ell )}{}^{\mathrm{T}}A\varPi _{i}^{(\ell )}\Vert \le \Vert A\Vert \) in (9). Similarly, \(|{\widehat{\theta }}_{i}^{(\ell )}|\le \Vert A\Vert \ (i=1,\ldots ,k, \ell =0,1,\ldots )\) can be proved.

References

  1. Aishima, K.: On convergence of iterative projection methods for symmetric eigenvalue problems. J. Comput. Appl. Math. 311, 513–521 (2017)

    Article  MathSciNet  Google Scholar 

  2. Aishima, K.: Global convergence of the restarted Lanczos and Jacobi-Davidson methods for symmetric eigenvalue problems. Numer. Math. 131, 405–423 (2015)

    Article  MathSciNet  Google Scholar 

  3. Aishima, K.: A note on the Rayleigh quotient iteration for symmetric eigenvalue problems. Jpn. J. Ind. Appl. Math. 31, 575–581 (2014)

    Article  MathSciNet  Google Scholar 

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

    Book  Google Scholar 

  5. Crouzeix, M., Philippe, B., Sadkane, M.: The Davidson method. SIAM J. Sci. Comput. 15, 62–76 (1994)

    Article  MathSciNet  Google Scholar 

  6. 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, 87–94 (1975)

    Article  MathSciNet  Google Scholar 

  7. Golub, G.H., Van Loan, C.F.: Eigenvalue computation in the 20th century. J. Comput. Appl. Math. 123, 35–65 (2000)

    Article  MathSciNet  Google Scholar 

  8. Golub, G.H., Van Loan, C.F.: Matrix Computations, 4th edn. Johns Hopkins University, Baltimore (2013)

    MATH  Google Scholar 

  9. Jia, Z.: The convergence of harmonic Ritz values, harmonic Ritz vectors, and refined harmonic Ritz vectors. Math. Comp. 74, 1441–1456 (2005)

    Article  MathSciNet  Google Scholar 

  10. Jia, Z.: On convergence of the inexact Rayleigh quotient iteration with MINRES. J. Comput. Appl. Math. 236, 4276–4295 (2012)

    Article  MathSciNet  Google Scholar 

  11. Lanczos, C.: An iteration method for the solution of the eigenvalue problem of linear differential and integral operators. J. Res. Nat. Bur. Stand. 45, 255–282 (1950)

    Article  MathSciNet  Google Scholar 

  12. Lehoucq, R.B., Meerbergen, K.: Using generalized Cayley transformations within an inexact rational Krylov sequence method. SIAM J. Matrix Anal. Appl. 20, 131–148 (1998)

    Article  MathSciNet  Google Scholar 

  13. Mach, T., Pranić, M.S., Vandebril, R.: Computing approximate (block) rational Krylov subspaces without explicit inversion with extensions to symmetric matrices. Electron. Trans. Numer. Anal. 43, 100–124 (2014)

    MathSciNet  MATH  Google Scholar 

  14. Meerbergen, K.: Changing poles in the rational Lanczos method for the Hermitian eigenvalue problem. Numer. Linear Algebra Appl. 8, 33–52 (2001)

    Article  MathSciNet  Google Scholar 

  15. Morgan, R.B.: Computing interior eigenvalues of large matrices. Linear Algebra Appl. 154–156, 289–309 (1991)

    Article  MathSciNet  Google Scholar 

  16. Morgan, R.B., Scott, D.S.: Generalizations of Davidson’s method for computing eigenvalues of sparse symmetric matrices. SIAM J. Sci. Statist. Comput. 7, 817–825 (1986)

    Article  MathSciNet  Google Scholar 

  17. Notay, Y.: Convergence analysis of inexact Rayleigh quotient iteration. SIAM J. Matrix Anal. Appl. 24, 627–644 (2003)

    Article  MathSciNet  Google Scholar 

  18. Paige, C.C., Parlett, B.N., van der Vorst, H.A.: Approximate solutions and eigenvalue bounds from Krylov subspaces. Numer. Linear Algebra Appl. 2, 115–133 (1995)

    Article  MathSciNet  Google Scholar 

  19. Parlett, B.N.: The Symmetric Eigenvalue Problem. Prentice-Hall, Englewood Cliffs (1980)

    MATH  Google Scholar 

  20. Ruhe, A.: Rational Krylov: A practical algorithm for large sparse nonsymmetric matrix pencils SIAM. J. Sci. Comput. 19, 1535–1551 (1998)

    MathSciNet  MATH  Google Scholar 

  21. Saad, Y.: Numerical Methods for Large Eigenvalue Problems, Revised edn. SIAM, Philadelphia (2011)

    Book  Google Scholar 

  22. Simoncini, V., Eldén, L.: Inexact Rayleigh quotient-type methods for eigenvalue computations. BIT 42, 159–182 (2002)

    Article  MathSciNet  Google Scholar 

  23. Sleijpen, G.L.G., van den Eshof, J.: On the use of harmonic Ritz pairs in approximating internal eigenpairs. Linear Algebra Appl. 358, 115–137 (2003)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  25. Sorensen, D.C.: Implicit application of polynomial filters in a \(k\)-step Arnoldi method. SIAM J. Matrix Anal. Appl. 13, 357–385 (1992)

    Article  MathSciNet  Google Scholar 

  26. Szyld, D.B., Xue, F.: Efficient preconditioned inner solves for inexact Rayleigh quotient iteration and their connections to the single-vector Jacobi-Davidson method. SIAM J. Matrix Anal. Appl. 32, 993–1018 (2011)

    Article  MathSciNet  Google Scholar 

  27. Vecharynski, E.: A generalization of Saad’s bound on harmonic Ritz vectors of Hermitian matrices. Linear Algebra Appl. 494, 219–235 (2016)

    Article  MathSciNet  Google Scholar 

  28. Vecharynski, E., Knyazev, A.: Preconditioned locally harmonic residual method for computing interior eigenpairs of certain classes of Hermitian matrices. SIAM J. Sci. Comput. 37, S3–S29 (2015)

    Article  MathSciNet  Google Scholar 

  29. Wu, G.: The convergence of Harmonic Ritz vectors and Harmonic Ritz values, revisited. SIAM J. Matrix Anal. Appl. 38, 118–133 (2017)

    Article  MathSciNet  Google Scholar 

  30. Xue, F., Elman, H.C.: Convergence analysis of iterative solvers in inexact Rayleigh quotient iteration. SIAM J. Matrix Anal. Appl. 31, 877–899 (2009)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The author thanks the reviewers for their valuable comments. This work was supported by JSPS Grant-in-Aid for Young Scientists (Grant Number 17K14143).

Author information

Authors and Affiliations

  1. Faculty of Computer and Information Sciences, Hosei University, 3–7–2 Kajino-cho, Koganei, Tokyo, 184–8584, Japan

    Kensuke Aishima

Authors
  1. Kensuke Aishima
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Kensuke Aishima.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work was supported by JSPS Grant-in-Aid for Young Scientists (Grant Number 17K14143).

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Aishima, K. Convergence proof of the Harmonic Ritz pairs of iterative projection methods with restart strategies for symmetric eigenvalue problems. Japan J. Indust. Appl. Math. 37, 415–431 (2020). https://doi.org/10.1007/s13160-019-00402-1

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13160-019-00402-1

Keywords

Mathematics Subject Classification

Search

Navigation