Skip to main content
SpringerLink
Log in

An Elementary Derivation of the Projection Method for Nonlinear Eigenvalue Problems Based on Complex Contour Integration

  • Conference paper
  • First Online:
Eigenvalue Problems: Algorithms, Software and Applications in Petascale Computing (EPASA 2015)

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 117))

  • 636 Accesses

Abstract

The Sakurai-Sugiura (SS) projection method for the generalized eigenvalue problem has been extended to the nonlinear eigenvalue problem A(z)w = 0, where A(z) is an analytic matrix valued function, by several authors. To the best of the authors’ knowledge, existing derivations of these methods rely on canonical forms of an analytic matrix function such as the Smith form or the theorem of Keldysh. While these theorems are powerful tools, they require advanced knowledge of both analysis and linear algebra and are rarely mentioned even in advanced textbooks of linear algebra. In this paper, we present an elementary derivation of the SS-type algorithm for the nonlinear eigenvalue problem, assuming that the wanted eigenvalues are all simple. Our derivation uses only the analyticity of the eigenvalues and eigenvectors of a parametrized matrix A(z), which is a standard result in matrix perturbation theory. Thus we expect that our approach will provide an easily accessible path to the theory of nonlinear SS-type methods.

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

Access this chapter

Log in via an institution

Institutional subscriptions

References

  1. Amako, T., Yamamoto, Y., Zhang, S.-L.: A large-grained parallel algorithm for nonlinear eigenvalue problems and its implementation using OmniRPC. In: Proceedings of IEEE International Conference on Cluster Computing, 2008, pp. 42–49. IEEE Press (2008)

    Google Scholar 

  2. Asakura, J., Sakurai, T., Tadano, H., Ikegami, T., Kimura, K.: A numerical method for nonlinear eigenvalue problems using contour integrals. JSIAM Lett. 1, 52–55 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  Google Scholar 

  4. Beyn, W.-J.: An integral method for solving nonlinear eigenvalue problems. Linear Algebra Appl. 436, 3839–3863 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  5. Gantmacher, F.R.: The Theory of Matrices. Chelsea, New York (1959)

    MATH  Google Scholar 

  6. Gohberg, I., Rodman, L.: Analytic matrix functions with prescribed local data. J. d’Analyse Mathématique 40, 90–128 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  7. Golub, G.H., Van Loan, C.F.: Matrix Computations, 3rd edn. Johns Hopkins University Press, Baltimore (1996)

    MATH  Google Scholar 

  8. Ikegami, T., Sakurai, T., Nagashima, U.: A filter diagonalization for generalized eigenvalue problems based on the Sakurai-Sugiura projection method. J. Comput. Appl. Math. 233, 1927–1936 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  9. Kato, T.: Perturbation Theory for Linear Operators, 2nd edn. Springer, Berlin (1976)

    MATH  Google Scholar 

  10. Kato, T.: A Short Introduction to Perturbation Theory for Linear Operators. Springer, New York (1982)

    Book  MATH  Google Scholar 

  11. Keldysh, M.V.: On the characteristic values and characteristic functions of certain classes of non-self-adjoint equations. Doklady Akad. Nauk SSSR (N. S.) 77, 11–14 (1951)

    Google Scholar 

  12. Keldysh, M.V.: The completeness of eigenfunctions of certain classes of nonselfadjoint linear operators. Uspehi Mat. Nauk 26(4(160)), 15–41 (1971)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  14. Ruhe, A.: Algorithms for the nonlinear eigenvalue problem. SIAM J. Numer. Anal. 10, 674–689 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  15. Saad, Y.: Numerical Methods for Large Eigenvalue Problems. Halsted Press, New York (1992)

    MATH  Google Scholar 

  16. Sakurai, T., Sugiura, H.: A projection method for generalized eigenvalue problems using numerical integration. J. Comput. Appl. Math. 159, 119–128 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  17. Sakurai, T., Tadano, H., Inadomi, Y., Nagashima, U.: A moment-based method for large-scale generalized eigenvalue problems. Appl. Numer. Anal. Comput. Math. 1, 516–523 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  18. Scott, D.S., Ward, R.C.: Solving symmetric-definite quadratic problems without factorization. SIAM J. Sci. Stat. Comput. 3, 58–67 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  19. Sloan, I.H., Joe, S.: Lattice Methods for Multiple Integration. Oxford University Press, New York (1994)

    MATH  Google Scholar 

  20. Trefethen, L.N., Weideman, J.A.C.: The exponentially convergent trapezoidal rule. SIAM Rev. 56, 385–458 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  21. Voss, H.: An Arnoldi method for nonlinear eigenvalue problems. BIT Numer. Math. 44, 387–401 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  22. Yokota, S., Sakurai, T.: A projection method for nonlinear eigenvalue problems using contour integrals. JSIAM Lett. 5, 41–44 (2013)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

We thank Professor Masaaki Sugihara for valuable comments. We are also grateful to the anonymous referees, whose suggestions helped us much in improving the quality of this paper. Prof. Akira Imakura brought reference [8] to our attention. This study is supported in part by the Ministry of Education, Science, Sports and Culture, Grant-in-Aid for Scientific Research (Nos. 26286087, 15H02708, 15H02709, 16KT0016, 17H02828, 17K19966) and the Core Research for Evolutional Science and Technology (CREST) Program “Highly Productive, High Performance Application Frameworks for Post Petascale Computing” of Japan Science and Technology Agency (JST).

Author information

Authors and Affiliations

  1. The University of Electro-Communications, 1-5-1 Chofugaoka, Chofu, Tokyo, 182-8585, Japan

    Yusaku Yamamoto

Authors
  1. Yusaku Yamamoto
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yusaku Yamamoto .

Editor information

Editors and Affiliations

  1. Department of Computer Science, University of Tsukuba, Tsukuba, Ibaraki, Japan

    Tetsuya Sakurai

  2. Department of Electrical Engineering and Computer Science, Nagoya University, Nagoya, Aichi, Japan

    Shao-Liang Zhang

  3. RIKEN Advanced Institute for Computational Science, Kobe, Hyogo, Japan

    Toshiyuki Imamura

  4. Department of Communication Engineering and Informatics, University of Electro-Communications, Graduate School of Informatics and Engineering, Tokyo, Japan

    Yusaku Yamamoto

  5. Department of Particle Physics, University of Tsukuba, Tsukuba, Ibaraki, Japan

    Yoshinobu Kuramashi

  6. Department of Applied Mathematics and Physics, Tottori University, Tottori, Japan

    Takeo Hoshi

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this paper

Cite this paper

Yamamoto, Y. (2017). An Elementary Derivation of the Projection Method for Nonlinear Eigenvalue Problems Based on Complex Contour Integration. In: Sakurai, T., Zhang, SL., Imamura, T., Yamamoto, Y., Kuramashi, Y., Hoshi, T. (eds) Eigenvalue Problems: Algorithms, Software and Applications in Petascale Computing. EPASA 2015. Lecture Notes in Computational Science and Engineering, vol 117. Springer, Cham. https://doi.org/10.1007/978-3-319-62426-6_16

Download citation

Publish with us

Policies and ethics

Search

Navigation