Skip to main content
SpringerLink
Log in

Successive mth approximation method for the nonlinear eigenvalue problem

  • Published:
Computational and Applied Mathematics Aims and scope Submit manuscript

Abstract

In this paper, we propose a successive mth (\(m\ge 2\)) approximation method for the nonlinear eigenvalue problem (NEP) and analyze its local convergence. Applying the partially orthogonal projection method to the successive mth approximation problem, we present the partially orthogonal projection method with the successive mth approximation for solving the NEP. Numerical experiments are reported to illustrate the effectiveness of the proposed methods.

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

References

  • Atkinson KE (1989) An Introduction to Numerical Analysis, 2nd edn. Wiley, New York

  • Bao L, Lin Y, Wei Y (2009) Restarted generalized Krylov subspace methods for solving large-scale polynomial eigenvalue problems. Numer Algorithm 50:17–32

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  Google Scholar 

  • Bramely JS, Dannis SCR (1982) The calculation of eigenvalues for the stationary perturbation of Poiseuille flow. J Comput Phys 47:179–198

    Article  Google Scholar 

  • Dai H, Bai Z-Z (2010) On smooth LU decompositions with applications to solutions of nonlinear eigenvalue problems. J Comput Math 28:745–766

    MathSciNet  MATH  Google Scholar 

  • Guo J-S, Lin W-W, Wang C-S (1995) Nonequivalence deflation for the solution of matrix talent value problems. Linear Algebra Appl 231:15–45

    Article  MathSciNet  MATH  Google Scholar 

  • Hoffnung L, Li R-C, Ye Q (2006) Krylov type subspace methods for matrix polynomials. Linear Algebra Appl 415:52–81

    Article  MathSciNet  MATH  Google Scholar 

  • Hu HY, Dowell EH, Virgin LN (1998) Stability estimation of high dimensional vibrating systems under state delay feedback control. J Sound Vib 214:497–511

    Article  MathSciNet  MATH  Google Scholar 

  • Jarlebring E (2012) Convergence factors of Newton methods for nonlinear eigenvalue problems. Linear Algebra Appl 436:3943–3953

    Article  MathSciNet  MATH  Google Scholar 

  • Kim CW, Bennighof JK (2007) Fast frequency response analysis of large-scale structures with nonproportional damping. Int J Numer Methods Eng 69:978–992

    Article  MATH  Google Scholar 

  • Kressner D (2009) A block Newton method for nonlinear eigenvalue problems. Numer Math 114:355–372

    Article  MathSciNet  MATH  Google Scholar 

  • Kublanovskaya VN (1970) On an approach to the solution of the generalized latent value problem for lambda-matrices. SIAM J Numer Anal 7:532–537

    Article  MathSciNet  MATH  Google Scholar 

  • Lancaster P, Markus AS, Zhou F (2003) Perturbation theory for analytic matrix functions: the semisimple case. SIAM J Matrix Anal Appl 25:606–626

    Article  MathSciNet  MATH  Google Scholar 

  • Li R-C (1989) QR decomposition and nonlinear eigenvalue problems (in Chinese). Math Numer Sin 11:374–385

    MATH  Google Scholar 

  • Mehrmann V, Voss H (2004) Nonlinear eigenvalue problems: a challenge for modern eigenvalue methods. GAMM Mitt 27:121–152

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Sun W, Liu K-M (2001) Iterative algorithms for nonlinear ordinary differential eigenvalue problems. Appl Numer Math 38:361–376

    Article  MathSciNet  MATH  Google Scholar 

  • Voss H (2003) Numerical methods for sparse nonlinear eigenproblems. In: Marek I (ed) Proceedings of the XVth Summer School on Software and Algorithms of Numerical Mathematics, Hejnice

  • Voss H (2004) An Arnoldi method for nonlinear eigenvalue problems. BIT 44:387–401

    Article  MathSciNet  MATH  Google Scholar 

  • Voss H (2007) A Jacobi–Davidson method for nonlinear and nonsymmetric eigenproblems. Comput Struct 85:1284–1292

    Article  MathSciNet  Google Scholar 

  • Voss H (2009) A minmax principle for nonlinear eigenproblems depending continuously on the eigenparameter. Numer Linear Algebra Appl 16:899–913

    Article  MathSciNet  MATH  Google Scholar 

  • Wei W, Dai H (2015) Partially orthogonal projection method and its variations for solving polynomial eigenvalue problem (in Chinese). Numer Math A J Chin Univ (to appear)

Download references

Acknowledgments

The authors would like to express their heartfelt thanks to Professor Jinyun Yuan and anonymous referees for their useful comments and suggestions which helped to improve the presentation of this paper. The work is supported by the National Natural Science Foundation of China under Grant No. 11571171 and No. 11401305, the Natural Science Foundation of Jiangsu Province of China under grant BK20141408, and the Fundamental Research Funds for the Central Universities No. NZ2014101. We would like to thank Dr. Jianhua Zhang for his helpful suggestions.

Author information

Authors and Affiliations

  1. College of Science, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, China

    Xiaoping Chen, Hua Dai & Wei Wei

  2. College of Jincheng, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, China

    Xiaoping Chen

Authors
  1. Xiaoping Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hua Dai
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Wei Wei
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xiaoping Chen.

Additional information

Communicated by Jinyun Yuan.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chen, X., Dai, H. & Wei, W. Successive mth approximation method for the nonlinear eigenvalue problem. Comp. Appl. Math. 36, 1009–1021 (2017). https://doi.org/10.1007/s40314-015-0277-5

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40314-015-0277-5

Keywords

Mathematics Subject Classification

Search

Navigation