Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-17T21:23:56.720Z Has data issue: false hasContentIssue false
  • English
  • Français

Solution of a Nonlinear Eigenvalue Problem Using Signed Singular Values

Part of: Equations and inequalities involving nonlinear operators Basic linear algebra Numerical linear algebra

Published online by Cambridge University Press:  31 January 2018

Kouhei Ooi ,
Yoshinori Mizuno ,
Tomohiro Sogabe ,
Yusaku Yamamoto  and
Shao-Liang Zhang
Kouhei Ooi
Affiliation:
Nagoya University, 1 Furo-cho, Chikusa, Nagoya, Aichi, 464-8603, Japan
Yoshinori Mizuno
Affiliation:
Meteorological Research Institute, 1-1 Nagamine, Tsukuba, Ibaraki, 305-0052, Japan
Tomohiro Sogabe
Affiliation:
Nagoya University, 1 Furo-cho, Chikusa, Nagoya, Aichi, 464-8603, Japan
Yusaku Yamamoto*
Affiliation:
The University of Electro-Communications, 1-5-1 Chofugaoka, Chofu, Tokyo, 182-8585, Japan
Shao-Liang Zhang
Affiliation:
Nagoya University, 1 Furo-cho, Chikusa, Nagoya, Aichi, 464-8603, Japan
*
*Corresponding author. Email address:yusaku.yamamoto@uec.ac.jp (Y. Yamamoto)
Get access

Abstract

We propose a robust numerical algorithm for solving the nonlinear eigenvalue problem A(ƛ)x = 0. Our algorithm is based on the idea of finding the value of ƛ for which A(ƛ) is singular by computing the smallest eigenvalue or singular value of A(ƛ) viewed as a constant matrix. To further enhance computational efficiency, we introduce and use the concept of signed singular value. Our method is applicable when A(ƛ) is large and nonsymmetric and has strong nonlinearity. Numerical experiments on a nonlinear eigenvalue problem arising in the computation of scaling exponent in turbulent flow show robustness and effectiveness of our method.

Keywords

Nonlinear eigenvalue problemsigned singular valuescaling exponentturbulent flow

MSC classification

Secondary: 15A18: Eigenvalues, singular values, and eigenvectors 47J10: Nonlinear spectral theory, nonlinear eigenvalue problems 65F15: Eigenvalues, eigenvectors
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 7 , Issue 4 , November 2017 , pp. 799 - 809
Copyright
Copyright © Global-Science Press 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Abdel-Aziz, M. R. H., Numerical safeguarded use of the implicit restarted Lanczos algorithm for solving nonlinear eigenvalue problems and itsmonotonicity analysis, PhD thesis, Rice University, USA, 1993.Google Scholar
[2] Amako, T., Yamamoto, Y., and Zhang, S.-L., A large-grained parallel algorithm for nonlinear eigenvalue problems and its implementation using OmniRPC, In Proceedings of the 2008 IEEE International Conference on Cluster Computing, pages 42–49, 2008.Google Scholar
[3] Asakura, J., Sakurai, T., Tadano, H., Ikegami, T. and Kimura, K., A numerical method for nonlinear eigenvalue problems using contour integrals, JSIAM Letters, 1 (2009), 52–55.Google Scholar
[4] Betcke, T. and Voss, H., A Jacobi-Davidson-type projection method for nonlinear eigenvalue problems, Future Generation Comput. Syst., 20 (2004), 363–372.Google Scholar
[5] Bunse-Gerstner, A., Byers, R., Mehrmann, V. and Nichols, N. K., Numerical computation of an analytic singular value decomposition of a matrix valued function, Numer. Math., 60 (1991), 1–39.Google Scholar
[6] Forsythe, G., Malcolm, M. and Moler, C., Computer Methods for Mathematical Computations. Englewood Cliffs, New Jersey, 1977.Google Scholar
[7] Garrett, C., Bai, Z. and Li, R. -C., A nonlinear QR algorithm for banded nonlinear eigenvalue problems, ACM Trans. Math. Software, 43 (2016), 4:1–4:19.Google Scholar
[8] Mizuno, Y., Ohi, K., Sogabe, T., Yamamoto, Y. and Kaneda, Y., Four-point correlation function of a passive scalar field in rapidly fluctuating turbulence: numerical analysis of an exact closure equation, Phys. Rev. E, 82 (2010), 036316–036324.Google Scholar
[9] Nemoshkalenko, V. V. and Antonov, N. V., Computational Methods in Solid State Physics. CRC Press, 1999.Google Scholar
[10] Neumaier, A., Residual inverse iteration for the nonlinear eigenvalue problem, SIAM J. Numer. Anal., 22 (1985), 914–923.Google Scholar
[11] Ruhe, A., Algorithms for the nonlinear eigenvalue problem, SIAM J. Numer. Anal., 10 (1973), 674–689.Google Scholar
[12] Szyld, D. and Xue, F., Local convergence analysis of several inexact Newton-type algorithms for general nonlinear eigenvalue problems, Numer. Math., 123 (2013), 333–362.Google Scholar
[13] Szyld, D. and Xue, F., Local convergence of Newton-like methods for degenerate eigenvalues of nonlinear eigenproblems. I. Classical algorithms, Numer. Math., 129 (2015), 353–381.Google Scholar
[14] Szyld, D. and Xue, F., Local convergence of Newton-like methods for degenerate eigenvalues of nonlinear eigenproblems: II. Accelerated algorithms, Numer. Math., 129 (2015), 383–403.Google Scholar
[15] Szyld, D. and Xue, F., Preconditioned eigensolvers for large-scale nonlinear Hermitian eigenproblems with variational characterizations. I. Extreme eigenvalues, Math. Comp., 85 (2016), 2887–2918.Google Scholar
[16] Tisseur, F. and Meerbergen, K., The quadratic eigenvalue problem, SIAM Rev., 43 (2001), 235–286.Google Scholar
[17] Voss, H., An Arnoldi method for nonlinear eigenvalue problems, BIT Numerical Mathematics, 44 (2004), 387–401.Google Scholar
[18] Voss, H., A Jacobi-Davidson method for nonlinear and nonsymmetric eigenproblems, Computers & Structures, 85 (2007), 1284–1292.Google Scholar
[19] Wright, K., Differential equations for the analytic singular value decomposion of a matrix, Numer. Math., 63 (1992), 283–295.Google Scholar
[20] Yamamoto, Y., An elementary derivation of the projection method for nonlinear eigenvalue problems based on complex contour integration, in em Eigenvalue Problems: Algorithms, Software and Applications in Petascale Computing, Proceedings of EPASA 2015, Springer (to appear).Google Scholar