Numerische Mathematik

, Volume 122, Issue 1, pp 169–195

# A linear eigenvalue algorithm for the nonlinear eigenvalue problem

• Elias Jarlebring
• Wim Michiels
• Karl Meerbergen
Article

## Abstract

The Arnoldi method for standard eigenvalue problems possesses several attractive properties making it robust, reliable and efficient for many problems. The first result of this paper is a characterization of the solutions to an arbitrary (analytic) nonlinear eigenvalue problem (NEP) as the reciprocal eigenvalues of an infinite dimensional operator denoted $${\mathcal {B}}$$ . We consider the Arnoldi method for the operator $${\mathcal {B}}$$ and show that with a particular choice of starting function and a particular choice of scalar product, the structure of the operator can be exploited in a very effective way. The structure of the operator is such that when the Arnoldi method is started with a constant function, the iterates will be polynomials. For a large class of NEPs, we show that we can carry out the infinite dimensional Arnoldi algorithm for the operator $${\mathcal {B}}$$ in arithmetic based on standard linear algebra operations on vectors and matrices of finite size. This is achieved by representing the polynomials by vector coefficients. The resulting algorithm is by construction such that it is completely equivalent to the standard Arnoldi method and also inherits many of its attractive properties, which are illustrated with examples.

65F15 65H17

## References

1. 1.
Arnoldi W.: The principle of minimized iterations in the solution of the matrix eigenvalue problem. Q. Appl. Math. 9, 17–29 (1951)
2. 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)Google Scholar
3. 3.
Bai Z., Su Y.: SOAR: A second-order Arnoldi method for the solution of the quadratic eigenvalue problem. SIAM J. Matrix Anal. Appl. 26(3), 640–659 (2005)
4. 4.
Betcke, T., Higham, N.J., Mehrmann, V., Schröder, C., Tisseur, F.: NLEVP: a collection of nonlinear eigenvalue problems. Technical report, University of Manchester (2010)Google Scholar
5. 5.
Beyn, W.J.: An integral method for solving nonlinear eigenvalue problems. Linear Algebra Appl. (2011, in press). doi:
6. 6.
Breda D., Maset S., Vermiglio R.: Pseudospectral approximation of eigenvalues of derivative operators with non-local boundary conditions. Appl. Numer. Math. 56, 318–331 (2006)
7. 7.
Fassbender H., Mackey D., Mackey N., Schröder C.: Structured polynomial eigenproblems related to time-delay systems. Electron. Trans. Numer. Anal. 31, 306–330 (2008)
8. 8.
Gohberg I., Lancaster P., Rodman L.: Matrix polynomials. Academic press, New York (1982)
9. 9.
Hale J., Verduyn Lunel S.M.: Introduction to functional differential equations. Springer, Berlin (1993)
10. 10.
Hochstenbach M.E., Sleijpen G.L.: Harmonic and refined Rayleigh–Ritz for the polynomial eigenvalue problem. Numer. Linear Algebra Appl. 15(1), 35–54 (2008)
11. 11.
Jarlebring E., Meerbergen K., Michiels W.: A Krylov method for the delay eigenvalue problem. SIAM J. Sci. Comput. 32(6), 3278–3300 (2010)
12. 12.
Kressner D.: A block Newton method for nonlinear eigenvalue problems. Numer. Math. 114(2), 355–372 (2009)
13. 13.
Kressner D., Schröder C., Watkins D.S.: Implicit QR algorithms for palindromic and even eigenvalue problems. Numer. Algorithms 51(2), 209–238 (2009)
14. 14.
Lehoucq R., Sorensen D., Yang C.: ARPACK user’s guide. Solution of large-scale eigenvalue problems with implicitly restarted Arnoldi methods. SIAM publications, Philadelphia (1998)
15. 15.
Liao B.-S., Bai Z., Lee L.-Q., Ko K.: Nonlinear Rayleigh–Ritz iterative method for solving large scale nonlinear eigenvalue problems. Taiwan. J. Math. 14(3), 869–883 (2010)
16. 16.
Mackey D., Mackey N., Mehl C., Mehrmann V.: Numerical methods for palindromic eigenvalue problems: Computing the anti-triangular Schur form. Numer. Linear Algebra 16, 63–86 (2009)
17. 17.
Meerbergen K.: The quadratic Arnoldi method for the solution of the quadratic eigenvalue problem. SIAM J. Matrix Anal. Appl. 30(4), 1463–1482 (2008)
18. 18.
Mehrmann V., Voss H.: Nonlinear eigenvalue problems: a challenge for modern eigenvalue methods. GAMM Mitteilungen 27, 121–152 (2004)
19. 19.
Neumaier A.: Residual inverse iteration for the nonlinear eigenvalue problem. SIAM J. Numer. Anal. 22, 914–923 (1985)
20. 20.
Peters G., Wilkinson J.: Inverse iterations, ill-conditioned equations and Newton’s method. SIAM Rev. 21, 339–360 (1979)
21. 21.
Ruhe A.: Algorithms for the nonlinear eigenvalue problem. SIAM J. Numer. Anal. 10, 674–689 (1973)
22. 22.
Sleijpen G.L., Booten A.G., Fokkema D.R., van der Vorst H.A.: Jacobi–Davidson type methods for generalized eigenproblems and polynomial eigenproblems. BIT 36(3), 595–633 (1996)
23. 23.
Su, Y., Bai, Z.: Solving rational eigenvalue problems via linearization. Technical report, Department of Computer Science and Mathematics, University of California, Davis (2008)Google Scholar
24. 24.
Trefethen L.N.: Spectral Methods in MATLAB. SIAM Publications, Philadelphia (2000)
25. 25.
Unger, H.: Nichtlineare Behandlung von Eigenwertaufgaben. Z. Angew. Math. Mech. 30, 281–282 (1950). English translation: http://www.math.tu-dresden.de/~schwetli/Unger.html
26. 26.
Voss H.: An Arnoldi method for nonlinear eigenvalue problems. BIT 44, 387–401 (2004)

## Authors and Affiliations

• Elias Jarlebring
• 1
• Wim Michiels
• 2
• Karl Meerbergen
• 2
1. 1.Royal Institute of Technology (KTH)StockholmSweden
2. 2.Katholieke Universiteit LeuvenLeuvenBelgium