BIT Numerical Mathematics

, Volume 51, Issue 4, pp 937–957

Analyzing the convergence factor of residual inverse iteration

• Elias Jarlebring
• Wim Michiels
Article

Abstract

We will establish here a formula for the convergence factor of the method called residual inverse iteration, which is a method for nonlinear eigenvalue problems and a generalization of the well-known inverse iteration. The formula for the convergence factor is explicit and involves quantities associated with the eigenvalue to which the iteration converges, in particular the eigenvalue and eigenvector. Residual inverse iteration allows for some freedom in the choice of a vector w k and we can use the formula for the convergence factor to analyze how it depends on the choice of w k . We also use the formula to illustrate the convergence when the shift is close to the eigenvalue. Finally, we explain the slow convergence for double eigenvalues by showing that under generic conditions, the convergence factor is one, unless the eigenvalue is semisimple. If the eigenvalue is semisimple, it turns out that we can expect convergence similar to the simple case.

Keywords

Nonlinear eigenvalue problems Residual inverse iteration Convergence factors

Mathematics Subject Classification (2000)

65F15 65H17 15A18

Preview

Unable to display preview. Download preview PDF.

References

1. 1.
Anselone, P., Rall, L.: The solution of characteristic value-vector problems by Newton’s method. Numer. Math. 11, 38–45 (1968)
2. 2.
Betcke, M.: Iterative projection methods for symmetric nonlinear eigenvalue problems with applications. Ph.D. thesis, Technical University Hamburg-Harburg (2007) Google Scholar
3. 3.
Betcke, M., Voss, H.: Stationary Schrödinger equations governing electronic states of quantum dots in the presence of spin-orbit splitting. Appl. Math. 52, 267–284 (2007)
4. 4.
Betcke, T., Higham, N.J., Mehrmann, V., Schröder, C., Tisseur, F.: NLEVP: A collection of nonlinear eigenvalue problems. University of Manchester, MIMS EPrint 2010.98 (2010) Google Scholar
5. 5.
Gohberg, I., Lancaster, P., Rodman, L.: Matrix Polynomials. Academic Press, San Diego (1982)
6. 6.
Jarlebring, E., Michiels, W.: Invariance properties in the root sensitivity of time-delay systems with double imaginary roots. Automatica 46, 1112–1115 (2010)
7. 7.
Kressner, D.: A block Newton method for nonlinear eigenvalue problems. Numer. Math. 114(2), 355–372 (2009)
8. 8.
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)
9. 9.
Meerbergen, K.: The quadratic Arnoldi method for the solution of the quadratic eigenvalue problem. SIAM J. Matrix Anal. Appl. 30(4), 1463–1482 (2008)
10. 10.
Mehrmann, V., Voss, H.: Nonlinear eigenvalue problems: A challenge for modern eigenvalue methods. Mitteilungen - Ges. Angew. Math. Mech. 27, 121–152 (2004)
11. 11.
Neumaier, A.: Residual inverse iteration for the nonlinear eigenvalue problem. SIAM J. Numer. Anal. 22, 914–923 (1985)
12. 12.
Peters, G., Wilkinson, J.: Inverse iterations, ill-conditioned equations and Newton’s method. SIAM Rev. 21, 339–360 (1979)
13. 13.
Rogers, E.: A minimax theory for overdamped systems. Arch. Ration. Mech. Anal. 16, 89–96 (1964)
14. 14.
Rott, O., Jarlebring, E.: An iterative method for the multipliers of periodic delay-differential equations and the analysis of a PDE milling model. In: Proceedings of the 9th IFAC Workshop on Time-Delay Systems, Prague, pp. 1–6 (2010) Google Scholar
15. 15.
Ruhe, A.: Algorithms for the nonlinear eigenvalue problem. SIAM J. Numer. Anal. 10, 674–689 (1973)
16. 16.
Schreiber, K.: Nonlinear eigenvalue problems: Newton-type methods and nonlinear Rayleigh functionals. Ph.D. thesis, TU Berlin (2008) Google Scholar
17. 17.
Trefethen, L.N., Bau, D.I.: Numerical Linear Algebra. SIAM, Philadelphia (1997)
18. 18.
Voss, H.: An Arnoldi method for nonlinear eigenvalue problems. BIT Numer. Math. 44, 387–401 (2004)
19. 19.
Voss, H.: Numerical methods for sparse nonlinear eigenvalue problems. In: Proc. XVth Summer School on Software and Algorithms of Numerical Mathematics, Hejnice, Czech Republic (2004). Report 70. Arbeitsbereich Mathematik, TU Hamburg-Harburg Google Scholar
20. 20.
Voss, H.: Iterative projection methods for computing relevant energy states of a quantum dot. J. Comput. Phys. 217(2), 824–833 (2006)