Skip to main content
SpringerLink
Log in

Convergence of inexact inverse iteration with application to preconditioned iterative solves

  • Published:
BIT Numerical Mathematics Aims and scope Submit manuscript

Abstract

In this paper we study inexact inverse iteration for solving the generalised eigenvalue problem A x=λM x. We show that inexact inverse iteration is a modified Newton method and hence obtain convergence rates for various versions of inexact inverse iteration for the calculation of an algebraically simple eigenvalue. In particular, if the inexact solves are carried out with a tolerance chosen proportional to the eigenvalue residual then quadratic convergence is achieved. We also show how modifying the right hand side in inverse iteration still provides a convergent method, but the rate of convergence will be quadratic only under certain conditions on the right hand side. We discuss the implications of this for the preconditioned iterative solution of the linear systems. Finally we introduce a new ILU preconditioner which is a simple modification to the usual preconditioner, but which has advantages both for the standard form of inverse iteration and for the version with a modified right hand side. Numerical examples are given to illustrate the theoretical results.

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

  1. E. L. Allgower and H. Schwetlick, A general view of minimally extended systems for simple bifurcation points, ZAMM Z. Angew. Math. Mech., 77 (1997), pp. 83–98.

    MATH  MathSciNet  Google Scholar 

  2. J. Berns-Müller, I. G. Graham, and A. Spence, Inexact inverse iteration for symmetric matrices, Linear Algebra Appl., 416 (2006), pp. 389–413.

    Article  MATH  MathSciNet  Google Scholar 

  3. J. Berns-Müller and A. Spence, Inexact inverse iteration and GMRES, 2006. Submitted.

  4. J. Berns-Müller and A. Spence, Inexact inverse iteration and the nonsymmetric generalized eigenvalue problem, 2006. To appear in SIAM J. Matrix Anal. Appl.

  5. J. W. Demmel, Applied Numerical Linear Algebra, SIAM, Philadelphia, 1997.

    MATH  Google Scholar 

  6. J. E. Dennis and R. B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Classics in Applied Mathematics, vol. 16, SIAM, Philadelphia, PA, 1996. Unabridged, corrected reprint of the 1983 original.

    MATH  Google Scholar 

  7. P. Deuflhard, Newton Methods for Nonlinear Problems, Springer Series in Computational Mathematics, vol. 35, Springer, Berlin, Heidelberg, 2004.

    MATH  Google Scholar 

  8. A. Edelman, T. Arias, and S. T. Smith, The geometry of algorithms with orthogonality constraints, SIAM J. Matrix Anal. Appl., 20 (1999), pp. 303–353.

    Article  MathSciNet  Google Scholar 

  9. G. H. Golub and Q. Ye, Inexact inverse iteration for generalized eigenvalue problems, BIT, 40 (2000), pp. 671–684.

    Article  MATH  MathSciNet  Google Scholar 

  10. H. B. Keller, Numerical solution of bifurcation and nonlinear eigenvalue problems, in Applications of Bifurcation Theory, P. H. Rabinowitz, ed., Academic Press, New York, 1977, pp. 359–384.

    Google Scholar 

  11. C. T. Kelley, Iterative Methods for Linear and Nonlinear Equations, Frontiers in Applied Mathematics, vol. 16, SIAM, Philadelphia, PA, 1995.

    MATH  Google Scholar 

  12. A. V. Knyazev, Toward the optimal preconditioned eigensolver: Locally optimal block preconditioned conjugate gradient method, SIAM J. Sci. Comput., 23 (2001), pp. 517–541.

    Article  MATH  MathSciNet  Google Scholar 

  13. Y.-L. Lai, K.-Y. Lin, and L. Wen-Wei, An inexact inverse iteration for large sparse eigenvalue problems, Numer. Linear Algebra Appl., 1 (1997), pp. 1–13.

    Google Scholar 

  14. K. Neymeyr, A geometric theory for preconditioned inverse iteration I: Extrema of the Rayleigh quotient, Linear Algebra Appl., 322 (2001), pp. 61–85.

    Article  MATH  MathSciNet  Google Scholar 

  15. G. Peters and J. Wilkinson, Inverse iteration, ill-conditioned equations and Newton’s method, SIAM Rev., 21 (1979), pp. 339–360.

    Article  MATH  MathSciNet  Google Scholar 

  16. A. Ruhe and T. Wiberg, The method of conjugate gradients used in inverse iteration, BIT, 12 (1972), pp. 543–554.

    Article  MATH  MathSciNet  Google Scholar 

  17. R. Scheichl, Parallel Solution of the Transient Multigroup Neutron Diffusion Equations with Multi-Grid and Preconditioned Krylov-Subspace Methods, Master’s thesis, Johannes Kepler Universität Linz, Austria, 1997.

  18. H. Schwetlick and R. Lösche, A generalized Rayleigh quotient iteration for computing simple eigenvalues of nonnormal matrices, ZAMM Z. Angew. Math. Mech., 80 (2000), pp. 9–25.

    Article  MATH  MathSciNet  Google Scholar 

  19. V. Simoncini and L. Eldén, Inexact Rayleigh quotient-type methods for eigenvalue computations, BIT, 42 (2002), pp. 159–182.

    Article  MATH  MathSciNet  Google Scholar 

  20. P. Smit and M. H. C. Paardekooper, The effects of inexact solvers in algorithms for symmetric eigenvalue problems, Linear Algebra Appl., 287 (1999), pp. 337–357.

    Article  MATH  MathSciNet  Google Scholar 

  21. H. Symm and J. Wilkinson, Realistic error bounds for a simple eigenvalue and its associated eigenvector, Numer. Math., 35 (1980), pp. 113–126.

    Article  MATH  MathSciNet  Google Scholar 

  22. H. Unger, Nichtlineare Behandlung von Eigenwertaufgaben, ZAMM Z. Angew. Math. Mech., 30 (1950), pp. 281–282.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, University of Bath, Claverton Down, BA2 7AY, United Kingdom

    M. A. Freitag & A. Spence

Authors
  1. M. A. Freitag
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. Spence
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. Spence.

Additional information

AMS subject classification (2000)

65F15, 65F10

Rights and permissions

Reprints and permissions

About this article

Cite this article

Freitag, M., Spence, A. Convergence of inexact inverse iteration with application to preconditioned iterative solves . Bit Numer Math 47, 27–44 (2007). https://doi.org/10.1007/s10543-006-0100-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10543-006-0100-1

Key words

Search

Navigation