BIT Numerical Mathematics

, Volume 47, Issue 1, pp 27–44 | Cite as

Convergence of inexact inverse iteration with application to preconditioned iterative solves

Article

Abstract

In this paper we study inexact inverse iteration for solving the generalised eigenvalue problem AxMx. 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.

Key words

inverse iteration Newton’s method preconditioned iterative methods 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 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.MATHMathSciNetGoogle Scholar
  2. 2.
    J. Berns-Müller, I. G. Graham, and A. Spence, Inexact inverse iteration for symmetric matrices, Linear Algebra Appl., 416 (2006), pp. 389–413.MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    J. Berns-Müller and A. Spence, Inexact inverse iteration and GMRES, 2006. Submitted.Google Scholar
  4. 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.Google Scholar
  5. 5.
    J. W. Demmel, Applied Numerical Linear Algebra, SIAM, Philadelphia, 1997.MATHGoogle Scholar
  6. 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.MATHGoogle Scholar
  7. 7.
    P. Deuflhard, Newton Methods for Nonlinear Problems, Springer Series in Computational Mathematics, vol. 35, Springer, Berlin, Heidelberg, 2004.MATHGoogle Scholar
  8. 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.CrossRefMathSciNetGoogle Scholar
  9. 9.
    G. H. Golub and Q. Ye, Inexact inverse iteration for generalized eigenvalue problems, BIT, 40 (2000), pp. 671–684.MATHCrossRefMathSciNetGoogle Scholar
  10. 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. 11.
    C. T. Kelley, Iterative Methods for Linear and Nonlinear Equations, Frontiers in Applied Mathematics, vol. 16, SIAM, Philadelphia, PA, 1995.MATHGoogle Scholar
  12. 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.MATHCrossRefMathSciNetGoogle Scholar
  13. 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. 14.
    K. Neymeyr, A geometric theory for preconditioned inverse iteration I: Extrema of the Rayleigh quotient, Linear Algebra Appl., 322 (2001), pp. 61–85.MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    G. Peters and J. Wilkinson, Inverse iteration, ill-conditioned equations and Newton’s method, SIAM Rev., 21 (1979), pp. 339–360.MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    A. Ruhe and T. Wiberg, The method of conjugate gradients used in inverse iteration, BIT, 12 (1972), pp. 543–554.MATHCrossRefMathSciNetGoogle Scholar
  17. 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.Google Scholar
  18. 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.MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    V. Simoncini and L. Eldén, Inexact Rayleigh quotient-type methods for eigenvalue computations, BIT, 42 (2002), pp. 159–182.MATHCrossRefMathSciNetGoogle Scholar
  20. 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.MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    H. Symm and J. Wilkinson, Realistic error bounds for a simple eigenvalue and its associated eigenvector, Numer. Math., 35 (1980), pp. 113–126.MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    H. Unger, Nichtlineare Behandlung von Eigenwertaufgaben, ZAMM Z. Angew. Math. Mech., 30 (1950), pp. 281–282.MATHGoogle Scholar

Copyright information

© Springer Science + Business Media B.V. 2006

Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of BathClaverton DownUnited Kingdom

Personalised recommendations