Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 

 

A geometric theory for preconditioned inverse iteration applied to a subspace


Author: Klaus Neymeyr
Journal: Math. Comp. 71 (2002), 197-216
MSC (2000): Primary 65N30, 65N25; Secondary 65F10, 65F15
Published electronically: September 17, 2001
MathSciNet review: 1862995
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The aim of this paper is to provide a convergence analysis for a preconditioned subspace iteration, which is designated to determine a modest number of the smallest eigenvalues and its corresponding invariant subspace of eigenvectors of a large, symmetric positive definite matrix. The algorithm is built upon a subspace implementation of preconditioned inverse iteration, i.e., the well-known inverse iteration procedure, where the associated system of linear equations is solved approximately by using a preconditioner. This step is followed by a Rayleigh-Ritz projection so that preconditioned inverse iteration is always applied to the Ritz vectors of the actual subspace of approximate eigenvectors. The given theory provides sharp convergence estimates for the Ritz values and is mainly built on arguments exploiting the geometry underlying preconditioned inverse iteration.


References [Enhancements On Off] (What's this?)

  • 1. James H. Bramble, Joseph E. Pasciak, and Andrew V. Knyazev, A subspace preconditioning algorithm for eigenvector/eigenvalue computation, Adv. Comput. Math. 6 (1996), no. 2, 159–189 (1997). MR 1431791, 10.1007/BF02127702
  • 2. Françoise Chatelin, Eigenvalues of matrices, John Wiley & Sons, Ltd., Chichester, 1993. With exercises by Mario Ahués and the author; Translated from the French and with additional material by Walter Ledermann. MR 1232655
  • 3. Ernest R. Davidson, The iterative calculation of a few of the lowest eigenvalues and corresponding eigenvectors of large real-symmetric matrices, J. Computational Phys. 17 (1975), 87–94. MR 0381271
  • 4. E. G. D′yakonov, Some iteration methods in eigenvalue problems, Mat. Zametki 34 (1983), no. 6, 897–912 (Russian). MR 731332
  • 5. Eugene G. D′yakonov, Optimization in solving elliptic problems, CRC Press, Boca Raton, FL, 1996. Translated from the 1989 Russian original; Translation edited and with a preface by Steve McCormick. MR 1396083
  • 6. E. G. D′yakonov and A. V. Knyazev, A group iteration method for finding the lowest eigenvalues, Vestnik Moskov. Univ. Ser. XV Vychisl. Mat. Kibernet. 2 (1982), 29–34, 81 (Russian). MR 663544
  • 7. E. G. D′yakonov and A. V. Knyazev, On an iterative method for finding lower eigenvalues, Russian J. Numer. Anal. Math. Modelling 7 (1992), no. 6, 473–486. MR 1202649, 10.1515/rnam.1992.7.6.473
  • 8. E. G. D′jakonov and M. Ju. Orehov, Minimization of computational work in finding the first eigenvalues of differential operators, Mat. Zametki 27 (1980), no. 5, 795–812, 831 (Russian). MR 578262
  • 9. S. K. Godunov, V. V. Ogneva, and G. P. Prokopov, The convergence of the modified steepest descent method in the computation of eigenvalues, Partial differential equations (Proc. Sympos.) (Russian), Izdat. “Nauka”, Moscow, 1970, pp. 77–80 (Russian). MR 0324886
  • 10. Gene H. Golub and Charles F. Van Loan, Matrix computations, 3rd ed., Johns Hopkins Studies in the Mathematical Sciences, Johns Hopkins University Press, Baltimore, MD, 1996. MR 1417720
  • 11. I. Ipsen. A history of inverse iteration, volume in Helmut Wielandt, Mathematische Werke, Mathematical Works, Vol. 2: Linear Algebra and Analysis, pages 464-472. Walter de Gruyter, Berlin, 1996.
  • 12. Ilse C. F. Ipsen, Computing an eigenvector with inverse iteration, SIAM Rev. 39 (1997), no. 2, 254–291. MR 1453320, 10.1137/S0036144596300773
  • 13. Andrew V. Knyazev, Preconditioned eigensolvers—an oxymoron?, Electron. Trans. Numer. Anal. 7 (1998), 104–123 (electronic). Large scale eigenvalue problems (Argonne, IL, 1997). MR 1667642
  • 14. K. Neymeyr. A posteriori error estimation for elliptic eigenproblems. Sonderforschungsbereich 382, Universität Tübingen, Report 132, 1999.
  • 15. K. Neymeyr. A geometric theory for preconditioned inverse iteration. I: Extrema of the Rayleigh quotient. Linear Algebra Appl. 322(1-3):61-85, 2001. CMP 2001:06
  • 16. K. Neymeyr. A geometric theory for preconditioned inverse iteration. II: Convergence estimates. Linear Algebra Appl. 322(1-3):87-104, 2001. CMP 2001:06
  • 17. Beresford N. Parlett, The symmetric eigenvalue problem, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1980. Prentice-Hall Series in Computational Mathematics. MR 570116
  • 18. W. V. Petryshyn, On the eigenvalue problem 𝑇𝑢-𝜆𝑆𝑢=0 with unbounded and nonsymmetric operators 𝑇 and 𝑆, Philos. Trans. Roy. Soc. London Ser. A 262 (1967/1968), 413–458. MR 0222697
  • 19. B.A. Samokish. The steepest descent method for an eigenvalue problem with semi-bounded operators. Izv. Vyssh. Uchebn. Zaved. Mat., 5:105-114, 1958.

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 65N30, 65N25, 65F10, 65F15

Retrieve articles in all journals with MSC (2000): 65N30, 65N25, 65F10, 65F15


Additional Information

Klaus Neymeyr
Affiliation: Mathematisches Institut der Universität Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany
Email: neymeyr@na.uni-tuebingen.de

DOI: https://doi.org/10.1090/S0025-5718-01-01357-6
Keywords: Symmetric eigenvalue problem, subspace iteration, preconditioning, multigrid, inverse iteration
Received by editor(s): July 27, 1999
Published electronically: September 17, 2001
Article copyright: © Copyright 2001 American Mathematical Society