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New estimates for Ritz vectors

Author: Andrew V. Knyazev
Journal: Math. Comp. 66 (1997), 985-995
MSC (1991): Primary 65F35
MathSciNet review: 1415802
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Abstract: The following estimate for the Rayleigh-Ritz method is proved:

\begin{displaymath}| \tilde \lambda - \lambda | |( \tilde u , u )| \le { \| A \tilde u - \tilde \lambda \tilde u \| } \sin \angle \{ u ; \tilde U \}, \ \| u \| =1. \end{displaymath}

Here $A$ is a bounded self-adjoint operator in a real Hilbert/euclidian space, $\{ \lambda , u \}$ one of its eigenpairs, $\tilde U$ a trial subspace for the Rayleigh-Ritz method, and $\{ \tilde \lambda , \tilde u \}$ a Ritz pair. This inequality makes it possible to analyze the fine structure of the error of the Rayleigh-Ritz method, in particular, it shows that $ |( \tilde u , u )| \le C \epsilon ^2, $ if an eigenvector $u$ is close to the trial subspace with accuracy $\epsilon $ and a Ritz vector $\tilde u$ is an $\epsilon $ approximation to another eigenvector, with a different eigenvalue. Generalizations of the estimate to the cases of eigenspaces and invariant subspaces are suggested, and estimates of approximation of eigenspaces and invariant subspaces are proved.

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  • 1. I. Babuska and J. Osborn, Eigenvalue problems. In P. G. Ciarlet and J. L. Lions, editors, Handbook of Numerical Analysis, Vol. II, pages 642-787. Elsevier Science Publishers, North-Holland, 1991.
  • 2. James H. Bramble, Andrew Knyazev and Joseph E. Pasciak, A subspace preconditioning algorithm for eigenvector/eigenvalue computation. Technical Report UCD/CCM Report 66, Center for Computational Mathematics, University of Colorado at Denver, 1995. Submitted to Advances in Computational Mathematics.
  • 3. Françoise Chatelin, Spectral approximation of linear operators, Computer Science and Applied Mathematics, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1983. With a foreword by P. Henrici; With solutions to exercises by Mario Ahués. MR 716134
  • 4. Chandler Davis and W. M. Kahan, The rotation of eigenvectors by a perturbation. III, SIAM J. Numer. Anal. 7 (1970), 1–46. MR 0264450
  • 5. Jean Descloux, Nabil Nassif, and Jacques Rappaz, On spectral approximation. I. The problem of convergence, RAIRO Anal. Numér. 12 (1978), no. 2, 97–112, iii (English, with French summary). MR 0483400
  • 6. Ralf Gruber and Jacques Rappaz, Finite element methods in linear ideal magnetohydrodynamics, Springer Series in Computational Physics, Springer-Verlag, Berlin, 1985. MR 800851
  • 7. Tosio Kato, On the upper and lower bounds of eigenvalues, J. Phys. Soc. Japan 4 (1949), 334–339. MR 0038738
  • 8. Tosio Kato, Perturbation theory for linear operators, 2nd ed., Springer-Verlag, Berlin-New York, 1976. Grundlehren der Mathematischen Wissenschaften, Band 132. MR 0407617
  • 9. A. V. Knyazev. Computation of eigenvalues and eigenvectors for mesh problems: algorithms and error estimates. Dept. Numerical Math. USSR Academy of Sciences, Moscow, 1986. In Russian.
  • 10. A. V. Knyazev, Sharp a priori error estimates for the Rayleigh-Ritz method with no assumptions on fixed sign or compactness, Mat. Zametki 38 (1985), no. 6, 900–907, 958 (Russian). MR 823428
  • 11. A. V. Knyazev, Convergence rate estimates for iterative methods for a mesh symmetric eigenvalue problem, Soviet J. Numer. Anal. Math. Modelling 2 (1987), no. 5, 371–396. Translated from the Russian. MR 915330
  • 12. A. V. Knyazev. New estimates for Ritz vectors. Technical Report 677, CIMS NYU, New York, 1994.
  • 13. M. A. Krasnosel′skiĭ, G. M. Vaĭnikko, P. P. Zabreĭko, Ya. B. Rutitskii, and V. Ya. Stetsenko, Approximate solution of operator equations, Wolters-Noordhoff Publishing, Groningen, 1972. Translated from the Russian by D. Louvish. MR 0385655
  • 14. Beresford N. Parlett, The symmetric eigenvalue problem, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1980. Prentice-Hall Series in Computational Mathematics. MR 570116
  • 15. Youcef Saad, Numerical methods for large eigenvalue problems, Algorithms and Architectures for Advanced Scientific Computing, Manchester University Press, Manchester; Halsted Press [John Wiley & Sons, Inc.], New York, 1992. MR 1177405
  • 16. Gilbert Strang and George J. Fix, An analysis of the finite element method, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1973. Prentice-Hall Series in Automatic Computation. MR 0443377
  • 17. H. F. Weinberger, Error bounds in the Rayleigh-Ritz approximation of eigenvectors, J. Res. Nat. Bur. Standards Sect. B 64B (1960), 217–225. MR 0129121
  • 18. H. F. Weinberger. Variational Methods for Eigenvalue Approximation. SIAM, 1974.

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Additional Information

Andrew V. Knyazev
Affiliation: Department of Mathematics, University of Colorado at Denver, Denver, Colorado 80217

Keywords: Eigenvalue problem, Rayleigh--Ritz method, approximation, error estimate
Received by editor(s): May 10, 1995
Received by editor(s) in revised form: September 5, 1995, and June 3, 1996
Additional Notes: This research was supported by the National Science Foundation under grant NSF-CCR-9204255 and was performed while the author was visiting the Courant Institute.
Article copyright: © Copyright 1997 American Mathematical Society