Optimal a priori error bounds for the Rayleigh-Ritz method

Sleijpen, Gerard; van den Eshof, Jasper; Smit, Paul

doi:10.1090/S0025-5718-02-01435-7

Optimal a priori error bounds for the Rayleigh-Ritz method
HTML articles powered by AMS MathViewer

by Gerard L. G. Sleijpen, Jasper van den Eshof and Paul Smit PDF

Math. Comp. 72 (2003), 677-684 Request permission

Abstract:

We derive error bounds for the Rayleigh-Ritz method for the approximation to extremal eigenpairs of a symmetric matrix. The bounds are expressed in terms of the eigenvalues of the matrix and the angle between the subspace and the eigenvector. We also present a sharp bound.

References

Ernest R. Davidson, The iterative calculation of a few of the lowest eigenvalues and corresponding eigenvectors of large real-symmetric matrices, J. Comput. Phys. 17 (1975), 87–94. MR 381271, DOI 10.1016/0021-9991(75)90065-0
Zhongxiao Jia and G. W. Stewart, An analysis of the Rayleigh-Ritz method for approximating eigenspaces, Math. Comp. 70 (2001), no. 234, 637–647. MR 1697647, DOI 10.1090/S0025-5718-00-01208-4
P. Lorenzen, Die Definition durch vollständige Induktion, Monatsh. Math. Phys. 47 (1939), 356–358. MR 38, DOI 10.1007/BF01695507
Andrew V. Knyazev, New estimates for Ritz vectors, Math. Comp. 66 (1997), no. 219, 985–995. MR 1415802, DOI 10.1090/S0025-5718-97-00855-7
Beresford N. Parlett, The symmetric eigenvalue problem, Classics in Applied Mathematics, vol. 20, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1998. Corrected reprint of the 1980 original. MR 1490034, DOI 10.1137/1.9781611971163
Y. Saad, On the rates of convergence of the Lanczos and the block-Lanczos methods, SIAM J. Numer. Anal. 17 (1980), no. 5, 687–706. MR 588755, DOI 10.1137/0717059
Paul Smit, The approximation of an eigenvector by ritzvectors, Technical Report FEW 684, Center for Economic Research, University of Tilburg, Tilburg, The Netherlands, 1995.
—, Numerical analysis of eigenvalue algorithms based on subspace iterations, Ph.D. thesis, Center for Economic Research, Tilburg University, Tilburg, The Netherlands, July 1997.