NDLI: RayleighRitz Majorization Error Bounds with Applications to FEM

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Knyazev, Andrew V. Argentati, Merico E.
Copyright Year	2010
Abstract	The RayleighRitz (RR) method finds the stationary values, called Ritz values, of the Rayleigh quotient on a given trial subspace as approximations to eigenvalues of a Hermitian operator A. If the trial subspace is A-invariant, the Ritz values are exactly some of the eigenvalues of A. Given two subspaces $\mathcal{X}$ and $\mathcal{Y}$ of the same finite dimension, such that $\mathcal{X}$ is A-invariant, the absolute changes in the Ritz values of A with respect to $\mathcal{X}$ compared to the Ritz values with respect to $\mathcal{Y}$ represent the RR absolute eigenvalue approximation error. Our first main result is a sharp majorization-type RR error bound in terms of the principal angles between $\mathcal{X}$ and $\mathcal{Y}$ for an arbitrary A-invariant $\mathcal{X}$, which was a conjecture in [SIAM J. Matrix Anal. Appl., 30 (2008), pp. 548559]. Second, we prove a novel type of RR error bound that deals with the products of the errors, rather than the sums. Third, we establish majorization bounds for the relative errors. We extend our bounds to the case $\dim\mathcal{X}\leq\dim\mathcal{Y}<\infty$ in Hilbert spaces and apply them in the context of the finite element method.
Starting Page	1521
Ending Page	1537
Page Count	17
File Format	PDF
ISSN	08954798
DOI	10.1137/08072574X
e-ISSN	10957162
Journal	SIAM Journal on Matrix Analysis and Applications (SJMAEL)
Issue Number	3
Volume Number	31
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2010-01-08
Access Restriction	Subscribed
Subject Keyword	angles error analysis perturbation RayleighRitz Inequalities involving eigenvalues and eigenvectors subspaces finite element method multiplicative bounds relative error bounds Norms of matrices, numerical range, applications of functional analysis to matrix theory Ritz values majorization projection Matrix norms, conditioning, scaling Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods eigenvalue
Content Type	Text
Resource Type	Article
Subject	Analysis

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