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RAYLEIGHRITZ MAJORIZATION ERROR BOUNDS WITH APPLICATIONS TO FEM AND SUBSPACE ITERATIONS
, 2008
"... The RayleighRitz 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 Ainvariant, the Ritz values are exactly some of the eigenvalues of A. Given two subspace ..."
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The RayleighRitz 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 Ainvariant, the Ritz values are exactly some of the eigenvalues of A. Given two subspaces X and Y of the same finite dimension, such that X is Ainvariant, the absolute changes in the Ritz values of A with respect to X compared to the Ritz values with respect to Y represent the absolute eigenvalue approximation error. A recent paper [SIAM J. Matrix Anal. Appl., 30 (2008), pp. 548559] by M. Argentati et al. bounds the error in terms of the principal angles between X and Y using weak majorization, e.g., a sharp bound is proved if X corresponds to a contiguous set of extreme eigenvalues of A. In this paper, we extend this sharp bound to dimX ≤ dimY and to the general case of an arbitrary Ainvariant subspace X, which was a conjecture in this previous paper. We present our RayleighRitz majorization error bound in the context of the finite element method (FEM), and show how it can improve known FEM eigenvalue error bounds. We derive a new majorizationtype convergence rate bound of subspace iterations and combine it with the previous result to obtain a similar bound for the block Lanczos method.
Angles between infinite dimensional subspaces with applications to the RayleighRitz and alternating projectors methods
, 2007
"... This paper is dedicated to our families. Abstract. We define angles fromto and between infinite dimensional subspaces of a Hilbert space using the spectra of the product of corresponding orthogonal projectors. Our definition is inspired by the work of E. J. Hannan, 1961/1962, who had suggested such ..."
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Cited by 6 (5 self)
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This paper is dedicated to our families. Abstract. We define angles fromto and between infinite dimensional subspaces of a Hilbert space using the spectra of the product of corresponding orthogonal projectors. Our definition is inspired by the work of E. J. Hannan, 1961/1962, who had suggested such an approach for general canonical correlations of stochastic processes. Subsequent later developments for the infinite dimensional case, however, have been mostly limited to the gap and the minimum gap between subspaces despite the active work concerning all the angles in finite dimensional spaces. Our paper is intended to revive the interest in the angles between infinite dimensional subspaces and provides a foundation for work in applications, e.g., on canonical correlations for functional data. We use the spectral theory of selfadjoint operators to investigate the properties of the angles, e.g., to establish connections between the angles corresponding to orthogonal complements. We express classical quantities: the gap and the minimum gap between subspaces, in terms of the angles. We
MAJORIZATION BOUNDS FOR RITZ VALUES OF HERMITIAN MATRICES
, 2008
"... Given an approximate invariant subspace we discuss the effectiveness of majorization bounds for assessing the accuracy of the resulting RayleighRitz approximations to eigenvalues of Hermitian matrices. We derive a slightly stronger result than previously for the approximation of k extreme eigenvalu ..."
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Given an approximate invariant subspace we discuss the effectiveness of majorization bounds for assessing the accuracy of the resulting RayleighRitz approximations to eigenvalues of Hermitian matrices. We derive a slightly stronger result than previously for the approximation of k extreme eigenvalues, and examine some advantages of these majorization bounds compared with classical bounds. From our results we conclude that the majorization approach appears to be advantageous, and that there is probably much more work to be carried out in this direction.
Angles Between Infinite Dimensional Subspaces with Applications to the RayleighRitz and Alternating Projectors Methods ✩
"... We define angles fromto and between infinite dimensional subspaces of a Hilbert space using the spectra of the product of corresponding orthogonal projectors. Our definition is inspired by the work of E. J. Hannan, 1961/1962, who had suggested such an approach for general canonical correlations of ..."
Abstract
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We define angles fromto and between infinite dimensional subspaces of a Hilbert space using the spectra of the product of corresponding orthogonal projectors. Our definition is inspired by the work of E. J. Hannan, 1961/1962, who had suggested such an approach for general canonical correlations of stochastic processes. Subsequent later developments for the infinite dimensional case, however, have been mostly limited to the gap and the minimum gap between subspaces despite the active work concerning all the angles in finite dimensional spaces. Our paper is intended to revive the interest in the angles between infinite dimensional subspaces and provides a foundation for work in applications, e.g., on canonical correlations for functional data. We use the spectral theory of selfadjoint operators to investigate the properties A preliminary version is available at