796 research outputs found
Optimal a priori error bounds for the Rayleigh-Ritz method
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
Rayleigh-Ritz majorization error bounds of the mixed type
The absolute change in the Rayleigh quotient (RQ) for a Hermitian matrix with
respect to vectors is bounded in terms of the norms of the residual vectors and
the angle between vectors in [\doi{10.1137/120884468}]. We substitute
multidimensional subspaces for the vectors and derive new bounds of absolute
changes of eigenvalues of the matrix RQ in terms of singular values of residual
matrices and principal angles between subspaces, using majorization. We show
how our results relate to bounds for eigenvalues after discarding off-diagonal
blocks or additive perturbations.Comment: 20 pages, 1 figure. Accepted to SIAM Journal on Matrix Analysis and
Application
Angles Between Infinite Dimensional Subspaces with Applications to the Rayleigh-Ritz and Alternating Projectors Methods
We define angles from-to and between infinite dimensional subspaces of a
Hilbert space, inspired by the work of E. J. Hannan, 1961/1962 for general
canonical correlations of stochastic processes. The spectral theory of
selfadjoint operators is used to investigate the properties of the angles,
e.g., to establish connections between the angles corresponding to orthogonal
complements. The classical gaps and angles of Dixmier and Friedrichs are
characterized in terms of the angles. We introduce principal invariant
subspaces and prove that they are connected by an isometry that appears in the
polar decomposition of the product of corresponding orthogonal projectors.
Point angles are defined by analogy with the point operator spectrum. We bound
the Hausdorff distance between the sets of the squared cosines of the angles
corresponding to the original subspaces and their perturbations. We show that
the squared cosines of the angles from one subspace to another can be
interpreted as Ritz values in the Rayleigh-Ritz method, where the former
subspace serves as a trial subspace and the orthogonal projector of the latter
subspace serves as an operator in the Rayleigh-Ritz method. The Hausdorff
distance between the Ritz values, corresponding to different trial subspaces,
is shown to be bounded by a constant times the gap between the trial subspaces.
We prove a similar eigenvalue perturbation bound that involves the gap squared.
Finally, we consider the classical alternating projectors method and propose
its ultimate acceleration, using the conjugate gradient approach. The
corresponding convergence rate estimate is obtained in terms of the angles. We
illustrate a possible acceleration for the domain decomposition method with a
small overlap for the 1D diffusion equation.Comment: 22 pages. Accepted to Journal of Functional Analysi
Accurate approximations to eigenpairs using the harmonic Rayleigh Ritz Method
The problem in this paper is to construct accurate approximations from a subspace to eigenpairs for symmetric matrices using the harmonic Rayleigh-Ritz method. Morgan introduced this concept in [14] as an alternative forRayleigh-Ritz in large scale iterative methods for computing interior eigenpairs. The focus rests on the choice and in uence of the shift and error estimation. We also give a discussion of the dierences and similarities with the rened Ritz approach for symmetric matrices. Using some numerical experiments we compare dierent conditions for selecting appropriate harmonic Ritz vectors
On Temple--Kato like inequalities and applications
We give both lower and upper estimates for eigenvalues of unbounded positive
definite operators in an arbitrary Hilbert space. We show scaling robust
relative eigenvalue estimates for these operators in analogy to such estimates
of current interest in Numerical Linear Algebra. Only simple matrix theoretic
tools like Schur complements have been used. As prototypes for the strength of
our method we discuss a singularly perturbed Schroedinger operator and study
convergence estimates for finite element approximations. The estimates can be
viewed as a natural quadratic form version of the celebrated Temple--Kato
inequality.Comment: submitted to SIAM Journal on Numerical Analysis (a major revision of
the paper
On the Convergence of Ritz Pairs and Refined Ritz Vectors for Quadratic Eigenvalue Problems
For a given subspace, the Rayleigh-Ritz method projects the large quadratic
eigenvalue problem (QEP) onto it and produces a small sized dense QEP. Similar
to the Rayleigh-Ritz method for the linear eigenvalue problem, the
Rayleigh-Ritz method defines the Ritz values and the Ritz vectors of the QEP
with respect to the projection subspace. We analyze the convergence of the
method when the angle between the subspace and the desired eigenvector
converges to zero. We prove that there is a Ritz value that converges to the
desired eigenvalue unconditionally but the Ritz vector converges conditionally
and may fail to converge. To remedy the drawback of possible non-convergence of
the Ritz vector, we propose a refined Ritz vector that is mathematically
different from the Ritz vector and is proved to converge unconditionally. We
construct examples to illustrate our theory.Comment: 20 page
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