The Rayleigh-Ritz method is widely used for eigenvalue approximation. Given a matrix $X$ with columns that form an orthonormal basis for a subspace $\X$, and a Hermitian matrix $A$, the eigenvalues of $X^HAX$ are called Ritz values of $A$ with respect to $\X$. If the subspace $\X$ is $A$-invariant then the Ritz values are some of the eigenvalues of $A$. If the $A$-invariant subspace $\X$ is perturbed to give rise to another subspace $\Y$, then the vector of absolute values of changes in Ritz values of $A$ represents the absolute eigenvalue approximation error using $\Y$. We bound the error in terms of principal angles between $\X$ and $\Y$. We capitalize on ideas from a recent paper [DOI: 10.1137/060649070] by A. Knyazev and M. Argentati, where the vector of absolute values of differences between Ritz values for subspaces $\X$ and $\Y$ was weakly (sub-)majorized by a constant times the sine of the vector of principal angles between $\X$ and $\Y$, the constant being the spread of the spectrum of $A$. In that result no assumption was made on either subspace being $A$-invariant. It was conjectured there that if one of the trial subspaces is $A$-invariant then an analogous weak majorization bound should only involve terms of the order of sine squared. Here we confirm this conjecture. Specifically we prove that the absolute eigenvalue error is weakly majorized by a constant times the sine squared of the vector of principal angles between the subspaces $\X$ and $\Y$, where the constant is proportional to the spread of the spectrum of $A$. For many practical cases we show that the proportionality factor is simply one, and that this bound is sharp. For the general case we can only prove the result with a slightly larger constant, which we believe is artificial.
We propose a verified computation method for eigenvalues in a region and the corresponding eigenvectors of generalized Hermitian eigenvalue problems. The proposed method uses complex moments to extract the eigencomponents of interest from a random matrix and uses the Rayleigh$\unicode{x2013}$Ritz procedure to project a given eigenvalue problem into a reduced eigenvalue problem. The complex moment is given by contour integral and approximated using numerical quadrature. We split the error in the complex moment into the truncation error of the quadrature and rounding errors and evaluate each. Th…
Convergence rates of block iterations for solving eigenvalue problems typically measure errors of Ritz values approximating eigenvalues. The errors of the Ritz values are commonly bounded in terms of principal angles between the initial or iterative subspace and the invariant subspace associated with the target eigenvalues. Such bounds thus cannot be applied repeatedly as needed for restarted block eigensolvers, since the left- and right-hand sides of the bounds use different terms. They must be combined with additional bounds which could cause an overestimation. Alternative repeatable bounds …
Many popular eigensolvers for large and sparse Hermitian matrices or matrix pairs can be interpreted as accelerated block preconditioned gradient (BPG) iterations in order to analyze their convergence behavior by composing known estimates. An important feature of BPG is the cluster robustness, i.e., reasonable performance for computing clustered eigenvalues is ensured by a sufficiently large block size. This feature can easily be explained for exact-inverse (exact shift-inverse) preconditioning by adapting classical estimates on nonpreconditioned eigensolvers, whereas the existing results for …
This paper describes a set of rational filtering algorithms to compute a few eigenvalues (and associated eigenvectors) of non-Hermitian matrix pencils. Our interest lies in computing eigenvalues located inside a given disk, and the proposed algorithms approximate these eigenvalues and associated eigenvectors by harmonic Rayleigh-Ritz projections on subspaces built by computing range spaces of rational matrix functions through randomized range finders. These rational matrix functions are designed so that directions associated with non-sought eigenvalues are dampened to (approximately) zero. Var…
Convergence analysis of block iterative solvers for Hermitian eigenvalue problems and the closely related research on properties of matrix-based signal filters are challenging, and attract increasing attention due to their recent applications in spectral data clustering and graph-based signal processing. We combine majorization-based techniques pioneered for investigating the Rayleigh-Ritz method in [SIAM J. Matrix Anal. Appl., 31 (2010), pp. 1521-1537] with tools of classical analysis of the block power method by Rutishauser [Numer. Math., 13 (1969), pp. 4-13] to derive convergence rate bound…
In this paper, we propose a unified approach for solving structure-preserving eigenvalue embedding problem (SEEP) for quadratic regular matrix polynomials with symmetry structures. First, we determine perturbations of a quadratic matrix polynomial, unstructured or structured, such that the perturbed polynomials reproduce a desired invariant pair while maintaining the invariance of another invariant pair of the unperturbed polynomial. If the latter is unknown, it is referred to as no spillover perturbation. Then we use these results for solving the SEEP for structured quadratic matrix polynomia…
