Bounds on changes in Ritz values for a perturbed invariant subspace of a Hermitian matrix

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.Comment: 12 pages. Accepted to SIAM Journal on Matrix Analysis and Applications (SIMAX

Angle-free cluster robust Ritz value bounds for restarted block eigensolvers

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 that are angle-free and depend only on the errors of the Ritz values have been pioneered for Hermitian eigenvalue problems in doi:10.1515/rnam.1987.2.5.371 but only for a single extreme Ritz value. We extend this result to all Ritz values and achieve robustness for clustered eigenvalues by utilizing nonconsecutive eigenvalues. Our new bounds cover the restarted block Lanczos method and its modifications with shift-and-invert and deflation, and are numerically advantageous.Comment: 24 pages, 4 figure

Restarting parallel Jacobi-Davidson with both standard and harmonic Ritz values

We study the Jacobi-Davidson method for the solution of large generalized eigenproblems as they arise in MagnetoHydroDynamics. We have combined Jacobi-Davidson (using standard Ritz values) with a shift and invert technique. We apply a complete LU decomposition in which reordering strategies based on a combination of block cyclic reduction and domain decomposition result in a well-parallelizable algorithm. Moreover, we describe a variant of Jacobi-Davidson in which harmonic Ritz values are used. In this variant the same parallel LU decomposition is used, but this time as a preconditioner to solve the `correction` equation. The size of the relatively small projected eigenproblems which have to be solved in the Jacobi-Davidson method is controlled by several parameters. The influence of these parameters on both the parallel performance and convergence behaviour will be studied. Numerical results of Jacobi-Davidson obtained with standard and harmonic Ritz values will be shown. Executions have been performed on a Cray T3E

Ritz values and Arnoldi convergence for non-Hermitian matrices

This thesis develops ways of localizing the Ritz values of non-Hermitian matrices. The restarted Arnoldi method with exact shifts, useful for determining a few desired eigenvalues of a matrix, employs Ritz values to refine eigenvalue estimates. In the Hermitian case, using selected Ritz values produces convergence due to interlacing. No generalization of interlacing exists for non-Hermitian matrices, and as a consequence no satisfactory general convergence theory exists. To study Ritz values, I propose the inverse field of values problem for k Ritz values, which asks if a set of k complex numbers can be Ritz values of a matrix. This problem is always solvable for k = 1 for any complex number in the field of values; I provide an improved algorithm for finding a Ritz vector in this case. I show that majorization can be used to characterize, as well as localize, Ritz values. To illustrate the difficulties of characterizing Ritz values, this work provides a complete analysis of the Ritz values of two 3 × 3 matrices: a Jordan block and a normal matrix. By constructing conditions for localizing the Ritz values of a matrix with one simple, normal, sought-after eigenvalue, this work develops sufficient conditions that guarantee convergence of the restarted Arnoldi method with exact shifts. For general matrices, the conditions provide insight into the subspace dimensions that ensure that shifts do not cluster near the wanted eigenvalue. As Ritz values form the basis for many iterative methods for determining eigenvalues and solving linear systems, an understanding of Ritz value behavior for non-Hermitian matrices has the potential to inform a broad range of analysis

Absolute Variation of Ritz Values, Principal Angles, and Spectral Spread

Let A be a d×d complex self-adjoint matrix, X,Y⊂Cd be k-dimensional subspaces and let X be a d×k complex matrix whose columns form an orthonormal basis of X. We construct a d×k complex matrix Yr whose columns form an orthonormal basis of Y and obtain sharp upper bounds for the singular values s(X∗AX−Y∗rAYr) in terms of submajorization relations involving the principal angles between X and Y and the spectral spread of A. We apply these results to obtain sharp upper bounds for the absolute variation of the Ritz values of A associated with the subspaces X and Y, that partially confirm conjectures by Knyazev and Argentati.Fil: Massey, Pedro Gustavo. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; ArgentinaFil: Stojanoff, Demetrio. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; ArgentinaFil: Zarate, Sebastian Gonzalo. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentin