in this paper for a given prescribed ritz values that satisfy in the some special conditions, we nd a symmetric nonnegative matrix, such that the given set be its ritz values.

Many inequality relations between real vector quantities can be succinctly expressed as “weak (sub)majorization” relations using the symbol ≺w. We explain these ideas and apply them in several areas: angles between subspaces, Ritz values, and graph Laplacian spectra, which we show are all surprisingly related. Let Θ(X ,Y) be the vector of principal angles in nondecreasing order between subspace...

FOM and GMRES are Krylov subspace iterative methods for solving nonsymmetric linear systems. The Ritz values are approximate eigenvalues, which can be computed cheaply within these algorithms. In this paper, we generalise the concept of Harmonic Ritz values, introduced by Paige et al. for symmetric matrices, to nonsymmetric matrices. We show that the zeroes of the residual polynomials of FOM an...

New bounds on the canonical angles between an invariant subspace of A and an approximating subspace by the differences between Ritz values and the targeted eigenvalues are obtained. From this result, various bounds are readily available to estimate how accurate the Ritz vectors computed from the approximating subspace may be, based on information on approximation accuracies in the Ritz values. ...

Rayleigh–Ritz eigenvalue estimates for Hermitian matrices obey Cauchy interlacing, which has helpful implications for theory, applications, and algorithms. In contrast, few results about the Ritz values of non-Hermitian matrices are known, beyond their containment within the numerical range. To show that such Ritz values enjoy considerable structure, we establish regions within the numerical ra...

Eigenvalue estimates that are optimal in some sense have selfevident appeal and leave estimators with a sense of virtue and economy. So, it is natural that ongoing searches for effective strategies for difficult tasks such as estimating matrix eigenvalues that are situated well into the interior of the spectrum revisit from time to time methods that are known to yield optimal bounds. This artic...

This paper concerns a harmonic projection method for computing an approximation to an eigenpair (λ, x) of a large matrix A. Given a target point τ and a subspace W that contains an approximation to x, the harmonic projection method returns an approximation (μ + τ, x̃) to (λ, x). Three convergence results are established as the deviation of x from W approaches zero. First, the harmonic Ritz value...