The convergence of harmonic Ritz values, harmonic Ritz vectors and refined harmonic Ritz vectors

Abstract

Abstract. 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,x) to (λ, x). Three convergence results are established as the deviation of x from W approaches zero. First, the harmonic Ritz value µ + τ converges to λ if a certain Rayleigh quotient matrix is uniformly nonsingular. Second, the harmonic Ritz vector x converges to x if the Rayleigh quotient matrix is uniformly nonsingular and µ + τ remains well separated from the other harmonic Ritz values. Third, better error bounds for the convergence of µ + τ are derived whenx converges. However, we show that the harmonic projection method can fail to find the desired eigenvalue λ-in other words, the method can miss λ if it is very close to τ . To this end, we propose to compute the Rayleigh quotient ρ of A with respect tox and take it as a new approximate eigenvalue. ρ is shown to converge to λ oncex tends to x, no matter how τ is close to λ. Finally, we show that if the Rayleigh quotient matrix is uniformly nonsingular, then the refined harmonic Ritz vector, or more generally the refined eigenvector approximation introduced by the author, converges. We construct examples to illustrate our theory.

A global harmonic Arnoldi method for large non-Hermitian eigenproblems with an application to multiple eigenvalue problems

“…This shows that the global harmonic Arnoldi method inherits convergence properties of the usual harmonic Arnoldi method. However, similar to the harmonic projection methods [21], we will see that the global harmonic Arnoldi method may fail to find a desired eigenvalue λ if it is very close to τ ; that is, the method may miss λ if it is very close to τ . To this end, we propose computing the F -Rayleigh quotient of A with respect to the harmonic F -Ritz vector as a new approximate eigenvalue.…”

Computation of Complex Eigenmodes for Resonators Filled With Gyrotropic Materials

“…For the computation of extremal eigenvalues, the well-known Rayleigh-Ritz approach [51,80,81] is recommendable. Imposing the Galerkin-condition • The Ritz values θ i are optimal approximations of the eigenvalues of A in the given subspace [73].…”

“…Nonetheless, if the residual norm r i approaches zero, convergence of the Ritz vector to an eigenvector is ensured [81].…”

“…By requiring W to be an orthonormal system, it can be reduced to the standard eigenvalue problem [74] , which (under certain conditions, see [84]) converge towards eigenvalues of A near the target τ. Note also that it is not necessary to explicitly invert the matrix (A − τ1) as one can compute the system matrix of the small projected problem from the right hand side of equation (3.153) provided that the orthogonal transformations performed on W are also applied to V in order to maintain the relation W = (A − τ1) V [74].…”

Harmonic and refined Rayleigh–Ritz for the polynomial eigenvalue problem

“…We note that if x is part of a larger subspace U the harmonic approach may still have difficulties selecting the right vector if there are (nearly) equal harmonic Ritz values, see [8]. This, however, rarely forms a problem in practical processes, since we can just continue the subspace method by expanding the search space and performing a new extraction process.…”