2016/03/06 by Gang Wu, Wu, Gang
Computer Science · Mathematics · Physics and Astronomy · #65F10 #65F15 #Advanced Optimization Algorithms Research #Electromagnetic Scattering and Analysis #FOS: Mathematics #Matrix Theory and Algorithms #Numerical Analysis (math.NA)
paper · pdf · doi:10.48550/arxiv.1603.01785
openalex publication_date 2016/03/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We are interested in computing a simple eigenpair (λ,\bf x) of a large non-Hermitian matrix A, by a general harmonic Rayleigh-Ritz projection method. Given a search subspace K and a target point τ, we focus on the convergence of the harmonic Ritz vector \widetilde\bf x and harmonic Ritz value \widetildeλ. In [Z. Jia, \em The convergence of harmonic Ritz values, harmonic Ritz vectors, and refined harmonic Ritz vectors, Math. Comput., 74 (2004), pp. 1441--1456.], Jia showed that for the convergence of harmonic Ritz vector and harmonic Ritz value, it is essential to assume certain Rayleigh quotient matrix being \it uniformly nonsingular as ∠(\bf x,K)→ 0. However, this assumption can not be guaranteed theoretically for a general matrix A, and the Rayleigh quotient matrix can be singular or near singular even if τ is not close to λ. In this paper, we abolish this constraint and derive new bounds for the convergence of harmonic Rayleigh-Ritz projection methods. We show that as the distance between \bf x and K tends to zero and τ is satisfied with the so-called \it uniform separation condition, the harmonic Ritz value converges, and the harmonic Ritz vector converges as (1)/(λ-τ) is well separated from other Ritz values of (A-τI)-1 in the orthogonal complement of (A-τI)\widetilde\bf x with respect to (A-τI)K.