The convergence of harmonic Ritz values, harmonic Ritz vectors and refined harmonic Ritz vectors
Zhongxiao Jia · Mathematics of Computation · 2004
This paper concerns a harmonic projection method for computing an approximation to an eigenpair ( λ , x ) (\lambda , x) of a large matrix A A . Given a target point τ \tau and a subspace W \mathcal {W} that contains an approximation to x x , the harmonic projection method returns an approximation ( μ + τ , x ~ ) (\mu +\tau , \tilde x) to ( λ , x ) (\lambda ,x) . Three convergence results are established as the deviation ϵ \epsilon of x x from W \mathcal {W} approaches zero. First, the harmonic Ritz value μ + τ \mu +\tau converges to λ \lambda if a certain Rayleigh quotient matrix is uniformly nonsingular. Second, the harmonic Ritz vector x ~ \tilde x converges to