The Harmonic Ritz Values of A from the Subspace Spanned by Those Wanted Refined Harmonic Ritz Vectors

In this paper,it was studied how to choose the approximated eigenvalues for the harmonic Rayleigh-Ritz projection.Generally,the refined harmonic Ritz vectors have minimizing residual norms and the subspace spanned by the refined harmonic Ritz vectors shall include more information about the desired eigenvectors,so the harmonic Ritz values in this subspace shall approach the desired eigenvalues better.Guided with this idea,this paper investigates how to compute the harmonic Ritz values θ_i of A from the subspace spanned by the refined Ritz vectors and use them as the approximation to those desired eigenvalues.For a Krylov subspace,a priori theoretical error bounds between θ_i and the harmonic Ritz values are given.Finally,we give a new refined harmonic Arnoldi algorithm used θ_i,(i=1,2,…,e),as the approximated eigenvalues and carry on the numerical experments.Numerical results confirm efficiency of the new algorithm.