On the use of harmonic Ritz pairs in approximating internal eigenpairs (Q1855434)

The paper concerns real symmetric matrices and focuses on the extraction of a good approximation of an eigenvectors of an internal eigenpair from an iteratively obtained subspace. Harmonic Rayleight-Ritz was proposed in 1991 by \textit{R. B. Morgan} [Linear Algebra Appl. 154-156, 289-309 (1991; Zbl 0734.65029)] and named in 1995 by \textit{C. C. Paige}, \textit{B. N. Parlett} and \textit{H. A. van der Vorst} [Numer. Linear Algebra Appl. 2, No. 2, 115-133 (1995; Zbl 0831.65036)]. In this paper it is compared with refined Rayleigh-Ritz (a block Arnoldi process) in the sense of \textit{Z. Jia} [Linear Algebra Appl. 270, 171-189 (1998; Zbl 0896.65035)], with emphasis on insight rather than on algorithms throughout this paper. Further topics studied for the same purpose are error analysis (a-priori as well as a-posteriori) and the selection of a suitable harmonic Ritz vector with respect to the shift.