author: Jia, Zhongxiao
shorttitle: Orhogonal projection eigenproblems
source: Sci. China, Ser. A 42, No 6, 577-585 (1999)
rsclass: 49R50; 47A52; 65K05
keywords: Ritz method, orthogonal bases, refined projection
revtext: Within the Ritz or Galerkin variational framework, an efficient partial numerical diagonalization of a Hermitian operator A can be performed within a suitable truncated m-dimensional basis. In the parallel non-Hermitian analyses a key difficulty emerges in connection with the possible mathematical failure of the variational background of the whole scheme. An explanation can be provided when we compare the implementations of the recipe in the two different bases: The projection of the eigenvalue problem (generically, on a subspace of the whole Hilbert space) may happen to be badly ill-conditioned. The key idea of the author is to employ his/her own results on the general Arnoldi algorithm [cf., e.g., his/her latest publication in J. Comp. Math. 18 (2000) 265 - 276 (Zentralblatt abstract Nr. DE01507134X) for an updated reference], emphasizing the fact that a bad convergence of the eigenvectors themselves is usually the very core of computational difficulties (i.a., Wilkinson (ref. ) recommends the inverse iterations strongly). Here, an alternative ``refined" recipe is analysed, re-calculating the eigenvectors via a minimization technique. In such a setting the paper offers a number of estimates (a priori error bounds) and conditions of convergence.