On optimal approximations in the eigenvalue problem for the Ritz and Bubnov-Galerkin methods

Abstract

A method is described for finding the best (in a certain sense) approximations of the eigenvalues for linear operator equations of the type Au=λBu , when they are solved by the Ritz and the Bubnov-Galerkin methods. The problem of optimal approximations is stated thus: given the system of coordinate functions { n }, it is required to find, among all the coordinate elements, the k elements for which the divergence δ (k) between the exact absolute value of the eigenvalue |λ| and its k -th approximation |λ (k) | is minimal, i. e. |λ (k) −|λ|=minδ (k) .