Optimal approximations in the eigenvalue problem for the Ritz and Bubnov–Galerkin methods

A method is described for finding the best (in a certain sense) approximations of the eigenvalues for linear operator equations of the type $Au=\lambda Bu$, 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 $\{\varphi_n\}$, it is required to find, among all the coordinate elements, the $k$ elements for which the divergence $\delta^{(k)}$ between the exact absolute value of the eigenvalue $|\lambda|$ and its $k$-th approximation $|\lambda^{(k)}|$ is minimal, i. e. $|\lambda^{(k)}|-|\lambda|=\min\delta^{(k)}$.