On the calculation of eigenvalues of a symmetric matrix

An algorithm is proposed for a relatively fast and highly accurate calculation of several eigenvalues and the corresponding eigenvectors of a large symmetric matrix. Results of numerical experiments are presented in which the nine lowest eigenvalues were calculated for the minus-Laplace operator with zero boundary conditions discretized on various two-dimensional regions using the five-point stencil and a grid with the number of nodes exceeding one million. The calculation of a part of the spectrum of an arbitrary square matrix is discussed.