On the computation of the eigenvalues of a symmetric matrix

Let A be a symmetric matrix of dimension N. The subject of this paper is a method for numerically approximating the mleq N consecutive smallest eigenvalues and corresponding eigenvectors of A. The method uses a special algorithm proposed by the author for computing the bounds of the spectrum of a symmetric matrix [see M. F. Sukhinin, Zh. Vychisl. Mat. Mat. Fiz. {bf 42} (2002), no.~11, 1619--1623; [msn] MR1967962 (2004b:15037) [/msn]]. par First, the smallest and the largest eigenvalues of A, lambda_1 and lambda_N, have to be computed with the help of the method mentioned above. Having lambda_1 and lambda_N, the "shift" lambda=lambda_N+|lambda_N|+{lambda_N-lambda_1} is to be computed. Suppose we know the eigenvalues lambda_1,lambda_2,dots,lambda_{k-1} and the corresponding orthonormal eigenvectors e_1,e_2,dots,e_{k-1}. Now we can compute the constants M_j=sqrt{lambda-lambda_j}, j=1,2,dots,k-1, and the new "deflated" matrix A^{(k)}=A+sum_{j=1}^{k-1}a_ja^*_j, where a_j=M_je_j. The eigenvalue lambda_k is now the smallest eigenvalue of A^{(k)} and has to be computed together with the corresponding eigenvector e_k with the help of the author's method. par The paper contains six numerical examples of the discrete Laplacian with homogeneous boundary conditions on six different plane domains. par Certain suggestions concerning the case of an arbitrary complex matrix A are added.