Many engineering and scientific applications require the computation of eigenvalues (and eigenvectors) of very large symmetric or Hermitian matrices. We describe a Lanczos procedure which allows us to compute either few or many eigenvalues of such matrices in any intervals specified by the user. This procedure can even be used to compute all of the eigenvalues. The desired eigenvalues are computed as eigenvalues of an associated symmetric tridiagonal matrix Tm. whose order depends upon the distribution of the eigenvalues in the given matrix A and upon which portions of the spectrum of A are desired. The storage requirements depend linearly upon the order of A, if the the storage required to generate the products Ax is also linear in the order. The amount of computation required depends directly upon the distribution of the desired eigenvalues and upon the cost of computing Ax. Numerical results for a very large matrix of order 4900 demonstrate that this procedure can be used on very large matrices. © 1981.