Tracking the Progress of the Lanczos Algorithm for Large Symmetric Eigenproblems

A reliable and efficient algorithm for finding all or part of the spectrum (set of eigenvalues without regard to multiplicity) of a large symmetric matrix A may be based on the Lanczos algorithm, by tracking the progress of the eigenvalues of the Lanczos tridiagonal matrices towards the eigenvalues of A. Rather than using routines for computing eigenvalues of tridiagonal matrices, we run recurrences on sets of points within and near the wanted part of the spectrum. Interpolation procedures are used at these points in order to estimate the actual positions of eigenvalues. New points are added and old ones are discarded according to the way the eigenvalues converge. The goal is to recognize convergence automatically and keep small the number of Lanczos steps, each of which demands access to the whole of the large matrix A. Results of computer runs are reported.