Tracking the Progress of the Lanczos Algorithm for Large Symmetric Eigenproblems

A reliable and efficient algorithm for finding all or part ofthe 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 tridiagonalmatrices towards the eigenvalues of A. Rather than using routinesfor computing eigenvalues of tridiagonal matrices, we run recurrenceson sets of points within and near the wanted part of the spectrum.Interpolation procedures are used at these points in order toestimate the actual positions of eigenvalues. New points areadded and old ones are discarded according to the way the eigenvaluesconverge. The goal is to recognize convergence automaticallyand keep small the number of Lanczos steps, each of which demandsaccess to the whole of the large matrix A. Results of computerruns are reported.