Which Eigenvalues Are Found by the Lanczos Method?

When discussing the convergence properties of the Lanczos iteration method for the real symmetric eigenvalue problem, Trefethen and Bau noted that the Lanczos method tends to find eigenvalues in regions that have too little charge when compared to an equilibrium distribution. In this paper a quantitative version of this rule of thumb is presented. We describe, in an asymptotic sense, the region containing those eigenvalues that are well approximated by the Ritz values. The region depends on the distribution of eigenvalues and on the ratio between the size of the matrix and the number of iterations, and it is characterized by an extremal problem in potential theory which was first considered by Rakhmanov. We give examples showing the connection with the equilibrium distribution.