Experiences with a Lanczos Eigensolver in High-Precision Arithmetic

We investigate the behavior of the Lanczos process when it is used to find all the eigenvalues of large sparse symmetric matrices. We study the convergence of classical Lanczos (i.e., without reorthogonalization) to the point where there is a cluster of Ritz values around each eigenvalue of the input matrix A. At that point, convergence to all the eigenvalues can be ascertained if A has no multiple eigenvalues. To eliminate multiple eigenvalues, we disperse them by adding to A a random matrix with a small norm; using high-precision arithmetic, we can perturb the eigenvalues and still produce accurate double-precision results. Our experiments indicate that the speed with which Ritz clusters form depends on the local density of eigenvalues and on the unit roundoff, which implies that we can accelerate convergence by using high-precision arithmetic in computations involving the Lanczos iterates.