Sharpness in rates of convergence for the symmetric Lanczos method

The Lanczos method is often used to solve a large and sparse symmetric matrix eigenvalue problem. There is a well-established convergence theory that produces bounds to predict the rates of convergence good for a few extreme eigenpairs. These bounds suggest at least linear convergence in terms of the number of Lanczos steps, assuming there are gaps between individual eigenvalues. In practice, often superlinear convergence is observed. The question is “do the existing bounds tell the correct convergence rate in general?”. An affirmative answer is given here for the two extreme eigenvalues by examples whose Lanczos approximations have errors comparable to the error bounds for all Lanczos steps.