Abstract

While discussing the convergence of the Lanczos method, Trefethen and Bau observed a relationship with electric charge distributions, and claimed that the Lanczos iteration tends to converge to eigenvalues in regions of ``too little charge'' for an equilibrium distribution. Recently, Kuijlaars found a theoretical justification for this phenomenon by considering the Lanczos method applied to a suitable sequence of matrices with similar spectra which may occur for instance in the discretization of PDEs while varying the parameter of discretization. The aim of the present note is to improve the result of Kuijlaars: we obtain a better rate of convergence under weaker regularity assumptions, and show that this new rate of convergence is sharp.