Summary: Krylov subspace iterations are among the best-known and most widely used numerical methods for solving linear systems of equations and for computing eigenvalues of large matrices. These methods are polynomial methods whose convergence behavior is related to the behavior of polynomials on the spectrum of the matrix. This leads to an extremal problem in polynomial approximation theory: How small can a monic polynomial of a given degree be on the spectrum?
This survey gives an introduction to a recently developed technique to analyze this extremal problem in the case of symmetric matrices. It is based on global information on the spectrum in the sense that the eigenvalues are assumed to be distributed according to a certain measure. Then, depending on the number of iterations, the Lanczos method for the calculation of eigenvalues finds those eigenvalues that lie in a certain region, which is characterized by means of a constrained equilibrium problem from potential theory. The same constrained equilibrium problem also describes the superlinear convergence of conjugate gradients and other iterative methods for solving linear systems.