On the sharpness of the weighted Bernstein-Walsh inequality, with applications to the superlinear convergence of conjugate gradients

TL;DR

This paper proves the sharpness of the weighted Bernstein-Walsh inequality with a new discretization technique, and applies it to establish convergence rates for conjugate gradient methods based on eigenvalue distributions.

Contribution

It introduces a novel discretization method for logarithmic potentials and confirms a conjecture relating Green functions to convergence rates in conjugate gradient algorithms.

Findings

Weighted Bernstein-Walsh inequality is sharp up to a universal constant.

A new mean value property for cumulative distribution functions is developed.

Convergence rate inequalities are established for specific eigenvalue distributions.

Abstract

In this paper we show that the weighted Bernstein-Walsh inequality in logarithmic potential theory is sharp up to some new universal constant, provided that the external field is given by a logarithmic potential. Our main tool for such results is a new technique of discretization of logarithmic potentials, where we take the same starting point as in earlier work of Totik and of Levin \& Lubinsky, but add an important new ingredient, namely some new mean value property for the cumulative distribution function of the underlying measure. As an application, we revisit the work of Beckermann \& Kuijlaars on the superlinear convergence of conjugate gradients. These authors have determined the asymptotic convergence factor for sequences of systems of linear equations with an asymptotic eigenvalue distribution. There was some numerical evidence to let conjecture that the integral mean of Green functions occurring in their work should also allow to give inequalities for the rate of convergence if one makes a suitable link between measures and the eigenvalues of a single matrix of coefficients. We prove this conjecture , at least for a class of measures which is of particular interest for applications.