Green equilibrium measures and representations of an external field
Journal of Approximation Theory
Stability of the notion of approximating class of sequences and applications
Journal of Computational and Applied Mathematics
A numerical solution of the constrained energy problem
Journal of Computational and Applied Mathematics
SIAM Journal on Matrix Analysis and Applications
On the Convergence of Rational Ritz Values
SIAM Journal on Matrix Analysis and Applications
An Error Analysis for Rational Galerkin Projection Applied to the Sylvester Equation
SIAM Journal on Numerical Analysis
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We give a theoretical explanation for superlinear convergence behavior observed while solving large symmetric systems of equations using the conjugate gradient method or other Krylov subspace methods. We present a new bound on the relative error after $n$ iterations. This bound is valid in an asymptotic sense when the size $N$ of the system grows together with the number of iterations. The bound depends on the asymptotic eigenvalue distribution and on the ratio $n/N$. Under appropriate conditions we show that the bound is asymptotically sharp.Our findings are related to some recent results concerning asymptotics of discrete orthogonal polynomials. An important tool in our investigations is a constrained energy problem in logarithmic potential theory.The new asymptotic bounds for the rate of convergence are illustrated by discussing Toeplitz systems as well as a model problem stemming from the discretization of the Poisson equation.