Bernhard Beckermann
Laboratoire d'Analyse Num'erique et d'Optimisation
UFR IEEA  M3, UST Lille, F59655 Villeneuve d'Ascq CEDEX,
France
email: bbecker@ano.univlille1.fr
Given a sequence of uniformly bounded discrete sets E_{N} (N = 0,1,...) in the complex plane, # E_{N} = N, we will study the functions

Our work is partially motivated by the investigation of the rate of convergence of the conjugate gradient method (CG) applied to large symmetric positive systems A_{N} x = b_{N}: it is wellknown that the relative error in the energy norm at the nth iteration is bounded above by 1/F_{n}(0,E_{N}), where E_{N} is the spectrum of A_{N}. As a typical model problem, for the 2D discretized Poisson equation on a uniform grid one obtains after rescaling as asymptotic eigenvalue distribution the complete elliptic integral

In the present talk we will give lower and upper bounds for the nth root limit of f_{n}(z,E_{N}) in terms of the unique measure m_{t} which under all probability measures m satisfying the constraint tm ú s has minimal logarithmic energy I(m). This constrained energy problem has been investigated by Rakhmanov (1996), DragnevSaff (1997), and several others in connection with the asymptotic behavior of discrete orthogonal polynomials. Our lower bound is valid for all complex z which do not attract ``too many'' of the elements of the sets E_{N}. For establishing upper bounds, we need to impose some separation condition for the elements of E_{N}. Here we show that it is sufficient to require that the discrete logarithmic energy of the sets E_{N} tends to I(s) for N « Ñ, as conjectured by Rakhmanov (1998). An important tool in our investigation are discrete Fekete points, i.e., subsets of E_{N} which have minimal discrete logarithmic energy. We finally show that the new resulting CG error bounds fit quite well the corresponding CG error curves for a number of examples.
References:
[1] B. Beckermann, On a conjecture of E.A. Rakhmanov.
To appear in Constructive Approximation (1999).
[2] B. Beckermann & A.B.J. Kuijlaars, Superlinear Convergence of Conjugate Gradients. Manuscript (1999).