Extremal Point Methods

Abstract

The fact that the weighted equilibrium potential simultaneously solves a certain Dirichlet problem on connected components of C\S _w coupled with the fact that the Fekete points are distributed according to the equilibrium distribution leads to a numerical method for determining Dirichlet solutions. However, the determination of the Fekete points is a hard problem, so first we consider an associated sequence a _n that is adaptively generated from earlier points according to the law: a _n is a point where the weighted polynomial expression

$$ |\left( {z - {{a}_{0}}} \right)\left( {z - {{a}_{1}}} \right) \cdots \left( {z - {{a}_{{n - 1}}}} \right)\omega {{\left( z \right)}^{n}}| $$

takes its maximum on ∑. These so-called Leja points are again distributed like the equilibrium distribution, so we can use them in place of weighted Fekete points.