# Extremal Point Methods

• Edward B. Saff
• Vilmos Totik
Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 316)

## 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.

## Authors and Affiliations

• Edward B. Saff
• 1
• Vilmos Totik
• 2
• 3
1. 1.Department of MathematicsUniversity of South Florida Institute for Constructive MathematicsTampaUSA
2. 2.Bolyai InstituteJozsef Attila UniversitySzegedHungary
3. 3.Department of MathematicsUniversity of South FloridaTampaUSA