Extremal Point Methods

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 % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfKttLearuavP1wzZbItLDhis9wBH5garm % Wu51MyVXgaruWqVvNCPvMCaebbnrfifHhDYfgasaacH8srps0lbbf9 % q8WrFfeuY-Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0-yr0RYxir % -Jbba9q8aq0-yq-He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGa % aeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbiabcYha8naabmaapaqaa8 % qacqWG6bGEcqGHsislcqWGHbqypaWaaSbaaSqaa8qacqaIWaama8aa % beaaaOWdbiaawIcacaGLPaaadaqadaWdaeaapeGaemOEaONaeyOeI0 % Iaemyyae2damaaBaaaleaapeGaeGymaedapaqabaaak8qacaGLOaGa % ayzkaaGaeS47IW0aaeWaa8aabaWdbiabdQha6jabgkHiTiabdggaH9 % aadaWgaaWcbaWdbiabd6gaUjabgkHiTiabigdaXaWdaeqaaaGcpeGa % ayjkaiaawMcaaiabeM8a3naabmaapaqaa8qacqWG6bGEaiaawIcaca % GLPaaapaWaaWbaaSqabeaapeGaemOBa4gaaOGaeiiFaWhaaa!51BD! $$ |\left( {z - {{a}_{0}}} \right)\left( {z - {{a}_{1}}} \right) \cdots \left( {z - {{a}_{{n - 1}}}} \right)\omega {{\left( z \right)}^{n}}| $$ 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.