The conformal `bratwurst' maps and associated Faber polynomials
type: Item
instance of: scholarly article
title: The conformal `bratwurst' maps and associated Faber polynomials
zbMATH Open document ID: 0960.30006
author: Tino Koch
author: Jörg Liesen
published in: Numerische Mathematik
publication date: 2001-05-20T00:00:00Z
review text: In the construction of polynomial iteration methods for the solution of a linear system \(Ax= b\), it is necessary to find inclusion sets \(\Omega\) which contain the spectrum of \(A\) but \(0\not\in\Omega\). The main object of this paper is to construct explicitly a class of non-convex inclusion sets. These are obtained by deforming an ellipse with foci at \(\pm 1\) so that the real axis is mapped onto the unit circle. The map \(\psi\) of \(\{w:|w|> 1\}\) onto the complement of \(\Omega\) is of the form \(\psi= \psi_2\circ J\circ \psi_1\) where \(\psi_1\) and \(\psi_2\) are linear maps and \(J\) is a Joukowski map. The coefficients of the expansion of \(\psi\) at \(\infty\) can be given explicitly from which the Faber polynomials for \(\Omega\) (suitably normalized) follow. The transfinite diameter of \(\Omega\) and the asymptotic convergence factor of the iteration method are also given explicitly. Several numerical examples are given.
Mathematics Subject Classification ID: 30C20
Mathematics Subject Classification ID: 65F10
zbMATH DE Number: 1491756
zbMATH Keywords: iteration methods
zbMATH Keywords: Faber polynomials
zbMATH Keywords: Joukowski map
reviewed by: Dieter Gaier
MaRDI profile type: Publication
DOI: 10.1007/S002110000141
description: scientific article; zbMATH DE number 1491756