The conformal `bratwurst' maps and associated Faber polynomials
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.
scientific article; zbMATH DE number 1491756