Faber versus minimal polynomials on annular sectors

We are investigating the size of minimal polynomials and Faber polynomials on annular sectors. The size is the uniform norm of the corresponding polynomials on the underlying sector. If the size is less than one, the corresponding polynomial is suitable for the use in so-called polynomial-based-iteration-schemes for solving linear systems. For intrinsic reasons all polynomials p to be considered must satisfy the normalization condition p(0) = 1. The annular sectors play the role of an inclusion set for the eigenvalues of the underlying matrix in the linear system to be solved. The minimal polynomials are those with less uniform norm in this class. The computation of minimal polynomials requires a Remez- type algorithm for complex cases. Faber polynomials are computed by means of Schwarz-Christoffel-formulae. Here singular integrals of different types occur. The annular sectors have the remarkable property that they are invariant under monomial mappings, which also allows the use of several unconnected sectors of the same size distributed equiangularly over the annulus. This is of great help in certain applications, where the mentioned eigenvalues are located in a disconnected set. There are comparisons in graphical form of both types of polynomials with respect to degree and to the different types of sectors. Most of the numerical work is included in these figures. (orig.)Available from TIB Hannover: RN 5999(94) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman