The Faber polynomials for annular sectors and an application to the iterative solution of linear systems of equations

Abstract

A conformal mapping of the exterior of the unit circle to the exterior of a region of the complex plane determines the Faber polynomials for that region. These polynomials are of interest in providing near-optimal polynomial approximations in a wide variety of contexts. The work of this thesis concerns the Faber polynomials for an annular sector {z : R ≤ |z| ≤ 1,0 ≤ | arg z| ≤ π}, with 0 < 0 < π and is contained in two main parts. In the first part the required conformal map is derived, and the first few Faber polynomials for the annular sector are given in terms of the transfinite diameter, p, of the region and two parameters a and b. These three numbers are determined numerically. We also give the Faber series for 1/z and improve upon a bound given in the literature for the norm of the Faber projection, ||xn||- In the second part of the thesis we give a new hybrid method for the iterative solution of linear systems of equations, Ax = b, where the coefficient matrix, A, is large, sparse, nonsingular and non-Hermitian. The method begins with a few steps of the Arnoldi method to produce some information on the location of the spectrum of A. Our method then switches to an iterative method based on the Faber polynomials for an annular sector placed around these eigenvalue estimates. An annular sector is thought to be a useful region because it can be scaled and rotated to enclose any eigenvalue estimates bounded away from zero. Some examples will be exhibited and we will compare existing methods with ours