×

Faber versus minimal polynomials on annular sectors. (English) Zbl 0871.65010

Ali, Rosihan M. (ed.) et al., Computational methods and function theory 1994. Proceedings of the conference, Penang, Malaysia, March 21–25, 1994. Singapore: World Scientific. Ser. Approx. Decompos. 5, 243-265 (1995).
Summary: 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 least 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. Various Gauss-formulae with high knot numbers are employed. 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.
For the entire collection see [Zbl 0863.00033].

MSC:

65D20 Computation of special functions and constants, construction of tables
65E05 General theory of numerical methods in complex analysis (potential theory, etc.)
30C10 Polynomials and rational functions of one complex variable
65F10 Iterative numerical methods for linear systems
PDFBibTeX XMLCite