On Clenshaws's method and a generalisation to faber series

Abstract

Summary. The method of expanding the solution of linear ordinary differential equations in power or Laurent series is classical, and is usually associated with the name of Frobenius. In the early days of electronic computation, it was appreciated that expansions in Chebyshev series are often of more practical use, and the necessary techniques developed by Clenshaw. (This is usually carried out in order to approximate special functions defined by ordinary differential equations, rather than as a technique for actually solving such equations in general, for which finite difference methods are to be preferred.) In this paper we show that by means of a conformal map the problem of expansion in Chebyshev series can in fact be reduced to that of expansion in a Laurent series, yielding a method which is usually computationally simpler. Moreover, the method can be generalised to the case of Faber expansions on regions of the complex plane. A non-trivial example is explored in order to illustrate the method, and we also use the technique to generalise an identity relating Chebyshev polynomials to the Faber case.

A numerical method for the computation of Faber polynomials for starlike domains

“…They choose, instead, to compute the coefficients of the Faber polynomials directly from a standard integral representation that makes use of their formula for ψ. ( We note, however, that their recurrence formula was subsequently used by Ellacott and Saff [6], for computing Faber polynomials in circular sectors, by means of (1.3).) In [5], Ellacott suggests computing the coefficients of Faber polynomials, from their integral representation in terms of the mapping function Ф, by using the fast Fourier transform (FFT).…”