Computation of polynomial and rational approximations in complex domains by the $$\tau $$ -method

We investigate numerical methods for computation of polynomial and rational approximations of functions in complex domains based on Faber polynomials and the Lanczos \(\tau \)-method. Our interest is motivated by applications in fractional partial differential equations. We give an overview of previous results related to the basis of Faber polynomials associated with a complex domain, Faber expansion, and the Lanczos \(\tau \)-method. We also collect numerical algorithms for the computational realization of these concepts. Our main new contribution is a \(\tau \)-method for rational approximation in complex domains which uses Faber polynomials in the perturbation term. We realize it via a novel hybrid symbolic-numeric algorithm which can be applied to arbitrary functions satisfying a suitable differential equation. We present some numerical examples, where we use sectors lying in the complex plane as our domains of interest. We compare results for the various polynomial and rational approximation techniques outlined above; in particular, we observe exponential convergence with respect to the rational degree for our proposed method.