Chebyshev rational interpolation (Q1370341)

Denote by \(E_\rho\), \(0 \leq \rho \leq 1\), the closed ellipse in \(\mathbb C\) with the boundary \[ \partial E_\rho = \{ t+ \rho t^{-1}:\;|t|=1 \}, \] and \[ s_j(z)=w^j+(\rho/w)^j, \quad z=w+\rho/w, \quad j=2, \dots, \] the corresponding Chebyshev polynomials. The paper faces the problem of the efficient computation of all the rational functions \[ \displaystyle \sum_{i=0}^m a_i s_i(z) \over \displaystyle \sum_{i=0}^n b_i s_i(z) \tag{*} \] which assume given values \(f_0,\dots,f_{n+m}\) (\(f_i \neq 0\)) at pairwise distinct points \(z_i, 0 \leq i \leq n+m\). A modification of a general look-ahead version of the Lanczos procedure in order to compute the coefficients of the denominator of the interpolating function (*) is derived; the same algorithm is applied to the interpolation data \((z_i,1/f_i)\), which gives the coefficients of the numerator of (*). The proposed approach is particularly suited for the rational interpolation at nodes on the boundary of \(E_\rho\). Several computational examples are considered, with \(f_i\) generated at random or representing values of a function analytic in \(E_\rho\).