Chebyshev rational interpolation

The Lanczos method and its variants can be used to solve efficiently the rational interpolation problem. In this paper we present a suitable fast modification of a general look-ahead version of the Lanczos process in order to deal with polynomials expressed in the Chebyshev orthogonal basis. The proposed approach is particularly suited for approximating analytic functions by means of rational interpolation at certain nodes located on the boundary of an elliptical region of the complex plane. In fact, in this case it overcomes some of the numerical difficulties which limited the applicability of the look-ahead Lanczos process for determining the coefficients both of the numerators and of the denominators with respect to the standard power basis.