Abstract. We provide an algorithm to compute the nodes and weights for Gauss-Chebyshev quadrature formulas integrating exactly in spaces of rational functions with arbitrary real poles outside [−1, 1]. Contrary to existing rational quadrature formulas, the computational effort is very low, even for extremely high degrees, and under certain conditions on the poles it can be shown that the complexity is of order O(n). This method is based on the derivation of explicit expressions for Chebyshev orthogonal rational functions, which are (thus far) the only examples of explicitly known orthogonal rational functions on [−1, 1] with arbitrary real poles outside this interval.

In this paper we provide an extension of the Chebyshev orthogonal rational functions with arbitrary real poles outside [−1, 1] to arbitrary complex poles outside [−1, 1]. The zeros of these orthogonal rational functions are not necessarily real anymore. By using the related para-orthogonal functions, however, we obtain an expression for the nodes and weights for rational Gauss-Chebyshev quadrature formulas integrating exactly in spaces of rational functions with arbitrary complex poles outside [−1, 1].

A new method for interpolation by rational functions of prescribed numerator and denominator degrees is presented. When the interpolation nodes are roots of unity or Chebyshev points, the algorithm is particularly simple and relies on discrete Fourier transform matrices, which results in a fast implementation using the Fast Fourier Transform. The method is generalised for arbitrary grids, which requires the construction of polynomials orthogonal on the set of interpolation nodes. The appearance of common factors in the numerator and denominator due to finite precision arithmetic is explained by the behaviour of the singular values of the linear system associated to rational interpolation problem. The new algorithm has connections with other methods, particularly the work of Jacobi and Kronecker, Berrut and Mittelmann, and Egecioglu and Koç. Short codes in Matlab and numerical experiments are included.

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