Fast and Stable Rational Interpolation in Roots of Unity and Chebyshev Points

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 Eg̃eciog̃lu and Koç. Short codes in Matlab and numerical experiments are included.