From numerical quadrature to Padé approximation

The paper reviews the relation between Padé-type approximants of a power series and interpolatory quadrature formulae with free nodes, and between Padé approximants and Gaussian quadrature methods. Then, it is shown how the Kronrod procedure and the anti-Gaussian quadrature methods could be used for estimating the error in Padé approximation. The ε-algorithm for accelerating the convergence of sequences, and computing recursively Padé approximants is evoked, and its error estimated by the same procedures. Finally, the case of series of functions is considered. Considerations on further research topics end the paper.