From numerical quadrature to Padé approximation

The paper reviews the relation between Padé-type approximants of a power series and interpolatory quadrature formulas with free nodes, and that between Padé approximants and Gaussian quadrature methods. Quadrature methods are well-known. They are used for obtaining an approximate value of a definite integral, and are described in any book of numerical analysis. In this talk, we will show that Padé–type approximants could be interpreted as quadrature formulas with free nodes for the special function g(x) = 1/(1 − xt), and that Padé approximants are, in fact, Gaussian quadratures for the same function g. Thus Kronrod procedure and antiGaussian quadrature formulas could be used for estimating their accuracy. Then, Padé approximation for series of functions will be discussed. The talk will end by some perspectives for future researches.