Nested multivariate Padé approximants

A new class of multivariate Padé approximants is introduced. When dealing with two variables x and y, the approach consists in applying the Padé approximation with respect to y to the coefficients of the Padé approximation with respect to x. This technique has a natural extension to n variables, and is well adapted to programming in any usual language. The advantage of the method is that the algorithm uses only univariate Padé approximation. As a consequence, only small systems need to be solved, and the computation of the approximant is much faster: if M is the number of derivatives with respect to x and y, the algorithm requires O(M4) operations instead of O(M6) when matrix methods are used for solving the coefficients problem, or O(M3) operations instead of O(M4) if staircase methods are used. Another important feature is that the reliable methods which have been developed for choosing the degrees of the numerator and the denominator in the univariate case can be straightforwardly used in the multivariate case.