A review of multivariate Padé approximation theory
It is well known that univariate Padé approximants (PA), which are a special case of univariate rational interpolants (RI) by letting all the interpolation points coincide, can be obtained in several equivalent ways: one can write down the system of linear equations that must be satisfied by the numerator and denominator coefficients, one can start a recursive computation scheme, one can consider convergents of corresponding continued fractions. Each of these three defining techniques has been generalized to the multivariate case. The equivalence between the defining techniques is however lost by the way in which the generalization was formulated. When the technique of the defining equations or the continued fraction approach is used to define multivariate Padé approximants (MPA), then the set of equations or the form of the continued fraction can be chosen so that many of the univariate properties carry over to the multivariate case. In fact you can force your approximant to satisfy a certain property by adding the right equations to your system or the right terms to your continued fraction. But it is not possible to give a linear system and a continued fraction expansion that generate the same rational approximant. Depending on what sort of approximant you want, you have to make your choice. On the other hand, if a recursive scheme is the starting point for the generalization, then it is possible to establish a link with the other two approaches. Very recently the author could prove that the MPA defined by means of a linear system of equations for their numerator and denominator coefficients, can after all be obtained as the successive convergents of a continued fraction, but then this continued fraction has a somewhat different form than the ones used up to now in the multivariate theory: the continued fraction is not branched. Also the author has generalized the recursive epsilon-algorithm so that MPA that result from a defining system of linear equations can now be computed recursively [Multivariate Padé approximants revisited. BIT, to appear]. It is to be expected that the equivalence of these three main defining techniques creates a whole lot of new algorithmic possibilities.
scientific article; zbMATH DE number 3912938