Operator Pade Approximants: Some ideas behind the theory and a numerical illustration

Abstract

Section 1 will be devoted to the discussion of some generalizations of the concept of Pade-approximant for multivariate functions, based on the interpolation property of a Pade-approximant. Most of those generalizations preserve, under some conditions, a number of properties of the univariate Pade-approximant. In Section 2 we will repeat the recursive schemes used for the computation of the univariate Pade-approximant: the ε-algorithm and the qd-algorithm. We will also show that, if a generalizing definition is based on these recursive algorithms, then much more interesting properties remain valid for the generalization. In Section 3 we will illustrate the approximation power of this type of Pade approximants on a numerical example. Other applications are: the solution of nonlinear systems of equations [7], the solution of nonlinear differential and integral equations [3], the acceleration of convergence [4]. Since those applications have already been treated extensively, they will not be mentioned here; the interested reader is referred to the literature.