A Multivariate Qd-like Algorithm

The problem of constructing a univariate rational interpolant or Pad~ approximant for given data can be solved in various equivMent ways: one can compute the explicit solution of the system of interpolation or approximation conditions, or one can start a recursive algorithm, or one can obtain the rational function as the convergent of an interpolating or corresponding continued fraction. In ease of multivariate functions general order systems of interpolation conditions for a multivariate rational interpolant and general order systems of approximation conditions for a multivariate Pad~ approximant were respectively solved in [6] and [9]. Equivalent recursive computation schemes were given in [3] for the rational interpolation case and in [5] for the Pad6 approximation ease. At that moment we stated that the next step was to write the general order rational interpolants and Pad6 approximants as the convergent of a multivariate continued fraction so that the univariate equivalence of the three main defining techniques was also established for the multivariate ease: algebraic relations, recurrence relations, continued fractions. In this paper a multivariate qd-like algorithm is developed that serves this purpose. AMS Subject classification8 : 65D05, 41A21. 1. A l g e b r a i c r e l a t i o n s a n d R e c u r r e n c e r e l a t i o n s . Let us restrict everything to the case of two variables for the sake of simplicity. Fur the rmore we assume tha t the finite in te rpola t ion set I = {(i,j)lfij is given a t (z~, yj)} is s t ruc tured so tha t it satisfies the inclusion property. This means .that if a po in t belongs to the da t a set, t hen the rec tangular subset of points Received October 1986. Revised November 1987. *) Senior Research Assis tant N F W O A MULTIVARIATE QD-LIKE ALGORITHM 99 emanating from the origin with the given point as its furthermost corner also lies in the data set. How this can be achieved in a lot of situations is explained in [6]. If none of the points in {(zl ,Yj)}( i j )ei coincide then we are dealing with a rational interpolation problem and the values in {flJ}(id)ei are function values. If all the interpolation points coincide then the problem is one of Pad4 approximation and it is well-known that the given data are not function values but Taylor coefficients. If some of the points coincide and some do not then the problem is of a mixed type and it is called a Hermite interpolation problem or a Newton-Pad~ approximation problem. In [6] is indicated how one should interpret the data flj: some of them are derivatives and some of them are function values. In the sequel of the text it does not play a role whether one is dealing with coalescent points or not since all the formulas remain valid in both cases and in the mixed case. Nevertheless we shall sometimes indicate how the formulas are to be read if some of the interpolation points coincide. Consider the following set of basis functions for the real-valued polynomials in two variables i-1 i-1 Bij(z,V) = I I ( x xk) I ] ( Y Y')" k=O t = O Clearly Bij(x, y) is a bivariate polynomial of degree i + j . Given the fir, we can write in a purely formal manner f ( z , y ) = E foi,ojBij(z,y) ( i , j ) e / V ~ where foi,oj are the bivariate divided differences Yo~,o~ = / [ so , . . . , s~] [yo,. • . , ys] given by y [~o, . . . , :~d [yo,. . , us] = "f [ * ~ " ' " sd [yo, . . . , yj] / [~o , . . . , s~_~] [yo . . . . , ys] X i ~ 0