Singular rules for a multivariate quotient-difference algorithm

Abstract

Using the multivariateqdg-algorithm developed in [5], it is possible to compute the partial numerators and denominators of a continued fraction representation associated with a descending staircase in a table of multivariate rational interpolants, more precisely, multivariate Newton-Padé approximants. The algorithm is only applicable if every three successive elements on the staircase are different. If a singularity occurs in the defining system of equations for the multivariate rational interpolant then singular rules must be developed. For the univariate Newton-Padé approximant this was done in [3] by Claessens and Wuytack. The idea to perturb the initial staircase and walk around the block structure in the table in order to avoid the singularity, is explored now in a multivariate setting. Another approach would be to use block bordering methods in combination with reverse bordering [4] in order to solve the rank deficient linear system of interpolation conditions (Newton-Padé approximation conditions) recursively. Since this last technique can also be used for scattered multivariate data exhibiting near-singularity, we describe the second approach in a separate paper [7]. Here we deal only with partially grid-structured data (satisfying the so-called rectangle rule or inclusion property).