Detecting discontinuity points from spectral data with the quotient-difference (qd) algorithm
Using the multivariateqdg-algorithm developed in , 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  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  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 . Here we deal only with partially grid-structured data (satisfying the so-called rectangle rule or inclusion property).