Composite schemes for multivariate blending rational interpolation

It is demonstrated that Newtonʹs interpolation polynomials and Thieleʹs interpolating continued fractions can be incorporated to generate various interpolation schemes based on rectangular grids, among them are two kinds of bivariate blending rational interpolants. However, blending rational interpolants strongly depend on the existence of so-called blending differences, which means that for some grids of data, one may fail to find out the corresponding rational interpolants as a whole. In this paper, we offer a solution scheme by adopting composite interpolation over triangular sub-grids. Characterization theorem is given, error estimation is worked out and vector valued case as well as matrix valued case is discussed.