A Criterion for Existence of Bivariate Vector Valued Rational Interpolants

Abstract:

      In this paper, a necessary and sufficient condition for the existence of a kind of bivariate vector valued rational interpolants over rectangular grids is given. This criterion is an algebraic method, i.e., by solving a system of equations based on the given data, we can directly test whether the relevant interpolant exists or not. By coming up with our method, the problem of how to deal with scalar equations and vector equations in the same system of equations is solved. After testing existence, an expression of the corresponding bivariate vector-valued rational interpolant can be constructed consequently. In addition, the way to get the expression is different from the one by making use of Thiele-type bivariate branched vector-valued continued fractions and Samelson inverse which are commonly used to construct the bivariate vector-valued rational interpolants. Compared with the Thiele-type method, the one given in this paper is more direct. Finally, some numerical examples are given to illustrate the result.