Stable monomial basis for multivariate Birkhoff interpolation problems

For a given node set Z and the corresponding interpolation conditions, one of the key problems in multivariate Birkhoff interpolation is determining a monomial basis that spans the interpolation space. However, a slight perturbation of the node set will result in a different monomial basis. We propose a more general multivariate Birkhoff interpolation scheme and provide a numerical algorithm that can determine the stable monomial basis to ensure that the interpolation polynomial always exists in the space spanned by the monomial basis for any perturbation of the node set within a given error e .