Peng Xia, Shugong Zhang, Na Lei
In this paper, we first apply the Fitzpatrick algorithm to osculatory rational interpolation. Then based on Fitzpatrick algorithm, we present a Neville-like algorithm for Cauchy interpolation. With this algorithm, we can determine the value of the interpolating function at a single point without computing the rational interpolating function.
Yihe Gong, Xue Jiang, Zhe Li et al.
In this paper we study the algebraic structure of error formulas for ideal interpolation. We introduce the so-called "normal" error formulas and prove that the lexicographic order reduced Gröbner basis admits such a formula for all ideal interpolation. This formula is a generalization of the "good" error formula proposed by Carl de Boor. Finally, we discuss a Shekhtman's example and give an explicit form of "normal" error formula for this example.
Xue Jiang, Shugong Zhang
We demonstrate the equivalence of two classes of $D$-invariant polynomial subspaces introduced in [8] and [9], i.e., these two classes of subspaces are different representations of the breadth-one $D$-invariant subspace. Moreover, we solve the discrete approximation problem in ideal interpolation for the breadth-one $D$-invariant subspace. Namely, we find the points, such that the limiting space of the evaluation functionals at these points is the functional space induced by the given $D$-invariant subspace, as the evaluation points all coalesce at one point.
Zhe Li, Shugong Zhang, Tian Dong
In this paper, we verify Carl de Boor's conjecture on ideal projectors for real ideal projectors of type partial derivative by proving that there exists a positive $η\in \mathbb{R}$ such that a real ideal projector of type partial derivative $P$ is the pointwise limit of a sequence of Lagrange projectors which are perturbed from $P$ up to $η$ in magnitude. Furthermore, we present an algorithm for computing the value of such $η$ when the range of the Lagrange projectors is spanned by the Gröbner éscalier of their kernels w.r.t. lexicographic order.
Zhe Li, Shugong Zhang, Tian Dong
In this paper, we focus on two classes of D-invariant polynomial subspaces. The first is a classical type, while the second is a new class. With matrix computation, we prove that every ideal projector with each D-invariant subspace belonging to either the first class or the second is the pointwise limit of Lagrange projectors. This verifies a particular case of a C. de Boor's conjecture asserting that every complex ideal projector is the pointwise limit of Lagrange projectors. Specifically, we provide the concrete perturbation procedure for ideal projectors of this type.
Tian Dong, Xiaoying Wang, Shugong Zhang et al.
As we all known, there is still a long way for us to solve arbitrary multivariate Lagrange interpolation in theory. Nevertheless, it is well accepted that theories about Lagrange interpolation on special point sets should cast important lights on the general solution. In this paper, we propose a new type of bivariate point sets, quasi-tower sets, whose geometry is more natural than some known point sets such as cartesian sets and tower sets. For bivariate Lagrange interpolation on quasi-tower sets, we construct the associated degree reducing interpolation monomial and Newton bases w.r.t. common monomial orderings theoretically. Moreover, by inputting these bases into Buchberger-Möller algorithm, we obtain the reduced Gröbner bases for vanishing ideals of quasi-tower sets much more efficiently than before.
Zhe Li, Shugong Zhang, Tian Dong
In this paper, we focus on a special class of ideal projectors. With the aid of algebraic geometry, we prove that for this special class of ideal projectors, there exist "good" error formulas as defined by C. de Boor. Furthermore, we completely analyze the properties of the interpolation conditions matched by this special class of ideal projectors, and show that the ranges of this special class of ideal projectors are the minimal degree interpolation spaces with regard to their associated interpolation conditions.
Xiaoying Wang, Shugong Zhang, Tian Dong
For the last almost three decades, since the famous Buchberger-Möller(BM) algorithm emerged, there has been wide interest in vanishing ideals of points and associated interpolation polynomials. Our paradigm is based on the theory of bivariate polynomial interpolation on cartesian point sets that gives us related degree reducing interpolation monomial and Newton bases directly. Since the bases are involved in the computation process as well as contained in the final output of BM algorithm, our paradigm obviously simplifies the computation and accelerates the BM process. The experiments show that the paradigm is best suited for the computation over finite prime fields that have many applications.