Discrete Dynamics in Nature and Society

Volume 2016, Article ID 7515876, 11 pages

http://dx.doi.org/10.1155/2016/7515876

## On the Singularity of Multivariate Hermite Interpolation of Total Degree

^{1}School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China^{2}School of Mathematics and Statistics, Taishan College, Taian 271021, China^{3}School of Software, Dalian University of Technology, Dalian 116620, China

Received 4 March 2016; Accepted 2 August 2016

Academic Editor: Guang Zhang

Copyright © 2016 Zhongyong Hu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We study the singularity of multivariate Hermite interpolation of type total degree on nodes with . We first check the number of the interpolation conditions and the dimension of interpolation space. And then the singularity of the interpolation schemes is decided for most cases. Also some regular interpolation schemes are derived, a few of which are proved due to theoretical argument and most of which are verified by numerical method. There are some schemes to be decided and left open.

#### 1. Introduction

Let be the space of all polynomials in variables, and let be the subspace of polynomials of total degree at most . Let be a set of pairwise distinct points in and be a set of nonnegative integers. The Hermite interpolation problem to be considered in this paper is described as follows: Find a (unique) polynomial satisfyingfor given values , where the numbers and are assumed to satisfy

Following [1, 2], such kind of problem is called Hermite interpolation of type total degree. The interpolation problem is called regular if the above equation has a unique solution for each choice of values . Otherwise, the interpolation problem is singular. As shown in [3], the regularity of Hermite interpolation problem implies that it is regular for almost all with . Hence, in this paper, we will call almost -regular if is regular for some . Otherwise, we call -singular. With no confusion, we also call it almost regular or singular for convenience. If is a nontrivial polynomial satisfying (1) with zero interpolation condition, we call a vanishing polynomial with respect to and . Obviously, being singular is equivalent to the existence of a vanishing polynomial of degree no more than .

The research of regularity of multivariate Hermite interpolation is more difficult than Lagrange case, although the latter is also difficult. One of the main reasons is that (2) does not hold in some cases. About the results of multivariate Hermite interpolation, one can refer to [1–10] and the references therein. Most recently, authors [5] made further development and gave complete description for the regularity of the interpolation problem on nodes, which is an extension of the results mentioned in [1, 2]. Besides, not any other results appeared for a big number of nodes. This paper is an extension of [5] and we will investigate the singularity of Hermite interpolation for with .

This paper is organized as follows. In Section 2, we consider the singularity of the Hermite interpolation of type total degree and present the main results. In Section 3, we present theoretical proofs for some regular schemes. Finally, in Section 4, we conclude our results.

#### 2. Singularity of Interpolation Schemes

In this section, we will investigate the singularity of Hermite interpolation of type total degree and (2) is always assumed to hold. Hermite interpolation of type total degree is affinely invariant in the sense that the interpolation is singular or regular for one choice of nodes. In what follows, we assume . In this case, since , there must exist a nontrivial quadratic polynomial which vanishes at . Also there exists a nontrivial linear polynomial vanishing at . Given and , if there are vanishing polynomials with respect to and , we will denote by one vanishing polynomial of them.

Here we always assume that no interpolation happens at if the th component of is . Obviously, vanishing polynomials always exist if since the number of the equations is less than the number of the unknowns.

For convenience, we always order with . In [5], authors showed that the inequalitymust hold if is regular, which gives evaluation of in (2). The following theorem implies that inequality (4) is very sharp.

Theorem 1. *Given and , if , then is singular.*

*Proof. *We first assume . Then is a vanishing polynomial with respect to and , and Thus, in this case is singular.

If , then . Thus is a vanishing polynomial with respect to and , and Collecting two cases, we complete the proof.

Next, we assume .

Lemma 2. *Given and , if , then is singular.*

*Proof. *Let . Then is a vanishing polynomial with respect to and . Moreover This completes the proof.

If , we easily get . The following theorem is due to [5], which will be used in next lemma.

Theorem 3 (see [5]). *Assume . Given and , ifthen the Hermite interpolation of type total degree is singular.*

This theorem implies that there exists a vanishing polynomial of degree no more than with respect to and if (8) holds.

Lemma 4. *Given and , if and , then is singular.*

*Proof. *Let . Then together with all of its partial derivatives of order up to vanishes at the points. For points , since it follows from Theorem 3 that there exists a polynomial with , together with all of its partial derivatives of order up to vanishing at for . Let . Then, and all of its partial derivatives of order up to vanish at for , and Thus, the interpolation is singular.

In what follows, we only need to consider and , which includes

Lemma 5. *Given and , suppose that (11) is satisfied; then Hermite interpolation of type total degree is almost regular if and only if and*

*Proof. *Set ; then and . We first check (2). Let Since for and , then We will show that implies . Note that Thus if , then Hencewhere Since for , is monotonically decreasing about . Thus, for due to and .

Since , then for , which means that (2) does not hold for . Thus we only need to consider . In this case Hence, for , (2) does not hold for and holds for only if (14) holds. We will show that it is almost regular in next section. For and , (2) holds only in the case of since (2) holds for . We can show that is singular if . In fact, we can take with as the vanishing polynomial. Obviously . The proof is completed.

Lemma 6. *Given and . Suppose that (12) is satisfied and that . Then *(i)*for , (2) holds only for one form and Hermite interpolation of type total degree is almost regular for ;*(ii)*for , if , it is singular; if , (2) has three positive integer solutions and corresponding interpolation schemes are almost regular;*(iii)*for ,(a) if , it is singular;(b)if and , it is also singular;(c)if and , (2) has four positive integer solutions and in these cases are almost regular;(d)if , it is singular;(e)if , (2) holds only for one form and it is almost regular.*

*Proof. *Set and ; then . If , it is easy to check that (2) holds if and only if and . This scheme is almost regular, which will be proved in Theorem 15 of the next section.

For , we first check (2). Let Since for and , thenBy the same argument with Lemma 5, one can show that implies that . The following facts can be checked easily:Since for , (2) does not hold for .

For and , (2) has three positive integer solutions and . These three schemes are almost regular, which can be verified by numerical method; see Remark 7.

Since for , (2) does not hold for and . Similarly, for means that (2) does not hold for and .

For and , (2) has four positive integer solutions which are shown to be almost regular by numerical method presented in Remark 7.

Let us consider the case of .

From the definition of , we obtainThen, (2) does not hold for , .

For , , (2) holds for the form Indeed, this form is the only one since . This scheme is almost regular, which will be proved in the next section.

Finally, we consider the case of . We can show that is singular by taking with for and with for as vanishing polynomial with respect to and . Here we use the fact that if which can be obtained by a simple calculation.

The proof is completed.

*Remark 7. *Generally speaking, it is difficult to judge the regularity of the interpolation schemes theoretically. For a given , one possible way to decide the regularity is based on numerical method: calculating the vanishing ideal (see [5] for details) or the corresponding Vandermonde matrix, where the points can be selected randomly. However, the former method needs to do symbolic calculation which is little useful for big , and . The latter one needs to judge the singularity of the matrix, which is also difficult if the order is very big. Although so, it is a good way for moderate , , and , which is employed in this paper for some simple cases.

*Lemma 8. Given and , suppose that (13) is satisfied and that . Then (i)if , (2) never holds;(ii)if , (2) has finite positive integer solutions listed in Table 2.*

*Proof. *We first check (2). Let Since for and , then In the same way with Lemma 5, one can show that implies that and . By a simple calculation, we have implies that holds for all . Thus we must take to ensure (2) if . Hence, let and we again have implying and . It is easy to get Thus we can obtain the possible pairs satisfying (2); see Figure 1.

By detailed analysis and computation, the solution of (2) can be obtained and is listed in Table 2; see Appendix.