Lagrange Multivariate Polynomial Interpolation: A Random Algorithmic Approach

. The problems of polynomial interpolation with several variables present more di ﬃ culties than those of one-dimensional interpolation. The ﬁ rst problem is to study the regularity of the interpolation schemes. In fact, it is well-known that, in contrast to the univariate case, there is no universal space of polynomials which admits unique Lagrange interpolation for all point sets of a given cardinality, and so the interpolation space will depend on the set Z of interpolation points. Techniques of univariate Newton interpolating polynomials are extended to multivariate data points by di ﬀ erent generalizations and practical algorithms. The Newton basis format, with divided-di ﬀ erence algorithm for coe ﬃ cients, generalizes in a straightforward way when interpolating at nodes on a grid within certain schemes. In this work, we propose a random algorithm for computing several interpolating multivariate Lagrange polynomials, called RLMVPIA (Random Lagrange Multivariate Polynomial Interpolation Algorithm), for any ﬁ nite interpolation set. We will use a Newton-type polynomials basis, and we will introduce a new concept called ð Z , z Þ -partition. All the given algorithms are tested on examples. RLMVPIA is easy to implement and requires no storage.


Introduction
Let K be a commutative field and p, n ∈ ℕ * . By Π p = K½x 1 , ⋯, x p , we denote the algebra of all polynomials in p variables, and we denote by Π p d the subspace of all polynomials of total degree less than or equal to d, where d is a nonnull integer.
Given a finite interpolation set Z n = fz 1 , ⋯z n g ⊂ K p of distincts nodes, the Lagrange interpolation problem consists of finding, for a given data vector R = ðr z : z ∈ Z n Þ ∈ K Z n , a polynomial P ∈ Π p such that P Z n ð Þ= R, that is, P z ð Þ = r z , z ∈ Z n : ð1Þ We will then say that P is an interpolating polynomial for R on Z. More precisely, Z n is called poised or correct or unisolvent [1][2][3] for a subspace P of Π p ; if the Lagrange interpolation problem (1) has a unique interpolating polynomial in P for any given data vector R = ðr z : z ∈ Z n Þ ∈ K Z n , it means, in other words, that the function is a linear isomorphism. Then, it is necessary that dim P = n. The problem of researching such subspaces will be denoted P ðZ n Þ.
In this article, we construct a random algorithm for finding several sub-spaces solutions of the problem P ðZ n Þ.
It is well-known that in the univariate case (p = 1) the Lagrange interpolation problem with respect to n distinct points is always uniquely solvable, if one takes P to be the space of all polynomials of degree less than or equal to n − 1.
In several variables, however, the situation is much more difficult. In order to successfully interpolate Z n on Π p d , we must have And even if this is the case, there can be the problem that the points lie on some algebraic surface of degree d; i.e., there is some polynomial Q of total degree at most d which vanishes on Z n . For example, take p = 2, d = 1, and Z = fð−1, 0Þ, ð0, 0Þ, ð1, 0Þg; it is easy to see that the set Z is not poised on the space Π 2 1 = spanf1, x, yg (since P = y vanishes on Z).
So the poisedness of multivariate polynomial interpolation depends on the geometric structure of the interpolation set Z n . Tensor product interpolation is the oldest particular case extending the univariate theory where the interpolation set and space are obtained by tensor products of the univariate ones. The Lagrange formula and the Newton formula with divided differences are easily extended to this problem, as can be found in [4][5][6][7][8][9][10][11][12][13][14][15].
In a recent publication [7], the author proposed a generalization of the univariate program of Newton form basis and divided-difference algorithm in K p . The required interpolation sets are those which admit an indexation having a regular structure as triangular, rectangular, or more generally a lower set [2,7,8,[16][17][18]. He shows how the index set structure appropriately determines the interpolation space. When the sum of indices is bounded by d, there is a unique interpolation with a polynomial of degree ≤ d.
In [5], by using the Schur complements and the Sylvester identity, the authors established the RMVPIA (Recursive MultiVariate Polynomial Interpolation Algorithm) when the interpolation set is a full grid.
Polynomial interpolation with several variables occurs in several topics of applied mathematics and engineering [19][20][21][22], hence the interest in seeking consistent and simple to implement polynomial interpolation algorithms.
In this work, we propose a new algorithm for computing several interpolation spaces for any finite interpolation set. This algorithm which is called RLMVPIA (Random Lagrange MultiVariate Polynomial Interpolation Algorithm) is based on a recursive random scheme. RLMVPIA allows us to simultaneously determine interpolating polynomials. For that, we will use a Newton-type polynomials basis, and we will introduce a new concept called ðZ, zÞ-partition. All the given algorithms are tested on examples. As RMVPIA, RLMVPIA is easy to implement and requires no storage.
The principle of our approach is to solve P ðZ n Þ knowing a solution of P ðZ n−1 Þ, where Z n−1 = fz 1 , ⋯, z n−1 g is a subset of Z n with n − 1 nodes. More precisely, if P n−1 is an interpolation space for Z n−1 , and P n−1 ∈ P n−1 verifying P n−1 ðzÞ = r z , z ∈ Z n−1 ; then, we construct, in a way, a polynomial Q n verifying Q n z ð Þ = 0,∀z ∈ Z n−1 , So P n = P n−1 + KQ n is a solution of P ðZ n Þ, and the solution of (1) in P n is a polynomial in the form where s n is a scalar to compute. This work is organized as follows: in Section 2 we present the notion of ðZ, zÞ-partition. In Section 3, we give the algorithm RLMVPIA. In Section 4, we illustrate our algorithms by different examples.

ðZ, zÞ-Partition Concept
This new approach takes into consideration the distribution of the nodes of the considered interpolation set Z n by introducing a new concept described in the following. We define a notion of ðZ, zÞ-partition, and we present a random algorithm for computing the polynomials Q i , for i ∈ ½½1 ; n, which will be used for giving the algorithm RLMVPIA.
In this section, Z is a finite set of K p , p ∈ ℕ * , and z ∈ K p \ Z. For k ∈ f1, ⋯, pg, we note x k the canonical coordinated form, defined as and x k ðZÞ = fx k ðtÞ: t ∈ Zg.
Definition 1. For k ∈ ½½1 ; p, let I k be a subset of x k ðZÞ.
(1) It will be said that ðI 1 , ⋯, In this case, ∑ p k=1 card ðI k Þ is called the length of the Z -partition ðI 1 , ⋯, I p Þ.
For all z ∈ K p \ Z, a ðZ, zÞ-partition still exists.

Journal of Applied Mathematics
Proof. To prove that, we show how to construct a ðZ, zÞ -partition. Let ϕ be the function and for k ∈ ½½1 ; p, we take Then ðI 1 , ⋯, I p Þ is a ðZ, zÞ-partition.
To illustrate the proof of the proposition, we give below some examples in the cases p = 2 and p = 3 We have ϕðZÞ = f1, 2g and It follows that Example 2. We take with z = ð11, 5Þ, and we have ϕðZÞ = f2g, so we obtain Example 3. We take with z = ð2, 3, 1Þ, and we have ϕðZÞ = f1, 3g, so Now, one can easily get the following result.
Proposition 6. Let I = ðI 1 , ⋯, I p Þ be a ðZ, zÞ-partition, and let Q I be the polynomial associated, given by with the convention that the product is equal to 1 when the set of indices is empty. So we have ∀t ∈ Z, The following algorithm called ZPNA (ðZ, zÞ-Partition Newton Algorithm) randomly constructs a ðZ, zÞ-partition and the associated polynomial.
and z = ð11, 5Þ. Applying the algorithm ZPNA several times, we get the following ðZ, zÞ-partitions and the associated polynomials.
Theorem 8. The algorithm ZPNA is correct.
Proof. For proving that, we use loop invariant to help us understand why an algorithm is correct. We must show three things about a loop invariant: (i) Initialization: it is true prior to the first iteration of the loop.
(ii) Maintenance: if it is true before an iteration of the loop, it remains true before the next iteration.
(iii) Termination: when the loop ends, the invariant gives us a useful property showing that the algorithm is correct.
For the ZPNA algorithm, we note for k ∈ ½½1 ; n, z 1 , ⋯z k the elements treated in the k first iterations, Z k = fz 1 , ⋯z k g et For initialization, we start by showing that the loop invariant holds before the first loop iteration: when k = 1, the I k−1 1 , ⋯, I k−1 p are empty, and no item is treated, we take Z 0 = ∅, so by convention, the I k−1 1 , ⋯, I k−1 p is a ðZ k−1 , zÞ-partition. For maintenance, next, we tackle the second property, showing that each iteration maintains the loop invariant. Assume that the I k−1 1 , ⋯, I k−1 p is a ðZ k−1 , zÞ-partition, we note z k the element chosen at the start of the iteration k, as z ∉ Z, the loop (while x i ðtÞ = x i ðzÞ) ends, so we note i the index founded such as x i ðz k Þ ≠ x i ðzÞ. The analysis of the sequence of the iteration k makes it possible without difficulty to assert that I k j = I k−1 j if j ≠ i and for j = i two cases arise: either In both cases, we have I k 1 , ⋯, I k p is a ðZ k , zÞ-partition: for t ∈ Z k , if t ∈ Z k−1 , then there is a j ∈ ½½1 ; n such as x j ðtÞ ∈ I j because I k−1 1 , ⋯, I k−1 p is a ðZ k−1 , zÞ-partition; if t = z k by construction, we also have the result. On the other hand, as x j ðzÞ is not in any of the I k−1 j , j ∈ ½½1 ; n, and as by construction x i ðz k Þ ≠ x i ðzÞ, it follows that x i ðzÞ ∉ I k i , hence the result. For termination, we examine finally what will happen when the loop terminates. When the loop terminates (for t in Z (a random choice)), the set Z n = Z, with the invariant of the loop, we have I n 1 , ⋯, I n p (which are the sets returned by the algorithm) which is a ðZ n = Z, zÞ-partition. Therefore, the algorithm is correct.

RLMVPIA Random Approach
For solving, recursively, the problem P ðZ n Þ, we start by choosing P 1 = spanfð1Þg as an obvious solution of the problem P ðZ 1 Þ, since we have, for all r 1 ∈ K, the constant polynomial which is the interpolating polynomial of R 1 = ðr 1 Þ on Z 1 = fz 1 g in P 1 : So we take Q 1 = 1 as a basis of P 1 and we will use the notion of ðZ, zÞ-partition for computing the polynomials Q i , for i ∈ ½½2 ; n, in order to give the algorithm RLMVPIA.
The following result shows how the solution of the problem P ðZ n Þ can be constructed recursively using the relationship (6).
be the associated polynomial. Then, the space P k = spanfQ 1 , ⋯Q k g is a solution of the problem P ðZ k Þ. More precisely, giving R k = ðr i : i = 1, ⋯, kÞ a data vector, the interpolating 4 Journal of Applied Mathematics polynomial for R k on Z k in P k is given by P k : where P k−1 is the interpolating polynomial for R k−1 = ðr i : i = 1, ⋯, k − 1Þ on Z k−1 in P k−1 = spanfQ 1 , ⋯, Q k−1 g and Proof. Let R k = ðr i : i = 1, ⋯, kÞ be a given data vector, and let us consider P k−1 the interpolating polynomial for R k−1 = we obtain P k ðz k Þ = r k : We conclude that so P k is an interpolating polynomial for R k on Z k in P k = spanfQ 1 , ⋯, Q k g. We deduce that the linear mapping is surjective. But as dim P k ≤ k, we conclude that the mapping is a linear isomorphism and that P k is an interpolation space for Z k , hence the result.

Remark 10.
(1) The interpolating polynomials P k obtained by the previous theorem depend on the indexation choice of the nodes of Z n (2) For constructing a solution of P ðZ n Þ, the following algorithm RLMVPIA chooses a random indexation of the interpolation nodes 4. Examples 4.1. Examples for the Case p = 2. We will give two examples: the first one concerns the particular case where the interpolation set is a full grid. We will see that the RLMVPIA and the RMVPIA [5] are equivalent. The second one is for a random configuration.
4.1.1. Example 1: Grid Case. When the interpolation set is a full grid, RLMVPIA gives a similar result to the one obtained in [5,7]. In the following example already studied in [5] where the set of nodes, Z n = Z ðn 1 +1Þðn 2 +1Þ is presented in Figure 1. We take by applying the random RLMVPIA, several times; we obtain the same interpolating polynomial given in [5].
For a random configuration using the RLMVPIA, we obtain different solutions, as can be seen in the following example.

Example 2.
In this example, we take the interpolation set ( Figure 2) Input: Interpolation set Z and interpolation values R n = lenght(Z) P 0 =0 Z 0 = ½ for k = 1 to n :

Conclusion
In conclusion, this work contributes to solve the problem of Lagrange multivariate polynomial interpolation with any finite set of interpolation nodes, using a recursive algorithm RLMVPIA with a random approach based on the ðZ, zÞ -partition concept. This study shows that RLMVPIA is easy to implement and requires no storage.
Currently, we are interested firstly, in refining the random approach of the algorithm, to build, in a more deterministic way, optimal solutions of smaller degree. The problem of optimal solution and the study of the numerical stability of the RLMVPIA are under investigation. One can also find natural applications of RLMVPIA in different topics of applied mathematics and engineering as the numerical resolution of PDEs, computer-aided design (CAD), cryptography, etc.

Data Availability
Our research does not use any archived datasets.

Conflicts of Interest
The authors declare that they have no conflicts of interest.