Abstract
The problems of polynomial interpolation with several variables present more difficulties than those of one-dimensional interpolation. The first 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 of interpolation points. Techniques of univariate Newton interpolating polynomials are extended to multivariate data points by different generalizations and practical algorithms. The Newton basis format, with divided-difference algorithm for coefficients, 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 finite interpolation set. We will use a Newton-type polynomials basis, and we will introduce a new concept called -partition. All the given algorithms are tested on examples. RLMVPIA is easy to implement and requires no storage.
1. Introduction
Let be a commutative field and . By , we denote the algebra of all polynomials in variables, and we denote by the subspace of all polynomials of total degree less than or equal to , where is a nonnull integer.
Given a finite interpolation set of distincts nodes, the Lagrange interpolation problem consists of finding, for a given data vector , a polynomial such that
We will then say that is an interpolating polynomial for on . More precisely, is called poised or correct or unisolvent [1–3] for a subspace of ; if the Lagrange interpolation problem (1) has a unique interpolating polynomial in for any given data vector , it means, in other words, that the function is a linear isomorphism. Then, it is necessary that . The problem of researching such subspaces will be denoted .
In this article, we construct a random algorithm for finding several sub-spaces solutions of the problem .
It is well-known that in the univariate case () the Lagrange interpolation problem with respect to distinct points is always uniquely solvable, if one takes to be the space of all polynomials of degree less than or equal to . In several variables, however, the situation is much more difficult. In order to successfully interpolate on , we must have
since
And even if this is the case, there can be the problem that the points lie on some algebraic surface of degree ; i.e., there is some polynomial of total degree at most which vanishes on . For example, take , , and ; it is easy to see that the set is not poised on the space (since vanishes on ).
So the poisedness of multivariate polynomial interpolation depends on the geometric structure of the interpolation set . 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 [4–15].
In a recent publication [7], the author proposed a generalization of the univariate program of Newton form basis and divided-difference algorithm in . 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, 16–18]. He shows how the index set structure appropriately determines the interpolation space. When the sum of indices is bounded by , there is a unique interpolation with a polynomial of .
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 [19–22], 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 -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 knowing a solution of , where is a subset of with nodes. More precisely, if is an interpolation space for , and verifying ; then, we construct, in a way, a polynomial verifying
So is a solution of , and the solution of (1) in is a polynomial in the form where is a scalar to compute.
This work is organized as follows: in Section 2 we present the notion of -partition. In Section 3, we give the algorithm RLMVPIA. In Section 4, we illustrate our algorithms by different examples.
2. -Partition Concept
This new approach takes into consideration the distribution of the nodes of the considered interpolation set by introducing a new concept described in the following. We define a notion of -partition, and we present a random algorithm for computing the polynomials , for , which will be used for giving the algorithm RLMVPIA.
In this section, is a finite set of , , and . For , we note the canonical coordinated form, defined as and .
Definition 1. For , let be a subset of . (1)It will be said that is a -partition ifIn this case, is called the length of the -partition . (2)Let . It is said that is a -partition if(a) is a -partition(b)
Proposition 2. For all , a -partition still exists.
Proof. To prove that, we show how to construct a -partition. Let be the function
is well defined because . We set
and for , we take
Then is a -partition.
To illustrate the proof of the proposition, we give below some examples in the cases and
2.1. Case
Example 1. We set We have and It follows that
Example 2. We take with and we have , so we obtain
2.2. Case
Example 3. We take with , and we have , so Now, one can easily get the following result.
Proposition 6. Let be a -partition, and let 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
The following algorithm called ZPNA (-Partition Newton Algorithm) randomly constructs a -partition and the associated polynomial.
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Remark 7. (1)All operations in the algorithm ZPNA have constant costs, so the complexity depends linearly on the length of (2)For a -partition, the total degree of the associated polynomial is equal to the length of the partition which is less than (3)The algorithm ZPNA is not deterministic. If we apply the algorithm several times, one can obtain several -partitions from which we can choose those of minimum lengthPoint 3 of the precedent remark can be illustrated by the following example: and . Applying the algorithm several times, we get the following -partitions and the associated polynomials. (1) and (2) and (3) and (4) and
It is clear that the first solution is the best.
Theorem 8. The algorithm 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 , the elements treated in the first iterations, et the values of in the iteration . So the following property is a loop invariant: At the start of each iteration of the for loop, the is a -partition.
For initialization, we start by showing that the loop invariant holds before the first loop iteration: when , the are empty, and no item is treated, we take , so by convention, the is a -partition.
For maintenance, next, we tackle the second property, showing that each iteration maintains the loop invariant. Assume that the is a -partition, we note the element chosen at the start of the iteration , as , the loop (while ) ends, so we note the index founded such as . The analysis of the sequence of the iteration makes it possible without difficulty to assert that if and for two cases arise: either if is already present in one of the otherwise . In both cases, we have is a -partition: for , if , then there is a such as because is a -partition; if by construction, we also have the result. On the other hand, as is not in any of the , and as by construction , it follows that , hence the result.
For termination, we examine finally what will happen when the loop terminates. When the loop terminates (for in (a random choice)), the set , with the invariant of the loop, we have (which are the sets returned by the algorithm) which is a -partition. Therefore, the algorithm is correct.
3. RLMVPIA Random Approach
For solving, recursively, the problem , we start by choosing as an obvious solution of the problem , since we have, for all , the constant polynomial which is the interpolating polynomial of on in So we take as a basis of and we will use the notion of -partition for computing the polynomials , for , in order to give the algorithm RLMVPIA.
The following result shows how the solution of the problem can be constructed recursively using the relationship (6).
Theorem 9. For , let be a -partition, and let be the associated polynomial. Then, the space is a solution of the problem . More precisely, giving a data vector, the interpolating polynomial for on in is given by : where is the interpolating polynomial for on in and
Proof. Let be a given data vector, and let us consider the interpolating polynomial for on in . For , as is a -partition, then . So the polynomial defined by the expression (25) above verifies . On the other hand, we have by definition , ; then, . So taking we obtain We conclude that so is an interpolating polynomial for on in . We deduce that the linear mapping is surjective. But as , we conclude that the mapping is a linear isomorphism and that is an interpolation space for , hence the result.
Remark 10. (1)The interpolating polynomials obtained by the previous theorem depend on the indexation choice of the nodes of (2)For constructing a solution of , the following algorithm RLMVPIA chooses a random indexation of the interpolation nodes
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4. Examples
4.1. Examples for the Case
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, 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.
4.1.2. Example 2
In this example, we take the interpolation set (Figure 2) and for interpolation values by applying the random RLMVPIA several times; we give among the solutions obtained the two following ones. The first one and the second one
4.2. Examples for the Case
In this section, we also give two examples for a full grid and for a random configuration.
4.2.1. Example 3: Grid Case
In this example, we take the full grid of considered in [5] (Figure 3) and the interpolation values by applying the random RLMVPIA, several times; we obtain the same interpolating polynomial given in [5]. which coincides with the solution obtained by RMVPIA in [5].
4.2.2. Example 4
In this example, we consider another configuration of the interpolation set in (Figure 4) and we take them as interpolation values
Applying the random RLMVPIA several times, we give among the solutions obtained the two following ones. The first one
The second one with a lower degree
Remark 11. The last example shows the importance of the random approach, in some cases, to get solutions of fairly low degree, which can influence the cost of evaluations.
5. 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 -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.
Acknowledgments
We are grateful to our Professor A. Messaoudi for his help and support.