A new algorithm for a recursive construction of the minimal interpolation space

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Abstract

In this work, we introduce a new interpolation algorithm, based on a recursive method for computing Lagrange interpolants. This algorithm allows to construct recursively the minimal interpolation space (see [1]) with respect to a finite set of points. We also extend this recursive method to the osculatory interpolation problem.

Introduction

Let v0,,vn be n+1 linearly independent continuous functions on Rd. Setting Vi:=span{v0,,vi}, for i=0,,n. For a given continuous function f defined on Rd, and a set of n+1 distinct points Xn={x0,,xn} of Rd, the Lagrange interpolation problem is defined as follows:findpVnsuch thatp(xi)=f(xi),i=0,,n.

The solution p of the above problem can be obtained by solving a linear system of equations, or by using the recursive method proposed in [1]. Some important examples of the spaces Vn are the subspaces of the space of polynomials of d variables and of degree less than or equal to n denoted by Pn(Rd). It is well-known that a multivariate interpolation problem is much more difficult than the corresponding univariate one. Many authors have been interested in the bivariate polynomial interpolation (see [2], [3], [13], [14], [15], [16], for instance). In order to successfully interpolate with a unique function in the space of polynomials of two variables and of degree less than or equal to n at the points of a set XR2, Liang [13] proved that it is necessary and sufficient that the number of nodes |X| is equal to ((n+1)(n+2)/2) and X is not a subset of any algebraic variety of degree less than or equal to n. In practice, this condition leads to solve the interpolation problem as a linear system of equations. In order to simplify the resolution of the interpolation problem, Chung and Yao [4] showed that if a set of nodes satisfies the Geometric Characterization (GC), then the Lagrange interpolation problem in Pn(R2) is unisolvent, and each element of the Lagrange basis is the product of n bivariate polynomials of total degree at most one. The GC condition has been studied by several authors, in particular Busch 1990, Carnicer and Gasca 2001. This approach is very useful and gives rise to explicit and simple formulae for the Lagrange basis functions. However, it becomes much complicated when the number of nodes increases. When XRd and |X|=N, there always exists a subspace VNPN(Rd) such that the Lagrange interpolation problem with respect to X is poised, i.e., uniquely solvable. For example, the Kergin interpolation, see [12]. de Boor and Ron [6], [7] introduced an interpolation algorithm for constructing a particular subspace VN called the least choice. Another interpolation algorithm has been developed by Sauer [20] in order to determine a subspace VN called a minimal degree interpolation space. Recently, a new interpolation algorithm has been introduced in [1], which is based on the recursive method for computing interpolants. This algorithm permits to construct a particular subspace VN called minimal interpolation space which satisfies the properties of minimal degree interpolation space. All these interpolation algorithms use globally the set of interpolation points Xn to construct the interpolation space Vn. If this space can not give a good interpolant of f at the set of points Xn, the interpolation algorithms in question can not improve recursively the interpolant of f by adding simply a point xn+1 to the set Xn. Indeed, the construction of the space Vn+1 requires a new implementation of these algorithms on the whole set Xn+1={x0,,xn,xn+1}.

In this work, we propose a new algorithm, which permits to construct recursively and iteratively the subspaces Vi, i=0,,n such that the following problemfindpVisuch thatp(xj)=f(xj),j=0,,iadmits a unique solution. We prove that each space Vi is the minimal interpolation space with respect to the set Xi and satisfies some important properties. We also extend the recursive method for computing Lagrange interpolants to the osculatory interpolation problem.

The paper is organized as follows. In Section 2, we give a recursive solution of a Lagrange interpolation problem, where the interpolation space V is spanned by some continuous functions defined on Rd. From this, we derive another algorithm which allows to compute the coefficients of the interpolant related to the basis of V. In Section 3, we apply this technique to the multivariate polynomial spaces, and we introduce a recursive algorithm which allows us to simplify the resolution of the multivariate interpolation problem. By using this algorithm, for every set Xn of n+1 points x0,,xn in Rd, we can construct recursively an interpolation subspaces ViPi(Rd), i=0,,n, corresponding to the sets of points Xi={x0,,xi}. We can also determine recursively the interpolant IifVi of a function f at the set Xi. The explicit form of Iif in the canonical basis of Vi can be given by the second algorithm introduced in Section 2. In Section 4, we introduce a new recursive method for computing osculatory interpolation problem. Some numerical examples are analyzed in Section 6. Finally, in Section 7 we give a conclusion.

Section snippets

A recursive construction of Lagrange interpolants

Let v0,,vn be n+1 linearly independent continuous functions on Rd. Setting Vi:=span{v0,,vi}, for i=0,,n. For a given continuous function f defined on Rd, and n+1 distinct points x0,,xn of Rd, we suppose that the interpolation problem:findpVnsuch thatp(xi)=f(xi),i=0,,nhas a unique solution, i.e., the following matrixA:=v0(x0)v1(x0)vn(x0)v0(x1)v1(x1)vn(x1)v0(xn)v1(xn)vn(xn)is nonsingular. The solution p of the above problem can be obtained by solving a linear system of equations or

Recursive construction of the minimal interpolation space.

Let f:RdR be a continuous function and x0,,xn be n+1 distinct points in Rd. In this section we will construct an interpolation space Vi:=span{v0,,vi}Pi(Rd) suitable for interpolating f, using the formula (2.2), at the points x0,,xi, i=0,,n. Let vr be a monomial in Pn(Rd). We denote by 1vr the space spanned by {1,,vr}. For example, if vr=xy and d=2, then 1vr=span{1,x,y,x2,xy}.

Definition 3

Let Xn be a given finite set composed by n+1 distinct points. If v0,,vn are monomials of Pn(Rd) given in

A recursive construction of osculatory interpolants in R2

let A0,A1,,An be n+1 points in R2. We suppose that, for each point Ai we haveAi=Ai0=Ai1==Aiki,i=0,,n.

To study the osculatory interpolation problem, in particular the Hermite interpolation problem, we suppose that we know in each knot Ai the value of f and its following derivatives:f(Ai0),Dmi1f(Ai1),Dmi2f(Ai2),,Dmiki(Aiki),i=0,,n,

wheremij=(αij,βij)N×N,with|mij|=αij+βijj,

andDmijf(Aij)=αij+βijxαijyβijf(Aij).

Put N=n+i=0nki and consider v00,v01,,v0k0,v10,,v1k1,,vn0,,vnknN+1 linearly

Numerical tests

In this section we give some numerical examples for testing the performance of Algorithm 2 and Algorithm 3. For this we have selected two bivariate test functions defined in Ω=[0,1]×[0,1] byf(x,y)=5exp(xy(1x)(1y)(xy))andg(x,y)=cosx2y2.

Example 1

Our aim in this example is to determine recursively the minimal interpolation space Vi which satisfies||fIif||Ωɛ,withɛ=2×103.

We apply Algorithm 2 to the set:X18={(0,0);(0,0.5);(0,1);(0.25,0.25);(0.5,0);(0.5,0.5);(0.5,1);(1,0);(1,0.5);(1,1);(0.15,0.75);(

Conclusion

In this paper, we have described a new algorithm to construct recursively the minimal interpolation space Vn with respect to a given set of points Xn={x0,,xn}. This algorithm is based on the recursive method proposed in [1] to compute the Lagrange interpolant of a function f. We also have extended this method to the osculatory interpolation problem. As the Lagrange interpolation, this extended recursive method can be used to construct an interpolation subspace Vn suitable to a set Xn such that

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Research supported in part by PROTARS III, D11/18.

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