A new algorithm for a recursive construction of the minimal interpolation space☆
Introduction
Let be linearly independent continuous functions on . Setting , for . For a given continuous function f defined on , and a set of distinct points of , the Lagrange interpolation problem is defined as follows:
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 are the subspaces of the space of polynomials of d variables and of degree less than or equal to n denoted by . 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 , Liang [13] proved that it is necessary and sufficient that the number of nodes is equal to 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 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 and , there always exists a subspace 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 called the least choice. Another interpolation algorithm has been developed by Sauer [20] in order to determine a subspace 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 called minimal interpolation space which satisfies the properties of minimal degree interpolation space. All these interpolation algorithms use globally the set of interpolation points to construct the interpolation space . If this space can not give a good interpolant of f at the set of points , the interpolation algorithms in question can not improve recursively the interpolant of f by adding simply a point to the set . Indeed, the construction of the space requires a new implementation of these algorithms on the whole set .
In this work, we propose a new algorithm, which permits to construct recursively and iteratively the subspaces , such that the following problemadmits a unique solution. We prove that each space is the minimal interpolation space with respect to the set 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 . 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 of points in , we can construct recursively an interpolation subspaces , , corresponding to the sets of points . We can also determine recursively the interpolant of a function f at the set . The explicit form of in the canonical basis of 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 be linearly independent continuous functions on . Setting , for . For a given continuous function f defined on , and distinct points of , we suppose that the interpolation problem:has a unique solution, i.e., the following matrixis 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 be a continuous function and be distinct points in . In this section we will construct an interpolation space suitable for interpolating f, using the formula (2.2), at the points , . Let be a monomial in . We denote by the space spanned by . For example, if and , then .
Definition 3 Let be a given finite set composed by distinct points. If are monomials of given in
A recursive construction of osculatory interpolants in
let be points in . We suppose that, for each point we have
To study the osculatory interpolation problem, in particular the Hermite interpolation problem, we suppose that we know in each knot the value of f and its following derivatives:
where
and
Put and consider 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 by
Example 1 Our aim in this example is to determine recursively the minimal interpolation space which satisfies We apply Algorithm 2 to the set:
Conclusion
In this paper, we have described a new algorithm to construct recursively the minimal interpolation space with respect to a given set of points . 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 suitable to a set such that
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Research supported in part by PROTARS III, D11/18.