Lagrange Multivariate Polynomial Interpolation: A Random Algorithmic Approach
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...
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Wiley
2022-01-01
|
| Series: | Journal of Applied Mathematics |
| Online Access: | http://dx.doi.org/10.1155/2022/8227086 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1832551257677496320 |
|---|---|
| author | A. Essanhaji M. Errachid |
| author_facet | A. Essanhaji M. Errachid |
| author_sort | A. Essanhaji |
| collection | DOAJ |
| description | 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 Z 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 Z,z-partition. All the given algorithms are tested on examples. RLMVPIA is easy to implement and requires no storage. |
| format | Article |
| id | doaj-art-6fbdf0390f3740ca9e475e3f9a2d70ec |
| institution | Kabale University |
| issn | 1687-0042 |
| language | English |
| publishDate | 2022-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Applied Mathematics |
| spelling | doaj-art-6fbdf0390f3740ca9e475e3f9a2d70ec2025-02-03T06:01:53ZengWileyJournal of Applied Mathematics1687-00422022-01-01202210.1155/2022/8227086Lagrange Multivariate Polynomial Interpolation: A Random Algorithmic ApproachA. Essanhaji0M. Errachid1Laboratory of MathematicsLaboratory of MathematicsThe 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 Z 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 Z,z-partition. All the given algorithms are tested on examples. RLMVPIA is easy to implement and requires no storage.http://dx.doi.org/10.1155/2022/8227086 |
| spellingShingle | A. Essanhaji M. Errachid Lagrange Multivariate Polynomial Interpolation: A Random Algorithmic Approach Journal of Applied Mathematics |
| title | Lagrange Multivariate Polynomial Interpolation: A Random Algorithmic Approach |
| title_full | Lagrange Multivariate Polynomial Interpolation: A Random Algorithmic Approach |
| title_fullStr | Lagrange Multivariate Polynomial Interpolation: A Random Algorithmic Approach |
| title_full_unstemmed | Lagrange Multivariate Polynomial Interpolation: A Random Algorithmic Approach |
| title_short | Lagrange Multivariate Polynomial Interpolation: A Random Algorithmic Approach |
| title_sort | lagrange multivariate polynomial interpolation a random algorithmic approach |
| url | http://dx.doi.org/10.1155/2022/8227086 |
| work_keys_str_mv | AT aessanhaji lagrangemultivariatepolynomialinterpolationarandomalgorithmicapproach AT merrachid lagrangemultivariatepolynomialinterpolationarandomalgorithmicapproach |