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Numerical Algorithms

, Volume 80, Issue 1, pp 253–278 | Cite as

GRPIA: a new algorithm for computing interpolation polynomials

  • Abderrahim MessaoudiEmail author
  • Mohammed Errachid
  • Khalide Jbilou
  • Hassane Sadok
Original Paper

Abstract

Let x0,x1, ⋯ , xn, be a set of n + 1 distinct real numbers (i.e., xmxj, for mj) and let ym,k, for m = 0, 1, ⋯ , n, and k = 0, 1, ⋯ , rm, with rmIN, be given real numbers. It is known that there exists a unique polynomial pN− 1 of degree N − 1 with \(N={\sum }_{m = 0}^{n}(r_{m}+ 1)\), such that \(p_{N-1}^{(k)}(x_{m})=y_{m,k}\), for m = 0, 1, ⋯ , n and k = 0, ⋯ , rm. pN− 1 is the Hermite interpolation polynomial for the set {(xm, ym,k), m = 0, 1, ⋯ , n, k = 0, 1, ⋯ , rm}. The polynomial pN− 1 can be computed by using the Lagrange generalized polynomials. Recently, Messaoudi et al. (2017) presented a new algorithm for computing the Hermite interpolation polynomial called the Matrix Recursive Polynomial Interpolation Algorithm (MRPIA), for a particular case where rm = μ = 1, for m = 0, 1, ⋯ , n. In this paper, we will give a new formulation of the Hermite polynomial interpolation problem and derive a new algorithm, called the Generalized Recursive Polynomial Interpolation Algorithm (GRPIA), for computing the Hermite polynomial interpolation in the general case. A new result of the existence of the polynomial pN− 1 will also be established, cost and storage of this algorithm will also be studied, and some examples will be given.

Keywords

Schur complement Sylvester’s identity Polynomial interpolation General Hermite polynomial interpolation Recursive polynomial interpolation algorithm Matrix recursive polynomial interpolation algorithm 

Mathematics Subject Classificaion (2010)

MSC 65F MSC 15A 

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Notes

Acknowledgments

We are grateful to the Professor C. Brezinski for his help and encouragement. We would like to thank the referee for his helpful comments and valuable suggestions.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • Abderrahim Messaoudi
    • 1
    Email author
  • Mohammed Errachid
    • 2
  • Khalide Jbilou
    • 3
  • Hassane Sadok
    • 3
  1. 1.Laboratory of Mathematics, Computing and Applications - Information Security (LabMiA-SI), Ecole Normale SupérieureMohammed V University in RabatTakaddoum, RabatMorocco
  2. 2.Laboratory of Mathematics, Computing and Applications - Information Security (LabMiA-SI), Centre Régional de Métier de l’Enseignement et de la Formation (CRMEF) de RabatMohammed V University in RabatRabatMorocco
  3. 3.LMPAUniversité du Littoral Côte d’OpaleCalais CedexFrance

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