Numerical Algorithms

, Volume 77, Issue 4, pp 1069–1092 | Cite as

New algorithm for computing the Hermite interpolation polynomial

Original Paper
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Abstract

Let x 0, x 1,⋯ , x n , be a set of n + 1 distinct real numbers (i.e., x i x j , for ij) and y i, k , for i = 0,1,⋯ , n, and k = 0 ,1 ,⋯ , n i , with n i ≥ 1, be given of real numbers, we know that there exists a unique polynomial p N − 1(x) of degree N − 1 where \(N={\sum }_{i=0}^{n}(n_{i}+1)\), such that \(p_{N-1}^{(k)}(x_{i})=y_{i,k}\), for i = 0,1,⋯ , n and k = 0,1,⋯ , n i . P N−1(x) is the Hermite interpolation polynomial for the set {(x i , y i, k ), i = 0,1,⋯ , n, k = 0,1,⋯ , n i }. The polynomial p N−1(x) can be computed by using the Lagrange polynomials. This paper presents a new method for computing Hermite interpolation polynomials, for a particular case n i = 1. We will reformulate the Hermite interpolation polynomial problem and give a new algorithm for giving the solution of this problem, the Matrix Recursive Polynomial Interpolation Algorithm (MRPIA). Some properties of this algorithm will be studied and some examples will also be given.

Keywords

Polynomial interpolation Hermite interpolation polynomials Schur complement Matrix Sylvester identity Recursive polynomial interpolation algorithm Matrix recursive interpolation algorithm 

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Notes

Acknowledgments

We are grateful to the Professor C. Brezinski for his helpful 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 New York 2017

Authors and Affiliations

  1. 1.Ecole Normale SupérieureMohammed V University in RabatRabatMorocco
  2. 2.LMPAUniversité du Littoral Côte d’OpaleCalaisFrance

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