GRPIA: a new algorithm for computing interpolation polynomials

Let x 0 , x 1 , ⋯ , x n , be a set of n + 1 distinct real numbers (i.e., x m ≠ x j , for m ≠ j ) and let y m , k , for m = 0, 1, ⋯ , n , and k = 0, 1, ⋯ , r m , with r m ∈ I N , be given real numbers. It is known that there exists a unique polynomial p N − 1 of degree N − 1 with N = ∑ m = 0 n ( r m...

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Bibliographic Details
Published in:Numerical algorithms Vol. 80; no. 1; pp. 253 - 278
Main Authors: Messaoudi, Abderrahim, Errachid, Mohammed, Jbilou, Khalide, Sadok, Hassane
Format: Journal Article
Language:English
Published: New York Springer US 01.01.2019
Springer Nature B.V
Springer Verlag
Series:Extrapolation and Fixed Points in Memoriam Peter Wynn (1931-2017)
Subjects:
Algebra
Algorithms
Computation
Computer Science
Hermite polynomials
Interpolation
Mathematics
Numeric Computing
Numerical Analysis
Original Paper
Real numbers
Theory of Computation
Recursive polynomial interpolation algorithm
Sylvester’s identity
General Hermite polynomial interpolation
MSC 65F
Polynomial interpolation
Matrix recursive polynomial interpolation algorithm
MSC 15A
Schur complement
ISSN:1017-1398, 1572-9265
Online Access:Get full text
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Summary:Let x 0 , x 1 , ⋯ , x n , be a set of n + 1 distinct real numbers (i.e., x m ≠ x j , for m ≠ j ) and let y m , k , for m = 0, 1, ⋯ , n , and k = 0, 1, ⋯ , r m , with r m ∈ I N , be given real numbers. It is known that there exists a unique polynomial p N − 1 of degree N − 1 with N = ∑ 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, ⋯ , r m . p N − 1 is the Hermite interpolation polynomial for the set {( x m , y m , k ), m = 0, 1, ⋯ , n , k = 0, 1, ⋯ , r m }. The polynomial p N − 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 r m = 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 p N − 1 will also be established, cost and storage of this algorithm will also be studied, and some examples will be given.
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ISSN:1017-1398
1572-9265
DOI:10.1007/s11075-018-0543-x