GRPIA: a new algorithm for computing interpolation polynomials

• 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
• DOI: 10.1007/s11075-018-0543-x

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.