GRPIA: a new algorithm for computing interpolation polynomials - Messaoudi, Abderrahim; Errachid, Mohammed; Jbilou, Khalide; Sadok, Hassane (Numerical algorithms, 2019)
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.