GRPIA: a new algorithm for computing interpolation polynomials

Abstract

Let x₀,x₁, ⋯ , 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 ∈ IN, be given real numbers. It is known that there exists a unique polynomial p_N− 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, ⋯ , 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.