GRPIA: a new algorithm for computing interpolation polynomials

Abstract

Let x0,x1, ⋯ , xn, be a set of n + 1 distinct real numbers (i.e., xm ≠ xj, for m ≠ j) and let ym,k, for m = 0, 1, ⋯ , n, and k = 0, 1, ⋯ , rm, with rm ∈ IN, be given real numbers. It is known that there exists a unique polynomial pN− 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, ⋯ , rm. pN− 1 is the Hermite interpolation polynomial for the set {(xm, ym,k), m = 0, 1, ⋯ , n, k = 0, 1, ⋯ , rm}. The polynomial pN− 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 rm = 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 pN− 1 will also be established, cost and storage of this algorithm will also be studied, and some examples will be given.