New algorithm for computing the Hermite interpolation polynomial

Abstract

Let x ₀, x ₁,⋯ , x _n , be a set of n + 1 distinct real numbers (i.e., x _i ≠ x _j , for i ≠ j) and y _{i, k} , for i = 0,1,⋯ , n, and k = 0 ,1 ,⋯ , n _i , with n _i ≥ 1, be given of real numbers, we know that there exists a unique polynomial p _{N − 1}(x) of degree N − 1 where $N={\sum }_{i=0}^{n}(n_{i}+1)$, such that $p_{N-1}^{(k)}(x_{i})=y_{i,k}$, for i = 0,1,⋯ , n and k = 0,1,⋯ , n _i . P _N−1(x) is the Hermite interpolation polynomial for the set {(x _i , y _{i, k} ), i = 0,1,⋯ , n, k = 0,1,⋯ , n _i }. The polynomial p _N−1(x) can be computed by using the Lagrange polynomials. This paper presents a new method for computing Hermite interpolation polynomials, for a particular case n _i = 1. We will reformulate the Hermite interpolation polynomial problem and give a new algorithm for giving the solution of this problem, the Matrix Recursive Polynomial Interpolation Algorithm (MRPIA). Some properties of this algorithm will be studied and some examples will also be given.