Matrix recursive polynomial interpolation algorithm: An algorithm for computing the interpolation polynomials

摘要

Let x(0), x(1),... , x(n), be a set of n+1 distinct real numbers (i.e., x(i) not equal x(j), for i not equal j) and y(m,k), for m = 0, 1,..., n, and k = 0, 1,..., n(m), with n(m) is an element of N, be given of real numbers, we know that there exists a unique polynomial p(N-1) of degree N - 1 where N = Sigma(n)(i=0) n(i) + 1), such that p(N-1)((k)) (chi(m)) = y(m,k), for m = 0, 1,..., n and k = 0, 1,.., n(m). p(N-1) is the Hermite interpolation polynomial for the set {(x(m), Y-m,Y-k), m = 0, 1,..., n, k = 0, 1,..., n(m)}. The polynomial p(N-1) can be computed by using the Lagrange generalized polynomials. Recently Messaoudi et al. (2018) presented a new algorithm for computing the Hermite interpolation polynomials, for a general case, called Generalized Recursive Polynomial Interpolation Algorithm (GRPIA), this algorithm has been developed without using the Matrix Recursive Interpolation Algorithm (bilou and Messaoudi, 1999). Messaoudi et al. (2017) presented also a new algorithm called Matrix Recursive Polynomial Interpolation Algorithm (MRPIA), for a particular case where n(m) = mu = 1, for m = 0, 1,, n. In this paper we will give the version of the MRPIA for a particular case n(m) = mu >= 0, for m = 0, 1,,,,, n. We will recall the result of the existence of the polynomial p(N-1) for this case, some of its properties will also be given. Using the MRPIA, a method will be proposed for the general case, where n(m), for some m, are different and some examples will also be given. (C) 2019 Elsevier B.V. All rights reserved.