Let \(\sigma _0, \sigma _1, \cdots , \sigma _n\) be a set of n+1 distinct real numbers (i.e., \(\sigma _i\ne \sigma _j\) , for \(i \ne j\) ) and \(F_{m,k}\) , for \(m=0,1,\cdots ,n\) and \(k=0,1,\cdots ,n_m\) , be given \(s\times r\) real matrices, with \(n_m\in \hbox { I\hspace{-2.0pt}N}\) . It is known that there exists a unique \(s\times r\) matrix polynomial \(P_N\) of degree N with \(N=\sum _{m=0}^n(n_m+1)-1\) , such that \(P_N^{(k)}(\sigma _m)=F_{m,k}\) , for \(m = 0, 1, \cdots , n\) and \(k=0,1,\cdots ,n_m\) . \(P_N\) is the Hermite matrix interpolation polynomial for the set \(\{(\sigma _m, F_{m,k}), m= 0, 1, \cdots , n, \; k=0,1,\cdots ,n_m\}\) . The matrix polynomial \(P_N\) can be computed by using the Lagrange generalized polynomials. Recently Messaoudi and Sadok, in [11], presented a new algorithm for computing the Lagrange matrix interpolation polynomial called the Recursive Matrix Polynomial Interpolation Algorithm (RMPIA), for a particular case where \(n_m=0\) , for \(m=0,1,\cdots ,n\) . In this paper we will give a new formulation of the Hermite matrix polynomial interpolation problem and derive a new algorithm, called the Generalized Recursive Matrix Polynomial Interpolation Algorithm (GRMPIA), for computing the Hermite matrix polynomial interpolation in the general case. A new result of the existence of the polynomial \( P_N\) will also be established, cost and storage of this algorithm will also be studied and some examples will be given.
